Abstract

Electromagnetic theories provide a tool to detect the origin of scattering in optical multilayers. Illumination and observation conditions that cause surface and bulk scatterings to have different behaviors are pointed out. Angular, wavelength, and polarization dependences are investigated for the location of structural irregularities at interfaces or in the bulk of a multilayer. Specific experiments can be designed.

© 1993 Optical Society of America

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References

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  1. 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]
  2. C. Amra, J. H. Apfel, E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992).
    [CrossRef] [PubMed]
  3. C. Amra, “From light scattering to the microstructure of thin film multilayers,” Appl. Opt. 32, 5481–5491 (1993).
    [CrossRef] [PubMed]
  4. J. M. Elson, J. P. Rahn, J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
    [CrossRef] [PubMed]
  5. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).
  6. This paper was originally presented at the joint session on scattering of the First Topical Meeting on Surface Roughness and Scattering and the Fifth Topical Meeting on Optical Interference Coatings that was held in Tucson, Ariz., June 1992.
  7. J. M. Elson, “Angle resolved light scattering from composite optical surfaces,” in Periodic Structures, Gratings, Moire Patterns, and Diffraction Phenomena I, C. H. Chin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–306 (1980).
  8. P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
    [CrossRef]
  9. 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).
  10. J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
    [CrossRef]
  11. 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 Media 1, 207–221 (1991).
    [CrossRef]
  12. S. Kassam, A. Duparré, K. Hehl, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
    [CrossRef] [PubMed]
  13. C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. A 10, 365–374 (1993).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), p. 453.
  15. J. M. Elson, J. P. Rahn, J. M. Bennett, “Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation-length, and roughness cross-correlation properties,” Appl. Opt. 22, 3207–3219 (1983).
    [CrossRef] [PubMed]
  16. R. D. Jacobson, S. R. Wilson, G. A. Al-Jumaily, J. R. McNeil, J. M. Bennett, L. Mattsson, “Microstructure characterization by angle-resolved scatter and comparison to measurements made by other techniques,” Appl. Opt. 31, 1426–1435 (1992).
    [CrossRef] [PubMed]
  17. H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1986).
    [CrossRef]
  18. A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor-deposited thin films,” Thin Solid Films 47, 219–223 (1977).
    [CrossRef]
  19. 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]
  20. Results from surface scattering were compared by these authors and show high agreement for multilayers at oblique incidence.
  21. P. Croce, L. Prod'homme, “Etude par diffusion lumineuse de la nature des surfaces de verre poli,” J. Opt. (Paris) 7, 121–132 (1976).
  22. J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D. dissertation (University of Rochester, Rochester, New York, 1974).
  23. H. E. Bennett, J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
    [CrossRef]
  24. C. Amra, “Minimizing scattering in multilayers: technique for searching optimal realization conditions,” in Laser Induced Damage in Optical Materials, 756, 265–271 (1987).
  25. C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 25, 2695–2702 (1986).
    [CrossRef] [PubMed]
  26. C. Amra, “Scattering distribution from multilayer mirrors— Theoretical research of a design for minimal losses,” in Laser Induced Damage in Optical Materials, 752, 594–602 (1986).

1993 (2)

1992 (3)

1991 (1)

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 Media 1, 207–221 (1991).
[CrossRef]

1987 (2)

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, “Minimizing scattering in multilayers: technique for searching optimal realization conditions,” in Laser Induced Damage in Optical Materials, 756, 265–271 (1987).

1986 (2)

C. Amra, “Scattering distribution from multilayer mirrors— Theoretical research of a design for minimal losses,” in Laser Induced Damage in Optical Materials, 752, 594–602 (1986).

C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 25, 2695–2702 (1986).
[CrossRef] [PubMed]

1984 (1)

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

1983 (1)

1981 (1)

1980 (1)

1977 (1)

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor-deposited thin films,” Thin Solid Films 47, 219–223 (1977).
[CrossRef]

1976 (1)

P. Croce, L. Prod'homme, “Etude par diffusion lumineuse de la nature des surfaces de verre poli,” J. Opt. (Paris) 7, 121–132 (1976).

1975 (1)

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]

1961 (1)

Albrand, G.

Al-Jumaily, G. A.

Amra, C.

C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. A 10, 365–374 (1993).
[CrossRef]

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

C. Amra, J. H. Apfel, E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992).
[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, “Minimizing scattering in multilayers: technique for searching optimal realization conditions,” in Laser Induced Damage in Optical Materials, 756, 265–271 (1987).

C. Amra, “Scattering distribution from multilayer mirrors— Theoretical research of a design for minimal losses,” in Laser Induced Damage in Optical Materials, 752, 594–602 (1986).

C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 25, 2695–2702 (1986).
[CrossRef] [PubMed]

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).

Apfel, J. H.

Bennett, H. E.

Bennett, J. M.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), p. 453.

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).

Bussemer, P.

S. Kassam, A. Duparré, K. Hehl, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
[CrossRef] [PubMed]

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 Media 1, 207–221 (1991).
[CrossRef]

Croce, P.

P. Croce, L. Prod'homme, “Etude par diffusion lumineuse de la nature des surfaces de verre poli,” J. Opt. (Paris) 7, 121–132 (1976).

Dirks, A. G.

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor-deposited thin films,” Thin Solid Films 47, 219–223 (1977).
[CrossRef]

Duparré, A.

Eastman, J. M.

J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D. dissertation (University of Rochester, Rochester, New York, 1974).

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. M. Bennett, “Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation-length, and roughness cross-correlation properties,” Appl. Opt. 22, 3207–3219 (1983).
[CrossRef] [PubMed]

J. M. Elson, J. P. Rahn, J. M. 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, Moire Patterns, and Diffraction Phenomena I, C. H. Chin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–306 (1980).

Flory, F.

Hehl, K.

S. Kassam, A. Duparré, K. Hehl, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
[CrossRef] [PubMed]

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 Media 1, 207–221 (1991).
[CrossRef]

Jacobson, R. D.

Kassam, S.

S. Kassam, A. Duparré, K. Hehl, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
[CrossRef] [PubMed]

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 Media 1, 207–221 (1991).
[CrossRef]

Leamy, H. J.

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor-deposited thin films,” Thin Solid Films 47, 219–223 (1977).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 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.

McNeil, J. R.

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]

Neubert, J.

Pelletier, E.

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

Porteus, J. O.

Prod'homme, L.

P. Croce, L. Prod'homme, “Etude par diffusion lumineuse de la nature des surfaces de verre poli,” J. Opt. (Paris) 7, 121–132 (1976).

Rahn, J. P.

Roche, P.

Wilson, S. R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), p. 453.

Appl. Opt. (7)

J. Opt. (Paris) (1)

P. Croce, L. Prod'homme, “Etude par diffusion lumineuse de la nature des surfaces de verre poli,” J. Opt. (Paris) 7, 121–132 (1976).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B. (1)

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]

Laser Induced Damage in Optical Materials (2)

C. Amra, “Minimizing scattering in multilayers: technique for searching optimal realization conditions,” in Laser Induced Damage in Optical Materials, 756, 265–271 (1987).

C. Amra, “Scattering distribution from multilayer mirrors— Theoretical research of a design for minimal losses,” in Laser Induced Damage in Optical Materials, 752, 594–602 (1986).

Phys. Rev. B (2)

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]

Thin Solid Films (1)

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapor-deposited thin films,” Thin Solid Films 47, 219–223 (1977).
[CrossRef]

Waves Random Media (1)

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 Media 1, 207–221 (1991).
[CrossRef]

Other (8)

H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1986).
[CrossRef]

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

This paper was originally presented at the joint session on scattering of the First Topical Meeting on Surface Roughness and Scattering and the Fifth Topical Meeting on Optical Interference Coatings that was held in Tucson, Ariz., June 1992.

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

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).

Results from surface scattering were compared by these authors and show high agreement for multilayers at oblique incidence.

J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D. dissertation (University of Rochester, Rochester, New York, 1974).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), p. 453.

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Figures (22)

Fig. 1
Fig. 1

Schematic view of (A) interface roughness and (B) bulk inhomogeneity inside a multilayer stack. Medium i lies between interfaces i − 1 and i, where 0 is the incident air medium.

Fig. 2
Fig. 2

Angular surface and bulk scatterings calculated for a semi-infinite glass medium at normal incidence (i = 0°). The interface roughness is δ = 1.12 nm, and the bulk inhomogeneity is Δn/n = 9.56 × 10−3. The illumination wavelength is λ = 633 nm. The spectra parameters are given in the text.

Fig. 3
Fig. 3

Same as Fig. 2, but illumination is at oblique incidence (i = 60°) with p-polarized light. The angular behaviors are strongly different for surface and bulk phenomena.

Fig. 4
Fig. 4

Wavelength variations of optical factor C (θ = 0°, λ) calculated at normal illumination (i = 0°) for surface and bulk scatterings relative to a semi-infinite glass medium. The spectral range is 400–800 nm. The scattering direction is θ = i = 0° (retroscattering). The vertical units are arbitrary, except for specular reflection, which is also plotted.

Fig. 5
Fig. 5

Surface and bulk scatterings calculated at normal illumination (i = 0°) for a 2H ZnS layer with a design wavelength of λ0 = 633 nm and a refractive index of nH = 2.3. The illumination wavelength is λ = λ0 = 633 nm. Surface scattering is calculated for the two extreme cases of correlated (α01 = 1) and uncorrelated (α01 = 0) surfaces. Roughnesses and inhomogeneity are δ0 = δ1 = 1.12 nm and Δn/n = 9.56 × 10−3, respectively. Partial correlation between roughnesses would allow us to superimpose the surface and the bulk curves.

Fig. 6
Fig. 6

Bulk scattering calculated for different half-wave layers (2L, 6L, 10L, and 20L) of a low-index (cryolite) material with a refractive index of nL = 1.3. 2L indicates an optical thickness equal to λ0/2, where λ0 = 633 nm is the design wavelength. The illumination wavelength is λ = λ0 = 633 nm. Bulk scattering increases with thickness.

Fig. 7
Fig. 7

Same as Fig. 6, but calculation is for the correlated (αij = 1) surface scattering that does not increase with thickness.

Fig. 8
Fig. 8

Same as Fig. 6, but calculation is for the uncorrelated (αij = 0) surface scattering for which the stack is almost nonexistent.

Fig. 9
Fig. 9

Angular surface and bulk scatterings calculated for a 2L SiO2 layer at oblique incidence (i = 56°) for pp polarization. The increase of scattering near specular reflection is characteristic of a bulk effect. The layer refractive index is nL = 1.49.

Fig. 10
Fig. 10

Same as Fig. 9, but calculation is for the angular polarization ratio η(θ) = IPP(θ)/ISS(θ). The origin of scattering can be easily detected, whatever the surface cross-correlation parameters.

Fig. 11
Fig. 11

Wavelength variations of surface and bulk retroscatterings (θ = 0°) calculated for a 10H ZnS layer at normal illumination. Specular reflection is also plotted. Correlated surface retroscattering is in phase with reflection and in phase opposition with uncorrelated surface scattering. Bulk scattering is in phase with uncorrelated surface scattering and may be confused with it.

Fig. 12
Fig. 12

Wavelength variations of polarization ratio η(θ = 60°, λ) calculated at normal illumination (i = 0°) for bulk scattering and uncorrelated surface scattering. The scattering direction under study is θ = 60°. The design is a 10L cryolite layer. Detection of the origin of scattering can be easily performed for low-index materials.

Fig. 13
Fig. 13

Angular surface and bulk scatterings calculated at normal illumination (i = 0°) for a 13-layer quarter-wave mirror with ZnS/cryolite materials. The design wavelength is equal to the illumination wavelength: λ0 = λ = 633 nm. Surface and bulk scatterings are calculated for the uncorrelated case (αij = 0). The interface roughnesses are equal to δ = δj = 1.12 nm. The bulk inhomogeneities are equal to Δn/n = (Δn/n)j = 9.56 × 10−3. The angular ranges 0°–90° and 90°–180° correspond to scattering by reflection and transmission, respectively.

Fig. 14
Fig. 14

Same as Fig. 13, but calculation is for oblique incidence (i = 60°) and for polarization ratio η(θ). Correlated surface scattering can be easily detected. On the other hand, uncorrelated surface scattering can be confused with bulk scattering.

Fig. 15
Fig. 15

Wavelength variations of correlated and uncorrelated surface retroscatterings (θ = i = 0°) from a 13-layer mirror at normal illumination. Specular reflection is also plotted and is in phase with correlated scattering. Correlated and uncorrelated surface scatterings are in phase opposition.

Fig. 16
Fig. 16

Same as Fig. 15, but calculation is for bulk scattering. Contrary to surface scattering, the spectral shape of bulk scattering is similar, whatever the cross-correlation parameters between inhomogeneities. Moreover, it is similar to the uncorrelated surface scattering.

Fig. 17
Fig. 17

Angular surface scattering calculated for a narrow-band filter at normal illumination (i = 0°) and for two polarization states (SS and PP) of the incident and the scattered lights. The design is HLHLH 6L HLHLH. The illumination wavelength is equal to the design wavelength: λ = λ0 = 633 nm. Each interface supports a roughness δj = δ = 0.29 nm. (A) Correlated roughnesses, (B) uncorrelated roughnesses. The low level at θ ≈ 0° of the correlated surface scattering is due to the fact that the scattered fields are in phase opposition by pairs.

Fig. 18
Fig. 18

Same as Fig. 17 but calculation is for bulk scattering. Cross correlation is relative to inhomogeneities that are given by (Δn/n)j = Δn/n = 3.02 × 10−4 in each layer.

Fig. 19
Fig. 19

Angular polarization ratio η(θ) calculated for surface and uncorrelated bulk scatterings at the illumination wavelength λ = λ0 = 633 nm. The design is the narrow-band filter of Figs. 17 and 18. The illumination incidence is i = 60°.

Fig. 20
Fig. 20

Wavelength variations of correlated and uncorrelated bulk scatterings from a narrow-band filter at normal illumination (i = 0°). The design is identical to that of Figs. 17 and 18. Specular reflection is also plotted. The spectral shape of bulk scattering is similar, whatever the cross-correlation parameters between inhomogeneities.

Fig. 21
Fig. 21

Same as Fig. 20, but calculation is for surface scattering whose spectral shape is strongly dependent on cross-correlation parameters.

Fig. 22
Fig. 22

Wavelength variations of polarization ratio η(θ = 57.5°, λ.) calculated for bulk and uncorrelated surface scatterings in a narrow-band filter at oblique illumination (i = 60°). The design is identical to that of Figs. 17 and 18. The origin of scattering can be detected.

Equations (52)

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

| h ( r ) / λ | 1 ,
| grad h | 1
FT ( grad h ) = j σ ĥ ,
| FT ( grad h ) | = σ | ĥ ( σ ) | 2 π λ | ĥ ( σ ) | 1 .
rot E = j ω μ H + M δ ( z ) ,
rot H = j ω E + J δ ( z ) .
( Y i Y i ) Ê i = Ĵ i Y i z M ̂ i = R ̂ i ,
( Y i Y i ) Ê i + = Ĵ i Y i z M ̂ i = R ̂ i + ,
R ̂ i ( S S ) = R ̂ i + ( S S ) = a i cos ϕ A y i 0 ,
R ̂ i ( S P ) = R ̂ i + ( S P ) = a i sin ϕ A y i 0 ,
R ̂ i ( P S ) = 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 ,
R ̂ i + ( P P ) = a i cos ϕ A x i 0 Y i b i σ A z i 0 ,
a i = j ω Δ i ĥ i = 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 .
| Δ / | = p ( r , z ) | 1 .
p ( r , z ) = p ( r ) exp ( υ z ) .
rot E = j ω μ H ,
rot H = j ω E + J .
( 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 ) ,
| Ê S S * ( 0 ) Ê S S * ( e i ) = p ̂ i cos ϕ ( 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 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 ( 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 ( ½ α i ) ( α i 2 cos ϕ + 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 ϒ i ± e i ) 1 ] / ϒ i ± ,
ξ i + = α i + α i 0 j υ i ,
ξ i = α i α i 0 j υ i ,
ϒ i + = ( α i α i 0 ) j υ i ,
ϒ i = ( α i + α i 0 ) j υ i ,
α i 0 = ( k i 2 σ 0 2 ) 1 / 2 .
G p + 1 ± = F p + 1 = 0 , F p + 1 + = 1 / ξ p + 1 + , F p + 1 = F p + 1 + A p + 1 + 0 .
I ± ( θ , ϕ ) = 4 π 2 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 ± α i j γ j j ,
δ j 2 = σ γ j j ( σ ) d σ .
( δ j / n j ) 2 = ( ¼ ) σ γ j j ( σ ) d σ .
γ j j ( σ ) = FT { δ g 2 exp [ ( τ / L g ) 2 ] + δ e 2 exp ( | τ / L e | ) } γ j j ( σ ) = ( 1 / 4 π ) ( δ g L g ) 2 exp [ ( σ L g / 2 ) 2 ] + ( 1 / 2 π ) ( δ e L e ) 2 [ 1 + ( σ L e ) 2 ] 3 / 2 .
α H H = α L L = 1 , α L H = α H L = 0 ,
α H H = α L L = α H L = α L H = | 0 for α i j = 0 1 for α i j = 1 .
δ e = 1 nm , L e = 2000 nm , δ g = 0.5 nm , L g = 200 nm , for the surface spectrum , δ e = 0.0171 , L e = 2000 nm , δ g = 0.00856 , L g = 200 nm , for the bulk spectrum .
ϕ = 0 , θ = n 2 ( n 2 sin 2 i ) n 2 + ( n 4 1 ) sin 2 i .
I ( σ ) = C ( σ ) γ ( σ ) , σ = 2 π λ sin θ .
δ m 2 = δ g 2 { 1 exp [ ( π L g / λ ) 2 ] }
D = 2 π θ = 0 π / 2 I ( θ ) sin θ d θ ( 1 / k 2 ) C ( σ = 0 ) δ m 2 .
L / λ 1 δ / λ s 1 .
I ( θ ) = ( 64 π 2 / λ 4 ) n 0 4 cos 2 ϕ γ ( θ ) f ( θ ) ,
f ( θ ) = | cos 2 θ for S S polarization 1 for P P polarization .
I 0 ( ) = 0 , I 1 ( ) = I .

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