Abstract

A new perturbation method for the diffraction of a plane wave by a grating with periodic imperfections is presented. The originality of the method lies in the fact that the perturbation occurs on a reference profile that is not a plane but a grating. First, the diffraction by a reference grating is treated. At this stage Maxwell’s equations are used in covariant form written in a nonorthogonal coordinate system fitted to the surface geometry. Second, the periodic errors are taken into account. The tensorial formalism permits the elaboration of this two-roughness-level model. The grating profile appears only through two fundamental functions. The variations of these functions under the effect of variation in the profile are expanded in power series of the perturbation parameter v1. v1 represents the maximum of the derivative of the function describing the perturbation. By using this formalism, we determine the efficiencies.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Breidne, D. Maystre, “Variational theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. 72, 499–506 (1982).
    [CrossRef]
  2. N. A. Finkelstein, C. H. Brawley, R. J. Meltzer, “The reduction of ghosts on diffraction spectra,” J. Opt. Soc. Am. 42, 121–126 (1952).
    [CrossRef]
  3. H. A. Rowland, “Gratings in theory and practice,” Philos. Mag. 35, 397 (1893).
  4. A. Maréchal, “La diffusion résiduelle des surfaces polies et des réseaux,” Opt. Acta 5, 70–74 (1958).
    [CrossRef]
  5. A. Lohman, “Contrast transfer in the grating spectrograph,” Opt. Acta 6, 175–185 (1959).
    [CrossRef]
  6. E. Ingelstam, E. Djurle, “The study of diffraction grating characteristics by simplified phase contrast methods,” J. Opt. Soc. Am. 43, 572–580 (1953).
    [CrossRef]
  7. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Heidelberg, 1980).
    [CrossRef]
  8. E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).
  9. J. Chandezon, “Les équations de Maxwell sous forme co-variante—application à l’étude de la propagation dans les guides périodiques et à la diffraction par les réseaux,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1979).
  10. M. Lichnerowicz, Eléments de Calcul Tensoriel (Armand Colin, Paris, 1950).
  11. J. M. Elson, R. H. Ritchie, “Photon interaction at a rough metal surface,” Phys. Rev. B 4, 4129–4138 (1971).
    [CrossRef]
  12. E. Marcelin, “Application des équations de Maxwell co-variantes à l’étude de la propagation dans les guides d’ondes coudés,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1992).
  13. 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]
  14. 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]
  15. A. Benali, J. Chandezon, J. Fontaine, “A new theory for scattering of electromagnetic waves from conducting or dielectric rough surfaces,” IEEE Trans. Antennas Propag. 40, 141–148 (1992).
    [CrossRef]
  16. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
    [CrossRef]
  17. R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B 262, 468–471 (1966).
  18. P. M. van den Berg, “Reflection by a grating: Rayleigh methods,” J. Opt. Soc. Am. 71, 1224–1229 (1981).
    [CrossRef]
  19. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [CrossRef]
  20. P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
    [CrossRef]
  21. N. Nieto-Vesperinas, “Depolarization of EM waves scattered from slightly rough random surfaces: a study by means of the extinction theorem,” J. Opt. Soc. Am. 72, 539–547 (1982).
    [CrossRef]
  22. J. M. Soto-Crespo, M. Nieto-Vesperinas, A. T. Friberg, “Scattering from slightly rough random surfaces: a detailed study on the validity of the small perturbation method,” J. Opt. Soc. Am. A 7, 1185–1201 (1990).
    [CrossRef]
  23. R. Dusséaux, “Etude de la diffraction d’une onde plane par un réseau—equations de Maxwell covariantes et méthodes de perturbation,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1993).
  24. V. P. Maslov, Théorie des Perturbations et des Méthodes Asymptotiques (Dunod, Paris, 1972).
  25. G. Zepp, Mécanique Quantique, Exercices avec Solutions (Vuibert, Paris, 1975).
  26. G. Burns, Introduction to Group Theory with Applications (Academic, New York, 1977).
  27. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
  28. D. Maystre, “General study of grating anomalies from electromagnetic surfaces modes,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), Chap. 17.
  29. M. Saillard, “Etude théorique et numérique de la diffraction de la lumière par des surfaces rugueuses diélectriques et conductrices,” Ph.D. dissertation (Université Aix-Marseille III, Marseille, France, 1990).
  30. D. Maystre, J. P. Rossi, “Implementation of a rigorous vector theory of speckle for two-dimensional microrough surfaces,” J. Opt. Soc. Am. A 3, 1276–1282 (1986).
    [CrossRef]

1992 (1)

A. Benali, J. Chandezon, J. Fontaine, “A new theory for scattering of electromagnetic waves from conducting or dielectric rough surfaces,” IEEE Trans. Antennas Propag. 40, 141–148 (1992).
[CrossRef]

1990 (1)

1986 (1)

1982 (3)

1981 (1)

1980 (1)

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]

1975 (1)

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

1971 (1)

J. M. Elson, R. H. Ritchie, “Photon interaction at a rough metal surface,” Phys. Rev. B 4, 4129–4138 (1971).
[CrossRef]

1966 (1)

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B 262, 468–471 (1966).

1959 (1)

A. Lohman, “Contrast transfer in the grating spectrograph,” Opt. Acta 6, 175–185 (1959).
[CrossRef]

1958 (1)

A. Maréchal, “La diffusion résiduelle des surfaces polies et des réseaux,” Opt. Acta 5, 70–74 (1958).
[CrossRef]

1953 (1)

1952 (1)

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

1907 (1)

Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

1893 (1)

H. A. Rowland, “Gratings in theory and practice,” Philos. Mag. 35, 397 (1893).

Benali, A.

A. Benali, J. Chandezon, J. Fontaine, “A new theory for scattering of electromagnetic waves from conducting or dielectric rough surfaces,” IEEE Trans. Antennas Propag. 40, 141–148 (1992).
[CrossRef]

Brawley, C. H.

Breidne, M.

Burns, G.

G. Burns, Introduction to Group Theory with Applications (Academic, New York, 1977).

Cadilhac, M.

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B 262, 468–471 (1966).

Chandezon, J.

A. Benali, J. Chandezon, J. Fontaine, “A new theory for scattering of electromagnetic waves from conducting or dielectric rough surfaces,” IEEE Trans. Antennas Propag. 40, 141–148 (1992).
[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]

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 co-variante—application à l’étude de la propagation dans les guides périodiques et à la diffraction par les réseaux,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1979).

Cornet, G.

Djurle, E.

Dupuis, M. T.

Dusséaux, R.

R. Dusséaux, “Etude de la diffraction d’une onde plane par un réseau—equations de Maxwell covariantes et méthodes de perturbation,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1993).

Elson, J. M.

J. M. Elson, R. H. Ritchie, “Photon interaction at a rough metal surface,” Phys. Rev. B 4, 4129–4138 (1971).
[CrossRef]

Finkelstein, N. A.

Fontaine, J.

A. Benali, J. Chandezon, J. Fontaine, “A new theory for scattering of electromagnetic waves from conducting or dielectric rough surfaces,” IEEE Trans. Antennas Propag. 40, 141–148 (1992).
[CrossRef]

Friberg, A. T.

Ingelstam, E.

Lichnerowicz, M.

M. Lichnerowicz, Eléments de Calcul Tensoriel (Armand Colin, Paris, 1950).

Lohman, A.

A. Lohman, “Contrast transfer in the grating spectrograph,” Opt. Acta 6, 175–185 (1959).
[CrossRef]

Marcelin, E.

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

Maréchal, A.

A. Maréchal, “La diffusion résiduelle des surfaces polies et des réseaux,” Opt. Acta 5, 70–74 (1958).
[CrossRef]

Maslov, V. P.

V. P. Maslov, Théorie des Perturbations et des Méthodes Asymptotiques (Dunod, Paris, 1972).

Maystre, D.

Meltzer, R. J.

Nieto-Vesperinas, M.

Nieto-Vesperinas, N.

Petit, R.

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B 262, 468–471 (1966).

Post, E. J.

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

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]

Rayleigh,

Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Ritchie, R. H.

J. M. Elson, R. H. Ritchie, “Photon interaction at a rough metal surface,” Phys. Rev. B 4, 4129–4138 (1971).
[CrossRef]

Rossi, J. P.

Rowland, H. A.

H. A. Rowland, “Gratings in theory and practice,” Philos. Mag. 35, 397 (1893).

Saillard, M.

M. Saillard, “Etude théorique et numérique de la diffraction de la lumière par des surfaces rugueuses diélectriques et conductrices,” Ph.D. dissertation (Université Aix-Marseille III, Marseille, France, 1990).

Soto-Crespo, J. M.

van den Berg, P. M.

Waterman, P. C.

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

Zepp, G.

G. Zepp, Mécanique Quantique, Exercices avec Solutions (Vuibert, Paris, 1975).

C. R. Acad. Sci. B (1)

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B 262, 468–471 (1966).

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. Benali, J. Chandezon, J. Fontaine, “A new theory for scattering of electromagnetic waves from conducting or dielectric rough surfaces,” IEEE Trans. Antennas Propag. 40, 141–148 (1992).
[CrossRef]

J. Acoust. Soc. Am. (1)

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

J. Opt. (Paris) (1)

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. Opt. Soc. Am. (6)

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

Opt. Acta (2)

A. Maréchal, “La diffusion résiduelle des surfaces polies et des réseaux,” Opt. Acta 5, 70–74 (1958).
[CrossRef]

A. Lohman, “Contrast transfer in the grating spectrograph,” Opt. Acta 6, 175–185 (1959).
[CrossRef]

Philos. Mag. (2)

H. A. Rowland, “Gratings in theory and practice,” Philos. Mag. 35, 397 (1893).

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

Phys. Rev. B (1)

J. M. Elson, R. H. Ritchie, “Photon interaction at a rough metal surface,” Phys. Rev. B 4, 4129–4138 (1971).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Other (11)

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

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Heidelberg, 1980).
[CrossRef]

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

J. Chandezon, “Les équations de Maxwell sous forme co-variante—application à l’étude de la propagation dans les guides périodiques et à la diffraction par les réseaux,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1979).

M. Lichnerowicz, Eléments de Calcul Tensoriel (Armand Colin, Paris, 1950).

D. Maystre, “General study of grating anomalies from electromagnetic surfaces modes,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), Chap. 17.

M. Saillard, “Etude théorique et numérique de la diffraction de la lumière par des surfaces rugueuses diélectriques et conductrices,” Ph.D. dissertation (Université Aix-Marseille III, Marseille, France, 1990).

R. Dusséaux, “Etude de la diffraction d’une onde plane par un réseau—equations de Maxwell covariantes et méthodes de perturbation,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1993).

V. P. Maslov, Théorie des Perturbations et des Méthodes Asymptotiques (Dunod, Paris, 1972).

G. Zepp, Mécanique Quantique, Exercices avec Solutions (Vuibert, Paris, 1975).

G. Burns, Introduction to Group Theory with Applications (Academic, New York, 1977).

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 (8)

Fig. 1
Fig. 1

Perfectly conducting grating. S separates the air [y > a(x)] from a perfectly conducting metal [ya(x)].

Fig. 2
Fig. 2

Description of the coupling (D1 = 3D0). The coupling between wave functions Ψ1n and Ψ2n is not described. (a) Unperturbed problem. Diffracted field: reference waves combination; real incidence θ: 2M + 1 reference waves Ψ 0 n ( 0 ); fictive incidence θ1: 2M supplementary waves Ψ 1 n ( 0 ); fictive incidence θ2: 2M supplementary waves Ψ 1 n ( 0 ). (b) Perturbed problem. Diffracted field: reference waves and supplementary waves combination.

Fig. 3
Fig. 3

Supplementary efficiencies ξ−1 and ξ1 versus angle of incidence for the E|| polarization case; λ = 0.85. The circles and the solid curve show the first and minus-first orders, respectively, obtained from the RDM. The crosses and the asterisks give the same efficiencies calculated with the PM. a(x) = h0 cos(K0x) + h1 cos(K1x), (h0; D0) = (0.5; 1), (h1; D1) = (0.04; 2). Reference orders: −4, −2, 0, 2. Orders −2, −1, 0, 1, 2: θ ∈ [0°, 8.6°]. Orders −2, −1, 0, 1: θ ∈ [8.6°, 16°]. Orders −3, −2, −1, 0, 1: θ ∈ [16°, 35.1°]. Orders −3, −2, −1, 0: θ ∈ [35.1°, 44.4°]. Orders −4, −3, −2, − 1, 0: θ ∈ [44.4°, 90°].

Fig. 4
Fig. 4

Supplementary efficiencies ξ−1 and ξ1 versus angle of incidence for the H|| polarization case; λ = 0.85. Symbols and parameters as for Fig. 3.

Fig. 5
Fig. 5

Supplementary efficiencies ξ−1 and ξ1 versus angle of incidence for the E|| polarization case; λ = 1.1. Symbols and parameters as for Fig. 3. Reference orders: −2, 0. Orders −1, 0, 1: θ ∈ [0°, 5.7°]. Orders −2, −1, 0, 1: θ ∈ [5.7°, 26.7°]. Orders −2, −1, 0: θ ∈ [26.7°, 40.5°]. Orders −2, − 1, 0: θ ∈ [40.5°, 90°].

Fig. 6
Fig. 6

Supplementary efficiencies ξ−1 and ξ1 versus angle of incidence for the H|| polarization case; λ = 1.1. Symbols and parameters as for Fig. 3.

Fig. 7
Fig. 7

Supplementary efficiencies ξ−1 and ξ1 versus angle of incidence for the E|| polarization case; λ = 1.35. Symbols and parameters as for Fig. 3. Reference orders: −2, 0. Orders − 1, 0, 1: θ ∈ [0°, 18.9°]. Orders −1, 0: θ ∈ [18.9°, 20.5°]. Orders −2, −1, 0: θ ∈ [20.5°, 90°]

Fig. 8
Fig. 8

Supplementary efficiencies ξ−1 and ξ1 versus angle of incidence for the H|| polarization case; λ = 1.35. Symbols and parameters as for Fig. 3.

Tables (6)

Tables Icon

Table 1 Relations of Normalization

Tables Icon

Table 2 Periods of Different Surfaces

Tables Icon

Table 3 Limiting Values h1,l for D = 2a

Tables Icon

Table 4 Limiting Values h1,l for D = 2a

Tables Icon

Table 5 Sum of Supplementary Efficiencies for E|| Polarizationa

Tables Icon

Table 6 Sum of Supplementary Efficiencies for H || Polarizationa

Equations (83)

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

{ E z ( E ) i Z H z ( H ) i } = F i ( x , y ) = F 0 exp ( - i α 0 x + i β 0 y ) , Z = μ 0 / 0 = 120 π ,             F 0 = 1 ,             α 0 = k sin θ , β 0 = k cos θ ,             k = 2 π / λ .
- 1 < sin θ n = sin θ + n λ D < 1.
ξ i j k j E k = - B i t , ξ i j k j H k = D i t + J i , i D i = ρ , i B i = 0 , i , j , k { 1 , 2 , 3 } , i = / x i .
i a i j = i = 1 3 i a i j .
E i = A i i E i ,             B i = A i i - 1 A i i B i , H i = A i i H i             D i = A i i - 1 A i i D i , A i i = x i / x i ,             A i i = x i / x i ,             i , i = 1 , 2 , 3.
D i = i j E j ,             B i = μ i j H j .
D i = E j ,             B i = μ H j ,
i j = g g i j ,             μ i j = μ g g i j ,             g = g i j .
( x 1 = x x 2 = y x 3 = z ) ( x 1 = x x 2 = u = y - a ( x ) x 3 = z ) .
L ψ ( x , u ) = 1 i k Ψ ( x , u ) u ,             Ψ ( x , u ) = [ F ( x , u ) G ( x , u ) ] , L = [ L 11 L 12 L 21 L 22 ]
= [ d ( x ) i k x - c ( x ) - 1 k 2 { x [ c ( x ) x ] + k 2 } 1 i k x [ d ( x ) ] ] ,
c ( x ) = 1 1 + a ( x ) 2 ,             a ( x ) = d a ( x ) d x , d ( x ) = a ( x ) 1 + a ( x ) 2 .
F ( x , u ) = E z ( x , u ) ,             G ( x , u ) = Z H x ( x , u ) , Z H u ( x , u ) = - i c ( x ) k F ( x , u ) x + d ( x ) G ( x , u ) , E x ( x , u ) = E u ( x , u ) = H z ( x , u ) = 0 ;
F ( x , u ) = Z H z ( x , u ) ,             G ( x , u ) = - E x ( x , u ) , E u ( x , u ) = i c ( x ) k F ( x , u ) x - d ( x ) G ( x , u ) , H x ( x , u ) = H u ( x , u ) = E z ( x , u ) = 0.
F ( x , u = 0 ) = E z ( x , u = 0 ) = 0 ;
G ( x , u = 0 ) = - E x ( x , u = 0 ) = 0.
Ψ ( x , u ) = ϕ ( x ) exp ( i k r u ) , L ϕ ( x ) = r ϕ ( x ) ,             ϕ ( x ) = [ f ( x ) g ( x ) ] .
ϕ ( x ) = m = - m = + ϕ m exp ( - i α m x ) , ϕ m = [ f m g m ] ,             α m = k sin θ + 2 π m / D .
[ exp ( - i α n x ) , exp ( - i α m x ) ] = 1 D 0 D exp ( - i α n x ) * exp ( - i α m x ) d x = δ n m .
c ( x ) = q = - q = + c q exp ( - i q K x ) , d ( x ) = q = - q = + d q exp ( - i q K x ) ,             ,             K = 2 π / D .
[ L ] ϕ = r ϕ [ [ L 11 ] [ L 12 ] [ L 21 ] [ L 22 ] ] ( f g ) = r ( f g ) ,
[ L 11 ] m q = - α q k d m - q , [ L 12 ] m q = - c m - q , [ L 21 ] m q = α m α q k 2 c m - q - δ m q , [ L 22 ] m q = - α m k d m - q ,
f = ( · f - m · f 0 · f m · ) ,             g = ( · g - m · g 0 · g m · ) .
f m = g m = 0             if m > M .
Ψ ( x , u ) = n = 1 4 M + 2 C n [ m = - M M ( ϕ n ) m exp ( - i α m x ) ] exp ( i k r n u ) .
r n 2 = cos θ n 2 = 1 - ( sin θ + n λ / D ) 2 .
( r n * - r n ) ϕ n , ϕ n = 0 ,
ϕ n , ϕ n = [ g n t f n t ] [ f n , g n ] = 1 D 0 D [ g n * ( x ) f n ( x ) + f n * ( x ) g n ( x ) ] d x = m = - M m = M ( g n ) m * ( f n ) m + ( f n ) m * ( g n ) m .
ϕ n , ϕ n = 0.
ϕ n , ϕ n = η n ,             η n C _ .
n = 1 2 M + 1 C n ( f n ) m = - ( f i ) m , n = 1 2 M + 1 C n ( g n ) m = - ( g i ) m ,             m [ - M , M ] .
ξ n = - C n 2 ϕ n , ϕ n ϕ i , ϕ i ,
ξ n = C n 2 .
L ( x , v 1 ) = L ( 0 ) ( x ) + n = 1 + L ( n ) ( x , v 1 ) .
L ( 1 ) = [ L 11 ( 1 ) L 12 ( 1 ) L 21 ( 1 ) L 22 ( 1 ) ] = [ v 1 i k d 1 ( x ) x - v 1 c 1 ( x ) - v 1 k 2 x [ c 1 ( x ) x ] v 1 i k x [ d 1 ( x ) ] ] ,
c 1 ( x ) = [ c ( x , v 1 ) v 1 ] v 1 = 0 = 2 c 0 ( x ) d 0 ( x ) b 1 ( x ) , d 1 ( x ) = [ d ( x , v 1 ) v 1 ] v 1 = 0 = [ d 0 ( x ) 2 - c 0 ( x ) 2 ] b 1 ( x ) ,
c 0 ( x ) = 1 1 + h 0 2 a 0 ( x ) 2 ,             d 0 ( x ) = h 0 a 0 ( x ) 1 + h 0 2 a 0 ( x ) 2 .
L ( 0 ) ϕ n ( 0 ) ( x ) = r n ( 0 ) ϕ n ( 0 ) ( x ) .
ϕ n ( 0 ) ( x ) = ϕ 0 n ( 0 ) ( x ) + p = 1 n 0 - 1 ϕ p n ( 0 ) ( x ) ,
ϕ 0 n ( 0 ) ( x ) = m = - M M ( ϕ 0 n ( 0 ) ) m exp ( - i α n 0 m x ) , ϕ p 0 n ( 0 ) ( x ) = m = - M M - 1 ( ϕ p n ( 0 ) ) m exp ( - i α n 0 m + p x ) , α n 0 m + p = α 0 + ( n 0 m + p ) K .
[ L p ( 0 ) ] ϕ p n ( 0 ) = r p n ( 0 ) ϕ p n ( 0 ) ,             [ L p ( 0 ) ] = [ [ L 11 p ( 0 ) ] [ L 12 p ( 0 ) ] [ L 21 p ( 0 ) ] [ L 22 p ( 0 ) ] ] ,
ϕ p n ( 0 ) = [ f p n ( 0 ) g p n ( 0 ) ] ,             p [ 0 , n 0 - 1 ] ,
[ L i j p ( 0 ) ] q r = 1 D 0 D exp ( - i α n 0 q + p x ) * L i j ( 0 ) exp ( - i α n 0 r + p x ) d x ,             i , j { 1 , 2 } .
α m n 0 + p = k [ sin θ + ( m n 0 + p ) λ / D ] = k [ ( sin θ + p λ / D ) + m λ / D 0 ] .
r p n ( 0 ) 2 = 1 - [ sin θ + ( n n 0 + p ) λ / D ] 2 = 1 - [ ( sin θ + p λ / D ) + n λ / D 0 ] 2 ,             n U .
Ψ ( x , u ) = Ψ i ( 0 ) ( x , u ) + Ψ d ( 0 ) ( x , u ) , Ψ d ( 0 ) ( x , u ) = n = 1 2 M + 1 C 0 n ( 0 ) Ψ 0 n ( 0 ) ( x , u ) .
ξ 0 n ( 0 ) = C 0 n ( 0 ) 2 , n U ξ 0 n ( 0 ) = 1 , ξ t n ( 0 ) = C t n ( 0 ) 2 = 0 , t 0.
( L ( 0 ) + L ( 1 ) ) ϕ t n ( x ) = r t n ϕ t n ( x ) ;
ϕ t n ( x ) = ϕ t n ( 0 ) ( x ) + ϕ t n ( 1 ) ( x ) , r t n = r t n ( 0 ) + r t n ( 1 ) ,             t [ 0 , n 0 - 1 ] , ψ t n ( x , u ) = [ ϕ t n ( 0 ) ( x ) + ϕ t n ( 1 ) ( x ) ] exp [ i k ( r t n ( 0 ) + r t n ( 1 ) ) u ] .
ϕ t n ( 1 ) ( x ) = m = 1 4 M + 2 a 0 m t n ( 1 ) ϕ 0 m ( 0 ) ( x ) + p = 1 n 0 - 1 [ m = 1 4 M a p m t n ( 1 ) ϕ p m ( 0 ) ( x ) ] ,             t [ 0 , n 0 - 1 ] .
Ψ ( x , u ) = Ψ 0 i ( x , u ) + Ψ d ( x , u ) ,
Ψ d ( x , u ) = n = 1 2 M + 1 C 0 n ( 0 ) ϕ 0 n ( 0 ) ( x ) exp [ i k ( r 0 n ( 0 ) + r 0 n ( 1 ) ) u ] + n = 1 2 M + 1 [ C 0 n ( 0 ) ϕ 0 n ( 1 ) ( x ) + C 0 n ( 1 ) ϕ 0 n ( 0 ) ( x ) ]
× exp [ i k ( r 0 n ( 0 ) + r 0 n ( 1 ) ) u ] + t = 1 n 0 - 1 { n = 1 2 M C t n ( 1 ) ϕ t n ( 0 ) ( x ) exp [ i k ( r t n ( 0 ) + r t n ( 1 ) ) u ] } .
[ L i j p t ( 1 ) ] q r = 1 D 0 D exp ( - i α n 0 q + p x ) * L i j ( 1 ) exp ( - i α n 0 r + t x ) d x ,             i , j = 1 , 2.
r 0 n ( 1 ) = ϕ 0 n , [ L 00 ( 1 ) ] ϕ 0 n ( 0 ) ϕ 0 n , ϕ 0 n ( 0 ) ,             n [ 1 , 2 M + 1 [             or             n = i ,
If r p m ( 0 ) r 0 n ( 0 ) ,             a p m 0 n ( 1 ) = ϕ p m , [ L p 0 ( 1 ) ] ϕ 0 n ( 0 ) ( r p m ( 0 ) - r 0 n ( 0 ) ) < ϕ p m , ϕ p m ( 0 ) ,             n [ 1 , 2 M + 2 ]             or             n = i .
n = 1 2 M + 1 C 0 n ( 1 ) f 0 n ( 0 ) ( x ) + t = 1 n 0 - 1 [ n = 1 2 M c t n ( 1 ) f t n ( 0 ) ( x ) ] = - n = 1 2 M + 1 c 0 n ( 0 ) f 0 n ( 1 ) ( x ) - f 0 i ( 1 ) ( x ) .
n = 1 2 M + 1 C 0 n ( 1 ) ( f 0 n ( 0 ) ) q = - m = 1 4 M + 2 a 0 m 0 i ( 1 ) ( f 0 m ( 0 ) ) q - n = 1 2 M + 1 C 0 n ( 0 ) m = 1 4 M + 2 a 0 m 0 n ( 1 ) ( f 0 m ( 0 ) ) q , q [ - M , M ] ,
n = 1 2 M C t n ( 1 ) ( f t n ( 0 ) ) q = - m = 1 4 M a t m 0 i ( 1 ) ( f t m ( 0 ) ) q - n = 1 2 M + 1 C 0 n ( 0 ) m = 1 4 M a t m 0 n ( 1 ) ( f t m ( 0 ) ) q , q [ - M , M - 1 ] ,             t [ 1 , n 0 - 1 ] .
ξ t n = ξ t n ( 2 ) = C t n ( 1 ) 2 ,             t 0.
ξ n 0 = ξ n 0 ( 2 ) = 4 ( k h 1 ) 2 cos θ cos θ n ( a n ) n 2 ,             ξ 0 = 1 - n U ξ n 0 .
ξ n 0 = ξ n 0 ( 2 ) = 4 ( k h 1 ) 2 cos θ cos θ n ( 1 - sin θ sin θ n ) 2 ( a 1 ) n 2 ,             ξ 0 = 1 - n U ξ n 0 .
v 1 1 ,             h 1 / D 1 1 / 2 π .
Δ g = 2 h 1 cos ( 2 π x / n 0 D 0 + π / n 0 ) sin ( π / n 0 ) 2 π h 1 / n 0 ,
Δ d = 2 h 1 K 1 sin ( 2 π x / n 0 D 0 + π / n 0 ) sin ( π / n 0 ) 4 π 2 h 1 / n 0 2 D 0 .
L ( 0 ) ϕ t n ( 1 ) ( x ) + L ( 1 ) ϕ t n ( 0 ) ( x ) = r t n ( 0 ) ϕ t n ( 1 ) ( x ) + r t n ( 1 ) ϕ t n ( 0 ) ( x ) ,             t [ 0 , n 0 - 1 ] .
m = 1 4 M + 2 a 0 m t n ( 1 ) ( r 0 m ( 0 ) - r t n ( 0 ) ) ϕ 0 m ( 0 ) ( x ) + p = 1 n 0 - 1 [ m = 1 4 M a p m t n ( 1 ) ( r p m ( 0 ) - r t n ( 0 ) ) ϕ p m ( 0 ) ( x ) ] + L ( 1 ) ϕ t n ( 0 ) ( x ) = r t n ( 1 ) ϕ t n ( 0 ) ( x ) ,             t [ 0 , n 0 - 1 ] .
m = 1 4 M + 2 a 0 m 0 n ( 1 ) ( r 0 m ( 0 ) - r 0 n ( 0 ) ) ϕ 0 m ( 0 ) + [ L 00 ( 1 ) ] ϕ 0 n ( 0 ) = r 0 n ( 1 ) ϕ 0 n ( 0 ) ,             n [ 1 , 2 M + 1 ]             or             n = i ,
m = 1 4 M a p m 0 n ( 1 ) ( r p m ( 0 ) - r 0 n ( 0 ) ) ϕ p m ( 0 ) + [ L p 0 ( 1 ) ] ϕ 0 n ( 0 ) = 0 ,             n [ 1 , 2 M + 1 ]             or             n = i ,             p [ 1 , n 0 - 1 ] ,
m = 1 4 M + 2 a 0 m t n ( 1 ) ( r 0 m ( 0 ) - r t n ( 0 ) ) ϕ 0 m ( 0 ) + [ L 0 t ( 1 ) ] ϕ t n ( 0 ) = 0 ,             n [ 1 , 2 M ] ,             t [ 1 , n 0 - 1 ] ,
m = 1 4 M a p m t n ( 1 ) ( r p m ( 0 ) - r t n 0 ) ϕ p m ( 0 ) + [ L p t ( 1 ) ] ϕ t n ( 0 ) = r t n ( 1 ) ϕ t n ( 0 ) ,             n [ 1 , 2 M ] ,             ( p , t ) [ 1 , n 0 - 1 ] 2 .
d 1 ( x ) = q r l q ( b 1 ) r exp [ - i ( q n 0 + r n 1 ) K x ] ,
c 1 ( x ) = q r k q ( b 1 ) r exp [ - i ( q n 0 + r n 1 ) K x ] ,
l q = r ( d 0 ) r ( d 0 ) q - r - ( c 0 ) r ( c 0 ) q - r ,
k q = 2 r ( c 0 ) r ( d 0 ) q - r .
[ L p t ( 1 ) ] = [ [ L 11 p t ( 1 ) ] [ L 12 p t ( 1 ) ] [ L 21 p t ( 1 ) ] [ L p t 22 ( 1 ) ] ] .
1 = 1 D 0 D exp ( - i α n 0 q + p x ) L i j ( 1 ) exp ( - i α n 0 r + t x ) d x .
[ L 11 p t ( 1 ) ] q r = - v 1 k α n 0 r + t w l u 0 + n 1 w ( b 1 ) u 1 - n 0 w , [ L 12 p t ( 1 ) ] q r = - v 1 w k u 0 + n 1 w ( b 1 ) u 1 - n 0 w , [ L 21 p t ( 1 ) ] q r = v 1 k 2 α n 0 r + t α n 0 q + p w k u 0 + n 1 w ( b 1 ) u 1 - n 0 w , [ L p t 22 ( 1 ) ] q r = - v 1 k α n 0 q + p w l u 0 + n 1 w ( b 1 ) u 1 - n 0 w ,
u 0 n 0 + u 1 n 1 = n 0 ( q - r ) + p - t .
m = 1 4 M + 2 a 0 m 0 n ( 1 ) ( r 0 m ( 0 ) - r 0 n ( 0 ) ) ϕ 0 m ( 0 ) , ϕ 0 m ( 0 ) + ϕ 0 m ( 0 ) , [ L 00 ( 1 ) ] ϕ 0 n ( 0 ) = r 0 n ( 1 ) ϕ 0 m ( 0 ) , ϕ 0 n ( 0 ) , n [ 1 , 2 M + 1 ]             or             n = i ,             m [ 1 , 4 M + 2 ] ,
m = 1 4 M a p m 0 n ( 1 ) ( r p m ( 0 ) - r 0 n ( 0 ) ) ϕ p m ( 0 ) , ϕ p m ( 0 ) + ϕ p m ( 0 ) , [ L p 0 ( 1 ) ] ϕ 0 n ( 0 ) = 0 , n [ 1 , 2 M + 1 ]             or             n = i , p [ 1 , n 0 - 1 ] ,             m [ 1 , 4 M ] .
ϕ 0 n ( 0 ) + ϕ 0 n ( 1 ) , ϕ 0 n ( 0 ) + ϕ 0 n ( 1 ) = 1 ,             n [ 1 , 2 M + 1 ] .
a 0 n 0 n ( 1 ) + a 0 n 0 n ( 1 ) * = 0.

Metrics