Optical Coatings

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Fundamental Optics Material Properties Optical Specifications Gaussian Beam Optics Optical Coatings 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. APPLICATION NOTE 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. 1.20 1 Visit Us OnLine! www.mellesgriot.com
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 Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings Visit Us Online! www.mellesgriot.com 1 1.21
Page 1 and 2: Fundamental Optics 1 Introduction 1
Page 3 and 4: Paraxial Formulas SIGN CONVENTIONS
Page 5 and 6: object F 2 Figure 1.4 image Example
Page 7 and 8: Figure 1.6 Figure 1.7 f f 2 object
Page 9 and 10: Figure 1.8 Figure 1.9 INDIVIDUAL EL
Page 11 and 12: Performance Factors After paraxial
Page 13 and 14: third-order, monochromatic, spheric
Page 15 and 16: Positive lens elements usually have
Page 17 and 18: Lens Shape Aberrations described in
Page 19: ay f-numbers 2.7 3.3 4.4 6.7 13.3 S
Page 23 and 24: Lens Selection Having discussed the
Page 25 and 26: Example 4: Diffraction-Limited Perf
Page 27 and 28: Aberration Balancing To improve sys
Page 29 and 30: Definition of Terms FOCAL LENGTH (f
Page 31 and 32: infinity (which is a comfortable vi
Page 33 and 34: For symmetric lenses (r 2 = 4r 1 ),
Page 35 and 36: Separation of Nodal Point from Corr
Page 37 and 38: Gaussian Beam Optics 2 Introduction
Page 39 and 40: Even if a Gaussian TEM 00 laser-bea
Page 41 and 42: eam expander INCORPORATING M 2 INTO
Page 43 and 44: M 2 AND THE LENS EQUATION For real-
Page 45 and 46: employed when laser power must be c
Page 47 and 48: Figure 2.13 Keplerian beam expander
Page 49 and 50: Optical Specifications 3 Wavefront
Page 51 and 52: Centration The mechanical axis and
Page 53 and 54: PERFECT RECTANGULAR LENS The MDMTF
Page 55 and 56: The permissible number of maximum-s
Page 57 and 58: Material Properties 4 Material Prop
Page 59 and 60: Introduction Glass manufacturers pr
Page 61 and 62: maximum value for homogeneity withi
Page 63 and 64: Melles Griot Lens Materials Melles
Page 65 and 66: Refractive Index of Five Schott Gla
Page 67 and 68: Synthetic Fused Silica Fused silica
Page 69 and 70: PERCENT INTERNAL TRANSMITTANCE 99.9
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Low-Expansion Borosilicate Glass Th
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ZERODUR ® Many optical application
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Optical Coatings 5 Optical Coatings
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OEM and Special Coatings Melles Gri
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PERCENT REFLECTANCE 100 90 80 70 60
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will depend on the ratio of optical
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v = angle of incidence PERCENT REFL
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PERCENT REFLECTANCE PER SURFACE 2.0
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Two-Layer Coatings of Arbitrary Thi
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ION-ASSISTED BOMBARDMENT Ion-assist
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Single-Layer MgF 2 Antireflection C
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PERCENT REFLECTANCE 5 4 3 2 1 norma
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EXTENDED-RANGE HEBBAR ANTIREFLECTIO
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V-Coatings V-coatings are multilaye
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Metallic High-Reflection Coatings W
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INTERNAL SILVER (/036) $ For intern
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Dielectric High-Reflection Coatings
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PERCENT REFLECTANCE 100 80 60 40 20
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MAXBRIte Coatings MAXBRIte (multi
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Laser-Line MAX-R Coatings These mu
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Ultrafast Coating Melles Griot has
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