Analytical solutions for zero-phase-shift transparent coatings on metallic reflectors at 10.6 μm

E. Cojocaru

doi:10.1364/AO.32.004843

Applied Optics
Vol. 32,
Issue 25,
pp. 4843-4845
(1993)
•https://doi.org/10.1364/AO.32.004843

Analytical solutions for zero-phase-shift transparent coatings on metallic reflectors at 10.6 μm

E. Cojocaru

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E. Cojocaru, "Analytical solutions for zero-phase-shift transparent coatings on metallic reflectors at 10.6 μm," Appl. Opt. 32, 4843-4845 (1993)

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Abstract

The relationships for refractive indices for zero-phase-shift quarter-wave coatings on metallic reflectors are given. Small variations in λ/4-layer phase thicknesses are also considered.

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Relations for Refractive Indices Deduced from the Zero-Phase-Shift Condition	Layer Phase Thicknesses (β)
Single-Layer Coating
(1.1) n₀p₁² = n₁²p₀	β₁ = π/2
Double-Layer Coating
(2.1) n₀n₁²p₂² = n₂²p₀p₁²	β₁ = β₂ = π/2
(2.2) n₀p_i² = n_i²p₀	β_i = π/2, β_j = π, i, j = 1, 2
Triple-Layer Coating
(3.1) n₀n₂²p₁²p₃² = n₁²n₃²p₀p₂²	β₁ = β₂ = β₃ = π/2
(3.2) n₀p_k² = n_k²p₀	β_i = β_j = π, β_k = π/2
	i, j, k = l, 2, 3
(3.3) n₀n_i²p_j² = n_j²p₀p_i²	β_i = β_j = π/2, β_k = π
	i, j, k = l, 2, 3 j > i

Layer Phase Thicknesses	Approximate Relations for the Small Deviations ∊ = (∊ = a tan t)
Single-Layer Coating
(1.1) β₁ = (π/2) − ∊₁	t₁ = a₁/(p₀q₁ − q₀p₁)
Double-Layer Coating
(2.1) β₁ = (π/2) − ∊₁,	t₁ = a₂/(p₀p₁q₂² − p₂²q₀q₁)
β₂ = π/2
(2.2) β₁ = π/2,	t₂ = a₂/(p₀p²q₂² − p₂²q₀q₂)
β₂ = (π/2) − ∊₂
Triple-Layer Coating
(3.1) β₁ = (π/2) − ∊₁,	t₁ = a₃/(p₀p₂²q₁q₃² − p₁p₃²q₀q₂²)
β₂ = β₃ = π/2
(3.2) β₁ = β₃ = π/2,	t₂ = a₃/(p₀p₂²q₂q₃² − p₂p₃²q₀q₂²)
β₂ = (π/2) − ∊₂
(3.3) β₁ = β₂ = π/2,	t₃ = a₃/(p₀p₂²q₂²q₃ − p₂²p₃q₀q₂²)
β₃ = (π/2) − ∊₃

Relations for Refractive Indices Deduced from the Zero-Phase-Shift Condition	Layer Phase Thicknesses (β)
Single-Layer Coating
(1.1) n₀p₁² = n₁²p₀	β₁ = π/2
Double-Layer Coating
(2.1) n₀n₁²p₂² = n₂²p₀p₁²	β₁ = β₂ = π/2
(2.2) n₀p_i² = n_i²p₀	β_i = π/2, β_j = π, i, j = 1, 2
Triple-Layer Coating
(3.1) n₀n₂²p₁²p₃² = n₁²n₃²p₀p₂²	β₁ = β₂ = β₃ = π/2
(3.2) n₀p_k² = n_k²p₀	β_i = β_j = π, β_k = π/2
	i, j, k = l, 2, 3
(3.3) n₀n_i²p_j² = n_j²p₀p_i²	β_i = β_j = π/2, β_k = π
	i, j, k = l, 2, 3 j > i

Layer Phase Thicknesses	Approximate Relations for the Small Deviations ∊ = (∊ = a tan t)
Single-Layer Coating
(1.1) β₁ = (π/2) − ∊₁	t₁ = a₁/(p₀q₁ − q₀p₁)
Double-Layer Coating
(2.1) β₁ = (π/2) − ∊₁,	t₁ = a₂/(p₀p₁q₂² − p₂²q₀q₁)
β₂ = π/2
(2.2) β₁ = π/2,	t₂ = a₂/(p₀p²q₂² − p₂²q₀q₂)
β₂ = (π/2) − ∊₂
Triple-Layer Coating
(3.1) β₁ = (π/2) − ∊₁,	t₁ = a₃/(p₀p₂²q₁q₃² − p₁p₃²q₀q₂²)
β₂ = β₃ = π/2
(3.2) β₁ = β₃ = π/2,	t₂ = a₃/(p₀p₂²q₂q₃² − p₂p₃²q₀q₂²)
β₂ = (π/2) − ∊₂
(3.3) β₁ = β₂ = π/2,	t₃ = a₃/(p₀p₂²q₂²q₃ − p₂²p₃q₀q₂²)
β₃ = (π/2) − ∊₃

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Applied Optics