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.

John Lekner J. Opt. Soc. Am. A 2(2) 186-188 (1985)

References

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Ratio of p-Polarization Absorption to s-Polarization Absorption^{a}

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

A_{p}^{(metal)}/A_{s}^{(metal)} ≃ n_{si}^{2} for bare metals at grazing angles of incidence.

Table 2

Critical Fraction of p-Polarized Light, f_{p}, Needed before the Quarter-Wave Reflector Is Nonoptimal^{a}

Number of Layer Pairs (N)

High-Index Layer Number Being Adjusted (n)

Critical Fraction of p-Polarized Light (f_{p})

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(SiO_{2}/ZrO_{2})^{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°

A_{S}

A_{p}

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(Al_{2}O_{3}/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°

A_{s}

A_{p}

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(Al_{2}O_{3}/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°

A_{s}

A_{p}

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(Al_{2}/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°

A_{s}

A_{p}

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(Al_{2}O_{3}/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(SiO_{2}/ZrO_{2})^{N} at ϕ = 89° and f_{s} = 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[f_{s}/n_{si}^{2} + (1 − f_{s})]n_{sr}α gives 2.65 × 10^{−4}.

Table 8

Comparison of Absorption of MLDC Ag(SiO_{2}/ZrO_{2})^{N} at ϕ = 89° and f_{s} = 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[f_{s}/n_{si}^{2} + (1 − f_{s})]n_{sr}α gives 0.327 × 10^{−3}.

Table 9

Ag(SiO_{2}/ZrO_{2})^{N} Layer Thicknesses As Found Using Eq. (27) for ϕ = 89^{a}

N

Δ_{H}^{(1)}/π/2

1

0.2101

2

0.2594

3

0.3167

4

0.3815

5

0.4517

Quarter-wave thickness of ZrO_{2}, 0.1435 μm; quarter-wave thickness of SiO_{2}, 0.2704 μm; thickness of SiO_{2} layer adjacent to Ag, 0.2224 μm; λ = 1.06 μm; f_{s} = 0.999.

Table 10

Coating Designs for Computer-Generated Optimum Ag(SiO_{2}/ZrO_{2})^{N} Coatings for λ = 1.06 and ϕ = 89°^{a}

Layer Number (N)

f_{s} = 0.994

f_{s} = 0.999

1

2

3

1

2

3

t_{1}

0.9498

0.9498

0.9498

0.9554

0.9521

0.9521

t_{2}

1.2139

1.2139

1.2139

1.1240

1.1042

1.1042

t_{3}

–

0.9535

0.9535

–

1.0466

1.0466

t_{4}

–

1.2098

1.2098

–

1.1111

1.1111

t_{5}

–

–

0.9535

–

–

1.0484

t_{6}

–

–

1.3024

–

–

1.1171

Thicknesses are given in units of quarter-wave thicknesses. ZrO_{2} (quarter-wave) = 0.1435 μm, SiO_{2} (quarter-wave) = 0.2704 μm.

Table 11

Comparison of Computer-Generated and Analytic Values for the Nominal and Average Absorptions^{a}

N

Computer Experiment

Analysis

A_{c}

$$\frac{{\u1fb9}_{c}-{A}_{c}}{{A}_{c}}$$

A_{a}

$$\frac{{\u1fb9}_{a}-{A}_{a}}{{A}_{a}}$$

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. (n_{0} = 3.6, n_{S} = 3.4, n_{L} = 3.4, n_{H} = 3.6, low-index layer adjacent to superstrate.)

Tables (11)

Table 1

Ratio of p-Polarization Absorption to s-Polarization Absorption^{a}

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

A_{p}^{(metal)}/A_{s}^{(metal)} ≃ n_{si}^{2} for bare metals at grazing angles of incidence.

Table 2

Critical Fraction of p-Polarized Light, f_{p}, Needed before the Quarter-Wave Reflector Is Nonoptimal^{a}

Number of Layer Pairs (N)

High-Index Layer Number Being Adjusted (n)

Critical Fraction of p-Polarized Light (f_{p})

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(SiO_{2}/ZrO_{2})^{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°

A_{S}

A_{p}

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(Al_{2}O_{3}/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°

A_{s}

A_{p}

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(Al_{2}O_{3}/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°

A_{s}

A_{p}

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(Al_{2}/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°

A_{s}

A_{p}

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(Al_{2}O_{3}/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(SiO_{2}/ZrO_{2})^{N} at ϕ = 89° and f_{s} = 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[f_{s}/n_{si}^{2} + (1 − f_{s})]n_{sr}α gives 2.65 × 10^{−4}.

Table 8

Comparison of Absorption of MLDC Ag(SiO_{2}/ZrO_{2})^{N} at ϕ = 89° and f_{s} = 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[f_{s}/n_{si}^{2} + (1 − f_{s})]n_{sr}α gives 0.327 × 10^{−3}.

Table 9

Ag(SiO_{2}/ZrO_{2})^{N} Layer Thicknesses As Found Using Eq. (27) for ϕ = 89^{a}

N

Δ_{H}^{(1)}/π/2

1

0.2101

2

0.2594

3

0.3167

4

0.3815

5

0.4517

Quarter-wave thickness of ZrO_{2}, 0.1435 μm; quarter-wave thickness of SiO_{2}, 0.2704 μm; thickness of SiO_{2} layer adjacent to Ag, 0.2224 μm; λ = 1.06 μm; f_{s} = 0.999.

Table 10

Coating Designs for Computer-Generated Optimum Ag(SiO_{2}/ZrO_{2})^{N} Coatings for λ = 1.06 and ϕ = 89°^{a}

Layer Number (N)

f_{s} = 0.994

f_{s} = 0.999

1

2

3

1

2

3

t_{1}

0.9498

0.9498

0.9498

0.9554

0.9521

0.9521

t_{2}

1.2139

1.2139

1.2139

1.1240

1.1042

1.1042

t_{3}

–

0.9535

0.9535

–

1.0466

1.0466

t_{4}

–

1.2098

1.2098

–

1.1111

1.1111

t_{5}

–

–

0.9535

–

–

1.0484

t_{6}

–

–

1.3024

–

–

1.1171

Thicknesses are given in units of quarter-wave thicknesses. ZrO_{2} (quarter-wave) = 0.1435 μm, SiO_{2} (quarter-wave) = 0.2704 μm.

Table 11

Comparison of Computer-Generated and Analytic Values for the Nominal and Average Absorptions^{a}

N

Computer Experiment

Analysis

A_{c}

$$\frac{{\u1fb9}_{c}-{A}_{c}}{{A}_{c}}$$

A_{a}

$$\frac{{\u1fb9}_{a}-{A}_{a}}{{A}_{a}}$$

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. (n_{0} = 3.6, n_{S} = 3.4, n_{L} = 3.4, n_{H} = 3.6, low-index layer adjacent to superstrate.)