Optical Coatings

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