Abstract

I report significant improvements to the differential method of Chandezon et al. [ J. Opt. Soc. Am. 72, 839– 846 ( 1982)]. The R-matrix propagation algorithm is used to remove completely the previously existing limitations on the total coating thickness and on the total number of coated layers. I analyze the symmetry properties of the eigenvalue problem that arises in the differential formalism and use them to speed up the numerical computation. The time needed for computing the eigensolutions of coated gratings in a conical mount is reduced to little more than what is needed for gratings in a classical mount. For gratings with dielectric coatings or gratings with symmetrical profiles the need to invert certain matrices appearing in the formalism is eliminated. Numerical results show that it is possible to make nearly 100% efficient surface-relief reflection gratings in a Littrow mount by the use of dielectric materials only.

© 1994 Optical Society of America

Full Article  |  PDF Article

Corrections

Lifeng Li, "Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited: errata," J. Opt. Soc. Am. A 13, 543-543 (1996)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-13-3-543

References

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  1. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,”J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  3. D. Maystre, “A new general integral theory for dielectric coated gratings,”J. Opt. Soc. Am. 68, 490–495 (1978).
    [CrossRef]
  4. D. Maystre, “A new theory for multiprofile, buried gratings,” Opt. Commun. 26, 127–132 (1978).
    [CrossRef]
  5. L. F. DeSandre, J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
    [CrossRef]
  6. 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]
  7. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1984), Vol. 21, pp. 1–67.
    [CrossRef]
  8. J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,”J. Opt. (Paris) 11, 235–241 (1980).
    [CrossRef]
  9. E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,”J. Opt. (Paris) 17, 175–180 (1986).
    [CrossRef]
  10. S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
    [CrossRef]
  11. E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
    [CrossRef]
  12. E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings—non-sinusoidal profile,” Opt. Acta 34, 155–158 (1987).
  13. L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
    [CrossRef]
  14. E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962), Chap. 3, p. 47.
  15. E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).
  16. J. Chandezon, “Les équations de Maxwell sous forme covariante, application a l’étude de la propagation dans les guides périodiques et a la diffraction par les réseaux,” Ph.D. dissertation (University of Clermont-Ferrand II, France, 1979).
  17. See, for example, R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, U.K., 1985), Chap. 1, Section 4, pp. 59–64.
  18. S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
    [CrossRef]
  19. D. Maystre, J. P. Laude, P. Gacoin, D. Lepere, J. P. Priou, “Gratings for tunable lasers: using multidielectric coatings to improve their efficiency,” Appl. Opt. 19, 3099–3102 (1980).
    [CrossRef] [PubMed]
  20. See, for example, H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Adam Hilger, Bristol, U.K., 1986).
    [CrossRef]

1993 (1)

1991 (2)

L. F. DeSandre, J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
[CrossRef]

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

1987 (1)

E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings—non-sinusoidal profile,” Opt. Acta 34, 155–158 (1987).

1986 (3)

E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,”J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

1984 (1)

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

1982 (3)

1980 (2)

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

D. Maystre, J. P. Laude, P. Gacoin, D. Lepere, J. P. Priou, “Gratings for tunable lasers: using multidielectric coatings to improve their efficiency,” Appl. Opt. 19, 3099–3102 (1980).
[CrossRef] [PubMed]

1978 (2)

D. Maystre, “A new general integral theory for dielectric coated gratings,”J. Opt. Soc. Am. 68, 490–495 (1978).
[CrossRef]

D. Maystre, “A new theory for multiprofile, buried gratings,” Opt. Commun. 26, 127–132 (1978).
[CrossRef]

Bryan-Brown, G. P.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Chandezon, J.

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]

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

J. Chandezon, “Les équations de Maxwell sous forme covariante, application a l’étude de la propagation dans les guides périodiques et a la diffraction par les réseaux,” Ph.D. dissertation (University of Clermont-Ferrand II, France, 1979).

Chuang, S. L.

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Cornet, G.

DeSandre, L. F.

Dupuis, M. T.

Elson, J. M.

Elston, S. J.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Gacoin, P.

Gaylord, T. K.

Kong, J. A.

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Laude, J. P.

Lepere, D.

Li, L.

Macleod, H. A.

See, for example, H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Adam Hilger, Bristol, U.K., 1986).
[CrossRef]

Mashev, L.

E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings—non-sinusoidal profile,” Opt. Acta 34, 155–158 (1987).

E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,”J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

Maystre, D.

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]

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

D. Maystre, J. P. Laude, P. Gacoin, D. Lepere, J. P. Priou, “Gratings for tunable lasers: using multidielectric coatings to improve their efficiency,” Appl. Opt. 19, 3099–3102 (1980).
[CrossRef] [PubMed]

D. Maystre, “A new general integral theory for dielectric coated gratings,”J. Opt. Soc. Am. 68, 490–495 (1978).
[CrossRef]

D. Maystre, “A new theory for multiprofile, buried gratings,” Opt. Commun. 26, 127–132 (1978).
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1984), Vol. 21, pp. 1–67.
[CrossRef]

Moharam, M. G.

Popov, E.

E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings—non-sinusoidal profile,” Opt. Acta 34, 155–158 (1987).

E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,”J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

Post, E. J.

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962), Chap. 3, p. 47.

Priou, J. P.

Raoult, G.

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

Sambles, J. R.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Appl. Opt. (1)

J. Opt. (Paris) (2)

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

E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,”J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

J. Opt. Commun. (1)

E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Opt. Acta (2)

E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings—non-sinusoidal profile,” Opt. Acta 34, 155–158 (1987).

Opt. Commun. (2)

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

D. Maystre, “A new theory for multiprofile, buried gratings,” Opt. Commun. 26, 127–132 (1978).
[CrossRef]

Phys. Rev. B (1)

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Radio Sci. (1)

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Other (5)

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1984), Vol. 21, pp. 1–67.
[CrossRef]

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962), Chap. 3, p. 47.

J. Chandezon, “Les équations de Maxwell sous forme covariante, application a l’étude de la propagation dans les guides périodiques et a la diffraction par les réseaux,” Ph.D. dissertation (University of Clermont-Ferrand II, France, 1979).

See, for example, R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, U.K., 1985), Chap. 1, Section 4, pp. 59–64.

See, for example, H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Adam Hilger, Bristol, U.K., 1986).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Multilayer-coated grating in a conical mount.

Fig. 2
Fig. 2

Notation for the media and medium interfaces of a multilayer-coated grating.

Fig. 3
Fig. 3

First-order Littrow mount diffraction efficiency of a coated sinusoidal grating. The parameters are λ = 0.59 μm, d = 0.3333 μm, h = 0.12 μm, Q = 15, n(+1) = 1.0, n(−1) = 1.46, nj = 2.37 with odd j, nj = 1.35 with even j, and TM polarization.

Fig. 4
Fig. 4

Same as Fig. 3 except that n(−1) = 1.15 + i7.15, nj = 2.37 with even j, n = 1.35 with odd j, and Q = 8.

Fig. 5
Fig. 5

Dependencies of the peak first-order Littrow mount diffraction efficiencies of the two gratings in Figs. 3 and 4 on the number of coated layers. The normalized thickness is fixed at ρ = 0.304.

Fig. 6
Fig. 6

Angular dependencies of the peak first-order Littrow mount diffraction efficiencies of the two gratings in Figs. 3 and 4. The normalized thickness is fixed at ρ = 0.304.

Tables (2)

Tables Icon

Table 1 Diffraction Efficiencies of Bare and Metal-Coated Sinusoidal Gratingsa

Tables Icon

Table 2 Diffraction Efficiencies of Bare and Metal-Coated Sinusoidal Gratingsa

Equations (92)

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e j = { y 0 - y ( - 1 ) j = 0 y j - y j - 1 1 j Q y ( + 1 ) - y Q j = Q + 1 ,
e ( j ) = { y ( j ) - y ( j - 1 ) + 2 j L y ( j + 1 ) - y ( j ) p j - 2 .
y = y j + a ( x ) ,             0 j Q .
k j 2 = j μ j κ 2 ,             0 j Q + 1 ,
k ( j ) 2 = ( j ) μ ( j ) κ 2 ,             + 1 j L + 1 or P - 1 j - 1 ,
k in = x ^ α 0 - y ^ β 0 ( L + 1 ) + z ^ k z ,
k in = x ^ α 0 - y ^ β 0 ( L + 1 ) ,
k ˜ j 2 = k j 2 - k z 2 ,
τ 1 ( j ) = i κ 2 ( j ) k ˜ ( j ) 2 ,             τ 2 ( j ) = i κ 2 μ ( j ) k ˜ ( j ) 2 ,             τ 3 ( j ) = i κ k z k ˜ ( j ) 2 ;
β m ( j ) = [ k ˜ ( j ) 2 - α m 2 ] 1 / 2 ,             Re [ β m ( j ) ] + Im [ β m ( j ) ] 0 ;
C ( x ) = 1 1 + a ˙ 2 ( x ) = m = - C m exp ( i m K x ) , C m = 1 d C ( x ) exp ( - i m K x ) d x ;
D ( x ) = a ˙ ( x ) 1 + a ˙ 2 ( x ) = m = - D m exp ( i m K x ) , D m = 1 d D ( x ) exp ( - i m K x ) d x .
F ( y ) = ( E z m , H z m , κ i H x m , κ i E x m ) T
F ( y ) = W ϕ ( y - y ) W - 1 F ( y ) ,
ϕ ( y ) = exp ( + i β m y ) exp ( + i β m y ) exp ( - i β m y ) exp ( - i β m y ) ,
W = [ 1 0 1 0 0 1 0 1 - τ 1 β m τ 3 α m τ 1 β m τ 3 α m τ 3 α m τ 2 β m τ 3 α m - τ 2 β m ] ,
u = y - a ( x ) .
E z u = D ( x ) E z x + i κ C ( x ) ( k ˜ 2 H x - i k z H z x ) ,
H z u = D ( x ) H z x - i κ μ C ( x ) ( k ˜ 2 E x - i k z E z x ) ,
H z u = x [ D ( x ) H x ] + i κ E z - i κ μ x [ C ( x ) ( i k z E x - E z x ) ] ,
E x u = x [ D ( x ) E x ] - i κ μ H z + i κ x [ C ( x ) ( i k z H x - H z x ) ] ,
Φ ( x , u ) = m = - Φ m ( u ) exp ( i α m x ) ,
1 i d d u F = M F ,
M = [ D m - n α n τ 3 τ 1 C m - n α n - 1 τ 1 C m - n 0 - τ 3 τ 2 C m - n α n D m - n α n 0 1 τ 2 C m - n 1 i μ ( k 2 δ m n - α m C m - n α n ) 0 α m D m - n τ 3 τ 2 α m C m - n 0 - 1 i ( k 2 δ m n - α m C m - n α n ) - τ 3 τ 1 α m C m - n α m D m - n ] ,
F ( u ) = ( E z m , H z m , κ i H x m , κ i E x m ) T .
f q = ζ q exp ( i λ q u ) b q ,
F ( u ) = W ϕ ( u - u ) W - 1 F ( u ) ,
ϕ ( u ) = δ p q exp ( i λ q u ) ,
F ( L + 1 ) [ y ( L ) ] = W ( L + 1 ) [ R ˜ n ( e ) , R ˜ n ( h ) , I ˜ n ( e ) , I ˜ n ( h ) ] T ,
F ( p - 1 ) [ y ( P ) ] = W ( P - 1 ) [ J ˜ n ( e ) , J ˜ n ( h ) , T ˜ n ( e ) , T ˜ n ( h ) ] T ,
R ˜ n ( s ) = R n ( s ) exp [ i β n ( L + 1 ) y ( L ) ] , I ˜ n ( s ) = I n ( s ) exp [ - i β n ( L + 1 ) y ( L ) ] ,
J ˜ n ( s ) = J n ( s ) exp [ i β n ( P - 1 ) y ( P ) ] , T ˜ n ( s ) = T n ( s ) exp [ - i β n ( P - 1 ) y ( P ) ] .
F 0 ( u ) = Z f ( - 1 ) ϕ ( - 1 ) ( u - y ) Z g ( - 1 ) - 1 F ( - 1 ) ( y ) ,
ϕ ( - 1 ) ( η ) = exp [ + i β m ( - 1 ) η ] exp [ + i β m ( - 1 ) η ] exp [ - β m ( - 1 ) η ] exp [ - i β m ( - 1 ) η ] exp [ i λ q ( - 1 ) η ] ,
F 0 ( u = y 0 ) = Z f ( - 1 ) [ J ˜ m ( e ) , J ˜ m ( h ) , T ˜ m ( e ) , T ˜ m ( h ) , b ˜ q ( - 1 ) ] T ,
F ( + 1 ) ( y ) = Z g ( + 1 ) ϕ ( + 1 ) ( y - u ) Z f ( + 1 ) - 1 F Q + 1 ( u ) ,
ϕ ( + 1 ) ( η ) = exp [ + i β m ( + 1 ) η ] exp [ + i β m ( + 1 ) η ] exp [ i λ q ( + 1 ) η ] exp [ - i β m ( + 1 ) η ] exp [ - i β m ( + 1 ) η ] ,
F Q + 1 ( u = y Q ) = Z f ( + 1 ) [ R ˜ m ( e ) , R ˜ m ( h ) , b ˜ q ( + 1 ) , I ˜ m ( e ) , I ˜ m ( h ) ] T ,
A X = b ,
W ( L + 1 ) [ R ˜ n ( e ) , R ˜ n ( h ) , I ˜ n ( e ) , I ˜ n ( h ) ] T = T W ( P - 1 ) [ J ˜ n ( e ) , J ˜ n ( h ) , T ˜ n ( e ) , T ˜ n ( h ) ] T ,
T = { j = 1 Q W ( j ) ϕ ( j ) [ e ( j ) ] W ( j ) - 1 } { Z g ( + 1 ) ϕ ( + 1 ) [ e ( + 1 ) ] Z f ( + 1 ) - 1 } × [ j = 1 Q W j ϕ j ( e j ) W j - 1 ] { Z f ( - 1 ) ϕ ( - 1 ) [ e ( - 1 ) ] Z g ( - 1 ) - 1 } × { j = P - 2 W ( j ) ϕ ( j ) [ e ( j ) ] W ( j ) - 1 } ,
Z f ( + 1 ) [ R ˜ n ( e ) , R ˜ n ( h ) , b ˜ q ( + 1 ) , I ˜ n ( e ) , I ˜ n ( h ) ] T = T Z f ( - 1 ) [ J ˜ m ( e ) , J ˜ m ( h ) , T ˜ m ( e ) , T ˜ m ( h ) , b ˜ p ( - 1 ) ] T ,
T = j = 1 Q W j ϕ j ( e j ) W j - 1 ,
T = l = 1 l t ( l ) ,
( Ω l X l ) = t ( l ) ( Ω l - 1 X l - 1 ) ,
( Ω l - 1 Ω l ) = r ( l ) ( X l - 1 X l ) ,
[ r 11 ( l ) r 12 ( l ) r 21 ( l ) r 22 ( l ) ] = [ - t 21 ( l ) - 1 t 22 ( l ) t 21 ( l ) - 1 t 12 ( l ) - t 11 ( l ) t 21 ( l ) - 1 t 22 ( l ) t 11 ( l ) t 21 ( l ) - 1 ] .
( Ω 0 Ω l ) = R ( l ) ( X 0 X l ) .
R 11 ( l ) = R 11 ( l - 1 ) + R 12 ( l - 1 ) Z ( l ) R 21 ( l - 1 ) , R 12 ( l ) = - R 12 ( l - 1 ) Z ( l ) r 12 ( l ) , R 21 l = r 21 ( l ) Z ( l ) R 21 ( l - 1 ) , R 22 ( l ) = r 22 ( l ) - r 21 ( l ) Z ( l ) r 12 ( l ) ,
Z ( l ) = [ r 11 ( l ) - R 22 ( l - 1 ) ] - 1 .
T = W Q j = 2 Q [ ϕ ( e j ) W j - 1 W j - 1 ] ϕ ( e 1 ) W 1 - 1 .
t = [ exp ( i λ m + e ) 0 0 exp ( i λ m - e ) ] [ A 11 A 12 A 21 A 22 ] ,
r = [ - A 21 - 1 A 22 A 21 - 1 exp ( - i λ - e ) exp ( i λ + e ) ( A 12 - A 11 A 21 - 1 A 22 ) exp ( i λ + e ) A 11 A 21 - 1 exp ( - i λ - e ) ] .
E x = - τ 2 κ [ a ˙ H z x - ( 1 + a ˙ 2 ) H z u ] + τ 3 κ E z x ,
H x = τ 1 κ [ a ˙ E z x - ( 1 + a ˙ 2 ) E z u ] + τ 3 κ H z x .
[ 2 x 2 + ( 1 + a ˙ 2 ) 2 u 2 - 2 a ˙ 2 x u - a ¨ u + k ˜ 2 ] ( E z H z ) = ( 0 0 ) .
F ( e ) = ( E z m ( e ) 0 - i τ 1 [ a ˙ E z ( e ) x - ( 1 + a ˙ 2 ) E z ( e ) u ] m τ 3 α m E z m ( e ) ) ,
F ( h ) = ( 0 H z m ( h ) τ 3 α m H z m ( h ) i τ 2 [ a ˙ H z ( h ) x - ( 1 + a ˙ 2 ) H z ( h ) u ] m ) .
1 i d d u ( f m g m ) = M ( f m g m ) ,
M = [ D m - n α n i μ C m - n 1 i μ ( k ˜ 2 δ m n - α m C m - n α n ) α m D m - n ] .
g m = 1 μ [ a ˙ f x - ( 1 + a ˙ 2 ) f u ] m .
F ( e ) = ( f m 0 - i τ 1 μ g m τ 3 α m f m ) ,             F ( h ) = ( 0 f m τ 3 α m f m i τ 2 μ g m ) .
1 p N 0 - 1 ,             N 0 q N ,             1 r N ,
{ λ p ( e ) + } , { λ q ( e ) + } , { λ p ( h ) + } , { λ q ( h ) + } , { λ p ( e ) - } , { λ q ( e ) - } , { λ p ( h ) - } , { λ q ( h ) - } ,
λ p ( s ) ± 0 ,             Im [ λ q ( s ) ± ] 0 ,
1 a < b N 0 - 1             if λ a ( s ) ± λ b ( s ) ± ,
N 0 a < b N             if Im [ λ a ( s ) ± ] Im [ λ b ( s ) ± ] or if Im [ λ a ( s ) ± ] = Im [ λ b ( s ) ± ] and Re [ λ a ( s ) ± ] < Re [ λ b ( s ) ± ] .
W = [ ν 1 p ( e ) + ν 1 q ( e ) + ν 1 p ( h ) + ν 1 q ( h ) + ν 1 p ( e ) - ν 1 q ( e ) - ν 1 p ( h ) - ν 1 q ( h ) - ν 2 p ( e ) + ν 2 q ( e ) + ν 2 p ( h ) + ν 2 q ( h ) + ν 2 p ( e ) - ν 2 q ( e ) - ν 2 p ( h ) - ν 2 q ( h ) - ν 3 p ( e ) + ν 3 q ( e ) + ν 3 p ( h ) + ν 3 q ( h ) + ν 3 p ( e ) - ν 3 q ( e ) - ν 3 p ( h ) - ν 3 q ( h ) - ν 4 p ( e ) + ν 4 q ( e ) + ν 4 p ( h ) + ν 4 q ( h ) + ν 4 p ( e ) - ν 4 q ( e ) - ν 4 p ( h ) - ν 4 q ( h ) - ] ,
W ˜ - 1 W = D ,
W ˜ - 1 = [ ν 3 p ( e ) + ν 3 q ( e ) - ν 3 p ( h ) + ν 3 q ( h ) - ν 3 p ( e ) - ν 3 q ( e ) + ν 3 p ( h ) - ν 3 q ( h ) + - ν 4 p ( e ) + - ν 4 q ( e ) - - ν 4 p ( h ) + - ν 4 q ( h ) - - ν 4 p ( e ) - - ν 4 q ( e ) + - ν 4 p ( h ) - - ν 4 q ( h ) + - ν 1 p ( e ) + - ν 1 q ( e ) - ν 1 p ( h ) + - ν 1 q ( h ) - - ν 1 p ( e ) - - ν 1 q ( e ) + - ν 1 p ( h ) - - ν 1 q ( h ) + ν 2 p ( e ) + ν 2 q ( e ) - ν 2 p ( h ) + ν 2 q ( h ) - ν 2 p ( e ) - ν 2 q ( e ) + ν 2 p ( h ) - ν 2 q ( h ) + ] * .
W - 1 = D - 1 W ˜ - 1 .
W ˜ - 1 = [ - ν 3 r ( e ) - - ν 3 r ( h ) - - ν 3 r ( e ) + - ν 3 r ( h ) + - ν 4 r ( e ) - - ν 4 r ( h ) - - ν 4 r ( e ) + - ν 4 r ( h ) + ν 1 r ( e ) - ν 1 r ( h ) - ν 1 r ( e ) + ν 1 r ( h ) + ν 2 r ( e ) - ν 2 r ( h ) - ν 2 r ( e ) + ν 2 r ( h ) + ] T .
ρ j = e j n j λ ,
L m ( η ) = 1 d exp [ i η a ( x ) - i m K x ] d x ,
A m n ( η ) = α m L m - n ( η ) ,
B m n ( ± 1 ) ( η ) = k ˜ ( ± 1 ) 2 - α m α n β n ( ± 1 ) K m - n ( η ) ,
K 1 m q ( ± 1 ) = l = 1 N L m - l [ - λ q ( ± 1 ) ] W 1 l q ( ± 1 ) ,
K 2 m q ( ± 1 ) = l = 1 N L m - l [ - λ q ( ± 1 ) ] W 2 l q ( ± 1 ) ,
K 3 m q ( ± 1 ) = l = 1 N L m - l [ - λ q ( ± 1 ) ] × [ - τ 1 ( ± 1 ) λ q ( ± 1 ) W 1 l q ( ± 1 ) + τ 3 ( ± 1 ) α m W 2 l q ( ± 1 ) ] ,
K 4 m q ( ± 1 ) = l = 1 N L m - l [ - λ q ( ± 1 ) ] × [ + τ 3 ( ± 1 ) α m W 1 l q ( ± 1 ) + τ 2 ( ± 1 ) λ q ( ± 1 ) W 2 l q ( ± 1 ) ] ,
Z f ( + 1 ) = [ L m - l [ β l ( + 1 ) ] 0 W 1 m q ( + 1 ) L m - n [ - β n ( + 1 ) ] 0 0 L m - l [ β l ( + 1 ) ] W 2 m q ( + 1 ) 0 L m - n [ - β n ( + 1 ) ] - τ 1 ( + 1 ) B m l ( + 1 ) [ β l ( + 1 ) ] τ 3 ( + 1 ) A m l [ β l ( + 1 ) ] W 3 m q ( + 1 ) - τ 1 ( + 1 ) B m n ( + 1 ) [ - β n ( + 1 ) ] τ 3 ( + 1 ) A m n [ - β n ( + 1 ) ] τ 3 ( + 1 ) A m l [ β l ( + 1 ) ] τ 2 ( + 1 ) B m l ( + 1 ) [ β l ( + 1 ) ] W 4 m q ( + 1 ) τ 3 ( + 1 ) A m n [ - β n ( + 1 ) ] τ 2 ( + 1 ) B m n ( + 1 ) [ - β n ( + 1 ) ] ] ,
l U ( + 1 ) ,             m U ,             n U ,             q V ( + 1 ) ;
Z f ( - 1 ) = [ L m - l [ β l ( - 1 ) ] 0 L m - n [ - β n ( - 1 ) ] 0 W 1 m q ( - 1 ) 0 L m - l [ β l ( - 1 ) ] 0 L m - n [ - β n ( - 1 ) ] W 2 m q ( - 1 ) - τ 1 ( - 1 ) B m l ( - 1 ) [ β l ( - 1 ) ] τ 3 ( - 1 ) A m l [ β l ( - 1 ) ] - τ 1 ( - 1 ) B m n ( - 1 ) [ - β n ( - 1 ) ] τ 3 ( - 1 ) A m n [ - β n ( - 1 ) ] W 3 m q ( - 1 ) τ 3 ( - 1 ) A m l [ β l ( - 1 ) ] τ 2 ( - 1 ) B m l ( - 1 ) [ β l ( - 1 ) ] τ 3 ( - 1 ) A m n [ - β n ( - 1 ) ] τ 2 ( - 1 ) B m n ( - 1 ) [ - β n ( - 1 ) ] W 4 m q ( - 1 ) ] ,
l U ,             m U ,             n U ( - 1 ) ,             q V ( - 1 ) ;
Z g ( + 1 ) = [ δ m l 0 K 1 m q ( + 1 ) δ m n 0 0 δ m l K 2 m q ( + 1 ) 0 δ m n - τ 1 ( + 1 ) B m ( + 1 ) δ m l τ 3 ( + 1 ) α m δ m l K 3 m q ( + 1 ) τ 1 ( + 1 ) β m ( + 1 ) δ m n τ 3 ( + 1 ) α m δ m n τ 3 ( + 1 ) α m δ m l τ 2 ( + 1 ) β m ( + 1 ) δ m l K 4 m q ( + 1 ) τ 3 ( + 1 ) α m δ m n - τ 2 ( + 1 ) β m ( + 1 ) δ m n ] ,
l U ( + 1 ) ,             m U ,             n U ,             q V ( + 1 ) ;
Z g ( - 1 ) = [ δ m l 0 δ m n 0 K 1 m q ( - 1 ) 0 δ m l 0 δ m n K 2 m q ( - 1 ) - τ 1 ( - 1 ) B m ( - 1 ) δ m l τ 3 ( - 1 ) α m δ m l τ 1 ( - 1 ) β m ( - 1 ) δ m n τ 3 ( - 1 ) α m δ m n K 3 m q ( - 1 ) τ 3 ( - 1 ) α m δ m l τ 2 ( - 1 ) β m ( - 1 ) δ m l τ 3 ( - 1 ) α m δ m n - τ 2 ( - 1 ) β m ( - 1 ) δ m n K 4 m q ( - 1 ) ] ,
l U ,             m U ,             n U ( - 1 ) ,             q V ( - 1 ) .
a ( x ) = h 2 [ 1 - cos ( K x ) ]
C m = { 0 m odd 1 ( 1 + p 2 ) 1 / 2 [ p 1 + ( 1 + p 2 ) 1 / 2 ] m m even ,
D m = { 0 m even sgn ( m ) - i ( 1 + p 2 ) 1 / 2 [ p 1 + ( 1 + p 2 ) 1 / 2 ] m m odd ,
L m ( η ) = i m exp ( i h 2 η ) J m ( - h 2 η ) ,

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