Simple expressions for predicting substrate, volume, and interface absorption and reflective phase shifts in high-reflectance quarter-wave stacks at oblique angles of incidence
Jeffrey B. Shellan, "Simple expressions for predicting substrate, volume, and interface absorption and reflective phase shifts in high-reflectance quarter-wave stacks at oblique angles of incidence," J. Opt. Soc. Am. 73, 1272-1281 (1983)
Approximate closed-form expressions are developed for predicting the substrate absorption, volume absorption, and interface absorption at oblique angles of incidence for quarter-wave stack reflectors. At normal incidence the results for S- and P-polarized radiation reduce to those obtained by Sparks [
M. Sparks,
J. Opt. Soc. Am. 67,
1590 (
1977)] and Bennett and Burge [
H. E. Bennett and D. K. Burge,
J. Opt. Soc. Am. 70,
268 (
1980)]. The significant difference in the interface absorption for S- and P-polarized light at large angles of incidence is of interest in damage threshold theory, especially damage due to surface defects or contamination. Simple expressions are also derived for the S- and P-polarization phase shift caused by layer-thickness errors or operation at non-quarter-wave wavelengths. A difference in the S and P phase shifts is of importance in annular resonator design and in assessing the uncorrectable beam degradation due to uneven mirror heating. Exact computer results based on the matrix methods are compared with the predictions of the simple expressions, and agreement to well within 1% is found in most cases of interest.
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Complex Index of Refraction at 2.8 μm of Al2O3, ZnS, Si, and Ag Used in the Analysis [β = (4π/λ)k]
Material
Real Index (n)
Imaginary Index (k)
Al2O3
1.56
0.0004
ZnS
2.25
0.0001
Si
3.415
0.0007
Ag
1.23
20.1
Table 2
Comparison of Substrate Absorption, A(S), for S- and P-Polarized Light for the Quarter-Wave Stack Design Si(Al2O3/ZnS)N and Light Incident at 30°
ϕ = 30°
Number of Layers (2N)
AS(S)
AP(S)
Approx.
Exact
Approx.
Exact
2
0.4653
0.3734
0.6814
0.4975
6
0.09580
0.09136
0.1767
0.1621
10
0.01972
0.01953
0.04584
0.04481
14
4.061 × 10−3
4.052 × 10−3
0.01189
0.01182
18
8.361 × 10−4
8.36 × 10−4
0.003083
0.003079
Table 3
Volume Absorption for S- and P-Polarized Light Incident at 30° as a Function of the Number of Layers in a Si(Al2O3/ZnS)N Quarter-Wave Stack
ϕ = 30°
Number of Layers (2N)
AS(V) (×10−3)
AP(V) (×10−3)
Approx.
Exact
Approx.
Exact
2
0.5933
0.313
0.7881
0.323
6
0.9848
0.863
1.394
1.106
10
1.065
1.037
1.551
1.459
14
1.082
1.075
1.592
1.566
18
1.085
1.084
1.602
1.594
Table 4
Substrate Absorption, A(S), for S- and P-Polarized Light Incident at ϕ = 60° as a Function of the Number of Layers in a Si(Al2O3/ZnS)N Quarter-Wave Stack
ϕ = 60°
Number of Layers (2N)
AS(S)
AP(S)
Approx.
Exact
Approx.
Exact
2
0.2363
0.2107
1.341
0.7522
6
0.03601
0.03538
0.4701
0.3763
10
0.005488
0.005475
0.1647
0.1519
14
8.363 × 10−4
8.37 × 10−4
0.05773
0.05606
18
1.274 × 10−4
1.27 × 10−4
0.02023
0.02001
Table 5
Volume Absorption, A(V), for S- and P-Polarized Light Incidence at ϕ = 60° as a Function of the Number of Layers in a Si(Al2O3/ZnS)N Quarter-Wave Stack
ϕ = 60°
Number of Layers (2N)
AS(V) (×10−3)
AP(V) (×10−3)
Approx.
Exact
Approx.
Exact
2
0.4292
0.294
1.739
0.357
6
0.6622
0.626
3.377
1.934
10
0.6977
0.692
3.952
3.242
14
0.7031
0.701
4.153
3.867
18
0.7039
0.703
4.224
4.114
Table 6
Comparison of the Substrate Absorption, A(S), for S- and P-Polarized Light for the Quarter-Wave Stack Ag(Al2O3/ZnS)N and Light Incident at 30° (from the Normal)
ϕ = 30°
Number of Layers (2N)
AS(S) (×10−3)
AP(S) (×10−3)
Approx.
Exact
Approx.
Exact
2
4.737
4.726
7.201
7.069
4
2.150
2.147
3.667
3.607
6
0.9754
0.975
0.868
1.838
8
0.4426
0.442
0.9511
0.937
10
0.2008
0.201
0.4844
0.477
Table 7
Comparison of Volume Absorption, A(V), for S- and P-Polarized Light for the Quarter-Wave Stack Ag(Al2O3/ZnS)N and Light Incident at 30°
ϕ = 30°
Number of Layers (2N)
AS(V) (×10−3)
AP(V) (×10−3)
Approx.
Exact
Approx.
Exact
2
0.5933
0.590
0.7881
0.775
4
0.8626
0.860
1.189
1.180
6
0.9848
0.983
1.394
1.389
8
1.040
1.040
1.498
1.494
10
1.065
1.064
1.551
1.549
Table 8
Substrate Absorption, A(S), for S- and P-Polarized Light as Calculated Exactly and in the Standing-Wave Approximation for Light Incident at ϕ = 60° [Ag(Al2O3/ZnS)N Quarter-Wave Design]
ϕ = 60°
Number of Layers (2N)
AS(S) (×10−3)
AP(S) (×10−3)
Approx.
Exact
Approx.
Exact
2
2.352
2.349
14.28
14.18
4
0.9181
0.918
8.455
8.416
6
0.3584
0.358
5.005
4.99
8
0.1399
0.140
2.963
2.957
10
0.05461
0.055
1.754
1.751
Table 9
Comparison of Volume Absorption, A(V), for S- and P-Polarized Light for the Quarter-Wave Stack Ag(Al2O3/ZnS)N and Light Incident at ϕ = 60°
ϕ = 60°
Number of Layers (2N)
AS(V) (×10−3)
AP(V) (×10−3)
Approx.
Exact
Approx.
Exact
2
0.4292
0.4280
1.739
1.660
4
0.5968
0.5960
2.768
2.710
6
0.6622
0.6620
3.378
3.336
8
0.6877
0.6870
3.738
3.709
10
0.6977
0.6970
3.952
3.930
Table 10
Interface Absorption as a Function of Incident Angle ϕ for S and P Polarization for an Ag(Al2O3/ZnS)4 Quarter-Wave Stack (N = 4); αHL(S) = 2.250 × 10−3
ϕ
AS(I) (2N, ¼)(×10−3)
AP(I) (2N, ¼)(×10−3)
Approx.
Exact
Approx.
Exact
0°
3.241
3.234
3.241
3.234
30°
2.839
2.834
3.709
3.700
45°
2.342
2.340
4.542
4.527
60°
1.672
1.670
6.512
6.471
Table 11
Interface Absorption as a Function of Incident Angle ϕ for S and P Polarization for an Ag(Al2O3/ZnS)4 Quarter-Wave Stack (N = 4)a
ϕ
AS(I) (2N, ¼)
AP(I) (2N, ¼)
Approx.
Exact
Approx.
Exact
0°
0
−1. × 10−6
0
1. × 10−6
30°
0
1. × 10−6
9.10 × 10−5
8.9 × 10−5
45°
0
0
2.23 × 10−4
2.19 × 10−4
60°
0
−1. × 10−6
4.73 × 10−4
4.64 × 10−4
Only surface absorption is present; αOH(P) = 7.880 × 10−5.
Table 12
Difference in S- and P-Polarization Phase Shift, δS–P, as Calculated Exactly, as Found in the High-Reflectivity Limit Approximation and as Found in the High-Reflectivity Limit with Corrections Made for the Bottom-Layer Thicknessa
Difference in S- and P-Polarization Phase Shift, δS–P
Ag(Al2O3/ZnS)2
Ag(Al2O3/ZnS)4
f (λ = fλ0)
Approx.
Revised Approx.
Exact
Approx.
Revised Approx.
Exact
0.90
16.496°
16.850°
17.915°
23.234°
23.327°
29.518°
0.92
12.910°
13.264°
13.717°
18.183°
18.276°
20.864°
0.94
9.476°
9.830°
9.963°
13.347°
13.441°
14.365°
0.96
6.186°
6.540°
6.536°
8.713°
8.806°
9.036°
0.98
3.030°
3.384°
3.349°
4.267°
4.361°
4.375°
0.99
1.500°
1.854°
1.824°
2.112°
2.206°
2.199°
1.00
0°
0.354°
0.337°
0°
0.0937°
0.088°
1.01
−1.470°
−1.116°
−1.120°
−2.070°
−1.976°
−1.980°
1.02
−2.911°
−2.557°
−2.551°
−4.100°
−4.006°
−4.024°
1.04
−5.710°
−5.356°
−5.355°
−8.042°
−7.949°
−8.117°
1.06
−8.404°
−8.050°
−8.108°
−11.836°
−11.742°
−12.334°
1.08
−10.997°
−10.643°
−10.839°
−15.489°
−15.395°
−16.829°
1.10
−13.497°
−13.143°
−13.573°
−19.010°
−18.916°
−21.793°
The wavelength is given by λ = fλ0, where λ0 is the central wavelength for the four- and eight-layer quarter-wave stack designed for 45° angle of incidence (λ0 = 2.8 μm).
Tables (12)
Table 1
Complex Index of Refraction at 2.8 μm of Al2O3, ZnS, Si, and Ag Used in the Analysis [β = (4π/λ)k]
Material
Real Index (n)
Imaginary Index (k)
Al2O3
1.56
0.0004
ZnS
2.25
0.0001
Si
3.415
0.0007
Ag
1.23
20.1
Table 2
Comparison of Substrate Absorption, A(S), for S- and P-Polarized Light for the Quarter-Wave Stack Design Si(Al2O3/ZnS)N and Light Incident at 30°
ϕ = 30°
Number of Layers (2N)
AS(S)
AP(S)
Approx.
Exact
Approx.
Exact
2
0.4653
0.3734
0.6814
0.4975
6
0.09580
0.09136
0.1767
0.1621
10
0.01972
0.01953
0.04584
0.04481
14
4.061 × 10−3
4.052 × 10−3
0.01189
0.01182
18
8.361 × 10−4
8.36 × 10−4
0.003083
0.003079
Table 3
Volume Absorption for S- and P-Polarized Light Incident at 30° as a Function of the Number of Layers in a Si(Al2O3/ZnS)N Quarter-Wave Stack
ϕ = 30°
Number of Layers (2N)
AS(V) (×10−3)
AP(V) (×10−3)
Approx.
Exact
Approx.
Exact
2
0.5933
0.313
0.7881
0.323
6
0.9848
0.863
1.394
1.106
10
1.065
1.037
1.551
1.459
14
1.082
1.075
1.592
1.566
18
1.085
1.084
1.602
1.594
Table 4
Substrate Absorption, A(S), for S- and P-Polarized Light Incident at ϕ = 60° as a Function of the Number of Layers in a Si(Al2O3/ZnS)N Quarter-Wave Stack
ϕ = 60°
Number of Layers (2N)
AS(S)
AP(S)
Approx.
Exact
Approx.
Exact
2
0.2363
0.2107
1.341
0.7522
6
0.03601
0.03538
0.4701
0.3763
10
0.005488
0.005475
0.1647
0.1519
14
8.363 × 10−4
8.37 × 10−4
0.05773
0.05606
18
1.274 × 10−4
1.27 × 10−4
0.02023
0.02001
Table 5
Volume Absorption, A(V), for S- and P-Polarized Light Incidence at ϕ = 60° as a Function of the Number of Layers in a Si(Al2O3/ZnS)N Quarter-Wave Stack
ϕ = 60°
Number of Layers (2N)
AS(V) (×10−3)
AP(V) (×10−3)
Approx.
Exact
Approx.
Exact
2
0.4292
0.294
1.739
0.357
6
0.6622
0.626
3.377
1.934
10
0.6977
0.692
3.952
3.242
14
0.7031
0.701
4.153
3.867
18
0.7039
0.703
4.224
4.114
Table 6
Comparison of the Substrate Absorption, A(S), for S- and P-Polarized Light for the Quarter-Wave Stack Ag(Al2O3/ZnS)N and Light Incident at 30° (from the Normal)
ϕ = 30°
Number of Layers (2N)
AS(S) (×10−3)
AP(S) (×10−3)
Approx.
Exact
Approx.
Exact
2
4.737
4.726
7.201
7.069
4
2.150
2.147
3.667
3.607
6
0.9754
0.975
0.868
1.838
8
0.4426
0.442
0.9511
0.937
10
0.2008
0.201
0.4844
0.477
Table 7
Comparison of Volume Absorption, A(V), for S- and P-Polarized Light for the Quarter-Wave Stack Ag(Al2O3/ZnS)N and Light Incident at 30°
ϕ = 30°
Number of Layers (2N)
AS(V) (×10−3)
AP(V) (×10−3)
Approx.
Exact
Approx.
Exact
2
0.5933
0.590
0.7881
0.775
4
0.8626
0.860
1.189
1.180
6
0.9848
0.983
1.394
1.389
8
1.040
1.040
1.498
1.494
10
1.065
1.064
1.551
1.549
Table 8
Substrate Absorption, A(S), for S- and P-Polarized Light as Calculated Exactly and in the Standing-Wave Approximation for Light Incident at ϕ = 60° [Ag(Al2O3/ZnS)N Quarter-Wave Design]
ϕ = 60°
Number of Layers (2N)
AS(S) (×10−3)
AP(S) (×10−3)
Approx.
Exact
Approx.
Exact
2
2.352
2.349
14.28
14.18
4
0.9181
0.918
8.455
8.416
6
0.3584
0.358
5.005
4.99
8
0.1399
0.140
2.963
2.957
10
0.05461
0.055
1.754
1.751
Table 9
Comparison of Volume Absorption, A(V), for S- and P-Polarized Light for the Quarter-Wave Stack Ag(Al2O3/ZnS)N and Light Incident at ϕ = 60°
ϕ = 60°
Number of Layers (2N)
AS(V) (×10−3)
AP(V) (×10−3)
Approx.
Exact
Approx.
Exact
2
0.4292
0.4280
1.739
1.660
4
0.5968
0.5960
2.768
2.710
6
0.6622
0.6620
3.378
3.336
8
0.6877
0.6870
3.738
3.709
10
0.6977
0.6970
3.952
3.930
Table 10
Interface Absorption as a Function of Incident Angle ϕ for S and P Polarization for an Ag(Al2O3/ZnS)4 Quarter-Wave Stack (N = 4); αHL(S) = 2.250 × 10−3
ϕ
AS(I) (2N, ¼)(×10−3)
AP(I) (2N, ¼)(×10−3)
Approx.
Exact
Approx.
Exact
0°
3.241
3.234
3.241
3.234
30°
2.839
2.834
3.709
3.700
45°
2.342
2.340
4.542
4.527
60°
1.672
1.670
6.512
6.471
Table 11
Interface Absorption as a Function of Incident Angle ϕ for S and P Polarization for an Ag(Al2O3/ZnS)4 Quarter-Wave Stack (N = 4)a
ϕ
AS(I) (2N, ¼)
AP(I) (2N, ¼)
Approx.
Exact
Approx.
Exact
0°
0
−1. × 10−6
0
1. × 10−6
30°
0
1. × 10−6
9.10 × 10−5
8.9 × 10−5
45°
0
0
2.23 × 10−4
2.19 × 10−4
60°
0
−1. × 10−6
4.73 × 10−4
4.64 × 10−4
Only surface absorption is present; αOH(P) = 7.880 × 10−5.
Table 12
Difference in S- and P-Polarization Phase Shift, δS–P, as Calculated Exactly, as Found in the High-Reflectivity Limit Approximation and as Found in the High-Reflectivity Limit with Corrections Made for the Bottom-Layer Thicknessa
Difference in S- and P-Polarization Phase Shift, δS–P
Ag(Al2O3/ZnS)2
Ag(Al2O3/ZnS)4
f (λ = fλ0)
Approx.
Revised Approx.
Exact
Approx.
Revised Approx.
Exact
0.90
16.496°
16.850°
17.915°
23.234°
23.327°
29.518°
0.92
12.910°
13.264°
13.717°
18.183°
18.276°
20.864°
0.94
9.476°
9.830°
9.963°
13.347°
13.441°
14.365°
0.96
6.186°
6.540°
6.536°
8.713°
8.806°
9.036°
0.98
3.030°
3.384°
3.349°
4.267°
4.361°
4.375°
0.99
1.500°
1.854°
1.824°
2.112°
2.206°
2.199°
1.00
0°
0.354°
0.337°
0°
0.0937°
0.088°
1.01
−1.470°
−1.116°
−1.120°
−2.070°
−1.976°
−1.980°
1.02
−2.911°
−2.557°
−2.551°
−4.100°
−4.006°
−4.024°
1.04
−5.710°
−5.356°
−5.355°
−8.042°
−7.949°
−8.117°
1.06
−8.404°
−8.050°
−8.108°
−11.836°
−11.742°
−12.334°
1.08
−10.997°
−10.643°
−10.839°
−15.489°
−15.395°
−16.829°
1.10
−13.497°
−13.143°
−13.573°
−19.010°
−18.916°
−21.793°
The wavelength is given by λ = fλ0, where λ0 is the central wavelength for the four- and eight-layer quarter-wave stack designed for 45° angle of incidence (λ0 = 2.8 μm).
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