Angular scattering from optical interference coatings: scalar scattering predictions and measurements

James M. Zavislan

doi:10.1364/AO.30.002224

Applied Optics
Vol. 30,
Issue 16,
pp. 2224-2244
(1991)
•https://doi.org/10.1364/AO.30.002224

Angular scattering from optical interference coatings: scalar scattering predictions and measurements

James M. Zavislan

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James M. Zavislan, "Angular scattering from optical interference coatings: scalar scattering predictions and measurements," Appl. Opt. 30, 2224-2244 (1991)

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Abstract

A scalar scattering theory is developed that predicts the angular distribution of light scattered and the total integrated scatter from a randomly rough or inhomogeneous optical interference coating. Three types of random variation are considered: uncorrelated roughness, additive roughness, and uncorrelated index inhomogeneity. The scattering calculations are formulated so that the output of any conventional thin film analysis program along with a coating’s surface or index statistics could be used to calculate the scattering distribution of a coating. The scattering calculations are compared to experimental measurements from a sixteen-layer high reflector coating with small additive roughness σ = 2.4 Å and large correlated roughness σ = 93 Å.

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Reflected Scattering Contribution	Transmitted Scattering Contribution	Description
DFRU_i(θ₁,θ₂,θ₃,λ)	DFTU_i(θ₁,θ₂,θ₃,λ)	Contribution from uncorrelated roughness at ij^th interface
DFRI_i(θ₁,θ₂,θ₃,λ)	DFTI_i(θ₁,θ₂,θ₃,λ)	Contribution from index inhomogeneity in i^th layer
DFRA_i(θ₁,θ₂,θ₃,λ)	DFTA_i(θ₁,θ₂,θ₃,λ)	Contribution from thickness variation in i^th layer
DFRA_S(θ₁,θ₂,θ₃,λ)	DFTA_S(θ₁,θ₂,θ₃,λ)	Contribution from correlated substrate roughness

Coefficient	Formula
	Reflected Scattered Light
ΔR_i	$∣ ρ_{0} (θ_{1}, λ) ∣^{2} - ∣ [ρ_{0} (θ_{1}, λ) + ({{}_{i}a}_{2} + i {{}_{i}b}_{2}) σ_{i}^{2} + 3 ({{}_{i}a}_{4} + i {{}_{i}b}_{4}) σ_{i}^{4}] ∣^{2}$
${{}_{i}A}_{1}^{refl .}$	$\begin{array}{l} [({{}_{i}a}_{1}^{2} + {{}_{i}b}_{1}^{2}) σ_{i}^{2} + 6 ({{}_{i}a}_{1} {{}_{i}a}_{3} + {{}_{i}b}_{1} {{}_{i}b}_{3}) σ_{i}^{4} + 30 ({{}_{i}a}_{1} {{}_{i}a}_{5} + {{}_{i}b}_{1} {{}_{i}b}_{5}) σ_{i}^{6} \\ + 9 ({{}_{i}a}_{3}^{2} + {{}_{i}b}_{3}^{2}) σ_{i}^{6} + 90 ({{}_{i}a}_{3} {{}_{i}a}_{5} + {{}_{i}b}_{3} {{}_{i}b}_{5}) σ_{i}^{8} + 225 ({{}_{i}a}_{5}^{2} + {{}_{i}b}_{5}^{2}) σ_{i}^{10}] \end{array}$
${{}_{i}A}_{2}^{refl .}$	$[2 ({{}_{i}a}_{2}^{2} + {{}_{i}b}_{2}^{2}) σ_{i}^{4} + 24 ({{}_{i}a}_{2} {{}_{i}a}_{4} + {{}_{i}b}_{2} {{}_{i}b}_{4}) σ_{i}^{6} + 72 ({{}_{i}a}_{4}^{2} + {{}_{i}b}_{4}^{2}) σ_{i}^{8}]$
${{}_{i}A}_{3}^{refl .}$	$[6 ({{}_{i}a}_{3}^{2} + {{}_{i}b}_{3}^{2}) σ_{i}^{6} + 120 ({{}_{i}a}_{3} {{}_{i}a}_{5} + {{}_{i}b}_{3} {{}_{i}b}_{5}) σ_{i}^{8} + 600 ({{}_{i}a}_{5}^{2} + {{}_{i}b}_{5}^{2}) σ_{i}^{10}]$
${{}_{i}A}_{4}^{refl .}$	$[24 ({{}_{i}a}_{4}^{2} + {{}_{i}b}_{4}^{2}) σ_{i}^{8}]$	_ia_j and _ib_j are calculated from amplitude reflection data at θ₁ and λ.
${{}_{i}A}_{5}^{refl .}$	$[120 ({{}_{i}a}_{5}^{2} + {{}_{i}b}_{5}^{2}) σ_{i}^{10}]$
k_α	$\frac{2 π n_{0}}{λ} \sqrt{{sin}^{2} θ_{1} + {sin}^{2} θ_{2} - 2sin θ_{1} sin θ_{2} cos θ_{3}}$

Coefficient	Formula
	Transmitted Scattered Light
ΔT_i	$∣ τ_{0} (θ_{1}, λ) ∣^{2} - ∣ [τ_{0} (θ_{1}, λ) + ({{}_{i}a}_{2} + i {{}_{i}b}_{2}) σ_{i}^{2} + 3 ({{}_{i}a}_{4} + i {{}_{i}b}_{4}) σ_{i}^{4}] ∣^{2}$
${{}_{i}A}_{1}^{tran .}$		_ia_j and _ib_j are calculated from amplitude transmission data at θ₁ and λ.
⋮	same as ${{}_{i}A}_{j}^{refl .}$
${{}_{i}A}_{5}^{tran .}$
k_α	$\frac{2 π}{λ} \sqrt{n_{0}^{2} {sin}^{2} θ_{1} + n_{s}^{2} {sin}^{2} θ_{2} {- 2n}_{0} n_{s} sin θ_{1} sin θ_{2} cos θ_{3}}$

	SiO₂		Ta₂O₅
λ (Å)	n	κ	n	κ
3000	1.50	0	2.19	0.0022
3500	1.48	0	2.16	0.0019
4000	1.47	0	2.13	0.0014
5000	1.46	0	2.08	0.0009
6000	1.46	0	2.06	4×10−6
7000	1.45	0	2.03	4×10−6
10000	1.45	0	1.99	6×10−6

Substrate roughness
Sample	Average rms roughness	Autocorrelation function
Y3*	93 ± 3 Å	Gaussian: σ₁² = 5184 Å², L₁ = 0.3 &;mu;m
Y3*	93 ± 3 Å	Gaussian: σ₂² = 4096 Å², L₂ = 1.2 &;mu;m
Z4	3.9 ± 0.1 Å	Exponential: σ₁² = 8.8 Å², L₁ = 0.23 &;mu;m
		Exponential: σ₂² = 7.4 Å², L₂ = 2.3 &;mu;m
		Gaussian: σ₃² = 2.9 Å², L₃ = 41.6 &;mu;m
Top interface roughness
Sample	Average rms roughness	Autocorrelation function

Y3* (correlated)	93 ± 3 Å	Gaussian: σ₁² = 5184 Å², L₁ = 0.3 &;mu;m
Y3* (correlated)	93 ± 3 Å	Gaussian: σ₂² = 4096 Å², L₂ = 1.2 &;mu;m
Z4 (additive)	9.6 ± 0.8 Å	Gaussian: σ₁² = 94.1 Å², L₁ = 0.17 &;mu;m
		Exponential: σ₂² = 8.8 Å², L₂ = 0.23 &;mu;m
		Exponential: σ₃² = 7.4 Å², L₃ = 2.3 &;mu;m
		Gaussian: σ₄² = 2.9 Å², L₄ = 41.6 &;mu;m

p-polarized light, 15° angle of incidence. 64Å substrate roughness
Real coefficient	Imaginary coefficient
_sa₀ = 2.6433 × 10⁻¹	_sb₀ = −9.4824 × 10⁻¹
_sa₁ = −1.8189 × 10⁻³	_sb₁ = −5.0704 × 10⁻⁴
_sa₂ = −4.8628 × 10⁻⁷	_sb₂ = 1.7444 × 10⁻⁶
_sa₃ = 1.1154 × 10⁻⁹	_sb₃ = 3.1092 × 10⁻¹⁰
_sa₄ = 1.4784 × 10⁻¹³	_sb₄ = −5.3040 × 10⁻¹³
_sa₅ = −2.0363 × 10⁻¹⁶	_sb₅ = −5.6772 × 10⁻¹⁷

Layer 2:
p-polarized light, 15° angle of incidence. 2.4 Å additive roughness.
Real coefficient	Imaginary coefficient
₂a₀ = 5.6739 × 10⁻¹	₂b₀ = −8.0394 × 10⁻¹
₂a₁ = 1.5040 × 10⁻³	₂b₁ = 1.0697 × 10⁻³
₂a₂ = −3.6820 × 10⁻⁷	₂b₂ = 1.8684 × 10⁻⁶
₂a₃ = −1.1545 × 10⁻⁹	₂b₃ = 1.0023 × 10⁻⁹
₂a₄ = −1.5020 × 10⁻¹¹	₂b₄ = 1.3129 × 10⁻¹¹
₂a₅ = 3.0066 × 10⁻¹³	₂b₅ = −8.8652 × 10⁻¹⁴

Abstract

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Applied Optics