Abstract

We present a simple method to calculate the angular scattering that is due to bulk inhomogeneities in an optical multilayer. We obtain formulas that are in the same form as those obtained for the roughness scattering.

© 1993 Optical Society of America

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References

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  1. C. Amra, J. H. Apfel, E. Pelletier, “The role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992).
    [CrossRef] [PubMed]
  2. J. M. Elson, J. P. Rahn, J. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
    [CrossRef] [PubMed]
  3. C. Amra, P. Roche, E. Pelletier, “Interface roughness cross-correlation laws deduced from scattering diagram measurements on optical multilayers: effect of the material grain size,” J. Opt. Soc. Am. B 4, 1087–1093 (1987).
    [CrossRef]
  4. C. Amra, “Calculs et mesures de diffusion appliqués à l’étude de la rugosité dans les traitements optiques multicouches,” J. Optics (Paris) 21, 83–98 (1990).
    [CrossRef]
  5. C. Amra, “Scattering characterization of materials in thin film form,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newman, M. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1438, 309–323 (1989).
  6. C. Amra, D. Torricini, Y. Boucher, E. Pelletier, “Scattering from optical surfaces and coatings: an easy investigation of microroughness,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 72–81 (1990).
    [CrossRef]
  7. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).
  8. C. Amra, C. Grèzes-Besset, P. Roche, E. Pelletier, “Description of a scattering apparatus: application to the problems of characterization of opaque surfaces,” Appl. Opt. 28, 2723–2730 (1989).
    [CrossRef] [PubMed]
  9. P. Roche, E. Pelletier, “Characterizations of optical surfaces by measurement of scattering distribution,” Appl. Opt. 23, 3561–3566 (1984).
    [CrossRef] [PubMed]
  10. P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
    [CrossRef]
  11. C. Amra, P. Bousquet, “Scattering from surfaces and multilayer coatings: recent advances for a better investigation of experiment,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 82–97 (1988).
    [CrossRef]
  12. J. M. Elson, “Angle resolved light scattering from composite optical surfaces,” in Periodic Structures, Gratings, Moiré Patterns, and Diffraction Phenomena I, C. H. Chin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–306 (1980).
    [CrossRef]
  13. A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
    [CrossRef]
  14. J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
    [CrossRef]
  15. P. Bussemer, K. Hehl, S. Kassam, “Theory of light scattering from rough surfaces and interfaces and from volume inhomogeneities in an optical layer stack,” Waves Random Med. 1, 207–221 (1991).
    [CrossRef]
  16. Numerical calculation issued from this bulk model was presented by C. Amra, L. Bruel, F. Cleva, “Theory and measurement of surface and bulk scattering in optical multilayers used as classical filters or waveguides,” in Optical Society of America 1991 Annual Meeting, Vol. 17 of 1991 Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 58.
  17. H. A. Macleod, in Thin-Film Optical Filters, 2nd ed., W. T. Welford, ed. (Hilger, Bristol, UK, 1986).
    [CrossRef]

1992

1991

P. Bussemer, K. Hehl, S. Kassam, “Theory of light scattering from rough surfaces and interfaces and from volume inhomogeneities in an optical layer stack,” Waves Random Med. 1, 207–221 (1991).
[CrossRef]

1990

C. Amra, “Calculs et mesures de diffusion appliqués à l’étude de la rugosité dans les traitements optiques multicouches,” J. Optics (Paris) 21, 83–98 (1990).
[CrossRef]

1989

1987

1984

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

P. Roche, E. Pelletier, “Characterizations of optical surfaces by measurement of scattering distribution,” Appl. Opt. 23, 3561–3566 (1984).
[CrossRef] [PubMed]

1981

1980

1975

A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
[CrossRef]

Amra, C.

C. Amra, J. H. Apfel, E. Pelletier, “The role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992).
[CrossRef] [PubMed]

C. Amra, “Calculs et mesures de diffusion appliqués à l’étude de la rugosité dans les traitements optiques multicouches,” J. Optics (Paris) 21, 83–98 (1990).
[CrossRef]

C. Amra, C. Grèzes-Besset, P. Roche, E. Pelletier, “Description of a scattering apparatus: application to the problems of characterization of opaque surfaces,” Appl. Opt. 28, 2723–2730 (1989).
[CrossRef] [PubMed]

C. Amra, P. Roche, E. Pelletier, “Interface roughness cross-correlation laws deduced from scattering diagram measurements on optical multilayers: effect of the material grain size,” J. Opt. Soc. Am. B 4, 1087–1093 (1987).
[CrossRef]

C. Amra, “Scattering characterization of materials in thin film form,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newman, M. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1438, 309–323 (1989).

C. Amra, D. Torricini, Y. Boucher, E. Pelletier, “Scattering from optical surfaces and coatings: an easy investigation of microroughness,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 72–81 (1990).
[CrossRef]

C. Amra, P. Bousquet, “Scattering from surfaces and multilayer coatings: recent advances for a better investigation of experiment,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 82–97 (1988).
[CrossRef]

Numerical calculation issued from this bulk model was presented by C. Amra, L. Bruel, F. Cleva, “Theory and measurement of surface and bulk scattering in optical multilayers used as classical filters or waveguides,” in Optical Society of America 1991 Annual Meeting, Vol. 17 of 1991 Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 58.

Apfel, J. H.

Bennett, J.

Bennett, J. M.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Boucher, Y.

C. Amra, D. Torricini, Y. Boucher, E. Pelletier, “Scattering from optical surfaces and coatings: an easy investigation of microroughness,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 72–81 (1990).
[CrossRef]

Bousquet, P.

P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
[CrossRef]

C. Amra, P. Bousquet, “Scattering from surfaces and multilayer coatings: recent advances for a better investigation of experiment,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 82–97 (1988).
[CrossRef]

Bruel, L.

Numerical calculation issued from this bulk model was presented by C. Amra, L. Bruel, F. Cleva, “Theory and measurement of surface and bulk scattering in optical multilayers used as classical filters or waveguides,” in Optical Society of America 1991 Annual Meeting, Vol. 17 of 1991 Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 58.

Bussemer, P.

P. Bussemer, K. Hehl, S. Kassam, “Theory of light scattering from rough surfaces and interfaces and from volume inhomogeneities in an optical layer stack,” Waves Random Med. 1, 207–221 (1991).
[CrossRef]

Cleva, F.

Numerical calculation issued from this bulk model was presented by C. Amra, L. Bruel, F. Cleva, “Theory and measurement of surface and bulk scattering in optical multilayers used as classical filters or waveguides,” in Optical Society of America 1991 Annual Meeting, Vol. 17 of 1991 Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 58.

Elson, J. M.

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

J. M. Elson, J. P. Rahn, J. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
[CrossRef] [PubMed]

J. M. Elson, “Angle resolved light scattering from composite optical surfaces,” in Periodic Structures, Gratings, Moiré Patterns, and Diffraction Phenomena I, C. H. Chin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–306 (1980).
[CrossRef]

Flory, F.

Grèzes-Besset, C.

Hehl, K.

P. Bussemer, K. Hehl, S. Kassam, “Theory of light scattering from rough surfaces and interfaces and from volume inhomogeneities in an optical layer stack,” Waves Random Med. 1, 207–221 (1991).
[CrossRef]

Kassam, S.

P. Bussemer, K. Hehl, S. Kassam, “Theory of light scattering from rough surfaces and interfaces and from volume inhomogeneities in an optical layer stack,” Waves Random Med. 1, 207–221 (1991).
[CrossRef]

Macleod, H. A.

H. A. Macleod, in Thin-Film Optical Filters, 2nd ed., W. T. Welford, ed. (Hilger, Bristol, UK, 1986).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
[CrossRef]

Mattsson, L.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Mills, D. L.

A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
[CrossRef]

Pelletier, E.

Rahn, J. P.

Roche, P.

Torricini, D.

C. Amra, D. Torricini, Y. Boucher, E. Pelletier, “Scattering from optical surfaces and coatings: an easy investigation of microroughness,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 72–81 (1990).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

J. Optics (Paris)

C. Amra, “Calculs et mesures de diffusion appliqués à l’étude de la rugosité dans les traitements optiques multicouches,” J. Optics (Paris) 21, 83–98 (1990).
[CrossRef]

Phys. Rev. B

A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
[CrossRef]

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

Waves Random Med.

P. Bussemer, K. Hehl, S. Kassam, “Theory of light scattering from rough surfaces and interfaces and from volume inhomogeneities in an optical layer stack,” Waves Random Med. 1, 207–221 (1991).
[CrossRef]

Other

Numerical calculation issued from this bulk model was presented by C. Amra, L. Bruel, F. Cleva, “Theory and measurement of surface and bulk scattering in optical multilayers used as classical filters or waveguides,” in Optical Society of America 1991 Annual Meeting, Vol. 17 of 1991 Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 58.

H. A. Macleod, in Thin-Film Optical Filters, 2nd ed., W. T. Welford, ed. (Hilger, Bristol, UK, 1986).
[CrossRef]

C. Amra, “Scattering characterization of materials in thin film form,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newman, M. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1438, 309–323 (1989).

C. Amra, D. Torricini, Y. Boucher, E. Pelletier, “Scattering from optical surfaces and coatings: an easy investigation of microroughness,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 72–81 (1990).
[CrossRef]

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

C. Amra, P. Bousquet, “Scattering from surfaces and multilayer coatings: recent advances for a better investigation of experiment,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 82–97 (1988).
[CrossRef]

J. M. Elson, “Angle resolved light scattering from composite optical surfaces,” in Periodic Structures, Gratings, Moiré Patterns, and Diffraction Phenomena I, C. H. Chin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–306 (1980).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Inhomogeneous layer i inside a multilayer structure.

Fig. 2
Fig. 2

Definition of the scattering angles. θ is from the sample normal, and ϕ is a polar angle.

Fig. 3
Fig. 3

Polarization components of the scattered wave.

Equations (152)

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i = i [ 1 + p i ( ρ ) ] ,
rot E = j ω μ H ,
rot H = j ω E ,
= { j in medium j i i = i ( 1 + p i ) in medium i .
rot E 0 = j ω μ H 0 ,
rot H 0 = j ω 0 E 0 ,
( E d , H d ) = ( E , H ) ( E 0 , H 0 ) ,
rot E d = j ω μ H d ,
rot H d = j ω 0 E d + J [ ( z z i 1 ) ( z z i ) ] ,
J = j ω i p i E .
| p i | 1 J j ω i p i E 0 .
Δ X j + k j 2 X j = 0 ,
z 2 X ̂ j + α j 2 X ̂ j = 0 X ̂ j = A j , x + exp ( j α j z ) + A j , x exp ( j α j z ) ,
X ̂ j ( σ , z ) = 1 4 π 2 r X j ( r , z ) exp ( j σ · r ) d r .
X j ( ρ ) = σ X j ( σ , ρ ) d σ ,
k j ± ( σ ) = σ ± α j ( σ ) z ,
X j = A j , x + exp ( j k j + · ρ ) + A j , x exp ( j k j · ρ ) .
σ = σ [ ( cos ϕ ) x + ( sin ϕ ) y ] ,
Δ X i + k i 2 X i = S i X ,
S i E = j ω μ J i + 1 j ω i grad ( div J i ) ,
S i H = rot J i ,
J i = j ω i p i E i 0 .
X i * = g i * S i X ,
Δ g i + k i 2 g i = δ ( ρ ) ,
X ̂ i * ( σ , z ) = 4 π 2 z = 0 e i ĝ i ( σ , z z ) Ŝ i X ( σ , z ) d z .
z 2 ĝ i + α i 2 ĝ i = 1 4 π 2 δ ( z ) ĝ i ( σ , z ) = 1 4 π 2 1 2 j α i exp ( j α i | z | ) ,
X ̂ i * ( σ , z ) = 1 2 j α i z = 0 e i Ŝ i X ( σ , z ) exp ( j α i | z z | ) d z .
[ X ̂ i ( 0 ) + X ̂ i * ( 0 ) ] X ̂ i 1 ( e i ) = 0 at interface ( i 1 ) , X ̂ i + 1 ( 0 ) [ X ̂ i ( e i ) + X ̂ i * ( e i ) ] = 0 at interface i .
X ̂ i ( 0 ) X ̂ i 1 ( e i ) = X ̂ i * ( 0 ) ,
X ̂ i ( e i ) X ̂ i + 1 ( 0 ) = X ̂ i * ( e i ) .
Ĥ i 1 ( e i ) = Y i 1 z Ê i 1 ( e i ) ,
Ĥ i + 1 ( 0 ) = Y i z Ê i + 1 ( 0 ) .
z Ê i ( 0 ) Ĥ i ( 0 ) } = M i 1 , i { z Ê i ( e i ) Ĥ i ( e i ) ,
M i 1 , i = [ a i b i c i a i ]
z Ê i + 1 ( 0 ) Ĥ i + 1 ( 0 ) } = M i , p { z Ê s + ( 0 ) ñ s z Ê s + ( 0 ) ,
z Ê i ( 0 ) Ĥ i ( 0 ) } = M i 1 , i { z Ê i * ( e i ) Ĥ i * ( e i ) + M i 1 , p { z Ê s + ñ s ( z Ê s + ) ,
F i ( e i ) = a i z Ê i * ( e i ) + b i Ĥ i * ( e i ) ,
G i ( e i ) = c i z Ê i * ( e i ) + a i Ĥ i * ( e i ) .
Ĥ i ( 0 ) = Y i 1 z Ê i ( 0 ) G i ( e i ) + Y i 1 F i ( e i ) ,
Ĥ i ( e i ) = Y i z Ê i ( e i ) G i ( 0 ) + Y i F i ( 0 ) ,
F i ( 0 ) = a i z Ê i * ( 0 ) b i Ĥ i * ( 0 ) ,
G i ( 0 ) = c i z Ê i * ( 0 ) + a i Ĥ i * ( 0 ) .
z Ê i 1 ( e i ) Ĥ i 1 ( e i ) } = M 0 , i 1 1 { z Ê 0 ( 0 ) ñ 0 z Ê 0 ( 0 ) .
z Ê i 1 ( e i ) = [ G i ( e i ) + Y i 1 F i ( e i ) + Ĥ i * ( 0 ) Y i 1 z Ê i * ( 0 ) ] / ( Y i 1 Y i 1 ) ,
z Ê i + 1 ( 0 ) = [ G i ( 0 ) Y i F i ( 0 ) Ĥ i * ( e i ) + Y i z Ê i * ( e i ) ] / ( Y i Y i ) .
( Y i 1 Y i 1 ) z Ê i 1 = ( c i + a i Y i 1 ) z Ê i * ( e i ) + ( a i + b i Y i 1 ) Ĥ i * ( e i ) + Ĥ i * ( 0 ) Y i 1 z Ê i * ( 0 ) ,
( Y i Y i ) z Ê i = ( c i + a i Y i ) z Ê i * ( 0 ) + ( a i + b i Y i ) Ĥ i * ( 0 ) Ĥ i * ( e i ) + Y i z Ê i * ( e i ) ,
z Ê i d = C i 1 z Ê i 1 ,
z Ê i d + = C i + z Ê i ,
C i 1 = m = 1 i 1 [ ( cos ψ m ) j ( sin ψ m ) ( Y m / ñ m ) ] ,
C i + = m = i + 1 p [ ( cos ψ m ) + j ( sin ψ m ) ( Y m 1 / ñ m ) ] .
u = σ σ = ( cos ϕ ) x + ( sin ϕ ) y , v = β σ ,
β = d σ d ϕ .
Ê β = β σ · Ê , Ê σ = σ σ · Ê
Ê β = Ê S , Ê σ = ( Ê p ) tan
rot E i * = j ω μ H i * ,
rot H i * = j ω i E i * + J i ,
J i = j ω i p i E i 0 .
z z Ê i * + j σ Ê i * = j ω μ Ĥ i * .
z Ê i * = 1 2 z = 0 e i Ŝ i E exp ( j α i | z z | ) sgn ( z z ) d z ,
sgn ( z z ) = { 1 for z > z 1 for z < z ·
( z Ê i * ) z = 0 = j α i Ê i * ( 0 ) ,
( z Ê i * ) z = e i = j α i Ê i * ( e i ) .
Ĥ i * ( 0 ) = 1 ω μ k i Ê i * ( 0 ) ,
Ĥ i * ( e i ) = 1 ω μ k i + Ê i * ( e i ) .
div Ê i * = 0 z Ê z , i * + j σ · Ê i * = 0 ,
Ê z , i * ( 0 ) = ( σ / α i ) Ê σ , i * ( 0 ) ,
Ê z , i * ( e i ) = ( σ / α i ) Ê σ , i * ( e i ) .
( Y i 1 Y i 1 ) Ê i 1 = Ê i * ( 0 ) ( ñ i + Y i 1 ) + Ê i * ( e i ) × exp ( j α i e i ) ( ñ i Y i 1 ) ,
( Y i Y i ) Ê i = Ê i * ( 0 ) exp ( j α i e i ) ( ñ i + Y i ) + Ê i * ( e i ) ( ñ i Y i ) ,
ñ i = { α i / ω μ for S polarization α i k i 2 / ω μ for P polarization
Ê i * ( 0 ) = 1 2 j α i z = 0 e i Ŝ i E ( σ , z ) exp ( j α i z ) d z ,
Ê i * ( e i ) = 1 2 j α i exp ( j α i e i ) z = 0 e i Ŝ i E ( σ , z ) exp ( j α i z ) d z .
( S i E ) tan = k i 2 p i ( E i 0 ) tan [ grad ( E i 0 · grad p i ) ] tan .
E i 0 ( ρ ) = A i 0 ( z ) exp ( j σ 0 · r ) ,
σ 0 = { σ 0 ( cos α ) σ 0 ( sin α ) , σ 0 = k 0 ( sin i 0 ) .
f = A i 0 · grad p i ,
( Ŝ i E ) tan = δ ( σ σ 0 ) * T ,
T = k i 2 p ̂ i ( A i 0 ) tan + j ( σ + σ 0 ) f ̂ ,
( Ŝ i E ) tan = k i 2 ( A i 0 ) tan p ̂ i ( σ σ 0 ) j σ f ̂ ( σ σ 0 ) .
Ŝ β i = k i 2 p ̂ i ( β σ · A i 0 ) ,
Ŝ σ i = α i 2 p ̂ i ( σ σ · A i 0 ) σ p ̂ i ( σ 0 · A i 0 ) j σ ( z p ̂ i ) ( z · A i 0 ) .
A i 0 ( S ) = β 0 σ 0 · A i 0 = A y i 0 for α = 0 .
Ŝ β i ( S ) = k i 2 p ̂ i [ cos ( ϕ α ) ] A i 0 ( S ) ,
Ŝ σ i ( S ) = α i 2 p ̂ i [ sin ( ϕ α ) ] A i 0 ( S ) .
A i 0 ( P ) = σ 0 σ 0 · A i 0 = A x i 0 for α = 0 .
Ŝ β i ( P ) = k i 2 p ̂ i [ sin ( ϕ α ) ] A i 0 ( P ) ,
Ŝ σ i ( P ) = { α i 2 [ cos ( ϕ α ) ] + σ σ 0 } p ̂ i A i 0 ( P ) j σ ( z p ̂ i ) A z i 0 .
A i 0 ( z ) = A i 1 + 0 exp ( j α i 0 z ) + A i 1 0 exp ( j α i 0 z )
α i 0 = ( k i 2 σ 0 2 ) 1 / 2 .
I i ( α ) = j z = 0 e i p ̂ i ( σ , z ) exp ( j α z ) d z .
J i ( α ) = A i 1 + 0 I i ( α + α i 0 ) + A i 1 0 I i ( α α i 0 ) ,
K i ( α ) = A i 1 + 0 I i ( α + α i 0 ) A i 1 0 I i ( α α i 0 ) ,
L i ( α ) = p ̂ i ( σ , e i ) exp ( j α e i ) A i 0 ( e i ) p ̂ i ( σ , 0 ) A i 0 ( 0 ) .
Ê S S * ( 0 ) Ê S S * ( e i ) } = ( k i 2 / 2 α i ) [ cos ( ϕ α ) ] { J y i ( α i ) exp ( j α i e i ) J y i ( α i ) ,
Ê S P * ( 0 ) Ê S P * ( e i ) } = ( α i / 2 ) [ sin ( ϕ α ) ] { J y i ( α i ) exp ( j α i e i ) J y i ( α i ) ,
Ê P S * ( 0 ) Ê P S * ( e i ) } = ( k i 2 / 2 α i ) [ sin ( ϕ α ) ] { J x i ( α i ) exp ( j α i e i ) J x i ( α i ) ,
Ê P P * ( 0 ) Ê P P * ( e i ) } = ( 1 / 2 α i ) { α i 2 [ cos ( ϕ α ) ] + σ σ 0 } { J x i ( α i ) exp ( j α i e i ) J x i ( α i ) + ( σ / 2 α i ) { α i J z i ( α i ) + α i 0 K z i ( α i ) L z i ( α i ) exp ( j α i e i ) [ α i J z i ( α i ) + α i 0 K z i ( α i ) L z i ( α i ) ] .
p i ( r , z ) = p i ( r ) p i ( z ) .
p i ( r , z ) = p i ( r ) exp ( υ i z ) .
I i ( α ) = p ̂ i ( σ ) exp [ j ( α j υ i ) e i ] 1 α j υ i ,
Ê S S * ( 0 ) Ê S S * ( e i ) } = p ̂ i [ cos ( ϕ α ) ] ( k i 2 / 2 α i ) { F y i exp ( j α i e i ) G y i ,
Ê S P * ( 0 ) Ê S P * ( e i ) } = p ̂ i [ sin ( ϕ α ) ] ( α i / 2 ) { F y i exp ( j α i e i ) G y i ,
Ê P S * ( 0 ) Ê P S * ( e i ) } = p ̂ i [ sin ( ϕ α ) ] ( k i 2 / 2 α i ) { F x i exp ( j α i e i ) G x i ,
Ê P P * ( 0 ) Ê P P * ( e i ) } = p ̂ i ( 1 / 2 α i ) { α i 2 [ cos ( ϕ α ) ] + σ σ 0 } × { F x i exp ( j α i e i ) G x i + j p ̂ i υ i σ 2 α i { F z i exp ( j α i e i ) G z i ,
F i = F i + A i 1 + 0 + F i A i 1 0 ,
G i = G i + A i 1 + 0 + G i A i 1 0 ,
F i ± = [ exp ( j ξ i ± e i ) 1 ] / ξ i ± ,
G i ± = [ exp ( j γ i ± e i ) 1 ] / γ i ±
ξ i + = α i + α i 0 j υ i ,
ξ i = α i α i 0 j υ i ,
γ i + = ( α i α i 0 ) j υ i ,
γ i = ( α i + α i 0 ) j υ i .
G p + 1 ± = F p + 1 = 0 , F p + 1 + = 1 / ξ p + 1 + , F 0 ± = 0 , G 0 ± = 1 / γ 0 ± .
j = j [ 1 + p j ( ρ ) ] , | p j | 1 .
Ê d ± = j = 1 p Ê j d ± ,
ϕ d = 1 2 ω μ σ α 0 ( σ ) | Ê d ( σ ) | 2 d σ in air ,
ϕ d + = 1 2 ω μ σ α s ( σ ) | Ê d + ( σ ) | 2 d σ in substrate .
ϕ d ± = k 3 2 ω μ θ , ϕ | Ê d ± ( θ , ϕ ) | 2 ( cos 2 θ ) ( sin θ ) d θ d ϕ ,
0 θ π 2 , 0 ϕ 2 π ,
( k , θ ) = { ( k 0 , θ 0 ) for ϕ d ( k s , θ s ) for ϕ d + .
I ± ( θ , ϕ ) = ( d ϕ d Ω ) ± = k 3 2 ω μ | Ê d ± ( θ , ϕ ) | 2 cos 2 θ .
ϕ 0 + = 1 2 ω μ | A 0 + | 2 k 0 ( cos i 0 ) ,
I ± ( θ , ϕ ) = k 3 k 0 | Ê d + A 0 + ( θ , ϕ ) | 2 f ( θ , i 0 ) ,
f = { ( cos 2 θ ) / ( cos i 0 ) for S S polarization ( cos 2 θ ) ( cos i 0 ) for P S polarization 1 / ( cos i 0 ) for S P polarization ( cos i 0 ) for P P polarization .
I ± ( θ , ϕ ) = i , j C i j ± ( θ , ϕ ) γ i j ( θ , ϕ ) ,
δ j 2 = | n j / 2 | 2 σ γ j j ( σ ) d σ ,
k + = { σ α ( σ ) , σ = { σ ( cos ϕ ) σ ( sin ϕ ) , σ = k sin θ , α = k 2 σ 2 = k ( cos θ ) , k = 2 π n λ ,
H tan + = ñ z E tan + ,
ñ = { ( 1 / η 0 μ r ) n α / k for S polarization ( 1 / η 0 μ r ) n k / α for P polarization .
H t = Y z E t ,
Y = ñ 1 r 1 + r
Y i 1 = j ñ i ( sin ψ i ) + Y i ( cos ψ i ) ( cos ψ i ) j Y i ( sin ψ i ) / ñ i ,
Y i 1 = C i 1 + D i 1 ñ s A i 1 + B i 1 ñ s ,
[ A i 1 B i 1 C i 1 D i 1 ]
z E i = z E i 1 [ ( cos ψ i ) + j ( sin ψ i ) Y i 1 / ñ i ]
z E i 1 = z E i [ ( cos ψ i ) j ( sin ψ i ) Y i / ñ i ] .
( A i 0 ) tan = r i ( A i + 0 ) tan ( A i + 0 ) tan = ( A i 0 / 1 + r i ) tan , r i = ( ñ i 0 Y i 0 ) / ( ñ i 0 + Y i 0 ) , A z , i 0 = r i A z , i 0 + A z , i 0 + = A z , i / ( 1 r i ) A z , i = σ 0 α i 0 Y i 0 ñ i 0 ( A i 0 ) tan .
Y i = Y i 1 ( cos ψ i ) + j ñ i ( sin ψ i ) ( cos ψ i ) + j ( sin ψ i ) Y i 1 / ñ i ,
Y i = C i D i ñ 0 A i B i ñ 0 ,
M 0 , i 1 = [ A i B i C i D i ]
z E i 1 = z E i [ ( cos ψ i ) j ( sin ψ i ) Y i / ñ i ] .
( Y i Y i ) Ê i = J ̂ i Y i z M ̂ i = R ̂ i ,
( Y i Y i ) Ê i + = J ̂ i Y i z M ̂ i = R ̂ i + .
J ̂ i = a i ( A i 0 ) tg , M ̂ i = b i A z i 0 ( σ z ) ,
a i = j ω Δ i ĥ i = 1 η 0 j 2 π λ ( n i + 1 2 n i 2 ) ĥ i ,
b i = j Δ i i + 1 ĥ i = j n i + 1 2 n i 2 n i + 1 2 ĥ i .
R ̂ i ( S S ) = a i ( cos ϕ ) A y i 0 ,
R ̂ i ( S P ) = a i ( sin ϕ ) A y i 0 ,
R ̂ i ( P S ) = a i ( sin ϕ ) A x i 0 ,
R ̂ i ( P P ) = a i ( cos ϕ ) A x i 0 Y i b i σ A z i 0 .
I ± ( θ , ϕ ) = i , j C i j ± ( θ , ϕ ) γ i j ( θ , ϕ ) ,

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