Abstract

New formulas for calculating the coherent reflectance, coherent transmittance, and scattering losses that characterize nonabsorbing multilayer systems with randomly rough boundaries are presented. These formulas were derived within the framework of the scalar theory of diffraction. The principal attention is paid to the approximate formulas related to a multilayer system with boundaries mutually uncorrelated from the statistical point of view. The formulas for a system with identically rough boundaries and for a system equivalent to a single thin film with uncorrelated boundaries are also presented. For all the models of a multilayer system with randomly rough boundaries it is assumed that slopes on all boundaries are small. Spectral dependences of the reflectance, transmittance, and scattering losses of some multilayer systems calculated with the derived formulas are presented. Some numerical results are compared with those published elsewhere.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Blazey, “Light scattering by laser mirrors,” Appl. Opt. 6, 831–836 (1967).
    [Crossref] [PubMed]
  2. K. H. Guenther, H. L. Gruber, H. K. Pulker, “Morphology and light scattering of dielectric multilayer systems,” Thin Solid Films 34, 363–367 (1976).
    [Crossref]
  3. H. K. Pulker, “Optical losses in dielectric films,” Thin Solid Films 34, 343–347 (1976).
    [Crossref]
  4. O. Arnon, “Loss mechanism in dielectric optical interference devices,” Appl. Opt. 16, 2147–2151 (1977).
    [Crossref] [PubMed]
  5. J. Ebert, H. Pannhorst, H. Küster, H. Welling, “Scatter losses of broadband interference coatings,” Appl. Opt. 18, 818–822 (1979).
    [Crossref] [PubMed]
  6. J. M. Eastman, “Scattering by all-dielectric multilayer bandpass filters and mirrors for lasers,” in The Physics of Thin Films, G. Hass, M. H. Francombe, eds. (Academic, New York, 1978), Vol. 10, pp. 167–226.
  7. J. M. Elson, “Infrared light scattering from surfaces covered with multiple dielectric overlayers,” Appl. Opt. 16, 2872–2881 (1977).
    [Crossref] [PubMed]
  8. J. M. Elson, “Diffraction and diffuse scattering from dielectric multilayers,” J. Opt. Soc. Am. 69, 48–54 (1979).
    [Crossref]
  9. 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]
  10. S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
    [Crossref]
  11. P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
    [Crossref]
  12. 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]
  13. C. Amra, “Calculs et mesures de diffusion appliqué s a l’étude de la rugositeé dans les traitements optiques multicouches,” J. Opt. (Paris) 21, 83–98 (1990).
    [Crossref]
  14. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).
  15. I. Ohlífdal, “Reflectance of multilayer systems with randomly rough boundaries,” Opt. Commun. 71, 323–326 (1989).
    [Crossref]
  16. I. Ohlífdal, K. Navrátil, F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film—nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. 61, 1630–1639 (1971).
    [Crossref]
  17. I. Ohlífdal, K. Navrátil, F. Lukeš, “Reflection of light on a system of nonabsorbing isotropic film—nonabsorbing isotropic substrate with rough boundaries,” Opt. Commun. 3, 40–44 (1971).
    [Crossref]
  18. I. Ohlífdal, F. Lukeš, K. Navrátil, “The problem of surface roughness in ellipsometry and reflectometry,” J. Phys. (Paris) 38, C5-77–C5-88 (1977).
    [Crossref]
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  20. P. J. Chandley, W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
    [Crossref]
  21. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).
  22. A. Vašíč, Optics of Thin Films (North-Holland, Amsterdam, 1960).
  23. Z. Knittl, Optics of Thin Films (Wiley, London, 1976).
  24. I. Ohlídal, K. Navrátil, “Spectroscopic methods for optical analysis of thin films,” Folia Fac. Sci. Natur. Univ. Purkyn. Brunensis 25, Physica37, 5–83 (1984).
  25. I. Ohlídal, F. Lukeš, K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974).
    [Crossref]
  26. H. E. Bennett, “Specular reflectance of aluminized ground glass and the height distribution of surface irregularities,” J. Opt. Soc. Am. 53, 1389–1394 (1963).
    [Crossref]
  27. C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
    [Crossref]

1990 (1)

C. Amra, “Calculs et mesures de diffusion appliqué s a l’étude de la rugositeé dans les traitements optiques multicouches,” J. Opt. (Paris) 21, 83–98 (1990).
[Crossref]

1989 (1)

I. Ohlífdal, “Reflectance of multilayer systems with randomly rough boundaries,” Opt. Commun. 71, 323–326 (1989).
[Crossref]

1987 (1)

1984 (1)

I. Ohlídal, K. Navrátil, “Spectroscopic methods for optical analysis of thin films,” Folia Fac. Sci. Natur. Univ. Purkyn. Brunensis 25, Physica37, 5–83 (1984).

1981 (1)

1980 (1)

1979 (4)

S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
[Crossref]

J. Ebert, H. Pannhorst, H. Küster, H. Welling, “Scatter losses of broadband interference coatings,” Appl. Opt. 18, 818–822 (1979).
[Crossref] [PubMed]

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

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

1977 (3)

1976 (2)

K. H. Guenther, H. L. Gruber, H. K. Pulker, “Morphology and light scattering of dielectric multilayer systems,” Thin Solid Films 34, 363–367 (1976).
[Crossref]

H. K. Pulker, “Optical losses in dielectric films,” Thin Solid Films 34, 343–347 (1976).
[Crossref]

1975 (1)

P. J. Chandley, W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
[Crossref]

1974 (1)

I. Ohlídal, F. Lukeš, K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974).
[Crossref]

1971 (2)

I. Ohlífdal, K. Navrátil, F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film—nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. 61, 1630–1639 (1971).
[Crossref]

I. Ohlífdal, K. Navrátil, F. Lukeš, “Reflection of light on a system of nonabsorbing isotropic film—nonabsorbing isotropic substrate with rough boundaries,” Opt. Commun. 3, 40–44 (1971).
[Crossref]

1967 (1)

1963 (1)

Amra, C.

C. Amra, “Calculs et mesures de diffusion appliqué s a l’étude de la rugositeé dans les traitements optiques multicouches,” J. Opt. (Paris) 21, 83–98 (1990).
[Crossref]

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]

Arnon, O.

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

Bennett, H. E.

Bennett, J. M.

Blazey, R.

Bousquet, P.

Carniglia, C. K.

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

Chandley, P. J.

P. J. Chandley, W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
[Crossref]

Eastman, J. M.

J. M. Eastman, “Scattering by all-dielectric multilayer bandpass filters and mirrors for lasers,” in The Physics of Thin Films, G. Hass, M. H. Francombe, eds. (Academic, New York, 1978), Vol. 10, pp. 167–226.

Ebert, J.

Elson, J. M.

Flory, F.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Gourley, S. J.

S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
[Crossref]

Gruber, H. L.

K. H. Guenther, H. L. Gruber, H. K. Pulker, “Morphology and light scattering of dielectric multilayer systems,” Thin Solid Films 34, 363–367 (1976).
[Crossref]

Guenther, K. H.

K. H. Guenther, H. L. Gruber, H. K. Pulker, “Morphology and light scattering of dielectric multilayer systems,” Thin Solid Films 34, 363–367 (1976).
[Crossref]

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976).

Küster, H.

Lissberger, P. H.

S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
[Crossref]

Lukeš, F.

I. Ohlífdal, F. Lukeš, K. Navrátil, “The problem of surface roughness in ellipsometry and reflectometry,” J. Phys. (Paris) 38, C5-77–C5-88 (1977).
[Crossref]

I. Ohlídal, F. Lukeš, K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974).
[Crossref]

I. Ohlífdal, K. Navrátil, F. Lukeš, “Reflection of light on a system of nonabsorbing isotropic film—nonabsorbing isotropic substrate with rough boundaries,” Opt. Commun. 3, 40–44 (1971).
[Crossref]

I. Ohlífdal, K. Navrátil, F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film—nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. 61, 1630–1639 (1971).
[Crossref]

Navrátil, K.

I. Ohlídal, K. Navrátil, “Spectroscopic methods for optical analysis of thin films,” Folia Fac. Sci. Natur. Univ. Purkyn. Brunensis 25, Physica37, 5–83 (1984).

I. Ohlífdal, F. Lukeš, K. Navrátil, “The problem of surface roughness in ellipsometry and reflectometry,” J. Phys. (Paris) 38, C5-77–C5-88 (1977).
[Crossref]

I. Ohlídal, F. Lukeš, K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974).
[Crossref]

I. Ohlífdal, K. Navrátil, F. Lukeš, “Reflection of light on a system of nonabsorbing isotropic film—nonabsorbing isotropic substrate with rough boundaries,” Opt. Commun. 3, 40–44 (1971).
[Crossref]

I. Ohlífdal, K. Navrátil, F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film—nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. 61, 1630–1639 (1971).
[Crossref]

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

Ohlídal, I.

I. Ohlídal, K. Navrátil, “Spectroscopic methods for optical analysis of thin films,” Folia Fac. Sci. Natur. Univ. Purkyn. Brunensis 25, Physica37, 5–83 (1984).

I. Ohlídal, F. Lukeš, K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974).
[Crossref]

Ohlífdal, I.

I. Ohlífdal, “Reflectance of multilayer systems with randomly rough boundaries,” Opt. Commun. 71, 323–326 (1989).
[Crossref]

I. Ohlífdal, F. Lukeš, K. Navrátil, “The problem of surface roughness in ellipsometry and reflectometry,” J. Phys. (Paris) 38, C5-77–C5-88 (1977).
[Crossref]

I. Ohlífdal, K. Navrátil, F. Lukeš, “Reflection of light on a system of nonabsorbing isotropic film—nonabsorbing isotropic substrate with rough boundaries,” Opt. Commun. 3, 40–44 (1971).
[Crossref]

I. Ohlífdal, K. Navrátil, F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film—nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. 61, 1630–1639 (1971).
[Crossref]

Pannhorst, H.

Pelletier, E.

Pulker, H. K.

H. K. Pulker, “Optical losses in dielectric films,” Thin Solid Films 34, 343–347 (1976).
[Crossref]

K. H. Guenther, H. L. Gruber, H. K. Pulker, “Morphology and light scattering of dielectric multilayer systems,” Thin Solid Films 34, 363–367 (1976).
[Crossref]

Rahn, J. P.

Roche, P.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

Vašíc, A.

A. Vašíč, Optics of Thin Films (North-Holland, Amsterdam, 1960).

Welford, W. T.

P. J. Chandley, W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
[Crossref]

Welling, H.

Appl. Opt. (5)

Folia Fac. Sci. Natur. Univ. Purkyn. Brunensis (1)

I. Ohlídal, K. Navrátil, “Spectroscopic methods for optical analysis of thin films,” Folia Fac. Sci. Natur. Univ. Purkyn. Brunensis 25, Physica37, 5–83 (1984).

J. Opt. (Paris) (1)

C. Amra, “Calculs et mesures de diffusion appliqué s a l’étude de la rugositeé dans les traitements optiques multicouches,” J. Opt. (Paris) 21, 83–98 (1990).
[Crossref]

J. Opt. Soc. Am. (4)

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

J. Phys. (Paris) (1)

I. Ohlífdal, F. Lukeš, K. Navrátil, “The problem of surface roughness in ellipsometry and reflectometry,” J. Phys. (Paris) 38, C5-77–C5-88 (1977).
[Crossref]

Opt. Acta (1)

S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
[Crossref]

Opt. Commun. (2)

I. Ohlífdal, K. Navrátil, F. Lukeš, “Reflection of light on a system of nonabsorbing isotropic film—nonabsorbing isotropic substrate with rough boundaries,” Opt. Commun. 3, 40–44 (1971).
[Crossref]

I. Ohlífdal, “Reflectance of multilayer systems with randomly rough boundaries,” Opt. Commun. 71, 323–326 (1989).
[Crossref]

Opt. Eng. (1)

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

Opt. Quantum Electron. (1)

P. J. Chandley, W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
[Crossref]

Surf. Sci. (1)

I. Ohlídal, F. Lukeš, K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974).
[Crossref]

Thin Solid Films (2)

K. H. Guenther, H. L. Gruber, H. K. Pulker, “Morphology and light scattering of dielectric multilayer systems,” Thin Solid Films 34, 363–367 (1976).
[Crossref]

H. K. Pulker, “Optical losses in dielectric films,” Thin Solid Films 34, 343–347 (1976).
[Crossref]

Other (6)

J. M. Eastman, “Scattering by all-dielectric multilayer bandpass filters and mirrors for lasers,” in The Physics of Thin Films, G. Hass, M. H. Francombe, eds. (Academic, New York, 1978), Vol. 10, pp. 167–226.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

A. Vašíč, Optics of Thin Films (North-Holland, Amsterdam, 1960).

Z. Knittl, Optics of Thin Films (Wiley, London, 1976).

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 (14)

Fig. 1
Fig. 1

Schematic diagram of the detection of light flux reflected and scattered from a multilayer system with randomly rough boundaries: L, lens; f, focal length; S, upper boundary of the system; M, mean plane of the upper boundary; O, origin of the coordinate plane xy; 2X, dimension of the irradiated rectangle in the mean plane M corresponding to the x axis; P(xf,yf), a point in the focal plane of lens L; R0, distance between lens L and the mean plane of the upper boundary M; and k1 and k2 are the wave vector of the incident and/or scattered wave.

Fig. 2
Fig. 2

Schematic diagram of the nonabsorbing UMS: (a) general view, (b) local view on the part of the UMS (djL denotes the local thickness of the jth thin film).

Fig. 3
Fig. 3

Schematic diagram of the nonabsorbing IMS: (a) general view, (b) local view of the part of the IMS.

Fig. 4
Fig. 4

Schematic diagram of a nonabsorbing multilayer system consisting of two identical multilayer systems separated by non-absorbing uncorrelated space thin film: (a) general view, (b) the uncorrelated thin-film equivalent to the system plotted in (a).

Fig. 5
Fig. 5

Schematic diagram of the nonabsorbing UTF: (a) general view, (b) local view of two parts of this film.

Fig. 6
Fig. 6

Spectral dependences of the transmittances T 0 and Tc together with the spectral dependence of the scattering losses S for the system G(HL)32H(LH)3G: T 0 is the transmittance of the ideally smooth system, Tc and S are the coherent transmittance and scattering losses corresponding to σk = 5 nm (k = 1,2,…, 14) (for details see the text).

Fig. 7
Fig. 7

Spectral dependences of the reflectances ℛ0 and Rc together with the spectral dependence of the scattering losses S for the system (HL)9HG: ℛ0 is the reflectance of an ideally smooth system; Rc and S are the coherent reflectance and scattering losses corresponding to σk = 5 nm (k = 1,2,…, 20) (for details see the text).

Fig. 8
Fig. 8

Spectral dependences of ΔR = Rc − ℛ0 for the system 1(HL)5HG with σk = 2.5 nm (k = 1,2,…, 12): dashed curve, ΔR of Eastman6; solid curve, ΔR calculated by means of the formulas presented here (for details see the text).

Fig. 9
Fig. 9

Spectral dependences of ΔR = Rc − ℛ0 for the system 1LG with σk = 2 nm (k = 1,2) n L d ¯ L = 250 nm, nL = 1.38, n = 1.5, and n0 = 1. Crosses, ΔR of Carniglia27R calculated by Eastman’s formula6); circles, ΔR calculated by means of the formulas presented here (1LG is the UMS).

Fig. 10
Fig. 10

Spectral dependences of Tc and S for the system 1(HL)32H(LH)31 with σk = 0.2 nm (k =1,2,…, 14), n H d ¯ H = n L d ¯ L = λ 0 / 4, λ0 = 500 nm, nH = 2.3, nL = 1.35, n0 = n = 1: solid curve, Tc of Eastman6; medium-dashed curve, Tc calculated by the formulas presented here; short-dashed curve, S of Eastman6; long-dashed curve, S calculated by the formulas presented here [1(HL)32H(LH)31 system corresponds to the UMS].

Fig. 11
Fig. 11

Spectral dependence of the relative error ΔTc/Tc for the system 1(HL)32H(LH)31 with σk = 0.2 nm (k = 1,2,…, 14). The values of the optical parameters of this system are the same as above (see the caption of Fig. 10).

Fig. 12
Fig. 12

Spectral dependences of the relative error ΔRc/Rc for the systems 1(HL)5HG (solid curve) and 1(HL)3HG (dashed curve). The values of σk and the optical parameters are the same as above (see the caption of Fig. 8).

Fig. 13
Fig. 13

Spectral dependences of ΔRc and |ΔR | for the rough system 1(HL)5HG. The values of the optical parameters and σk of this system are the same as above (see the caption of Fig. 8).

Fig. 14
Fig. 14

Spectral dependences of Rc and ΔR/;Δℛ0 for the same system, 1(HL)5HG (for details see the caption of Fig. 8).

Equations (67)

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

Ê ( x f , y f ) = K ̂ X X Y Y Û ( x , y ) exp [ i ( 2 π / λ f ) ( x x f + y y f ) ] d x d y ,
K ̂ = i exp [ i ( π / λ f ) ( 1 R 0 / f ) ( x f 2 + y f 2 ) ] λ f ,
Û ( x , y ) = R ̂ A 0 exp [ i ( 4 π / λ ) n 0 η 1 ( x , y ) ] ,
R c = ( 4 X Y A 0 2 ) 1 Ê ( x f , y f ) Ê * ( x f , y f ) d x f d y f ,
R c = R ̂ exp [ i υ η 1 ( x , y ) ] R ̂ * exp [ i υ η 1 ( x , y ) ] ,
R ̂ = r 1 + r ¯ ̂ 2 exp ( i x 1 ) 1 + r 1 r ¯ ̂ 2 exp ( i x 1 ) ,
r ¯ 2 = r 2 + r ¯ ̂ 3 exp ( i x 2 ) 1 + r 2 r ¯ ̂ 3 exp ( i x 2 ) , r ¯ ̂ k = r k + r ¯ ̂ k + 1 exp ( i x k ) 1 + r k r ¯ ̂ k + 1 exp ( i x k ) , r ¯ ̂ N = r N + r N + 1 exp ( i x N ) 1 + r N r N + 1 exp ( i x N ) , r k = n k 1 n k n k 1 + n k ,
R ̂ = r 1 + r 2 exp ( i x 1 ) + r 3 exp i ( x 1 + x 2 ) + + r N + 1 exp i ( x 1 + x 2 + + x N ) .
R ̂ exp [ i υ η 1 ( x , y ) ] = r 1 exp ( υ 2 σ 2 / 2 ) + j = 1 N r j + 1 exp ( i q = 1 j x ¯ q ) exp ( 1 2 p = 1 j + 1 Q p 2 σ p 2 ) ,
exp ( i Q 1 η 1 ) = exp ( i Q 1 η 1 ) w ( η 1 ) d η 1 = exp ( Q 1 2 σ 1 2 2 ) ,
R c = r 1 2 exp ( υ 2 σ 1 2 ) + j = 1 N r j + 1 2 exp ( p = 1 j + 1 Q p 2 σ p 2 ) + 2 r 1 exp ( υ 2 σ 2 2 ) j = 1 N r j + 1 exp ( 1 2 p = 1 j + 1 Q p 2 σ p 2 ) × cos ( q = 1 j x ¯ q ) + 2 s = j + 1 N j = 1 N r j + 1 r s + 1 × exp [ ( p = 1 j + 1 Q p 2 σ p 2 + 1 2 p = j + 2 s + 1 Q p 2 σ p 2 ) ] cos ( q = j + 1 s x ¯ q ) .
T c = n n 0 T ̂ exp [ i υ η N + 1 ( x , y ) ] T ̂ * exp [ i υ η N + 1 ( x , y ) ] ,
T ̂ = t 1 t 2 t N + 1 exp [ i ( x 1 + x 2 + + x N ) / 2 ] .
t k = 2 n k 1 n k 1 + n k ,
T c = n n 0 p = 1 N + 1 t p 2 exp ( l = 1 N + 1 Q ¯ l 2 σ l 2 ) ,
S = 1 R c T c .
| 0 0 | < 1 ,
| T 0 T 0 | < 2 ,
R ̂ exp [ i υ η 1 ( x , y ) ] = r 1 exp [ i υ η 1 ( x , y ) ] + r ¯ ̂ 2 exp { i [ x 1 + υ η 1 ( x , y ) ] } 1 + r 1 r ¯ ̂ 2 exp ( i x 1 ) .
R ̂ exp [ i υ η 1 ( x , y ) ] = r 1 exp [ i υ η 1 ( x , y ) ] + r ¯ ̂ 2 exp { i [ x 1 + υ η 1 ( x , y ) ] } 1 + r 1 r ¯ ̂ 2 exp ( i x 1 ) .
R ̂ exp [ i υ η 1 ( x , y ) ] = R ̂ 0 , 0 + R ¯ ̂ 1 , 0 1 + r 1 R ¯ ̂ 1 , 1 ,
R ̂ 0 , 0 = r 1 exp [ i υ η 1 ( x , y ) ] = r 1 exp [ σ 1 2 υ 2 / 2 ] ,
R ¯ ̂ 1 , 0 = r ¯ ̂ 2 exp { i [ x 1 + υ η 1 ( x , y ) ] } = R ̂ 1 , 0 + R ¯ ̂ 2 , 0 1 + r 2 R ¯ ̂ 2 , 2 ,
R ¯ ̂ 1 , 1 = r ¯ ̂ 2 exp ( i x 1 ) = R ̂ 1 , 1 + R ¯ ̂ 2 , 1 1 + r 2 R ¯ ̂ 2 , 2 , R ¯ ̂ j , k = r ¯ ̂ j + 1 exp ( i p = k j x p ) = R ̂ j , k + R ¯ ̂ j + 1 , k 1 + r j + 1 R ¯ ̂ j + 1 , j + 1 ,
R ̂ j , k = r j + 1 exp ( i p = k j x p ) , j = 1 , 2 , , N , k = j , j 1 , , 0 .
R ̂ j , k = r j + 1 exp ( i q = k j x ¯ q ) exp ( 1 2 p = k j + 1 Q p 2 σ p 2 ) ,
T ̂ = t 1 t ¯ ̂ 2 exp ( i x 1 / 2 ) 1 + r 1 r ¯ ̂ 2 exp ( i x 1 ) ,
t ¯ ̂ 2 = t 2 t ¯ ̂ 3 exp ( i x 2 / 2 ) 1 + r 2 r ¯ ̂ 3 exp ( i x 2 ) , t ¯ ̂ k = t k t ¯ ̂ k + 1 exp ( i x k / 2 ) 1 + r k r ¯ ̂ k + 1 exp ( i x k ) , t ¯ ̂ N = t N t N + 1 exp ( i x N / 2 ) 1 + r N r N + 1 exp ( i x N ) .
T ̂ exp [ i υ η N + 1 ( x , y ) ] = t 1 T ¯ ̂ 1 1 + r 1 R ¯ ̂ 1 , 1 ,
T ¯ ̂ l = t ¯ ̂ l + 1 exp { i [ 1 2 p = 1 1 x p + υ η N + 1 ( x , y ) ] } = t l + 1 T ¯ ̂ l + 1 1 + r l + 1 R ¯ ̂ l + 1 , l + 1 , l = 1 , 2 , , N 1 .
T ¯ ̂ = T ̂ N = t N + 1 exp { i [ 1 2 p = 1 N x p + υ η N + 1 ( x , y ) ] } = t N + 1 exp ( i 2 p = 1 N x ¯ p ) exp ( 1 2 l = 1 N + 1 Q ¯ l 2 σ l 2 ) .
δ ̂ = Â / Ĵ ( Â / Ĵ ) ,
R ¯ c = Â / J Â / Ĵ * .
R ¯ c = R c + | δ ̂ | 2 + 2 Re [ δ ̂ * ( Â / Ĵ ) ] ,
Δ R c = δ ̂ | 2 + 2 Re [ δ * ( Â / Ĵ ) ] | .
δ ̂ = 1 Ĵ Â 1 + Ĵ Ĵ Ĵ Â Ĵ Â Ĵ Â Ĵ Ĵ 2 .
 Ĵ = r 1 exp ( i υ η 1 ) + r ¯ ̂ 2 exp [ i ( x 1 + υ η 1 ) ] + r 1 2 r ¯ ̂ 2 exp [ i ( x 1 + υ η 1 ) ] + r 1 r ¯ ̂ 2 2 exp [ i ( 2 x 1 + υ η 1 ) ] = R ̂ 0 , 0 + ( 1 + r 1 2 ) R ¯ ̂ 1 , 0 + r 1 ρ ¯ ̂ 1 , 0 ,
ρ ¯ ̂ 1 , 0 = r ¯ ̂ 2 2 exp [ i ( 2 x 1 + υ η 1 ) ] = ρ ̂ 1 , 0 + 2 r 2 R ¯ ̂ 2 , 2 , 1 , 0 + ρ ¯ ̂ 2 , 0 1 + 2 r 1 + R ¯ ̂ 2 , 2 + r 2 2 + ρ ¯ ̂ 2 , 2 ,
ρ ̂ 1 , 0 = r 1 2 exp [ i ( 2 x 1 + η 1 ) ] , R ¯ ̂ 2 , 2 , 1 , 0 = r ¯ ̂ 3 exp [ i ( x 2 + 2 x 1 + υ η 1 ) ] = R ̂ 2 , 2 , 1 , 0 + R ¯ ̂ 3 , 2 , 1 , 0 1 + r 3 R ¯ ̂ 3 , 3 , ρ ¯ ̂ 2 , 2 = r ¯ 3 3 exp ( i 2 x 2 ) = ρ ̂ 2 , 2 + 2 r 3 R ¯ ̂ 3 , 3 , 2 , 2 + ρ ¯ ̂ 3 , 2 1 + 2 r 3 R ¯ ̂ 3 , 3 + r 3 ρ ¯ ̂ 3 , 3 , ρ ¯ ̂ 2 , 0 = r ¯ 3 2 exp [ i ( 2 x 1 + 2 x 2 + υ η 1 ) = ρ ̂ 2 , 0 + 2 r 3 R ¯ ̂ 3 , 3 , 2 , 0 + ρ ̂ 3 , 0 1 + 2 r 3 R ¯ ̂ 3 , 3 + r 3 2 ρ ̂ 3 , 3 , R ¯ ̂ j 1 , j 2 + 1 , j 2 , k = r ¯ ̂ j 1 + 1 exp [ i ( q = j 2 + 1 j 1 x q + 2 s = k j 2 x s ) ] = R ̂ j 1 , j 2 + 1 , j 2 , k + R ¯ ̂ j 1 + 1 , j 2 + 1 , j 2 , k 1 + r j + 1 R ¯ ̂ j 1 + 1 , j 1 + 1 , j 1 = 2 , 3 , N , j 2 = 1 , 2 , , j 1 1 , k = 0 , 1 , 2 , , j 2 ; ρ ¯ ̂ j , k = r ¯ ̂ j 1 + 1 2 exp ( i 2 q = k j x q ) = ρ ̂ j , k + 2 r j + 1 + R ¯ ̂ j + 1 , j + 1 , j , k + ρ ¯ ̂ j + 1 , k 1 + 2 r j + 1 R ¯ ̂ j + 1 , j + 1 + r j + 1 2 ρ ¯ ̂ j + 1 , j + 1 , j = 2 , 3 , , N , k = 0 , j ; R ¯ ̂ N , j 2 + 1 , j 2 , k = R ̂ N , j 2 + 1 , j 2 , k , ρ ¯ ̂ N , k = ρ ¯ ̂ N , k R ¯ ̂ j 1 , j 2 + 1 , j 2 , k = r j 1 + 1 exp [ i ( q = j 2 + 1 j 1 x q + 2 s = k j 2 x s ) ] = r j 1 + 1 exp [ i ( q = j 2 + 1 j 1 x ¯ q + 2 s = k j 2 x ¯ s ) ] × exp ( 1 2 q = j 2 + 1 j 1 + 1 Q q 2 σ q 2 ) × exp ( 1 2 s = k j 2 + 1 D s 2 σ s 2 ) ,
ρ ̂ j , k = r j + 1 2 exp ( i 2 q = k j x q ) = r j + 1 2 exp ( i 2 q = k j x ¯ q ) exp ( 1 2 p = k j + 1 D p 2 σ p 2 )
Δ T c = | T ¯ c T c | = δ 1 | 2 + 2 Re [ δ 1 * ( Ĉ / Ĵ ) ] | ,
Ĉ = t 1 t ¯ ̂ 1 exp ( i x 1 / 2 ) exp ( i υ η N + 1 ) , δ ̂ 1 = Ĉ / Ĵ ( Ĉ / Ĵ ) .
δ ̂ 1 Ĉ Ĵ Ĉ Ĵ Ĵ 2 .
Ĉ Ĵ = t 1 T ¯ ̂ 1 + t 1 r 1 t 2 T ¯ ̂ 2 , 1 + t 2 τ ¯ ̂ 2 1 + 2 r 2 R ¯ ̂ 2 , 2 + r 2 2 ρ ¯ ̂ 2 , 2 ,
T ¯ ̂ 2 , 1 = t ¯ ̂ 3 exp { i [ ½ ( x 2 + 3 x 1 ) + υ η N + 1 ] } = t 3 T ¯ ̂ 3 , 1 1 + r 3 R ¯ ̂ 3 , 3 , τ ¯ ̂ 2 = t ¯ ̂ 3 r ¯ ̂ 3 exp { i 3 / 2 ( x 2 + x 1 ) + υ η N + 1 } = t 3 r 3 T ¯ ̂ 3 , 2 + t 3 τ ¯ ̂ 3 1 + 2 r 3 R ¯ ̂ 3 , 3 + r 3 2 ρ ¯ ̂ 3 , 3 , T ¯ ̂ j , k = t ¯ ̂ j + 1 exp { i [ 1 2 s = k + 1 j s x + 3 q = 1 k x q + υ η N + 1 ] } = t j + 1 T ¯ ̂ j + 1 , k 1 + r j + 1 R ¯ ̂ j + 1 , j + 1 , τ ¯ ̂ j = t ¯ ̂ j + 1 r ¯ ̂ j + 1 exp [ i ( 3 2 q = 1 j x q + υ η N + 1 ) ] = t j + 1 r j + 1 T ¯ ̂ j + 1 , j + t j + 1 + τ ¯ ̂ j + 1 1 + 2 r j + 1 R ¯ ̂ j + 1 , j + 1 + r j + 1 2 ρ ¯ ̂ j + 1 , j + 1 , j = 2 , 3 , , N 1 , k = 1 , 2 , , j 1 , τ ¯ ̂ N = τ ̂ N , T ¯ ̂ N , k = T ̂ N , k ; T ¯ ̂ N , k = t N + 1 exp [ i 1 2 ( p = k N x ¯ p + 3 q = 1 k = 1 x ¯ q ) ] × exp ( 1 2 p = k N + 1 Q ¯ p 2 σ p 2 ) exp ( 1 2 q = 1 k S q 2 σ q 2 ) , S q = 3 Q ¯ q , Q ¯ k = 2 π λ n k , S k = 6 π λ n k 1 , τ ̂ N = t N + 1 r N + 1 exp ( i 3 2 p = 1 N x ¯ p ) exp ( 1 2 l = 1 N + 1 S l 2 σ l 2 ) .
Ĉ Ĵ = t 1 T ̂ 1 + t 1 r 1 τ ̂ 1 .
x j = 4 π λ n j d ¯ j cos α ,
cos α = ( 1 + η x 2 + η y 2 ) 1 / 2 η x = η ( x , y ) x , η y = η ( x , y ) y ,
x j = 4 π λ n j d ¯ j = x ¯ j .
R ̂ exp [ i υ η ( x , y ) ] = R ̂ 0 exp ( υ 2 σ 2 / 2 λ 2 ) ,
T ̂ exp [ i υ η ( x , y ) ] = T ̂ 0 exp ( υ 2 σ 2 / 2 λ 2 ) ,
R c = 0 exp ( 16 π 2 n 0 2 σ 2 / λ 2 ) ,
T c = T 0 exp ( 4 π 2 ( n n 0 ) 2 σ 2 / λ 2 ) .
R ̂ = r ¯ ̂ I + r ¯ ̂ II ( t ¯ ̂ I t ¯ ̂ I r ¯ ̂ I r ¯ ̂ I ) exp ( i x ) 1 r ¯ ̂ II r ¯ ̂ I exp ( i x ) = r ¯ ̂ I + t ¯ ̂ I t ¯ ̂ I m = 0 r ¯ ̂ I m r ¯ ̂ II m + 1 exp [ i ( m + 1 ) x ] ,
x = x ¯ + 4 π λ n s [ η 1 ( x , y ) η 2 ( x , y ) ] ,
r ¯ ̂ II = r N 1 + 2 r ¯ ̂ N 1 + 3 exp ( i x ¯ N 1 + 2 ) 1 + r N 1 + 2 r ¯ ̂ N 1 + 3 exp ( i x ¯ N 1 + 2 ) ,
r ¯ ̂ N 1 + 3 = r N 1 + 3 r ¯ ̂ N 1 + 4 exp ( i x ¯ N 1 + 3 ) 1 + r N 1 + 3 r ¯ ̂ N 1 + 4 exp ( i x ¯ N 1 + 3 ) , r ¯ ̂ N 1 + N 2 + 1 = r N 1 + N 2 + 1 r N 1 + N 2 + 2 exp ( i x ¯ N 1 + N 2 + 1 ) 1 + r N 1 + N 2 + 1 r N 1 + N 2 + 2 exp ( i x ¯ N 1 + N 2 + 1 ) , r k = n k 1 n k n k 1 + n k ,
r ¯ ̂ I = r N 1 + 1 r ¯ ̂ N 1 exp ( i x ¯ N 1 ) 1 + r N 1 + 1 r ¯ ̂ N 1 exp ( i x ¯ N 1 ) ,
r ¯ ̂ N = r N 1 + r ¯ ̂ N 1 1 exp ( i x ¯ N 1 1 ) 1 + r N 1 r ¯ ̂ N 1 1 exp ( i x ¯ N 1 1 ) , r ¯ ̂ 2 = r 2 + r 1 exp ( i x ¯ 1 ) 1 + r 2 r 1 exp ( i x ¯ 1 ) , r k = r k for k = 1 , 2 , , N 1 + 1 .
t ¯ ̂ I = t N 1 + 1 t ¯ ̂ N 1 exp ( i x ¯ N 1 / 2 ) 1 + r N 1 + 1 r ¯ ̂ N 1 exp ( i x ¯ N 1 ) ,
t ¯ ̂ N 1 = t N 1 t ¯ ̂ N 1 1 exp ( i x ¯ N 1 1 / 2 ) 1 + r N 1 r ¯ ̂ N 1 1 exp ( i x ¯ N 1 1 ) , t ¯ ̂ 2 = t 2 t 1 exp ( i x ¯ 1 / 2 ) 1 + r 2 r 1 exp ( i x ¯ 1 ) , t k = 2 n k n k + n k 1 for k = N 1 + 1 , N 1 , , 1 .
R c = | r ¯ ̂ I | 2 exp ( υ 2 σ 1 2 ) + | t ̂ I t ̂ I | 2 m = 0 | r ̂ I | 2 m | r ¯ ̂ II | 2 ( m + 1 ) exp ( B m 2 σ 1 2 ) exp ( M m 2 σ 2 2 ) + 2 | t ¯ ̂ I t ̂ I | 2 l = m + 1 m = 0 | r ¯ ̂ I | m + l | r ¯ ̂ II | m + l + 2 exp [ ( B m 2 + B l 2 ) σ 1 2 / 2 ] exp [ ( M m 2 + M l 2 ) σ 2 2 / 2 ] × cos [ ( m l ) x ¯ + ( m l ) ( α I + α II ) ] + 2 | r ¯ ̂ I | | t ¯ ̂ I t ¯ ̂ I | exp ( υ 2 σ 2 / 2 ) m = 0 | r ¯ ̂ I | m | r ¯ ̂ II | m + 1 × exp ( B m 2 σ 1 2 / 2 ) exp ( M m 2 σ 2 2 / 2 ) cos [ ( m + 1 ) x ¯ + m α I + ( m + 1 ) α II α I + β I ] ,
T ̂ = t ¯ ̂ I t ¯ ̂ II exp ( i x / 2 ) 1 r ¯ ̂ II r ¯ ̂ I exp ( i x ) = t ¯ ̂ I t ¯ ̂ II m = 0 r ¯ ̂ II m r ¯ ̂ I m exp [ i ( m + ½ ) x ] ,
T c = n n 0 { | t ¯ ̂ I | 2 | t ¯ ̂ II | 2 m = 0 | r ¯ ̂ II | 2 m | r ¯ ̂ I | 2 m × exp ( A m 2 σ 1 2 ) exp ( c m 2 σ 2 2 ) + 2 | t ¯ ̂ I | 2 | t ¯ ̂ II | 2 l = m + 1 m = 0 | r ¯ ̂ II | m + l | r ¯ ̂ I | m + l × exp [ ( A m 2 + A l 2 ) σ 1 2 / 2 ] exp [ ( c m 2 + c l 2 ) σ 2 2 / 2 ] × cos [ ( m l ) x ¯ + ( m l ) ( α II + α I ) ] } ,
A m = 4 π λ ( m + 1 2 ) n s , C m = 4 π λ [ ( m + 1 2 ) n s + ( n 0 n ) / 2 ] .
η 1 ( x , y ) = η 2 ( x , y ) + d [ 1 + η 2 x 2 ( x , y ) + η 2 y 2 ( x , y ) ] 1 / 2 ,
η 1 ( x , y ) = η 2 ( x , y ) + d ,

Metrics