Abstract

A novel first-order nonparaxial scalar theory for calculating the angular scattering that is caused by the interface roughness in an optical multilayer was proposed. As in the case that the interface roughness is moderate, the analytic expressions of angular-resolved scattering for a typical p-layer design were derived. Notably, these formulas are general because they do not depend on the prior restrictive hypothesis for the correlation degree of the various interfaces in a stack. In order to verify the theory, the formulas in the case of single-surface are presented and are exactly identical to those of the generalized Harvey-Shack theory. Also, their smooth-surface approximations are the same in form as those given by the typical first-order vector perturbation theories and are validated by numerically comparing with the typical vector theory for three representative multilayer design types with slightly rough interfaces. In addition, the usability of the novel theory in the case of moderate roughness is discussed by comparing this theory to the typical theories for optical coatings at different roughness levels.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
First-order nonparaxial scalar theory of surface and bulk scattering for high-quality optical coatings

Kepeng Zhang, Renshuai Huang, Xiaoxi Tian, Yinhua Zhang, Wei Huang, and ChunLin Guan
J. Opt. Soc. Am. A 35(11) 1823-1831 (2018)

Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles

Andrey Krywonos, James E. Harvey, and Narak Choi
J. Opt. Soc. Am. A 28(6) 1121-1138 (2011)

Scattering from multilayer thin films: theory and experiment

P. Bousquet, F. Flory, and P. Roche
J. Opt. Soc. Am. 71(9) 1115-1123 (1981)

References

  • View by:
  • |
  • |
  • |

  1. V. K. Sakharov, “Model of lock-in in a ring laser and a semiconductor laser gyro,” Tech. Phys. 56(8), 1135–1141 (2011).
    [Crossref]
  2. H. Miao, H. Yang, R. X. Adhikari, and Y. Chen, “Quantum limits of interferometer topologies for gravitational radiation detection,” Class. Quantum Gravity 31(16), 165010 (2014).
    [Crossref]
  3. S. Zeidler, T. Akutsu, Y. Torii, E. Hirose, Y. Aso, and R. Flaminio, “Calculation method for light scattering caused by multilayer coated mirrors in gravitational wave detectors,” Opt. Express 25(5), 4741–4760 (2017).
    [Crossref] [PubMed]
  4. S. Schröder, A. Duparré, and A. Tünnermann, “Roughness evolution and scatter losses of multilayers for 193 nm optics,” Appl. Opt. 47(13), C88–C97 (2008).
    [Crossref] [PubMed]
  5. J. A. Thornton, “Structure and topography of sputtered coatings,” Annu. Rev. Mater. Sci. 7, 239–260 (1977).
    [Crossref]
  6. W. M. Tong and R. S. Williams, “Kinetics of surface growth: phenomenology, scaling, and mechanisms of smoothening and roughening,” Annu. Rev. Chem. 45(1), 401–438 (1994).
    [Crossref]
  7. P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Am. 71(9), 1115–1123 (1981).
    [Crossref]
  8. J. M. Elson, “Diffraction and diffuse scattering from dielectric multilayers,” J. Opt. Soc. Am. 69(1), 48–54 (1979).
    [Crossref]
  9. J. M. Elson, “Surface scattering in optical interference coatings,” J. Opt. Soc. Am. 66, 230–234 (1974).
  10. C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. A 10(2), 365–374 (1993).
    [Crossref]
  11. S. Schröder, T. Herffurth, H. Blaschke, and A. Duparré, “Angle-resolved scattering: an effective method for characterizing thin-film coatings,” Appl. Opt. 50(9), C164–C171 (2011).
    [Crossref] [PubMed]
  12. C. Amra, J. H. Apfel, and E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31(16), 3134–3151 (1992).
    [Crossref] [PubMed]
  13. C. Amra, C. Grèzes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. 32(28), 5492–5503 (1993).
    [Crossref] [PubMed]
  14. C. Amra, “From light scattering to the microstructure of thin-film multilayers,” Appl. Opt. 32(28), 5481–5491 (1993).
    [Crossref] [PubMed]
  15. C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. A 11(1), 197–210 (1994).
    [Crossref]
  16. C. Deumié, H. Giovannini, and C. Amra, “Ellipsometry of light scattering from multilayer coatings,” Appl. Opt. 35(28), 5600–5608 (1996).
    [Crossref] [PubMed]
  17. S. Maure, G. Albrand, and C. Amra, “Low-level scattering and localized defects,” Appl. Opt. 35(28), 5573–5582 (1996).
    [Crossref] [PubMed]
  18. C. Deumié, R. Richier, P. Dumas, and C. Amra, “Multiscale roughness in optical multilayers: atomic force microscopy and light scattering,” Appl. Opt. 35(28), 5583–5594 (1996).
    [Crossref] [PubMed]
  19. G. Xu, P. G. Stegmann, S. D. Brooks, and P. Yang, “Modeling the single and multiple scattering properties of soot-laden mineral dust aerosols,” Opt. Express 25(24), A990–A1008 (2017).
    [Crossref] [PubMed]
  20. J. Zhang, H. Wu, H. Jiao, S. Schröder, M. Trost, Z. Wang, and X. Cheng, “Reducing light scattering in high-reflection coatings through destructive interference at fully correlated interfaces,” Opt. Lett. 42(23), 5046–5049 (2017).
    [Crossref] [PubMed]
  21. P. Beckmann and A. Spizzichino, “The Scattering of Electromagnetic Waves from Rough Surfaces,” (Pergamon Press, 1963).
  22. J. M. Eastman, “Scattering by All- Dielectric Multilayer Bandpass Filters and Mirrors for Lasers,” in Physics of Thin Films. Advances in Research and Development 10, 167–226 (1978).
  23. J. M. Eastman, “Surface Scattering in Optical Interference Coatings,” J. Opt. Soc. Am. 66, 230–234 (1974).
  24. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18(2), 104–115 (1979).
    [Crossref]
  25. J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
    [Crossref]
  26. A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28(6), 1121–1138 (2011).
    [Crossref] [PubMed]
  27. R. A. Craig, G. J. Exarhos, W. T. Pawlewicz, and R. E. Williford, “Interference-enhanced Raman scattering from Ti02/Si02 multilayers: measurement and theory,” Appl. Opt. 26(19), 4193–4197 (1987).
    [Crossref] [PubMed]
  28. J. M. Elson, J. P. Rahn, and J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19(5), 669–679 (1980).
    [Crossref] [PubMed]
  29. S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, and J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Express 19(10), 9820–9835 (2011).
    [Crossref] [PubMed]
  30. H. Giovannini and C. Amra, “Scattering-reduction effect with overcoated rough surfaces: theory and experiment,” Appl. Opt. 36(22), 5574–5579 (1997).
    [Crossref] [PubMed]

2017 (3)

2014 (1)

H. Miao, H. Yang, R. X. Adhikari, and Y. Chen, “Quantum limits of interferometer topologies for gravitational radiation detection,” Class. Quantum Gravity 31(16), 165010 (2014).
[Crossref]

2011 (4)

2008 (1)

2007 (1)

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[Crossref]

1997 (1)

1996 (3)

1994 (2)

C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. A 11(1), 197–210 (1994).
[Crossref]

W. M. Tong and R. S. Williams, “Kinetics of surface growth: phenomenology, scaling, and mechanisms of smoothening and roughening,” Annu. Rev. Chem. 45(1), 401–438 (1994).
[Crossref]

1993 (3)

1992 (1)

1987 (1)

1981 (1)

P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Am. 71(9), 1115–1123 (1981).
[Crossref]

1980 (1)

1979 (2)

K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18(2), 104–115 (1979).
[Crossref]

J. M. Elson, “Diffraction and diffuse scattering from dielectric multilayers,” J. Opt. Soc. Am. 69(1), 48–54 (1979).
[Crossref]

1978 (1)

J. M. Eastman, “Scattering by All- Dielectric Multilayer Bandpass Filters and Mirrors for Lasers,” in Physics of Thin Films. Advances in Research and Development 10, 167–226 (1978).

1977 (1)

J. A. Thornton, “Structure and topography of sputtered coatings,” Annu. Rev. Mater. Sci. 7, 239–260 (1977).
[Crossref]

1974 (2)

J. M. Elson, “Surface scattering in optical interference coatings,” J. Opt. Soc. Am. 66, 230–234 (1974).

J. M. Eastman, “Surface Scattering in Optical Interference Coatings,” J. Opt. Soc. Am. 66, 230–234 (1974).

Adhikari, R. X.

H. Miao, H. Yang, R. X. Adhikari, and Y. Chen, “Quantum limits of interferometer topologies for gravitational radiation detection,” Class. Quantum Gravity 31(16), 165010 (2014).
[Crossref]

Akutsu, T.

Albrand, G.

Amra, C.

Apfel, J. H.

Aso, Y.

Bennett, J. M.

Blaschke, H.

Bousquet, P.

P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Am. 71(9), 1115–1123 (1981).
[Crossref]

Brooks, S. D.

Bruel, L.

Carniglia, K.

K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18(2), 104–115 (1979).
[Crossref]

Chen, Y.

H. Miao, H. Yang, R. X. Adhikari, and Y. Chen, “Quantum limits of interferometer topologies for gravitational radiation detection,” Class. Quantum Gravity 31(16), 165010 (2014).
[Crossref]

Cheng, X.

Choi, N.

Coriand, L.

Craig, R. A.

Deumié, C.

Dumas, P.

Duparré, A.

Eastman, J. M.

J. M. Eastman, “Scattering by All- Dielectric Multilayer Bandpass Filters and Mirrors for Lasers,” in Physics of Thin Films. Advances in Research and Development 10, 167–226 (1978).

J. M. Eastman, “Surface Scattering in Optical Interference Coatings,” J. Opt. Soc. Am. 66, 230–234 (1974).

Elson, J. M.

Exarhos, G. J.

Flaminio, R.

Flory, F.

P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Am. 71(9), 1115–1123 (1981).
[Crossref]

Giovannini, H.

Grèzes-Besset, C.

Harvey, J. E.

Herffurth, T.

Hirose, E.

Jiao, H.

Krywonos, A.

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28(6), 1121–1138 (2011).
[Crossref] [PubMed]

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[Crossref]

Maure, S.

Miao, H.

H. Miao, H. Yang, R. X. Adhikari, and Y. Chen, “Quantum limits of interferometer topologies for gravitational radiation detection,” Class. Quantum Gravity 31(16), 165010 (2014).
[Crossref]

Pawlewicz, W. T.

Pelletier, E.

Penalver, D. H.

Rahn, J. P.

Richier, R.

Roche, P.

P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Am. 71(9), 1115–1123 (1981).
[Crossref]

Sakharov, V. K.

V. K. Sakharov, “Model of lock-in in a ring laser and a semiconductor laser gyro,” Tech. Phys. 56(8), 1135–1141 (2011).
[Crossref]

Schröder, S.

Stegmann, P. G.

Thornton, J. A.

J. A. Thornton, “Structure and topography of sputtered coatings,” Annu. Rev. Mater. Sci. 7, 239–260 (1977).
[Crossref]

Tong, W. M.

W. M. Tong and R. S. Williams, “Kinetics of surface growth: phenomenology, scaling, and mechanisms of smoothening and roughening,” Annu. Rev. Chem. 45(1), 401–438 (1994).
[Crossref]

Torii, Y.

Trost, M.

Tünnermann, A.

Vernold, C. L.

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[Crossref]

Wang, Z.

Williams, R. S.

W. M. Tong and R. S. Williams, “Kinetics of surface growth: phenomenology, scaling, and mechanisms of smoothening and roughening,” Annu. Rev. Chem. 45(1), 401–438 (1994).
[Crossref]

Williford, R. E.

Wu, H.

Xu, G.

Yang, H.

H. Miao, H. Yang, R. X. Adhikari, and Y. Chen, “Quantum limits of interferometer topologies for gravitational radiation detection,” Class. Quantum Gravity 31(16), 165010 (2014).
[Crossref]

Yang, P.

Zeidler, S.

Zhang, J.

Annu. Rev. Chem. (1)

W. M. Tong and R. S. Williams, “Kinetics of surface growth: phenomenology, scaling, and mechanisms of smoothening and roughening,” Annu. Rev. Chem. 45(1), 401–438 (1994).
[Crossref]

Annu. Rev. Mater. Sci. (1)

J. A. Thornton, “Structure and topography of sputtered coatings,” Annu. Rev. Mater. Sci. 7, 239–260 (1977).
[Crossref]

Appl. Opt. (11)

R. A. Craig, G. J. Exarhos, W. T. Pawlewicz, and R. E. Williford, “Interference-enhanced Raman scattering from Ti02/Si02 multilayers: measurement and theory,” Appl. Opt. 26(19), 4193–4197 (1987).
[Crossref] [PubMed]

J. M. Elson, J. P. Rahn, and J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19(5), 669–679 (1980).
[Crossref] [PubMed]

H. Giovannini and C. Amra, “Scattering-reduction effect with overcoated rough surfaces: theory and experiment,” Appl. Opt. 36(22), 5574–5579 (1997).
[Crossref] [PubMed]

S. Schröder, A. Duparré, and A. Tünnermann, “Roughness evolution and scatter losses of multilayers for 193 nm optics,” Appl. Opt. 47(13), C88–C97 (2008).
[Crossref] [PubMed]

S. Schröder, T. Herffurth, H. Blaschke, and A. Duparré, “Angle-resolved scattering: an effective method for characterizing thin-film coatings,” Appl. Opt. 50(9), C164–C171 (2011).
[Crossref] [PubMed]

C. Amra, J. H. Apfel, and E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31(16), 3134–3151 (1992).
[Crossref] [PubMed]

C. Amra, C. Grèzes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. 32(28), 5492–5503 (1993).
[Crossref] [PubMed]

C. Amra, “From light scattering to the microstructure of thin-film multilayers,” Appl. Opt. 32(28), 5481–5491 (1993).
[Crossref] [PubMed]

C. Deumié, H. Giovannini, and C. Amra, “Ellipsometry of light scattering from multilayer coatings,” Appl. Opt. 35(28), 5600–5608 (1996).
[Crossref] [PubMed]

S. Maure, G. Albrand, and C. Amra, “Low-level scattering and localized defects,” Appl. Opt. 35(28), 5573–5582 (1996).
[Crossref] [PubMed]

C. Deumié, R. Richier, P. Dumas, and C. Amra, “Multiscale roughness in optical multilayers: atomic force microscopy and light scattering,” Appl. Opt. 35(28), 5583–5594 (1996).
[Crossref] [PubMed]

Class. Quantum Gravity (1)

H. Miao, H. Yang, R. X. Adhikari, and Y. Chen, “Quantum limits of interferometer topologies for gravitational radiation detection,” Class. Quantum Gravity 31(16), 165010 (2014).
[Crossref]

in Physics of Thin Films. Advances in Research and Development (1)

J. M. Eastman, “Scattering by All- Dielectric Multilayer Bandpass Filters and Mirrors for Lasers,” in Physics of Thin Films. Advances in Research and Development 10, 167–226 (1978).

J. Opt. Am. (1)

P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Am. 71(9), 1115–1123 (1981).
[Crossref]

J. Opt. Soc. Am. (3)

J. M. Elson, “Diffraction and diffuse scattering from dielectric multilayers,” J. Opt. Soc. Am. 69(1), 48–54 (1979).
[Crossref]

J. M. Elson, “Surface scattering in optical interference coatings,” J. Opt. Soc. Am. 66, 230–234 (1974).

J. M. Eastman, “Surface Scattering in Optical Interference Coatings,” J. Opt. Soc. Am. 66, 230–234 (1974).

J. Opt. Soc. Am. A (3)

Opt. Eng. (2)

K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18(2), 104–115 (1979).
[Crossref]

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Tech. Phys. (1)

V. K. Sakharov, “Model of lock-in in a ring laser and a semiconductor laser gyro,” Tech. Phys. 56(8), 1135–1141 (2011).
[Crossref]

Other (1)

P. Beckmann and A. Spizzichino, “The Scattering of Electromagnetic Waves from Rough Surfaces,” (Pergamon Press, 1963).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 Schematic illustration of the irregularity at interface i in a multilayer. The z axis is normal to all interfaces. The top-surface, interface 0, is assumed to be in air, and the substrate is semi-infinite.
Fig. 2
Fig. 2 Directions of waves of incidence, reflection and transmission scattering. θ0 is the polar incident angle. θs and ϕs characterize a particular scattering direction. θs is from the sample normal and ϕs is the azimuthal angle.
Fig. 3
Fig. 3 Directions of the initially scattered fields at interface i. θi and θsi are the propagation angles (given by Snell’s law) of the main fields and the scattered fields in medium i, respectively.
Fig. 4
Fig. 4 Illustration for notation of the initially scattered field propagation in a multilayer.
Fig. 5
Fig. 5 Angular surface scattering calculated for (a) a narrow-band filter, (b) an anti-reflective coating, and (c) a 24-layer mirror. The substrate and the incident medium have indices ns = 1.52 and n0 = 1.0. The design wavelength λ0 is equal to the illumination wavelength: λ0 = λ = 633 nm. The incident angles θ0 are in Fig. 5(a), 0°; Fig. 5(b), 0°; Fig. 5(c), 30°. The design angles are equal to the incident angles: i0 = θ0. The angular range from −90° to 90°corresponds to reflection scattering and the region of 90° < θs < 180° and −180° < θs < −90°correspond to transmission scattering. Note that all interfaces within a design have the same roughness.
Fig. 6
Fig. 6 Drawings for the ARS of the 24-layer reflector predicted by the NSS and BVS theories.
Fig. 7
Fig. 7 Angular surface scattering in reflection for a single layer coating at normal illumination.
Fig. 8
Fig. 8 Results of (a) ARS and (b) normalized ARS calculations for the single layer coating with different interfacial structures represented as different roughness values.
Fig. 9
Fig. 9 Results of (a) ARS and (b) normalized ARS calculations for the 24-layer mirror.

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

{ E i ( s ) = E i 1 ( s ) { E i 1 ( 0 ) } + E i 2 ( s ) { E i 4 ( 0 ) } E i ( s ) + = E i 3 ( s ) { E i 1 ( 0 ) } + E i 4 ( s ) { E i 4 ( 0 ) } ,
{ E i ( d ) = P i 1 { E i ( s ) + } + P i 2 { E i ( s ) } E i ( d ) + = P i 3 { E i ( s ) + } + P i 4 { E i ( s ) } ,
E i k ( s ) ( θ s , ϕ s ; θ 0 ) = a i k ( θ 0 ) ρ i k ( f x , f y ) ,
ρ i k ( f x , f y ) = 1 A s F { exp [ i 2 π λ η i k h i ( x , y ) ] } ,
f x = sin θ s cos θ s sin θ 0 λ , f y = sin θ s sin ϕ s λ ,
{ a i 1 = r i 1 ( 0 ) E i 1 ( 0 ) a i 2 = t i 2 ( 0 ) E i 4 ( 0 ) a i 3 = t i 1 ( 0 ) E i 1 ( 0 ) a i 4 = r i 2 ( 0 ) E i 4 ( 0 ) , { η i 1 = N i ( 0 ) + N i η i 2 = ( N j ( 0 ) N i ) η i 3 = N i ( 0 ) N j η i 4 = ( N j ( 0 ) + N j ) ,
r i 1 ( 0 ) = N i ( 0 ) N j ( 0 ) N j ( 0 ) + N i ( 0 ) , r i 2 ( 0 ) = N j ( 0 ) N i ( 0 ) N j ( 0 ) + N i ( 0 ) ,
t i 1 ( 0 ) = 2 N i ( 0 ) N j ( 0 ) + N i ( 0 ) , t i 2 ( 0 ) = 2 N j ( 0 ) N j ( 0 ) + N i ( 0 ) ,
N i ( 0 ) = ( n i 2 n 0 2 sin 2 θ 0 ) 1 / 2 , N i = ( n i 2 n 0 2 sin 2 θ s ) 1 / 2 .
[ E i 1 ( 0 ) E i 2 ( 0 ) ] = I i ( 0 ) [ E i 3 ( 0 ) E i 4 ( 0 ) ] = I i ( 0 ) m = i + 1 p T m ( 0 ) I m ( 0 ) [ E s ( 0 ) 0 ] ,
T m ( 0 ) = [ exp ( i φ m ( 0 ) ) 0 0 exp ( i φ m ( 0 ) ) ] ,
I m ( 0 ) = 1 d m 3 t m 1 ( 0 ) [ 1 d m 4 r m 1 ( 0 ) d m 1 r m 1 ( 0 ) d m 3 d m 2 t m 2 ( 0 ) t m 1 ( 0 ) + d m 4 d m 1 r m 1 ( 0 ) r m 1 ( 0 ) ]
d m 1 = exp { [ 4 π λ N m ( 0 ) σ m , r e l ] 2 } ,
d m 2 = d m 3 = exp { [ 2 π λ ( N m ( 0 ) N m + 1 ( 0 ) ) σ m , r e l ] 2 } ,
d m 4 = exp { [ 4 π λ N m + 1 ( 0 ) σ m , r e l ] 2 } .
σ m , r e l = [ 1 / λ 1 / λ f y = 1 / λ 2 f x 2 1 / λ 2 f x 2 P S D m ( f x , f y ) d f x d f y ] 1 / 2 .
P i k ( E ) = b i k E ,
b i 1 = { 1 i = 0 1 A i W i i 0 , b i 2 = { D j A j C j W j exp ( i φ j ) i p 0 i = p ,
b i 4 = { 1 C j W j i p 1 i = p , b i 3 = { 0 i = 0 B i A i C i W i exp ( i φ i ) i 0 .
W i = 1 B i D i A i C i exp ( i 2 φ i ) ,
[ A i B i ] = m = 1 i 1 1 r i m [ exp ( i φ i m + 1 ) 0 0 exp ( i φ i m + 1 ) ] [ 1 r i m r i m 1 ] ,
[ C i D i ] = m = i p 1 1 + r m [ exp ( i φ m ) 0 0 exp ( i φ m ) ] [ 1 r m r m 1 ] ,
E i ( d ) ± = k = 1 4 c i k ± ρ i k ,
c i 1 = a i 1 b i 1 , c i 2 = a i 2 b i 1 , c i 3 = a i 3 b i 2 , c i 4 = a i 4 b i 2 , c i 1 + = a i 1 b i 3 , c i 2 + = a i 2 b i 3 , c i 3 + = a i 3 b i 4 , c i 4 + = a i 4 b i 4 .
E ( d ) ± = i = 0 p E i ( d ) ± = i = 0 p k = 1 4 c i k ± ρ i k ,
| E ( d ) ± | 2 = | i = 0 p k = 1 4 c i k ± ρ i k | 2 .
| E ( d ) ± | 2 = | F { E ˜ ( d ) ± } | 2 = F { E ˜ ( d ) ± E ˜ ( d ) ± } = F { i = 0 p j = 0 p k = 1 4 l = 1 4 E ˜ i k ( d ) ± E ˜ j l ( d ) ± } ,
E ˜ i k ( d ) ± = c i k ± F 1 { ρ i k } = 1 A s c i k ± exp [ i 2 π λ η i k h i ( x , y ) ] ,
ε { | E ( d ) ± | 2 } = F { i = 0 p j = 0 p k = 1 4 l = 1 4 ε { E ˜ i k ( d ) ± E ˜ j l ( d ) ± } } ,
ε { E ˜ i k ( d ) ± E ˜ j l ( d ) ± } = 1 A s c i k ± c j l ± ε { exp { i 2 π λ [ η i k h i + η j l h j ] } } ,
H i k ; j l = ε { exp [ i 2 π λ ( η i k h i + η j l h j ) ] } = exp { i 2 π λ [ η i k ε { h i } + η j l ε { h j } ] } exp [ 2 π 2 λ 2 ( η i k 2 σ i , r e l 2 2 η i k η j l A C V i j + η j l 2 σ j , r e l 2 ) ] ,
H i k ; j l ( x , y ; θ 0 , θ s ) = A i k ; j l ( θ 0 , θ s ) + B i k ; j l G i k ; j l ( x , y ; θ 0 , θ s ) ,
A i k ; j l = exp [ 2 π 2 ( η i k 2 σ i , r e l 2 + η j l 2 σ j , r e l 2 ) / λ 2 ] ,
B i k ; j l = 1 exp [ 2 π 2 ( η i k 2 σ i , r e l 2 + η j l 2 σ j , r e l 2 ) / λ 2 ] ,
G i k ; j l = exp { 2 π 2 λ 2 η i k 2 σ i , r e l 2 + η j l 2 σ j , r e l 2 η i k 2 σ i , t o t 2 + η j l 2 σ j , t o t 2 η i k η j l A C V i j } 1 exp { 2 π 2 ( η i k 2 σ i , r e l 2 + η j l 2 σ j , r e l 2 ) / λ 2 } 1 ,
ε { | E ( d ) ± | 2 } = 1 A s { i = 0 p j = 0 p k = 1 4 l = 1 4 c i k ± c j l ± F { B i k ; j l G i k ; j l } } .
A R S ± ( θ s , ϕ s ; θ i ) = cos θ s { i = 0 p j = 0 p k = 1 4 l = 1 4 K i k c i k ± K j l c j l ± F { B i k ; j l G i k ; j l } } ,
K i k ( θ i ) = B i k ; i k ( θ i ) ( f x = 1 / λ 1 / λ f y = 1 f x 2 1 f x 2 F { B i k ; i k G i k ; i k } d f x d f y ) 1 ,
A R S ( θ s , ϕ s ) = cos θ s | r 01 ( 0 ) E 01 ( 0 ) | 2 K 01 K 01 F { B 01 ; 01 G 01 ; 01 } = cos θ s R P i K F { B 01 ; 01 G 01 ; 01 } ,
B 01 ; 01 = 1 exp [ 4 π 2 λ 2 n 0 2 ( cos θ i + cos θ s ) 2 σ 0 , r e l 2 ] ,
G 01 ; 01 = exp { 4 π 2 λ 2 n 0 2 ( cos θ i + cos θ s ) 2 σ 0 , r e l 2 σ 0 , t o t 2 A C V 00 } 1 exp { π 2 n 0 2 ( cos θ i + cos θ s ) 2 σ 0 , r e l 2 / λ 2 } 1 .
A R S ± ( θ s , ϕ s ) = 4 π 2 cos θ s λ 2 { i = 0 p j = 0 p C i ± C j ± P S D i j ( f x , f y ) } ,
C i ± = k = 1 4 K i k η i k c i k ± ,
A R S ± | i = 0 p C i ± | 2 P S D ( f x , f y ) ,
A C V ( x , y ) = σ s 1 2 exp [ ( r / l c 1 ) 2 ] + σ s 2 2 exp ( | r / l c 2 | ) ,
A C V ( x , y ) = σ s 1 2 exp [ ( r / l c 1 ) 2 ] + σ s 2 2 exp [ ( r / l c 2 ) 2 ] ,
A C V ( x ) = σ t o t 2 exp [ ( x / l c ) 2 ] ,

Metrics