Abstract

We present a differential theory to describe the optical response of multilayer bigrating structures. The formalism is based on an extension of the coordinate transformation method used by Chandezon et al. [ J. Opt. Soc. Am. A 72, 839 ( 1982)] and is a rigorous differential theory capable of modeling highly modulated multilayer bigratings. The method is used to model experimental reflectivity data taken from bigratings and shows good agreement.

© 1996 Optical Society of America

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  1. C. R. Lawrence, N. J. Geddes, N. Furlong, J. R. Sambles, “Surface plasmon resonance studies of immunoreactions utilizing disposable diffraction gratings,” Biosensors Bioelectron. 11, 389–400 (1996).
    [CrossRef]
  2. S. N. Koreshev, S. G. Seregin, “Variable sensitivity interferometer with a diffraction beam splitter,” Opt. Spectrosk. 77, 991–997 (1994).
  3. S. A. Kaufman, A. A. Liberman, E. M. Yankevich, “A diffraction grating optical beam splitter for a working standard of mean laser-emission power,” Meas. Tech. USSR 36, 176–179 (1993).
    [CrossRef]
  4. J. Schmidt, R. Vokel, W. Stork, J. T. Sheridan, J. Schwider, N. Streibl, F. Durst, “Diffractive beam splitter for laser doppler velocimetry,” Opt. Lett. 17, 1240–1242 (1992).
    [CrossRef] [PubMed]
  5. M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-based optical sensor using acousto-optics,” J. Phys. E 6, 1193–1200 (1995).
  6. G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
    [CrossRef]
  7. N. Kawatsuki, M. Uetsuki, “Crossed grating beam splitter for magnetooptical pickup head,” Jpn. J. Appl. Phys. 29, 2420–2423 (1990).
    [CrossRef]
  8. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  9. D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–309 (1978).
    [CrossRef]
  10. P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
    [CrossRef]
  11. R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
    [CrossRef]
  12. R. C. McPhedran, G. H. Derrick, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 227.
    [CrossRef]
  13. G. H. Derrick, R. C. McPhedran, “Coated crossed gratings,” J. Opt. (Paris) 15, 69–81 (1984).
    [CrossRef]
  14. N. E. Glass, A. A. Maradudin, V. Celli, “Theory of surface-polariton resonances and field enhancements in light scattering from bigratings,” J. Opt. Soc. Am. 73, 1240–1248 (1983).
    [CrossRef]
  15. J. Greffet, C. Baylard, P. Versaevel, “Diffraction of electromagnetic waves by crossed gratings: a series solution,” Opt. Lett. 17, 1740–1742 (1992).
    [CrossRef] [PubMed]
  16. B. Laks, D. L. Mills, A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4976 (1981).
    [CrossRef]
  17. G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
    [CrossRef]
  18. J. B. Harris, E. L. Wood, T. W. Preist, J. R. Sambles, “Conical diffraction from multilayer gratings containing uniaxial materials,” J. Opt. Soc. Am. A 13, 803–810 (1996).
    [CrossRef]
  19. J. B. Harris, T. W. Preist, J. R. Sambles, “A differential method for multilayer diffraction gratings made with uniaxial materials,” J. Opt. Soc. Am. A 12, 1965–1973 (1995).
    [CrossRef]
  20. D. Y. K. Ko, J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
    [CrossRef]
  21. N. P. K. Cotter, T. W. Preist, J. R. Sambles, “A scattering matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [CrossRef]
  22. S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
    [CrossRef]
  23. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]

1996

C. R. Lawrence, N. J. Geddes, N. Furlong, J. R. Sambles, “Surface plasmon resonance studies of immunoreactions utilizing disposable diffraction gratings,” Biosensors Bioelectron. 11, 389–400 (1996).
[CrossRef]

J. B. Harris, E. L. Wood, T. W. Preist, J. R. Sambles, “Conical diffraction from multilayer gratings containing uniaxial materials,” J. Opt. Soc. Am. A 13, 803–810 (1996).
[CrossRef]

1995

N. P. K. Cotter, T. W. Preist, J. R. Sambles, “A scattering matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
[CrossRef]

J. B. Harris, T. W. Preist, J. R. Sambles, “A differential method for multilayer diffraction gratings made with uniaxial materials,” J. Opt. Soc. Am. A 12, 1965–1973 (1995).
[CrossRef]

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-based optical sensor using acousto-optics,” J. Phys. E 6, 1193–1200 (1995).

1994

S. N. Koreshev, S. G. Seregin, “Variable sensitivity interferometer with a diffraction beam splitter,” Opt. Spectrosk. 77, 991–997 (1994).

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

1993

S. A. Kaufman, A. A. Liberman, E. M. Yankevich, “A diffraction grating optical beam splitter for a working standard of mean laser-emission power,” Meas. Tech. USSR 36, 176–179 (1993).
[CrossRef]

1992

1991

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

1990

N. Kawatsuki, M. Uetsuki, “Crossed grating beam splitter for magnetooptical pickup head,” Jpn. J. Appl. Phys. 29, 2420–2423 (1990).
[CrossRef]

1988

1984

G. H. Derrick, R. C. McPhedran, “Coated crossed gratings,” J. Opt. (Paris) 15, 69–81 (1984).
[CrossRef]

1983

1982

1981

B. Laks, D. L. Mills, A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4976 (1981).
[CrossRef]

1979

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

1978

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–309 (1978).
[CrossRef]

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Baylard, C.

Bradberry, G. W.

M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-based optical sensor using acousto-optics,” J. Phys. E 6, 1193–1200 (1995).

Bryan-Brown, G. P.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

Cann, P. S.

M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-based optical sensor using acousto-optics,” J. Phys. E 6, 1193–1200 (1995).

Celli, V.

Chandezon, J.

Cornet, G.

Cotter, N. P. K.

Derrick, G. H.

G. H. Derrick, R. C. McPhedran, “Coated crossed gratings,” J. Opt. (Paris) 15, 69–81 (1984).
[CrossRef]

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

R. C. McPhedran, G. H. Derrick, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 227.
[CrossRef]

Dupuis, M. T.

Durst, F.

Elston, S. J.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

Furlong, N.

C. R. Lawrence, N. J. Geddes, N. Furlong, J. R. Sambles, “Surface plasmon resonance studies of immunoreactions utilizing disposable diffraction gratings,” Biosensors Bioelectron. 11, 389–400 (1996).
[CrossRef]

Geddes, N. J.

C. R. Lawrence, N. J. Geddes, N. Furlong, J. R. Sambles, “Surface plasmon resonance studies of immunoreactions utilizing disposable diffraction gratings,” Biosensors Bioelectron. 11, 389–400 (1996).
[CrossRef]

Glass, N. E.

Granet, G.

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

Greffet, J.

Harris, J. B.

Jory, M. J.

M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-based optical sensor using acousto-optics,” J. Phys. E 6, 1193–1200 (1995).

Kaufman, S. A.

S. A. Kaufman, A. A. Liberman, E. M. Yankevich, “A diffraction grating optical beam splitter for a working standard of mean laser-emission power,” Meas. Tech. USSR 36, 176–179 (1993).
[CrossRef]

Kawatsuki, N.

N. Kawatsuki, M. Uetsuki, “Crossed grating beam splitter for magnetooptical pickup head,” Jpn. J. Appl. Phys. 29, 2420–2423 (1990).
[CrossRef]

Ko, D. Y. K.

Koreshev, S. N.

S. N. Koreshev, S. G. Seregin, “Variable sensitivity interferometer with a diffraction beam splitter,” Opt. Spectrosk. 77, 991–997 (1994).

Laks, B.

B. Laks, D. L. Mills, A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4976 (1981).
[CrossRef]

Lawrence, C. R.

C. R. Lawrence, N. J. Geddes, N. Furlong, J. R. Sambles, “Surface plasmon resonance studies of immunoreactions utilizing disposable diffraction gratings,” Biosensors Bioelectron. 11, 389–400 (1996).
[CrossRef]

Li, L.

Liberman, A. A.

S. A. Kaufman, A. A. Liberman, E. M. Yankevich, “A diffraction grating optical beam splitter for a working standard of mean laser-emission power,” Meas. Tech. USSR 36, 176–179 (1993).
[CrossRef]

Maradudin, A. A.

Maystre, D.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

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

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–309 (1978).
[CrossRef]

McPhedran, R. C.

G. H. Derrick, R. C. McPhedran, “Coated crossed gratings,” J. Opt. (Paris) 15, 69–81 (1984).
[CrossRef]

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

R. C. McPhedran, G. H. Derrick, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 227.
[CrossRef]

Mills, D. L.

B. Laks, D. L. Mills, A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4976 (1981).
[CrossRef]

Nevière, M.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–309 (1978).
[CrossRef]

Preist, T. W.

Sambles, J. R.

Schmidt, J.

Schwider, J.

Seregin, S. G.

S. N. Koreshev, S. G. Seregin, “Variable sensitivity interferometer with a diffraction beam splitter,” Opt. Spectrosk. 77, 991–997 (1994).

Sheridan, J. T.

Stork, W.

Streibl, N.

Uetsuki, M.

N. Kawatsuki, M. Uetsuki, “Crossed grating beam splitter for magnetooptical pickup head,” Jpn. J. Appl. Phys. 29, 2420–2423 (1990).
[CrossRef]

Versaevel, P.

Vincent, P.

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Vokel, R.

Wood, E. L.

Yankevich, E. M.

S. A. Kaufman, A. A. Liberman, E. M. Yankevich, “A diffraction grating optical beam splitter for a working standard of mean laser-emission power,” Meas. Tech. USSR 36, 176–179 (1993).
[CrossRef]

Appl. Phys.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Biosensors Bioelectron.

C. R. Lawrence, N. J. Geddes, N. Furlong, J. R. Sambles, “Surface plasmon resonance studies of immunoreactions utilizing disposable diffraction gratings,” Biosensors Bioelectron. 11, 389–400 (1996).
[CrossRef]

J. Opt. (Paris)

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–309 (1978).
[CrossRef]

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, “Coated crossed gratings,” J. Opt. (Paris) 15, 69–81 (1984).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. E

M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-based optical sensor using acousto-optics,” J. Phys. E 6, 1193–1200 (1995).

Jpn. J. Appl. Phys.

N. Kawatsuki, M. Uetsuki, “Crossed grating beam splitter for magnetooptical pickup head,” Jpn. J. Appl. Phys. 29, 2420–2423 (1990).
[CrossRef]

Meas. Tech. USSR

S. A. Kaufman, A. A. Liberman, E. M. Yankevich, “A diffraction grating optical beam splitter for a working standard of mean laser-emission power,” Meas. Tech. USSR 36, 176–179 (1993).
[CrossRef]

Opt. Commun.

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Opt. Lett.

Opt. Spectrosk.

S. N. Koreshev, S. G. Seregin, “Variable sensitivity interferometer with a diffraction beam splitter,” Opt. Spectrosk. 77, 991–997 (1994).

Phys. Rev. B

B. Laks, D. L. Mills, A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4976 (1981).
[CrossRef]

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

Pure Appl. Opt.

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

Other

R. C. McPhedran, G. H. Derrick, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 227.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Diagrammatic representation of the structure to be modeled.

Fig. 2
Fig. 2

Comparison of the product of efficiencies calculated with a previous code for two separate gratings of profiles s1(v) and s2(w) with the efficiency calculated for a bigrating of profile s1(v) + s2(w). The overall agreement between the two obtained by use of the parameters in Table 1 can be regarded as a first-order test of our code.

Fig. 3
Fig. 3

Fit to experimental data of zero-order efficiencies taken at azimuthal angle ϕ = 90° from a gold bigrating of crossing angle 60° for (a) Rpp and (b) Rss polarizations. The parameters that we used in calculating the theoretical curves are given in Table 3.

Tables (4)

Tables Icon

Table 1 Bigrating Parameters Used to Calculate the Theoretical Curves in Fig. 2 and the Efficiencies Shown in Table 2

Tables Icon

Table 2 Selection of Zero-Order Efficiencies Calculated with the Parameters in Table 1

Tables Icon

Table 3 Comparison of Efficiencies Calculated by Graneta and Those Calculated in This Work

Tables Icon

Table 4 Parameters Determined in Comparing Theory with Experimental Data in Fig. 3

Equations (49)

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y = d j + s ( x , z ) ,             d Q + 1 = 0 , d j = ( e Q + e Q - 1 + + e j ) ,             j = 1 , , Q + 1 ,
v = x ,             u = y - s ( x , z ) , w = z = x cos η + z sin η .
e 1 = i + s v j - cot η k ,             e 1 = i , e 2 = j ,             e 2 = - ( s v + s w cos η ) i + j - s w sin η k , e 3 = s w j + csc η k ,             e 3 = cos η i + sin η k ,
curl A = sin η | e 1 e 2 e 3 v u w A 1 A 2 A 3 | .
sin η [ E 3 u - w ( D 1 E 1 + C E 2 + D 3 E 3 ) ] = i μ ( F w H 1 - D 1 H 2 + F v w H 3 ) ,
sin η ( E 1 w - E 3 v ) = i μ H 2 ,
sin η [ v ( D 1 E 1 + C E 2 + D 3 E 3 ) - E 1 u ] = i μ ( F v w H 1 - D 3 H 2 + F v H 3 ) ,
sin η [ H 3 u - w ( D 1 H 1 + C H 2 + D 3 H 3 ) ] = - i ɛ ( F w E 1 - D 1 E 2 + F v w E 3 ) ,
sin η ( H 1 w - H 3 v ) = - i ɛ E 2 ,
sin η [ v ( D 1 H 1 + C H 2 + D 3 H 3 ) - H 1 u ] = - i ɛ ( F v w E 1 - D 3 E 2 + F v E 3 ) .
A 1 = F w A 1 - D 1 A 2 + F v w A 3 . A 2 = D 1 A 1 + C A 2 + D 3 A 3 , A 3 = F v w A 1 - D 3 A 2 + F v A 3 .
C = 1 1 + s v 2 + s w 2 + 2 s v s w cos η , D 1 = ( s v + s w cos η ) C ,             D 3 = ( s w + s v cos η ) C , F v = ( 1 + s v 2 sin 2 η ) C ,             F w = ( 1 + s w 2 sin 2 η ) C , F v w = ( cos η - s v s w sin 2 η ) C .
E 1 u = v ( D 1 E 1 ) + D 3 E 1 ω + ( D 3 v ) E 3 + i sin η ɛ v [ C ( H 1 ω - H 3 v ) ] - i μ sin η ( F v w H 1 + F v H 3 ) ,
E 3 u = ( D 1 ω ) E 1 + D 1 E 3 v + ω ( D 3 E 3 ) + i sin η ɛ ω [ C ( H 1 ω - H 3 v ) ] + i μ sin η ( F w H 1 + F v w H 3 ) ,
H 1 u = - i sin η μ v [ C ( E 1 ω - E 3 v ) ] + i ɛ sin η ( F v w E 1 + F v E 3 ) + v ( D 1 H 1 ) + D 3 H 1 ω + ( D 3 v ) H 3 ,
H 3 u = - i sin η μ ω [ C ( E 1 ω - E 3 v ) ] - i ɛ sin η ( F w E 1 + F v w E 3 ) + ( D 1 ω ) H 1 + D 1 H 3 v + ω ( D 3 H 3 ) .
Ψ ( v , u , w ) = m , n Ψ m , n ( u ) exp i ( α m v + γ n w ) , α m = ( ɛ Q + 1 ) 1 / 2 sin θ ( cos ϕ - sin ϕ cot η ) + m K v , γ n = ( ɛ Q + 1 ) 1 / 2 sin θ sin ϕ sin η + n K w ,             m , n = 0 , ± 1 , ± 2 , ,
Φ ( v , w ) = p , q Φ p , q exp i ( p K v v + q K w w ) ,             p , q = 0 , ± 1 , ± 2 , ,
- i E 1 ( a , b ) u = m , n { ( α a D 1 ( p , q ) + γ n D 3 ( p , q ) ) E 1 ( m , n ) + p K v D 3 ( p , q ) E 3 ( m , n ) - ( α a γ n sin η ɛ C ( p , q ) + μ sin η F v w ( p , q ) ) H 1 ( m , n ) + ( α a α m sin η ɛ C ( p , q ) - μ sin η F v ( p , q ) ) H 3 ( m , n ) ,
- i E 3 ( a , b ) u = m , n { q K ω D 1 ( p , q ) E 1 ( m , n ) + ( α m D 1 ( p , q ) + γ b D 3 ( p , q ) ) E 3 ( m , n ) - ( γ n γ b sin η ɛ C ( p , q ) - μ sin η F w ( p , q ) ) H 1 ( m , n ) + ( α m γ b sin η ɛ C ( p , q ) + μ sin η F v w ( p , q ) ) H 3 ( m , n ) ,
- i H 1 ( a , b ) u = m , n { ( γ n α a sin η μ C ( p , q ) + ɛ sin η F v w ( p , q ) ) E 1 ( m , n ) + ( - α a α m sin η μ C ( p , q ) + ɛ sin η F v ( p , q ) ) E 3 ( m , n ) + ( α a D 1 ( p , q ) + γ n D 3 ( p , q ) ) H 1 ( m , n ) + p K v D 3 ( p , q ) H 3 ( m , n ) ,
- i H 1 ( a , b ) u = m , n { ( γ n γ b sin η μ C ( p , q ) - ɛ sin η F w ( p , q ) ) E 1 ( m , n ) - ( α m γ b sin η μ C ( p , q ) + ɛ sin η F v w ( p , q ) ) E 3 ( m , n ) + q K ω D 1 ( p , q ) H 1 ( m , n ) + ( α m D 1 ( p , q ) + γ b D 3 ( p , q ) ) H 3 ( m , n ) .
- i ξ ( u ) u = T ξ ( u ) ,
T = [ T 11 T 12 T 13 T 14 T 21 T 22 T 23 T 24 T 31 T 32 T 33 T 34 T 41 T 42 T 43 T 44 ] .
T 11 : α a D 1 ( p , q ) + γ n D 3 ( p , q ) ,             T 12 : p K v D 3 ( p , q ) , T 13 : - α a γ n sin η ɛ C ( p , q ) - μ sin η F v w ( p , q ) , T 14 : α a α m sin η ɛ C ( p , q ) - μ sin η F v ( p , q ) , T 21 : q K ω D 1 ( p , q ) ,             T 22 : α m D 1 ( p , q ) + γ b D 3 ( p , q ) , T 23 : - γ n γ b sin η ɛ C ( p , q ) + μ sin η F w ( p , q ) , T 24 : α m γ b sin η ɛ C ( p , q ) + μ sin η F v w ( p , q ) , T 32 : α a γ n sin η μ C ( p , q ) + ɛ sin η F v w ( p , q ) , T 32 : - α a α m sin η μ C ( p , q ) + ɛ sin η F v ( p , q ) , T 33 = T 11 ,             T 34 = T 12 , T 41 : γ n γ b sin η μ C ( p , q ) - ɛ sin η F w ( p , q ) , T 42 : - α m γ b sin η μ C ( p , q ) - ɛ sin η F v w ( p , q ) , T 43 = T 21 ,             T 44 = T 22 .
ξ ( u ) = q = 1 4 ( 2 M + 1 ) ( 2 N + 1 ) b q V q exp i r q u ,
ξ j ( u ) = M j ϕ j ( u ) b j ,
β 0 , 0 = - [ μ Q + 1 ɛ Q + 1 - ( α 0 + γ 0 cos η ) 2 - ( γ 0 sin η ) 2 ] 1 / 2 ,
β a , b = [ μ Q + 1 ɛ Q + 1 - ( α a + γ b cos η ) 2 - ( γ b sin η ) 2 ] 1 / 2 ,
β a , b = - [ μ 0 ɛ 0 - ( α a + γ b cos η ) 2 - ( γ b sin η ) 2 ] 1 / 2 .
E 1 = f ( g 1 - g 3 cos η - β a , b ɛ h 3 sin η ) ,
E 3 = f ( g 3 - g 1 cos η + β a , b ɛ h 1 sin η ) ,
H 1 = f ( h 1 - h 3 cos η + β a , b μ g 3 sin η ) ,
H 3 = f ( h 3 - h 1 cos η - β a , b μ g 1 sin η ) ,
f - 1 = sin 2 η ( 1 - β a , b 2 μ ɛ ) , g 1 = sin η i ɛ H 2 w + ( s v + s w cos η ) E 2 ,
g 3 = - sin η i ɛ H 2 v + ( s w + s v cos η ) E 2 ,
h 1 = - sin η i μ H 2 w + ( s v + s w cos η ) H 2 ,
h 3 = sin η i μ H 2 v + ( s w + s v cos η ) H 2 .
ψ a , b v = i ( α a + β a , b s v ) ψ a , b , ψ a , b w = i ( γ a + β a , b s w ) ψ a , b
β a , b s v ψ a , b = m , n ψ m , n a , b ( m - a ) K v L m - a , n - b ( - β a , b ) × exp i ( α m v + β a , b u + γ n w ) , β a , b s w ψ a , b = m , n ψ m , n a , b ( n - b ) K w L m - a , n - b ( - β a , b ) × exp i ( α m v + β a , b u + γ n w ) ,
( E 1 ) m , n a , b = { γ b + α a cos η ɛ sin η ( 1 - β a , b 2 μ ɛ ) H 2 ( a , b ) - [ p K v β a , b + β a , b α a μ ɛ ( 1 - β a , b 2 μ ɛ ) ] E 2 ( a , b ) } L - p , - q ( - β a , b ) ,
( E 3 ) m , n a , b = { - ( α a + γ b cos η ) ɛ sin η ( 1 - β a , b 2 μ ɛ ) H 2 ( a , b ) - [ q K ω β a , b + γ b β a , b μ ɛ ( 1 - β a , b 2 μ ɛ ) ] E 2 ( a , b ) } L - p , - q ( - β a , b ) .
( H 1 ) m , n a , b = { - [ p K v β a , b + β a , b α a μ ɛ ( 1 - β a , b 2 μ ɛ ) ] H 2 ( a , b ) - γ b + α a cos η μ sin η ( 1 - β a , b 2 μ ɛ ) E 2 ( a , b ) } L - p , - q ( - β a , b ) .
( H 3 ) m , n a , b = { - [ q K ω β a , b + β a , b γ b μ ɛ ( 1 - β a , b 2 μ ɛ ) ] H 2 ( a , b ) + α a + γ b cos η μ sin η ( 1 - β a , b 2 μ ɛ ) E 2 ( a , b ) } L - p , - q ( - β a , b ) ,
R ^ = [ M ^ 11 - ( M 11 S 12 + M 12 ) ( M 21 S 12 + M 22 ) - 1 M ^ 21 ] - 1 × [ ( M 11 S 12 + M 12 ) ( M 21 S 12 + M 22 ) - 1 L - L ] ,
T ^ = QS 22 ( M 21 S 12 + M 22 ) - 1 ( M ^ 21 R ^ + L ) ,
[ M ^ 11 0 M ^ 12 0 M ^ 21 0 M ^ 22 0 ] - 1 [ M 11 0 M 12 0 M 21 0 M 22 0 ] = [ I 0 0 Q ] .
R m n ( s ) R m [ s 1 ( v ) ] R n [ s 2 ( w ) ] R ( 0 ) ,
s ( v , w ) = h 4 [ sin ( 2 π v λ g ) + sin ( 2 π w λ g ) ] ,

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