Abstract

This article presents an original method for the theoretical analysis of the intensity radiated by a dielectric waveguide with rough walls. The method is based on Maxwell’s equations under their covariant form written in nonorthogonal coordinate systems adapted to the geometry of the waveguide. The solutions are found by using a perturbation method starting from a guide with smooth walls. The statistical characteristics of the radiant intensity, the mean value, and the probability density function are analytically determined.

© 2010 Optical Society of America

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References

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  1. D. Marcuse, “Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function,” Bell Syst. Tech. J. 48, 3233–3242 (1969).
  2. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972).
  3. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).
  4. D. G. Hall, “Scattering of optical guided waves by waveguide surface roughness: A three-dimensional treatment,” Opt. Lett. 6, 601–603 (1981).
    [CrossRef] [PubMed]
  5. D. G. Hall, “In-plane scattering in planar optical waveguides: refractive-index fluctuations and surface roughness,” J. Opt. Soc. Am. A 2, 747–752 (1985).
    [CrossRef]
  6. G. H. Ames and D. G. Hall, “Attenuation in planar optical waveguides: comparison of theory and experiment,” IEEE J. Quantum Electron. 19, 845–853 (1983).
    [CrossRef]
  7. J. P. R. Lacey and F. P. Payne, “Radiation loss from planer waveguides with random wall imperfections,” IEE Proc.-J: Optoelectron. 137, 282–288 (1990).
    [CrossRef]
  8. F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. Quantum Electron. 26, 977–986 (1994).
    [CrossRef]
  9. P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  10. J. M. Elson, “Propagation in planar waveguides and the effects of wall roughness,” Opt. Express 9, 461–475 (2001).
    [CrossRef] [PubMed]
  11. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  12. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
    [CrossRef]
  13. J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings, a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  14. S. Afifi, “Propagation et diffraction d’une onde électromagnétique dans des structures apériodiques,” Ph.D. dissertation (Université Blaise Pascal, 1986).
  15. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]
  16. N. P. K. Cotter, T. W. Preist, and J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [CrossRef]
  17. G. Granet, J. P. Plumey, and J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
    [CrossRef]
  18. G. Granet, “Analysis of diffraction by surface-relief crossed grating with use of Chandezon method: application to multilayer crossed grating,” J. Opt. Soc. Am. A 15, 1121–1131 (1998).
    [CrossRef]
  19. E. Marcelin, “Application des équations de Maxwell covariantes à l’étude de la propagation dans les guides d’ondes coudés,” Ph.D. dissertation (Université Blaise Pascal, 1992).
  20. R. Dusséaux, P. Cornet, and P. Chambelin, “Etude de transformateurs plan-E dans un système de coordonnées non orthogonales,” Ann. Telecommun. 5–6, 311–323 (1999).
  21. R. Dusséaux and C. Faure, “Analyse de composants plan-E symétriques en guides d’onde à section rectangulaire,” Ann. Telecommun. 9–10, 834–855 (2002).
  22. R. Dusséaux and C. Faure, “Telegraphist’s equations for rectangular waveguides and analysis in nonorthogonal coordinates,” Prog. Electromagn. Res. PIER 88, 53–71 (2008).
    [CrossRef]
  23. F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.: Optoelectron. 141, 242–246 (1994).
    [CrossRef]
  24. T. M. Elfouhaily and C. A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
    [CrossRef]
  25. R. Dusséaux, C. Faure, J. Chandezon, and F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 12, 1271–1282 (1995).
    [CrossRef]
  26. S. Afifi and M. Diaf, “Scattering by random rough surfaces: study of direct and inverse problem,” Opt. Commun. 265, 11–17 (2006).
    [CrossRef]
  27. R. Dusséaux and R. de Oliveira, “Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: analytical formulae derived from the small perturbation method,” Waves Random Complex Media 17, 305–320 (2007).
    [CrossRef]
  28. S. Afifi and R. Dusséaux, “Statistical study of the radiation losses due to surface roughness for a dielectric slab deposited on a metal substrate,” Opt. Commun. 281, 4663–4670 (2008).
    [CrossRef]
  29. S. Afifi, R. Dusséaux, and R. de Oliveira, “Statistical distribution of the field scattered by rough layered interfaces: formulae derived from the small perturbation method,” Waves Random Complex Media 20, 1–22 (2010).
    [CrossRef]

2010 (1)

S. Afifi, R. Dusséaux, and R. de Oliveira, “Statistical distribution of the field scattered by rough layered interfaces: formulae derived from the small perturbation method,” Waves Random Complex Media 20, 1–22 (2010).
[CrossRef]

2008 (2)

S. Afifi and R. Dusséaux, “Statistical study of the radiation losses due to surface roughness for a dielectric slab deposited on a metal substrate,” Opt. Commun. 281, 4663–4670 (2008).
[CrossRef]

R. Dusséaux and C. Faure, “Telegraphist’s equations for rectangular waveguides and analysis in nonorthogonal coordinates,” Prog. Electromagn. Res. PIER 88, 53–71 (2008).
[CrossRef]

2007 (1)

R. Dusséaux and R. de Oliveira, “Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: analytical formulae derived from the small perturbation method,” Waves Random Complex Media 17, 305–320 (2007).
[CrossRef]

2006 (1)

S. Afifi and M. Diaf, “Scattering by random rough surfaces: study of direct and inverse problem,” Opt. Commun. 265, 11–17 (2006).
[CrossRef]

2004 (1)

T. M. Elfouhaily and C. A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

2002 (1)

R. Dusséaux and C. Faure, “Analyse de composants plan-E symétriques en guides d’onde à section rectangulaire,” Ann. Telecommun. 9–10, 834–855 (2002).

2001 (1)

1999 (1)

R. Dusséaux, P. Cornet, and P. Chambelin, “Etude de transformateurs plan-E dans un système de coordonnées non orthogonales,” Ann. Telecommun. 5–6, 311–323 (1999).

1998 (1)

1996 (1)

1995 (3)

1994 (3)

L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
[CrossRef]

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.: Optoelectron. 141, 242–246 (1994).
[CrossRef]

F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. Quantum Electron. 26, 977–986 (1994).
[CrossRef]

1992 (1)

E. Marcelin, “Application des équations de Maxwell covariantes à l’étude de la propagation dans les guides d’ondes coudés,” Ph.D. dissertation (Université Blaise Pascal, 1992).

1990 (1)

J. P. R. Lacey and F. P. Payne, “Radiation loss from planer waveguides with random wall imperfections,” IEE Proc.-J: Optoelectron. 137, 282–288 (1990).
[CrossRef]

1986 (1)

S. Afifi, “Propagation et diffraction d’une onde électromagnétique dans des structures apériodiques,” Ph.D. dissertation (Université Blaise Pascal, 1986).

1985 (1)

1983 (1)

G. H. Ames and D. G. Hall, “Attenuation in planar optical waveguides: comparison of theory and experiment,” IEEE J. Quantum Electron. 19, 845–853 (1983).
[CrossRef]

1982 (1)

1981 (1)

1980 (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

1974 (1)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).

1972 (1)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972).

1969 (1)

D. Marcuse, “Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function,” Bell Syst. Tech. J. 48, 3233–3242 (1969).

1941 (1)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Afifi, S.

S. Afifi, R. Dusséaux, and R. de Oliveira, “Statistical distribution of the field scattered by rough layered interfaces: formulae derived from the small perturbation method,” Waves Random Complex Media 20, 1–22 (2010).
[CrossRef]

S. Afifi and R. Dusséaux, “Statistical study of the radiation losses due to surface roughness for a dielectric slab deposited on a metal substrate,” Opt. Commun. 281, 4663–4670 (2008).
[CrossRef]

S. Afifi and M. Diaf, “Scattering by random rough surfaces: study of direct and inverse problem,” Opt. Commun. 265, 11–17 (2006).
[CrossRef]

S. Afifi, “Propagation et diffraction d’une onde électromagnétique dans des structures apériodiques,” Ph.D. dissertation (Université Blaise Pascal, 1986).

Ames, G. H.

G. H. Ames and D. G. Hall, “Attenuation in planar optical waveguides: comparison of theory and experiment,” IEEE J. Quantum Electron. 19, 845–853 (1983).
[CrossRef]

Chambelin, P.

R. Dusséaux, P. Cornet, and P. Chambelin, “Etude de transformateurs plan-E dans un système de coordonnées non orthogonales,” Ann. Telecommun. 5–6, 311–323 (1999).

Chandezon, J.

G. Granet, J. P. Plumey, and J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

R. Dusséaux, C. Faure, J. Chandezon, and F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 12, 1271–1282 (1995).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings, a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
[CrossRef]

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Cornet, G.

Cornet, P.

R. Dusséaux, P. Cornet, and P. Chambelin, “Etude de transformateurs plan-E dans un système de coordonnées non orthogonales,” Ann. Telecommun. 5–6, 311–323 (1999).

Cotter, N. P. K.

de Oliveira, R.

S. Afifi, R. Dusséaux, and R. de Oliveira, “Statistical distribution of the field scattered by rough layered interfaces: formulae derived from the small perturbation method,” Waves Random Complex Media 20, 1–22 (2010).
[CrossRef]

R. Dusséaux and R. de Oliveira, “Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: analytical formulae derived from the small perturbation method,” Waves Random Complex Media 17, 305–320 (2007).
[CrossRef]

Diaf, M.

S. Afifi and M. Diaf, “Scattering by random rough surfaces: study of direct and inverse problem,” Opt. Commun. 265, 11–17 (2006).
[CrossRef]

Dupuis, M. T.

Dusséaux, R.

S. Afifi, R. Dusséaux, and R. de Oliveira, “Statistical distribution of the field scattered by rough layered interfaces: formulae derived from the small perturbation method,” Waves Random Complex Media 20, 1–22 (2010).
[CrossRef]

R. Dusséaux and C. Faure, “Telegraphist’s equations for rectangular waveguides and analysis in nonorthogonal coordinates,” Prog. Electromagn. Res. PIER 88, 53–71 (2008).
[CrossRef]

S. Afifi and R. Dusséaux, “Statistical study of the radiation losses due to surface roughness for a dielectric slab deposited on a metal substrate,” Opt. Commun. 281, 4663–4670 (2008).
[CrossRef]

R. Dusséaux and R. de Oliveira, “Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: analytical formulae derived from the small perturbation method,” Waves Random Complex Media 17, 305–320 (2007).
[CrossRef]

R. Dusséaux and C. Faure, “Analyse de composants plan-E symétriques en guides d’onde à section rectangulaire,” Ann. Telecommun. 9–10, 834–855 (2002).

R. Dusséaux, P. Cornet, and P. Chambelin, “Etude de transformateurs plan-E dans un système de coordonnées non orthogonales,” Ann. Telecommun. 5–6, 311–323 (1999).

R. Dusséaux, C. Faure, J. Chandezon, and F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 12, 1271–1282 (1995).
[CrossRef]

Elfouhaily, T. M.

T. M. Elfouhaily and C. A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

Elson, J. M.

Faure, C.

R. Dusséaux and C. Faure, “Telegraphist’s equations for rectangular waveguides and analysis in nonorthogonal coordinates,” Prog. Electromagn. Res. PIER 88, 53–71 (2008).
[CrossRef]

R. Dusséaux and C. Faure, “Analyse de composants plan-E symétriques en guides d’onde à section rectangulaire,” Ann. Telecommun. 9–10, 834–855 (2002).

R. Dusséaux, C. Faure, J. Chandezon, and F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 12, 1271–1282 (1995).
[CrossRef]

Granet, G.

G. Granet, “Analysis of diffraction by surface-relief crossed grating with use of Chandezon method: application to multilayer crossed grating,” J. Opt. Soc. Am. A 15, 1121–1131 (1998).
[CrossRef]

G. Granet, J. P. Plumey, and J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Guérin, C. A.

T. M. Elfouhaily and C. A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

Hall, D. G.

Lacey, J. P. R.

F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. Quantum Electron. 26, 977–986 (1994).
[CrossRef]

J. P. R. Lacey and F. P. Payne, “Radiation loss from planer waveguides with random wall imperfections,” IEE Proc.-J: Optoelectron. 137, 282–288 (1990).
[CrossRef]

Ladouceur, F.

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.: Optoelectron. 141, 242–246 (1994).
[CrossRef]

Lalanne, P.

Li, L.

Love, J. D.

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.: Optoelectron. 141, 242–246 (1994).
[CrossRef]

Marcelin, E.

E. Marcelin, “Application des équations de Maxwell covariantes à l’étude de la propagation dans les guides d’ondes coudés,” Ph.D. dissertation (Université Blaise Pascal, 1992).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972).

D. Marcuse, “Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function,” Bell Syst. Tech. J. 48, 3233–3242 (1969).

Maystre, D.

J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings, a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
[CrossRef]

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Molinet, F.

Morris, G.

Payne, F. P.

F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. Quantum Electron. 26, 977–986 (1994).
[CrossRef]

J. P. R. Lacey and F. P. Payne, “Radiation loss from planer waveguides with random wall imperfections,” IEE Proc.-J: Optoelectron. 137, 282–288 (1990).
[CrossRef]

Plumey, J. P.

G. Granet, J. P. Plumey, and J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Preist, T. W.

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Sambles, J. R.

Senden, T. J.

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.: Optoelectron. 141, 242–246 (1994).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Ann. Telecommun. (2)

R. Dusséaux, P. Cornet, and P. Chambelin, “Etude de transformateurs plan-E dans un système de coordonnées non orthogonales,” Ann. Telecommun. 5–6, 311–323 (1999).

R. Dusséaux and C. Faure, “Analyse de composants plan-E symétriques en guides d’onde à section rectangulaire,” Ann. Telecommun. 9–10, 834–855 (2002).

Bell Syst. Tech. J. (1)

D. Marcuse, “Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function,” Bell Syst. Tech. J. 48, 3233–3242 (1969).

IEE Proc.-J: Optoelectron. (1)

J. P. R. Lacey and F. P. Payne, “Radiation loss from planer waveguides with random wall imperfections,” IEE Proc.-J: Optoelectron. 137, 282–288 (1990).
[CrossRef]

IEE Proc.: Optoelectron. (1)

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.: Optoelectron. 141, 242–246 (1994).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. H. Ames and D. G. Hall, “Attenuation in planar optical waveguides: comparison of theory and experiment,” IEEE J. Quantum Electron. 19, 845–853 (1983).
[CrossRef]

J. Opt. (Paris) (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Commun. (2)

S. Afifi and M. Diaf, “Scattering by random rough surfaces: study of direct and inverse problem,” Opt. Commun. 265, 11–17 (2006).
[CrossRef]

S. Afifi and R. Dusséaux, “Statistical study of the radiation losses due to surface roughness for a dielectric slab deposited on a metal substrate,” Opt. Commun. 281, 4663–4670 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. Quantum Electron. 26, 977–986 (1994).
[CrossRef]

Prog. Electromagn. Res. (1)

R. Dusséaux and C. Faure, “Telegraphist’s equations for rectangular waveguides and analysis in nonorthogonal coordinates,” Prog. Electromagn. Res. PIER 88, 53–71 (2008).
[CrossRef]

Pure Appl. Opt. (1)

G. Granet, J. P. Plumey, and J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Waves Random Complex Media (2)

S. Afifi, R. Dusséaux, and R. de Oliveira, “Statistical distribution of the field scattered by rough layered interfaces: formulae derived from the small perturbation method,” Waves Random Complex Media 20, 1–22 (2010).
[CrossRef]

R. Dusséaux and R. de Oliveira, “Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: analytical formulae derived from the small perturbation method,” Waves Random Complex Media 17, 305–320 (2007).
[CrossRef]

Waves Random Media (1)

T. M. Elfouhaily and C. A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

Other (5)

S. Afifi, “Propagation et diffraction d’une onde électromagnétique dans des structures apériodiques,” Ph.D. dissertation (Université Blaise Pascal, 1986).

E. Marcelin, “Application des équations de Maxwell covariantes à l’étude de la propagation dans les guides d’ondes coudés,” Ph.D. dissertation (Université Blaise Pascal, 1992).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).

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Figures (7)

Fig. 1
Fig. 1

Asymmetric dielectric waveguide with nonparallel rough walls.

Fig. 2
Fig. 2

Statistical average of the total radiation intensity as a function of L.

Fig. 3
Fig. 3

Statistical average of the total radiation intensity as a function of θ 0 - E mode.

Fig. 4
Fig. 4

Statistical average of the total radiation intensity as a function of θ 0 - H mode.

Fig. 5
Fig. 5

Statistical average of the total radiation intensity as a function of q.

Fig. 6
Fig. 6

Statistical average of the radiation intensity as a function of L.

Fig. 7
Fig. 7

Probability density of radiation intensity for various values of the correlation coefficient q - H polarization, l a = 0.25 λ , l b = 0.2 λ , θ 0 = 72 ° , and θ = 60 ° .

Tables (1)

Tables Icon

Table 1 Boundary Conditions

Equations (131)

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R a a ( x ) = σ a 2   exp ( | x | l a ) ,
R b b ( x ) = σ b 2   exp ( | x | l b ) ,
R a b ( x ) = 2 q σ a σ b l a l b l a + l b [ exp ( x l a ) U ( x ) + exp ( x l b ) U ( + x ) ] .
R ̂ a a ( α ) = σ a 2 2 l a 1 + ( l a α ) 2 ,
R ̂ b b ( α ) = σ b 2 2 l b 1 + ( l b α ) 2 ,
R ̂ a b ( α ) = 2 q σ a σ b l a l b ( 1 + j l a α ) ( 1 j l b α ) .
{ E k x j E j x k = j k Z g j = 1 3 g i j H j Z H k x j Z H j x k = j k g j = 1 3 g i j E j } ,     ( i , j , k ) = ( 1 , 2 , 3 ) , ( 2 , 3 , 1 ) , ( 3 , 1 , 2 ) ,
x = x ,     u = u 0 y a ( x ) b ( x ) a ( x ) + u 0 ,     z = z .
y = a ( x ) u = 0 ,
y = b ( x ) + u 0 u = u 0 .
[ v x v u v z ] = A [ v x v y v z ] .
A = [ x / x y / x z / x x / u y / u z / u x / z y / z z / z ] = [ 1 d ( x , u ) 0 0 1 + h ( x ) 0 0 0 1 ] ,
h ( x ) = [ b ( x ) a ( x ) ] / u 0 ,     d ( x , u ) = u h ̇ ( x ) + a ̇ ( x ) ,    
h ̇ ( x ) = d h ( x ) / d x ,     a ̇ ( x ) = d a ( x ) / d x .
[ v x v u v z ] = G 1 [ v x v u v z ] .
G = A G c A t = [ g x x g x u g x z g u x g u u g u z g z x g z u g z z ] = [ 1 + d ( x , u ) 2 [ 1 + h ( x ) ] d ( x , u ) 0 [ 1 + h ( x ) ] d ( x , u ) [ 1 + h ( x ) ] 2 0 0 0 1 ] ,
G 1 = [ g x x g x u g x z g u x g u u g u z g z x g z u g z z ] = [ 1 d ( x , u ) / [ 1 + h ( x ) ] 0 d ( x , u ) / [ 1 + h ( x ) ] [ 1 + d ( x , u ) 2 ] / [ 1 + h ( x ) ] 2 0 0 0 1 ] .
g = det ( G ) = [ 1 + h ( x ) ] 2 .
[ Z H x Z H u ] = j g k [ g u u g x u g x u 1 ] [ u E z x E z ] ,
[ E x E u ] = j g k [ g u u g x u g x u 1 ] [ u Z H z x Z H z ] ,
u { g [ g u u u F ( x , u ) + g x u x F ( x , u ) ] } + x { g [ g x u u F ( x , u ) + x F ( x , u ) ] } + k 2 g F ( x , u ) = 0 ,
k G ( x , u ) = j g g u u u F ( x , u ) + j g g x u x F ( x , u ) ,
F ( x , u ) = F ( 0 ) ( x , u ) + F ( 1 ) ( x , u ) ,
G ( x , u ) = G ( 0 ) ( x , u ) + G ( 1 ) ( x , u ) .
R ( 0 ) F ( 0 ) ( x , u ) = 0 ,
k G ( 0 ) ( x , u ) = £ ( 0 ) F ( 0 ) ( x , u ) ,
R ( 0 ) = u 2 + x 2 + k 2 ,
£ ( 0 ) = j u .
R ( 1 ) F ( 0 ) ( x , u ) + R ( 0 ) F ( 1 ) ( x , u ) = 0 ,
k G ( 1 ) ( x , u ) = £ ( 1 ) F ( 0 ) ( x , u ) + £ ( 0 ) F ( 1 ) ( x , u ) .
R ( 1 ) = a ( x ) b ( x ) u 0 u ( u + x ) + x ( { u u 0 [ a ̇ ( x ) b ̇ ( x ) ] a ̇ ( x ) } u a ( x ) b ( x ) u 0 x ) k 2 a ( x ) b ( x ) u 0 ,
£ ( 1 ) = j a ( x ) b ( x ) u 0 u + j { u u 0 [ a ̇ ( x ) b ̇ ( x ) ] a ̇ ( x ) } x .
t g ( χ 20 u 0 ) = e 1 2 ( x ̃ 10 / χ 20 ) + e 3 2 ( x ̃ 30 / χ 20 ) 1 e 1 2 e 3 2 ( x ̃ 10 / χ 20 ) ( x ̃ 30 / χ 20 ) ,
x ̃ 10 = β 0 2 k 1 2 ,     χ 20 = k 2 2 β 0 2 ,    
x ̃ 30 = β 0 2 k 3 2 ,     β 0 = k 2   sin   θ 0 .
e 1 , 3 = 1     in   E   polarization ,
e 1 , 3 = n 2 / n 1 , 3     in   H   polarization .
F ̂ 1 ( 0 ) ( β , u ) = 2 π e 1   cos ( χ 2 u 0 φ ) exp [ x ̃ 1 ( u u 0 ) ] δ ( β β 0 ) ,
F ̂ 3 ( 0 ) ( β , u ) = 2 π e 3   cos   φ   exp ( x ̃ 3 u ) δ ( β β 0 ) ,
x ̃ 1 = β 2 k 1 2 ,     χ 2 = k 2 2 β 2 ,     x ̃ 3 = β 2 k 3 2 ,
cos   φ = [ 1 + e 3 4 ( x ̃ 3 2 / χ 2 2 ) ] 1 / 2 ,    
cos ( χ 2 u 0 φ ) = [ 1 + e 1 4 ( x ̃ 1 2 / χ 2 2 ) ] 1 / 2 ,
F ̂ 2 ( 0 ) ( β , u ) = 2 π   cos ( χ 2 u φ ) δ ( β β 0 ) .
N 2 x ¯ = Re ( N 2 x ) = 1 2 Re ( E 2 , u H 2 , z E 2 , z H 2 , u ) ,
N 2 x ¯ ( x , u ) = β 0 2 k 0 Z 0 | F 2 ( 0 ) ( x , u ) | 2 ,
P t = 0 1 + N 2 x ¯ ( x , u ) d u d z .
P t = β 0 4 k 0 Z 0 U 0 ,
U 0 = [ u 0 + e 1 2 1 x ̃ 10 1 + ( x ̃ 10 2 / χ 20 2 ) 1 + e 1 4 ( x ̃ 10 2 / χ 20 2 ) + e 3 2 1 x ̃ 30 1 + ( x ̃ 30 2 / χ 20 2 ) 1 + e 3 4 ( x ̃ 30 2 / χ 20 2 ) ] .
F ̂ m ( 1 ) ( β , u ) = K ̂ a , m ( β , u ) a ̂ ( β m β 0 ) + K ̂ b , m ( β , u ) b ̂ ( β m β 0 ) .
K ̂ a , 1 ( β , u ) = A ̂ a , 1 ( β ) exp ( j χ 1 u ) + e 1 x ̃ 10 ( u u 0 1 ) cos ( χ 20 u 0 φ 0 ) exp [ x ̃ 10 ( u u 0 ) ] ,
K ̂ b , 1 ( β , u ) = A ̂ b , 1 ( β ) exp ( j χ 1 u ) e 1 u u 0 x ̃ 10 cos ( χ 20 u 0 φ 0 ) exp [ x ̃ 10 ( u u 0 ) ] ,
K ̂ a , 3 ( β , u ) = A ̂ a , 3 ( β ) exp ( + j χ 3 u ) e 3 x ̃ 30 ( u u 0 1 ) cos   φ 0   exp ( x ̃ 30 u ) ,
K ̂ b , 3 ( β , u ) = A ̂ b , 3 ( β ) exp ( + j χ 3 u ) + e 3 u u 0 x ̃ 30   cos   φ 0   exp ( x ̃ 30 u ) .
K ̂ a , 2 ( β , u ) = A ̂ a , 2 ( ) ( β ) exp ( + j χ 2 u ) + A ̂ a , 2 ( + ) ( β ) exp ( j χ 2 u ) + χ 20 ( u u 0 1 ) sin ( χ 20 u φ 0 ) ,
K ̂ b , 2 ( β , u ) = A ̂ b , 2 ( ) ( β ) exp ( + j χ 2 u ) + A ̂ b , 2 ( + ) ( β ) exp ( j χ 2 u ) u u 0 χ 20   sin ( χ 20 u φ 0 ) ,
χ m ( β ) = k m 2 β 2 .
N m u ¯ = Re ( N m u ) = 1 2 Re ( E m , z H m , x E m , x H m , z ) .
N m u ¯ ( x , u ) = n m 2 Z 0 Re [ F m ( x , u ) G m ( x , u ) ] .
P r , m = n m 2 Z 0 Re [ + F m ( 1 ) ( x , u ) G m ( 1 ) ( x , u ) d x ] .
P r , m = n m 4 π Z 0 Re [ + F ̂ m ( 1 ) ( β , u ) G ̂ m ( 1 ) ( β , u ) d β ] .
F ̂ 1 ( 1 ) ( β , u ) = A ̂ 1 ( β ) exp ( j χ 1 u ) ,
F ̂ 3 ( 1 ) ( β , u ) = A ̂ 3 ( β ) exp ( + j χ 3 u ) ,
G ̂ 1 ( 1 ) ( β , u ) = χ 1 ( β ) k 1 F ̂ 1 ( 1 ) ( β , u ) ,
G ̂ 3 ( 1 ) ( β , u ) = χ 3 ( β ) k 3 F ̂ 3 ( 1 ) ( β , u ) ,
A ̂ m ( β ) = A ̂ a , m ( β ) a ̂ ( β β 0 ) + A ̂ b , m ( β ) b ̂ ( β β 0 ) .
P r = 1 4 π Z 0 k 0 π / 2 + π / 2 k 1 2 cos 2 θ 1 | F ̂ 1 ( 1 ) ( k 1   sin   θ 1 , u ) | 2 d θ 1 + 1 4 π Z 0 k 0 π / 2 + π / 2 k 3 2 cos 2 θ 3 | F ̂ 3 ( 1 ) ( k 3   sin   θ 3 , u ) | 2 d θ 3 .
P r = 1 4 π Z 0 k 0 π / 2 + π / 2 k 1 2 cos 2 θ 1 | A ̂ 1 ( 1 ) ( k 1   sin   θ 1 ) | 2 d θ 1 + 1 4 π Z 0 k 0 π / 2 + π / 2 k 3 2 cos 2 θ 3 | A ̂ 3 ( 1 ) ( k 3   sin   θ 3 ) | 2 d θ 3 .
I r , m ( θ m ) = n m 2 k 0 cos 2 θ m 4 π Z 0 | A ̂ m ( 1 ) ( k m   sin   θ m ) | 2 .
η r , m ( θ m ) = I r , m ( θ m ) P t / Δ α = T m ( θ m ) L | A ̂ m ( 1 ) ( k m   sin   θ m ) | 2 ,
T m ( θ m ) = 2 n m 2 k 0 cos 2 θ m n 2 U 0   sin   θ 0 .
A ̂ m ( 1 ) ( β ) = A ̂ a , m ( β ) a ̂ ( β β 0 ) + A ̂ b , m ( β ) b ̂ ( β β 0 ) = 0.
σ R m 2 ( β ) = Re 2 [ A ̂ m ( 1 ) ( β ) ] = Re 2 [ A ̂ a , m ( β ) ] Re 2 [ a ̂ ( β β 0 ) ] + Im 2 [ A ̂ a , m ( β ) ] Im 2 [ a ̂ ( β β 0 ) ] + Re 2 [ A ̂ b , m ( β ) ] Re 2 [ b ̂ ( β β 0 ) ] + Im 2 [ A ̂ b , m ( β ) ] Im 2 [ b ̂ ( β β 0 ) ] + 2   Re [ A ̂ a , m ( β ) ] Re [ A ̂ b , m ( β ) ] Re [ a ̂ ( β β 0 ) ] Re [ b ̂ ( β β 0 ) ] + 2   Im [ A ̂ a , m ( β ) ] Im [ A ̂ b , m ( β ) ] Im [ a ̂ ( β β 0 ) ] Im [ b ̂ ( β β 0 ) ] + 2   Im [ A ̂ a , m ( β ) A ̂ b , m ( β ) ] Re [ a ̂ ( β β 0 ) ] Im [ b ̂ ( β β 0 ) ] ,
σ I m 2 ( β ) = Im 2 [ A ̂ m ( 1 ) ( β ) ] = Re 2 [ A ̂ a , m ( β ) ] Im 2 [ a ̂ ( β β 0 ) ] + Im 2 [ A ̂ a , m ( β ) ] Re 2 [ a ̂ ( β β 0 ) ] + Re 2 [ A ̂ b , m ( β ) ] Im 2 [ b ̂ ( β β 0 ) ] + Im 2 [ A ̂ b , m ( β ) ] Re 2 [ b ̂ ( β β 0 ) ] + 2   Im [ A ̂ a , m ( β ) ] Im [ A ̂ b , m ( β ) ] Re [ a ̂ ( β β 0 ) ] Re [ b ̂ ( β β 0 ) ] + 2   Re [ A ̂ a , m ( β ) ] Re [ A ̂ b , m ( β ) ] Im [ a ̂ ( β β 0 ) ] Im [ b ̂ ( β β 0 ) ] + 2   Im [ A ̂ a , m ( β ) A ̂ b , m ( β ) ] Re [ a ̂ ( β β 0 ) ] Im [ b ̂ ( β β 0 ) ] ,
Γ R m I m ( β ) = Re [ A ̂ m ( 1 ) ( β ) ] Im [ A ̂ m ( 1 ) ( β ) ] = Re [ A ̂ a , m ( β ) ] Im [ A ̂ a , m ( β ) ] { Re 2 [ a ̂ ( β β 0 ) ] Im 2 [ a ̂ ( β β 0 ) ] } + Re [ A ̂ b , m ( β ) ] Im [ A ̂ b , m ( β ) ] { Re 2 [ b ̂ ( β β 0 ) ] Im 2 [ b ̂ ( β β 0 ) ] } + Im [ A ̂ a , m ( β ) A ̂ b , m ( β ) ] { Re [ a ̂ ( β β 0 ) ] Re [ b ̂ ( β β 0 ) ] Im [ a ̂ ( β β 0 ) ] Im [ b ̂ ( β β 0 ) ] } .
Re 2 [ c ̂ ( β ) ] = 1 2 L + L ( L | x | ) cos ( β x ) R c c ( x ) d x + 1 2 L + L ( L | x | ) sinc [ β ( L | x | ) ] R c c ( x ) d x ,
Im 2 [ c ̂ ( β ) ] = 1 2 L + L ( L | x | ) cos ( β x ) R c c ( x ) d x 1 2 L + L ( L | x | ) sinc [ β ( L | x | ) ] R c c ( x ) d x ,
Re [ a ̂ ( β ) ] Re [ b ̂ ( β ) ] = 1 2 L + L ( L | x | ) cos ( β x ) R a b ( x ) d x + 1 2 L + L ( L | x | ) sinc [ β ( L | x | ) ] R a b ( x ) d x ,
Im [ a ̂ ( β ) ] Im [ b ̂ ( β ) ] = 1 2 L + L ( L | x | ) cos ( β x ) R a b ( x ) d x 1 2 L + L ( L | x | ) sinc [ β ( L | x | ) ] R a b ( x ) d x ,
Re [ a ̂ ( β ) ] Im [ b ̂ ( β ) ] = Im [ a ̂ ( β ) ] Re [ b ̂ ( β ) ] = 1 2 L + L ( L | x | ) sin ( β x ) R a b ( x ) d x .
η r , m ( θ m ) = T m ( θ m ) L | A ̂ m ( 1 ) ( k m   sin   θ m ) | 2 .
η r , m ( θ m ) = T m ( θ m ) L | A ̂ a , m ( k m   sin   θ m ) | 2 L + L ( L | x | ) R a a ( x ) cos [ ( k m   sin   θ m k 2   sin   θ 0 ) x ] d x + T m ( θ m ) L | A ̂ b , m ( k m   sin   θ m ) | 2 L + L ( L | x | ) R b b ( x ) cos [ ( k m   sin   θ m k 2   sin   θ 0 ) x ] d x + 2 T m ( θ m ) L Re { A ̂ a , m ( k m   sin   θ m ) A ̂ b , m ( k m   sin   θ m ) L + L ( L | x | ) exp [ j ( k m   sin   θ m k 2   sin   θ 0 ) x ] R a b ( x ) d x } .
η r , m ( θ m ) = T m ( θ m ) | A ̂ a , m ( k m   sin   θ m ) | 2 R ̂ a a ( k m   sin   θ m k 2   sin   θ 0 ) + T m ( θ m ) | A ̂ b , m ( k m   sin   θ m ) | 2 R ̂ b b ( k m   sin   θ m k 2   sin   θ 0 ) + 2 T m ( θ m ) Re [ A ̂ a , m ( k m   sin   θ m ) A ̂ b , m ( k m   sin   θ m ) R ̂ a b ( k m   sin   θ m k 2   sin   θ 0 ) ] .
η t = π / 2 + π / 2 η r , 1 ( θ 1 ) d θ 1 + π / 2 + π / 2 η r , 3 ( θ 3 ) d θ 3 .
η t = k 0 2 ( n 2 2 n 1 2 ) χ 20 2 x ̃ 10 k 2   sin   θ 0 ( 1 + x ̃ 10 u 0 2 ) π / 2 + π / 2 R ̂ a a ( k 1   sin   θ k 2   sin   θ 0 ) d θ ,
η t = ( n 2 2 n 1 2 ) χ 20 2 x ̃ 10 k 2   sin   θ 0 4 ( 1 + u 0 x ̃ 10 2 n 2 2 n 1 2 n 1 4 χ 20 2 + n 2 4 x ̃ 10 2 χ 20 2 + x ̃ 10 2 ) π / 2 + π / 2 ( n 2 2 + n 1 2 n 2 2 n 1 2 n 1 4 + n 2 4 n 1 4 n 2 4 sin 2 θ ) ( x ̃ 10 2 β 0 2 + n 1 2 sin 2 θ n 2 2 n 1 2 sin 2 θ ) R ̂ a a ( k 1   sin   θ k 2   sin   θ 0 ) d θ .
p R m I m ( x , y ) = 1 2 π σ R m σ I m 1 ρ m 2 exp [ 1 2 ( 1 ρ m 2 ) ( x 2 σ R m 2 2 ρ m x y σ R m σ I m + y 2 σ I m 2 ) ] ,
Re [ A ̂ m ( 1 ) ( β m ) ] = M   cos   Ψ ,
Im [ A ̂ m ( 1 ) ( β m ) ] = M   sin   Ψ ,
p M ψ ( m , ψ ) = m p R I ( m   cos   ψ , m   sin   ψ ) ,
p M ( m ) = π + π m p R I ( m   cos   ψ , m   cos   ψ ) d ψ .
p η r , m ( w ) = 1 2 w T m ( θ m ) / L p M ( w L / T m ( θ m ) ) ,     with   w 0.
p η r 1 , 3 ( w ) = exp [ w 4 T m ( θ m ) ( 1 ρ m 2 ) ( 1 σ ̃ R m 2 + 1 σ ̃ I m 2 ) ] 2 σ ̃ R m σ ̃ I m T m ( θ m ) 1 ρ m 2 I 0 ( w 4 T m ( θ m ) ( 1 ρ m 2 ) ( 1 σ ̃ R m 2 1 σ ̃ I m 2 ) 2 + 4 ρ m 2 σ ̃ R m 2 σ ̃ I m 2 ) ,
σ ̃ R m 2 = σ R m 2 / L ,
σ ̃ I m 2 = σ I m 2 / L ,
η r , m = T m ( θ m ) ( σ ̃ R m 2 + σ ̃ I m 2 ) ,
η r , m 2 ( θ m ) = T m 2 ( θ m ) [ 3 ( σ ̃ R m 4 + σ ̃ I m 4 ) + 2 ( 1 + 2 ρ m 2 ) σ ̃ R m 2 σ ̃ I m 2 ] ,
σ η r , m ( θ m ) = T m ( θ m ) 2 ( σ ̃ R m 4 + σ ̃ I m 4 + 2 ρ m 2 σ ̃ R m 2 σ ̃ I m 2 ) .
σ ̃ R , m 2 = σ ̃ I , m 2 = 1 2 | A ̂ a , m ( k m   sin   θ m ) | 2 R ̂ a a ( k m   sin   θ m k 2   sin   θ 0 ) + 1 2 | A ̂ b , m ( k m   sin   θ m ) | 2 R ̂ b b ( k m   sin   θ m k 2   sin   θ 0 ) + 2   Re [ A ̂ a , m ( k m   sin   θ m ) A ̂ b , m ( k m   sin   θ m ) R ̂ a b ( k m   sin   θ m k 2   sin   θ 0 ) ] .
p η r , m ( w ) = 1 2 σ ̃ R m 2 T m ( θ m ) exp [ w 2 σ ̃ R m 2 T m ( θ m ) ] , with     w 0 ,
p η r , m ( w ) = 1 η r , m exp ( w η r , m ) ,     with     w 0.
η r , m p ( θ m ) = η r , m ( θ m ) p p .
η r , m 2 ( θ m ) = 2 η r , m ( θ m ) 2 ,     σ η r , m 2 ( θ m ) = η r , m ( θ m ) 2 ,    
σ η r , m ( θ m ) = η r , m ( θ m ) .
k 1 G ̂ 1 ( 0 ) ( β , u ) = j 2 π e 1 x ̃ 1   cos ( χ 2 u 0 φ ) exp [ x ̃ 1 ( u u 0 ) ] δ ( β β 0 ) ,
k 3 G ̂ 3 ( 0 ) ( β , u ) = j 2 π e 3 x ̃ 3   cos   φ   exp ( x ̃ 3 u ) δ ( β β 0 ) .
k 2 G ̂ 2 ( 0 ) ( β , u ) = j 2 π χ 2   cos ( χ 2 u φ ) δ ( β β 0 ) .
k m G ̂ m ( 1 ) ( β , u ) = Q ̑ a , m ( β , u ) a ̂ ( β m β 0 ) + Q ̑ b , m ( β , u ) b ̂ ( β m β 0 ) .
Q ̑ a , 1 ( β , u ) = χ 1 A ̂ a , 1 ( β ) exp ( j χ 1 u ) + j e 1 ( k 1 2 β 0 β ) ( u u 0 1 ) cos ( χ 20 u 0 φ 0 ) exp [ x ̃ 10 ( u u 0 ) ] ,
Q ̑ b , 1 ( β , u ) = χ 1 A ̂ b , 1 ( β ) exp ( j χ 1 u ) j u u 0 e 1 ( k 1 2 β 0 β ) cos ( χ 20 u 0 φ 0 ) exp [ x ̃ 1 ( u u 0 ) ] ,
Q ̑ a , 3 ( β , u ) = χ 3 R 3 ( 10 ) ( β ) exp ( + j χ 3 u ) + j e 3 ( u u 0 1 ) ( k 3 2 β 0 β ) cos   φ 0   exp ( x ̃ 30 u ) ,
Q ̑ b , 3 ( β , u ) = χ 3 A ̂ b , 1 ( β ) exp ( + j χ 3 u ) j e 3 ( k 3 2 β 0 β ) u u 0 cos   φ 0   exp ( x ̃ 30 u ) .
Q ̑ a , 2 ( β , u ) = χ 2 A ̂ a , 2 ( ) ( β ) exp ( + j χ 2 u ) + χ 2 A ̂ a , 2 ( + ) ( β ) exp ( j χ 2 u ) + j ( u u 0 1 ) ( k 2 2 β 0 β ) cos ( χ 20 u φ 0 ) ,
Q ̑ b , 2 ( β , u ) = χ 2 A ̂ b , 2 ( ) ( β ) exp ( + j χ 2 u ) + χ 2 A ̂ b , 2 ( + ) ( β ) exp ( j χ 2 u ) j u u 0 ( k 2 2 β 0 β ) cos ( χ 20 u φ 0 ) .
A ̂ a , 1 ( E ) ( β ) = j k 0 2 χ 2 ( n 2 2 n 3 2 ) cos   φ 0 r E ( β ) exp ( + j χ 1 u 0 ) ,
A ̂ a , 1 ( H ) ( β ) = ( j β β 0 n 2 2 n 3 2 χ 3 x ̃ 30 ) n 1 n 2 χ 2 ( n 2 2 n 3 2 ) cos   φ 0 r H ( β ) exp ( + j χ 1 u 0 ) ,
A ̂ b , 1 ( E ) ( β ) = k 0 2 [ χ 3   sin ( χ 2 u 0 ) j χ 2   cos ( χ 2 u 0 ) ] ( n 2 2 n 1 2 ) r E ( β ) cos ( χ 20 u 0 φ 0 ) exp ( + j χ 1 u 0 ) ,
A ̂ b , 1 ( H ) ( β ) = { n 2 2 χ 3 [ β β 0   sin ( χ 2 u 0 ) χ 2 x ̃ 10   cos ( χ 2 u 0 ) ] j n 3 2 χ 2 [ β β 0   cos ( χ 2 u 0 ) + χ 2 x ̃ 10   sin ( χ 2 u 0 ) ] } n 2 ( n 2 2 n 1 2 ) cos ( χ 20 u 0 φ 0 ) n 1 r H ( β ) exp ( + j χ 1 u 0 ) ,
A a , 3 ( E ) ( β ) = k 0 2 ( n 2 2 n 3 2 ) [ j χ 2   cos ( χ 2 u 0 ) χ 1   sin ( χ 2 u 0 ) ] cos   φ 0 r E ( β ) ,
A a , 3 ( H ) ( β ) = { n 2 2 χ 1 [ x ̃ 30 χ 2   cos ( χ 2 u 0 ) β β 0   sin ( χ 2 u 0 ) ] + j n 1 2 χ 2 [ x ̃ 30 χ 2   sin ( χ 2 u 0 ) + β β 0   cos ( χ 2 u 0 ) ] } n 2 ( n 2 2 n 3 2 ) cos   φ 0 n 3 r H ( β ) ,
A b , 3 ( E ) ( β ) = j k 0 2 χ 2 ( n 2 2 n 1 2 ) r E ( β ) cos ( χ 20 u 0 φ 0 ) ,
A b , 3 ( H ) ( β ) = ( n 2 2 n 1 2 χ 1 x ̃ 10 j β β 0 ) n 2 n 3 χ 2 ( n 2 2 n 1 2 ) cos ( χ 20 u 0 φ 0 ) r H ( β ) .
A a , 2 ( E , ) ( β ) = j k 0 2 χ 2 χ 1 2 ( n 2 2 n 3 2 ) cos   φ 0 r E ( β ) exp ( j χ 2 u 0 ) ,
A a , 2 ( H , ) ( β ) = ( j β β 0 n 2 2 n 3 2 χ 3 x ̃ 30 ) n 2 ( n 1 2 χ 2 n 2 2 χ 1 ) ( n 2 2 n 3 2 ) cos   φ 0 2 r H ( β ) exp ( j χ 2 u 0 ) ,
A b , 2 ( E , ) ( β ) = j k 0 2 χ 2 + χ 3 2 ( n 2 2 n 1 2 ) r E ( β ) cos ( χ 20 u 0 φ 0 ) ,
A b , 2 ( H , ) ( β ) = ( n 2 2 n 1 2 χ 1 x ̃ 10 j β β 0 ) ( n 3 2 χ 2 + n 2 2 χ 3 ) ( n 2 2 n 1 2 ) cos ( χ 20 u 0 φ 0 ) 2 r H ( β ) ,
A a , 2 ( E , + ) ( β ) = j k 0 2 χ 2 + χ 1 2 ( n 2 2 n 3 2 ) cos   φ 0 r E ( β ) exp ( + j χ 2 u 0 ) ,
A a , 2 ( H , + ) ( β ) = ( j β β 0 n 2 2 n 3 2 χ 3 x ̃ 30 ) ( n 1 2 χ 2 + n 2 2 χ 1 ) ( n 2 2 n 3 2 ) cos   φ 0 2 r H ( β ) exp ( + j χ 2 u 0 ) ,
A b , 2 ( E , + ) ( β ) = j k 0 2 χ 3 χ 2 2 ( n 2 2 n 1 2 ) r E ( β ) cos ( χ 20 u 0 φ 0 ) ,
A b , 2 ( H , + ) ( β ) = ( n 2 2 n 1 2 χ 1 x ̃ 10 j β β 0 ) n 2 ( n 3 2 χ 2 n 2 2 χ 3 ) ( n 2 2 n 1 2 ) cos ( χ 20 u 0 φ 0 ) 2 r H ( β ) .
r E ( β ) = χ 2 ( χ 3 + χ 1 ) cos ( χ 2 u 0 ) + j ( χ 1 χ 3 + χ 2 2 ) sin ( χ 2 u 0 ) ,
r H ( β ) = n 2 2 χ 2 ( n 1 2 χ 3 + n 3 2 χ 1 ) cos ( χ 2 u 0 ) + j ( n 1 2 n 3 2 χ 2 2 + n 2 4 χ 1 χ 3 ) sin ( χ 2 u 0 ) .

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