Optical Coatings

Fundamental Optics
Material Properties Optical Specifications Gaussian Beam Optics
Optical Coatings
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.
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