Although quarter-wave multilayer dielectric coatings (QW MLDC’s) usually offer the highest reflectivity possible at a single wavelength, this may not be the case at grazing angles of incidence. For incident angles greater than the Brewster angle, the p-polarization absorption can easily be several orders of magnitude higher than the s-polarization absorption because of the destructive interference between the reflection from the superstrate–top-layer interface and the rest of the coating. An analytic approach is developed for designing optimum reflectivity coatings at grazing angles of incidence once the fraction of s- and p-polarized light is given. These coatings can give at least an order-of-magnitude reduction over QW MLDC’s in total coating absorption, even for the case when less than 0 1% of the incident radiation is p polarized. The absorption of the designs found from the analytic results compares favorably with designs generated using numerical methods of nonlinear optimization Expressions are also developed for calculating the sensitivity of high-reflectivity quarter-wave stacks to coating thickness error.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
Ratio of p-Polarization Absorption to s-Polarization Absorptiona
Wavelength (μm)
Material
Ag
Al
Au
Cu
0.5
8.3
23
5.0
5.6
1.0
45
77
45
40
2.0
150
270
190
130
Ap(metal)/As(metal) ≃ nsi2 for bare metals at grazing angles of incidence.
Table 2
Critical Fraction of p-Polarized Light, fp, Needed before the Quarter-Wave Reflector Is Nonoptimala
Number of Layer Pairs (N)
High-Index Layer Number Being Adjusted (n)
Critical Fraction of p-Polarized Light (fp)
1
1
2.13 × 10−7
2
1
8.54 × 10−8
2
1.74 × 10−7
3
1
3.43 × 10−8
2
6.98 × 10−8
3
1.42 × 10−7
4
1
1.38 × 10−8
2
2.80 × 10−8
3
5.72 × 10−8
4
1.17 × 10−7
5
1
5.53 × 10−9
2
1.13 × 10−8
3
2.30 × 10−8
4
4.70 × 10−8
5
9.70 × 10−8
Ag(SiO2/ZrO2)N, λ = 1.06 μm, ϕ = 89°.
Table 3
Substrate Absorption for s- and p-Polarized Light As Calculated Exactly and in the Standing-Wave Approximation for Light Incident at ϕ = 80°a
Number of Layers (2N)
ϕ = 80°
AS
Ap
Exact
Approx.
Exact
Approx.
2
0.07317
0.07597
0.9995
4.173
6
9.671 × 10−3
9.718 × 10–3
0.8454
1.742
10
1.242 × 10−3
1.243 × 10–3
0.5207
0.7272
14
1.594 × 10−4
1.590 × 10–4
0.2623
0.3036
18
–
–
0.1191
0.1267
Si(Al2O3/ZnS)N quarter-wave design.
Table 4
Substrate Absorption for s- and p-Polarized Light As Calculated Exactly and in the Standing-Wave Approximation for Light Incident at ϕ = 80° a
Number of Layers (2N)
ϕ = 80°
As
Ap
Exact
Approx.
Exact
Approx.
2
0.8055
10.31
0.5499
0.7880
6
0.7459
1.319
0.2809
0.3289
10
0.1553
0.1687
0.1284
0.1373
14
0.02135
0.02158
0.05572
0.05733
18
2.755 × 10−3
2.760 × 10−3
0.02365
0.02393
Si(Al2O3/ZnS)N quarter-wave design, except top layer is one-half-wave thick.
Table 5
Substrate Absorption for s- and p-Polarized Light As Calculated Exactly and in the Standing-Wave Approximation for Light Incident at ϕ = 88° a
Number of Layers (2N)
ϕ = 88°
As
Ap
Exact
Approx.
Exact
Approx.
2
0.2727
50.36
0.1491
0.1613
6
0.9509
6.278
0.06678
0.06911
10
0.5475
0.7827
0.02917
0.02960
14
0.09299
0.09758
0.01260
0.01268
18
0.01209
0.01216
5.417 × 10−3
5.432 × 10−3
Si(Al2/ZnS)N quarter-wave design, except top layer is one-half-wave thick.
Table 6
Substrate Absorption for s- and p-Polarized Light As Calculated Exactly and in the Standing-Wave Approximation for Light Incident at ϕ = 88° a
Number of Layers (2N)
ϕ = 88°
As
Ap
Exact
Approx. 1
Approx. 2
Exact
Approx. 1
Approx. 2
2
1.488 × 10−4
1.490 × 10−4
1.484 × 10−4
0.1151
0.1224
0.2260
4
–
–
–
0.1033
0.1090
0.1479
6
–
–
–
0.08094
0.08440
0.09683
The Ag(Al2O3/ZnS)N quarter-wave stack is the simpler design. Approx. 1 takes into account error in bottom layer, whereas Approx. 2 does not.
Table 7
Comparison of Absorption of MLDC Ag(SiO2/ZrO2)N at ϕ = 89° and fs = 0.999 for Quarter Wave, Quarter Wave with Optimum Top-Layer Thickness, Numerically Generated Optimum, and Bare Silver
Substrate-compensated design for p-polarized light (exact results). Equation (35).
Exact results; the equation A = 4[fs/nsi2 + (1 − fs)]nsrα gives 2.65 × 10−4.
Table 8
Comparison of Absorption of MLDC Ag(SiO2/ZrO2)N at ϕ = 89° and fs = 0.994 for Quarter Wave, Quarter Wave with Optimum Top-Layer Thickness, Numerically Generated Optimum, and Bare Silver
Substrate-compensated design for p-polarized light (exact results). Equation (35).
Exact results; the equation A = 4[fs/nsi2 + (1 − fs)]nsrα gives 0.327 × 10−3.
Table 9
Ag(SiO2/ZrO2)N Layer Thicknesses As Found Using Eq. (27) for ϕ = 89a
N
ΔH(1)/π/2
1
0.2101
2
0.2594
3
0.3167
4
0.3815
5
0.4517
Quarter-wave thickness of ZrO2, 0.1435 μm; quarter-wave thickness of SiO2, 0.2704 μm; thickness of SiO2 layer adjacent to Ag, 0.2224 μm; λ = 1.06 μm; fs = 0.999.
Table 10
Coating Designs for Computer-Generated Optimum Ag(SiO2/ZrO2)N Coatings for λ = 1.06 and ϕ = 89°a
Layer Number (N)
fs = 0.994
fs = 0.999
1
2
3
1
2
3
t1
0.9498
0.9498
0.9498
0.9554
0.9521
0.9521
t2
1.2139
1.2139
1.2139
1.1240
1.1042
1.1042
t3
–
0.9535
0.9535
–
1.0466
1.0466
t4
–
1.2098
1.2098
–
1.1111
1.1111
t5
–
–
0.9535
–
–
1.0484
t6
–
–
1.3024
–
–
1.1171
Thicknesses are given in units of quarter-wave thicknesses. ZrO2 (quarter-wave) = 0.1435 μm, SiO2 (quarter-wave) = 0.2704 μm.
Table 11
Comparison of Computer-Generated and Analytic Values for the Nominal and Average Absorptionsa
N
Computer Experiment
Analysis
Ac
Aa
30
0.12160
0.041
0.12241
0.051
35
0.07058
0.052
0.06912
0.060
40
0.04048
0.064
0.03903
0.070
45
0.02306
0.074
0.02204
0.080
50
0.01309
0.085
0.01244
0.090
The subscripts c and a designate computer and analytic results, respectively. (n0 = 3.6, nS = 3.4, nL = 3.4, nH = 3.6, low-index layer adjacent to superstrate.)
Tables (11)
Table 1
Ratio of p-Polarization Absorption to s-Polarization Absorptiona
Wavelength (μm)
Material
Ag
Al
Au
Cu
0.5
8.3
23
5.0
5.6
1.0
45
77
45
40
2.0
150
270
190
130
Ap(metal)/As(metal) ≃ nsi2 for bare metals at grazing angles of incidence.
Table 2
Critical Fraction of p-Polarized Light, fp, Needed before the Quarter-Wave Reflector Is Nonoptimala
Number of Layer Pairs (N)
High-Index Layer Number Being Adjusted (n)
Critical Fraction of p-Polarized Light (fp)
1
1
2.13 × 10−7
2
1
8.54 × 10−8
2
1.74 × 10−7
3
1
3.43 × 10−8
2
6.98 × 10−8
3
1.42 × 10−7
4
1
1.38 × 10−8
2
2.80 × 10−8
3
5.72 × 10−8
4
1.17 × 10−7
5
1
5.53 × 10−9
2
1.13 × 10−8
3
2.30 × 10−8
4
4.70 × 10−8
5
9.70 × 10−8
Ag(SiO2/ZrO2)N, λ = 1.06 μm, ϕ = 89°.
Table 3
Substrate Absorption for s- and p-Polarized Light As Calculated Exactly and in the Standing-Wave Approximation for Light Incident at ϕ = 80°a
Number of Layers (2N)
ϕ = 80°
AS
Ap
Exact
Approx.
Exact
Approx.
2
0.07317
0.07597
0.9995
4.173
6
9.671 × 10−3
9.718 × 10–3
0.8454
1.742
10
1.242 × 10−3
1.243 × 10–3
0.5207
0.7272
14
1.594 × 10−4
1.590 × 10–4
0.2623
0.3036
18
–
–
0.1191
0.1267
Si(Al2O3/ZnS)N quarter-wave design.
Table 4
Substrate Absorption for s- and p-Polarized Light As Calculated Exactly and in the Standing-Wave Approximation for Light Incident at ϕ = 80° a
Number of Layers (2N)
ϕ = 80°
As
Ap
Exact
Approx.
Exact
Approx.
2
0.8055
10.31
0.5499
0.7880
6
0.7459
1.319
0.2809
0.3289
10
0.1553
0.1687
0.1284
0.1373
14
0.02135
0.02158
0.05572
0.05733
18
2.755 × 10−3
2.760 × 10−3
0.02365
0.02393
Si(Al2O3/ZnS)N quarter-wave design, except top layer is one-half-wave thick.
Table 5
Substrate Absorption for s- and p-Polarized Light As Calculated Exactly and in the Standing-Wave Approximation for Light Incident at ϕ = 88° a
Number of Layers (2N)
ϕ = 88°
As
Ap
Exact
Approx.
Exact
Approx.
2
0.2727
50.36
0.1491
0.1613
6
0.9509
6.278
0.06678
0.06911
10
0.5475
0.7827
0.02917
0.02960
14
0.09299
0.09758
0.01260
0.01268
18
0.01209
0.01216
5.417 × 10−3
5.432 × 10−3
Si(Al2/ZnS)N quarter-wave design, except top layer is one-half-wave thick.
Table 6
Substrate Absorption for s- and p-Polarized Light As Calculated Exactly and in the Standing-Wave Approximation for Light Incident at ϕ = 88° a
Number of Layers (2N)
ϕ = 88°
As
Ap
Exact
Approx. 1
Approx. 2
Exact
Approx. 1
Approx. 2
2
1.488 × 10−4
1.490 × 10−4
1.484 × 10−4
0.1151
0.1224
0.2260
4
–
–
–
0.1033
0.1090
0.1479
6
–
–
–
0.08094
0.08440
0.09683
The Ag(Al2O3/ZnS)N quarter-wave stack is the simpler design. Approx. 1 takes into account error in bottom layer, whereas Approx. 2 does not.
Table 7
Comparison of Absorption of MLDC Ag(SiO2/ZrO2)N at ϕ = 89° and fs = 0.999 for Quarter Wave, Quarter Wave with Optimum Top-Layer Thickness, Numerically Generated Optimum, and Bare Silver
Substrate-compensated design for p-polarized light (exact results). Equation (35).
Exact results; the equation A = 4[fs/nsi2 + (1 − fs)]nsrα gives 2.65 × 10−4.
Table 8
Comparison of Absorption of MLDC Ag(SiO2/ZrO2)N at ϕ = 89° and fs = 0.994 for Quarter Wave, Quarter Wave with Optimum Top-Layer Thickness, Numerically Generated Optimum, and Bare Silver
Substrate-compensated design for p-polarized light (exact results). Equation (35).
Exact results; the equation A = 4[fs/nsi2 + (1 − fs)]nsrα gives 0.327 × 10−3.
Table 9
Ag(SiO2/ZrO2)N Layer Thicknesses As Found Using Eq. (27) for ϕ = 89a
N
ΔH(1)/π/2
1
0.2101
2
0.2594
3
0.3167
4
0.3815
5
0.4517
Quarter-wave thickness of ZrO2, 0.1435 μm; quarter-wave thickness of SiO2, 0.2704 μm; thickness of SiO2 layer adjacent to Ag, 0.2224 μm; λ = 1.06 μm; fs = 0.999.
Table 10
Coating Designs for Computer-Generated Optimum Ag(SiO2/ZrO2)N Coatings for λ = 1.06 and ϕ = 89°a
Layer Number (N)
fs = 0.994
fs = 0.999
1
2
3
1
2
3
t1
0.9498
0.9498
0.9498
0.9554
0.9521
0.9521
t2
1.2139
1.2139
1.2139
1.1240
1.1042
1.1042
t3
–
0.9535
0.9535
–
1.0466
1.0466
t4
–
1.2098
1.2098
–
1.1111
1.1111
t5
–
–
0.9535
–
–
1.0484
t6
–
–
1.3024
–
–
1.1171
Thicknesses are given in units of quarter-wave thicknesses. ZrO2 (quarter-wave) = 0.1435 μm, SiO2 (quarter-wave) = 0.2704 μm.
Table 11
Comparison of Computer-Generated and Analytic Values for the Nominal and Average Absorptionsa
N
Computer Experiment
Analysis
Ac
Aa
30
0.12160
0.041
0.12241
0.051
35
0.07058
0.052
0.06912
0.060
40
0.04048
0.064
0.03903
0.070
45
0.02306
0.074
0.02204
0.080
50
0.01309
0.085
0.01244
0.090
The subscripts c and a designate computer and analytic results, respectively. (n0 = 3.6, nS = 3.4, nL = 3.4, nH = 3.6, low-index layer adjacent to superstrate.)