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Trapped light scattering within optical coatings: a multilayer roughness-coupling process

Claude Amra, Myriam Zerrad, and Michel Lequime

Open Access Open Access

Abstract

Despite numerous works devoted to light scattering in multilayer optics, trapped scattering has not been considered until now. This consists in a roughness-coupling process at each interface of the multilayer, giving rise to electromagnetic modes traveling within the stack. Such a modal scattering component is today necessary for completing the energy balance within high-precision optics including mirrors for gyro-lasers and detection of gravitational waves, where every ppm (part per million) must be accounted for. We show how to calculate this trapped light and compare its order of magnitude with the free space scattering component emerging outside the multilayer.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Data availability

No data were generated or analyzed in the presented research.

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Figures (21)
Fig. 1.
Fig. 1. Schematic diagram of classical and modal absorption. Illumination is given by the blue arrows, with the classical absorption region in blue. The guided modes are in red, and the transverse dimensions of the modal absorption region are illustrated in red.

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Fig. 2.
Fig. 2. Geometry of the multilayer cavity, with refractive indices ni and thicknesses ei.

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Fig. 3.
Fig. 3. Relationship between spatial pulsation $\vec{\sigma }\; $ and wave direction (θ, φ). The wave vector is denoted $\vec{k}$ .

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Fig. 4.
Fig. 4. Schematic diagram of bandwidths associated with plane, modal and evanescent waves, (case k0 < ks).

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Fig. 5.
Fig. 5. Optical properties (reflection) of the 17 layer multi-dielectric mirror.

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Fig. 6.
Fig. 6. Scattering power density versus normalized frequency for SS and PP polarization. Case of a 17 layer mirror illuminated at normal incidence at the design wavelength λ = λ0 = 600 nm.

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Fig. 7.
Fig. 7. Variations of the scattering power density versus wavelength and spatial frequency. Case of the 17 layer mirror designed for normal incidence at 600 nm.

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Fig. 8.
Fig. 8. TE polarization. Amount $\eta _m^q$ versus wavelength of light trapped by each guided mode of the stack. Normalization is performed with respect to free space scattering in air. Case of a 17 layer mirror with design wavelength 600 nm.

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Fig. 9.
Fig. 9. TM polarization. Amount $\eta _m^q$ versus wavelength of light trapped by each guided mode of the stack. Normalization is performed with respect to free space scattering in air. Case of a 17 layer mirror with design wavelength 600 nm.

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Fig. 10.
Fig. 10. Total amount versus wavelength of light trapped in the form of guided modes in the stack ( ${\eta _m}$ ), or under total reflection ( ${\eta _s}$ ) in the substrate. Normalization is performed with respect to free space scattering in air. Case of a 17 layer mirror with design wavelength of 600 nm. Both polarizations are considered.

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Fig. 11.
Fig. 11. Optical properties (transmission) of the single-cavity narrow-band filter.

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Fig. 12.
Fig. 12. Scattering power density versus normalized frequency for SS and PP polarizations. Case of a single-cavity Fabry-Perot filter illuminated at normal incidence at the design wavelength λ = λ0 = 600 nm.

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Fig. 13.
Fig. 13. Variations of scattering power density versus wavelength and spatial frequency. Case of the single-cavity Fabry-Perot filter designed at 600 nm.

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Fig. 14.
Fig. 14. TE polarization. Amount $\eta _m^q$ versus wavelength of light trapped by each guided mode of the stack. Normalization is performed with respect to free space scattering in air. Case of the single-cavity Fabry-Perot filter with design wavelength 600 nm.

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Fig. 15.
Fig. 15. TM polarization. Amount $\eta _m^q$ versus wavelength of light trapped by each guided mode of the stack. Normalization is performed with respect to free space scattering in air. Case of the single-cavity Fabry-Perot filter with design wavelength 600 nm.

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Fig. 16.
Fig. 16. Total amount versus wavelength of light trapped in the form of guided modes in the stack ( ${\eta _m}$ ), or under total reflection ( ${\eta _s}$ ) in the substrate. Normalization is performed with respect to free space scattering in air. Case of the single-cavity Fabry-Perot filter with central wavelength 600 nm. Both polarizations are considered.

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Fig. 17.
Fig. 17. Optical properties (transmission) of a multiple cavity filter plotted on a logarithmic scale.

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Fig. 18.
Fig. 18. Power density of scattering versus normalized frequency for SS and PP polarizations. Case of a multiple cavity narrow-band filter designed for normal illumination at wavelength λ = λ0 = 600 nm.

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Fig. 19.
Fig. 19. TE polarization. Amount $\eta _m^q$ versus wavelength of light trapped by each guided mode of the stack. Normalization is performed with respect to free space scattering in air. Case of the multiple-cavity cavity narrow-band filter with design wavelength 600 nm.

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Fig. 20.
Fig. 20. TM polarization. Amount $\eta _m^q$ versus wavelength of light trapped by each guided mode of the stack. Normalization is performed with respect to free space scattering in air. Case of the multiple-cavity narrow-band filter with design wavelength 600 nm.

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Fig. 21.
Fig. 21. Total amount versus wavelength of light trapped in the form of guided modes in the stack ( ${\eta _m}$ ), or under total reflection ( ${\eta _s}$ ) in the substrate. Normalization is performed with respect to free space scattering in air. Case of the single-cavity narrow-band filter with central wavelength 600 nm. Both polarizations are considered.

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Tables (1)

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Table 1. Coupling Efficiency η m q for Each Mode and Polarization in the Case of a 17 Layer Mirror Illuminated at Normal Incidence at the Design Wavelength λ = λ0 = 600 nm. The Abscissa is the Mode Order q.

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Equations (52)

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J → ( r → , z ) = ∑ i = 0 p ⁡ J → i ( r → ) δ ( z − z i ) M → ( r → , z ) = ∑ i = 0 p ⁡ M → i ( r → ) δ ( z − z i ) ,
F = ( 1 2 ) Re { ∫ ( M → ∗ . H → − J → . E → ∗ ) d Ω } ,
f ( σ → ) = 2 π 2 σ Re { ∑ i = 0 p ⁡ { M ^ → i ∗ ( σ → ) . H ^ → i ( σ → ) − J ^ → i ( σ → ) . E ^ ′ → i ∗ ( σ → ) } } ,
f ( σ → ) = d F / d σ d φ ,
σ → = σ ( cos φ , sin φ ) = 2 π ν → ,
σ < k ⇒ σ = k sin ⁡ θ with k = 2 π n / λ ,
H → i = ∑ j = 0 p ⁡ H → j i E ′ → i = ∑ j = 0 p ⁡ E ′ → j i .
j = i ⇒ H → i i = Y i z → ∧ E → i i
j > i ⇒ H → j i = Y i ′ z → ∧ E → j i ′ = Y i ′ z → ∧ C j i ′ E → j j ′
j < i ⇒ H → j i = Y i z → ∧ E → j i = Y i z → ∧ C j i E → j j ,
f ( σ → ) = 2 π 2 σ Re ( f 0 + f 1 ) ,
f 0 ( σ → ) = ∑ i = 0 p ⁡ { α i | M ^ i | 2 + β i | J ^ i | 2 + γ i z → . ( M ^ i ∗ ∧ J ^ i ) } ,
α i = − Y i Y i ′ / Δ Y i β i = 1 / Δ Y i ∗
γ i = 2 j Im { Y i / Δ Y i } Δ Y i = Y i − Y i ′ .
f 1 ( σ → ) = ∑ i , j j > i ⁡ f i j + ∑ i , j j < i ⁡ f i j
f i j , j > i = C j i ′ Y i ′ Δ Y j [ z → . ( M i ∗ ∧ J j ) − Y j M i ∗ . M j ] + C j i ′ ∗ Δ Y j ∗ [ J i . J j ∗ − Y j ∗ z → . ( M j ∗ . J i ) ]
f i j , j < i = C j i Y i Δ Y j [ z → . ( M i ∗ ∧ J j ) − Y j ′ M i ∗ . M j ] + C j i ∗ Δ Y j ∗ [ J i . J j ∗ − Y j ′ ∗ z → . ( M j ∗ . J i ) ] .
f 0 ( σ ) = ∑ i = 0 p ⁡ ( Y i | E i | 2 − Y i ′ | E i ′ | 2 )
f i j , j > i = C j i ′ Y i ′ E j j ′ ( E i i ∗ − E i i ′ ∗ ) + C j i ′ ∗ E j j ′ ∗ ( Y i E i i − Y i ′ E i i ′ ) ]
f i j , j < i = C j i Y i E j j ( E i i ∗ − E i i ′ ∗ ) + C j i ′ ∗ E j j ′ ∗ ( Y i E i i − Y i ′ E i i ′ ) .
J i → ( r → ) = − h i ( r → ) z → ∧ Δ i [ ∂ H → 0 ( r → , z ) ∂ z ]
M i → ( r → ) = − h i ( r → ) z → ∧ Δ i [ ∂ E → 0 ( r → , z ) ∂ z ] + grad → ( h ) ∧ Δ i [ E → 0 ( r → , z ) ] ,
J ^ i → ( σ → ) = j ω ( ϵ i + 1 − ϵ i ) h ^ i ( σ → − σ → 0 ) A → t g 0 ( z i )
M ^ i → ( σ → ) = − j ω [ ϵ i + 1 − ϵ i ϵ i + 1 ] h ^ i ( σ → − σ → 0 ) A z , i 0 ( z i ) ( σ → ∧ z → ) ,
Φ = Re { ∫ P → ( r → , z ) . z → d x d y }  with  P → = ( 1 2 ) E → ∗ ∧ H → ,
d Φ − d σ d φ = g − ( → σ ) = 2 π 2 σ Re ( n ~ 0 ) | A 0 − | 2 in the incident medium
d Φ + d σ d φ = g + ( → σ ) = 2 π 2 σ Re ( n ~ s ) | A s + | 2 in the substrate medium .
Φ − = ∫ 0 k 0 g − ( → σ ) d σ d φ Φ + = ∫ 0 k s g + ( → σ ) d σ d φ .
σ > k 0 , s ⇒ Re ( n ~ 0 , s ) = 0.
max ( k 0 , k s ) < σ < max ( k i ) .
F m = ∫ max ( k 0 , k s ) max ( k i ) f ( → σ ) d σ d φ = 2 π 2 ∫ max ( k 0 , k s ) max ( k i ) σ R e { f 0 ( → σ ) + f 1 ( → σ ) } d σ d φ .
1 / R ( κ q ) = 0
F m = ∫ max ( k 0 , k s ) max ( k i ) f ( σ → ) d σ d φ = j π ∑ q ⁡ Re s ( f ) κ q .
lim A → 0 ⁡ F m ( A ) = lim A → 0 ⁡ { ∫ max ( k 0 , k s ) max ( k i ) f ( σ → , A ) d σ d φ } ≠ ∫ max ( k 0 , k s ) max ( k i ) { lim A → 0 ⁡ f ( σ → , A ) } d σ d φ .
η m = F m / F rad ,
F rad = ∫ 0 k 0 f ( σ ) d σ F m = ∫ k s k H f ( σ ) d σ .
η m = ∑ q ⁡ η m q .
Real [ κ q ] ≈ κ q 0 .
F s = ∫ k 0 k s f ( σ ) d σ .
η s = F s / F rad .
γ ( σ ) = ( 1 / 4 π ) δ 2 L 2 exp [ ( − σ L 2 ) 2 ] ,
L ≪ λ / n H ,
( n e ) H = ( n e ) L = λ 0 / 4.
ν ∗ = ( λ / 2 π ) σ .
ARS ( θ , φ ) = BRDF ( θ , φ ) cos θ = k f ( σ ) / t g θ ,
ν ∗ > n H ⇒ f ( ν ∗ , A = 0 ) = 0.
η s ≈ ∫ k 0 k s σ d σ ∫ 0 k 0 σ d σ = ( n s n 0 ) 2 − 1.
h ( x , y ) = h 0 ( x , y ) ∗ x ∑ q ⁡ δ ( x − q T )
⇒ h ^ ( σ x , σ y ) = ( 1 T ) ∑ q ⁡ h ^ 0 ( 2 π q T , σ y ) δ ( σ x − 2 π q / T ) .
h ^ ( σ x , σ y ) = ( 1 T ) ∑ q ⁡ h ^ 0 ( 2 π q T , σ y ) S ^ ( σ x − 2 π q / T ) .
h ^ ( σ x , σ y ) = ( 1 T x T y ) ∑ q , p ⁡ h ^ 0 ( 2 π q T x , 2 π p T y ) S ^ ( σ x − 2 π q T x , σ y − 2 π p T y )
h ( x , y ) = h ( r ) = a Circ ( r / b ) ⇒ h ^ ( σ ) = ( a 2 π ) ∫ 0 b r J 0 ( σ r ) d r ,

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