Optical Coatings

Fundamental Optics 1 

Introduction 1.2 

Paraxial Formulas 1.3 

Imaging Properties of Lens Systems 1.6 

Lens Combination Formulas 1.8 

Performance Factors 1.11 

Lens Shape 1.17 

Lens Combinations 1.18 

Diffraction Effects 1.20 

Lens Selection 1.23 

Spot Size 1.26 

Aberration Balancing 1.27 

Definition of Terms 1.29 

Paraxial Lens Formulas 1.32 

Principal-Point Locations 1.36 

1.1 1 

Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings

Fundamental Optics 

Material Properties Optical Specifications Gaussian Beam Optics 

Introduction 

Even though several thousand different optical components 

are listed in this catalog, performing a few simple calculations will 

usually determine the appropriate optics for an application or, at 

the very least, narrow the list of choices. 

The process of solving virtually any optical engineering problem 

can be broken down into two main steps. First, paraxial calculations 

(first order) are made to determine critical parameters such 

as magnification, focal length(s), clear aperture (diameter), and 

object and image position. These paraxial calculations are covered 

in the next section of this chapter. 

Second, actual components are chosen based on these paraxial 

values, and their actual performance is evaluated with special 

attention paid to the effects of aberrations. A truly rigorous 

performance analysis for all but the simplest optical systems 

generally requires computer ray tracing, but simple generalizations 

can be used, especially when the lens selection process is 

confined to a limited range of component shapes. 

In practice, the second step may reveal conflicts with design 

constraints, such as component size, cost, or product availability. 

System parameters may therefore require modification. 

Because some of the terms used in this chapter may not be 

familiar to all readers, a glossary of terms is provided beginning 

on page 1.29. 

Finally, it should be noted that the discussion in this chapter 

relates only to systems with uniform illumination; optical systems 

for Gaussian beams are covered in Chapter 2, Gaussian Beam 

Optics. 

ENGINEERING SUPPORT 

Melles Griot maintains a staff of knowledgeable, 

experienced applications engineers at each of our 

facilities worldwide. The information given in this 

chapter is sufficient to enable the user to select the 

most appropriate catalog lenses for the most 

commonly encountered applications. However, when 

additional optical engineering support is required, 

our applications engineers are available to provide 

assistance. Do not hesitate to contact us for help in 

product selection or to obtain more detailed 

specifications on Melles Griot products. 

THE OPTICAL 

ENGINEERING PROCESS 

Determine basic system 

parameters, such as 

magnification and 

object/image distances 

Using paraxial formulas 

and known parameters, 

solve for remaining values 

Pick lens components 

based on paraxially 

derived values 

Determine if chosen 

component values conflict 

with any basic 

system constraints 

Estimate performance 

characteristics of system 

Determine if performance 

characteristics meet 

original design goals 

Optical Coatings 

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Paraxial Formulas 

SIGN CONVENTIONS 

The validity of the paraxial lens formulas is dependent on adherence to the following sign conventions: 

For lenses: (refer to figure 1.1) 

s is 1 for object to left of H 

(the first principal point) 

s is 5 for object to right of H 

s″ is 1 for image to right of H″ 

(the second principal point) 

s″ is 5 for image to left of H″ 

m is 1 for an inverted image 

m is 5 for an upright image 

For mirrors: 

When using the thin-lens approximation, simply refer to the left and right of the lens. 

Figure 1.1 

h 

front focal point 

f object v 

H H″ 

F 

s 

f 

principal points 

f 

f is 1 for convex (diverging) mirrors 

f is 5 for concave (converging) mirrors 

s is 1 for object to left of H 

s is 5 for object to right of H 

s″ is 5 for image to right of H″ 

s″ is 1 for image to left of H″ 

m is 1 for an inverted image 

m is 5 for an upright image 

rear focal point 

Note location of object and image relative to front and rear focal points. 

f = lens diameter 

m = 

s″/s = h″/h = magnification or 

conjugate ratio, said to be infinite if 

either s″ or s is infinite 

v = arcsin (f/2s) 

h = object height 

h″ = image height 

Sign conventions 

F″ 

s″ 

image 

s = object distance, positive for object (whether real 

or virtual) to the left of principal point H 

s″ = image distance (s and s″ are collectively called 

conjugate distances, with object and image in 

conjugate planes), positive for image (whether real 

or virtual) to the right of the principal point H″ 

f = effective focal length (EFL) which may be positive 

(as shown) or negative. f represents both FH and 

H″F″, assuming lens to be surrounded by medium 

of index 1.0 

h″ 

Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 

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Typically, the first step in optical problem solving is to select a 

system focal length based on constraints such as magnification or 

conjugate distances (object and image distance). The relationship 

among focal length, object position, and image position is 

given by 

1 

f 

= 1 s 

1 

+ . 

s′′ 

m = s ′′ 

= h ′′ . 

(1.2) 

s h 

This relationship can be used to recast the first formula into the 

following forms: 

f = m (s + s ′′ ) 

2 

(m + 1) 

f = sm 

m + 1 

s + s′′ 

f = 

m + 2 + 1 m 

s (m + 1) = s + s′′ 

1 

= 1 1 

4 

s′′ 

f s 

1 1 1 

= 4 

s′′ 

50 200 

s ′′ = 66.7 mm 

m = s ′′ 

= 66.7 

s 200 = 0.33 

(or real image is 0.33 mm high and inverted). 

(1.1) 

This formula is referenced to figure 1.1 and the sign conventions 

given on page 1.3. 

By definition, magnification is the ratio of image size to object 

size or 

where (s + s″) is the approximate object-to-image distance. 

(1.3) 

(1.4) 

(1.5) 

(1.6) 

With a real lens of finite thickness, the image distance, object 

distance, and focal length are all referenced to the principal points, 

not to the physical center of the lens. By neglecting the distance 

between the lens’ principal points, known as the hiatus, s + s″ 

becomes the object-to-image distance. This simplification, called the 

thin-lens approximation, can speed up calculation when dealing 

with simple optical systems. 

Example 1: Object outside Focal Point 

A 1-mm-high object is placed on the optical axis, 200 mm left of the 

left principal point of a 01 LDX 103 (f = 50 mm). Where is the 

image formed, and what is the magnification? (See figure 1.2.) 

object 

Figure 1.2 

Figure 1.3 

F1 

200 66.7 

F 2 

image 

Example 1 (f = 50 mm, s = 200 mm, s″ = 66.7 mm) 

Example 2: Object inside Focal Point 

The same object is placed 30 mm left of the left principal point of 

the same lens. Where is the image formed, and what is the magnification? 

(See figure 1.3.) 

1 

= 1 1 

4 

s′′ 

50 30 

s ′′ = 475 mm 

m = s ′′ 475 

= = 42.5 

s 30 

(or virtual image is 2.5 mm high and upright). 

In this case, the lens is being used as a magnifier, and the image can 

be viewed only back through the lens. 

image 

F 1 F 2 

object 

Example 2 (f = 50 mm, s = 30 mm, s″ = 475 mm) 

Example 3: Object at Focal Point 

A 1-mm-high object is placed on the optical axis, 50 mm left of the 

first principal point of an 01 LDK 019 (f = 50 mm). Where is the 

image formed, and what is the magnification? (See figure 1.4.) 

1 1 1 

= 

s′′ 

450 

4 50 

s ′′ = 425 mm 

m = s ′′ 425 

= = 40.5 

s 50 

(or virtual image is 0.5 mm high and upright). 


object 

F 2 

Figure 1.4 

image 

Example 3 (f = 450 mm, s = 50 mm, s″ = 425 mm) 

A simple graphical method can also be used to determine paraxial 

image location and magnification. This graphical approach relies on 

two simple properties of an optical system. First, a ray that enters 

the system parallel to the optical axis crosses the optical axis at the 

focal point. Second, a ray that enters the first principal point of the 

system exits the system from the second principal point parallel to 

its original direction (i.e., its exit angle with the optical axis is the same 

as its entrance angle). This method has been applied to the three 

previous examples illustrated in figures 1.2 through 1.4. Note that by 

using the thin-lens approximation, this second property reduces to the 

statement that a ray passing through the center of the lens is undeviated. 

F-NUMBER AND NUMERICAL APERTURE 

The paraxial calculations used to determine necessary element 

diameter are based on the concepts of focal ratio (f-number or f/#) 

and numerical aperture (NA). The f-number is the ratio of the focal 

length of the lens to its clear aperture (effective diameter). 

f-number = f f . 

(1.7) 

To visualize the f-number, consider a lens with a positive focal 

length illuminated uniformly with collimated light. The f-number 

defines the angle of the cone of light leaving the lens which ultimately 

forms the image. This is an important concept when the throughput 

or light-gathering power of an optical system is critical, such as 

when focusing light into a monochromator or projecting a highpower 

image. 

The other term used commonly in defining this cone angle is 

numerical aperture. Numerical aperture is the sine of the angle made 

by the marginal ray with the optical axis. By referring to 

figure 1.5 and using simple trigonometry, it can be seen that 

f 

NA = sin v = (1.8) 

2f 

or 

1 

NA = 

2(f-number) . (1.9) 

F 1 

f 

2 

Figure 1.5 

principal surface 

F-number and numerical aperture 

Ray f-numbers can also be defined for any arbitrary ray if its 

conjugate distance and the diameter at which it intersects the 

principal surface of the optical system are known. 

NOTE 

Because the sign convention given previously is not 

used universally in all optics texts, the reader may 

notice differences in the paraxial formulas. However, 

results will be correct as long as a consistent set of 

formulas and sign conventions is used. 

f 

v 





Imaging Properties of Lens Systems 

THE OPTICAL INVARIANT 

To understand the importance of the numerical aperture, consider 

its relation to magnification. Referring to figure 1.6, 

f 

NA (object side) = sin v = 2s 

f 

NA " (image side) = sin v ′′ = 2s ′′ 

which can be rearranged to show 

and 

f = 2s sinv 

f = 2s ′′ sinv′′ 

leading to 

s′′ 

s 

= sin v 

sinv′′ 

= NA . 

NA" 

Since s ′′ 

is simply the magnification of the system, 

s 

we arrive at 

m = NA . 

NA" 

(1.10) 

(1.11) 

(1.12) 

(1.13) 

(1.14) 

(1.15) 

The magnification of the system is therefore equal to the ratio 

of the numerical apertures on the object and image sides of the 

system. This powerful and useful result is completely independent 

of the specifics of the optical system, and it can often be used to determine 

the optimum lens diameter in situations involving aperture 

constraints. 

When a lens or optical system is used to create an image of a 

source, it is natural to assume that, by increasing the diameter (f) 

of the lens, we will be able to collect more light and thereby produce 

a brighter image. However, because of the relationship between 

magnification and numerical aperture, there can be a theoretical limit 

beyond which increasing the diameter has no effect on lightcollection 

efficiency or image brightness. 

Since the numerical aperture of a ray is given by f/2s, once a 

focal length and magnification have been selected, the value of NA 

sets the value of f. Thus, if one is dealing with a system in which the 

numerical aperture is constrained on either the object or image 

side, increasing the lens diameter beyond this value will increase 

system size and cost but will not improve performance (i.e., throughput 

or image brightness). This concept is sometimes referred to as 

the optical invariant. 

Example: System with Fixed Input NA 

Two very common applications of simple optics involve coupling 

light into an optical fiber or into the entrance slit of a monochromator. 

Although these problems appear to be quite different, they 

both have the same limitation — they have a fixed numerical 

aperture. For monochromators, this limit is usually expressed in 

terms of the f-number. In addition to the fixed numerical aperture, 

they both have a fixed entrance pupil (image) size. 

Suppose it is necessary, using a singlet lens from this catalog, to 

couple the output of an incandescent bulb with a filament 1 mm in 

diameter into an optical fiber as shown in figure 1.7. Assume that the 

fiber has a core diameter of 100 mm and a numerical aperture of 0.25, 

and that the design requires that the total distance from the source 

to the fiber be 110 mm. Which lenses are appropriate? 

By definition, the magnification must be 0.1. Letting s + s″ total 

110 mm (using the thin-lens approximation), we can use equation 

1.3, 

f = m (s + s ′′ ) 

(m + 1) 2 

to determine that the focal length is 9.1 mm. To determine the 

conjugate distances, s and s″, we utilize equation 1.6, 

s (m + 1) = s + s ′′, 

and find that s = 100 mm and s″ = 10 mm. 

We can now use the relationship NA = Ω/2s or NA″ = Ω/2s″ to 

derive Ω, the optimum clear aperture (effective diameter) of the lens. 

With an image numerical aperture of 0.25 and an image distance 

(s″) of 10 mm, 

f 

0.25 = 20 

f 

= 5 mm. 

Accomplishing this imaging task with a single lens therefore 

requires an optic with a 9.1-mm focal length and a 5-mm diameter. 

Using a larger diameter lens will not result in any greater system 

throughput because of the limited input numerical aperture of the 

optical fiber. The singlet lenses in this catalog that meet these criteria 

are 01 LPX 003, which is plano-convex, and 01 LDX 003 and 

01 LDX 005, which are biconvex. 


SAMPLE CALCULATION 

To understand how to use this relationship between magnification 

and numerical aperture, consider the following example. 

Making some simple calculations has reduced our choice of 

lenses to just three. Chapter 2, Gaussian Beam Optics, discusses 

how to make a final choice of lenses based on various performance 

criteria. 


Figure 1.6 

Figure 1.7 

f 

f 

2 

object side 

magnification = 

h" 

= 

0.1 

= 0.1X 

h 1.0 

filament 

h = 1 mm 

v 

s 

Numerical aperture and magnification 

s + s" = 110 mm 

f 

NA = = 0.025 

2s 

s = 100 mm 

optical system 

f = 9.1 mm 

f = 5 mm 

f 

NA" = = 0.25 

2s" 

fiber core 

h" = 0.1 mm 

s" = 10 mm 

Optical system geometry for focusing the output of an incandescent bulb into an optical fiber 

s″ 

v″ 

image side 






Lens Combination Formulas 

PARAXIAL LENS COMBINATION FORMULAS 

Many optical tasks require several lenses in order to achieve an 

acceptable level of performance. One possible approach to lens 

combinations is to consider each image formed by each lens as the 

object for the next lens and so on. This is a valid approach, but it is 

time consuming and unnecessary. 

It is much simpler to calculate the effective (combined) focal 

length and principal-point locations and then use these results in 

any subsequent paraxial calculations (see figure 1.8). They can even 

be used in the optical invariant calculations described in the 

preceding section. 

EFFECTIVE FOCAL LENGTH 

The following formulas show how to calculate the effective focal 

length and principal-point locations for a combination of any two 

arbitrary components. The approach for more than two lenses is very 

simple: calculate the values for the first two elements, then perform 

the same calculation for this combination with the next lens. This is 

continued until all lenses in the system are accounted for. 

The expression for the combination focal length is the same 

whether lens separation distances are large or small and whether f 1 

and f 2 are positive or negative: 

f = 

1 

f 

ff 1 2 

f 1 + f2 4 . 

d 

This may be more familiar in the form 

s ′′ = 

2 

= 1 f 

+ 

1 

f 

4 d 

ff 

. 

1 2 1 2 

f 2 (f1 

4 d) 

. 

f + f 4 d 

1 2 

z = s 2′′ 4 f . 

(1.16) 

(1.17) 

Notice that the formula is symmetric with respect to interchange 

of the lenses (end-for-end rotation of the combination) at constant 

d. The next two formulas are not. 

COMBINATION FOCAL-POINT LOCATION 

For all cases, 

COMBINATION SECONDARY 

PRINCIPAL-POINT LOCATION 

(1.18) 

Because the thin-lens approximation is obviously highly invalid 

for most combinations, the ability to determine the location of the 

secondary principal point is vital for accurate determination of d when 

another element is added. The simplest formula for this calculates 

how far the secondary principal point of the final (second) element 

is moved by being part of the combination: 

(1.19) 

COMBINATION EXAMPLES 

It is possible for a lens combination or system to exhibit principal 

planes that are far removed from the system. When such systems 

are themselves combined, negative values of d may occur. Probably 

the simplest example of a negative d-value situation is shown in 

figure 1.9. Meniscus lenses with steep surfaces have external principal 

planes. When two of these lenses are brought into contact, a 

negative value of d can occur. Other combined-lens examples are 

shown in figures 1.10 through 1.13. 

SYMBOLS 

f 

f 1 

f 2 

d 

= combination focal length (EFL), positive if 

combination final focal point falls to right of 

combination secondary principal point, 

negative otherwise. 

= focal length (EFL) of first element. 

= focal length (EFL) of second element. 

= distance from secondary principal point of 

first element to primary principal point of 

second element (positive if primary principal 

point is to right of the secondary principal 

point, negative otherwise). 

s 2 ″ = distance from secondary principal point of 

second element to final combination focal 

point (location of final image for object at 

infinity to left), positive if the focal point is 

to right of second element secondary principal 

point. 

z 

= distance to combination secondary principal 

point measured from secondary principal 

point of second element, positive if 

combination secondary principal point is to 

right of secondary principal point of second 

element. 

Note: These paraxial formulas apply to coaxial 

combinations of both thick and thin lenses immersed 

in any fluid with refractive index independent of 

position. They assume that light propagates from left 

to right through an optical system. 


Figure 1.8 

Figure 1.9 

INDIVIDUAL ELEMENT 

1st element 

COMBINATION 

2 elements 

SUBSYSTEM 

d 

d 

2nd element 

z, from formula 

3rd element 

combination secondary principal plane 

(to find combination primary principal plane, 

apply procedure to reversed combination 

resulting from end-to-end rotation) 

subsystem secondary principal plane 

n-1 elements nth element to be added to complete the system 

d 

z, from formula 

system secondary 

COMPLETE SYSTEM 

principal plane 

principal planes 

not “crossed” 

system primary principal plane (secondary principal 

plane located by z formula for reversed system) 

lens combinations or systems may exhibit “crossed” principal planes; single lenses cannot 

SUBSYSTEM 

principal planes internal but “crossed” 

n-1 elements 

Generalization from combinations to systems 

1 2 3 4 

d 

subsystem secondary principal plane 

nth element to be added to complete the system 

subsystem primary principal plane 

3 4 1 2 

d>0 d


d 

s 2 ″ 

f 1 f 2 

z 

f

Performance Factors 

After paraxial formulas have been used to select values for component 

focal length(s) and diameter(s), the final step is to select 

actual lenses. As in any engineering problem, this selection process 

involves a number of tradeoffs, including performance, cost, weight, 

and environmental factors. 

The performance of real optical systems is limited by several 

factors, including lens aberrations and light diffraction. The magnitude 

of these effects can be calculated with relative ease. 

Numerous other factors, such as lens manufacturing tolerances 

and component alignment, impact the performance of an optical 

system. Although these are not considered explicitly in the following 

discussion, it should be kept in mind that if calculations indicate that 

a lens system only just meets the desired performance criteria, in 

practice it may fall short of this performance as a result of other 

factors. In critical applications, it is generally better to select a lens 

whose calculated performance is significantly better than needed. 

DIFFRACTION 

Diffraction, a natural property of light arising from its wave 

nature, poses a fundamental limitation on any optical system. Diffraction 

is always present, although its effects may be masked if 

the system has significant aberrations. When an optical system is 

essentially free from aberrations, its performance is limited solely 

by diffraction, and it is referred to as diffraction limited. 

In calculating diffraction, we simply need to know the focal 

length(s) and aperture diameter(s); we do not consider other lensrelated 

factors such as shape or index of refraction. 

Since diffraction increases with increasing f-number, and aberrations 

decrease with increasing f-number, determining optimum 

system performance often involves finding a point where the combination 

of these factors has a minimum effect. 

ABERRATIONS 

To determine the precise performance of a lens system, we can 

trace the path of light rays through it, using Snell’s law at each 

optical interface to determine the subsequent ray direction. This 

process, called ray tracing, is usually accomplished on a computer. 

When this process is completed, it is typically found that not all 

the rays pass through the points or positions predicted by paraxial 

theory. These deviations from ideal imaging are called lens 

aberrations. 

The direction of a light ray after refraction at the interface between 

two homogeneous, isotropic media of differing index of refraction is 

given by Snell’s law: 

n 1 sinß 1 = n 2 sinß 2 

( 1.20) 

where ß 1 is the angle of incidence, ß 2 is the angle of refraction, and 

both angles are measured from the surface normal as shown in figure 

1.14. 

material 1 

index n 1 

material 2 

index n2 

Figure 1.14 

wavelength l d 

Technical Assistance 

v 1 

v 2 

Refraction of light at a dielectric boundary 

APPLICATION NOTE 

Detailed performance analysis of an optical system 

is accomplished using computerized ray-tracing 

software. Melles Griot applications engineers have 

the capability to provide a ray-tracing analysis of 

simple catalog components systems. If you need 

assistance in determining the performance of your 

optical system, or in selecting optimum components 

for your particular application, please contact your 

nearest Melles Griot office. 

Alternately, a database containing prescription 

information for most of the components listed in this 

catalog is available on the catalog CD-ROM. If you 

would like to obtain a copy of this database, please 

contact your Melles Griot representative. 

For analysis of more complex optical systems, 

or the design of totally custom lenses, Melles Griot 

Optical Systems, located in Rochester, New York, can 

supply the necessary support. This group specializes 

in the design and fabrication of high-precision, 

multielement lens systems. For more information 

about their capabilities, please call your Melles Griot 

representative. 






Even though tools for precise analysis of an optical system are 

becoming easier to use and are readily available, it is still quite useful 

to have a method for quickly estimating lens performance. This 

not only saves time in the initial stages of system specification, but 

can also help achieve a better starting point for any further 

computer optimization. 

The first step in developing these rough guidelines is to realize 

that the sine functions in Snell’s law can be expanded in an infinite 

Taylor series: 

3 

1 1 1 

5 

1 

sin v = v 4 v /3! + v /5! 4 v /7! + v /9! 4. . . 

The first approximation we can make is to replace all sine functions 

with their arguments (i.e., replace sin ß 1 with ß 1 itself and so 

on). This is called first-order or paraxial theory because only the first 

terms of the sine expansions are used. Design of any optical system 

generally starts with this approximation using the paraxial formulas. 

The assumption that sinß = ß is reasonably valid for ß close to zero 

(i.e., high f-number lenses). With more highly curved surfaces (and 

particularly marginal rays), paraxial theory yields increasingly large 

deviations from real performance because sinß ≠ ß. These deviations 

are known as aberrations. Because a perfect optical system (one 

without any aberrations) would form its image at the point and to 

the size indicated by paraxial theory, aberrations are really a measure 

of how the image differs from the paraxial prediction. 

As already stated, exact ray tracing is the only rigorous way to 

analyze real lens surfaces. Before the advent of computers, this was 

excessively tedious and time consuming. Seidel addressed this issue 

by developing a method of calculating aberrations resulting from 

the ß 1 3 /3! term. The resultant third-order lens aberrations are therefore 

called Seidel aberrations. 

To simplify these calculations, Seidel put the aberrations of an 

optical system into several different classifications. In monochromatic 

light they are spherical aberration, astigmatism, field 

curvature, coma, and distortion. In polychromatic light there are 

also chromatic aberration and lateral color. Seidel developed 

methods to approximate each of these aberrations without actually 

tracing large numbers of rays using all the terms in the sine 

expansions. 

In actual practice, aberrations occur in combinations rather 

than alone. This system of classifying them, which makes analysis 

much simpler, gives a good description of optical system image 

quality. In fact, even in the era of powerful ray-tracing software, 

Seidel’s formula for spherical aberration is still widely used. 

7 

1 

9 

1 

SPHERICAL ABERRATION 

Figure 1.15 illustrates how an aberration-free lens focuses 

incoming collimated light. All rays pass through the focal point F ″. 

The lower figure shows the situation more typically encountered in 

single lenses. The farther from the optical axis the ray enters the 

lens, the nearer to the lens it focuses (crosses the optical axis). The 

distance along the optical axis between the intercept of the rays 

that are nearly on the optical axis (paraxial rays) and the rays that 

go through the edge of the lens (marginal rays) is called longitudinal 

spherical aberration (LSA). The height at which these rays 

intercept the paraxial focal plane is called transverse spherical 

aberration (TSA). These quantities are related by 

TSA = LSA ! tan u″. 

aberration-free lens 

u″ 

paraxial focal plane 

longitudinal spherical aberration 

LSA 

F″ 

F″ 

TSA 

transverse spherical aberration 

Figure 1.15 Spherical aberration of a plano-convex lens 

(1.21) 

Spherical aberration is dependent on lens shape, orientation, and 

conjugate ratio, as well as on the index of refraction of the materials 

present. Parameters for choosing the best lens shape and orientation 

for a given task are presented later in this chapter. However, the 


third-order, monochromatic, spherical aberration of a plano-convex 

lens used at infinite conjugate ratio can be estimated by 

spot size due to spherical aberration = 0.067 f . (1.22) 

f/# 3 

Theoretically, the simplest way to eliminate or reduce spherical 

aberration is to make the lens surface(s) with a varying radius of curvature 

(i.e., an aspheric surface) designed to exactly compensate for 

the fact that sin v ≠ v at larger angles. In practice, however, most lenses 

with high surface quality are manufactured by grinding and polishing 

techniques that naturally produce spherical or cylindrical surfaces. 

The manufacture of aspheric surfaces is more complex, and it is 

difficult to produce a lens of sufficient surface accuracy to eliminate 

spherical aberration completely. Fortunately, these aberrations 

can be virtually eliminated, for a chosen set of conditions, by combining 

the effects of two or more spherical (or cylindrical) surfaces. 

In general, simple positive lenses have undercorrected spherical 

aberration, and negative lenses usually have overcorrected spherical 

aberration. By combining a positive lens made from low-index glass 

with a negative lens made from high-index glass, it is possible to produce 

a combination in which the spherical aberrations cancel but 

the focusing powers do not. The simplest examples of this are 

cemented doublets, such as the 01 LAO series which produce 

minimal spherical aberration when properly used. 

object point 

Figure 1.16 

optical axis 

tangential plane 

tangential image 

(focal line) 

principal ray 

sagittal plane 

optical system 

Astigmatism represented by sectional views 

ASTIGMATISM 

When an off-axis object is focused by a spherical lens, the natural 

asymmetry leads to astigmatism. The system appears to have two 

different focal lengths. 

As shown in figure 1.16, the plane containing both optical axis 

and object point is called the tangential plane. Rays that lie in this 

plane are called tangential rays. Rays not in this plane are referred 

to as skew rays. The chief, or principal, ray goes from the object 

point through the center of the aperture of the lens system. The 

plane perpendicular to the tangential plane that contains the principal 

ray is called the sagittal or radial plane. 

The figure illustrates that tangential rays from the object come 

to a focus closer to the lens than do rays in the sagittal plane. When 

the image is evaluated at the tangential conjugate, we see a line in 

the sagittal direction. A line in the tangential direction is formed at 

the sagittal conjugate. Between these conjugates, the image is either 

an elliptical or a circular blur. Astigmatism is defined as the 

separation of these conjugates. 

The amount of astigmatism in a lens depends on lens shape only 

when there is an aperture in the system that is not in contact with the 

lens itself. (In all optical systems there is an aperture or stop, although 

in many cases it is simply the clear aperture of the lens element itself.) 

Astigmatism strongly depends on the conjugate ratio. 

paraxial 

focal plane 

sagittal image (focal line) 





COMA 

In spherical lenses, different parts of the lens surface exhibit different 

degrees of magnification. This gives rise to an aberration 

known as coma. As shown in figure 1.17, each concentric zone of 

a lens forms a ring-shaped image called a comatic circle. This causes 

blurring in the image plane (surface) of off-axis object points. An 

off-axis object point is not a sharp image point, but it appears as a 

characteristic comet-like flare. Even if spherical aberration is 

corrected and the lens brings all rays to a sharp focus on axis, a 

lens may still exhibit coma off axis. See figure 1.18. 

As with spherical aberration, correction can be achieved by 

using multiple surfaces. Alternatively, a sharper image may be 

produced by judiciously placing an aperture, or stop, in an optical 

system to eliminate the more marginal rays. 

FIELD CURVATURE 

Even in the absence of astigmatism, there is a tendency of optical 

systems to image better on curved surfaces than on flat planes. This 

effect is called field curvature (see figure 1.19). In the presence of astigmatism, 

this problem is compounded because there are two separate 

astigmatic focal surfaces that correspond to the tangential and 

sagittal conjugates. 

Field curvature varies with the square of field angle or the square 

of image height. Therefore, by reducing the field angle by one-half, 

it is possible to reduce the blur from field curvature to a value of 0.25 

of its original size. 

S 

1′ 

1 

1 

1′ 

1 

1′ 

P,O 

Figure 1.18 

Figure 1.19 

points on lens 

1 

4 2 

1′ 

4′ 2′ 

3 3′ 0 3′ 3 

2′ 4′ 

1′ 

2 

4 

1 

positive transverse coma 

focal plane 

Positive transverse coma 

spherical focal surface 

Field curvature 

corresponding 

points on S 

1 

4 

1′ 

2 

4′ 2′ 

3 

3′ 

S 

60° 


Figure 1.17 

Imaging an off-axis point source by a lens with positive transverse coma 


Positive lens elements usually have inward curving fields, and negative 

lenses have outward curving fields. Field curvature can thus 

be corrected to some extent by combining positive and negative 

lens elements. 

DISTORTION 

The image field not only may have curvature but may also be 

distorted. The image of an off-axis point may be formed at a 

location on this surface other than that predicted by the simple 

paraxial equations. This distortion is different from coma (where 

rays from an off-axis point fail to meet perfectly in the image 

plane). Distortion means that even if a perfect off-axis point image 

is formed, its location on the image plane is not correct. Furthermore, 

the amount of distortion usually increases with increasing 

image height. The effect of this can be seen as two different kinds 

of distortion: pincushion and barrel (see figure 1.20). Distortion 

does not lower system resolution; it simply means that the image 

shape does not correspond exactly to the shape of the object. 

Distortion is a separation of the actual image point from the 

paraxially predicted location on the image plane and can be 

expressed either as an absolute value or as a percentage of the 

paraxial image height. 

It should be apparent that a lens or lens system has opposite 

types of distortion depending on whether it is used forward or backward. 

This means that if a lens were used to make a photograph, 

and then used in reverse to project it, there would be no distortion 

in the final screen image. Also, perfectly symmetrical optical systems 

at 1:1 magnification have no distortion or coma. 

CHROMATIC ABERRATION 

The aberrations previously described are purely a function of the 

shape of the lens surfaces, and can be observed with monochromatic 

light. There are, however, other aberrations that arise when 

these optics are used to transform light containing multiple 

wavelengths. 

OBJECT 

PINCUSHION 

DISTORTION 

BARREL 

DISTORTION 

The index of refraction of a material is a function of wavelength. 

Known as dispersion, this is discussed in Chapter 4, Material 

Properties. From Snell’s law (see equation 1.20), it can be seen that 

light rays of different wavelengths or colors will be refracted at 

different angles since the index is not a constant. Figure 1.21 shows 

the result when polychromatic collimated light is incident on a positive 

lens element. Because the index of refraction is higher for 

shorter wavelengths, these are focused closer to the lens than the 

longer wavelengths. Longitudinal chromatic aberration is defined 

as the axial distance from the nearest to the farthest focal point. 

As in the case of spherical aberration, positive and negative 

elements have opposite signs of chromatic aberration. Once again, 

by combining elements of nearly opposite aberration to form a 

doublet, chromatic aberration can be partially corrected. It is necessary 

to use two glasses with different dispersion characteristics, 

so that the weaker negative element can balance the aberration of 

the stronger, positive element. 

Variations of Aberrations with Aperture, 

Field Angle, and Image Height 

Aberration 

Lateral Spherical 

Longitudinal Spherical 

Coma 

Astigmatism 

Field Curvature 

Distortion 

Chromatic 

white light ray 

Aperture 

blue light ray 

blue focal point 

red light ray 

red focal point 

Figure 1.20 Pincushion and barrel distortion Figure 1.21 Longitudinal chromatic aberration 

(Ω) 

Ω 3 

Ω 2 

Ω 2 

Ω 

Ω 

— 

— 

Field Angle 

(ß) 

— 

— 

ß 

ß 2 

ß 2 

ß 3 

— 

Image Height 

(y) 

— 

— 

y 

y 2 

y 2 

y 3 

— 

longitudinal 

chromatic 

aberration 





LATERAL COLOR 

Lateral color is the difference in image height between blue and 

red rays. Figure 1.22 shows the chief ray of an optical system 

consisting of a simple positive lens and a separate aperture. Because 

of the change in index with wavelength, blue light is refracted more 

strongly than red light, which is why rays intercept the image plane 

at different heights. Stated simply, magnification depends on color. 

Lateral color is very dependent on system stop location. 

For many optical systems, the third-order term is all that may 

be needed to quantify aberrations. However, in highly corrected 

systems or in those having large apertures or a large angular field 

of view, third-order theory is inadequate. In these cases, exact ray 

tracing is absolutely essential. 

aperture 

Figure 1.22 

Lateral color 

red light ray 

blue light ray 

focal plane 

lateral color 


Achromatic Doublets Are Superior 

to Simple Lenses 

Because achromatic doublets correct for spherical 

as well as chromatic aberration, they are often 

superior to simple lenses for focusing collimated 

light or collimating point sources, even in purely 

monochromatic light. 

Although there is no simple formula that can be 

used to estimate the spot size of a doublet, the 

tables on page 1.26 give sample values that can be 

used to estimate the performance of other catalog 

achromats. 



Lens Shape 

Aberrations described in the preceding section are highly 

dependent on application, lens shape, and material of the lens (or, 

more exactly, its index of refraction). The singlet shape that minimizes 

spherical aberration at a given conjugate ratio is called best-form. 

The criterion for best-form at any conjugate ratio is that the marginal 

rays are equally refracted at each of the lens/air interfaces. This 

minimizes the effect of sin v ≠ v. It is also the criterion for minimum 

surface-reflectance loss. Another benefit is that absolute coma is 

nearly minimized for best-form shape, at both infinite and unit 

conjugate ratios. 

To further explore the dependence of aberrations on lens shape, it 

is helpful to make use of the Coddington shape factor, q, defined as 

q = (r 2 + r 1 ) . (1.23) 

(r 4 r ) 

Figure 1.23 

2 1 

Figure 1.23 shows the transverse and longitudinal spherical 

aberration of a singlet lens as a function of the shape factor, q. In this 

particular instance, the lens has a focal length of 100 mm, operates 

at f/5, has an index of refraction of 1.518722 (BK7 at the mercury 

green line, 546.1 nm), and is being operated at the infinite conjugate 

ratio. It is also assumed that the lens itself is the aperture stop. An 

asymmetric shape that corresponds to a q-value of about 0.7426 for 

this material and wavelength is the best singlet shape for on-axis 

imaging. Best-form shapes are used in Melles Griot laser-line-focusing 

singlet lenses. It is important to note that the best-form shape is 

dependent on refractive index. For example, with a high-index 

material, such as silicon, the best-form lens for the infinite conjugate 

ratio is a meniscus shape. 

ABERRATIONS IN MILLIMETERS 

5 

4 

3 

2 

1 

42 

exact transverse spherical 

aberration (TSA) 

41.5 41 40.5 0 0.5 1 1.5 2 

SHAPE FACTOR (q) 

exact longitudinal spherical aberration (LSA) 

Aberrations of positive singlets at infinite conjugate ratio as a function of shape 

At infinite conjugate with a typical glass singlet, the plano-convex 

shape (q = 1), with convex side toward the infinite conjugate, performs 

nearly as well as the best-form lens. Because a plano-convex lens costs 

much less to manufacture than an asymmetric biconvex singlet, these 

lenses are quite popular. Furthermore, this lens shape exhibits nearminimum 

total transverse aberration and near-zero coma when used 

off axis, thus enhancing its utility. 

For imaging at unit magnification (s = s″ = 2f), a similar analysis 

would show that a symmetric biconvex lens is the best shape. Not 

only is spherical aberration minimized, but coma, distortion, and 

lateral chromatic aberration exactly cancel each other out. These 

results are true regardless of material index or wavelength, which 

explains the utility of symmetric convex lenses, as well as symmetrical 

optical systems in general. However, if a remote stop is present, 

these aberrations may not cancel each other quite as well. 

For wide-field applications, the best-form shape is definitely not 

the optimum singlet shape, especially at the infinite conjugate ratio, 

since it yields maximum field curvature. The ideal shape is determined 

by the situation and may require rigorous ray-tracing analysis. 

It is possible to achieve much better correction in an optical system 

by using more than one element. The cases of an infinite 

conjugate ratio system and a unit conjugate ratio system are 

discussed in the following section. 






Lens Combinations 

INFINITE CONJUGATE RATIO 

As shown in the previous discussion, the best-form singlet lens 

for use at infinite conjugate ratios is generally nearly plano-convex. 

Figure 1.24 shows a plano-convex lens (01 LPX 023) with 

incoming collimated light at a wavelength of 546.1 nm. This drawing, 

including the rays traced through it, is shown to exact scale. The 

marginal ray (ray f-number 1.5) strikes the paraxial focal plane significantly 

off the optical axis. 

This situation can be improved by using a two-element system. 

The second part of the figure shows a precision achromat (01 LAO 014), 

which consists of a positive low-index (crown glass) element cemented 

to a negative meniscus high-index (flint glass) element. This is drawn 

to the same scale as the plano-convex lens. No spherical aberration 

can be discerned in the lens. Of course, not all of the rays pass exactly 

through the paraxial focal point; however, in this case, the departure 

is measured in micrometers, rather than in millimeters, as in the case 

of the plano-convex lens. Additionally, chromatic aberration (not 

shown) is much better corrected in the doublet. Even though these 

lenses are known as achromatic doublets, it is important to remember 

that even with monochromatic light the doublet’s performance is 

superior. 

Figure 1.24 also shows the f-number at which singlet performance 

becomes unacceptable. The ray with f-number 7.5 practically intercepts 

the paraxial focal point, and the f/3.8 ray is fairly close. This useful 

drawing, which can be scaled to fit a plano-convex lens of any focal 

length, can be used to estimate the magnitude of its spherical aberration, 

although lens thickness affects results slightly. 

UNIT CONJUGATE RATIO 

Figure 1.25 shows three possible systems for use at the unit 

conjugate ratio. All are shown to the same scale and using the 

same ray f-numbers with a light wavelength of 546.1 nm. The first 

system is a symmetric biconvex lens (01 LDX 027), the best-form 

singlet in this application. Clearly, significant spherical aberration 

is present in this lens at f/2.7. Not until f/13.3 does the ray closely 

approach the paraxial focus. 

A dramatic improvement in performance is gained by using two 

identical plano-convex lenses with convex surfaces facing and nearly 

in contact. Those shown in figure 1.25 are both 01 LPX 081. The combination 

of these two lenses yields almost exactly the same focal 

length as the biconvex lens. To understand why this configuration 

improves performance so dramatically, consider that if the biconvex 

lens were split down the middle, we would have two identical 

plano-convex lenses, each working at an infinite conjugate ratio, 

but with the convex surface toward the focus. This orientation is 

opposite to that shown to be optimum for this shape lens. On the other 

hand, if these lenses are reversed, we have the system just described 

but with a better correction of the spherical aberration. 

ray f-numbers 

1.5 

1.9 

2.5 

3.8 

7.5 

1.5 

1.9 

2.5 

3.8 

7.5 

PLANO-CONVEX LENS 

ACHROMAT 

paraxial image plane 

01 LPX 023 

01 LAO 014 

Figure 1.24 Single-element plano-convex lens compared 

with a two-element achromat 

The previous examples indicate that an achromat is superior in 

performance to a singlet when used at the infinite conjugate ratio 

and at low f-numbers. Since the unit conjugate case can be thought 

of as two lenses, each working at the infinite conjugate ratio, the next 

step is to replace the plano-convex singlets with achromats, yielding 

a four-element system. The third part of figure 1.25 shows a system 

composed of two 01 LAO 037 lenses. Once again, spherical aberration 

is not evident, even in the f/2.7 ray. 


ay f-numbers 

2.7 

3.3 

4.4 

6.7 

13.3 

SYMMETRIC BICONVEX LENS 

IDENTICAL PLANO-CONVEX LENSES 

2.7 

3.3 

4.4 

6.7 

13.3 

2.7 

3.3 

4.4 

6.7 

13.3 

01 LDX 027 

01 LPX 081 

IDENTICAL ACHROMATS 

01 LAO 037 

Figure 1.25 Three possible systems for use at the unit conjugate ratio 

paraxial image plane 






Diffraction Effects 

In all light beams, some energy is spread outside the region predicted 

by rectilinear propagation. This effect, known as diffraction, 

is a fundamental and inescapable physical phenomenon. 

Diffraction can be understood by considering the wave nature 

of light. Huygen’s principle (figure 1.26) states that each point on 

a propagating wavefront is an emitter of secondary wavelets. The 

combined focus of these expanding wavelets forms the propagating 

wave. Interference between the secondary wavelets gives rise to a 

fringe pattern that rapidly decreases in intensity with increasing 

angle from the initial direction of propagation. Huygen’s principle 

nicely describes diffraction, but rigorous explanation demands a 

detailed study of wave theory. 

Diffraction effects are traditionally classified into either Fresnel 

or Fraunhofer types. Fresnel diffraction is primarily concerned 

with what happens to light in the immediate neighborhood of a 

diffracting object or aperture. It is thus only of concern when the 

illumination source is close to this aperture or object. Consequently, 

Fresnel diffraction is rarely important in most optical setups. 

Fraunhofer diffraction, however, is often very important. This is 

the light-spreading effect of an aperture when the aperture (or 

object) is illuminated with an infinite source (plane-wave illumination) 

and the light is sensed at an infinite distance (far-field) from 

this aperture. 

From these overly simple definitions, one might assume that 

Fraunhofer diffraction is important only in optical systems with 

infinite conjugate, whereas Fresnel diffraction equations should be 

considered at finite conjugate ratios. Not so. A lens or lens system 

of finite positive focal length with plane-wave input maps the farfield 

diffraction pattern of its aperture onto the focal plane; therefore, 

it is Fraunhofer diffraction that determines the limiting 

performance of optical systems. More generally, at any conjugate 

ratio, far-field angles are transformed into spatial displacements 

in the image plane. 


Rayleigh Criterion 

In imaging applications, spatial resolution is ultimately 

limited by diffraction. Calculating the maximum possible 

spatial resolution of an optical system requires an 

arbitrary definition of what is meant by resolving two 

features. In the Rayleigh criterion, it is assumed that 

two separate point sources can be resolved when the 

center of the Airy disc from one overlaps the first 

dark ring in the diffraction pattern of the second. In 

this case, the smallest resolvable distance, d, is 

0.61 l 

d = = 1.22 l f/# . 

N.A. 

CIRCULAR APERTURE 

Fraunhofer diffraction at a circular aperture dictates the 

fundamental limits of performance for circular lenses. It is important 

to remember that the spot size, caused by diffraction, of a circular 

lens is 

d = 2.44 l f/# 

(1.24) 

where d is the diameter of the focused spot produced from planewave 

illumination and l is the wavelength of light being focused. 

Notice that it is the f-number of the lens, not its absolute diameter, 

that determines this limiting spot size. 

The diffraction pattern resulting from a uniformly illuminated circular 

aperture actually consists of a central bright region, known as 

the Airy disc (see figure 1.27), which is surrounded by a number of much 

fainter rings. Each ring is separated by a circle of zero intensity. The 

irradiance distribution in this pattern can be described by 

I = I 

x 0 

Figure 1.26 

secondary 

wavelets 

wavefront 

aperture 

⎡ 2J 1(x) 

⎤ 

⎢ 

⎣ x 

⎥ 

⎦ 

where I 0 = peak irradiance in image 

J (x) = x ( 41) 

1 

Huygen’s principle 

2 

J (x) = Bessel function of the first kind of order unity 

1 

x = 

∞ 

∑ 

n=1 

πD sin v 

l 

n+1 

2n4 

2 

x 

(n 4 1)!n!2 

2n4 

1 

where l = wavelength 

D= aperture diameter 

v = angular radius from pattern maximum. 

some light diffracted 

into this region 

wavefront 

(1.25) 

This useful formula shows the far-field irradiance distribution from 

a uniformly illuminated circular aperture of diameter, D. 


AIRY DISC DIAMETER = 2.44 l f/# 

Figure 1.27 Center of a typical diffraction pattern for a 

circular aperture 

SLIT APERTURE 

A slit aperture, which is mathematically simpler, is useful in 

relation to cylindrical optical elements. The irradiance distribution 

in the diffraction pattern of a uniformly illuminated slit aperture is 

described by 

where 

I = I 

x 0 

I 

0 

x = 

Energy Distribution in the Diffraction Pattern of a Circular or Slit Aperture 

Ring or Band 

⎡sin x ⎤ 

⎢ ⎥ 

⎣⎢ 

x ⎦⎥ 

Central Maximum 

First Dark 

First Bright 

Second Dark 

Second Bright 

Third Dark 

Third Bright 

Fourth Dark 

Fourth Bright 

Fifth Dark 

Note: Position variable (x) is defined in the text. 

2 

= peak irradiance in image 

pw sin v 

l 

where l = wavelength 

w = slit width 

v = angular deviation from pattern maximum. 

Position 

(x) 

0.0 

1.22p 

1.64p 

2.23p 

2.68p 

3.24p 

3.70p 

4.24p 

4.71p 

5.24p 

(1.26) 

Circular Aperture 

Relative 

Intensity 

(I x /I 0 ) 

1.0 

0.0 

0.0175 

0.0 

0.0042 

0.0 

0.0016 

0.0 

0.0008 

0.0 

ENERGY DISTRIBUTION TABLE 

The table below shows the major features of pure (unaberrated) 

Fraunhofer diffraction patterns of circular and slit apertures. The 

table shows the position, relative intensity, and percentage of total 

pattern energy corresponding to each ring or band. It is especially 

convenient to characterize positions in either pattern with the same 

variable x. This variable is related to field angle in the circular 

aperture case by 

l 

sin v = x pD 

l 

sin v = x pw 

where w is the slit width, p has its usual meaning, and D, w, and l 

are all in the same units (preferably millimeters). 

Linear instead of angular field positions are simply found from 

r = s″ tan (v) 

(1.29) 

where s″ is the secondary conjugate distance. This last result is often 

seen in a different form, namely the diffraction-limited spot-size 

equation. For a circular lens that was stated at the outset of this 

section: 

d = 2.44 l f/# 

Energy 

in Ring 

(%) 

83.8 

7.2 

2.8 

1.5 

1.0 

Position 

(x) 

0.0 

1.00p 

1.43p 

2.00p 

2.46p 

3.00p 

3.47p 

4.00p 

4.48p 

5.00p 

Slit Aperture 

Relative 

Intensity 

(I x /I 0 ) 

1.0 

0.0 

0.0472 

0.0 

0.0165 

0.0 

0.0083 

0.0 

0.0050 

0.0 

(1.27) 

where D is the aperture diameter. For a slit aperture, this relationship 

is given by 

(1.28) 

(see 1.24) 

This value represents the smallest spot size that can be achieved 

by an optical system with a circular aperture of a given f-number. 

Energy 

in Band 

(%) 

90.3 

4.7 

1.7 

0.8 

0.5 




The graph in figure 1.28 shows the form of both circular and slit 

aperture diffraction patterns when plotted on the same normalized 

scale. Aperture diameter is equal to slit width so that patterns between 

x-values and angular deviations in the far-field are the same. 

when dealing with Gaussian beams, the location of the focused spot 

also departs from that predicted by the paraxial equations given 

in this chapter. This is also detailed in chapter 2. 


GAUSSIAN BEAMS 

Apodization, or nonuniformity of aperture irradiance, alters 

diffraction patterns. If pupil irradiance is nonuniform, the formulas 

and results given previously do not apply. This is important to 

remember because most laser-based optical systems do not have 

uniform pupil irradiance. The output beam of a laser operating 

in the TEM 00 mode has a smooth Gaussian irradiance profile. 

Formulas to determine the focused spot size from such a beam are 

discussed in Chapter 2, Gaussian Beam Optics. Furthermore, 

NORMALIZED PATTERN IRRADIANCE (y) 

CIRCULAR APERTURE 

1.0 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

slit 

aperture 

91.0% within first bright ring 

83.9% in Airy disc 

0.0 

48 47 46 45 44 43 42 41 0 1 2 3 4 5 6 7 8 

POSITION IN IMAGE PLANE (x) 

90.3% in 

central maximum 

95.0% within the two 

adjoining subsidiary maxima 

circular 

aperture 

y = 

c 

⎛ 2J 1(x) 

⎞ 

⎜ 

⎝ x 

⎟ 

⎠ 

2 

∞ 

x 

n+1 

where J 1(x) = x ∑( 4 1) 

(n 4 1)!n!2 

n=1 

Note : J (x) is the Bessel function 

1 

2n 4 2 

of the first kind of order unity. 

2n 4 1 

p 

x = D sinv 

l 

l = wavelength 

D = aperture diameter 

v = angular radius from pattern maximum 

2 

⎛ sin x ⎞ 

p 

ys 

= ⎜ , where x w sin 

x 

⎟ 

= v 

⎝ ⎠ 

l 

l = wavelength 

w = slit width 

v = angular deviation direction of pattern 

maximum 


SLIT APERTURE 

Figure 1.28 Fraunhofer diffraction pattern of a singlet slit superimposed on the Fraunhofer diffraction pattern of a 



Lens Selection 

Having discussed the most important factors that affect a lens or 

a lens system’s performance, we will now address the practical matter 

of selecting the optimum catalog components for a particular task. 

The following useful relationships are important to keep in mind 

throughout the selection process: 

$ Diffraction-limited spot size = 2.44 ¬ f/# 

$ Approximate on-axis spot size 

of a plano-convex lens at the infinite 

conjugate resulting from spherical aberration = 

$ Optical invariant = 

Example 1: Collimating an Incandescent Source 

Produce a collimated beam from a quartz halogen bulb having 

a 1-mm-square filament. Collect the maximum amount of light 

possible and produce a beam with the lowest possible divergence 

angle. 

This problem, illustrated in figure 1.29, involves the typical tradeoff 

between light-collection efficiency and resolution (where a beam 

is being collimated rather than focused, resolution is defined by beam 

divergence). To collect more light, it is necessary to work at a low 

f-number, but because of aberrations, higher resolution (lower divergence 

angle) will be achieved by working at a higher f-number. 

In terms of resolution, the first thing to realize is that the 

minimum divergence angle (in radians) that can be achieved using 

any lens system is the source size divided by system focal length. An 

off-axis ray (from the edge of the source) entering the first principal 

point of the system exits the second principal point at the same 

angle. Therefore, increasing system focal length improves this limiting 

divergence because the source appears smaller. 

An optic that can produce a spot size of 1 mm when focusing a 

perfectly collimated beam is therefore required. Since source size is 

inherently limited, it is pointless to strive for better resolution. This 

level of resolution can be achieved easily with a plano-convex lens. 

Figure 1.29 

m = NA 

NA" . 

Collimating an incandescent source 

f 

0.067 f 

f/# 3 

While angular divergence decreases with increasing focal length, 

spherical aberration of a plano-convex lens increases with increasing 

focal length. To determine the appropriate focal length, set the 

spherical aberration formula for a plano-convex lens equal to the 

source (spot) size: 

0.067 f 

3 

f/# 

= 1 mm. 

This ensures a lens that meets the minimum performance needed. 

To select a focal length, make an arbitrary f-number choice. As 

can be seen from the relationship, as we lower the f-number (increase 

collection efficiency), we decrease the focal length, which will worsen 

the resultant divergence angle (minimum divergence = 1 mm/f). 

In this example, we will accept f/2 collection efficiency, which gives 

us a focal length of about 120 mm. For f/2 operation we would 

need a minimum diameter of 60 mm. The 01 LPX 209 fits this 

specification exactly. Beam divergence would be about 8 mrad. 

Finally, we need to verify that we are not operating below the 

theoretical diffraction limit. In this example, the numbers (1-mm 

spot size) indicate that we are not, since 

diffraction-limited spot size = 2.44 ! 0.5 mm ! 2 = 2.44 mm. 

Example 2: Coupling an Incandescent Source into a Fiber 

On pages 1.6 and 1.7 we considered a system in which the output 

of an incandescent bulb with a filament of 1 mm in diameter was 

to be coupled into an optical fiber with a core diameter of 100 µm 

and a numerical aperture of 0.25. From the optical invariant and 

other constraints given in the problem, we determined that system 

focal length is 9.1 mm, diameter = 5 mm, s = 100 mm, s″ = 10 mm, 

NA″ = 0.25, and NA = 0.025 (or f/2 and f/20). The singlet lenses 

that match these specifications are the plano-convex 01 LPX 003 

or biconvex lenses 01 LDX 003 and 01 LDX 005. The closest 

achromat would be the 01 LAO 001. 

v min = 

source size 

f 

v min 

(see eq. 1.22) 





We can immediately reject the biconvex lenses because of 

spherical aberration. We can estimate the performance of the 

01 LPX 003 on the focusing side by using our spherical aberration 

formula: 

0.067 (10) 

spot size = = 84 mm. 

3 

2 

We will ignore, for the moment, that we are not working at the 

infinite conjugate. 

This is slightly smaller than the 100-µm spot size we’re trying 

to achieve. However, since we are not working at infinite conjugate, 

the spot size will be larger than given by our simple calculation. 

This lens is therefore likely to be marginal in this situation, 

especially if we consider chromatic aberration. A better choice is the 

achromat. Although a computer ray trace would be required to 

determine its exact performance, it is virtually certain to provide adequate 

performance. 

Example 3: Symmetric Fiber-to-Fiber Coupling 

Couple an optical fiber with an 8-µm core and a 0.15 numerical 

aperture into another fiber with the same characteristics. Assume 

a wavelength of 0.5 µm. 

This problem, illustrated in figure 1.30, is essentially a 1:1 imaging 

situation. We want to collect and focus at a numerical aperture of 

0.15 or f/3.3, and we need a lens with an 8-µm spot size at this 

f-number. Based on the lens combination discussion on page 1.8, 

our most likely setup is either a pair of identical plano-convex lenses 

or achromats, faced front to front. To determine the necessary focal 

length for a plano-convex lens, we again use the spherical aberration 

estimate formula: 

0.067 f 

3 

3.3 

= 0.008 mm. 

This formula yields a focal length of 4.3 mm and a minimum 

diameter of 1.3 mm. The 01 LPX 423 meets these criteria. The 

biggest problem with utilizing these tiny, short focal length lenses 

is the practical considerations of handling, mounting, and positioning 

them. Since using a pair of longer focal length singlets would 

result in unacceptable performance, the next step might be to 

use a pair of the slightly longer focal length, larger achromats, 

such as the 01 LAO 001. The performance data, given on page 1.26, 

shows that this combination does provide the required 8-mm spot 

diameter. 

Because fairly small spot sizes are being considered here, it is 

important to make sure that the system is not being asked to work 

below the diffraction limit: 

2.44 ! 0.5 mm ! 3.3 = 4 mm . 

Since this is half the spot size caused by aberrations, it can be 

safely assumed that diffraction will not play a significant role here. 

An entirely different approach to a fiber-coupling task such as 

this would be a pair of spherical ball lenses (06 LMS series), listed 

on page 15.15, or one of the gradient-index lenses (06 LGT series), 

listed on page 15.19. 

s = f 

s"= f 


Figure 1.30 

Symmetric fiber-to-fiber coupling 


Example 4: Diffraction-Limited Performance 

Determine at what f-number a plano-convex lens being used at 

an infinite conjugate ratio with 0.5-mm wavelength light becomes 

diffraction limited (i.e., the effects of diffraction exceed those caused 

by aberration). 

To solve this problem, set the equations for diffraction-limited spot 

size and third-order spherical aberration equal to each other. The 

result depends upon focal length, since aberrations scale with focal 

length, while diffraction is solely dependent upon f-number. Substituting 

some common focal lengths into this formula, we get f/8.6 

at f = 100 mm, f/7.2 at f = 50 mm, and f/4.8 at f = 10 mm. 

2.44 0.5 m f/# = 0.067 ! 

! m ! 

f 

3 

f/# 

or 

1/4 

f/# = (54.9 ! f) . 

When working with these focal lengths (and under the conditions 

previously stated), we can assume essentially diffraction-limited 

performance above these f-numbers. Keep in mind, however, that 

this treatment does not take into account manufacturing tolerances 

or chromatic aberration, which will be present in polychromatic 

applications. 

MELLES GRIOT LENS DATABASE 

A database containing prescription information 

for most of the optical components listed in this 

catalog is included in the Melles Griot catalog on 

CD-ROM. This database, in a Zemax format, 

facilitates the determination of 

• Spot size 

• Prescription information 

• Wavefront distortion. 

Please contact our sales department for your free 

Melles Griot Catalog on CD-ROM: 

Phone: 1-800-835-2626 / (949) 261-5600 

FAX: (949) 261-7790 

E-mail: mglit@irvine.mellesgriot.com 

Non-US customers should contact the nearest 

Melles Griot office (see back cover). 





Spot Size 

In general, the performance of a lens or lens system in a specific 

circumstance should be determined by an exact trigonometric ray 

trace. Melles Griot applications engineers can supply ray-trace 

data for particular lenses and systems of catalog components on 

request. However, for certain situations, some simple guidelines 

can be used for lens selection. The optimum working conditions 

for some of the lenses in this catalog have already been presented. 

The following tables give some quantitative results for a variety 

of simple and compound lens systems that can be constructed 

from standard catalog optics. 

In interpreting these tables, remember that these theoretical values 

obtained from computer ray tracing consider only the effects 

of ideal geometric optics. Effects of manufacturing tolerances have 

not been considered. Furthermore, remember that using more than 

one element provides a higher degree of correction but makes 

alignment more difficult. When actually choosing a lens or a lens 

system, it is important to note the tolerances and specifications 

clearly described for each Melles Griot lens in the product listings. 

The tables give spot size for a variety of lenses used at several different 

f-numbers. All the tables are for on-axis, uniformly illuminated, 

collimated input light at 632.8 nm. They assume that the lens is 

facing in the direction that produces a minimum spot size. When 

the spot size caused by aberrations is smaller or equal to the 

diffraction-limited spot size, the notation “DL’’ appears next to 

the entry. The shorter focal length lenses produce smaller spot sizes 

because aberrations increase linearly as a lens is scaled up. 

Focal Length = 10 mm 

f/2 

f/3 

f/5 

f/10 

01 LDX 005 01 LPX 005 01 LAO 001 

550 

120 

30 

15 (DL) 

*Diffraction-limited performance is indicated by DL. 


Spot Size (µm)* 

95 

25 

8 (DL) 

15 (DL) 

4 

5 (DL) 

8 (DL) 

15 (DL) 

The effect on spot size caused by spherical aberration is strongly 

dependent on f-number. For a plano-convex singlet, spherical 

aberration is inversely dependent on the cube of the f-number. For 

doublets, this relationship can be even higher. On the other hand, 

the spot size caused by diffraction increases linearly with f-number. 

Thus, for some lens types, spot size at first decreases and then 

increases with f-number, meaning that there is some optimum 

performance point where both aberrations and diffraction combine 

to form a minimum. 

Unfortunately, these results cannot be generalized to situations 

where the lenses are used off axis. This is particularly true of the 

achromat/aplanatic meniscus lens combinations because their 

performance degrades rapidly off axis. 


f/2 

f/3 

f/5 

f/10 

01 LPX 049 01 LAO 024 

350 

90 

17 

15 (DL) 




80 

11 

8 (DL) 

15 (DL) 

01 LAO 059 & 

01 LAM 059 

4 

5 (DL) 

8 (DL) 

15 (DL) 

01 LDX 123 01 LPX 127 01 LAO 079 01 LAO 126 & 01 LAM 126 

f/2 

f/3 

f/5 

f/10 

800 

225 

42 

15 (DL) 

600 

200 

30 

15 (DL) 

80 

35 

9 

15 (DL) 

6 

5 (DL) 

8 (DL) 

15 (DL) 




Aberration Balancing 

To improve system performance, optical designers make sure 

that the total aberration contribution from all surfaces taken together 

sums to nearly zero. Normally, such a process requires computerized 

analysis and optimization. However, there are some simple 

guidelines that can be used to achieve this with lenses available in 

this catalog. This approach can yield systems that operate at a much 

lower f-number than can usually be achieved with simple lenses. 

Specifically, we will examine how to null the spherical aberration 

from two or more lenses in collimated, monochromatic light. Thus, 

this technique will be most useful for laser beam focusing and 

expanding. 

Figure 1.31 shows the third-order longitudinal spherical 

aberration coefficients for four of the most common positive and 

negative lens shapes when used with parallel, monochromatic 

incident light. The plano-convex and plano-concave lenses both 

show minimum spherical aberration when oriented with their curved 

surface facing the incident parallel beam. All other configurations 

exhibit larger amounts of spherical aberration. With these lens types, 

it is now possible to show how various systems can be corrected for 

spherical aberration. 

A two-element laser beam expander is a good starting example. 

In this case, two lenses are separated by a distance which is the 

sum of their focal lengths, so that the overall system focal length is 

infinite. This system will not focus incoming collimated light, but 

it will change the beam diameter. By definition, each of the lenses 

is operating at the same f-number. 

The equation for longitudinal spherical aberration shows that 

for two lenses with the same f-number, aberration varies directly with 

the focal lengths of the lenses. The sign of the aberration is the same 

as focal length. Thus, it should be possible to correct the spherical 

Figure 1.31 

positive lenses 

negative lenses 

aberration 

coefficient 

(k) 

plano-convex (reversed) 01 LPX 

plano-concave (reversed) 01 LPK 

symmetric-convex 01 LDX 

symmetric-concave 01 LDK 

longitudinal spherical aberration (3rd order) = 

kf 

f/# 2 

plano-convex (normal) 01 LPX 

plano-concave (normal) 01 LPK 

1.069 0.403 0.272 

Third-order longitudinal spherical aberration of typical lens shapes 

aberration of this Galilean-type beam expander, which consists of 

a positive focal length objective and a negative diverging lens. 

If a plano-convex lens of focal length f 1 oriented in the normal 

direction is combined with a plano-concave lens of focal length f 2 

oriented in its reverse direction, the total spherical aberration of 

the system is 

LSA = 0.272 f 

2 

f/# 

f 

f 

1 

2 

1 

1.069 f2 

+ . 

2 

f/# 

After setting this equal to zero, we obtain 

1.069 

= 4 

0.272 = 4 3.93. 

(1.30) 

To make the magnitude of aberration contributions of the two 

elements equal so they will cancel out, and thus correct the system, 

select the focal length of the positive element to be 3.93 times that 

of the negative element. 

Figure 1.32 shows a beam-expander system made up of catalog 

elements, in which the focal length ratio is 4:1. This simple system is 

corrected to about 1/6 wavelength at 632.8 nm, even though the objective 

is operating at f/4 with a 20-mm aperture diameter. This is remarkably 

good wavefront correction for such a simple system; one would 

normally assume that a doublet objective would be needed and a 

complex diverging lens as well. This analysis does not take into 

account manufacturing tolerances. 

A beam expander of lower magnification can also be derived 

from this information. If a symmetric-convex objective is used 

together with a reversed plano-concave diverging lens, the aberration 

coefficients are in the ratio of 1.069/40.403 = 42.65. Figure 1.32 

shows a system of catalog lenses that provides a magnification of 





a) CORRECTED 4!BEAM EXPANDER 

f= 420 mm 

10-mm diameter 

plano-concave 

01 LPK 001 

b) CORRECTED 2.7x BEAM EXPANDER 

f= 420 mm 


plano-concave 

01 LPK 001 

f= 80 mm 

22.4-mm diameter 

plano-convex 

01 LPX 149 

c) SPHERICALLY CORRECTED 25-mm EFL f/2.0 OBJECTIVE 

f= 425 mm 


plano-concave 

01 LPK 003 

f= 54 mm 


symmetric-convex 

01 LDX 119 

f= 50 mm (2) 


plano-convex 

01 LPX 108 

2.7 (the closest possible given the available focal lengths). The 

maximum wavefront error in this case is only 1/4 wave, even though 

the objective is working at f/3.3. 

The relatively fast speed of these objectives is a great advantage 

in minimizing the length of these beam expanders. They would be 

particularly useful with Nd:YAG and argon-ion lasers, which tend 

to have large output beam diameters. 

These same principles can be utilized to create high numerical 

aperture objectives that might be used as laser focusing lenses. 

Figure 1.32 shows an objective consisting of an initial negative 

element, followed by two identical plano-convex positive elements. 

Again, all of the elements operate at the same f-number, so that 

their aberration contributions are proportional to their focal lengths. 

To obtain zero total spherical aberration from this configuration, 

we must satisfy 

or 

1.069 f + 0.272 f + 0.272 f = 0 

f 

f 

1 

2 

1 2 2 

= 40.51. 

Therefore, a corrected system should result if the focal length of 

the negative element is just about half that of each of the positive 

lenses. In this case, f 1 = 425 mm and f 2 = 50 mm yield a total system 

focal length of about 25 mm and an f-number of approximately 

f/2. This objective, corrected to 1/6 wave, has the additional advantage 

of a very long working distance. 

UV OPTICS 


Figure 1.32 

balancing 

Combining catalog lenses for aberration 

The material presented in this section is based on the work of John 

F. Forkner. 

Melles Griot now offers a selection of UV optics 

ranging from 193 to 355 nm. See Chapter 16, 

UV Optics, for details. 


Definition of Terms 

FOCAL LENGTH (f) 

Two distinct terms describe the focal lengths associated with 

every lens or lens system. The effective focal length (EFL) or 

equivalent focal length (denoted f in figure 1.33) determines 

magnification and hence the image size. The term f appears 

frequently in the lens formulas and tables of standard lenses. 

Unfortunately, f is measured with reference to principal points 

which are usually inside the lens so the meaning of f is not 

immediately apparent when a lens is visually inspected. 

The second type of focal length relates the focal plane positions 

directly to landmarks on the lens surfaces (namely the vertices) 

which are immediately recognizable. It is not simply related to image 

size but is especially convenient for use when one is concerned about 

correct lens positioning or mechanical clearances. Examples of this 

second type of focal length are the front focal length (FFL, denoted 

f f in figure 1.33) and the back focal length (BFL, denoted f b ). 

The convention in all of the figures (with the exception of a single 

deliberately reversed ray) is that light travels from left to right. 

secondary principal surface 

primary principal point 

t 

secondary principal point 

e 

primary principal surface 

ray from object at infinity 

rear (secondary) 

focal point 

ray from object at infinity 

r 

primary vertex A 

2 

optical axis 

1 

F 

H H″ 

A 2 secondary vertex 

F″ 

front (primary) 

r 1 

focal point 

reversed ray locates front focal 

point or primary principal surface 

front focal point 

A = front focus to front 

edge distance 

B = rear edge to rear 

focus distance 

Figure 1.33 Focal length and focal points 

f f 

A 

f 

f = effective focal length; 

may be positive (as shown) 

or negative 

f f = front focal length 

f b = back focal length 

t c 

FOCAL POINT (F OR F″) 

Rays that pass through or originate at either focal point must be, 

on the opposite side of the lens, parallel to the optical axis. This 

fact is the basis for locating both focal points. 

PRIMARY PRINCIPAL SURFACE 

Let us imagine that rays originating at the front focal point F (and 

therefore parallel to the optical axis after emergence from the opposite 

side of the lens) are singly refracted at some imaginary surface, 

instead of twice refracted (once at each lens surface) as actually 

happens. There is a unique imaginary surface, called the principal 

surface, at which this can happen. 

To locate this unique surface, consider a single ray traced from 

the air on one side of the lens, through the lens and into the air on 

the other side. The ray is broken into three segments by the lens. 

Two of these are external (in the air), and the third is internal (in 

the glass). The external segments can be extended to a common 

point of intersection (certainly near, and usually within, the lens). The 

t e = edge thickness 

t c = center thickness 

f b 

B 

f 

rear focal point 

r 1 = radius of curvature of first 

surface (positive if center of 

curvature is to right) 

r 2 = radius of curvature of second 

surface (negative if center of 

curvature is to left) 





principal surface is the locus of all such points of intersection of 

extended external ray segments. The principal surface of a perfectly 

corrected optical system is a sphere centered on the focal point. 

Near the optical axis, the principal surface is nearly flat, and 

for this reason, it is sometimes referred to as the principal plane. 

SECONDARY PRINCIPAL SURFACE 

This term is defined analogously to the primary principal surface, 

but it is used for a collimated beam incident from the left and focused 

to the rear focal point F ≤ on the right. Rays in that part of the 

beam nearest the axis can be thought of as once refracted at the 

secondary principal surface, instead of being refracted by both lens 

surfaces. 

PRIMARY PRINCIPAL POINT (H) 

OR FIRST NODAL POINT 

This point is the intersection of the primary principal surface with 

the optical axis. 

SECONDARY PRINCIPAL POINT (H≤) 

OR SECONDARY NODAL POINT 

This point is the intersection of the secondary principal surface 

with the optical axis. 

CONJUGATE DISTANCES (S AND S″) 

The conjugate distances are the object distance, s, and image 

distance, s″. Specifically, s is the distance from the object to H, and 

s″ is the distance from H″ to the image location. The term infinite 

conjugate ratio refers to the situation in which a lens is either focusing 

incoming collimated light, or being used to collimate a source (therefore 

either s or s″ is infinity). 

PRIMARY VERTEX (A 1 ) 

The primary vertex is the intersection of the primary lens surface 

with the optical axis. 

SECONDARY VERTEX (A 2 ) 

The secondary vertex is the intersection of the secondary lens 

surface with the optical axis. 

BACK FOCAL LENGTH (f b ) 

This length is the distance from the secondary vertex (A 2 ) to 

the rear focal point (F″ ). 

EDGE-TO-FOCUS DISTANCES (A AND B) 

A is the distance from the front focal point to the front edge of 

the lens. B is the distance from the rear edge of the lens to the rear 

focal point. Both distances are presumed always to be positive. 

REAL IMAGE 

A real image is one in which the light rays actually converge; 

if a screen were placed at the point of focus, an image would be 

formed on it. 

VIRTUAL IMAGE 

A virtual image does not represent an actual convergence of light 

rays. A virtual image can be viewed only by looking back through 

the optical system, such as in the case of a magnifying glass. 

F-NUMBER (F/#) 

The f-number (also known as the focal ratio, relative aperture, 

or speed) of a lens system is defined to be the effective focal length 

divided by system clear aperture. Ray f-number is the conjugate 

distance for that ray divided by the height at which it intercepts the 

principal surface. 

f/# = f φ . (1.31) 

NUMERICAL APERTURE (NA) 

The numerical aperture of a lens system is defined to be the sine 

of the angle, v 1 , that the marginal ray (the ray that exits the lens 

system at its outer edge) makes with the optical axis multiplied by 

the index of refraction (n) of the medium. The numerical aperture 

can be defined for any ray as the sine of the angle made by that ray 

with the optical axis multiplied by the index of refraction: 


EFFECTIVE FOCAL LENGTH (EFL, f) 

Assuming that the lens is surrounded by air or vacuum (refractive 

index 1.0), this is both the distance from the front focal point (F) to the 

primary principal point (H) and the distance from the secondary principal 

point (H″) to the rear focal point (F″). Later we use f to designate 

the paraxial effective focal length for the design wavelength (¬ 0 ). 

FRONT FOCAL LENGTH (f f ) 

This length is the distance from the front focal point (F) to the 

primary vertex (A 1 ). 

NA = n sin v. 

MAGNIFICATION POWER 

Often, positive lenses intended for use as simple magnifiers are 

rated with a single magnification, such as 4#. To create a virtual 

image for viewing with the human eye, in principle, any positive 

lens can be used at an infinite number of possible magnifications. 

However, there is usually a narrow range of magnifications that 

will be comfortable for the viewer. Typically, when the viewer adjusts 

the object distance so that the image appears to be essentially at 


infinity (which is a comfortable viewing distance for most individuals), 

magnification is given by the relationship 

magnification = 

Thus, a 25-mm focal length positive lens would be a 10! magnifier. 

DIOPTERS 

Diopter is a term used to define the reciprocal of the focal length, 

which is commonly used for ophthalmic lenses. The inverse focal 

length of a lens expressed in diopters is 

diopters = 1000 

f 

250 mm 

f 

Thus, the smaller the focal length, the larger the power in diopters. 

DEPTH OF FIELD AND DEPTH OF FOCUS 

(f in mm). (1.32) 

(f in mm). (1.33) 

In an imaging system, depth of field refers to the distance in 

object space over which the system delivers an acceptably sharp 

image. The criteria for what is acceptably sharp is arbitrarily chosen 

by the user; depth of field increases with increasing f-number. 

For an imaging system, depth of focus is the range in image 

space over which the system delivers an acceptably sharp image. In 

other words, this is the amount that the image surface (such as a 

screen or piece of photographic film) could be moved while maintaining 

acceptable focus. Again, criteria for acceptability are defined 

arbitrarily. 

In nonimaging applications, such as laser focusing, depth of 

focus refers to the range in image space over which the focused 

spot diameter remains below an arbitrary limit. 


Technical Reference 

For further reading about the definitions and 

formulas presented here, refer to the following 

publications: 

Rudolph Kingslake, Lens Design Fundamentals 

(Academic Press) 

Rudolph Kingslake, Optical System Design 

(Academic Press) 

Warren Smith, Modern Optical Engineering 

(McGraw Hill). 

If you need help with the use of definitions and 

formulas presented in this catalog, our applications 

engineers will be pleased to assist you. 





Paraxial Lens Formulas 

PARAXIAL FORMULAS FOR LENSES IN AIR 

The following formulas are based on the behavior of paraxial 

rays, which are always very close and nearly parallel to the optical 

axis. In this region, lens surfaces are always very nearly normal to 

the optical axis, and hence all angles of incidence and refraction 

are small. As a result, the sines of the angles of incidence and 

refraction are small (as used in Snell’s law) and can be approximated 

by the angles themselves (measured in radians). 

The paraxial formulas do not include effects of spherical 

aberration experienced by a marginal ray — a ray passing through 

the lens near its edge or margin. All effective focal length values (f) 

tabulated in this catalog are paraxial values which correspond to the 

paraxial formulas. 

The following paraxial formulas are valid for both thick and 

thin lenses unless otherwise noted. The refractive index of the lens 

glass, n, is the ratio of the speed of light in vacuum to the speed of 

light in the lens glass. All other variables are defined in figure 1.33. 

Focal Length 

1 

⎛ 1 1 ⎞ (n 41) 

= (n 41) 

4 + 

f 

⎜ 

⎝ r r 

⎟ 

⎠ n 

t c 

rr 

where n is the refractive index, t c is the center thickness, and the 

sign convention previously given for the radii r 1 and r 2 applies. For 

thin lenses, t c ≅ 0, and for plano lenses either r 1 or r 2 is infinite. In 

either case the second term of the above equation vanishes, and we 

are left with the familiar Lens Maker’s formula 

1 

f 

⎛ 

= (n 41) 

⎜ 4 

⎝ 

2 

1 1 ⎞ 

⎟ 

r1 r2⎠ 

(1.35) 

(1.34) 

1 2 

1 2 

s>0 

Surface Sagitta and Radius of Curvature 

(refer to figure 1.34) 

2 2 ⎛ d ⎞ 

r = (r 4s) + ⎜ ⎟ 

⎝ 2⎠ 

2 ⎛ d ⎞ 

s = r 4 r 4⎜ 

⎟ 

⎝ 2⎠ 

r 

1 

= 

f 

2 

= s 2 + d . 

8s 

2 

2 

> 0 

An often useful approximation is to neglect s/2. 

Symmetric Lens Radii (r 2 = 5r 1 ) 

With center thickness constrained, 

⎡ 

⎤ 

ft 

2 c 

r 1 = (n 4 1) 

⎢ ⎛ ⎞ 

f ± f 4 

⎥ 

⎢ ⎜ 

n 

⎟ 

⎝ ⎠ 

⎥ 

⎣⎢ 

⎦⎥ 

⎡ 

⎤ 

tc 

= (n 4 1) f 

⎢ ⎛ ⎞ 

1 + 14 

⎥ 

⎢ ⎜ 

⎝ nf 

⎟ 

⎠ 

⎥ 

⎣⎢ 

⎦⎥ 

2 (n41) 

(n41) 

4 

r nr 

1 

2 

⎡ ⎧⎪ 

⎛ f ⎞ ⎫⎪ 

⎤ 

⎢t + 2r ⎨14 

cos ⎜arcsin ⎝ 2r 

⎟ ⎬⎥ 

⎩⎪ 

1 

⎣ 

⎢ 

⎠ ⎭⎪ ⎦ 

⎥ 

(1.40) 

2 c 1 

1 

(1.36) 

(1.37) 

(1.38) 

(1.39) 

where, in the first form, the + sign is chosen for the square root if f is 

positive, but the 4 sign must be used if f is negative. In the second 

form, the + sign must be used regardless of the sign of f. With edge 

thickness constrained, the equation for r 1 becomes transcendental: 

where Ω is the lens diameter. This equation can be solved by numerical 

methods. 

Plano Lens Radius 

Since r 2 is infinite, 

r>0 

d 

2 

r 1 = (n 4 1) f. 

Principal-Point Locations (signed distances from vertices) 

(1.41) 


(r4s) 

Figure 1.34 Surface sagitta and radius of curvature 

A H ′′ = 

2 

AH = 

1 

4rt 

2 c 

n (r 4 r ) + t (n 4 1) 

2 1 c 

4rt 

1 c 

n (r 4 r ) + t (n 4 1) 

2 1 c 

where the above sign convention applies. 

(1.42) 

(1.43) 


For symmetric lenses (r 2 = 4r 1 ), 

AH = 4A H′′ 

1 2 

AH = 0 

1 

= 

and 

t c 

A2H ′′ = 4 . 

n 

rt 1 c 

. 

2nr 4 t (n 4 1) 

1 c 

⎧ 

⎪ f ⎡1 

(n 1) 

HH ′′ = t c ⎨1 

4 ⎢ 4 4 

n f n 

⎣⎢ 

⎩⎪ 

⎛ 1 ⎞ 

HH ′′ = t c ⎜1 

4 ⎟ . 

⎝ n ⎠ 

Q = 2 p(1 4cos v) 

⎛ v ⎞ 

2 

= 4 p sin ⎜ 

⎝ 2 

⎟ 

⎠ 

⎛ f ⎞ 

v = arctan ⎜ 

⎝ 2 s′′ 

⎟ 

⎠ 

2 

t ⎤⎫ 

c ⎪ 

⎥⎬ 

rr 1 2⎦⎥ 

⎭⎪ 

(1.44) 

If either r 1 or r 2 is infinite, l’Hôpital’s rule from calculus must be used. 

Thus, referring to page 1.27, for plano-convex lenses in the correct 

orientation, 

(1.45) 

For flat plates, by letting r 1 →∞ in a symmetric lens, we obtain 

A 1 H = A 2 H″ = t c /2n. These results are useful in connection with 

the following paraxial lens combination formulas. 

Hiatus or Interstitium (principal-point separation) 

(1.46) 

which, in the thin-lens approximation (exact for plano lenses), 

becomes 

(1.47) 

Solid Angle 

The solid angle subtended by a lens, for an observer situated at an 

on-axis image point, is 

where this result is in steradians, and where 

(1.48) 

is the apparent angular radius of the lens clear aperture. For an 

observer at an on-axis object point, use s instead of s″. To convert 

from steradians to the more intuitive sphere units, simply divide 

Q by 4p. If the Abbé sine condition is known to apply, ß may 

be calculated using the arc sine function instead of the arc 

tangent. 

Back Focal Length 

and 

f = f" + AH′′ 

b 2 

= f " 4 

t c 

f b = f " 4 

n . 

f = f 4 AH 

f 1 

= f + 

rt 2 c 

n(r 4 r ) + t (n 4 1) 

2 1 c 

rt 1 c 

n(r 4 r ) + t (n 4 1) 

2 1 c 

m = s ′′ 

s 

f 

= 

s 4 f 

= s ′′ 4 f . 

f 

PARAXIAL FORMULAS FOR 

LENSES IN ARBITRARY MEDIA 

(1.49) 

where the sign convention presented above applies to A 2 H″ and to 

the radii. If r 2 is infinite, l’Hôpital’s rule from calculus must be used, 

whereby 

Front Focal Length 

(1.50) 

(1.50) 

(1.51) 

where the sign convention presented above applies to A 1 H and to 

the radii. If r 1 is infinite, l’Hôpital’s rule from calculus must be used, 

whereby 

t c 

f f = f 4 

n . 

(1.52) 

Edge-to-Focus Distances 

For positive lenses, 

A = f + s 

f 1 

B = f + s 

b 2 

(1.53) 

(1.54) 

where s 1 and s 2 are the sagittas of the first and second surfaces. 

Bevel is neglected. 

Magnification or Conjugate Ratio 

(1.55) 

These formulas allow for the possibility of distinct and completely 

arbitrary refractive indices for the object space medium (refractive 

index n′), lens (refractive index n″), and image space medium (refractive 

index n). In this situation, the effective focal length assumes two 

distinct values, namely f in object space and f″ in image space. It is 

also necessary to distinguish the principal points from the nodal 

points. The lens serves both as a lens and as a window separating 

the object space and image space media. 





The situation of a lens immersed in a homogenous fluid (figure 

1.35) is included as a special case (n = n″). This case is of 

considerable practical importance. The two values f and f″ are again 

equal, so that the lens-combination formulas are applicable to 

systems immersed in a common fluid. The general case (two different 

fluids) is more difficult, and it must be approached by ray tracing on 

a surface-by-surface basis. 

LENS CONSTANT (k) 

This number appears frequently in the following formulas. It is 

an explicit function of the complete lens prescription (both radii, 

t c and n′ ) and both media indices (n and n″). This dependence is 

implicit anywhere that k appears. 

k = n ′ 4 n + n ′′ 4 n ′ t c(n′ 4n)(n′′ 4 n ′) 

4 . (1.56) 

r r 

nrr ′ 

f = n k 

n 

s + n ′′ 

s′′ 

1 2 

Effective Focal Lengths 

f ′′ = n ′′ 

k . 

Lens Formula (Gaussian form) 

= k. 

Lens Formula (Newtonian form) 

xx ′′ = ff ′′ = nn ′′ 

k 2 

where x = s4f and x″ = s″4f ″. 

index n = 1 (air or vacuum) 

f f 

1 2 

f 

(1.57) 

(1.58) 

(1.59) 

Principal-Point Locations 

nt c 

⎛ n ′′ 4n′ 

⎞ 

AH 1 = ⎜ ⎟ 

(1.60) 

k ⎝ nr ′ 2 ⎠ 

4n′′ tc 

⎛ n ′ 4 n⎞ 

A2H ′′ = ⎜ ⎟ . (1.61) 

k ⎝ nr ′ ⎠ 

Object-to-First-Principal-Point Distance 

s = 

s ′′ = 

ns′′ 

. 

ks ′′ 4 n′′ 

Second Principal-Point-to-Image Distance 

Magnification 

m = ns ′′ 

. 

n′′ 

s 

index n"= 1.333 (water) 

f″ 

f b 

1 

n′′ 

s 

. (1.63) 

ks 4 n 

Lens Maker’s Formula 

n 

f 

= n ′′ 

f ′′ 

AN 1 = AH+HN 

1 

A N ′′ = A H ′′ +H′′ N ′′. 

2 2 

= k. 

Nodal-Point Locations 

(1.62) 

(1.64) 

(1.65) 

(1.66) 

(1.67) 

A 1 A 2 

F 

H H″ N N″ 

F″ 


Figure 1.35 

Symmetric lens with disparate object and image space indices 

index n′ = 1.51872 (BK7) 


Separation of Nodal Point 

from Corresponding Principal Point 

HN = H″N″ = (n″4n)/k, positive for N to right of H 

and N″ to right of H″. 

Back Focal Length 

f b = f ′′ + AH. 2 ′′ (see eq. 1.49) 

Front Focal Length 

f f = f 4 A1H. 

(see eq. 1.51) 

Focal Ratios 

The focal ratios are f/f and f ″/f, where f is the diameter of the 

clear aperture of the lens. 

Numerical Apertures 

n sin v 

⎛ f ⎞ 

where v = arcsin ⎜ 

⎝ 2s 

⎟ 

⎠ 

and 

n′′ 

sin v" 

⎛ f ⎞ 

where v" = arcsin ⎜ 

⎝ 2s ′′ 

⎟ 

⎠ 

. 

Solid Angles (in steradians) 

Q = 2 p(1 4 cos v) 

⎛ v ⎞ 

2 

= 4p 

sin ⎜ 

⎝ 2 

⎟ 

⎠ 

⎛ f ⎞ 

where v = arctan ⎜ 

⎝ 2s 

⎟ 

⎠ 

(see eq.1.48) 

Q ″ = 2p 2 p 

(1 4 cos v″) 

′′) 

⎛ v ⎞ 

2 

= 4p 

sin ⎜ 

⎝ 2 

⎟ 

(1.68) 

⎠ 

⎛ f ⎞ 

where v′′ 

= arctan ⎜ 

⎝ 2s ′′ 

⎟ 

⎠ 

. 

To convert from steradians to spheres, simply divide by 4p. 


For Quick Approximations 

Much time and effort can be saved by ignoring the 

differences among f, f b , and f f in these formulas 

(assume f = f b = f f ) by thinking of s as the lens-toobject 

distance, by thinking of s″ as the lens-to-image 

distance, and by thinking of the sum of conjugate 

distances s + s″ as being the object-to-image distance. 

This is known as the thin-lens approximation. 


Physical Significance of the Nodal Points 

A ray directed at the primary nodal point N of a lens 

appears to emerge from the secondary nodal point 

N″ without change of direction. Conversely, a ray 

directed at N″ appears to emerge from N without 

change of direction. At the infinite conjugate ratio, 

if a lens is rotated about a rotational axis orthogonal 

to the optical axis at the secondary nodal point 

(i.e., if N″ is the center of rotation), the image 

remains stationary during the rotation. This fact 

is the basis for the nodal slide method for measuring 

nodal-point location. The nodal points coincide with 

their corresponding principal points when the image 

space and object space refractive indices are equal (n 

= n″). This makes the nodal slide method the most 

precise method of principal-point location. 




Principal-Point Locations 

Figure 1.36 indicates approximately where the principal points fall 

in relation to the lens surfaces for various standard lens shapes. The 

exact positions depend on the index of refraction of the lens material, 

and on the lens radii, and can be found by formula. In extreme 

meniscus lens shapes (short radii or steep curves), it is possible that 

both principal points will fall outside the lens boundaries. For 

symmetric lenses, the principal points divide that part of the optical 

axis between the vertices into three approximately equal segments. 

For plano lenses, one principal point is at the curved vertex, and the 

other is approximately one-third of the way to the plane vertex. 


F″ 

H″ F″ H″ 

H″ 

F″ 

F″ 

H″ 

F″ 

F″ 

H″ 

H″ 

F″ 

F″ 

H″ 

H″ 

H″ 

F″ 

F″ 

H″ 


Figure 1.36 

Principal points of common lenses 


Gaussian Beam Optics 2 

Introduction to Gaussian Beam Optics 2.2 

Transformation and Magnification by Simple Lenses 2.6 

Lens Selection 2.10 

2.1 1 





Introduction to Gaussian Beam Optics 

In most laser applications it is necessary to focus, modify, or 

shape the laser beam by using lenses and other optical elements. In 

general, laser-beam propagation can be approximated by assuming 

that the laser beam has an ideal Gaussian intensity profile, 

corresponding to the theoretical TEM 00 mode. Coherent Gaussian 

beams have peculiar transformation properties that require special 

consideration. In order to select the best optics for a particular laser 

application, it is important to understand the basic properties of 

Gaussian beams. Unfortunately, the output from real-life lasers is 

not truly Gaussian (although helium neon lasers and argon-ion 

lasers are a very close approximation). To accommodate this variance, 

a quality factor, M 2 (called the “M-square” factor), has been defined 

to describe the deviation of the laser beam from a theoretical 

Gaussian. For a theoretical Gaussian, M 2 =1; for a real laser beam, 

M 2 >1. Helium neon lasers typically have an M 2 factor that is less 

than 1.1. For ion lasers, the M 2 factor is typically between 1.1 and 

1.3. Collimated TEM 00 diode laser beams usually have an M 2 factor 

ranging from 1.1 to 1.7. For high-energy multimode lasers, the M 2 

factor can be as high as 3 or 4. In all cases, the M 2 factor, which 

varies significantly, affects the characteristics of a laser beam and 

cannot be neglected in optical designs. 

In the following discussion, we will first treat the characteristics 

of a theoretical Gaussian beam (M 2 = 1) and then show how these 

characteristics change as the beam deviates from the theoretical. In 

all cases, a circularly symmetric wavefront is assumed, as would be 

the case for a helium neon laser or an argon-ion laser. Diode laser 

beams are asymmetric and often astigmatic, which causes their 

transformation to be more complex. 

Although in some respects component design and tolerancing 

for lasers are more critical than they are for conventional optical 

components, the designs often tend to be simpler since many of 

the constants associated with imaging systems are not present. For 

instance, laser beams are nearly always used on axis, which eliminates 

the need to correct asymmetric aberration. Chromatic aberrations 

are of no concern in single-wavelength lasers, although they are 

critical for some tunable and multiline laser applications. In fact, the 

only significant aberration in most single-wavelength applications 

is primary (third-order) spherical aberration. 

Scatter from surface defects, inclusions, dust, or damaged coatings 

is of greater concern in laser-based systems than in incoherent 

systems. Speckle content arising from surface texture and beam 

coherence can limit system performance. 

Because laser light is generated coherently, it is not subject to 

some of the limitations normally associated with incoherent sources. 

All parts of the wavefront act as if they originate from the same 

point, and consequently the emergent wavefront can be precisely 

defined. Starting out with a well-defined wavefront permits more 

precise focusing and control of the beam than would otherwise be 

possible. 

In order to gain an appreciation of the principles and limitations 

of Gaussian beam optics, it is necessary to understand the nature of 

the laser output beam. In TEM 00 mode, the beam emitted from a laser 

is a perfect plane wave with a Gaussian transverse irradiance profile 

as shown in figure 2.1. The Gaussian shape is truncated at some 

diameter either by the internal dimensions of the laser or by some 

limiting aperture in the optical train. To specify and discuss propagation 

characteristics of a laser beam, we must define its diameter 

in some way. The commonly adopted definition is the diameter at 

which the beam irradiance (intensity) has fallen to 1/e 2 (13.5%) of its 

peak, or axial, value. 

BEAM WAIST AND DIVERGENCE 

Diffraction causes light waves to spread transversely as they 

propagate, and it is therefore impossible to have a perfectly collimated 

beam. The spreading of a laser beam is in precise accord with the 

predictions of pure diffraction theory; aberration is totally insignificant 

in the present context. Under quite ordinary circumstances, 

the beam spreading can be so small it can go unnoticed. The following 

formulas accurately describe beam spreading, making it 

easy to see the capabilities and limitations of laser beams. The 

notation is consistent with much of the laser literature, particularly 

with Siegman’s excellent Introduction to Lasers and Masers 

(McGraw-Hill). 

PERCENT IRRADIANCE 

100 

80 

60 

40 

20 

13.5 

Figure 2.1 

41.5w 4w 0 w 1.5w 

CONTOUR RADIUS 

Irradiance profile of a Gaussian TEM 00 mode 


Even if a Gaussian TEM 00 laser-beam wavefront were made 

perfectly flat at some plane, with all elements moving in precisely 

parallel directions, it would quickly acquire curvature and begin 

spreading in accordance with 

⎡ 

2 

⎛ 2 

p w ⎞ ⎤ 

0 

R(z) = z 

⎢ 

1 + 

⎥ 

⎢ ⎜ ⎟ 

⎝ lz 

⎥ 

⎠ 

⎣⎢ 

⎦⎥ 

and 

⎡ 

lz 

w(z) = w 0 + ⎛ 2 

⎞ ⎤ 

⎢1 

2 

⎝ ⎜ ⎥ 

⎢ ⎟ 

p w0 

⎠ ⎥ 

⎣ 

⎦ 

12 / 

where z is the distance propagated from the plane where the wavefront 

is flat, l is the wavelength of light, w 0 is the radius of 

the 1/e 2 irradiance contour at the plane where the wavefront is flat, w(z) 

is the radius of the 1/e 2 contour after the wave has propagated a 

distance z, and R(z) is the wavefront radius of curvature after 

propagating a distance z. R(z) is infinite at z = 0, passes through 

a minimum at some finite z, and rises again toward infinity as 

z is further increased, asymptotically approaching the value of z itself. 

The plane z = 0 marks the location of a Gaussian waist, or a place 

where the wavefront is flat, and w 0 is called the beam waist radius. 

A waist occurs naturally at the midplane of a symmetric confocal 

cavity. Another waist occurs at the surface of the planar mirror 

of the quasi-hemispherical cavity used in many Melles Griot lasers. 

The irradiance distribution of the Gaussian TEM 00 beam, 

namely, 

I (r) = I e = 

2P 

w e 2 

p 

42r 

2 / w 

2 42r 

2 / w 

2 

0 

where w = w(z) and P is the total power in the beam, is the same 

at all cross sections of the beam. The invariance of the form of the 

distribution is a special consequence of the presumed Gaussian 

distribution at z = 0. If a uniform irradiance distribution had been 

presumed at z = 0, the pattern at z = ∞ would have been the familiar 

Airy disc pattern given by a Bessel function, while the pattern at 

intermediate z values would have been enormously complicated. (See 

Born and Wolf, Principles of Optics, 2d ed, Pergamon/ Macmillan). 

Simultaneously, as R(z) asymptotically approaches z for large 

z, w(z) asymptotically approaches the value 

w(z) 

lz 

≅ 

p 

w 0 

where z is presumed to be much larger than pw 0 /l so that the 1/e 2 

irradiance contours asymptotically approach a cone of angular 

radius 

v 

= w(z) 

z 

l 

= . 

pw 0 

, 

(2.1) 

(2.2) 

(2.3) 

(2.4) 

(2.5) 

This value is the far-field angular radius of the Gaussian TEM 00 

beam. The vertex of the cone lies at the center of the waist (see 

figure 2.2). 

It is important to note that, for a given value of l, variations of 

beam diameter and divergence with distance z are functions of a 

single parameter. This is often chosen to be w 0 , or the beam waist 

radius. 

The direct relationship between beam waist and divergence 

(v ∝ 1/w 0 ) must always be considered when focusing a TEM 00 laser 

beam. Because of this relationship, the spectrally selective coating 

of the spherical output mirror of a Melles Griot laser is actually supported 

on the concave inner surface of a weak meniscus lens. In 

this paraxial, high f-number configuration, the lens introduces no 

significant aberration. A new beam waist, larger than the intracavity 

beam waist, is formed by this lens near its output pupil. The 

transformed beam has greatly reduced divergence, which is 

advantageous for most applications. Note that it is the 1/e 2 beam 

diameter of this extracavity waist that is published in this catalog. 

As an example to illustrate the relationship between beam waist 

and divergence, let us consider the real case of a Melles Griot red 

5-mW HeNe laser, 05 LHR 151, with a specified beam diameter of 

0.8 mm (i.e., w 0 = 0.4 mm). In the far-field region, 

l 

v = 

pw = 632.8 × 

( p) 

(0.4) 

Using the asymptotic approximation, at a distance of z = 100 m, 

w 

w 0 

w 0 

0 

w(z) = zv 

5 4 

= (10 )( 5.04 × 10 

4 ) 

= 50.4 mm 

1 

e 2 

6 

10 5 54 

irradiance surface 

v 

= 5.04 × 10 rad. 

which is approximately 126 times larger than w 0 . 

asymptotic cone 

z 

w 0 

Figure 2.2 Growth in 1/e 2 contour radius with distance 

propagated away from Gaussian waist 





Suppose instead that we decide to reduce the divergence 

by directing the laser into a beam expander (reversed telescope) 

of angular magnification m = 10, such as Melles Griot model 

09 LBM 013 (figure 2.3). Consider the case in which the expander 

is focused to form a waist of radius w 0 = 4.0 mm at the expander 

output lens. Since v ∝ 1/w 0 , by definition, v is reduced by a factor 

of 10; therefore, for z = 100 m, 

For the expanded beam, the ratio w(z)/w 0 is only a factor of 12.6 

for a distance of 100 m, but it is a factor of 126 for the same distance 

when the laser is used alone. 

OPTIMUM COLLIMATION 

Typically, one has a fixed value for w 0 and uses the previously given 

expression to calculate w(z) for an input value of z. However, one can 

also utilize this equation to see how final beam radius varies with starting 

beam radius at a fixed distance, z. Figure 2.4 shows the Gaussian 

beam propagation equation plotted as a function of w 0 , with the 

particular values of l = 632.8 nm and z = 100 m. 

The beam radius at 100 m reaches a minimum value for a starting 

beam radius of about 4.5 mm. Therefore, if we wanted to achieve 

the best combination of minimum beam diameter and minimum 

beam spread (or best collimation) over a distance of 100 m, our 

optimum starting beam radius would be 4.5 mm. Any other starting 

value would result in a larger beam at z = 100 m. 

We can find the general expression for the optimum starting 

beam radius for a given distance, z. Doing so yields 

w (optimum) = 

0 

5 5 4 

w(z) = (10 )( 504 . × 10 ) 

= 5.04 mm. 

10 

1/2 

⎛ lz 

⎞ 

⎜ 

⎝ p 

⎟ . (2.6) 

⎠ 

Using this optimum value of w 0 will provide the best combination 

of minimum starting beam diameter and minimum beam 

spread (ratio of w(z)/w 0 ) over the distance z. The previous example 

of z = 100 and l=632.8 nm gives w 0 (optimum) = 4.48 mm, shown 

graphically in figure 2.4. If we put this value for w 0 (optimum) back 

into the expression for w(z), w(z) = √}}2 w 0 . Thus, for this example, 

w(100) = √}}2 (4.48) = 6.3 mm. 

By turning this previous equation around, we can define a 

distance, called the Rayleigh range (z R ), over which the beam radius 

spreads by a factor of √}}2 as 

z R = 

with 

If we use beam-expanding optics (such as the 09 LBC, 09 LBX, 

09 LBZ, or 09 LCM series), which allow us to adjust the position 

of the beam waist, we can actually double the distance over which 

beam divergence is minimized. Figure 2.5 illustrates this situation, 

in which the beam starts off at a value of w(z R ) = (2lz/p) 1/2 , goes 

through a minimum value of w 0 = w(z R )/√}}2 , and then returns to 

w(z R ). By focusing the beam-expanding optics to place the beam 

waist at the midpoint, we can restrict beam spread to a factor of √}}2 

over a distance of 2z R , as opposed to just z R . 

This result can now be used in the problem of finding the starting 

beam radius that yields the minimum beam diameter and beam 

spread over 100 m. Using 2z R = 100, or z R = 50, and l = 632.8 nm, 

we get a value of w(z R ) = (2lz/p) 1/2 = 4.5 mm, and w 0 = 3.2 mm. 

Thus, the optimum starting beam radius is the same as previously 

calculated. However, by focusing the expander we achieve a final 

beam radius that is no larger than our starting beam radius, while 

still maintaining the √}}2 factor in overall variation. 

Alternately, if we started off with a beam radius of 6.3 mm 

(√}}2w 0 ), we could focus the expander to provide a beam waist of 

w 0 = 4.5 mm at 100 m, and a final beam radius of 6.3 mm at 200 m. 

FINAL BEAM RADIUS (mm) 

100 

80 

60 

40 

20 

pw 

l 

2 

0 

w(z ) = 2w 

. 

R 0 

0 

0 1 2 3 4 5 6 7 8 9 10 

STARTING BEAM RADIUS w 0 (mm) 

(2.7) 


Figure 2.3 

telescope) 

Laser beam expander 09 LBM 013 (reversed 

Figure 2.4 Beam radius at 100 m as a function of starting 

beam radius for a HeNe laser at 632.8 nm 


eam expander 

INCORPORATING M 2 INTO THE BASIC EQUATIONS 

The following discussion is taken from the analysis by Sun [Haiyin 

Sun, “Thin Lens Equation for a Real Laser Beam with Weak Lens 

Aperture Truncation,” Opt. Eng. 37, no. 11 (November 1998)]. From 

equation 2.5 we see that, for a theoretical Gaussian beam, the smallest 

possible value of the radius-divergence product is 

w 0 v = l/p. 

For a real laser beam, we have 

w 0M v M = M 2 l/p >l/p 

where w 0M and v M are the 1/e 2 intensity waist radius and the farfield 

half-divergent angle of the real laser beam, respectively, and 

M 2 factors into equations 2.1 and 2.2 as follows: 

w M (z) = w 0M [1+(zlM 2 /pw 0M 2 ) 2 ] 1/2 

R M (z) = z[1+(pw 0M 2 /zlM 2 ) 2 ] 

where w M and R M are the 1/e 2 intensity radius of the beam and the 

beam wavefront radius at z, respectively. 

The definition for the Rayleigh range (equation 2.7) remains 

the same for a real laser beam and becomes 

z R = pw 0R 2 /l. 

z R 

Together, equations 2.9, 2.10, and 2.11 form a complete set to 

denote the input of a real laser beam into a thin lens. 

w 0 

w(–z R ) = √2w 0 

w(z R ) = √2w 0 

Figure 2.5 Focusing a beam expander to minimize beam 

radius and spread over a specified distance 

z R 

(2.8) 

(2.9) 

(2.10) 

(2.11) 

LASERS AND LASER SYSTEMS 

Melles Griot manufactures many types of lasers and 

laser systems for laboratory and OEM applications. 

These, along with a wide variety of laser accessories, are 

found in Chapter 41 through 47. Laser types include 

helium neon (HeNe) and helium cadmium (HeCd) lasers; 

argon, krypton, and mixed gas (argon/krypton) ion 

lasers; diode lasers, and diode-pumped solid-state 

(DPSS) lasers. 






Transformation and Magnification by Simple Lenses 

It is already clear from the previous discussion that Gaussian 

beams transform in an unorthodox manner. Siegman uses matrix 

transformations to treat the general problem of Gaussian beam 

propagation with lenses and mirrors. A less rigorous, but in many 

ways more insightful, approach to this problem has been developed 

by Self [S.A. Self, “Focusing of Spherical Gaussian Beams,” Appl. 

Opt. 22, no. 5 (March 1983): 658]. Self shows a method to model 

transformations of a laser beam through simple optics, under 

paraxial conditions, by calculating the Rayleigh range and beam 

waist location following each individual optical element. These 

parameters are calculated using a formula analogous to the 

well-known standard lens formula. Melles Griot engineers have 

found this method to be particularly useful. The main points are as 

follows. 

The standard lens equation can be written in dimensionless 

form: 

1 

s/f 

+ 

1 

s ″/f 

1 

s + z s f) + 1 

= 1 2 

s f 

R /( 4 ″ 

or, in dimensionless form, 

1 

(s/f) + (z /f) /(s/f 1) + 1 

= 1. 

2 

4 (s″/f) 

R 

= 1. 

For Gaussian beams, Self has derived an analogous formula by 

assuming that the waist of the input beam represents the object, 

and the waist of the output beam represents the image. The formula 

is expressed in terms of the Rayleigh range of the input beam. 

In the regular form, 

In the far-field limit as z R → 0, this reduces to the geometric 

optics equation. A plot of (s/f) versus (s″/f) for various values of 

(z R /f) is shown in figure 2.6. There are three distinct regions of 

interest. For a positive thin lens, these correspond to real object 

and real image, real object and virtual image, and virtual object 

and real image. 

The main differences between Gaussian beam optics and 

geometric optics, highlighted in such a plot, can be summarized as 

follows: 

$ There is a maximum and minimum image distance for 

Gaussian beams. 

(2.12) 

(2.13) 

(2.14) 

$ The maximum image distance occurs at s = f + z R , rather than 

at s = f. 

$ There is a common point in the Gaussian beam expression 

at s/f = s″/f =1. For a simple positive lens, this is the point at 

which the incident beam has a waist at the front focus and the 

emerging beam has a waist at the rear focus. 

IMAGE DISTANCE (s"/f) 

5 

4 

3 

2 

1 

0 

41 

42 

43 

44 

45 

parameter 

44 

43 

z 

( 

R 

f ) 

42 

0 

0.25 

0.50 

1 2 

41 0 1 2 3 4 5 

OBJECT DISTANCE 

(s/f) 

Figure 2.6 Plot of the lens formula for Gaussian beams, 

with normalized Rayleigh range of the input beam as 

the parameter 

$ A lens appears to have a shorter focal length as z R /f increases 

from zero (i.e., there is a Gaussian focal shift). 

Self recommends calculating z R , w 0 , and the position of w 0 for 

each optical element in the system in turn so that the overall transformation 

of the beam can be calculated. To carry this out, it is 

also necessary to consider magnification: w 0 ″/w 0 . The magnification 

is given by 

m = w 0″ 

w = 1 

. 

0 ⎧ 2 2 

⎨[ 14 

(s/f)] 

+(z R/f) 

⎫ 

⎬ 

⎩ 

⎭ 

The Rayleigh range of the output beam depends on m 2 , as can 

be seen from the previous example, and is given by 

z 

″ = m z . 

R 

2 

R 

All the above formulas are written in terms of the Rayleigh range 

of the input beam. Unlike the geometric case, the formulas are not 

symmetric with respect to input and output beam parameters. For 

back tracing beams, it is useful to know the Gaussian beam formula 

in terms of the Rayleigh range of the output beam: 

1 

s + 1 

2 

s ″ + z ″ /(s″ 

4 f ) 

R 

= 1 f . 

(2.15) 

(2.16) 

(2.17) 


M 2 AND THE LENS EQUATION 

For real-world beams, the lens equation can be modified to 

incorporate M 2 . Equation 2.12 becomes 

1/[s+(z R /M 2 ) 2 /(s-f)]+1/2″ = 1/f, 

and equation 2.14 transforms to 

1/[(s/f)+(z R /M 2 f) 2 /(s/f-1)]+1/(s″/f) = 1. 

BEAM CONCENTRATION 

The spot size and focal position of a Gaussian beam can be 

determined from the previous equations. Two cases of particular 

interest occur when s = 0 (the input waist is at the first principal 

surface of the lens system) and s = f (the input waist is at the front 

focal point of the optical system). For s = 0, we get 

and 

s ″ = 

w = 

f 

1 + ( lf/ pw 

) 

2 2 

0 

lf/ pw0 

2 

[ 1 + ( lf/ pw0 

) ] 

/ 

2 12 

. 

For the case of s = f, the equations for image distance and waist 

size reduce to the following: 

and 

s ″ = f 

w = lf/ p w 0 

. 

Substituting typical values into these equations yields nearly 

identical results, and for most applications, the simpler, second set 

of equations can be used. 

In many applications, a primary aim is to focus the laser to a very 

small spot, as shown in figure 2.7, by using either a single lens or a 

combination of several lenses. Melles Griot has designed a series of 

single lenses optimized for this specific purpose. For example, by 

using a 05 LHR 151 laser and a focusing singlet, 01 LFS 033, the 

formula should be modified as follows: 

46 

4lf 

w(z) 

3 w = 4(632.8 × 10 )( 7) 

≅ 

p 

( 3)( 0. 

4p) 

43 

= 4.70 × 10 mm 

= 4.7 mm. 

(2.18) 

(2.19) 

(2.20) 

(2.21) 

The factor 4/3 arises because of the careful balance of spherical 

aberration and diffraction designed into the singlet. The ratio f/w 

is proportional to lens f-number, but is not equal to it. 

If a particularly small spot is desired, there is an advantage to 

using a well-corrected high-numerical-aperture microscope objective 

(see Chapter 29, Microscope Components, Spatial Filters and 

Apertures) to concentrate the laser beam. The principal advantage 

of the microscope objective over a simple lens is the diminished 

level of spherical aberration. Although microscope objectives are 

often used for this purpose, they are never designed for use at the 

infinite conjugate ratio. Suitably optimized lens systems, which 

Melles Griot can design and build on special request, are more 

effective in beam-concentration tasks. 

DEPTH OF FOCUS 

Depth of focus (±D z), that is, the range in image space over 

which the focused spot diameter remains below an arbitrary limit, 

can be derived from the formula 

⎡ 

2 

⎛ lz 

⎞ ⎤ 

w(z) = w ⎢ 

0 1 + ⎥ 

⎢ ⎜ 2 ⎟ 

⎝ pw0 

⎠ ⎥ 

⎣ 

⎦ 

Dz 

2 

0.32pw 0 

≈ ± . 

l 

12 / 

The first step in performing a depth-of-focus calculation is to set 

the allowable degree of spot size variation. If we choose a typical 

value of 5%, or w(z) = 1.05w 0 , and solve for z = D z, the result is 

By applying this result to the combination of the 05 LHR 151 

laser and laser-line focusing singlet 01 LFS 033, we find 

Dz = 0.32 p( 470 . × 10 ) 

± 

47 

6328 × 10 

= ± 35.1 mm. 

. 

43 

2 

(2.22) 

Since the depth of focus is proportional to the square of focal 

spot size, and focal spot size is directly related to f-number, the 

depth of focus is proportional to the square of the f-number of the 

focusing system. 

2w 0 

w 

1 

D beam 

e 2 

Figure 2.7 Concentration of a laser beam by a laser-line 

focusing singlet 






TRUNCATION 

In a diffraction-limited lens, the diameter of the image spot is 

d = K 

INTENSITY 

1.0 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

Figure 2.8 

plane 

INTENSITY 

1.0 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

Figure 2.9 

plane 

× l × 

f/# 

50% 

intensity 

13.5% 

intensity 

2.44 l (f-number) 

Airy disc intensity distribution at the image 

1.83 l (f-number) 

50% 

intensity 

13.5% 

intensity 

Gaussian intensity distribution at the image 

(2.23) 

where K is a constant dependent on truncation ratio and pupil 

illumination, l is the wavelength of light, and f/# is the speed of the 

lens at truncation. The intensity profile of the spot is strongly dependent 

on the intensity profile of the radiation filling the entrance 

pupil of the lens. For uniform pupil illumination, the image spot takes 

on an Airy disc intensity profile as shown in figure 2.8. If the pupil 

illumination is Gaussian in profile, an image spot of Gaussian 

profile results as shown in figure 2.9. When the pupil illumination 

is between these two extremes, a hybrid intensity profile results. 

In the case of the Airy disc, the intensity falls to zero at the 

point d zero = 2.44 ! l ! f/#, defining the diameter of the spot (see 

figure 2.8). When the pupil illumination is not uniform, the image 

spot intensity never falls to zero making it necessary to define the 

diameter at some other point. This is commonly done for two 

points: 

d FWHM = 50% intensity point 

and 

d 2 = 13.5% intensity point. 

1/e 

It is helpful to introduce the truncation ratio 

T = D D 

The k function, plotted in figure 2.10, permits calculation of 

on-axis spot diameter for any beam truncation ratio. 

The optimal choice for truncation ratio depends on the relative 

importance of spot size, peak spot intensity, and total power in the 

spot as demonstrated in the table below. The total power loss in 

the spot can be calculated by using 

42(D /D 

P = e t b ) 2 (2.27) 

L 

b 

t 

K = 1. 

029 + 

0.7125 

(T 4 0.2161) 

(2.24) 

where D b is the Gaussian beam diameter measured at the 1/e 2 

intensity point, and D t is the limiting aperture diameter of the lens. 

If T = 2, which approximates uniform illumination, the image spot 

intensity profile approaches that of the classic Airy disc. When 

T = 1, the Gaussian profile is truncated at the 1/e 2 diameter, and the 

spot profile is clearly a hybrid between an Airy pattern and a 

Gaussian distribution. When T = 0.5, which approximates the case 

for an untruncated Gaussian input beam, the spot intensity profile 

approaches a Gaussian distribution. 

Calculation of spot diameter for these or other truncation ratios 

requires that K be evaluated. This is done by using the formulas 

and 

0.6445 

4 

(T 4 0.2161) 

(2.25) 

FWHM 2.179 2.221 

K = 1.6449 + 

2 

1/e 

4 

0.6460 

0. 

5320 

4 

(T 0.2816) (T 4 0.2816) 

1.821 1.891 

for a truncated Gaussian beam. A good compromise between power 

loss and spot size is often a truncation ratio of one. When T = 2 

(approximately uniform illumination), fractional power loss is 60%. 

When T = 1, d 1/e 

2 is just 8.0% larger than when T = 2, while fractional 

power loss is down to 13.5%. Because of this large savings in power 

with relatively little growth in the spot diameter, truncation ratios 

of 0.7 to 1.0 are typically used. Ratios as low as 0.5 might be 

. 

(2.26) 


employed when laser power must be conserved. However, this low 

value often wastes too much of the available clear aperture of the 

lens. 

The mathematics of the effects of truncation on a real-world 

laser beam are beyond the scope of this chapter. Suffice it to say that 

truncation, in general, increases the M 2 factor of the beam. For an 

in-depth treatment of this problem, please refer to the 

aforementioned paper by Haiyin Sun as well as “Changes in 

Characteristics of a Gaussian Beam Weakly Diffracted by a Circular 

Aperture” by P. Belland and J. Crenn, App. Opt. 21 (1982). 

Spot Diameters and Fractional Power Loss 

for Three Values of Truncation 

Truncation Ratio d FWHM d 1/e 

2 d zero P L (%) 

Infinity 

2.0 

1.0 

0.5 

SPATIAL FILTERING 

1.03 

1.05 

1.13 

1.54 

1.64 

1.69 

1.83 

2.51 

2.44 

— 

— 

— 

100 

60 

13.5 

0.03 

Laser light scattered from dust particles residing on optical 

surfaces may produce interference patterns resembling holographic 

zone planes. Such patterns can cause difficulties in interferometric 

and holographic applications where they form a highly detailed, 

contrasting, and confusing background that interferes with desired 

information. Spatial filtering is a simple way of suppressing this 

interference and maintaining a very smooth beam irradiance distribution. 

The scattered light propagates in different directions from 

the laser light and hence is spatially separated at a lens focal plane. 

By centering a small aperture around the focal spot of the direct 

beam, it is possible to block scattered light while allowing the direct 

K FACTOR 

3.0 

2.5 

2.0 

1.5 

1.0 

0.5 

0 

Figure 2.10 

spot measured at 13.5% intensity level 

spot measured at 50% intensity level 

spot diameter = K ! l ! f-number 

1.0 2.0 3.0 4.0 

T(Db/D t ) 

K factors as a function of truncation ratio 

beam to pass unscathed. The result is a cone of light that has a very 

smooth irradiance distribution and can be refocused to form a 

collimated beam that is almost equally smooth (see figure 2.11). 

As a compromise between ease of alignment and complete 

spatial filtering, it is best that the aperture diameter be about two 

times the 1/e 2 beam contour at the focus, or about 1.33 times the 

99% throughput contour diameter. 

Figure 2.11 Spatial filtering smoothes the irradiance 

distribution 


Modular and Multiaxis Spatial Filters 

The Melles Griot range of spatial filters includes 

a three-axis unit with precision micrometers 

(07 SFM 001) and a compact five-axis version 

(07 SFM 003). These devices feature an open design 

that provides access to the beam as it passes 

through the instrument. Details of these products 

and standard microscope objectives and mounted 

pinholes that work with these spatial filters are 

described in Chapter 29, Microscope Components, 

Spatial Filters, and Apertures. 

For those who wish to fabricate their own spatial 

filters, unmounted pinholes can also be found in 

Chapter 29, Microscope Components, Spatial Filters, 

and Apertures. The precision individual pinholes are 

for general-purpose spatial-filtering tasks. The highenergy 

laser precision pinholes are constructed 

specifically to withstand irradiation from high-energy 

lasers. 





Lens Selection 

The most important relationships that we will use in the process tables list beam diameter, so remember to divide by 2). Assuming a 

of lens selection for Gaussian beam optical systems are as follows: collimated beam, we use the propagation formula to determine the 

spot size at 80 m: 

Focused spot radius 

⎡ 

2 

l 

w= f ⎛ 

3 ⎞ ⎤ 

12 / 

5 

⎢ 0.6328 × 10 × 80, 

000 ⎥ 

. 

(from 2.4) w (80 m) = 0.4⎢1 

+ ⎜ 

⎟ 

p w 0 

⎢ ⎜ 

⎝ ( p)( 04 . ) ⎟ 

⎥ 

2 ⎠ ⎥ 

⎣ 

⎦ 

Beam propagation 

= 40.3- mm beam radius 

or 80.6-mm beam diameter. This is just about exactly a factor of 10 

⎡ 

2 12 / 

z 

w(z) = w0 

1+ ⎛ ⎞ ⎤ 

⎢ 

l 

larger than we wanted. We can use the formula for w 

⎢ 2 

⎝ ⎜ ⎥ 

0 (optimum) 

(from 2.2) 

⎟ 

pw0 

⎠ ⎥ 

to determine the smallest collimated beam diameter we could 

⎣ ⎦ 

achieve at a distance of 80 m: 

12 / 

⎛ lz⎞ 

1/2 

3 

w 0 (optimum) = 

⎛ 

4 

⎜ ⎟ 

0.6328 × 10 × 80,000⎞ 

⎝ p ⎠ 

w 0 (optimum) = ⎜ 

⎟ = 4.0 mm. 

⎝ 

p 

and 

⎠ 

2 

pw 

This tells us that if we expand the beam by a factor of 10 

0 

(from 2.7) 

z R = . 

(4.0 mm/0.4 mm), we can produce a collimated beam 8 mm in 

l 

diameter, which, if focused at the midpoint (40 m), will again be 

We can also utilize the equation for the approximate on-axis 

8 mm in diameter at a distance of 80 m. This 10# expansion could 

spot size caused by spherical aberration for a plano-convex lens at 

be accomplished most easily with one of the Melles Griot beam 

the infinite conjugate: 

expanders, such as the 09 LBX 003 or 09 LBM 013. However, if there 

spot diameter (3rd -order spherical aberration) = 0.067 f is a space constraint and a need to perform this task with a system 

(f/#) . 3 

This formula is for uniform illumination, not a Gaussian intensity 

profile. However, since it yields a larger value for spot size than actually 

occurs, its use will provide us with conservative lens choices. 

Keep in mind that this formula is for spot diameter whereas the 

Gaussian beam formulas are all stated in terms of spot radius. 

Example 1: Obtain 8-mm spot at 80 m 

Using the Melles Griot HeNe laser 05 LHR 151, produce a spot 

8 mm in diameter at a distance of 80 m (see figure 2.12). 

The product tables in Chapter 44, Helium Neon Lasers, gives the 

output beam radius for the 25 LHR 151 as 0.4 mm (the product 

that is no longer than 50 mm, this can be accomplished by using 

catalog components. 

Figure 2.13 illustrates the two main types of beam expanders. The 

Keplerian type consists of two positive lenses which are positioned 

with their focal points nominally coincident. The Galilean type consists 

of a negative diverging lens, followed by a positive collimating 

lens, again positioned with their focal points nominally coincident. 

In both cases, the overall length of the optical system is given by 

overall length = f + f 1 2 

and the magnification is given by 

magnification = f f2 

1 

where a negative sign, in the Galilean system, indicates an inverted 

image (which is unimportant for laser beams). The Keplerian system, 

0.8 mm 

01 LDK 001 

01 LAO 059 

8 mm 


Figure 2.12 

45 mm 80 m 

Lens spacing adjusted empirically to achieve the desired spot size at 80 m 


Figure 2.13 

Keplerian beam expander 

f 1 f 2 

Galilean beam expander 

f 1 

f 2 

Two main types of beam expanders 

with its internal point of focus, allows one to utilize a spatial filter, 

while the Galilean system has the advantage of shorter length for 

a given magnification. 

In order to determine necessary focal lengths for an expander, 

we need to solve these two equations for the two unknowns. 

In this case, 

and 

f + f = 50 

1 2 

f 

f 

2 

1 

= 410. 

Using a negative value for the magnification will provide us 

with a Galilean expander. This yields values of f 2 = 55.5 mm and 

f 1 = 45.5 mm. 

Figure 2.14 

01 LDK 001 

01 LAO 059 01 LLP 017 

Ideally, a plano-concave diverging lens is used for minimum 

spherical aberration, but the shortest catalog focal length available is 

410 mm. There is, however, a biconcave lens with a focal length of 

45 mm (01 LDK 001). Even though this is not the optimum shape 

lens for this application, the extremely short focal length is likely to have 

negligible aberrations at this f-number. Ray tracing would confirm 

this. 

Now that we have selected a diverging lens with a focal length 

of 45 mm, we need to choose a collimating lens with a focal length 

of 50 mm. To determine whether a plano-convex lens is acceptable, 

check the spherical aberration formula: 

spot size resulting from spherical aberration 

0.067 × 50 

= =14 mm. 

3 

6.25 

The spot diameter resulting from diffraction is 

2w = 

0 

43 

2 (0.6328 × 10 ) 50 

= 5 mm. 

p4.0 

43 

0.6328 × 10 × 100 

w = = 50 mm. 

p0.4 

45 mm 95 mm 

Laser focusing system with long working distance 

Clearly, a plano-convex lens will not be adequate. The next choice 

would be an achromat, such as the 01 LAO 059. The data in the spot 

size charts on page 1.26 indicates that this lens is probably diffraction 

limited at this f-number. Our final system would therefore consist of 

the 01 LDK 001 spaced about 45 mm from the 01 LAO 059, which 

would have its flint element facing toward the laser. 

Example 2: Obtain 10 mm spot at > 100 mm 

Focus the output of an 05 LHR 151 to a spot diameter of 10 mm, 

but with the constraint that the last surface of the focusing optics 

is no closer than 100 mm to the focal point (see figure 2.14). 

Using a 100-mm-focal-length lens, the Gaussian beam focusing 

equation yields a spot radius of 

Thus, even a diffraction-limited focusing lens, with a 100-mm 

focal length, will produce a 100-µm-diameter focal spot with an 





0.8-mm-diameter input beam. In order to achieve the spot size 

wanted, the beam must first be expanded by a factor of 10 before 

it is focused. The 10# expander described in the previous example 

could perform the task, as could any of the standard 10# expanders 

offered by Melles Griot. 

For focusing, we now have an 8-mm-diameter beam going into 

the 100-mm-focal-length lens, so we are operating at f/12.5. At this 

f-number we can probably use a plano-convex lens, but it is a good 

idea to check the spherical aberration to make sure. 

0.067 × 100 

spot size (spherical aberration) = = 3 mm. 

3 

12.5 

The plano-convex lens, oriented with its convex surface toward 

the beam expander, will provide diffraction-limited performance in 

this case. 

Although the effects of manufacturing tolerances should always 

be taken into account when choosing a standard catalog lens, they 

are not significant for the input lens of this beam expander because 

the aperture is so small. With a diameter of 1 mm or less, virtually 

any of the lenses in this catalog introduce only a fraction of a wave 

of wavefront distortion as a result of manufacturing errors. However, 

with a larger beam, lens quality is a consideration. One of the 

precision-grade lenses, in this case the 01 LLP 017, should be used 

for this precision application. 

Example 3: Collimate a diode laser 

Collect and collimate the output of a diode laser to a 25-mmdiameter 

diffraction-limited beam. The output wavelength is 780 nm 

and has a full-angle divergence of 60°!20° (see figure 2.15). 

The first step is to determine the numerical aperture needed to 

collect all the light from a source with a 60-degree divergence angle. 

Since numerical aperture is defined to be the sine of the half angle 

of divergence, 

NA = sin 30º = 0.5. 

Stated in terms of f-number, 1/(2 NA), this is f/1. At this low 

f-number we can immediately rule out virtually any simple lens or 

achromat; even if a simple lens were available at this low 

f-number, it would not provide the performance level required. The 

best choice would be a highly corrected, multielement diode laser 

collimating lens, such as the 06 GLC 002, which has a numerical 

aperture of 0.5. 

The 06 GLC 002 yields a collimated elliptical beam with dimensions 

of 8 mm ! 2.7 mm. The smaller dimension of this beam must 

be expanded to match the larger dimension; otherwise, it will have 

a larger beam divergence because of diffraction. Since there is 

approximately a 3:1 ratio in the two dimensions, we will use a 3# 

anamorphic prism pair, 06 GPA 004, to accomplish the expansion. 

This will now yield a collimated beam 8 mm in diameter. 

The next step is to expand the beam by a factor of 3.125#in order 

to get to the desired 25-mm beam diameter. Since no constraint has 

been given on the length of our optical system, we’ll play it safe and 

operate our beam expander at a minimum of f/10. This virtually 

ensures diffraction-limited performance, even with singlets. 

At f/10 and an 8-mm-diameter input beam, we would need a 

focal length of 80 mm for the input lens of our collimator. Since we 

are looking for diffraction-limited performance, our best choice 

would be one of the precision diode laser singlets (06 LXP series). 

Once again, we choose a high-precision lens because our beam has 

a fairly large diameter and the effects of manufacturing tolerances 

must be considered. 

The closest focal length we have in this series of lenses is the 

06 LXP 009 with a focal length of 110 mm. Operating at f/13.75, 

we will have diffraction-limited performance, which can be verified 

by using the formula for spherical aberration. We now need a 

collimating lens with a focal length of 3.125 ! 110 mm = 344 mm. 

The best choice is probably the 01 LAO 277 because there is no 

precision singlet lens with the necessary focal length. The achromat 

is also manufactured to tighter tolerances. 

The final system would then consist of the 06 GLC 002 mated 

directly to the 06 GPA 004, followed by the 06 LXP 009 with its 

curved surface facing toward the diode laser. The spacing between 

the 06 LXP 009 and 06 GPA 004 is not critical. Finally, the 

01 LAO 277 would follow, spaced approximately 455 mm from the 

singlet, with its flint surface facing toward the diode laser. 

Since the standard coating supplied with the 01 LAO series 

achromats does not perform very well at 780 nm, this lens should 

be specified with a /076 coating, which is optimized for performance 

at 780 nm. 

06 GPA 004 

06 LXP 009 01 LAO 277 

06 GLC 002 


Figure 2.15 

1.1 mm 

455 mm 

Melles Griot diode laser components, showing how they may be used in relation to each other 


Optical Specifications 3 

Wavefront Distortion 3.2 

Centration 3.3 

Modulation Transfer Function 3.4 

Cosmetic Surface Quality – U.S. Military Specifications 3.6 

Surface Accuracy 3.8 

3.1 1 





Wavefront Distortion 

Sometimes the best specification for an optical component is 

its effect on the emergent wavefront. This is particularly true for 

optical flats, collimation lenses, mirrors, and retroreflectors where 

the presumed effect of the element is to transmit or reflect the 

wavefront without changing its shape. Wavefront distortion is often 

characterized by the peak-to-valley deformation of the emergent 

wavefront from its intended shape. Specifications are normally 

quoted in fractions of a wavelength. 

Consider a perfectly plane, monochromatic wavefront, incident 

at an angle normal to the face of a window. Deviation from perfect 

surface flatness, as well as inhomogeneity of the bulk material 

refractive index of the window, will cause a deformation of the 

transmitted wavefront away from the ideal plane wave. In a 

retroreflector, each of the faces plus the material will affect the 

emergent wavefront. Consequently, any reflecting or refracting 

element can be characterized by the distortions imparted to a perfect 

incident wavefront. 

INTERFEROMETER MEASUREMENTS 

Melles Griot measures wavefront distortion with a laser 

interferometer. The wavefront from a helium neon laser 

(l = 632.8 nm) is expanded and then divided into a reference 

wavefront and test wavefronts by using a partially transmitting 

reference surface. The reference wavefront is reflected back to the 

interferometer, and the test wavefront is transmitted through the 

surfaces to the test element. The reference surface is a known flat 

or spherical surface whose surface error is on the order of l/20. 

When the test wavefront is reflected back to the interferometer, 

either from the surface being tested or from another l/20 reference 

surface, the reference and test wavefronts recombine at the 

interferometer. Constructive and destructive interference occurs 

between the two wavefronts. A difference in the optical paths of 

the two wavefronts is caused by any error present in the test element 

and any tilt of one wavefront relative to the other. The fringe pattern 

is projected onto a viewing screen or camera system. 

A slight tilt of the test wavefront to the reference wavefront produces 

a set of fringes whose parallelism and straightness depend on 

the element under test. The distance between successive fringes 

(usually measured from dark band to dark band) represents one 

wavelength difference in the optical path traveled by the two 

wavefronts. In surface and transmitted wavefront testing, the test 

wavefront travels through an error in the test piece twice. Therefore, 

one fringe spacing represents one half wavelength of surface 

error or transmission error of the test element. 

A determination of the convexity or concavity of the error in the 

test element can be made if the zero-order direction of the interference 

cavity (the space between the reference and test surfaces) is 

known. The zero-order direction is the direction of the center of tilt 

between the reference and test wavefronts. 

Fringes that curve around the center of tilt (zero-order) are 

convex, as a result of a high area on the test surface. Conversely, 

fringes that curve away from the center of tilt (zero-order) are 

concave as a result of a low area on the test surface. 

By using a known tilt and zero-order direction, the amount and 

direction (convex or concave) of the error in the test element can be 

determined from the fringe pattern. Six fringes of tilt are introduced 

for typical examinations. Melles Griot uses wavefront distortion 

measurements to characterize achromats, windows, filters, beamsplitters, 

prisms, and many other optical elements. This testing 

method is consistent with the way in which these components are 

normally used. 

INTERFEROGRAM INTERPRETATION 

Melles Griot tests lenses with a noncontact phase-measuring 

interferometer. The interferometer has a zoom feature to increase 

resolution of the optic under test. The interferometric cavity length 

is modulated, and a computerized data analysis program is used 

to interpret the interferogram. This computerized analysis increases 

the accuracy and repeatability of each measurement and eliminates 

subjective operator interpretation. 


Centration 

The mechanical axis and optical axis exactly coincide in a 

perfectly centered lens. 

OPTICAL AND MECHANICAL AXES 

For a simple lens, the optical axis is defined as a straight line 

that joins the centers of lens curvature. For a plano-convex or planoconcave 

lens, the optical axis is the line through the center of 

curvature and perpendicular to the plano surface. 

The mechanical axis is determined by the way in which the lens 

will be mounted during use. There are typically two types of 

mounting configurations, edge mounting and surface mounting. 

With edge mounting, the mechanical axis is the centerline of the lens 

mechanical edge. Surface mounting uses one surface of the lens as 

the primary stability for lens tip and then encompasses the lens 

diameter for centering. The mechanical axis for this type of mounting 

is a line perpendicular to the mounting surface and centered on 

the entrapment diameter. 

Ideally, the optical and mechanical axes coincide. The tolerance 

on centration is the allowable amount of radial separation of these 

two axes, measured at the focal point of the lens. The centration 

angle is equal to the inverse tangent of the allowable radial separation 

divided by the focal length. 

MEASURING CENTRATION ERROR 

Centration error is measured by rotating the lens on its mechanical 

axis and observing the orbit of the focal point. To determine 

the centration error, the radius of this orbit is divided by the lens focal 

length and then converted to an angle. 

Figure 3.1 

optical axis 

mechanical axis 

C 2 

v 

H H″ 

Centration and orbit of apparent focus 

c 

edge grinding removes 

material outside imaginary cylinder 

DOUBLETS AND TRIPLETS 

It is more difficult to achieve a given centration specification 

for a doublet than it is for a singlet because each element must be 

individually centered to a tighter specification, and the two optical 

axes must be carefully aligned during the cementing process. 

Centration is even more complex for triplets because three optical 

axes must be aligned. The centration error of doublets and triplets 

is measured in the same manner as that of simple lenses. One 

method used to obtain precise centration in compound lenses is 

to align the elements optically and edge the combination. 

CYLINDRICAL OPTICS 

Cylindrical optics can be evaluated for centering error in a 

manner similar to simple lenses. The major difference is that 

cylindrical optics have mechanical and optical planes rather than 

axes. The mechanical plane is established by the expected mounting, 

which can be edge only or the surface-edge combination 

described above. The radial separation between the focal line and 

the established mechanical plane is the centering error and can 

be converted into an angular deviation in the same manner as for 

simple lenses. The centering error is measured by first noting the focal 

line displacement in one orientation, then rotating the lens 

180 degrees and noting the new displacement. The centering error 

angle is the inverse tangent of the total separation divided by twice 

the focal length. 

H″′ 

orbit of 

apparent focus 

true focus C 1 

F″ 






Modulation Transfer Function 

The modulation transfer function (MTF), a quantitative measure 

of image quality, is far superior to any classic resolution criteria. 

MTF describes the ability of a lens or system to transfer object 

contrast to the image. Curves can be associated with the subsystems 

that make up a complete electro-optical or photographic system. 

MTF data can be used to determine the feasibility of overall system 

expectations. 

Bar-chart resolution testing of lens systems is deceptive because 

almost 20% of the energy arriving at a lens system from a bar chart 

is modulated at the third harmonic and higher frequencies. Consider 

instead a sine-wave chart in the form of a positive transparency in 

which transmittance varies in one dimension. Assume that the 

transparency is viewed against a uniformly illuminated background. 

The maximum and minimum transmittances are T max and T min , 

respectively. A lens system under test forms a real image of the 

sine-wave chart, and the spatial frequency (u) of the image is 

measured in cycles per millimeter. Corresponding to the transmittances 

T max and T min are the image irradiances I max and I min . 

By analogy with Michelson’s definition of visibility of interference 

fringes, the contrast or modulation of the chart and image are 

defined, respectively, as 

and 

M = 

c 

M = 

i 

T 

T 

max 

max 

4 T 

+ T 

Imax 

4 I 

I + I 

max 

min 

min 

min 

min 

where M c is the modulation of the chart and M i is the modulation 

of the image. 

The modulation transfer function of the optical system at spatial 

frequency u is then defined to be 

MTF = MTF(u) = M /M i c . 

(3.1) 

(3.2) 

(3.3) 

The graph of MTF versus u is a modulation transfer function 

curve and is defined only for lenses or systems with positive focal 

length that form real images. 

It is often convenient to plot the magnitude of MTF (u) versus 

u. Changes in MTF curves are easily seen by graphical comparison. 

For example, for lenses, the MTF curves change with field 

angle positions and conjugate ratios. In a system with astigmatism 

or coma, different MTF curves are obtained that correspond to 

various azimuths in the image plane through a single image point. 

For cylindrical lenses, only one azimuth is meaningful. MTF 

curves can be either polychromatic or monochromatic. Polychromatic 

curves show the effect of any chromatic aberration that 

may be present. For a well-corrected achromatic system, 

polychromatic MTF can be computed by weighted averaging of 

monochromatic MTFs at a single image surface. MTF can also 

be measured by a variety of commercially available instruments. 

Most instruments measure polychromatic MTF directly. 

PERFECT CIRCULAR LENS 

The monochromatic, diffraction-limited MTF (or MDMTF) of 

a circular aperture (perfect aberration-free spherical lens) at an 

arbitrary conjugate ratio is given by the formula 

MDMTF(x) = 2 ⎡ 

⎢arc cos (x) 4 x 14 

x 

π ⎣ 

where the arc cosine function is in radians and x is the normalized 

spatial frequency defined by 

x = u 

u ic 

where u is the absolute spatial frequency and u ic is the incoherent 

diffraction cutoff spatial frequency. There are several formulas for 

u ic including 

u = 

ic 

= 

1.22 

r 

d 

⎛ 1.22l 

⎞ 

n ′′ D 14⎜ 

⎝ n ′′ D 

⎟ 

⎠ 

ls′′ 

2 

⎛ 1.22l 

⎞ 

2n ′′ sin(u ′′) 14⎜ 

n D 

⎟ 

⎝ ′′ ⎠ 

= 

l 

2n ′′ sin(u ′′) 

= 

l 

= n ′′ D 

(3.6) 

ls′′ 

where r d is the linear spot radius in the case of pure diffraction 

(Airy disc radius), D is the diameter of the lens clear aperture (or 

of a stop in near-contact), l is the wavelength, s″ is the secondary 

conjugate distance, u″ is the largest angle between any ray and the 

optical axis at the secondary conjugate point, the product n″ sin(u″) 

is by definition the image space numerical aperture, and n″ is the 

image space refractive index. It is essential that D, l, and s″ have 

consistent units (usually millimeters, in which case u and u ic will be 

in cycles per millimeter). The relationship 

sin(u ′′) = D 

2s′′ 

implies that the secondary principal surface is a sphere centered 

upon the secondary conjugate point. This means that the lens is 

completely free of spherical aberration and coma, and, in the special 

case of infinite conjugate ratio (s″ = f″), 

u = n 

D ic ′′ . l f 

2 

2 

⎤ 

⎥ 

⎦ 

(3.4) 

(3.5) 

(3.7) 

(3.8) 


PERFECT RECTANGULAR LENS 

The MDMTF of a rectangular aperture (perfect aberrationfree 

cylindrical lens) at arbitrary conjugate ratio is given by the 

formula 

MDMTF(x) = (14 

x) 

where x is again the normalized spatial frequency u/u ic , where, in 

the present cylindrical case, 

u = 

ic 

1 

r 

d 

and r d is one-half the full width of the central stripe of the diffraction 

pattern measured from first maximum to first minimum. This 

formula differs by a factor of 1.22 from the corresponding formula 

in the circular aperture case. The following applies to both circular 

and rectangular apertures: 

u = 

ic 

2n ′′ sin(u ′′) 

. 

l 

The remaining three expressions for u ic in the circular aperture case 

can be applied to the present rectangular aperture case provided that 

two substitutions are made. Everywhere the constant 1.22 formerly 

appeared, it must be replaced by 1.00. Also, the aperture diameter 

D must now be replaced by the aperture width w. The relationship 

sin(u″) = w/2s″ means that the secondary principal surface is a 

circular cylinder centered upon the secondary conjugate line. In 

the special case of infinite conjugate ratio, the incoherent cutoff 

frequency for cylindrical lenses is 

u = n 

w ic ′′ . l f 

(3.9) 

(3.10) 

(3.11) 

(3.12) 

IDEAL PERFORMANCE AND REAL LENSES 

In an ideal lens, the x-intercept and the MDMTF-intercept are 

at unity (1.0). MDMTF(x) for the rectangular case is a straight line 

between these intercepts. For the circular case, MDMTF(x) is a 

curve that dips slightly below the straight line. These curves are 

shown in figure 3.2. Maximum contrast (unity) is apparent when 

spatial frequencies are low (i.e., for large features). Poor contrast is 

apparent when spatial frequencies are high (i.e., small features). 

All examples are limited at high frequencies by diffraction effects. 

A normalized spatial frequency of unity corresponds to the 

diffraction limit. 

All real cylindrical, monochromatic MTF curves fall on or below 

the straight MDMTF(x) line. Similarly, all real spherical and monochromatic 

MTF curves fall on or below the circular MDMTF(x) 

curve. Thus the two ideal MDMTF(x) curves represent the perfect 

(ideal) optical performance. Optical element or system quality is 

measured by how closely the real MTF curve approaches the 

corresponding ideal MDMTF(x) curve (see figure 3.3). 

MTF is an extremely sensitive measure of image degradation. 

To illustrate this, consider a lens having a quarter wavelength of 

spherical aberration. This aberration, barely discernible by eye, 

would reduce the MTF by as much as 0.2 at the midpoint of the 

spatial frequency range. 

MDMTF 

1.0 

.8 

.6 

.4 

.2 


rectangular aperture 

0 .2 .4 .6 .8 1.0 

NORMALIZED SPATIAL FREQUENCY, X 

Figure 3.2 MDMTF(x) vs x, as a function of normalized 

spatial frequency, x 

MTF 

1.0 

.8 

.6 

.4 

.2 

lens with 

1 / 4 wavelength 

aberration 

MDMTF 

0 .2 .4 .6 .8 1.0 

NORMALIZED SPATIAL FREQUENCY, X 

Figure 3.3 MTF as a function of normalized spatial 

frequency, x 






Cosmetic Surface Quality — 

U.S. Military Specifications 

Cosmetic surface quality describes the level of defects that can 

be visually noted on the surface of an optical component. Specifically, 

it defines state of polish, freedom from scratches and digs, and 

edge treatment of components. These factors are important, not only 

because they affect the appearance of the component, but also 

because they scatter light, which adversely affects performance. 

Scattering can be particularly important in laser applications because 

of the intensity of the incident illumination. Unwanted diffraction 

patterns caused by scratches can lead to degraded system 

performance, and scattering of high-energy laser radiation can 

cause component damage. Overspecifying cosmetic surface quality, 

on the other hand, can be costly. Melles Griot components are 

tested at appropriate levels of cosmetic surface quality according 

to their intended application. 

The most common and widely accepted convention for specifying 

surface quality is the U.S. Military Surface Quality Specification, 

MIL-0-13830A, Amendment 3. The surface quality of all 

Melles Griot optics is tested in accordance with this specification. 

In Europe, an alternative specification, the DIN (Deutsche Industrie 

Norm) specification, DIN 3140, Sheet 7, is used. Melles Griot 

can also work to ISO-10110 requirements. 

SPECIFICATION STANDARDS 

As stated above, all optics in this catalog are referenced to MIL- 

0-13830A standards. These standards include scratches, digs, grayness, 

edge chips, and cemented interfaces. It is important to note that 

inspection of polished optical surfaces for scratches is accomplished 

by visual comparison to scratch standards. Thus, it is not the actual 

width of the scratch that is ascertained, but the appearance of the 

scratch as compared to these standards. A part is rejected if any 

scratches exceed the maximum size allowed. Digs, on the other hand, 

specified by actual defect size, can be measured quantitatively. 

Because of the subjective nature of this examination, it is critical 

to use trained inspectors who operate under standardized conditions 

in order to achieve consistent results. Melles Griot optics are 

compared by experienced quality assurance personnel using scratch 

and dig standards according to U.S. military drawing C7641866 

Rev L. Additionally, our inspection areas are equipped with lighting 

that meets the specific requirements of MIL-0-13830A. 

The scratch-and-dig designation for a component or assembly 

is specified by two numbers. The first defines allowable maximum 

scratch visibility, and the second refers to allowable maximum dig 

diameter, separated by a hyphen; for example, 

80–50 represents a commonly acceptable cosmetic standard. 

60–40 represents an acceptable standard for most scientific 

research applications. 

10–5 represents a precise standard for very demanding laser 

applications. 

SCRATCHES 

A scratch is defined as any marking or tearing of a polished 

optical surface. In principle, scratch numbers refer to the width 

of the reference scratch in ten thousandths of a millimeter. For 

example, an 80 scratch is equivalent to an 8-µm standard scratch. 

However, this equivalence is determined strictly by visual 

comparison, and the appearance of a scratch can depend upon the 

component material and the presence of any coatings. Therefore, 

a scratch on the test optic that appears equivalent to the 80 standard 

scratch is not necessarily 8 mm wide. 

If maximum visibility scratches are present (e.g., several 

60 scratches on a 60–40 lens), their combined lengths cannot exceed 

half of the part diameter. Even with some maximum visibility 

scratches present, MIL-0-13830A still allows many combinations 

of smaller scratch sizes and lengths on the polished surface. 

DIGS 

A dig is a pit or small crater on the polished optical surface. 

Digs are defined by their diameters, which are the actual sizes of the 

digs in hundredths of a millimeter. The diameter of an irregularly 

shaped dig is 1/2# (length plus width): 

50 dig = 0.5 mm in diameter 




10 dig = 0.1 mm in diameter. 


The permissible number of maximum-size digs shall be one per 

each 20 mm of diameter (or fraction thereof) on any single surface. 

The sum of the diameters of all digs, as estimated by the inspector, 

shall not exceed twice the diameter of the maximum size specified 

per any 20-mm diameter. Digs less than 25 micrometers are ignored. 

EDGE CHIPS 

Lens edge chips are allowed only outside the clear aperture of 

the lens. The clear aperture is 90% of the lens diameter unless 

otherwise specified. Chips smaller than 0.5 mm are ignored, and 

those larger than 0.5 mm are ground so that there is no shine to 

the chip. The sum of the widths of chips larger than 0.5 mm cannot 

exceed 30% of the lens perimeter. 

Prism edge chips outside the clear aperture are allowed. If the 

prism leg dimension is 25.4 mm or less, chips may extend inward 

1.0 mm from the edge. If the leg dimension is larger than 25.4 mm, 

chips may extend inward 2.0 mm from the edge. Chips smaller than 

0.5 mm are ignored, and those larger than 0.5 mm must be stoned 

or ground, leaving no shine to the chip. The sum of the widths of 

chips larger than 0.5 mm cannot exceed 30% of the length of the edge 

on which they occur. 

CEMENTED INTERFACES 

Because a cemented interface is considered a lens surface, specified 

surface quality standards apply. Edge separation at a cemented 

interface cannot extend into the element more than half the distance 

to the element clear aperture up to a maximum of 1.0 mm. The sum 

of edge separations deeper than 0.5 mm cannot exceed 10% of the 

element perimeter. 

BEVELS 

Although bevels are not specified in MIL-0-13830A, our 

standard shop practice specifies that element edges are beveled to 

a face width of 0.25 to 0.5 mm at an angle of 45°±15°. Edges meeting 

at angles of 135° or larger are not beveled. 

COATING DEFECTS 

Defects caused by an optical element coating, such as scratches, 

voids, pinholes, dust, or stains, are considered with the scratchand-dig 

specification for that element. Coating defects are allowed 

if their size is within the stated scratch-and-dig tolerance. Coating 

defects are counted separately form substrate defects. 





Surface Accuracy 

When attempting to specify how closely an optical surface 

conforms to its intended shape, a measure of surface accuracy is 

needed. Surface accuracy can be determined by interferometric 

techniques. Traditional techniques involve comparing the actual 

surface to a the test plate gage. In this approach, surface accuracy 

is measured by counting the number of rings or fringes and examining 

the regularity of the fringe. The accuracy of the fit between 

the lens and the test gage (as shown in figure 3.4) is described by the 

number of fringes seen when the gage is in contact with the lens. Test 

plates are made flat or spherical to within small fractions of a fringe. 

The accuracy of a test plate is only as good as the means used to 

measure its radii. Extreme care must be used when placing a test plate 

in contact with the actual surface to prevent damage to the surface. 

Modern techniques for measuring surface accuracy utilize phasemeasuring 

interferometry with advanced computer data analysis 

software. Removing operator subjectivity has made this approach 

considerably more accurate and repeatable. A zoom function can 

increase the resolution across the entire surface or a specific region 

to enhance the accuracy of the measurement. 

SURFACE FLATNESS 

Surface flatness is simply surface accuracy with respect to a 

plane reference surface. It is used extensively in mirror and optical 

flat specifications. 

POWER AND IRREGULARITY 

During manufacture, a precision component is frequently compared 

with a test plate that has an accurate polished surface that is 

the inverse of the surface under test. When the two surfaces are 

brought together and viewed in nearly monochromatic light, 

Newton’s rings (interference fringes caused by the near-surface 

standard surface in contact 

contact) appear. The number of rings indicates the difference in 

radius between the surfaces. This is known as power or sometimes 

as figure. It is measured in rings that are equivalent to half 

wavelengths. 

Beyond their number, the rings may exhibit distortion that 

indicates nonuniform shape differences. The distortion may be local 

to one small area, or it may be in the form of noncircular fringes 

over the whole aperture. All such nonuniformities are known 

collectively as irregularity. 

test surface 

maximum deviation 

air gap between surfaces 

reference surface 

surface accuracy 


Figure 3.4 

Surface accuracy 


Material Properties 4 

Material Properties Overview 4.2 

Introduction 4.3 

Optical Properties 4.4 

Mechanical and Chemical Properties 4.6 

Melles Griot Lens Materials 4.7 

Five Schott Glass Types 4.8 

Synthetic Fused Silica 4.11 

Optical Crown Glass 4.14 

Low-Expansion Borosilicate Glass 4.15 

Sapphire 4.16 

ZERODUR ® 4.17 

Calcium Fluoride 4.18 

4.1 1 



Material Properties Overview 

Material 

Usable Transmission Range 

Index of 

Refraction 

Features 

BK7 

BK7 

1.52 @ 

0.55 mm 

Excellent all-around lens material provides broad transmission with 

excellent mechanical characteristics 


LaSFN9 

SF11 

F2 

BaK1 

Optical-Quality 

Synthetic 

Fused Silica 

(OQSFS) 

UV-Grade 

Synthetic 

Fused Silica 

(UVGSFS) 

Optical Crown 

Glass 

Low-expansion 

borosilicate glass 

LEBG 

Sapphire 

Zinc Selenide 

Calcium 

Fluoride 

LaSFN9 

SF11 

F2 

BaK1 

OQSFS 

UVGSFS 

OPTICAL CROWN 

LEBG 

SAPPHIRE 

ZINC SELENIDE 

CALCIUM FLUORIDE 

1.86 @ 

0.55 mm 

1.79 @ 

0.55 mm 

1.62 @ 

0.55 mm 

1.57 @ 

0.55 mm 

1.46 @ 

0.55 mm 

1.46 @ 

0.55 mm 

1.52 @ 

0.55 mm 

1.48 @ 

0.55 mm 

1.77 @ 

0.55 mm 

2.40 @ 

10.6 mm 

1.399 @ 

5 mm 

High-refractive-index flint glass provides more power with less 

curvature needed 

High-refractive-index flint glass provides more power with less 

curvature needed 

Material represents a good compromise between higher index and 

acceptable mechanical characteristics 

Excellent all-around lens material, but has weaker chemical 

characteristics than BK7 

Material provides good UV transmission and superior mechanical 

characteristics 

Material provides excellent UV transmission and superior mechanical 

characteristics 

This lower tolerance glass can be used as a mirror substrate or in noncritical 

applications 

Excellent thermal stability, low cost, and homogeneity makes LEBG useful 

for high-temperature windows, mirror substrates, and condenser lenses 

Excellent mechanical and thermal characteristics make it a superior 

window material 

Zinc selenide is most popular for transmissive IR optics, transmits 

visible and IR, and has low absorption in the red end of the spectrum 

This popular UV excimer laser material is used for windows, lenses, 

and mirror substrates 

0.1 0.5 1.0 5.0 10.0 

WAVELENGTHS IN mm 



Introduction 

Glass manufacturers provide hundreds of different glass types 

with differing optical transmissibility and mechanical strengths. 

Melles Griot has simplified the task of selecting the right material 

for an optical component by offering each of our standard components 

in a single material, or in a small range of materials best 

suited to typical applications. 

There are, however, two instances in which one might need to 

know more about optical materials: one might need to determine 

the performance of a catalog component in a particular application, 

or one might need specific information to select a material for a 

custom component. The information given in this chapter is intended 

to help those in such situations. 

The most important material properties to consider in regard to 

an optical element are as follows: 

$ Transmission versus wavelength 

$ Index of refraction 

$ Thermal characteristics 

$ Mechanical characteristics 

$ Chemical characteristics 

$ Cost. 

Transmission versus Wavelength 

A material must be transmissive at the wavelength of interest if 

it is to be used for a transmissive component. A transmission curve 

allows the optical designer to estimate the attenuation of light, at 

various wavelengths, caused by internal material properties. For 

mirror substrates, the attenuation may be of no consequence. 

Index of Refraction 

The index of refraction, as well as the rate of change of index with 

wavelength (dispersion), might require consideration. High-index 

materials allow the designer to achieve a given power with less 

surface curvature, typically resulting in lower aberrations. On the 

other hand, most high-index flint glasses have higher dispersions, 

resulting in more chromatic aberration in polychromatic applications. 

They also typically have poorer chemical characteristics than lower 

index crown glasses. 

Thermal Characteristics 

The thermal expansion coefficient can be particularly important 

in applications in which the part is subjected to high temperatures, 

such as high-intensity projection systems. This is also of concern 

when components must undergo large temperature cycles, such as 

in optical systems used outdoors. 

Mechanical Characteristics 

The mechanical characteristics of a material are significant in 

many areas. They can affect how easy it is to fabricate the material 

into shape, which affects product cost. Scratch resistance is important 

if the component will require frequent cleaning. Shock and vibration 

resistance are important for military, aerospace, or certain 

industrial applications. Ability to withstand high pressure differentials 

is important for windows used in vacuum chambers. 

Chemical Characteristics 

The chemical characteristics of a material, such as acid or stain 

resistance, can also affect fabrication and durability. As with mechanical 

characteristics, chemical characteristics should be taken into 

account for optics used outdoors or in harsh conditions. 

Cost 

Cost is almost always a factor to consider when specifying 

materials. Furthermore, the cost of some materials, such as UVgrade 

synthetic fused silica, increases sharply with larger diameters 

because of the difficulty in obtaining large pieces of the material. 






Optical Properties 

The most important optical properties of a material are its 

internal and external transmittances, surface reflectances, and 

refractive indices. The formulas that connect these variables in the 

on-axis case are presented below. 

TRANSMISSION 

External transmittance is the single-pass irradiance transmittance 

of an optical element. Internal transmittance is the single-pass irradiance 

transmittance in the absence of any surface reflection losses 

(i.e., transmittance of the material). External transmittance is of 

paramount importance when selecting optics for an image-forming 

lens system because external transmittance neglects multiple 

reflections between lens surfaces. Transmittance measured with an 

integrating sphere will be slightly higher. Let T e denote the desired 

external irradiance transmittance (see equation 4.1), T i the 

corresponding internal transmittance, t 1 the single-pass transmittance 

of the first surface, and t 2 the single-pass transmittance of 

the second surface: 

4mt 

T e = t1t2T i = t1t2e 

c 

where e is the base of the natural system of logarithms, m is the 

absorption coefficient of the lens material, and t c is the lens center 

thickness. This allows for the possibility that the lens surfaces might 

have unequal transmittances (for example, one is coated and the 

other is not). Assuming that both surfaces are uncoated, 

t t = 14 

2r + r 

1 2 

where 

⎛ 

r = n 4 1 ⎞ 

⎜ 

⎝ n + 1 

⎟ 

⎠ 

2 

2 

is the single-surface single-pass irradiance reflectance at normal 

incidence as given by the Fresnel formula. The refractive index n must 

be known or calculated from the material dispersion formula 

(equation 4.6). These results are monochromatic. Both m and n are 

functions of wavelength. 

To calculate either T i or the T e for a lens at any wavelength of 

interest, first find the value of absorption coefficient m (equation 4.4). 

Typically, internal transmittance T i is tabulated as a function of wavelength 

for two distinct thicknesses T c1 and T c2 , and m must be found 

from these. Thus 

1 ⎡1n T i (t c1) 

1n T i (t c2) 

⎤ 

m = 4 ⎢ + ⎥ 

2 ⎣⎢ 

t c1 

t c2 ⎦⎥ 

where the bar denotes averaging. In portions of the spectrum where 

absorption is strong, a value for T i is typically given only for the lesser 

thickness. Then 

1 

m = 4 1n T i . 

t 

c 

(4.1) 

(4.2) 

(4.3) 

(4.4) 

(4.5) 

When it is necessary to find transmittance at wavelengths other 

than those for which T i is tabulated, use linear interpolation. 

The on-axis T e value is normally the most useful, but some 

applications require that transmittance be known along other ray 

paths, or that it be averaged over the entire lens surface. The method 

outlined above is easily extended to encompass such cases. Values 

of t 1 and t 2 must be found from complete Fresnel formulas for arbitrary 

angles of incidence. The angles of incidence will be different 

at the two surfaces; therefore, t 1 and t 2 will generally be unequal. 

Distance t c , which becomes the surface-to-surface distance along 

a particular ray, must be determined by ray tracing. It is necessary 

to account separately for the s- and p-planes of polarization, and 

it is usually sufficient to average results for both planes at the end 

of the calculation. 

REFRACTIVE INDEX AND DISPERSION 

The Schott Optical Glass catalog offers nearly 300 different 

optical glasses. For lens designers, the most important difference 

among these glasses is the index of refraction and dispersion (rate 

of change of index with wavelength). Typically, an optical glass is 

specified by its index of refraction at a wavelength in the middle of 

the visible spectrum, usually 587.56 nm (the helium d-line), and by 

the Abbé v-value, defined to be v d = (n d 41)/ (n F 4n C ). The designations 

F and C stand for 486.1 nm and 656.3 nm, respectively. Here, 

v d shows how the index of refraction varies with wavelength. The 

smaller v d is, the faster the rate of change is. Glasses are roughly 

divided into two categories: crowns and flints. Crown glasses are 

those with n d < 1.60 and v d > 55, or n d > 1.60 and v d > 50. The 

others are flint glasses. 

The refractive index of glass from 365 to 2300 nm can be 

calculated by using the following formula: 

n = 

⎛ 2 

2 

2 

B1l 

C + B2l 

C + B3l 

C + 1 

⎞ 

⎜ 2 

2 

2 ⎟ 

⎝ l 4 1 l 4 2 l 4 3 ⎠ 

Here l, the wavelength, must be in micrometers, and the constants 

B 1 through C 3 are given by the glass manufacturer. Our tabulation 

of these constants for the glasses used in our catalog 

components are presented on page 4.8. Values for other glasses 

can be obtained from the manufacturer’s literature. This equation 

yields an index value that is accurate to better than 1!10 45 over 

the entire transmission range, and even less in the visible spectrum. 

OTHER OPTICAL CHARACTERISTICS 

Homogeneity within Melt 

Homogeneity within melt is the amount of refractive index 

variation within the manufactured glass blank. Inhomogeneity of 

refractive index can result in transmitted wavefront distortion. The 

1/2 

(4.6) 


maximum value for homogeneity within melt for all Schott optical 

glasses used in Melles Griot catalog components is 1!10 44 . 

Striae Grade 

Striae are thread-like inclusions within an optical glass. Striae 

grades are specified in U.S. military specification MIL-G 174B. All 

Melles Griot catalog components that utilize Schott optical glass 

are specified to have striae that conform to MIL-G 174B grade A. 

Grade A means that no visible striae, streaks, or cords are present 

in the glass. 

Stress Birefringence 

Mechanical stress in optical glass leads to birefringence (anisotropy 

in index of refraction) which can impair the optical performance of 

a finished component. Optical glass is annealed (heated and cooled) 

to remove any residual stress left over from the original manufacturing 

process. Schott Glass defines fine annealed glass to have a 

maximum of 12 nm/cm of residual stress birefringence for blanks 

of up to 800 mm in diameter and 100 mm in thickness. 


Fused-Silica Optics 

Synthetic fused silica, described on page 4.11, is an 

ideal optical material for many laser applications. 

It is transparent from as low as 180 nm to over 

2.0 mm, has low coefficient of thermal expansion, 

and is resistant to scratching and thermal shock. 

For more information on some of the specific 

components manufactured from fused silica, see the 

following pages: Lenses, pages 6.22–6.29; Mirrors, 

9.12–9.17; Beamsplitters, 11.4–11.8. 






Mechanical and Chemical Properties 

Mechanical and chemical properties of glass are important to lens 

manufacturers. These properties can also be significant to the user, 

especially when the component will be used in a harsh environment. 

Different polishing techniques and special handling may be needed 

depending on whether the glass is hard or soft, or whether it is 

extremely sensitive to acid or alkali. 

To quantify the chemical properties of glasses, each glass is rated 

according to four categories: climatic resistance, stain resistance, acid 

resistance, and alkali and phosphate resistance. 

Climatic Resistance 

Humidity can cause a cloudy film to appear on the surface of 

some optical glass. Climatic resistance expresses the susceptibility 

of a glass to this process. In this test, glass is placed in a watervapor-saturated 

environment and subjected to a temperature cycle 

which alternately causes condensation and evaporation. The glass 

is given a rating from 1 to 4 depending on the amount of surface 

scattering induced by the test. A rating of 1 indicates little or no 

change after seven days of exposure; a rating of 4 means a significant 

change occurred in less than 30 hours. 

Stain Resistance 

Stain resistance expresses resistance to mildly acidic water 

solutions, such as fingerprints or perspiration. In this test, a few 

drops of a mild acid are placed on the glass. A colored stain, caused 

by interference, will appear if the glass starts to decompose. A rating 

from 1 to 5 is given to each glass, depending on how much time 

elapses before stains occur. A rating of 1 indicates no observed stain 

in 100 hours of exposure; a rating of 5 means that staining occurred 

in less than 0.2 hours. 

Acid Resistance 

Acid resistance quantifies the resistance of a glass to stronger 

acidic solutions. Acid resistance can be particularly important to 

lens manufacturers because acidic solutions are typically used to strip 

coatings from glass or to separate cemented elements. A rating 

from 1 to 4 indicates progressively less resistance to a pH 0.3 acid 

solution, and values from 51 to 53 are used for glass with too little 

resistance to be tested with such a strong solution. 

Alkali and Phosphate Resistance 

Alkali resistance is also important to the lens manufacturer 

since the polishing process usually takes place in an alkaline 

solution. Phosphate resistance is becoming more significant as 

users move away from cleaning methods that involve chlorofluorocarbons 

(CFCs) to those that may be based on traditional 

phosphate-containing detergents. In each case, a two-digit number 

is used to designate alkali or phosphate resistance. The first number, 

from 1 to 4, indicates the length of time that elapses before any 

surface change occurs in the glass, and the second digit reveals the 

extent of the change. 

Microhardness 

The most important mechanical property of glass is microhardness. 

A precisely specified diamond scribe is placed on the glass 

surface under a known force. The indentation is then measured. 

The Knoop and the Vickers microhardness tests are used to measure 

the hardness of a polished surface and a freshly fractured surface, 

respectively. 


Glass Manufacturers 

The catalogs of optical glass manufacturers contain 

products covering a very wide range of optical 

characteristics. However, it should be kept in mind 

that the glass types that exhibit the most desirable 

properties in terms of index of refraction and 

dispersion often have the least practical chemical and 

mechanical characteristics. Furthermore, poor 

chemical and mechanical attributes translate directly 

into increased component costs because working 

these sensitive materials increases fabrication time 

and lowers yield. Please contact us before specifying 

an exotic glass in an optical design so that we can 

advise you of the impact that that choice will have on 

part fabrication. 


Melles Griot Lens Materials 

Melles Griot simple lenses are made of synthetic fused silica, 

BK7 grade A fine annealed glass, and several other materials. The 

following table identifies the materials used in Melles Griot lenses. 

Some of these materials are also used in prisms, mirror substrates, 

and other products. 

Glass type designations and physical constants are the same as 

those published by Schott Glass. Melles Griot occasionally uses 

corresponding glasses made by other glass manufacturers but only 

when this does not result in a significant change in optical properties. 

The quality of performance of optical lenses and prisms depends 

on the quality of the material used. No amount of skill during 

manufacture can eradicate striae, bubbles, inclusions, or variations 

in index. Melles Griot takes considerable care in its material selection, 

using only first-class optical materials from reputable glass manufacturers. 

The result is reliable, repeatable, consistent performance. 

The following physical constant values are reasonable averages 

based on historical experience. Individual material specimens may 

deviate from these means. Materials having tolerances more 

restrictive than those published in the rest of this chapter, or materials 

traceable to specific manufacturers, are available only on special 

request. 

BK7 OPTICAL GLASS 

A borosilicate crown glass, BK7, is the material used in many 

Melles Griot products. BK7 performs well in chemical tests so that 

special treatment during polishing is not necessary. BK7, relatively 

hard glass, does not scratch easily and can be handled without special 

precautions. The bubble and inclusion content of BK7 is very 

low: the bubble and inclusion content cross-section totals less than 

0.029 mm 2 per 100 cm 3 . Another important characteristic of BK7 

is its excellent transmittance, as low as 350 nm. Because of these properties, 

BK7 is used widely throughout the optics industry. A variant 

of BK7, designated UBK7, has transmission almost as low as 

300 nm. This special glass is useful in applications requiring a high 

index of refraction, the desirable chemical properties of BK7, and 

transmission deeper into the ultraviolet range. 

Melles Griot Lens Materials 

Materials 

Synthetic Fused Silica, UV Grade 

Synthetic Fused Silica, Optical Quality 

BK7, Grade A Fine Annealed 

LaSF N9, Grade A Fine Annealed 

BaK1, Grade A Fine Annealed 

Optical Crown 

Low-Expansion Borosilicate Glass 

(LEBG) 

SF11, Grade A Fine Annealed 

SK11 and SF5, Grade A Fine Annealed 

Sapphire 

Zinc Selenide 

Various Glass Combinations 

Melles Griot reserves the right to make material changes or substitutions on any optical components without prior notice. 

Lens Product Numbers 

01 LQC 01 LQP 

01 LQD 01 LQS 

01 LQB 01 LQT 

01 LQF 

Selected 01 CMP series 

01 LCN 01 LMN 

01 LCP 01 LMP 

01 LDK 01 LPK 

01 LDX 01 LPX 

01 LFS 

01 LPX 401 01 LPX 411 

01 LPX 405 01 LPX 413 

01 LPX 407 06 LMS 

01 LPX 415 01 LPX 423 

01 LPX 421 

01 LAG 

Selected 01 CMP series 

06 LXP 

06 LAI 

01 LSX 

12 LNZ 12 LPZ 

01 LAL 04 EWR 001 

01 LAO 04 OAS 

01 LAT 04 OAP 

01 LBX 06 DDL 

04 ECW 06 DBF 

04 EHY 06 GLC 

04 EPP 06 GLR 

04 ERA 001 09 LBM 

04 EWA 09 LCM 

04 EWP 001 09 LSL 





Five Schott Glass Types 

The following tables list the most important optical and physical 

constants for Schott optical glass types BK7, SF11, LaSFN9, 

BaK1, and F2. These types are used in most Melles Griot simple 

lens products and prisms. Index of refraction and transmission, as well 

as the most commonly required chemical characteristics and mechanical 

constants, are listed. Further numerical data and a more detailed 

discussion of the various testing processes can be found in the Schott 

Optical Glass catalog. 

The index of refraction data were obtained by using the constants 

listed below together with the dispersion formula (equation 4.6). 

The constants were determined through the index-of-refraction 

measurements of a typical melt for each glass type. Note that the 

dispersion formula is valid only within the wavelength range 

Physical Constants of Five Schott Glasses 

Melt-to-Melt Mean Index Tolerance 

Homogeneity within Melt 

Striae Grade (MIL-G-174-A) 

Stress Birefringence, nm/cm, Yellow Light 

Abbé Factor (v d ) 

Constants of Dispersion Formula: 

B 1 

B 2 

B 3 

C 1 

C 2 

C 3 

Density (g /cm 43 ) 

Coefficient of Linear Thermal Expansion (a): 

430º to +70º (per ºC) 

+20º to +300º (per ºC) 

Transformation Temperature 

Young’s Modulus (dynes/mm 2 ) 

Climate Resistance 

Stain Resistance 

Acid Resistance 

Alkali Resistance 

Phosphate Resistance 

Knoop Hardness 

Poisson’s Ratio 

Glass Type 

BK7 SF11 LaSFN9 BaK1 F2 

±0.001 

±1!10 44 

A 

10 

64.17 

1.03961212 

2.31792344!10 41 

1.01046945 

6.00069867!10 43 

2.00179144!10 42 

1.03560653!10 2 

2.51 

7.1!10 46 

8.3!10 46 

557ºC 

8.20!10 9 

2 

0 

1.0 

2.0 

2.3 

610 

0.206 

±0.001 

±1!10 44 

A 

10 

25.76 

1.73848403 

3.11168974!10 41 

1.17490871 

1.36068604!10 42 

6.15960463!10 42 

1.21922711!10 2 

4.74 

6.1!10 46 

6.8!10 46 

505ºC 

6.60!10 9 

1 

0 

1.0 

1.2 

1.0 

450 

0.235 

listed. It can be used to interpolate refractive index at other wavelengths 

within this range (to a precision of 1!10 45 or better), but it 

should not be used to extrapolate to wavelengths beyond this range. 

Furthermore, the actual melt-to-melt tolerance on the index of refraction 

typically is about ±0.001. 

The internal transmittance values shown are melt-to-melt experimental 

means and may be affected by thermal history (coating, 

annealing, or tempering operations) after manufacture. 

For more detailed information of these materials, please refer to 

the Schott Optical Glass catalog. 

±0.002 

±1!10 44 

A 

10 

32.17 

1.97888194 

3.20435298!10 41 

1.92900751 

1.18537266!10 42 

5.27381770!10 42 

1.66256540!10 2 

4.44 

7.4!10 46 

8.4!10 46 

703ºC 

1.09!10 10 

2 

0 

2.0 

1.0 

1.0 

630 

0.286 

±0.001 

±1!10 44 

A 

10 

57.55 

1.12365662 

3.09276848!10 41 

8.81511957!10 41 

6.44742752!10 43 

2.22284402!10 42 

1.07297751!10 2 

3.19 

7.6!10 46 

8.6!10 46 

592ºC 

7.30!10 9 

2 

1 

3.3 

1.2 

2.0 

530 

0.252 

±0.001 

±1!10 44 

A 

10 

36.37 

1.34533359 

2.09073176!10 41 

9.37357162!10 41 

9.97743871!10 43 

4.70450767!10 42 

1.11886764!10 2 

3.61 

8.2!10 46 

9.2!10 46 

438ºC 

5.70!10 9 

1 

0 

1.0 

2.3 

1.3 

420 

0.220 



Refractive Index of Five Schott Glass Types 

Wavelength 

l 

Refractive Index, n 

Fraunhofer 

(nm) BK7 SF11 LaSFN9 BaK1 F2 

Designation Source Spectral Region 

351.1 

1.53894 — — 1.60062 1.67359 

Ar laser 

UV 

363.8 

1.53649 — — 1.59744 1.66682 

Ar laser 

UV 

404.7 

1.53024 1.84208 1.89844 1.58941 1.65064 

h 

Hg arc 

Violet 

435.8 

1.52668 1.82518 1.88467 1.58488 1.64202 

g 

Hg arc 

Blue 

441.6 

1.52611 1.82259 1.88253 1.58415 1.64067 

HeCd laser 

Blue 

457.9 

1.52461 1.81596 1.87700 1.58226 1.63718 

Ar laser 

Blue 

465.8 

1.52395 1.81307 1.87458 1.58141 1.63564 

Ar laser 

Blue 

472.7 

1.52339 1.81070 1.87259 1.58071 1.63437 

Ar laser 

Blue 

476.5 

1.52309 1.80946 1.87153 1.58034 1.63370 

Ar laser 

Blue 

480.0 

1.52283 1.80834 1.87059 1.58000 1.63310 

F′ 

Cd arc 

Blue 

486.1 

1.52238 1.80645 1.86899 1.57943 1.63208 

F 

H 2 arc 

Blue 

488.0 

1.52224 1.80590 1.86852 1.57927 1.63178 

Ar laser 

Blue 

496.5 

1.52165 1.80347 1.86645 1.57852 1.63046 

Ar laser 

Green 

501.7 

1.52130 1.80205 1.86524 1.57809 1.62969 

Ar laser 

Green 

514.5 

1.52049 1.79880 1.86245 1.57707 1.62790 

Ar laser 

Green 

532.0 

1.51947 1.79479 1.85901 1.57580 1.62569 

Nd laser 

Green 

546.1 

1.51872 1.79190 1.85651 1.57487 1.62408 

e 

Hg arc 

Green 

587.6 

1.51680 1.78472 1.85025 1.57250 1.62004 

d 

He arc 

Yellow 

589.3 

1.51673 1.78446 1.85002 1.57241 1.61989 

D 

Na arc 

Yellow 

632.8 

1.51509 1.77862 1.84489 1.57041 1.61656 

HeNe laser 

Red 

643.8 

1.51472 1.77734 1.84376 1.56997 1.61582 

C′ 

Cd arc 

Red 

656.3 

1.51432 1.77599 1.84256 1.56949 1.61503 

C 

H 2 arc 

Red 

694.3 

1.51322 1.77231 1.83928 1.56816 1.61288 

Ruby laser 

Red 

786.0 

1.51106 1.76558 1.83323 1.56564 1.60889 

IR 

821.0 

1.51037 1.76359 1.83142 1.56485 1.60768 

IR 

830.0 

1.51020 1.76311 1.83098 1.56466 1.60739 

GaAlAs laser 

IR 

852.1 

1.50980 1.76200 1.82997 1.56421 1.60671 

s 

Ce arc 

IR 

904.0 

1.50893 1.75970 1.82785 1.56325 1.60528 

GaAs laser 

IR 

1014.0 

1.50731 1.75579 1.82420 1.56152 1.60279 

t 

Hg arc 

IR 

1060.0 

1.50669 1.75445 1.82293 1.56088 1.60190 

Nd laser 

IR 

1300.0 

1.50370 1.74901 1.81764 1.55796 1.59813 

InGaAsP laser 

IR 

1500.0 

1.50127 1.74554 1.81412 1.55575 1.59550 

IR 

1550.0 

1.50065 1.74474 1.81329 1.55520 1.59487 

IR 

1970.1 

1.49495 1.73843 1.80657 1.55032 1.58958 

Hg arc 

IR 

2325.4 

1.48921 1.73294 1.80055 1.54556 1.58465 

Hg arc 

IR 

Visit Us Online! www.mellesgriot.com 1 4.9 



Internal Transmittance of Five Schott Glass Types 

Internal Transmittance (%) 

Wavelength 

l 

BK7 

Thickness (mm) 

SF11 


LaSFN9 


BaK1 


F2 


(nm) 5 25 5 25 5 25 5 25 5 25 


300 

310 

320 

330 

340 

350 

360 

370 

380 

390 

400 

420 

440 

460 

480 

500 

540 

580 

620 

660 

700 

0.26 

0.59 

0.81 

0.91 

0.96 

0.986 

0.991 

0.995 

0.996 

0.998 

0.998 

0.998 

0.999 

0.999 

0.999 

0.999 

0.999 

0.999 

0.999 

0.999 

0.999 

— 

0.07 

0.35 

0.65 

0.83 

0.93 

0.96 

0.974 

0.980 

0.989 

0.991 

0.993 

0.994 

0.994 

0.995 

0.996 

0.996 

0.996 

0.997 

0.997 

0.998 

— 

— 

— 

— 

— 

— 

— 

— 

0.13 

0.46 

0.73 

0.93 

0.97 

0.986 

0.991 

0.995 

0.998 

0.998 

0.998 

0.999 

0.999 

— 

— 

— 

— 

— 

— 

— 

— 

— 

0.02 

0.21 

0.69 

0.86 

0.93 

0.95 

0.976 

0.988 

0.992 

0.992 

0.993 

0.994 

— 

— 

— 

— 

— 

— 

— 

0.55 

0.70 

0.80 

0.86 

0.92 

0.94 

0.96 

0.972 

0.980 

0.990 

0.995 

0.996 

0.997 

0.997 

— 

— 

— 

— 

— 

— 

— 

0.05 

0.18 

0.34 

0.47 

0.66 

0.76 

0.83 

0.87 

0.91 

0.95 

0.975 

0.983 

0.986 

0.990 

0.64 

0.81 

0.89 

0.94 

0.97 

0.981 

0.990 

0.995 

0.996 

0.997 

0.998 

0.998 

0.998 

0.998 

0.998 

0.998 

0.999 

0.999 

0.999 

0.999 

0.999 

0.11 

0.34 

0.56 

0.73 

0.84 

0.91 

0.95 

0.976 

0.982 

0.987 

0.988 

0.989 

0.989 

0.990 

0.991 

0.991 

0.993 

0.994 

0.995 

0.996 

0.997 

— 

— 

— 

— 

0.81 

0.95 

0.973 

0.987 

0.992 

0.995 

0.996 

0.997 

0.998 

0.998 

0.999 

0.999 

0.999 

0.999 

0.999 

0.999 

0.999 

— 

— 

— 

— 

0.42 

0.78 

0.87 

0.94 

0.96 

0.973 

0.982 

0.987 

0.989 

0.991 

0.992 

0.993 

0.995 

0.995 

0.995 

0.995 

0.996 



Synthetic Fused Silica 

Fused silica is an ideal optical material for many applications. It 

is transparent over a wide spectral range, has a low coefficient of 

thermal expansion, and is resistant to scratching and thermal shock. 

Synthetic fused silica (amorphous silicon dioxide) is formed by 

chemical combination of silicon and oxygen. It is not to be confused 

with fused quartz, which is made by crushing and melting natural 

crystals, or by fusing silica sand, which results in a granular microstructure 

and bubble entrapment. Microstructure and impurities 

lead to local index variations and contribute, along with bubbles and 

opaque particles, to reduced transmission throughout the spectrum. 

Synthetic fused silica is far purer than fused quartz. This 

increased purity ensures higher ultraviolet transmission and freedom 

from striae or inclusions. The synthetic fused-silica materials used 

by Melles Griot are manufactured by flame hydrolysis to extremely 

high standards. The resultant material is colorless and non-crystalline, 

and it has an impurity content of only about one part per million. 

Controlling the purity of reactants and the conditions of reaction 

ensures the high quality of the synthetic fused silica from which our 

lenses are made. 

Synthetic fused-silica lenses offer a number of advantages over 

glass or fused quartz: 

$ Greater ultraviolet and infrared transmission 

$ Low coefficient of thermal expansion, which provides stability 

and resistance to thermal shock over large temperature 

excursions 

$ Wider thermal operating range 

$ Increased hardness and resistance to scratching 

$ Much higher resistance to radiation darkening from 

ultraviolet, X-rays, gamma rays, and neutrons. 

Optical-quality synthetic fused silica (OQSFS) lenses are ideally 

suited for applications in energy-gathering and imaging systems 

in the mid-ultraviolet, visible, and near-infrared spectral regions. 

The low dispersion of fused silica reduces chromatic aberration. 

UV-grade synthetic fused silica (UVGSFS) is selected to offer 

the highest transmission (especially in the deep ultraviolet) and very 

low fluorescence levels (approximately 0.1% that of fused natural 

quartz excited at 254 nm). UV-grade synthetic fused silica does not 

fluoresce in response to wavelengths longer than 290 nm. In deep 

ultraviolet applications, UV-grade synthetic fused silica is an ideal 

choice. Its tight index tolerance ensures highly predictable lens 

specifications. 

The left-hand table on page 4.13 shows the refractive index of a 

typical UV-grade synthetic fused silica versus wavelength at 20ºC. 

To obtain the index for optical-quality synthetic fused silica, round 

the values off to the fourth decimal place. 

Glass transmittances are affected by thermal history after manufacture, 

as well as during the manufacturing process. Depending on 

the manufacturer and subsequent thermal processing (coating, 

annealing, or tempering), it is possible for any optical glass, including 

BK7, to show internal transmittance reductions of several percent 

across the entire spectrum with external transmittance correspondingly 

affected. Transmittance of all glass is especially uncertain at 

wavelengths approaching the water absorption band at 2.7 mm. 

Synthetic fused silica also shows batch-to-batch transmittance 

variations, especially in deep ultraviolet and infrared. These 

variations are related to manufacture and impurity content rather 

than subsequent history. In the ultraviolet, these variations have 

been attributed to uncontrollable fluctuations in metallic impurity 

content at the parts per billion level. Ultraviolet transmittance is the 

basis for the classifications UV grade and optical quality. A 

specification of UV grade ensures that a specimen is represented by 

the broadest curve. Transmittance curves for optical quality may fall 

anywhere between the UVGSFS curve and the OQSFS curve shown 

in figure 4.1. 

Infrared batch-to-batch transmittance variations in synthetic 

fused silica are attributable to fluctuations in the OH chemical bond 

content. These variations are most pronounced at wavelengths near 

and beyond the water absorption band at 2.7 mm and are normally 

uncontrolled because ultraviolet transmittance is generally regarded 

as more important. High infrared transmittance can be ensured by 

appropriate manufacturing controls, but only at the sacrifice of 

ultraviolet transmittance. 

Visible spectrum batch-to-batch transmittance variations in 

synthetic fused silica are insignificant. The high ultraviolet internal 

transmittance of UV-grade synthetic fused silica is correlated with 

a visible internal transmittance that is so high it is beyond traditional 

methods of measurement. It is necessary to measure optical signal 

attenuation in fibers drawn of the material. 

Figure 4.2 shows a semilogarithmic comparison of the internal 

transmittances of UV-grade synthetic fused silica and BK7 glass. 

It is evident from this graph that UV-grade synthetic fused silica 

averages about two orders of magnitude less absorption loss than 

BK7 across the visible spectrum. In a sample thickness of 10 mm, 

the internal transmittance of UV-grade synthetic fused silica differs 

from unity only in the fifth decimal place. The high internal transmittance 

of such a material can be exploited by maintaining the optic 

at Brewster’s angle for the appropriate linear polarization, or with 

the assistance of high-efficiency antireflection coatings such as 

HEBBAR or one of the laser line V-coats. With these coatings it 

is possible to achieve external transmittances of 98.5% and 99.5%, 

respectively. Synthetic fused silica and HEBBAR are especially 

well suited to each other in visible spectrum applications. 




100 

a) LOWER LIMITS 

90 


PERCENT EXTERNAL TRANSMITTANCE 


80 

70 

60 

50 

40 

30 

20 

10 

0 

140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 

WAVELENGTH IN NANOMETERS 

b) UPPER LIMITS 

100 

90 

80 

70 

60 

50 

40 

30 

20 

UVGSFS 

OQSFS 

BK7 

BK7 

UVGSFS 

OQSFS 

10 


Figure 4.1 

0 

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 

WAVELENGTH IN MICROMETERS 

Comparison of uncoated external transmittances for UVGSFS, OQSFS, and BK7, all 10 mm in thickness 


PERCENT INTERNAL TRANSMITTANCE 

99.997 

99.995 

99.993 

99.991 

99.990 

99.97 

99.95 

99.93 

99.91 

99.90 

99.7 

99.5 

99.3 

99.1 

99.0 

97 

95 

93 


200 400 600 800 1000 1200 1400 

UVGSFS 

BK7 

OH bond 

resonance 

200 400 600 800 1000 1200 1400 


Figure 4.2 Semilogarithmic comparison of internal 

transmittances of UVGSFS and BK7 

The internal transmittance of UV-grade synthetic fused silica 

shows a pronounced dip at 950 nm, while the data for BK7 give 

no hint of a corresponding feature. It should be understood that BK7 

and UVGSFS are manufactured by very different processes. One 

of the many differences in these materials is that UVGSFS has a 

much higher content of OH chemical bonds (hydroxyl content) 

than does BK7. The dip in UVGSFS transmittance corresponds to 

the OH bond resonance. 

1 Malitson, I.H. “Interspecimen Comparison of the Refractive Index 

of Fused Silica,” Journal of the Optical Society of America 55, no. 10 

(October 1965): 1205–1209. 

SYNTHETIC FUSED-SILICA CONSTANTS 

Abbé Constant: 67.880.5 

Change of Refractive Index with Temperature (0º to 700ºC): 

1.28 ! 10 45 /ºC 

Homogeneity (maximum index variation over 10-cm aperture): 

2 ! 10 45 

Density (at 25ºC): 2.20 g/cc 

Continuous Operating Temperature: Maximum 900ºC 

Coefficient of Thermal Expansion: 5.5 ! 10 47 /ºC 

Specific Heat (25ºC): 0.177 cal/gºC 

Dispersion Formula 1 at 20ºC (l in mm): 

2 

2 

0.6961663l 

0.4079426l 

2 

n 4 1 = 

+ 

2 2 2 2 

l 4 (0.0684043) l 4 (0.1162414) 

2 

0.8974794l 

+ 

. (4.7) 

2 2 

l 4 (9.896161) 

Refractive Index of UV-Grade Synthetic Fused Silica* 

Wavelength 

(nm) 

180.0 

190.0 

200.0 

213.9 

226.7 

230.2 

239.9 

248.3 

265.2 

275.3 

280.3 

289.4 

296.7 

302.2 

330.3 

340.4 

351.1 

361.1 

365.0 

404.7 

435.8 

441.6 

457.9 

476.5 

486.1 

488.0 

496.5 

514.5 

*Accuracy 83!10 45 . 

Index of 

Refraction 

1.58529 

1.56572 

1.55051 

1.53431 

1.52275 

1.52008 

1.51337 

1.50840 

1.50003 

1.49591 

1.49404 

1.49099 

1.48873 

1.48719 

1.48054 

1.47858 

1.47671 

1.47513 

1.47454 

1.46962 

1.46669 

1.46622 

1.46498 

1.46372 

1.46313 

1.46301 

1.46252 

1.46156 

Wavelength 

(nm) 

532.0 

546.1 

587.6 

589.3 

632.8 

643.8 

656.3 

694.3 

706.5 

786.0 

820.0 

830.0 

852.1 

904.0 

1014.0 

1064.0 

1100.0 

1200.0 

1300.0 

1400.0 

1500.0 

1550.0 

1660.0 

1700.0 

1800.0 

1900.0 

2000.0 

2100.0 

Index of 

Refraction 

1.46071 

1.46008 

1.45846 

1.45840 

1.45702 

1.45670 

1.45637 

1.45542 

1.45515 

1.45356 

1.45298 

1.45282 

1.45247 

1.45170 

1.45024 

1.44963 

1.44920 

1.44805 

1.44692 

1.44578 

1.44462 

1.44402 

1.44267 

1.44217 

1.44087 

1.43951 

1.43809 

1.43659 





Optical Crown Glass 

In optical crown glass, a low-index commercial-grade glass, the 

index of refraction, transmittance, and homogeneity are not 

controlled as carefully as they are in optical-grade glasses such as 

BK7. Optical crown is suitable for applications in which component 

tolerances are fairly loose and as a substrate material for mirrors. 

Transmittance characteristics for optical crown are shown in 

figure 4.3. Relevant properties of optical crown are shown in the 

accompanying table. 

OPTICAL CROWN GLASS CONSTANTS 

Glass Type Designation: B270 

Abbé Constant: 

v d = 58.5 

Dispersion: (n F 4 n C ) = 0.0089 

Density: 2.55 g cm 43 at 23°C 

Young’s Modulus: 71.5 kN/mm 2 

Specific Heat: C p (20º to 100°C) = 0.184 cal/g°C 

Coefficient of Linear Expansion (20º to 300°C): 

93.3 ! 10 47 /°C 

Transformation Temperature: 521°C 

Softening Point: 708°C 


100 

90 

80 

70 

60 

50 

40 

30 

20 

10 

Refractive Index of Optical Crown Glass 

Wavelength 

(nm) 

Refractive 

Index, n 

Fraunhofer 

Designation Source 

Spectral 

Region 

435.8 

480.0 

486.1 

546.1 

587.6 

589.0 

643.8 

656.3 

1.53394 

1.52960 

1.52908 

1.52501 

1.52288 

1.52280 

1.52059 

1.52015 

g 

F′ 

F 

e 

d 

D 

C′ 

C 

Hg arc 

Cd arc 

H 2 arc 

Hg arc 

He arc 

Na arc 

Cd arc 

H 2 arc 

Blue 

Blue 

Blue 

Green 

Yellow 

Yellow 

Red 

Red 

Transmission Values for 6-mm-thick Sample 

300 nm = 0.3% 

310 nm = 7.5% 

320 nm = 30.7% 

330 nm = 56.6% 

340 nm = 73.6% 

350 nm = 83.1% 

360 nm = 87.2% 

380 nm = 88.8% 

400 nm = 90.6% 

450 nm = 90.9% 

500 nm = 91.4% 

600 nm = 91.5% 

Note: Transmission in visible region (including reflection loss) = 91.7% (t = 2 mm). 

300 400 500 

1000 2000 3000 


Figure 4.3 


External transmittance for 10-mm-thick uncoated optical crown glass 


Low-Expansion Borosilicate Glass 

The most well-known low-expansion borosilicate glass (LEBG) 

is Pyrex ® made by Corning. It is well suited for applications in 

which high temperature, thermal shock, or resistance to chemical 

attack are primary considerations. On the other hand, LEBG is 

typically less homogeneous and contains more striae and bubbles 

than optical glasses such as BK7. This material is ideally suited 

to such tasks as mirror substrates, condenser lenses for high-power 

illumination systems, or windows in high-temperature 

environments. Because of its low cost and excellent thermal stability, 

it is the standard material used in test plates and optical flats. As 

seen in figure 4.4, transmission of LEBG extends into the 

ultraviolet and well into the infrared. The index of refraction in 

this material varies considerably from batch to batch. Typical 

values are shown in the accompanying table. 

LOW-EXPANSION BOROSILICATE GLASS CONSTANTS 

Abbé Constant: v d = 66 

Density: 2.23 g cm 43 at 25°C 

Young’s Modulus: 5.98 !10 9 dynes/mm 2 

Poisson’s Ratio: 0.20 

Specific Heat at 25ºC: 0.17 cal/g°C 

Coefficient of Linear Expansion (0° to 300°C): 

3.25!10 46 /°C 

Softening Point: 820°C 

Melting Point: 1250°C 

Pyrex ® is a registered trademark of Corning, Inc. 


100 

80 

60 

40 

20 

0 

.2 .4 .6 .8 1 1.4 


2 2.4 2.8 

Figure 4.4 External transmittance for 8-mm-thick uncoated 

low-expansion borosilicate glass 

Refractive Index of Low-Expansion Borosilicate Glass 

Wavelength 

(nm) 

486.1 

514.5 

546.1 

587.6 

643.8 

Refractive 

Index, n 

1.479 

1.477 

1.476 

1.474 

1.472 

Fraunhofer 

Designation 

F 

e 

d 

C′ 

Source 

H 2 arc 

Ar laser 

Hg arc 

Na arc 

Cd arc 

Spectral 

Region 

Blue 

Green 

Green 

Yellow 

Red 






Sapphire 

Sapphire is a superior window material in many ways. Because 

of its extreme surface hardness, sapphire can be scratched by only 

a few substances (such as diamond or boron nitride) other than 

itself. Chemically inert and insoluble in almost everything except at 

highly elevated temperatures, sapphire can be cleaned with impunity. 

For example, even hydrogen fluoride fails to attack sapphire at 

temperatures below 300ºC. Sapphire exhibits high internal 

transmittance all the way from 150 nm (vacuum ultraviolet) to 

6000 nm (middle infrared). The external transmittance of sapphire 

is shown in figure 4.5. Because of its great strength, sapphire windows 

can safely be made much thinner than windows of other glass types, 

and therefore are useful even at wavelengths that are very close to 

their transmission limits. Because of the exceptionally high thermal 

conductivity of sapphire, thin windows can be very effectively cooled 

by forced air or other methods. Conversely, sapphire windows can 

easily be heated to prevent condensation. 

Sapphire is single-crystal aluminum oxide (Al 2 O 3 ). Because of 

its hexagonal crystalline structure, sapphire exhibits anisotropy in 

many optical and physical properties. The exact characteristics of an 

optical component made from sapphire depend on the orientation 

of the optic axis or c-axis relative to the element surface. Sapphire 

exhibits birefringence, a difference in index of refraction in orthogonal 

directions. The difference in index is 0.008 between light traveling 

along the optic axis and light traveling perpendicular to it. Malitson 1 

determined a dispersion relationship for the ordinary ray in sapphire. 

This formula, along with the appropriate constants is shown below 

(l in micrometers): 

A1l 

A2l 

A3l 

n2 

4 1 = 4 + 

+ 

2 4 

2 2 4 

2 2 

l l1 

l l2 

l 4 l 

where 

A 1 = 1.023798 

A 2 = 1.058264 

A 3 = 5.280792 

2 

l1 

= 0.00377588 

2 

l2 

= 0.0122544 

2 

l = 321.3616. 

3 

2 

The transmission of sapphire is limited primarily by losses caused 

by surface reflections. The high index of sapphire makes magnesium 

fluoride almost an ideal single-layer antireflection coating. When 

a single layer of magnesium fluoride is deposited on sapphire and 

optimized for 550 nm, total transmission of a sapphire component 

can be kept above 98% throughout the entire visible spectrum. 

1 Malitson, I.H. “Refraction and Dispersion of Synthetic Sapphire,” 

Journal of the Optical Society of America 525, no. 12 (Dec. 1967): 1377. 

2 

2 

2 

3 

(4.8) 


100 

80 

60 

40 

20 

Figure 4.5 

sapphire 

0 

.1 .2 .3 .5 1 1.5 


3 5 8 

External transmittance for 1-mm-thick uncoated 

SAPPHIRE CONSTANTS* 

Density: 3.98 g cm 43 at 25ºC 

Young’s Modulus*: 3.7 ! 10 10 dynes/mm 2 

Poisson’s Ratio*: 40.02 

Moh Hardness: 9 (by definition) 

Specific Heat at 25ºC: 0.18 cal/gºC 

Coefficient of Linear Expansion (0º to 500ºC): 

7.7 ! 10 46 /ºC 

Softening Point: 1800ºC 

*Sapphire is anisotropic in many of its properties which require tensor 

description. These values are averages over many directions. 

Refractive Index of Sapphire 

Wavelength 

(nm) 

265.2 

351.1 

404.7 

488.0 

514.5 

532.0 

546.1 

632.8 

1550.0 

2000.0 

Refractive Index 

n 

1.8337 

1.7970 

1.7858 

1.7754 

1.7731 

1.7718 

1.7708 

1.7660 

1.7462 

1.7377 


ZERODUR ® 

Many optical applications require a substrate material with a 

near-zero coefficient of thermal expansion and/or excellent thermal 

shock resistance. ZERODUR ® with its very small coefficient of 

thermal expansion at room temperature is such a material. 

ZERODUR, which belongs to the glass-ceramic composite 

class of materials, has both an amorphous (vitreous) component and 

a crystalline component. This Schott glass is subjected to special 

thermal cycling during manufacture so that approximately 75% of 

the vitreous material is converted to the crystalline quartz form. 

The crystals are typically only 50 nm in diameter, and ZERODUR 

appears reasonably transparent to the eye because the refractive 

indices of the two phases are almost identical. However, scattering 

at the grain boundaries precludes the use of ZERODUR for 

transmissive optics. 

Typical of amorphous substances, the vitreous phase has a 

positive coefficient of thermal expansion. The crystalline phase has 

a negative coefficient of expansion at room temperature. The overall 

linear thermal expansion coefficient of the combination is almost 

zero at useful temperatures. 

Figure 4.6 shows the variation of expansion coefficient with 

temperature for a typical sample. The actual performance varies very 

slightly, batch to batch, with the room temperature expansion 

coefficient in the range of 80.15 ! 10 46 /ºC. By design, this material 

exhibits a change in the sign of the coefficient near room temperature. 

A comparison of the thermal expansion coefficients of ZERODUR 

and fused silica is shown in the figure. ZERODUR, is markedly 

superior over a large temperature range, makes ideal mirror 

substrates for such stringent applications as multiple-exposure 

holography, holographic and general interferometry, manipulation 

of moderately powerful laser beams, and space-borne imaging 

systems. 

ZERODUR ® CONSTANTS 

Abbé Constant: v d = 66 

Dispersion: (n f – n c ) = 0.00967 

Density: 2.53 g cm 43 a 25ºC 

Young’s Modulus: 9.1 ! 10 9 dynes/mm 2 

Poisson’s Ratio: 0.24 

Specific Heat at 25ºC: 0.196 cal/gºC 

Coefficient of Linear Expansion (20º to 300ºC) : 

0.0580.10 ! 10 46 /ºC 

Maximum Temperature: 600ºC 

Zerodur ® is a registered trademark of Schott Glass Technologies. 

Refractive Index of ZERODUR ® 

Wavelength 

THERMAL EXPANSION COEFFICIENT (X 10 –6 / °K) 

(nm) 

656.3 

643.8 

587.6 

546.1 

486.1 

480.0 

435.8 

.8 

.6 

.4 

.2 

0 

–.2 

–.4 

–.6 

–.8 

–271 

ZERODUR 

fused silica 

–250 –150 –50 0 50 150 

TEMPERATURE IN DEGREES CENTIGRADE 

Figure 4.6 Comparison of thermal expansion coefficients 

of ZERODUR ® and fused silica 

MIRROR SUBSTRATES 

ZERODUR is commonly 

used as a substrate for 

l/20 mirrors with 

aluminum type coatings. 

See Chapter 9, Mirrors, 

for ZERODUR coated 

mirrors. 

Fraunhofer 

Designation 

C 

C′ 

d 

e 

F 

F′ 

g 


n 

1.5394 

1.5399 

1.5424 

1.5447 

1.5491 

1.5497 

1.5544 





Calcium Fluoride 

Calcium fluoride (CaF 2 ), a cubic single-crystal material, has 

widespread applications in the ultraviolet and infrared spectra. 

CaF 2 is an ideal material for use with excimer lasers. It can be 

manufactured into windows, lenses, prisms, and mirror substrates. 

CaF 2 transmits over the spectral range of about 130 nm to 

10 mm as shown in figure 4.7. Traditionally, it has been used primarily 

in the infrared, rather than in the ultraviolet. CaF 2 occurs naturally 

and can be mined. It is also produced synthetically using the 

Stockbarger method, which is a time- and energy-consuming process. 

Unfortunately, achieving acceptable deep ultraviolet transmission 

and damage resistance in CaF 2 requires much greater material 

purity than in the infrared, and it completely eliminates the possibility 

of using mined material. 

To meet the need for improved component lifetime and 

transmission at 193 nm and below, manufacturers have introduced 

a variety of inspection and processing methods to identify and 

remove various impurities at all stages of the production process, 

from incoming materials through crystallization. The needs for 

improved material homogeneity and stress birefringence have also 

caused producers to make alterations to the traditional Stockbarger 

approach. These changes allow tighter temperature control during 

crystal growth, as well as better regulation of vacuum and annealing 

process parameters. 

Excimer-grade CaF 2 provides the combination of deep ultraviolet 

transmission (for 193 nm and even 157 nm), high damage threshold, 

resistance to color center formation, low fluorescence, high 

homogeneity, and low stress birefringence characteristics required 

for the most demanding deep ultraviolet applications. 

CALCIUM FLUORIDE CONSTANTS 

Density: 3.18 gm cm 43 @ 25ºC 

Poisson Ratio: 0.26 

dN/dT: 410.6!10 46 /ºC 

Young’s Modulus: 1.75!10 7 psi 

Coefficient of Linear Expansion: 

18.9!10 46 /ºC (from 20ºC to 60ºC) 

Melting Point: 1360ºC 

Refractive Index of Calcium Fluoride 

Wavelength 


(mm) 

n 

0.193 

1.501 

0.248 

1.468 

0.257 

1.465 

0.266 

1.462 

0.308 

1.453 

0.355 

1.446 

0.486 

1.437 

0.587 

1.433 

0.65 

1.432 

0.7 

1.431 

1.0 

1.428 

1.5 

1.426 

2.0 

1.423 

2.5 

1.421 

3.0 

1.417 

4.0 

1.409 

5.0 

1.398 

6.0 

1.385 

7.0 

1.369 

8.0 

1.349 



PERCENT TRANSMITTANCE 

100 

80 

60 

40 

20 

0 

.2 .4 .6 .8 1.0 2 4.0 10 

Figure 4.7 

External transmittance for calcium fluoride 


Optical Coatings 5 

Optical Coatings 5.2 

OEM and Special Coatings 5.3 

The Reflection of Light 5.4 

Single-Layer Antireflection Coatings 5.8 

Multilayer Antireflection Coatings 5.12 

Thin-Film Production 5.14 

Single-Layer MgF 2 Antireflection Coatings 5.17 

HEBBAR Coatings 5.18 

V-Coatings 5.23 

High-Reflection Coatings 5.24 

Metallic High-Reflection Coatings 5.25 

Dielectric High-Reflection Coatings 5.29 

MAXBRIte Coatings 5.33 

Laser-Line MAX-R Coatings 5.35 

Ultrafast Coating 5.37 

5.1 1 






A comprehensive survey of all optical components currently in 

use would reveal that the vast majority are made of various types 

of glass. This survey would also reveal that a majority of these 

optics are coated with thin layers of material(s) different from the 

substrate. The purpose of these coatings is to modify the reflection 

and transmission properties at the surface of the optical element. 

Whenever light passes from one medium into a medium of 

different optical properties (most notably refractive index), part of 

the light (between 0% and 100%) is reflected and part of the light 

(between 100% and 0%) is transmitted. The intensity ratio of reflected 

and transmitted components is primarily a function of the difference 

in refractive index and the angle of incidence. For many uncoated 

optical glasses, reflected light typically represents a few percent of 

incident radiation. For designs using more than a few components, 

losses in transmitted light level can accumulate rapidly. More 

important are corresponding losses in image contrast or modulation 

caused by weakly reflected ghost images superimposed on the desired 

image. Such unwanted images are often defocused beyond recognition 

so that contrast reduction (rather than image confusion) is their 

primary effect. 

Applications generally require that the reflected portion of 

incident light approach 0% for transmitting optics (lenses) and 

100% for reflective optics (mirrors), or is at some fixed intermediate 

value for partial reflectors (beamsplitters). The only applications 

that do not require coated optics involve transmitting optics in 

which only a few surfaces are in the optical path, where transmission 

inefficiencies may be tolerable. 

In principle, the surface of any optical element can be coated with 

thin layers of various materials (called thin films) in order to ensure 

the desired reflection/transmission ratio. Unfortunately, with the 

exception of simple metallic coatings, this ratio depends on the 

nature of the material from which the optic is fabricated, as well as 

the wavelength and angle of incidence. There is also a polarization 

dependence to this ratio when the angle of incidence is not 0 degrees. 

A multilayer coating (sometimes more than 100 individual layers) 

can optimize the reflection/transmission ratio for several sets of 

conditions (wavelength and angle of incidence) or optimize it over 

a particular range of conditions. 

Melles Griot is the leading supplier of precision simple optics. 

Because optics for most applications require a coating of some sort, 

it would not have been possible to achieve this market-leading position 

without our extensive knowledge of thin-film coatings. With 

the state-of-the-art coating department located in Irvine, California, 

as well as other coating facilities in Japan; Rochester, New York; and 

the British Isles, Melles Griot is able not only to coat large volumes 

of catalog and special optics, but also to develop and evaluate new 

coatings for special customer requirements. 

With new and expanded coating capabilities, Melles Griot now 

offers the same high-quality coatings as a separate service to 

customers wishing to supply their own substrates. As with any 

special or OEM order, please contact Melles Griot to discuss your 

requirements with one of our qualified applications engineers. 

Today, dielectric coatings are remarkably hard and durable. 

With proper care and handling, they can have a long life. In fact, 

the surface of many high-index glasses that are prone to staining can 

be protected with a durable antireflection coating. Several factors 

influence coating durability. Coating designs should be optimized 

for minimal overall thickness to reduce mechanical stress. The most 

resilient materials should be used. Great care should be taken in coating 

fabrication to ensure high-quality, nongranular, even layers. 

Although we cannot prevent accidental abuse of coated optics, 

Melles Griot concentrates on these other factors to produce coatings 

that are as durable as possible. 

Although the Melles Griot optical-coating departments have 

many years of experience in designing and fabricating various types 

of dielectric and metallic coatings, the science of thin films is still 

developing rapidly. Melles Griot monitors and incorporates new 

technology so that we are always able to offer the most advanced 

coatings available. 

The Melles Griot range of coatings currently includes antireflection, 

metallic reflectors, all-dielectric reflectors, hybrid reflectors, 

partial reflectors (beamsplitters), and filters for monochromatic, 

dichroic, or broadband applications. Many of the coatings can be 

applied to the simple optics described in this catalog; some coatings 

can be applied only to a specific range of products; and some 

of the coatings are supplied only as an integral part of a specific product 

(e.g., cube beamsplitters). 

If you require a special coating not described in this catalog, please 

contact a Melles Griot applications engineer to discuss our special 

coating design services. 


OEM and Special Coatings 

Melles Griot maintains coating capabilities at each of its lens 

fabrication facilities worldwide, including the Irvine, California, 

Photonics Components facility. 

In the last few years, Melles Griot has expanded and improved 

this coating facility to take advantage of the latest developments in 

thin-film technology. The resulting operation can provide highvolume 

coatings at competitive prices to OEM customers, as well 

as specialized, high-performance coatings for the most demanding 

user. 

The most important aspect of our coating capabilities is our 

expert design and manufacturing staff. This group blends years of 

practical experience with recent academic research knowledge. 

With a thorough understanding of both design and production 

issues, Melles Griot excels at producing repeatable, high-quality 

coatings at competitive prices. 

USER-SUPPLIED SUBSTRATES 

Melles Griot not only coats catalog and custom optics with 

standard and special coatings, but also applies these coatings to 

user-supplied substrates. A significant portion of our coating 

business involves applying standard or slightly modified catalog 

coatings to special substrates. 

HIGH VOLUME 

The high-volume output capabilities of the Melles Griot coating 

departments result in very competitive pricing for large-volume 

special orders. Even the small-order customer benefits from this 

large volume. Small quantities of special substrates can be coated 

with popular catalog coatings during routine production runs at a 

very modest cost. 

CUSTOM DESIGNS 

A large portion of the work carried out at Melles Griot coating 

facilities is special coatings designed and manufactured to customer 

specifications. 

These designs cover a wide range of wavelengths, from infrared 

to ultraviolet, and applications ranging from basic research through 

the design and manufacture of industrial and medical products. The 

most common special coating requests are for modified catalog 

coatings, which usually involve a simple shift in the design wavelength. 

TECHNICAL SUPPORT 

Melles Griot applications engineers are available to discuss your 

system requirements at any stage. This can make a significant 

difference to overall coating cost. Often a simple modification to a 

system design can enable catalog components or coatings to be 

substituted for special designs at a reduced cost, without affecting 

performance. 






The Reflection of Light 

REFLECTIONS AT UNCOATED SURFACES 

Whenever light is incident on the boundary between two media, 

some light is reflected and some is transmitted (undergoing 

refraction) into the second medium. Several physical laws govern 

the direction, phase, and relative amplitude of the reflected light. 

For our purposes, it is necessary to consider only polished optical 

surfaces. Diffuse reflections from rough surfaces are not considered 

here. 

The law of reflection states that the angle of incidence equals the 

angle of reflection. This is illustrated in figure 5.1 which shows 

reflection of a light ray at a simple air/glass interface. The incident 

and reflected rays make an equal angle with the axis perpendicular 

to the interface between the two media. 

INTENSITY 

At a simple interface between two dielectric materials, the 

amplitude of reflected light is a function of the ratio of the refractive 

index of the two materials, polarization of the incident light, and 

the angle of incidence. 

When a beam of light is incident on a plane surface at normal 

incidence, the relative amplitude of the reflected light, as a proportion 

of the incident light, is given by 

(14 

p) 

(1 + p) 

where p is the ratio of the refractive indices of the two materials 

(n 1 /n 2 ). Intensity is the square of this expression. 

air n = 1.00 

glass n = 1.52 

Figure 5.1 

interface 

incident 

ray 

reflected 

ray 

v i v r 

v i = v r 

v t 

refracted 

ray 

sinv t n 

= air 

sinv i n glass 

Reflection and refraction at a simple air/glass 

(5.1) 

The amount of reflected light is therefore larger when the 

disparity between the two refractive indices is greater. For an air/glass 

interface with the glass having a refractive index of 1.5, the intensity 

of the reflected light will be 4% of the incident light. For an 

optical system containing ten such surfaces, this shows that the 

transmitted beam will be attenuated to 66% of the incident beam 

from reflection losses alone. 

INCIDENCE ANGLE 

The intensity of reflected and transmitted beams is also a function 

of the angle of incidence. Because of refraction effects, it is necessary 

to consider internal and external reflection separately at this point. 

External reflection is defined as reflection at an interface where the 

incident beam originates in the material of lower refractive index 

(i.e., air in the case of an air/glass or air/water interface). Internal 

reflection refers to the opposite case. 

EXTERNAL REFLECTION AT A DIELECTRIC BOUNDARY 

Fresnel’s laws of reflection precisely describe amplitude and 

phase relationships between reflected and incident light at a 

boundary between two dielectric media. It is convenient to think 

of incident radiation as the superposition of two plane-polarized 

beams, one with its electric field parallel to the plane of incidence 

(p-polarized) and the other with its electric field perpendicular 

to the plane of incidence (s-polarized). Fresnel’s laws can be 

summarized in the following two equations which give the reflectance 

of the s- and p-polarized components: 

( ) 

( ) 

⎡sin 

v14v 

⎤ 

2 

r s = ⎢ 

⎥ 

⎣⎢ 

sin v1 + v2 

⎦⎥ 

2 

⎡ 

v14v2) 

⎤ 

r p = ⎢ 

⎥ . 

⎣⎢ 

tan( v1 + v2) 

⎦⎥ 

⎛ 

r = n 4 1 ⎞ 

⎜ 

⎝ n + 1 

⎟ 

⎠ 

. 

2 

2 

(5.2) 

(5.3) 

In the limit of normal incidence in air, Fresnel’s laws reduce to 

the following simple equation: 

(5.4) 

It can easily be seen that, for a refractive index of 1.52 (crown 

glass), this gives a reflectance of 4%. This important result shows 

that about 4% of all illumination incident normal to an air-glass 

surface will be reflected. In a multielement lens systems, reflection 

losses would be very high if antireflection coatings were not used. 

The variation of reflectance with angle of incidence for both the 

s- and p-polarized components can be seen in figure 5.2. It can be 

seen that the reflectance remains close to 4% over about 30 degrees 

incidence, and that it rises rapidly to 100% at grazing incidence. In 

addition, note that the p-component vanishes at 56º 39′. This angle, 

called Brewster’s angle, is the angle at which the reflected light is 


PERCENT REFLECTANCE 

100 

90 

80 

70 

60 

50 

40 

30 

20 

10 

0 10 20 30 40 50 60 70 80 90 

Figure 5.2 External reflection at a glass surface (n = 1.52) 

showing s- and p-polarized components 

completely polarized (see figure 5.3). This situation occurs when 

the reflected and refracted rays are perpendicular to each other 

(v 1 + v 2 = 90º ). This leads to the expression for Brewster’s angle, v B: 

v 1 = v B = arctan (n 2 /n 1 ). 

s-plane 

p-plane 

ANGLE OF INCIDENCE IN DEGREES 

Under these conditions, electric dipole oscillations of the p- 

component will be along the direction of propagation and therefore 

cannot contribute to the reflected ray. At Brewster’s angle, reflectance 

of the s-component is about 15%. 

INTERNAL REFLECTION AT A DIELECTRIC BOUNDARY 

For light incident from a higher to a lower refractive index 

medium, we can apply the results of Fresnel’s laws in exactly the 

same way. The angle in the high-index material at which polarization 

occurs is smaller by the ratio of the refractive indices in accordance 

with Snell’s law. The internal polarizing angle is 33º2l′ for 

a refractive index of 1.52, corresponding to the Brewster angle 

(56º 39′) in the external medium as shown in figure 5.4. 

The angle at which the emerging refracted ray is at grazing incidence 

is called the critical angle (see figure 5.5). For an external 

medium of air or vacuum (n = 1), the critical angle is given by 

vc ( l ) = arc sin ⎛ 1 ⎞ 

⎜ ⎟ 

⎝ n( l) 

⎠ 

and depends on the refractive index n(l), which is a function of 

wavelength. For all angles of incidence higher than the critical angle, 

total internal reflection occurs. 

v p 

(5.5) 

air or vacuum 

index n 1 

p-polarized 

incident ray 

isotropic dielectric solid 

index n 2 

dipole axis 

direction 

e 1 

normal 

v 2 

v 1 

absent p-polarized 

reflected ray 

refracted ray 

dipole radiation 

pattern: sin 2 v 

p-polarized 

refracted ray 

Figure 5.3 Brewster’s angle (at this angle, the p-polarized 

component is completely absent in the reflected ray) 

n air 

n glass 

v c = critical angle 

d 

c 

b 

a 

vc 

Figure 5.4 Internal reflection at a glass surface (n = 1.52) 

showing s- and p-polarized components 


100 

90 

80 

70 

60 

50 

40 

30 

20 

10 

Brewster 

total reflection 

angle 

33° 

PRODUCT 

21' 

NUMBER A B 

07 PHT 501/07 PHF 501 10 3 

07 PHT 503/07 PHF 503 15 5 

07 PHT 505/07 PHF 505 20 5 

07 PHT 507/07 PHF 507 30 5 

07 PHT 509/07 PHF critical 509 angle 40 5 

r 

07 PHT s 

511/07 PHF 41° 5118' 

50 5 

r p 

0 10 20 30 40 50 60 70 80 90 

ANGLE OF INCIDENCE IN DEGREES 

Figure 5.5 Critical angle (at this angle, the emerging ray is at 

grazing incidence) 

a 

a 

b 

c 

b 

d 

c 





PHASE CHANGES ON REFLECTION 

There is another, more subtle difference between internal and 

external reflections. During external reflection, light waves undergo 

a 180-degree phase shift. No such phase shift occurs for internal 

reflection (except in total internal reflection). This is one of the 

important principles on which multilayer films operate. 

INTERFERENCE 

Quantum theory shows us that light has wave/particle duality. 

In most classical optics experiments, it is generally the wave properties 

that are most important. With the exception of certain laser systems 

and electro-optic devices, the transmission properties of light through 

an optical system can be well predicted and rationalized by wave 

theory. 

One consequence of the wave properties of light is that waves 

exhibit interference effects. Light waves that are in phase with each 

other undergo constructive interference, (see figure 5.6). Light waves 

that are exactly out of phase with each other (by 180 degrees or 

p radians) undergo destructive interference, and their amplitudes 

cancel. In intermediate cases, total amplitude is given by the vector 

resultant, and intensity is given by the square of amplitude. 

Various experiments and instruments demonstrate light 

interference phenomena. Some interference effects are possible 

only with coherent sources (i.e., lasers), but many are produced by 

incoherent light. Three of the best-known demonstrations of visible 

light interference are Young’s slits experiment, Newton’s rings, and 

the Fabry-Perot interferometer. These are described in most elementary 

optics and physics texts. 

In all of these demonstrations, light from a source is split in 

some way to produce two similar wavefronts. The wavefronts are 

recombined with a variable path difference between them. Whenever 

the path difference is an integral number of half wavelengths (and 

if the wavefronts are of equal intensity), they cancel by destructive 

interference (i.e., an intensity minimum is produced). An intensity 

minimum is still produced if the interfering wavefronts are of differing 

amplitude; the result is just non-zero. When the path difference is 

an integral number of wavelengths, their intensities sum by constructive 

interference, and an intensity maximum is produced. 

AMPLITUDE AMPLITUDE 

constructive interference 

TIME 

destructive interference 

zero amplitude 

TIME 

wave I 

wave II 

resultant 

wave 

wave I 

wave II 

resultant 

wave 

Figure 5.6 A simple representation of constructive and 

destructive wave interference 


THIN-FILM INTERFERENCE 

Thin-film coatings also rely on the principles of interference. 

Thin films are dielectric or metallic materials whose thickness is 

comparable to, or less than, the wavelength of light. 

When a beam of light is incident on a thin film, some of the 

light will be reflected at the front surface, and some of light will be 

reflected at the rear surface as shown in figure 5.7. The remainder 

will be transmitted. At this stage, we shall ignore multiple reflections. 

The two reflected wavefronts can interfere with each other. This 


will depend on the ratio of optical thickness of the material and 

the wavelength of the incident light (see figure 5.8). The optical 

thickness of an element is defined as the equivalent vacuum thickness 

(i.e., the distance that light would travel in vacuum in the same 

amount of time as it takes to traverse the optical element of interest). 

In other words, the optical thickness of a piece of material is the 

thickness of that material corrected for the apparent change of 

wavelength passing through it. 

air n 0 ~1.00 

air n 0 

l 

n 0 

t = 1.5l/n = 0.75l 

t op = tn = 1.5l 

front and back 

surface reflections 

homogeneous 

thin 

film 

refractive 

index = n 

t 

physical 

thickness 

dense 

medium 

n≈2.00 

t op 

optical thickness 

transmitted light 

optical thickness 

of film, t op = nt 

Figure 5.8 A schematic diagram showing the effects of 

lower light velocity in a dense medium (in this example, the 

velocity of light is halved in the dense medium n = n/n 0 , and the 

optical thickness of the medium is 2!the real thickness) 

t 

l 

n 

l 

n 0 

Figure 5.7 Front and back surface reflections for a thin 

film at near-normal incidence 

The optical thickness is given by t op = t ! n, where t is the 

physical thickness, and n is the ratio of the speed of light in the 

material to the speed of light in vacuum: 

n = c (vacuum) . 

v (medium) 

To a very good approximation, n is the refractive index of the 

material. 

Returning to the thin film at normal incidence, the phase 

difference between the reflected wavefronts is given by (t op /l) ! 

2p, where l is the wavelength of light, as usual, plus any phase 

differences caused by reflections at the surfaces. Clearly, if the 

wavelength of the incident light and the thickness of the film are 

such that a phase difference exists between reflections of p, then 

reflected wavefronts interfere destructively, and overall reflected 

intensity is a minimum. If the two reflections are of equal amplitude, 

then this amplitude (and hence intensity) minimum will be zero. 

In the absence of absorption or scatter, the principle of 

conservation of energy indicates all “lost” reflected intensity will 

appear as enhanced intensity in the transmitted beam. The sum 

of the reflected and transmitted beam intensities is always equal 

to the incident intensity. This important fact has been confirmed 

experimentally. 

Conversely, when the total phase shift between two reflected 

wavefronts is equal to zero (or multiples of 2p), then the reflected 

intensity will be a maximum, and the transmitted beam will be 

reduced accordingly. 

Spectrophotometer used to obtain a transmission and 

reflectance measurement from a optical coating 

(5.6) 





Single-Layer Antireflection Coatings 

The simple principles of single-layer antireflection coatings should 

now be clear. The substrate (glass, quartz, etc.) is coated with a thin 

layer of material so that reflections from the outer surface of the film 

and the outer surface of the substrate cancel each other by destructive 

interference. The intensity of the transmitted beam is correspondingly 

increased so that, ignoring scattering and absorption, 

incident energy = reflected energy + transmitted energy. 

Two requirements create an exact cancellation of reflected beams 

with a single-layer coating: The reflections are exactly 180 degrees 

(p radians) out of phase, and they have the same intensity, 

FILM THICKNESS 

The thickness of a single-layer antireflection film must be an odd 

number of quarter wavelengths in order to achieve the correct phase 

for cancellation. This requirement is shown in figure 5.9, which 

explains the mechanism of a hypothetically perfect single-layer antireflection 

coating. There is a p/2 phase shift for reflections at both interfaces 

because they are low to high index medium interfaces. These 

identical phase shifts cancel each other out. The net phase shift 

between the two reflections is therefore determined solely by the 

optical path difference 2t ! n c , where t is the physical thickness of 

the coating layer and n c is the refractive index of the coating material. 

The phase shift is therefore 2tn/l. 

Single-layer antireflection coatings are generally deposited with 

a thickness of l/4, where l is the desired wavelength for peak 

performance. The phase shift is 180 degrees (p radians), and the 

reflections are in a condition of exact destructive interference. 

air 

n 0 

thin 

film 

n 

glass 

n = 1.52 

wavelength 

= l 

If t op, the optical 

thickness (nt) = l/4, 

then reflections 

interfere destructively 

REFRACTIVE INDEX 

The intensity of a reflected beam from a single surface, at normal 

incidence, is given by 

[(1 4 p) / (1 + p)] 2 ! the incident intensity 

(5.7) 

where p is the ratio of the refractive indices of the two materials at 

the interface. 

For the two reflected beams to be equal in intensity, it is necessary 

that p, the refractive index ratio, be the same at both the interfaces 

n 

n 

air 

film 

= 

n 

n 

film 

substrate 

(i.e., the three refractive indices must form a geometric progression). 

Since the refractive index of air is 1.0, the thin antireflection 

film ideally should have a refractive index of √}}}}} n substrate }}}}}. Optical 

glasses typically have refractive indices of between 1.5 and 1.75. 

Unfortunately, there is no ideal material that can be deposited in 

durable thin layers with a low enough refractive index to satisfy 

this requirement exactly (n = 1.23 for an antireflection coating on 

crown glass). However, magnesium fluoride (MgF 2 ) is a good 

compromise because it forms high-quality, stable films and has a 

reasonably low refractive index, 1.38 at a wavelength of 550 nm. 

Magnesium fluoride is probably the most widely used thin-film 

material for optical coatings. Although its performance is not 

outstanding, it represents a significant improvement over an uncoated 

surface. Typical crown glass surfaces reflect from 4% to 5% of visible 

light at normal incidence. A high-quality MgF 2 coating can reduce 

this value to 1.5%. For many applications this improvement is sufficient, 

and sophisticated multilayer coatings are not necessary. 

Such coatings work extremely well over a wide range of wavelengths 

and angles of incidence, despite the fact that the theoretical 

target of 0% reflectance is achieved by a film of quarter wavelength 

optical thickness only for normal incidence, and only if the refractive 

index of the coating material is exactly the geometric mean of the 

substrate and air. In fact, the single layer of quarter-wave-thickness 

MgF 2 coating designed for normal incidence makes its most 

significant contribution to the transmission of steep surfaces, where 

most rays are incident at large angles (see figure 5.10). 

WAVELENGTH DEPENDENCE 

(5.8) 


t 

physical 

thickness 

resultant reflected 

intensity = zero 

Figure 5.9 Schematic representation of a single-layer 

antireflection coating 

As with any thin film, performance depends on the incident light 

wavelength for two reasons. First, at other than the design wavelength, 

film thickness is no longer the ideal l/4. This is taken into account by 

all thin-film design programs. A more subtle effect, which can be quite 

important, is caused by the change in refractive index of the coating 

and substrate with wavelength (i.e., dispersion). Only the most upto-date 

computer design packages, such as those used by Melles Griot, 

include this higher level of sophistication for multilayer coatings. For 

single-layer antireflection coatings, wavelength dependence of the 

coating performance can be evaluated from analytical expressions. 


v = angle of incidence 


(at 45° incidence) 

v 

MgF 2 

1 /4 wavelength optical thickness 

at 550 nm (n = 1.38) 

6 

5 

4 

3 

2 

1 

glass 

Figure 5.10 Performance of a normal incidence coating design for 550 nm working at 45 degrees compared with a 45 

degrees incidence coating working at 45 degrees. 


AT 550 NANOMETERS 

subscripts: R s = reflectance for s-polarization 

subscripts: R av = reflectance for average polarization 

subscripts: R p = reflectance for p-polarization 

40 

35 

30 

25 

20 

15 

10 

5 

uncoated glass 

single-layer 

MgF 2 

R s = (normal incidence coating at 45°) 

R s = (45° incidence coating) 

R av = (normal incidence coating at 45°) 

0 20 40 60 80 

ANGLE OF INCIDENCE IN AIR (IN DEGREES) 

R av = (45° incidence coating) 

400 500 600 700 


R p = (normal incidence coating at 45°) 

R p = (45° incidence coating) 






ANGLE OF INCIDENCE 

The irradiance reflectance of any thin-film coating varies with 

the angle of incidence. Two main effects lead to a complicated 

dependence of reflectance (hence transmission) on the angle of 

incidence. First, the path difference of the front and rear surface 

reflection from any layer is a function of angle. As the angle of incidence 

increases from zero (normal incidence), the optical path difference 

is decreased. The change in path difference results in a 

change of phase difference between the two interfering reflections 

in an identical manner to the phase change resulting from tilting a 

Fabry-Perot interferometer. 

The reflectance of any optical interface varies according to the 

angle of incidence as shown in figure 5.10. Thin-film performance 

evaluation at arbitrary angles of incidence is therefore quite complex, 

even for a simple one-layer antireflection coating. In short, the 

phase difference between the two pertinent reflections changes 

together with their relative amplitude. 

COATING FORMULAS (SINGLE LAYER) 

Because of the practical importance and wide usage of singlelayer 

coatings, especially at oblique incidence, it is valuable to have 

formulas from which coating reflectance curves, as functions of 

wavelength, angle of incidence, and polarization, can be calculated. 

COATING DISPERSION FORMULA 

The first step in evaluating performance of a single-layer antireflection 

coating is to calculate the refractive index of the film and 

substrate at the wavelength of interest. For optical purposes, a thin 

film may be considered to be perfectly homogeneous. The refractive 

index of MgF 2 , whether amorphous or crystalline, is connected to 

density with the Lorentz-Lorenz formula. The crystalline ordinary 

and extraordinary indices of refraction may be averaged for the 

amorphous phase. 

The formulas for crystalline MgF 2 are, respectively, 

43 

(3.5821) (10 ) 

n o = 1.36957 + 

( l40.14925) 

and 

43 

(3.7415) (10 ) 

n e = 1.381 + 

( l40.14947) 

for the ordinary and extraordinary rays, where l is the wavelength 

in microns. 

For the average of the ordinary and extraordinary indices of 

refraction, 

n = n( l) = 1 2 

(n o + n e). 

(5.9) 

(5.10) 

(5.11) 

The value 1.38 is the universally accepted amorphous film index 

for MgF 2 at a wavelength of 550 nanometers, which assumes a 

packing density of 100%. Real films, however, tend to be slightly 

porous. The refractive index of a real magnesium fluoride film is usually 

slightly lower than 1.38 because the packing density is rarely 

100% in practice. Because it is a complex function of the manufacturing 

process, packing density varies slightly from batch to 

batch. Air and water vapor can also settle in the film and affect its 

refractive index. For Melles Griot magnesium fluoride coatings, 

this will usually correspond to an effective refractive index between 

97% and 100% of the 1.38 theoretical value. 

COATED SURFACE 

REFLECTANCE AT NORMAL INCIDENCE 

Suppose that the coating is of quarterwave optical thickness for 

some wavelength l. Let n a denote the refractive index of the external 

medium at this wavelength (1.0 for air or vacuum), and let n f and n s , 

respectively, denote the film and substrate indices. For normal incidence 

at this wavelength (as shown in figure 5.11), the single-pass irradiance 

reflectance of the coated surface can be shown to be 

⎛ 

R = n n n 2 

a s4 

⎞ 

f 

⎜ 

2 ⎟ 

⎝ n n + n ⎠ 

a s f 

air or vacuum 

index n a 

wavelength l 

MgF 2 

antireflection 

coating 

index n f 

Figure 5.11 Reflectance at normal incidence 

2 

substrate 

index n s 

(5.12) 

regardless of the polarization state of the incident radiation. This 

function is shown in figure 5.12 

COATED SURFACE 

REFLECTANCE AT OBLIQUE INCIDENCE 

At oblique incidence, the situation is more complex. Let n 1 , n 2 , and 

n 3 , respectively, represent the wavelength-dependent refractive indices 

of the external medium (air or vacuum), coating film, and substrate 

as shown in figure 5.13. Assume that the coating exhibits a reflectance 

extremum of the first order for some wavelength l d and angle of 

incidence v 1d in the external medium. The coating is completely 

specified when v 1d and l d are known. One may then identify n 2 with 

the film index n f (1.38 for MgF 2 at 550 nm). The extremum is a 

minimum if n 2 is less than n 3 and a maximum if n 2 exceeds n 3 . The 

same formulas apply in either case. 


PERCENT REFLECTANCE PER SURFACE 

2.0 

1.8 

1.6 

1.4 

1.2 

1.0 

.8 

.6 

.4 

.2 

Figure 5.12 Reflectance at surface of substrate with 

index n g when coated with a quarter wavelength of 

magnesium fluoride (index n=1.38) 

air or vacuum index n 1 

wavelength l 1 

MgF 2 antireflection 

coating index n 2 

fused silica 

glass or silica substrate 

index n 3 

BK7 

optical path difference = 2n 2 b–n 1 a 

v 1 

a 

b v 2 

v 3 

SF11 

LaSFN9 

1.4 1.5 1.6 1.7 1.8 1.9 

REFRACTIVE INDEX (n g ) 

Figure 5.13 Reflectance at oblique incidence 

Corresponding to the angle of incidence v 1d is an angle of 

refraction in the film: 

v 

As v 1 is reduced from v 1d to zero, the reflectance extremum shifts 

in wavelength from l d to l n , where the subscript n denotes normal 

incidence. 

This wavelength is given by the equation 

l 

2d 

n 

⎛ v1d 

= arcsin sin ⎞ 

⎜ . 

⎝ n ( l ) 

⎟ 

⎠ 

2 d 

⎛ n 2 ( ln) 

⎞ ⎛ ld 

= ⎜ 

⎝ n ( l ) 

⎟ ⎜ 

⎠ ⎝ cos v 

2 d 

2d 

⎞ 

⎟ . 

⎠ 

b 

h 

(5.13) 

(5.14) 

Corresponding to the arbitrary angle of incidence v 1 and 

arbitrary wavelength l 1 are angles of refraction in the coating and 

substrate, given by 

v 

and 

v 

2 

3 

⎛ 

1 l1 v1 

= arcsin n ( ) sin ⎞ 

⎜ 

⎝ n ( l ) 

⎟ 

⎠ 

2 1 

⎛ 

1 l1 v1 

= arcsin n ( ) sin ⎞ 

⎜ 

⎝ n ( l ) 

⎟ 

⎠ . 

3 1 

Following are formulas for the single-interface amplitude 

reflectances for both the p- and s-polarizations: 

r = 

12p 

r = 

23p 

r = 

12s 

r = 

23s 

n cos v 4n cos v 

n cos v + n cos v 

2 1 1 2 

2 1 1 2 

n cos v 

n cos v 

4n cos v 

+ n cos v 

3 2 2 3 

3 2 2 3 

n cos v 4n cos v 

n cos v + n cos v 

1 1 2 2 

1 1 2 2 

n cos v 

n cos v 

4n cos v 

2 2 3 3 

2 2 

+ n cos v 

3 3 

The subscript “12p,” for example, means that the formula gives the 

amplitude reflectance for the p-polarization at the interface between 

the first and second media. 

The corresponding irradiance reflectances for the coated surface, 

accounting for both interferences and the phase differences between 

the reflected waves, are given by 

and 

R = 

p 

R = 

s 

2 

2 

r 12p + r 23p + 2r12pr 23p cos (2 b ) 

2 2 

1 + r r + 2r r cos (2 b ) 

2 

12p 23p 

2 

12p 23p 

r 12s + r 23s + 2r12sr 23s cos (2 b ) 

2 2 

1 + r r + 2r r cos (2 b ) 

12s 23s 

12s 23s 

where b is the phase difference (in the external medium) between 

waves reflected from the first and second surfaces of the coating. 

p 

b = 2 n 2 ( l1 

) h cos v2 

. 

l 

1 

The cosines must be in radians. The average reflectance is given by 

R = 1 2 (R p + R s ) . 

. 

(5.15) 

(5.16) 

(5.17) 

(5.18) 

(5.19) 

(5.20) 

(5.21) 

(5.22) 

(5.23) 

(5.24) 

With these formulas, reflectance curves can be calculated as functions 

of either wavelength l 1 or angle of incidence v 1 . 






Multilayer Antireflection Coatings 

Previously, we discussed basic principles of thin-film design and 

operation for a simple antireflection coating of magnesium fluoride. 

It is useful to discuss to also discuss layer antireflection coatings in 

order to understand the operation of multilayer coatings. It is beyond 

the scope of this chapter to cover all aspects of modern thin-film 

design and operation; however, it is hoped that this section will provide 

the reader with insight into thin films that will be useful when 

considering system designs and specifying cost-effective coatings. 

Two basic types of antireflection coating have been developed 

that are worth examining in detail: the quarter/quarter coating and 

the multilayer broadband coating. 

THE QUARTER/QUARTER COATING 

This coating is used as an alternative to the single-layer 

antireflection coating. It was developed because of the lack of suitable 

materials available to improve the performance of single-layer 

coatings. The basic problem of a single-layer antireflection coating 

is that the refractive index of the coating material is too high, 

resulting in too strong a reflection from the first surface which cannot 

be completely canceled by interference of the weaker reflection 

from the substrate surface. In a two-layer coating, the first reflection 

is canceled by interference with two weaker reflections. 

A quarter/quarter coating consists of two layers, both of which 

have an optical thickness of a quarter wave at the wavelength of interest. 

The outer layer is made of a low-refractive-index material, and the 

inner layer is made of a high-refractive-index material (compared to 

the substrate). As figure 5.14 shows, the second and third reflections 

are both exactly 180 degrees out of phase with the first reflection. 

As with any multilayer coating, performance and design are 

calculated in terms of relative amplitudes and phases which are 

then summed to give the overall (net) amplitude of the reflected 

beam. The overall amplitude is then squared to give the intensity. 

How does one calculate the required refractive index of the inner 

layer? Several methodologies have been developed over the last 40 

to 50 years to calculate thin-film coating properties and converge 

on optimum designs. The whole field has been revolutionized in 

recent years with the availability of powerful microcomputers. 

Among the most sophisticated and effective programs are those 

developed by Professor H. A. Macleod, which are used by 

Melles Griot. 

With a two-layer quarter/quarter coating optimized for one 

wavelength at normal incidence, the required refractive indices 

can easily be calculated by hand. The formula for exact zero 

reflectance for such a coating is 

2 

3 

= n 

2 0 

2 

nn 1 

n 

(5.25) 

where n 0 is the refractive index of air (approximated as 1.0), n 3 is 

the refractive index of the substrate material, and n 1 and n 2 are the 

refractive indices of the two film materials, as indicated in figure 5.14. 

If the substrate is crown glass with a refractive index of 1.52 and 

if the first layer is the lowest possible refractive index, 1.38 (MgF 2 ), 

the refractive index of the high-index layer needs to be 1.70. Either 

beryllium oxide or magnesium oxide could be used for the inner layer, 

but both are soft materials and will not produce very durable coatings. 

Although it allows some freedom in the choice of coating 

materials and can give very low reflectance, the quarter/quarter 

coating is very restrictive in its design. In principle, it is possible to 

deposit two materials simultaneously to achieve layers of almost any 

required refractive index, but such coatings are not very practical. 

As a consequence, thin-film engineers have developed multilayer 

antireflection coatings and two-layer coating designs to allow the 

refractive index of each layer to be chosen. 

quarter/quarter antireflection coating 

A B C 

AMPLITUDE 

Figure 5.14 

coating 

TIME 

air (n 0 = 1.0) 

low-index layer (n 1 = 1.38) 

high-index layer (n 2 = 1.70) 

substrate (n 3 = 1.52) 

wavefront A 

wavefront B 

wavefront C 

resultant 

wave 

Interference in a typical quarter/quarter 


Two-Layer Coatings of Arbitrary Thickness 

Interference is often thought of in terms of constructive or 

destructive interference, where the phase shift between interfering 

wavefronts is either 0 or 180 degrees. For two wavefronts to 

completely cancel, as in a single-layer antireflection coating, a phase 

shift of exactly 180 degrees is required. Where three or more reflecting 

surfaces are involved, complete cancellation can be achieved 

by carefully choosing arbitrary phase and relative intensities. This 

is the basis of a two-layer antireflection coating, where the layers are 

adjusted to suit the refractive index of available materials, instead 

of vice versa. For a given combination of materials, there are usually 

two combinations of layer thicknesses that will give zero reflectance 

at the design wavelength. These two combinations are of different 

overall thickness. For any type of thin-film coating, the thinnest 

possible overall coating is used since it will have better mechanical 

properties (less stress). In this case, the thinner combination is also 

less wavelength sensitive. 

Two-layer antireflection coatings are the simplest of the socalled 

V-coatings. The term V-coating arises from the shape of the 

reflectance curve as a function of wavelength, which is a skewed 

V shape with a reflectance minimum at the design wavelength (see 

figure 5.15). V-coatings are very popular, economical coatings for 

near monochromatic applications, such as optical systems using 

nontunable laser radiation (e.g., helium neon lasers at 632.8 nm). 

BROADBAND ANTIREFLECTION COATINGS 

Many optical systems (particularly imaging systems) use 

polychromatic (more than one wavelength) light. In order for the 

system to have a flat spectral response, transmitting optics are coated 

with a broadband or dichroic antireflection coating. The main 

technique used in designing antireflection coatings that are highly 

efficient at more than one wavelength is to use absentee layers within 

the coating. There are two additional techniques that can be used 

for shaping performance curves of high-reflectance coatings and 

wavelength-selective filters, but these are not applicable to antireflection 

coatings. 

Absentee Layers 

An absentee layer is a film of dielectric material that does not 

change the performance of the overall coating at one particular 

wavelength, usually the wavelength for which the coating is being 

principally optimized. This results from the fact that the coating has 

an optical thickness of a half wave at that wavelength. The effects 

of the “extra” reflections cancel out at the two interfaces since no 

additional phase shifts are introduced. In theory, the performance 

of the coating is the same at that wavelength whether the absentee 

layer is present or not. 

At other wavelengths, the absentee layer starts to have an effect, 

for two reasons. The ratio between physical thickness of the layer 

and the wavelength of light changes with wavelength. Also, dispersion 

of the coating material causes optical thickness to change with 

wavelength. 

Multilayer Broadband Antireflection Coatings 

The complex, computer-design techniques used by Melles Griot 

for multilayer antireflection coatings are based on the simple principles 

of interference and phase shifts described in the preceding text. 

All methods consider the combined effect of various film elements. 

Because of the extensive properties of coherent interference, it is 

meaningless to consider individual layers in a multilayer coating. 

Each layer is influenced by the optical properties of the layer next 

to it. The properties of that layer are influenced by its environment. 

Clearly, this represents at least a complex series of matrix multiplications, 

where each matrix corresponds to a single layer. 

An important aspect that is often overlooked in simple theory 

is that there are multiple “reflections” in the coatings. In the previous 

discussions, only first-order reflections have been considered. This 

oversimplified approach is unable to predict correctly the behavior 

of multilayer coatings. Second, third, and higher order terms must 

be considered if real behavior is to be modeled accurately. The exact 

behavior of an antireflection coating is clearly dependent on the 

refractive index of the substrate to which it is applied. In order to 

simplify the task of choosing and ordering coatings for optics of different 

glass types, Melles Griot has listed the coatings in this catalog 

according to performance. Actual coatings applied by Melles Griot 

are adjusted for different glass types in order to achieve the specified 

performance. 

REFLECTANCE 

l 0 

WAVELENGTH 

Figure 5.15 Characteristic performance curve of 

a V-coating 






Thin-Film Production 

VACUUM DEPOSITION 

Melles Griot manufactures thin films by a process known as vacuum 

deposition. Uncoated substrates are placed in a large vacuum 

chamber capable of achieving a vacuum of at least 10 46 torr. At 

the bottom of the chamber is a source of the film material to be 

vaporized, as shown in figure 5.16. The substrates are mounted on 

a series of rotating carousels, arranged so that each substrate sweeps 

in planetary style through the same time-averaged volume in the 

chamber. 

THERMAL EVAPORATION 

The source of vaporized material is usually one of two types. 

The simpler, older type relies on resistive heating of a thin folded 

strip (boat) of tungsten, tantalum, or molybdenum by a high direct 

current. Small amounts of the coating material are loaded into the 

substrates 

thermocouple 

quartz lamp 

(heating) 

baseplate 

power 

supply 

filter 

detector 

rotation motor 

monitoring 

plate substrates 

shutter 

quartz lamp 

vapor 

E-beam gun 

chopper 

light source 

water 

cooling 

reflection signal 

optical monitor 

power 

supply 

vacuum 

system 

Figure 5.16 Schematic view of a typical vacuum deposition 

chamber 

boat. A high current (10–100 A) is passed through the boat, which 

undergoes resistive heating. The coating material is then vaporized 

thermally. Because the chamber is at a greatly reduced pressure, 

there is a very long mean free path for the free atoms or molecules, 

and the heavy vapor is able to reach the moving substrates at the 

top of the chamber. Here it condenses back to the solid state, forming 

a thin, uniform film. 

Several problems are associated with thermal evaporation. Some 

useful substances can react with the hot boat, which can cause 

impurities to be deposited with the layers, changing optical 

properties. In addition, many materials, particularly metal oxides, 

cannot be vaporized this way, because the material of the boat 

(tungsten, tantalum, or molybdenum) melts at a lower temperature. 

Instead of a layer of zirconium oxide, a layer of tungsten would 

be deposited on the substrate. 

For the more volatile materials, thermal evaporation is still often 

the method of choice. Coatings of excellent quality can be produced 

if they are deposited on a hot substrate. 

SOFT FILMS 

Until the advent of electron bombardment as a superior 

alternative, only materials that melted at moderate temperatures 

(2000ºC) could be incorporated into thin-film coatings. Unfortunately, 

the more volatile materials also happen to be the softer 

materials, which produce less resilient films. Consequently, early 

multilayer coatings deteriorated fairly quickly and required undue 

amounts of care during cleaning. More important, sophisticated 

designs with performance specifications at several wavelengths 

could not be easily produced since these designs required many 

individual layers, and the softness of the layers made some of these 

films impractical. 

ELECTRON BOMBARDMENT 

Electron bombardment has become the accepted method of 

choice for optical thin-film fabrication. This method is capable of 

vaporizing even highly involatile materials, such as titanium oxide and 

zirconium oxide. Using large cooled crucibles precludes reaction of 

the heated material with the metal of the boat or crucible. 

A high-flux electron gun (1 A at 10 kV) is aimed at the film 

material contained in a large, water-cooled, copper crucible. Intense 

local heating melts and vaporizes some of the coating material in 

the center of the crucible without causing undue heating of the 

crucible itself. For particularly involatile materials, the electron gun 

can be focused to intensify its effects. 

Careful control of temperature and vacuum conditions ensures 

that most of the vapor is in the form of atoms or molecules, as 

opposed to clusters. This produces a more even coating with better 

optical characteristics and improved longevity. 


ION-ASSISTED BOMBARDMENT 

Ion-assisted bombardment is a coating technique that can offer 

unique benefits under certain circumstances. Ion assist during 

coating leads to a higher atomic or molecular packing density in the 

thin-film layers. This results in a higher refractive index and, most 

important, superior mechanical characteristics. 

Specifically, the lack of voids in the more efficiently packed 

film means that it is far less susceptible to water-vapor absorption. 

Water absorption by an optical coating can change the index of 

refraction of layers and, hence, the optical properties. Water absorption 

can also cause mechanical changes that can ultimately lead to 

failure. 

Ion-assisted coating can also be used for cold processing. 

Eliminating the need to heat parts allows cemented parts, such as 

achromats, to be safely coated. 

MONITORING AND CONTROLLING LAYER THICKNESS 

A chamber set up for multilayer deposition has several sources 

that are preloaded with various coating materials. The entire 

multilayer coating is deposited without opening the chamber. 

A source is heated, or the electron gun is turned on, until the 

source is stable. The shutter above the source is opened to expose 

the chamber to the vaporized material. When a particular layer is 

deposited to the correct thickness, the shutter is closed and the source 

is turned off. This process is repeated for the other sources. 

The most common method of monitoring the deposition process 

is optical monitoring. A monitor beam of light passes through the 

chamber and is incident on a blank monitor substrate. Reflected 

light is detected using photomultiplier and phase-sensitive detection. 

As each layer is deposited onto the reference blank, the intensity 

of reflected light from it oscillates in a pseudo sine wave (rectified). 

The turning points represent quarter- and half-wave thicknesses at 

the monitor wavelength, with intermediate thicknesses between. 

Deposition is automatically stopped as the reflectance of the reference 

surface passes through the appropriate point. 

SCATTERING 

Reflectance and transmittance are usually the most important 

optical properties specified for a thin film, closely followed by 

absorption. However, the degree of scattering caused by a coating 

is often the limiting factor in the ability of coated optics to perform 

in certain applications. Scattering is quite complex. The overall 

degree of scattering is determined by imperfections in layer interfaces 

and interference between photons of light scattered by these 

imperfections as shown in figure 5.17. It is also a function of the 

granularity of the layers. This is difficult to control as it is an inherent 

characteristic of the materials used. Careful modification of deposition 

conditions can make a considerable difference to this effect. 

The most notable examples of applications where scattering is 

critical are intracavity mirrors for low-gain lasers, such as certain 

helium neon laser lines and continuous-wave dye lasers. 

TEMPERATURE AND STRESS 

A major problem with thin films is caused by inherent mechanical 

stresses. Even with careful control of the vacuum, source 

temperature, and optimized positioning of the optics being coated, 

many thin-film materials do not deposit well on cold substrates. 

This is particularly true of involatile materials. Raising the substrate 

temperature a few hundred degrees improves the quality of these 

films, often making the difference between usable and useless film. 

The elevated temperature seems to allow freshly condensed atoms 

(or molecules) to undergo limited surface diffusion. 

Optics that have been given a multilayer thin-film coating at an 

elevated temperature require very slow cooling to room temperature. 

Thermal expansion coefficients of substrate and film materials are 

likely to be somewhat different. As cooling occurs, the coating 

contracts and produces stress in the layers. Many pairs of coating 

materials do not adhere particularly well to each other owing to 

different chemical properties and bulk packing characteristics. 

Temperature-induced stress and poor interlayer adhesion are 

the most common thickness limitations for optical thin films. Until 

new technologies, such as ion-assisted deposition, are developed 

into true production tools, stress must be reduced by minimizing 

overall coating thickness and by carefully controlling the production 

process. 

incident light 

Figure 5.17 Interface imperfections scattering light in a 

multilayer coating 






INTRINSIC STRESS 

Even in the absence of thermal-contraction-induced stress, the 

layers often are not mechanically stable because of intrinsic stress 

from interatomic forces. The homogeneous thin film is not the 

preferred phase for most coating materials. In the lowest energy, 

natural form of the material, molecules are aligned in a crystalline 

symmetric fashion. This is the form in which intermolecular forces 

are more nearly in equilibrium. 

In addition to intrinsic molecular forces, intrinsic stress results 

from poor packing. If packing density is considerably less than 

100%, the intermolecular binding may be sufficiently weakened to 

make the layer totally unstable. 

PRODUCTION CONTROL 

Two major factors are involved in producing a coating to perform 

to a particular set of specifications. First, sound design techniques 

must be used. If design procedures cannot accurately predict the 

behavior of a coating, there is little chance that satisfactory coatings 

will be produced. Second, if the manufacturing phase is not carefully 

controlled, the thin-film coatings produced may perform quite 

differently from the computer simulation. 

Melles Griot uses the latest computer design programs with 

exhaustive iterations to ensure that the final design is optimized. 

Manufacturing high-quality thin films is not trivial. At Melles Griot, 

more effort is expended on monitoring thin-film manufacture than 

on any other single manufacturing procedure. Without such careful 

monitoring, the tedious design and optimization phase would 

be wasted. 

Great care is taken in coating production at every level. Not only 

are all obvious precautions taken, such as thorough precleaning and 

controlled cool down, but even the smallest details of the manufacturing 

process are carefully controlled. Our thoroughness and 

attention to detail ensures that the customer will always be supplied 

with the best design, manufactured to the highest standards. 

QUALITY CONTROL 

All batches of Melles Griot coatings are rigorously and thoroughly 

tested for quality. Even with the most careful production control, 

this is necessary to ensure that only the highest quality parts are 

shipped. 

Our inspection system meets the stringent demands of 

MIL-I-45208A and our spectrophotometers are calibrated to 

standards traceable to the National Institute of Standards and 

Technology (NIST). Upon request, we can provide complete 

environmental and photometric testing to MIL-C-675 and 

MIL-M-13508. All are firm assurances of dependability and 

accuracy. 


Single-Layer MgF 2 

Antireflection Coatings 

Magnesium fluoride (MgF 2 ) is commonly used for single-layer 

antireflection coatings because of its almost ideal refractive index 

(1.38 at 550 nm) and high durability. These coatings are optimized 

for 550 nm (Melles Griot coating suffix /066) and 670 nm (/067) for 

normal incidence, but as can be seen from the reflectance curves, in 

figures 5.18 and 5.19, they are extremely insensitive to wavelength 

and incidence angle. Many standard lenses in stock are coated with 

MgF 2 . Our precision optimized achromats (01 LAO series) are 

supplied standard with the /066 coating. 

Should you wish to specify a different wavelength and incidence 

angle, it is no problem to shift the coating design. Please bear in 

mind, however, that additional delivery time is needed for special 

coatings, and that care should be taken in selecting the quantity of 

items coated to maximize the efficiency of the coating run. Partially 

filled chambers result in higher unit prices. 

Single-layer antireflection coatings are routinely available for 

almost any angle of incidence and any wavelength between 200 nm 

in the ultraviolet and 1.6 mm in the infrared. To obtain such coatings, 

simply specify (for each surface of each part) the precise wavelength 

and angle of incidence for which reflectance is to be minimized. As 

the 1.6-mm wavelength is approached, the angle-of-incidence range 

becomes restricted to near-normal incidence. This is because of 

practical limitations on physical coating thickness. It is usually 

inadvisable to request a MgF 2 coating for any wavelength greater 

than 1.6 mm. Thicker MgF 2 coatings are possible, but they tend to 

exhibit crazing, poor adhesion, and significantly increased scattering. 

Single-layer antireflection coatings for use on very steeply curved 

or short-radius surfaces should be specified for an angle of incidence 

approximately half as large as the largest angle of incidence 

encountered by the surface. Depending on the specific application, 

determination of the best wavelength for use in a coating specification 

may require ray and energy tracing of the optical system in its 

anticipated environment. 

The effectiveness of MgF 2 as an antireflection coating is increased 

dramatically with increasing refractive index of the component 

material. This means that, for use on high-index materials, there is 

often little point in using more complex coatings. The reflectance 

curves shown are for MgF 2 on BK7 optical glass. 

SINGLE VERSUS MULTILAYER COATINGS 

While MgF 2 does not offer the same performance as multilayer 

coatings, such as HEBBAR (described on the following page), it 

is preferred in some circumstances. Specifically, on lenses with very 

steep surfaces, such as our 01 LAG series aspherics, MgF 2 will 

actually perform better than HEBBAR near the edge of the lens 

because the performance of a coating shifts with the angle of 

incidence. The shifted MgF 2 will never be worse than an uncoated 

lens, but, at very high angles, HEBBAR can actually be shifted to 

a region where its performance is worse than if there were no coating 

at all. 


5 

4 

3 

2 

1 

normal and 45° incidence 

45° 

typical reflectance curves 

400 500 600 700 


Figure 5.18 Single-layer MgF 2 coating /066 

$ The most popular and versatile antireflection coating 

for visible wavelengths 

$ Highly durable and most economical 

$ Optimized for 550 nm, normal incidence 

$ Relatively insensitive to changes in incidence angle 

$ Damage threshold: 13.2 J/cm 2 810%, 10-nsec pulse 

(1050 MW/cm 2 ) at 532 nm 


Figure 5.19 

Wavelength 

Range 

(nm) 

5 

4 

3 

2 

1 

Single-Layer MgF 2 Antireflection Coating 

Normal Incidence 

On BK7 

(%) 

0° 

normal incidence 

Maximum Reflectance 

On Fused Silica 

(%) 

typical reflectance curve 

500 600 700 800 


/067 Single-layer MgF 2 , visible/IR 

$ Optimized for 670 nm, normal incidence 

$ Useful for most visible and near-infrared diode wavelengths 

$ Highly durable and insensitive to angle 

$ Damage threshold: see /066 (similar specifications) 

COATING 

SUFFIX 

400–700 

520–820 

2.0 

2.0 

2.25 

2.25 

/066 

/067 

Note: To order this coating, append coating suffix to product number and specify which 

surfaces are to be coated. 





HEBBAR Coatings 

Our HEBBAR (high-efficiency broadband antireflection) 

coatings provide a very low reflectance over a broad spectral 

bandwidth. These multilayer films, comprising alternate layers of 

various index materials, are combined to reduce overall reflectance 

to an extremely low level for the broad spectral range covered. 

These coatings exhibit a characteristic, double-minimum reflectance 

curve covering a range of some 300 nm in wavelength. The 

reflectance does not exceed 1.0% and is more typically below 0.6% 

over this entire range. Within a more limited spectral range on 

either side of the central peak, reflectance can be held well below 

0.4%. HEBBAR coatings are somewhat insensitive to angle of 

incidence. The effect of increasing the angle of incidence, however, 

is to shift the curve to slightly shorter wavelengths and to increase 

the long wavelength reflectance slightly. These coatings are extremely 

useful for high-numerical-aperture (low f-number) lenses or steeply 

curved surfaces. In these cases, incidence angle varies significantly 

over aperture. 

Six versions of HEBBAR coatings are offered (see figures 5.20 

through 5.25). Many of our components are carried in stock with 

a HEBBAR coating. The /078 covers most of the visible spectrum 

(415–700 nm) and is optimized for normal incidence. A 45-degreeincidence 

version of this coating (/079) is available. The infraredshifted 

/077 covers the range from 700 nm to 1100 nm. Two 

HEBBAR coatings, the /075 and the /076, are specifically optimized 

for diode laser wavelengths. The /074 is a modified HEBBAR 

coating intended for use in the range from 300 nm to 500 nm. Over 

this range, the reflectance is less than 1% and typically does not 

exceed 0.5%. Due to the special nature of the /074 coating, it is 

designed to be used only at an angle of incidence of 0±15 degrees, 

and it is not suitable for use on lenses with steeply curved surfaces. 

For these coatings, reflectance values apply to indices 1.47–1.55 

only. Other indices, while having their own designs, will have 

reflectance values approximately 20% higher for incidence angles 

from 0 to 15 degrees and 25% higher for incidence angles of 30 

degrees. The typical reflectance curves shown for the HEBBAR 

coatings are for BK7 substrates, except for /074 which is for fused 

silica. 

HEBBAR Coatings 

Wavelength 

Range 

(nm) 

440–660 

415–685 

415–700 

415–680 

632.8 

632.8 

425–670 

425–670 

440–670 

440–670 

750–1100 

700–1100 

700–1150 

700–1100 

650–850 

650–850 

780–850 

780–850 

725–875 

300–500 

300–500 

Maximum 

Reflectance 

(%) 

0.6 Abs 

0.4 Avg 

1.0 Abs 

1.0 Abs 

0.3 Abs 

0.4 Abs 

0.6 Avg 

1.0 Abs 

0.4 Abs 

1.0 Abs 

0.6 Abs 

0.4 Avg 

1.0 Abs 

1.0 Abs 

0.6 Avg 

1.0 Abs 

0.25 Abs 

0.40 Abs 

1.0 Abs 

1.0 Abs 

0.5 Avg 

Angle of 

Incidence 

(degrees) 

0–15 

0–15 

0–15 

0–30 

0–15 

0–30 

45 

45 

30–45 

45–50 

0–15 

0–15 

0–15 

0–30 

0–15 

0–15 

0–15 

0–30 

0–30 

0–15 

0–15 

COATING 

SUFFIX 

/078 

/079 

/077 

/075 

/076 

/074 

Note: To order, append coating suffix to product number and specify which surfaces are 

to be coated. 




5 

4 

3 

2 

1 


45° incidence 


400 500 600 700 


Figure 5.20 HEBBAR coating for visible /078 

$ Industry-standard multilayer, AR coating for 415 to 700 nm 

$ Excellent performance with HeNe and visible diode lasers 

$ Optimized for normal incidence 

$ R avg < 0.4%, R abs < 1.0% 

$ Damage threshold: 3.8 J/cm 2 810%, 

10-nsec pulse (230 MW/cm 2 ) at 532 nm 


5 

4 

3 

2 

1 


400 500 600 700 


Figure 5.21 HEBBAR coating for visible /079 


$ Optimized for 425–670 nm at 45-degree incidence 

$ Perfect for plate beamsplitting applications 

$ R avg < 0.6%, R abs < 1.0% 




5 

4 

3 

2 

1 

5 

4 

3 

2 

1 




700 800 900 1000 1100 1200 


Figure 5.22 HEBBAR coating for near-infrared /077 

$ Covers popular Ti:sapphire and diode laser wavelengths: 

750 to 1100 nm 

$ R avg < 0.4%, R abs < 0.6% 






500 550 600 650 700 750 800 850 900 


Figure 5.23 HEBBAR coating for near-infrared and diode 

wavelengths /075 

$ Optimized for performance from 660 to 835 nm 

$ Versatile for use with most diode lasers from visible 

to near-infrared wavelengths 

$ R avg < 0.5%, R abs < 1.0% 






5 


5 




4 

3 

2 

1 



500 600 700 800 900 1000 


Figure 5.24 HEBBAR coating for diode lasers /076 

$ Optimized for diode laser wavelengths, from 780 to 850 nm 

$ R avg < 0.25%, R abs < 0.4% 


LASER-INDUCED DAMAGE 


4 

3 

2 

1 



300 350 400 450 500 


Figure 5.25 HEBBAR coating for ultraviolet /074 

$ Excellent broadband coverage for 300 to 500 nm 

$ Covers HeCd and argon laser lines 

$ R abs < 1.0% 


10-nsec pulse (260 MW/cm 2 ) at 355 nm, on silica substrate 

Melles Griot conducts laser-induced damage testing of our optics at Big Sky Laser Technologies, Inc., in Bozeman, MT. 

Although the damage thresholds listed in this chapter do not constitute a performance guarantee, they are 

representative of the damage resistance of our coatings. Occasionally, in the damage threshold specifications, a 

reference is made to another coating because a suitable high-power laser is not available to test the coating within its 

design wavelength range. The damage threshold of the referenced coating should be an accurate representation of 

the coating in question. 

For each damage threshold specification, the information given is the peak fluence (energy per square centimeter), 

pulse width, peak irradiance (power per square centimeter), and test wavelength. The peak fluence is the total energy 

per pulse, the pulse width is the full width at half maximum (FWHM), and the test wavelength is the wavelength of the 

laser used to incur the damage. The peak irradiance is the energy of each pulse divided by the effective pulse length, 

which is from 12.5% to 25% longer than the pulse FWHM. All tests are performed at a repetition rate of 20 Hz for 

10 seconds at each test point. This is important because longer durations can cause damage at lower fluence levels, 

even at the same repetition rate. 

The damage resistance of any coating depends on substrate, wavelength, and pulse duration. Improper handling and 

cleaning can also reduce the damage resistance of a coating, as can the environment in which the optic is used. These 

damage threshold values are presented as guidelines and no warranty is implied. (See Chapter 14, High Energy Laser 

Optics for details on our guaranteed high-energy laser coatings.) 

When choosing a coating for its power-handling capabilities, some simple guidelines can make the decision process 

easier. First, the substrate material is very important. Higher damage thresholds can be achieved using fused silica 

instead of BK7. Second, consider the coating. Metal coatings have the lowest damage thresholds. Broadband dielectric 

coatings, such as the HEBBAR and MAXBRIte are better, but single-wavelength or laser-line coatings, such as the 

V and the MAX-R coatings, are better still. If even higher thresholds are needed, then high-energy laser (HEL) coatings 

are required, such as those listed in Chapter 14. If you have any questions or concerns regarding the damage levels 

involved in your applications, please contact a Melles Griot applications engineer. 


EXTENDED-RANGE HEBBAR ANTIREFLECTION COATINGS 

Many optical systems require transmission of several, quite disparate 

wavelengths or transmission over a very broad continuum of wavelengths. 

Examples include systems involving two types of lasers, a laser 

system producing fundamental and harmonic wavelengths, a multiplewave 

mixing experiment, stimulated Raman experiments, or a system 

using one laser for action and another for alignment. 

In these situations, normal broadband coatings will not suffice. 

For such cases, Melles Griot makes available extended-range antireflection 

coatings. These coatings offer either a single, extended 

performance band or two separate high-performance regions. In the 

latter case, one of the low-reflectance regions can be quite narrow, 

since many of the applications often involve at least one laser beam. 

Melles Griot manufactures many such coatings for a variety of customer 

specifications. The following special coatings are offered as 

standard catalog items. The typical reflectance curves shown for 

the extended-range HEBBAR coatings are for BK7 substrates, 

except for /072 which is for fused silica. 

Visible/1064 nm Coating 

This coating, designated /083 and shown in figure 5.26 is 

designed for broadband antireflectance in the visible, as well as at 

1064 nm, the wavelength of Nd-YAG lasers. With less than 1% 

reflectance between 450 and 680 nm, and less than 0.25% 

reflectance at 1064 nm, this coating will find many uses in any 

system using a visible source in conjunction with low to moderate 

power Nd:YAG laser fundamental radiation. Its high performance 

is guaranteed for incidence angles up to 15 degrees. Optics 

with this coating are therefore best used at normal incidence and 

can be used for both converging and diverging beams. 

Diode Laser Coating 

This coating, designated /084 and shown in figure 5.27, is 

designed to operate at two popular diode laser wavelengths. It is a 

modified broadband coating which works well through the nearinfrared 

spectrum with reflection minima between 780 nm and 

830 nm and also at 1300 nm. 

Performance is guaranteed for incidence angles up to 15 degrees. 

This coating can therefore be used even without complete collimation 

of the diode radiation. 

Extended-Range HEBBAR Antireflection Coatings 

Coating 

Visible / 1064 nm 

Diode Laser 

Extended Broadband 

UV Broadband 

Wavelength Range 

(nm) 

450–700 

1064 

780–830 

1300 

420–1100 

245–440 

Maximum Reflectance 

(%) 

1.25 

0.25 

0.5 

0.5 

1.75 

1.0 

Extended Broadband Coating 

This very broad coating, designated /073 and shown in figure 

5.28, offers good performance over the entire visible and nearinfrared 

spectral range. It is effective with broadband infrared 

sources, such as infrared LEDs, as well as systems using several, 

widely separated discrete laser lines. 

UV Broadband Coating 

The ultraviolet broadband antireflection coating, designated 

/072 and shown in figure 5.29, is designed for use on fused-silica 

substrates and provides less than 1% reflectance from 245 nm to 

440 nm. It is particularly useful with most popular excimer laser 

lines, as well as other ultraviolet sources, such as mercury lamps. 

The broad response of this coating allows it to perform well even 

with poorly collimated light, which can be especially advantageous 

when dealing with excimer laser sources. 

HIGH-ENERGY-LASER COATED OPTICS 

Optics and coatings specifically designed and 

manufactured to withstand laser-induced damage 

are recommended for high-power lasers, particularly 

pulsed lasers. 

See Chapter 14, High Energy Laser Optics, for an 

extensive listing of these products, together with a 

brief discussion of laser-induced damage. 

Average Reflectance 

(%) 


5 


5 






4 

3 

2 

1 

400 600 800 1000 1100 


5 

4 

3 

2 

1 


Figure 5.26 HEBBAR coating for visible and Nd:YAG 

wavelengths /083 

$ Extremely versatile extended antireflection coating from 

450 to 700 nm and 1064 nm 

$ Ideal for Nd:YAG laser fundamental and second harmonic 

$ Also performs well across the visible, including HeNe and 

visible diode laser lines 

$ Performance guaranteed from 0º–15º on BK7 substrate 

$ R abs < 1.25% @ 450–700 nm, R abs < 0.25% @ 1064 nm 


10-nsec pulse (107 MW/cm 2 ) at 532 nm; 

5.4 J/cm 2 810%, 20-nsec pulse (220 MW/cm 2 ) at 1064 nm on 

silica substrate 




750 900 1150 1350 1500 

Figure 5.27 HEBBAR coating for IR diode lasers /084 

$ Extended antireflection coating: 780 to 830 nm and 1300 nm 

$ Can be used with relatively uncollimated diode laser beams 

$ R abs < 0.5% @ 780 nm to 830 and 1300 nm 

$ Damage threshold: see /083 (similar specifications at 1064 nm) 



4 

3 

2 

1 

400 500 600 700 800 900 1000 1100 

5 

4 

3 

2 

1 



Figure 5.28 HEBBAR coating for visible and near-IR /073 

$ Extended antireflection coating for 420 to 1100 nm 

$ Excellent broadband coating, covering the visible and 

near-infrared regions 

$ R avg

V-Coatings 

V-coatings are multilayer antireflection coatings that reduce the 

reflectance of a component to near-zero for one very specific wavelength. 

Our V-coatings are intended for use at normal incidence, for 

maximum reflectances of not more than 0.25% at their design wavelength. 

They are extremely sensitive to both wavelength and angle of 

incidence. For example, a V-coating intended for the helium neon 

wavelength (632.8 nm) when used at 30-degree incidence will reflect 

about 0.8%. At 45-degree incidence, the same coating will reflect over 

2.5%. If your application involves other than normal angles of 

incidence or high-numerical-aperture (low f-number) optics, it may 

be better to use a HEBBAR coating. 

Experience shows that the maximum reflectance typically 

achieved by these coatings is often closer to 0.1% than the 0.25% 

we specify. Using V-coatings on fused-silica optics can therefore 

provide exceptionally high external transmittances. 

V-Coating, Normal Incidence 

Wavelength 

(nm) 

193 

248 

266 

308 

351 

364 

442 

458 

466 

473 

476 

488 

496 

502 

514 

532 

543 

633 

670 

694 

780 

830 

850 

904 

1064 

1300 

1523 

1550 

Laser Type 

ArF 

ArF 

YAG 3rd harm. 

XeCl 

Ar ion 

Ar ion 

HeCd 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

YAG 2nd harm. 

HeNe 

HeNe 

GaAlAs 

Ruby 

GaAlAs 

GaAlAs 

GaAlAs 

GaAs 

Nd:YAG 

InGaAsP 

HeNe 

InGaAsP 

Maximum 

Reflectance 

(%) 

0.5 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

0.25 

COATING 

SUFFIX 

/101 

/102 

/103 

/104 

/105 

/107 

/111 

/112 

/113 

/114 

/115 

/116 

/117 

/118 

/119 

/122 

/121 

/123 

/128 

/124 

/163 

/166 

/167 

/125 

/126 

/168 

/169 

/169 

The typical reflectance curve illustrated in figure 5.30 is for a 

V-coating on BK7 optical glass. Clearly the performance of such a 

coating is highly dependent on the refractive index of the component 

material. However, we specify all coatings by performance 

figures and not by design. This means that we will change the design 

to suit the material being coated. This makes ordering coatings 

simple: select the specification you want to achieve, tell us what to 

put it on, and we do the rest. This means that your reflectance curve 

may vary from the one shown on this page, but the coating will 

meet your requirements. In addition to the standard coatings listed 

here, Melles Griot can supply V-coatings at any center wavelength 

from 193 nm to 2000 nm. 

These coatings are not intended for use with high-energy lasers 

although the Nd:YAG coatings are suitable for many mediumpower 

applications including diagnostics. 


5 

4 

3 

2 

1 



550 600 

650 700 


Figure 5.30 Example of a V-coating for 632.8 nm /123 

$ Near-zero reflectance at one specific wavelength 

and incidence angle 

$ Maximum reflectance often less than 0.1% 

$ Standard coatings available for most laser lines 

$ Custom center wavelengths at specific angles of incidence 

available per request 


10-nsec pulse (361 MW/cm 2 ) at 532 nm for /122 on silica 

substrate; 10.6 J/cm 2 810%, 20-nsec pulse (480 MW/cm 2 ) 

at 1064 nm for /126 on BK7 substrate 






High-Reflection Coatings 

Melles Griot offers a wide variety of high-reflection coatings 

for mirrors, beamsplitters, polarizing beamsplitters, dichroic mirrors, 

bandpass filters, and rejection filters. Some of these coatings are 

applied to optics as requested; others are offered only as an integral 

part of specialized optical elements. 

High-reflection coatings are ordered in the same way as antireflection 

coatings, namely by appending the three-digit coating 

suffix to the catalog number of the part being ordered. 

High-reflection coatings may be applied to the outside of a 

component, such as a flat piece of glass, to produce a first-surface 

mirror. Alternately, they may be applied to an internal surface to 

produce a second-surface mirror, such as a prism. 

High-reflection coatings can be categorized as either metallic or 

dielectric coatings. 

METALLIC COATINGS 

Metallic coatings are used primarily for mirrors and are not 

classified as thin films in the strictest sense. They do not rely on 

principles of interference, but rather on the optical properties of 

the coating material. However, metallic coatings are often overcoated 

with thin dielectric films to increase reflectance over a desired 

range of wavelengths or angles of incidence. In these cases, the 

metallic coating is said to be “enhanced.” 

Overcoating metallic coatings with a hard, single, dielectric layer 

of half-wave optical thickness improves abrasion and tarnish resistance 

but only marginally affects optical properties. Depending on 

the dielectric used, such overcoated metals are referred to as durable, 

protected, or hard coated. 

The main advantages of metallic coatings are broadband spectral 

performance, insensitivity to angle of incidence and polarization, 

and low cost. Their primary disadvantages are lower durability, 

lower reflectance, and lower damage threshold. 

DIELECTRIC COATINGS 

High-reflectance dielectric layers work on the same principles 

as dielectric antireflection coatings. Quarter-wave thicknesses of 

alternately high- and low-refractive index materials are applied to 

the substrate to form a dielectric multilayer as shown in figure 5.31. 

By choosing materials of appropriate refractive indices, the various 

reflected wavefronts can be made to interfere constructively in order 

to produce a highly efficient reflector. 

The peak reflectance value is dependent upon the ratio of 

refractive indices of the two materials, as well as the number of 

layer pairs. Increasing either increases the reflectance. 

The width of the reflectance curve (versus wavelength) is also 

determined by the film index ratio. The larger the ratio, the wider 

the high-reflectance region. In contrast to antireflection coatings, 

the inherent shape of a high-reflection coating can be modified in 

several different ways. The two most effective ways of modifying the 

performance curve are to use two or more stacks centered at slightly 

shifted design wavelengths, or to slightly perturb the layer thickness 

within a stack. 

Reflectance of such films can easily be made to exceed the highest 

metallic reflectances over limited wavelength intervals. Such films 

are effective for both s- and p-polarization components and over 

a wide angle-of-incidence range. At oblique incidence, reflectance 

is markedly reduced. 

Because of the materials chosen for the multilayer, durability 

and abrasion resistance of such films are normally superior to those 

of metallic films. 

air 

substrate 

Figure 5.31 

quarter-wave thickness of high-index material 

quarter-wave thickness of low-index material 

A simple quarter-wave stack 


Metallic High-Reflection Coatings 

We offer eight forms of standard metallic high-reflection coatings 

formed by vacuum deposition. These coatings, which can be used 

at any angle of incidence, can be applied to most optical components. 

Simply append the coating suffix number to the component product 

number (see figures 5.32 through 5.39). 

Metallic reflective coatings are delicate and require care during 

cleaning. Dielectric overcoats substantially improve abrasion resistance, 

but they are not impervious to abrasive cleaning techniques. 

Clean, dry pressurized gas can be used to blow off loose particles, 

then clean, deionized water, a mild detergent, and alcohol can be 

used. Gentle cleaning with a swab is recommended. 

ALUMINUM (/016) 

$ The most widely used metallic mirror coating 

$ Provides consistently high reflectance throughout the 

near-ultraviolet, visible, and near-infrared regions 

$ R avg > 90% from 400 to 1200 nm 



0.3 J/cm 2 810%, 20-nsec pulse (14 MW/cm 2 ) at 1064 nm 

Aluminum, the most widely used metal for reflecting films, offers 

consistently high reflectance throughout the visible, near-infrared, 

and near-ultraviolet regions of the spectrum. While silver exhibits 

slightly higher reflectance than aluminum through most of the visible 

spectrum, the advantage is temporary because of oxidation 

tarnishing. Aluminum also oxidizes, though more slowly, and its 

oxide is tough and corrosion resistant. Oxidation significantly 

reduces aluminum reflectance in the ultraviolet and causes slight scattering 

throughout the spectrum. 


100 

95 

90 

85 

80 


400 600 800 1000 


Figure 5.32 Aluminum coating /016 


PROTECTED ALUMINUM (/011) 

$ The best general-purpose metallic reflector for visible to 

near-infrared 

$ Protective overcoat extends life of mirror and protects surface 

$ R avg > 87% from 400 to 800 nm 




Protected aluminum is the very best general-purpose, metallic 

coating for use as an external reflector in the visible and nearinfrared 

spectra. Unless we specify otherwise or you specifically 

request a different coating, our mirrors are coated with protected 

aluminum. Protected aluminum is coated with a dielectric film of 

disilicon trioxide (Si 2 O 3 ) of half-wavelength optical thickness at 

550 nm. The protective film arrests oxidation and helps maintain 

high reflectance. It is durable enough to protect the aluminum coating 

from minor abrasions. 


100 

95 

90 

85 


80 45° incidence 

s-plane 

p-plane 

400 450 500 550 600 650 700 750 


Figure 5.33 Protected aluminum coating /011 





ENHANCED ALUMINUM (/023) 

ULTRAVIOLET-ENHANCED ALUMINUM (/028) 


$ Enhanced performance in the mid-visible region 

$ Durability of protected aluminum 

$ R avg > 93% from 450 to 750 nm 


10-nsec pulse (33 MW/cm 2 ) at 532 nm, 


By coating the aluminum with a multilayer dielectric film, 

reflectance is increased over a wide range of wavelengths. The 

durable enhancing multilayer produces a peak reflectance 

of 95% with an average across the visible spectrum of 93%. 

This coating is well suited for applications requiring the durability 

and reliability of protected aluminum, but with higher reflectance 

in the mid-visible regions. The reflectance of enhanced aluminum 

peaks between 530 nm and 580 nm and is high from 400 nm to 

800 nm. 


100 

95 

90 

85 

80 



s-plane 

p-plane 


400 450 500 550 600 650 700 750 


$ Maintains reflectance in the ultraviolet region 

$ Dielectric overcoat prevents oxidation and increases abrasion 

resistance 

$ R avg > 86% from 250 to 400 nm 

$ R avg > 85% from 400 to 700 nm 


10-nsec pulse (5.7 MW/cm 2 ) at 355 nm 

By applying a film of an ultraviolet-transmitting dielectric (usually 

MgF 2 ), the reflectance of pure, bare aluminum can be preserved in 

the ultraviolet. The dielectric layer prevents oxidation of the aluminum 

surface and provides abrasion resistance. While the resulting surface 

is not as abrasion resistant as our protected aluminum, this coating 

may be cleaned with care. Reflectance averages over 86% from 250 

to 400 nm and over 86% throughout the visible spectrum. This 

coating can be applied to all Melles Griot mirrors. 


100 

90 

80 

70 

60 


200 250 300 350 400 



Figure 5.34 Enhanced aluminum coating /023 Figure 5.35 Ultraviolet-enhanced aluminum coating /028 



INTERNAL SILVER (/036) 

$ For internal reflection (second-surface) mirrors and prisms only 

$ Preferred for visible to near-infrared region 

$ Less polarization effects than aluminum 

$ R avg > 98% from 400 nm to 1200 nm 


Through most of the visible and near-infrared spectra, silver has 

higher reflectance than aluminum, at least for a short time following 

deposition. Rapid oxidation quickly causes unprotected silver coatings 

to deteriorate. In the internal silver coating, oxidation and tarnish are 

prevented by coating the external surface with an additional layer of 

either Inconel ® or copper. The Inconel or copper layers are subsequently 

painted to increase abrasion resistance. In this way, the high 

initial reflectance of silver is indefinitely preserved. 

Silver is frequently used in the near-infrared (the interval 

containing neodymium and gallium arsenide laser lines) because 

it avoids the small dip in reflectance exhibited by aluminum in this 

interval. In the near ultraviolet, silver has very low reflectance, and 

aluminum is a preferable choice. From the visible into the middleinfrared, 

silver offers the highest internal reflectance available from 

a metallic coating. Silver has less effect than aluminum on the 

polarization state in these regions of the spectrum. 


100 

95 

90 

85 

80 


400 450 500 550 600 650 700 


Figure 5.36 Internal silver coating /036 

Inconel ® is a registered trademark of EM Industries, Inc. 


PROTECTED SILVER (/038) 

$ Extremely versatile mirror coating 

$ Excellent performance for the visible to infrared region 

$ R avg > 95% from 400 nm to 20 mm 

$ Can be used for ultrafast Ti:Sapphire laser applications 


10-nsec pulse (75 MW/cm 2 ) at 532 nm, 


Melles Griot uses a proprietary coating and edge-sealing 

technology to offer a first-surface external protected silver coating. 

In recent tests, the protected silver coating has shown no broadening 

effect on a 52-femtosecond pulse. This information is presented 

as a guideline for femtosecond applications and no warranty is 

implied. Protected silver offers extremely broad performance, from 

400 nm to well into the infrared, with excellent durability. 

Due to the specialized tooling required to produce the protected 

silver coating, it is offered only on a limited range of substrates. 


100 

80 

60 

40 

20 

0 

400 



500 600 700 800 900 


Figure 5.37 Protected silver coating /038 




BARE GOLD (/045) 

PROTECTED GOLD (/055) 

$ Widely used in the near, middle, and far infrared 

$ Effectively controls thermal radiation 


$ Damage threshold: 1.1 J/cm 2 810% , 


$ The performance of the durability of dielectrics 

$ Protective overcoat extends coating life 






Because it combines good tarnish resistance with consistently 

high reflectance throughout the near, middle, and far infrared, gold 

is widely used in these regions. While it is possible to construct multilayer 

films that may surpass the reflectance of gold at specific 

wavelengths, the useful range of gold is unequaled. Gold is especially 

effective in controlling thermal radiation. Because bare gold is soft 

and scratches easily, bare-gold mirrors should be cleaned only by 

flow-washing with solvents and clean water or by blowing the surface 

clean with a low-pressure stream of dry air. 


100 

80 

60 

40 

20 

0 

400 

800 1200 1600 2000 2400 2800 


The Melles Griot proprietary protected gold mirror coating 

combines the natural spectral performance of gold with the 

durability of hard dielectrics. Protected gold provides over 96% 

average reflectance from 650 to 1700 nm, and over 98% average 

reflectance from 2 to 16 mm. As well as the damage threshold listed 

above, the /055 coating was tested for laser-induced damage and was 

found to withstand up to 18±2 J/cm 2 with a 260-µs pulse 

(0.4 MW/cm 2 ) at a wavelength of 3 µm. These mirrors can be cleaned 

regularly using standard organic solvents, such as alcohol or acetone. 


100 

80 

60 

40 

20 

0 



700 800 900 1000 1100 1200 1300 1400 1500 1600 


Figure 5.38 Bare gold coating /045 Figure 5.39 Protected gold coating /055 

Coating Type 

Aluminum 

Protected Aluminum 

Enhanced Aluminum 

UV-Enhanced Aluminum 

Internal Silver 

Protected Silver 

Bare Gold 

Protected Gold 


Metallic High-Reflection Coatings 

To order, append coating suffix to product number. 

Wavelength Range 

(nm) 

400–1200 

400–800 

450–750 

250–400 

400–1200 

400–20,000 

700–20,000 

650–16,000 


Average Reflectance 

(%) COATING SUFFIX 

90 

87 

93 

86 

98 

95 

99 

98 

/016 

/011 

/023 

/028 

/036 

/038 

/045 

/055 


Dielectric High-Reflection Coatings 

QUARTER-WAVE STACK 

The basic building block for any coating involving high levels of 

reflection is the quarter-wave stack— a stack of alternate layers of 

high- and low-refractive-index material. Each layer in the stack 

ideally has an optical thickness of a quarter wave at the design 

wavelength. Alternate reflections are phase shifted by 180 degrees 

because they occur at low- to high-index interfaces (external 

reflections). These phase shifts are exactly canceled by the 

180-degree phase shifts caused by the path difference between 

alternate reflecting surfaces. All reflected wavefronts are therefore 

exactly in phase and undergo only constructive interference. 

If the difference in the refractive index of the materials is large, 

then a quarter-wave stack containing only a few layers will have a 

very high reflectance. 

PERFORMANCE CURVE 

The reflection versus wavelength performance curve of a single 

dielectric stack has a characteristic flat top inverted V shape as 

shown in figure 5.40. Clearly, reflectance is a maximum at the wavelength 

for which both the high- and low-index layers of the multilayer 

are exactly one-quarter-wave thick. 

Outside the fairly narrow region of high reflectance, the 

reflectance slowly reduces toward zero in an oscillatory fashion. 

Width and height (i.e., peak reflectance) of the high-reflectance 

region are functions of the refractive-index ratio of the two materials 

used, together with the number of layers actually included in the 

stack. The peak reflectance can be increased by adding more layers, 

or by using materials with a higher refractive index ratio. Amplitude 

reflectivity at a single interface is given by 

(14p) 

(1+ p) 

where 

⎛ 

p = n ⎜ 

⎝ n 

H 

L 

n S is the index of the substrate, and n H and n L are the indices of the highand 

low-index layers. N is the total number of layers in the stack. 

The width of the high-reflectance part of the curve (versus wavelength) 

is also determined by the film index ratio. The higher the 

ratio, the wider is the high-reflectance region. 

SCATTERING 

N41 

2 

⎞ nH 

⎟ × (5.26) 

⎠ n , 

S 

The main parameters used to describe the performance of a 

thin film are reflectance and transmittance (plus absorptance, where 

applicable). Another less well-defined parameter is scattering. This 

is hard to define because of the inherent granular properties of the 

materials used in the films. Granularity causes some of the incident 

light to be “lost” by specular reflection. Often, it is scattering, not 

mechanical stress and weakness in the coating, that limits the 

maximum practical thickness of an optical coating. 


100 

80 

60 

40 

20 

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 

BROADBAND COATINGS 

RELATIVE WAVELENGTH 

Figure 5.40 Typical reflectance curve of an unmodified 

quarter-wave stack 

In contrast to antireflection coatings, the inherent shape of a 

high-reflectance coating can be modified in several different ways. 

The two most effective ways of modifying a performance curve are 

to use two or more stacks centered at slightly shifted design wavelengths, 

or to perturb the layer thicknesses within a stack. 

There is a subtle difference between multilayer antireflection 

coatings and multilayer high-reflection coatings, which allows the 

performance curves of the latter to be modified by using layer thicknesses 

designed for different wavelengths within a single coating. 

Consider a multilayer consisting of pairs, or stacks of layers, which 

are designed for different wavelengths. At any given wavelength, 

providing at least one of the layers is highly reflective for that wavelength, 

the overall coating will be highly reflective at that wavelength. 

Whether the other components transmit or are partially reflective 

at that wavelength is immaterial. Transmission of light of that wavelength 

will be blocked by reflection of a single component. 

On the other hand, in an antireflection coating, even if one of 

the stacks is exactly antireflective at a certain wavelength, the overall 

coating may still be quite reflective because of reflections by the 

other components (see figure 5.41). 

This can be summarized by an empirical rule. At any wavelength, 

the reflection of a multilayer coating consisting of several discrete 

components will be at least that of the most reflective component. 

Exceptions to this rule are coatings that have been designed to 

produce interference effects not just involving the surfaces within 

the two-layer or multilayer component stack, but also between the 

stacks themselves. Obvious examples are narrowband interference 

filters which are described in detail later and in Chapter 13, Filters. 






BROADBAND REFLECTION COATINGS 

The design procedure for a broadband reflection coating should 

now be apparent. Two design techniques are used. The most obvious 

approach is to use two quarter-wave stacks with their maximum 

reflectance wavelengths separated on either side of the design wavelength. 

This type of coating, however, tends to be too thick and 

often has poor scattering characteristics. The basic design is very 

useful for dichroic high reflectors, where the peak reflectances of 

two stacks are at different wavelengths. 

A more elegant approach to broadband dielectric coatings is 

to use a single modified quarter-wave stack. In this modified stack, 

the layers are not all of the same optical thickness. They are graded 

between the quarter-wave thickness for two wavelengths at either 

end of the intended broadband performance region. The optical 

thicknesses of the individual layers are usually chosen to follow a 

simple arithmetic or geometric progression. Using designs of this 

type, multilayer, broadband, high-reflectance coatings are possible 

with reflectances in excess of 99% over several hundred nanometers. 

The greatest impact of improved broadband reflector design 

and manufacturing technology has almost certainly been on dye laser 

design and applications. In many of these scanning systems, high 

reflectance over a large wavelength region is absolutely essential. In 

many non-laser instruments, all-dielectric coatings are favored over 

metallic coatings because of their high reflectance. Multilayer 

broadband coatings are available with high-reflectance regions 

spanning almost the entire visible spectrum. Such films are effective 

for both s- and p-polarization components, and over a wide range 

of incidence angles. At oblique incidence, reflectance is markedly 

reduced. 

Because of the materials chosen for the multilayer, durability 

and abrasion resistance of such films are superior to those of metallic 

films. Although the reflectance of dielectric coatings can easily be 

made to exceed the highest metallic reflectances over very large 

wavelength intervals, metallic coatings are still superior in terms of 

usable ranges of incidence angles and wavelengths for a single coating. 

POLARIZATION EFFECTS 

When light is incident on any optical surface at angles other 

than normal incidence, there is always a difference in the reflection/ 

transmission behavior of s- and p-polarization components. In 

some instances, this difference can be made extremely small. On 

the other hand, it is sometimes advantageous to design a thin-film 

coating that maximizes this effect (e.g., thin-film polarizers). 

Polarization effects are not normally considered for antireflection 

coatings since these are nearly always used at normal incidence 

where the two polarization components are equivalent. 

High-reflectance or partially reflecting coatings are frequently 

used away from normal incidence, particularly at 45 degrees, for 

beam steering or beam splitting. Polarization effects can therefore 

be quite important for these types of coating. 

effective broadband high-reflection coating 

incident 


NOTE: If at least one component is totally 

reflective, the coating will not transmit 

light at that wavelength. 

noneffective broadband antireflection coating 

incident 


NOTE: Unless every component is totally 

nonreflective, some reflection losses will occur. 

totally reflective component for l 0 

partially reflective component for l 0 

totally nonreflective component for l 0 

Figure 5.41 Schematic multicomponent coatings with 

only one component exactly matched to the incident 

wavelength, l . (the high-reflection coating is successful; the 

antireflection coating is not). 

At certain wavelengths, a multilayer dielectric coating shows a 

remarkable difference in its reflectance of the s- and p-polarization 

components (see figure 5.42). 

The basis for the effect is the difference in effective refractive 

index of the layers of film for s- and p-components of the incident 

beam, as the angle of incidence is increased from zero. This should 

not be confused with the phenomenon of birefringence in certain 

crystalline materials, most notably calcite. Unlike birefringence, it 

does not require the symmetric properties of a truly crystalline 

phase. It arises from the difference in magnetic and electric field 

asymmetries for s- and p-components of an electromagnetic wave 

at oblique incidence. 

The resultant difference in reflectance of the two polarization 

components is always in the same sense. Maximum s-polarization 

reflectance is always greater than the maximum p-polarization 



100 

80 

60 

40 

20 

p-plane 

0.8 0.9 1.0 1.1 1.2 

RELATIVE WAVELENGTH 

reflectance at oblique incidence. If the reflectance is plotted as a 

function of wavelength for some arbitrary incidence angle, the 

s-polarization high-reflectance peak always extends over a broader 

wavelength region than the p-polarization peak. 

Many dielectric coatings are used at peak reflectance wavelengths 

where polarization differences can be made negligible. In some 

cases, the polarization differences can be be put to good use. The 

“edge” region of the reflectance curve is a wavelength region in 

which the s-polarization reflectance is much higher than the 

p-polarization reflectance. This can be maximized in a design to 

produce a very efficient thin-film polarizer. 

EDGE FILTERS AND HOT OR COLD MIRRORS 

s-plane 

Figure 5.42 The s-polarization reflectance curve is always 

broader and higher than the p-polarization reflectance 

curve 

In many optical systems, it is necessary to have a wavelength 

filtering system that transmits all light of wavelengths longer than 

a reference wavelength or transmits light at shorter wavelengths 

than a reference wavelength. These types of filters are often called 

short-wavelength or long-wavelength cutoff filters. 

Traditionally, such filters were made from colored glasses. 

Melles Griot offers a range of these economical and useful filters in 

Chapter 13, Filters. Although they are adequate for many applications, 

they have two drawbacks: they function by absorbing unwanted 

wavelengths, which may be a problem in such high-power situations 

as projection optics, and the edge of the transmission curve may not 

be as sharp as necessary for some applications. 

Thin films acting as edge filters are now routinely manufactured 

using a modified quarter-wave stack as the basic building block. 

Melles Griot produces many edge filters specially designed to meet 

customers’ specifications. A selection suitable for various laser 

applications is offered as standard catalog items. 

This type of filter is used in high-power image-projection systems 

where the light source often generates intense amounts of heat 

(infrared and near-infrared radiation). Thin-film filters designed 

to separate visible and infrared radiation are known as hot or cold 

mirrors, depending on which wavelength region is rejected (reflected) 

and which is transmitted. Melles Griot offers both hot and cold 

mirrors as catalog items (see Chapter 13, Filters). 

INTERFERENCE FILTERS 

In many applications, particularly those in the field of resonance 

atomic or molecular spectroscopy, a filtering system is required 

that transmits only a very narrow range of wavelengths of 

incident light. For particularly high-resolution applications, monochromators 

may be used, but these have very poor throughputs. In 

instances where moderate resolution is required and where the 

desired region(s) is fixed, interference filters should be used. 

Interference filters are produced by applying a complex multilayer 

coating to a colored glass blank. The complex coating consists 

of a series of broadband quarter-wave stacks which act as a verythin 

multiple-cavity Fabry-Perot interferometer. The colored glass 

absorbs light that would be transmitted by higher order interferences. 

Figure 5.43 shows the transmission curve of a typical 

Melles Griot interference filter, the 550-nm filter from the visible- 

40 filter set (03 IFS 008). Notice the square shape of the transmission 

curve which dies away very quickly outside the high-transmission 

(low-reflectance) region. 

More information concerning the design and operation of such 

filters can be found together with product listings in Chapter 13, 

Filters. 

PERCENT TRANSMITTANCE 

100 

90 

80 

70 

60 

50 

40 

30 

20 

10 

Figure 5.43 

typical transmittance curve 

450 550 650 750 


Spectral performance of an interference filter 





PARTIALLY TRANSMITTING COATINGS 

In many applications, it is desirable to split a beam of light into 

two components with an arbitrary intensity ratio. This is performed 

by inserting an optical surface at some oblique angle (usually 45 

degrees) to produce a separate reflected and transmitted component. 

In most cases, a multilayer coating is applied to the surface in order 

to modify intensity and polarization ratios of the two beams. 

An alternative to the outdated metallic beamsplitter is a broadband 

(or narrowband) multilayer dielectric stack with a limited 

number of pairs of layers, which transmits a fixed amount of the incident 

light. Just as in the case of metallic beamsplitter coatings, the 

ratio of reflected and transmitted beams depends on the angle of 

incidence. Since the angle of incidence is normally fixed at 45 degrees, 

this does not present a significant problem. Unlike a metallic coating, 

a high-quality film will introduce negligible losses by either 

absorption or scattering. There are, however, two drawbacks to 

dielectric beamsplitters. The performance of these coatings is more 

wavelength sensitive than that of metallics, and the ratio of transmitted 

and reflected intensities may be quite different for the s- and 

p-polarization components of the incident beam. In polarizers, this 

can be used to advantage. The difference in partial polarization of 

the reflected and transmitted beams is not important, particularly 

where polarized lasers are used. In beamsplitters, this is usually a 

drawback. A hybrid metal-dielectric coating is often the best 

compromise. 

Melles Griot produces coated beamsplitters with designs ranging 

from broadband performance without polarization compensation, 

to broadband with some compensation for polarization, to a 

completely new range of cube beamsplitters that are virtually nonpolarizing 

at certain laser wavelengths. These nonpolarizing 

beamsplitters offer unparalleled performance with the reflected 

s- and p-components matched to better than 5%. 



MAXBRIte Coatings 

MAXBRIte (multilayer all-dielectric xerophilous broadband 

reflecting interference) coatings are, without a doubt, the best broadband 

mirror coatings commercially available. The /001 coating covers 

the visible spectrum from 480 nm to 700 nm, the /003 is useful 

for diode laser applications from 630 nm to 850 nm, and the /009 

coating offers enhanced blue response. They all reflect well over 

98% of incident laser radiation within their respective wavelength 

ranges. 

These coatings exhibit exceptionally high reflectances for both 

s- and p-polarizations. In each case, at the most important laser 

wavelengths and for angles of incidence as high as 45 degrees, the 

average of s- and p-reflectances exceeds 99%. For most applications, 

they are superior to metallic or enhanced metallic coatings. 

The /001 MAXBRIte coating (figure 5.44) is suitable for 

instrumental and external laser-beam manipulation tasks. It is the 

ideal choice for use with tunable dye and parametric oscillator 

systems. The structural design of this coating is such that flatness 

specifications as tight as l/10 can be maintained using low-expansion 

substrate materials. 

The /003 MAXBRIte coating (figure 5.45) covers all the 

important diode laser wavelengths from 630 to 850 nm; therefore, 

it can be used with both visible and near-infrared diode lasers. This 

broadband coating is ideal for applications employing nontemperature- 

stabilized diode lasers where wavelength drift is likely to 

occur. The /003 also makes it possible to use a HeNe laser to align 

diode systems. 

The /007 ultraviolet MAXBRIte coating (figure 5.46) offers 

superior performance for ultraviolet applications. It is ideal for use 

with many of the excimer lasers, as well as the third and fourth harmonics 

of most solid-state lasers. It is also particularly useful with 

broadband ultraviolet light sources, such as mercury and xenon 

lamps. Due to mechanical stresses within this intricate coating, it 

is limited to substrates of l/4 figure or less. 

The /009 extended MAXBRIte coating (figure 5.47) offers superior 

response below 500 nm, and it is particularly useful for helium 

cadmium lasers at 442 nm, or the blue lines of argon-ion lasers. Like 

the /007, mechanical stresses in this complex coating limit its use to 

substrates of l/4 figure or less. 

Because of the many applications for these superior coatings we 

stock a number of precoated substrates. 

MAXBRIte Coatings 

Wavelength 

Range 

(nm) 

480–700 

630–850 

245–390 

420–700 

Average 

Reflectance 

(%) 

98 

98 

98 

98 

Angle of 

Incidence 

(degrees) 

0±45 

0±45 

0±45 

0±45 

Note: To order, append coating suffix to product number. 

COATING 

SUFFIX 

/001 

/003 

/007 

/009 



100 

99 

98 

97 

96 

100 

99 

98 

97 

96 



500 600 700 800 


Figure 5.44 MAXBRIte /001 coating 



600 700 800 900 



$ Exceptional reflectance for s- and p- polarization 

$ Excellent performance for common visible lines 

$ Suitable for external laser-beam manipulation and many 

instrument applications 

$ R avg > 98% from 480 to 700 nm 





$ Useful with visible, near-infrared diode, and HeNe lasers 

$ Easily accommodates diode wavelength drift 

$ Ideal for pointing and alignment applications 

$ R avg > 98% from 630 to 850 nm 






100 


100 




80 

60 

40 

20 

0 



250 300 350 

400 



$ Excellent performance for excimer and YAG third- and 

fourth-harmonic lines, as well as broadband ultraviolet 

sources 

$ Superior reflectance from 0 to 45 degrees incidence for 

s- and p-polarizations 

$ R avg > 98% from 245 to 390 nm 


OPTICS GLASS CLEANING 


99 

98 

97 

96 



400 500 600 700 



$ Wavelength range extended even farther than /001 

MAXBRIte 

$ Outstanding performance from 0 to 45 degrees incidence 

$ R avg > 98% from 420 to 700 nm 


10-nsec pulse (35 MW/cm 2 ) at 532 nm on silica substrate 

Melles Griot removes all contamination from our substrates prior to coating with a high-volume, five-stage Interlab 

semi-aqueous glass cleaner. 


Laser-Line MAX-R Coatings 

These multilayer coatings achieve the highest possible reflectances 

at specific laser wavelengths and at particular angles of incidence. At 

these wavelengths and angles, laser-line MAX-R coatings outperform 

MAXBRIte . We offer coatings for angles of incidence at 

0 degrees (see figure 5.48) and 45 degrees (see figure 5.49). Because 

the layer thicknesses differ for these two angles, the coatings cannot 

be used interchangeably. Laser-line MAX-R coatings are intended 

for external beam-manipulation applications. Specified coating 

reflectances apply to p-polarization (except at normal incidence, 

where the p- and s-polarization states are indistinguishable). 

Reflectances for s-polarization generally exceed those for p- 

polarization. Other coatings can be supplied at any center wavelength 

from 193 nm to 1.6 mm. 


LASER-INDUCED DAMAGE 

The following laser damage threshold statistics do 

not constitute a performance guarantee but should 

be representative for MAX-R coatings. 

/205 2.4 J/cm 2 810%, 


/255 13.3 J/cm 2 810%, 


/225 12.6 J/cm 2 810%, 


/275 13.6 J/cm 2 810%, 


/241 2.4 J/cm 2 810%, 


/291 17.7 J/cm 2 810%, 



100 

80 

60 

40 

20 

Figure 5.48 


100 

Figure 5.49 



0.8 0.9 1.0 1.1 1.2 

RELATIVE WAVELENGTH, g = l 0 /l 

80 

60 

40 

20 

MAX-R coating, normal incidence 


s-polarization 

p-polarization 


0.8 0.9 1.0 1.1 1.2 

RELATIVE WAVELENGTH, g = l 0 /l 

MAX-R coating, 45-degree incidence 

$ Highest possible reflectance achieved at specific laser 

wavelengths and angles of incidence 

$ Standard MAX-R coatings available for popular 

laser wavelengths at both 0 degrees and 45 degrees 

$ Custom designs easily produced 

$ Coatings available from 193 nm to 1550 nm 

PLEASE NOTE FOR THE ABOVE CURVES: 

For g = l 0 /l, l 0 is the design wavelength and l is 

arbitrary. For example, l = l 0 /g = 1064 nm/0.9 = 

1182 nm, if one looks at g = 0.9 for the /291 coating. 

For values of g below 0.8 and above 1.2, the 

reflectance curves oscillate sinusoidally, in a manner 

that varies from coating to coating and run to run. 




Laser-Line MAX-R Coatings, Normal Incidence 

Laser-Line MAX-R Coatings, 45-Degree Incidence 

Minimum 

Reflectance R p (%) 

Minimum 

Reflectance R p (%) 

Wavelength 

(nm) 

Laser Type 

0º 

Incidence 

0º±15° 

Incidence 

COATING 

SUFFIX 

Wavelength 

(nm) 

Laser Type 

45º 

Incidence 

45º±10° 

Incidence 

COATING 

SUFFIX 


193 

248 

266 

308 

351 

364 

442 

458 

466 

473 

476 

488 

496 

502 

514 

532 

543 

633 

670 

694 

780 

830 

850 

904 

1064 

1300 

1523 

1550 

ArF 

KrF 

Nd:YAG 4th harm. 

XeCl 

Ar ion 

Ar ion 

HeCd 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Nd:YAG 2nd harm. 

HeNe 

HeNe 

GaAlAs 

Ruby 

GaAlAs 

GaAlAs 

GaAlAs 

GaAs 

Nd:YAG 

InGaAsP 

HeNe 

InGaAsP 

97.0 

98.0 

98.0 

99.0 

99.0 

99.0 

99.3 

99.5 

99.5 

99.5 

99.5 

99.5 

99.5 

99.5 

99.5 

99.5 

99.5 

99.5 

99.5 

99.3 

99.3 

99.3 

99.3 

99.3 

99.2 

99.2 

99.2 

99.2 

94.0 

95.0 

95.0 

96.0 

96.0 

96.0 

99.0 

99.3 

99.3 

99.3 

99.3 

99.3 

99.3 

99.3 

99.3 

99.3 

99.3 

99.3 

99.3 

99.0 

99.0 

99.0 

99.0 

99.0 

99.0 

99.0 

99.0 

99.0 


/201 

/202 

/203 

/204 

/205 

/207 

/209 

/211 

/213 

/215 

/217 

/219 

/221 

/222 

/223 

/225 

/226 

/229 

/228 

/231 

/233 

/237 

/238 

/239 

/241 

/245 

/247 

/247 

193 

248 

266 

308 

351 

364 

442 

458 

466 

473 

476 

488 

496 

502 

514 

532 

543 

633 

670 

694 

780 

830 

850 

904 

1064 

1300 

1523 

1550 

ArF 

KrF 

Nd:YAG 4th harm. 

XeCl 

Ar ion 

Ar ion 

HeCd 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Ar ion 

Nd: YAG 2nd harm. 

HeNe 

HeNe 

GaAlAs 

Ruby 

GaAlAs 

GaAlAs 

GaAlAs 

GaAs 

Nd:YAG 

InGaAsP 

HeNe 

InGaAsP 

97.0 

98.0 

98.0 

98.0 

98.0 

98.0 

99.0 

99.3 

99.3 

99.3 

99.3 

99.3 

99.5 

99.5 

99.5 

99.5 

99.5 

99.5 

99.0 

99.0 

99.0 

99.0 

99.0 

99.0 

99.0 

99.0 

99.0 

99.0 

94.0 

95.0 

95.0 

95.0 

96.0 

96.0 

98.0 

98.0 

98.5 

98.5 

98.5 

98.5 

98.5 

98.5 

98.5 

98.5 

98.5 

98.5 

98.5 

98.5 

98.5 

98.5 

98.5 

98.5 

98.0 

98.5 

98.5 

98.5 


/251 

/252 

/253 

/254 

/255 

/257 

/259 

/261 

/263 

/265 

/267 

/269 

/271 

/272 

/273 

/275 

/276 

/279 

/278 

/281 

/283 

/287 

/288 

/289 

/291 

/295 

/297 

/297 

SPHERICAL AND CYLINDRICAL MIRRORS 


Virtually any of our mirror coatings can be applied to standard optical components. Common examples include using 

plano-concave lenses to make spherical concave mirrors or coating plano-convex lenses to make secondary mirrors for 

Cassegrain beam expanders or telescopes. The radius of curvature of any of our simple lenses can be calculated using 

the data given in the product tables together with the formulas presented at the end of Chapter 1, Fundamental 

Optics. 


Ultrafast Coating 

Melles Griot has developed a new coating which is designed for 

ultrafast laser systems in the near-infrared. The ultrafast coating 

(/091, shown in figure 5.50), an all-dielectric coating, centered at 

800 nm, minimizes pulse broadening for ultrafast applications. The 

coating also offers exceptionally high reflectance for both 

s- and p-polarizations in the range from 750 nm to 870 nm. 

The ultrafast coating is ideal for high-power Ti:sapphire laser 

applications. This coating is superior to protected and enhanced 

metallic coatings because of its ability to handle higher powers. For 

lower power light sources, the protected silver /038 coating is strongly 

recommended for ultrafast applications because of its low pulsebroadening 

effect. 

Melles Griot offers the /091 coating standard on both a 12.5-mmand 

a 25.0-mm-diameter ultraviolet synthetic fused-silica substrate. 


100 

80 

60 

40 

20 

Ultrafast Coating (/091) 

Wavelength 

Range 

(nm) 

p-plane 

s-plane 

700 800 900 

1000 


Figure 5.50 Ultrafast coating /091 

Minimum 

Reflectance 

R p (%) 

Angle of 

Incidence 

(deg) 

Pulse 

Broadening 

(%) 


$ R p > 99% from 770 mm to 830 nm, Rs >99% from 750 nm 

to 870 nm 

$ High laser-damage threshold 

$ Low pulse broadening 

COATING 

SUFFIX 

770–830 99.0 45




SALT-FOG AND HUMIDITY 

TESTING FOR OPTICAL COATINGS 

Melles Griot has tested several standard optical 

coatings to ensure spectral performance under 

non-laboratory environmental conditions. The 

following list identifies optical coatings that passed 

environmental testing for salt- fog and humidity. 

The humidity testing was done per MIL-C-14806A 

paragraph 4.4.6. The salt-fog test was done per 

MIL-C-14806 paragraph 4.4.8 and MIL-STD-810C 

reference 1.3, method 509.1, procedure I. 

Humidity testing was done at 120ºF and the relative 

humidity was at 95-100% for 24 hours. Salt-fog 

testing was done at a temperature of 95ºF, pH 

solution of 6.5, collection rate of 1.18 ml/hr, and 

specific gravity of 1.034. 

Optical Coatings Passing Environmental Test 

for Salt-Fog and Humidity 

Antireflection 

Single layer MgF 2 /066 

Single layer MgF 2 /067 

HEBBAR /072 

HEBBAR /073 

HEBBAR /074 

HEBBAR /075 

HEBBAR /076 

HEBBAR /077 

HEBBAR /078 

HEBBAR /079 

HEBBAR /083 

HEBBAR /084 

V-coating at 248 nm 




V-coating at 1523–1550 nm 

Reflective 

MAXBRite /001 

MAXBRite /003 

MAXBRite /009 

MAX-R at 351 nm 



COATING CAPABILITIES 

Melles Griot uses a state-of-the-art, Eddy Company, 

SYS/48B ion-assisted coating chamber for our highprecision 

reflective and antireflection coatings. 

This fully automatic system produces multilayer 

dielectric coatings with excellent thin film quality, 

high damage thresholds, and low loss. In addition, 

the automatic coating process allows us to produce 

coatings with higher precision, better uniformity, 

and greater batch-to-batch repeatability. This benefits 

not only our research customers but also our OEM 

customers. To meet your specific needs Melles Griot 

can produce custom coatings in high volume or in 

prototype quantities. Contact your local Melles Griot 

sales office for more information. 

Our coating chamber is located in a Class-10,000 

clean room to ensure coatings of the highest quality. 

If required, we can also inspect coated optics in a 

clean room and then package them in special cleanroom 

packaging so that the parts can be shipped 

direct to and opened in a clean room environment. 

Filter/Beamsplitter 

Hot Mirror 

Cold Mirror 

UV plate beamsplitter 

03 BTF at 550 nm 

03 BTF at 850 nm 

03 BDS 001 beamsplitter State-of-the art Eddy SYS/48B coating chamber

Optical Coatings

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