Abstract

We consider the diffraction of a p -polarized electromagnetic wave incident from the vacuum side onto a one-dimensional grating ruled on the planar surface of a dielectric medium characterized by an isotropic, frequency-dependent dielectric constant (ω). The plane of incidence is assumed to be perpendicular to the grooves of the grating. On the basis of the Rayleigh hypothesis, integral equations are derived for the amplitudes of the reflected and transmitted beams. Formal solutions of these equations are obtained in the form of series of powers of the grating profile function. Recursion relations for determining the successive terms in these expansions are established. It is then pointed out that, if the series for the amplitudes of the reflected and transmitted waves are truncated after any finite number of terms, they have poles at the frequency of a surface polariton on a flat surface and not at the frequency of a surface polariton on the grating surface, in disagreement with the requirements of formal scattering theory. It is shown how the direct iterative series solutions for these coefficients can be rearranged into quotients of two series in powers of the surface-profile function in which the series in the denominator vanishes at the frequency of a surface polariton on the grating surface as required. Recursion relations are derived for the terms in both of these series. The solutions presented here are convenient in not requiring the inversion of large matrices and are well suited for machine computation in addition to possessing the required analytic properties.

© 1983 Optical Society of America

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References

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  1. C. Lopez, F. J. Yndurain, and N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970–972 (1978).
    [Crossref]
  2. N. Garcia, V. Celli, and M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
    [Crossref]
  3. J. Shen and A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
    [Crossref]
  4. N. Garcia, A. A. Maradudin, and V. Celli, “On the Debye–Waller factor in atom-surface scattering,” Philos. Mag. A 45, 287–298 (1982).
    [Crossref]
  5. N. Garcia and A. A. Maradudin, “The quantum Debye–Waller factor in atom surface scattering: angular and temperature dependence,” Surf. Sci. (to be published).
  6. N. Garcia, V. Celli, N. R. Hill, and N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
    [Crossref]
  7. R. K. Rosich and J. R. Wait, “A general perturbation solution for reflection from two-dimensional periodic surfaces,” Radio Sci. 12, 719–729 (1977).
    [Crossref]
  8. See, for example. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), Chap. 21.
  9. B. Laks, D. L. Mills, and A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4975 (1981).
    [Crossref]
  10. Rayleigh, Theory of Sound, 2nd. ed. (Dover, New York, 1945), Vol. II, p. 89.
  11. E. Wolf, in Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, 1973), p. 339.
    [Crossref]
  12. F. Toigo, A. Marvin, V. Celli, and N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
    [Crossref]
  13. N. R. Hill, “Integral-equation perturbative approach to optical scattering from rough surfaces,” Phys. Rev. B 24, 7112–7120 (1981).
    [Crossref]
  14. See, for example, D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
    [Crossref]
  15. A. A. Maradudin, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 405–510.
  16. See, for example, J. T. Cushing, Applied Analytical Mathematics for Physical Scientists (Wiley, New York, 1975), pp. 196–235.

1982 (1)

N. Garcia, A. A. Maradudin, and V. Celli, “On the Debye–Waller factor in atom-surface scattering,” Philos. Mag. A 45, 287–298 (1982).
[Crossref]

1981 (2)

B. Laks, D. L. Mills, and A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4975 (1981).
[Crossref]

N. R. Hill, “Integral-equation perturbative approach to optical scattering from rough surfaces,” Phys. Rev. B 24, 7112–7120 (1981).
[Crossref]

1980 (1)

J. Shen and A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[Crossref]

1979 (1)

N. Garcia, V. Celli, and M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
[Crossref]

1978 (2)

C. Lopez, F. J. Yndurain, and N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970–972 (1978).
[Crossref]

N. Garcia, V. Celli, N. R. Hill, and N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[Crossref]

1977 (2)

R. K. Rosich and J. R. Wait, “A general perturbation solution for reflection from two-dimensional periodic surfaces,” Radio Sci. 12, 719–729 (1977).
[Crossref]

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[Crossref]

1974 (1)

See, for example, D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
[Crossref]

Burstein, E.

See, for example, D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
[Crossref]

Cabrera, N.

N. Garcia, V. Celli, N. R. Hill, and N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[Crossref]

Celli, V.

N. Garcia, A. A. Maradudin, and V. Celli, “On the Debye–Waller factor in atom-surface scattering,” Philos. Mag. A 45, 287–298 (1982).
[Crossref]

N. Garcia, V. Celli, and M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
[Crossref]

N. Garcia, V. Celli, N. R. Hill, and N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[Crossref]

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[Crossref]

Cushing, J. T.

See, for example, J. T. Cushing, Applied Analytical Mathematics for Physical Scientists (Wiley, New York, 1975), pp. 196–235.

Garcia, N.

N. Garcia, A. A. Maradudin, and V. Celli, “On the Debye–Waller factor in atom-surface scattering,” Philos. Mag. A 45, 287–298 (1982).
[Crossref]

N. Garcia, V. Celli, and M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
[Crossref]

C. Lopez, F. J. Yndurain, and N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970–972 (1978).
[Crossref]

N. Garcia, V. Celli, N. R. Hill, and N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[Crossref]

N. Garcia and A. A. Maradudin, “The quantum Debye–Waller factor in atom surface scattering: angular and temperature dependence,” Surf. Sci. (to be published).

Hill, N. R.

N. R. Hill, “Integral-equation perturbative approach to optical scattering from rough surfaces,” Phys. Rev. B 24, 7112–7120 (1981).
[Crossref]

N. Garcia, V. Celli, N. R. Hill, and N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[Crossref]

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[Crossref]

Laks, B.

B. Laks, D. L. Mills, and A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4975 (1981).
[Crossref]

Lopez, C.

C. Lopez, F. J. Yndurain, and N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970–972 (1978).
[Crossref]

Maradudin, A. A.

N. Garcia, A. A. Maradudin, and V. Celli, “On the Debye–Waller factor in atom-surface scattering,” Philos. Mag. A 45, 287–298 (1982).
[Crossref]

B. Laks, D. L. Mills, and A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4975 (1981).
[Crossref]

J. Shen and A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[Crossref]

N. Garcia and A. A. Maradudin, “The quantum Debye–Waller factor in atom surface scattering: angular and temperature dependence,” Surf. Sci. (to be published).

A. A. Maradudin, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 405–510.

Marvin, A.

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[Crossref]

Merzbacher, E.

See, for example. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), Chap. 21.

Mills, D. L.

B. Laks, D. L. Mills, and A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4975 (1981).
[Crossref]

See, for example, D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
[Crossref]

Nieto-Vesperinas, M.

N. Garcia, V. Celli, and M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
[Crossref]

Rayleigh,

Rayleigh, Theory of Sound, 2nd. ed. (Dover, New York, 1945), Vol. II, p. 89.

Rosich, R. K.

R. K. Rosich and J. R. Wait, “A general perturbation solution for reflection from two-dimensional periodic surfaces,” Radio Sci. 12, 719–729 (1977).
[Crossref]

Shen, J.

J. Shen and A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[Crossref]

Toigo, F.

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[Crossref]

Wait, J. R.

R. K. Rosich and J. R. Wait, “A general perturbation solution for reflection from two-dimensional periodic surfaces,” Radio Sci. 12, 719–729 (1977).
[Crossref]

Wolf, E.

E. Wolf, in Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, 1973), p. 339.
[Crossref]

Yndurain, F. J.

C. Lopez, F. J. Yndurain, and N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970–972 (1978).
[Crossref]

Opt. Commun. (1)

N. Garcia, V. Celli, and M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
[Crossref]

Philos. Mag. A (1)

N. Garcia, A. A. Maradudin, and V. Celli, “On the Debye–Waller factor in atom-surface scattering,” Philos. Mag. A 45, 287–298 (1982).
[Crossref]

Phys. Rev. B (6)

J. Shen and A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[Crossref]

N. Garcia, V. Celli, N. R. Hill, and N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[Crossref]

B. Laks, D. L. Mills, and A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4975 (1981).
[Crossref]

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[Crossref]

N. R. Hill, “Integral-equation perturbative approach to optical scattering from rough surfaces,” Phys. Rev. B 24, 7112–7120 (1981).
[Crossref]

C. Lopez, F. J. Yndurain, and N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970–972 (1978).
[Crossref]

Radio Sci. (1)

R. K. Rosich and J. R. Wait, “A general perturbation solution for reflection from two-dimensional periodic surfaces,” Radio Sci. 12, 719–729 (1977).
[Crossref]

Rep. Prog. Phys. (1)

See, for example, D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
[Crossref]

Other (6)

A. A. Maradudin, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 405–510.

See, for example, J. T. Cushing, Applied Analytical Mathematics for Physical Scientists (Wiley, New York, 1975), pp. 196–235.

See, for example. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), Chap. 21.

Rayleigh, Theory of Sound, 2nd. ed. (Dover, New York, 1945), Vol. II, p. 89.

E. Wolf, in Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, 1973), p. 339.
[Crossref]

N. Garcia and A. A. Maradudin, “The quantum Debye–Waller factor in atom surface scattering: angular and temperature dependence,” Surf. Sci. (to be published).

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Equations (65)

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ζ ( x 1 ) = p = - ζ ˆ ( p ) exp ( i 2 π p a x 1 ) ,
( 2 x 1 2 + 2 x 3 2 + ω 2 c 2 ) H 2 > ( x 1 x 3 ω ) = 0 ,             x 3 > ζ ( x 1 ) ,
( 2 x 1 2 + 2 x 3 2 + ( ω ) ω 2 c 2 ) H 2 < ( x 1 x 3 ω ) = 0 ,             x 3 < ζ ( x 1 ) .
H 2 < ( x 1 x 3 ω ) | x 3 = ζ ( x 1 ) = H 2 > ( x 1 x 3 ω ) | x 3 = ζ ( x 1 ) ,
1 ( ω ) n H 2 < ( x 1 x 3 ω ) | x 3 = ζ ( x 1 ) = n H 2 > ( x 1 x 3 ω ) | x 3 = ζ ( x 1 ) ,
n = { 1 + [ ζ ( x 1 ) ] 2 } - 1 / 2 [ - ζ ( x 1 ) x 1 + x 3 ]
H 2 > ( x 1 x 3 ω ) = exp [ i k x 1 - i α 0 ( k ω ) x 3 ] + p = - A p ( k ω ) exp [ i k p x 1 + i α p ( k ω ) x 3 ]             x 3 > ζ max ,
H 2 < ( x 1 x 3 ω ) = p = - B p ( k ω ) exp [ i k p x 1 - i β p ( k ω ) x 3 ]             x 3 < ζ min ,
k p k + 2 π p a
α p ( k ω ) = ( ω 2 c 2 - k p 2 ) 1 / 2 ,             k p 2 < ω 2 c 2
= i ( k p 2 - ω 2 c 2 ) 1 / 2 ,             k p 2 > ω 2 c 2 ,
β p ( k ω ) = [ ( ω ) ω 2 c 2 - k p 2 ] 1 / 2 ,             Re β p ( k ω ) > 0 ,             Im β p ( k ω ) > 0.
p { - exp [ i α p ( k ω ) ζ ( x 1 ) + i k p x 1 ] A p + exp [ - i β p ( k ω ) ζ ( x 1 ) + i k p x 1 ] B p } = exp [ - i α 0 ( k ω ) ζ ( x 1 ) + i k x 1 ] ,
p { [ - i α p ( k ω ) + i k p ζ ( x 1 ) ] exp [ i α p ( k ω ) ζ ( x 1 ) + i k p x 1 ] A p + 1 ( ω ) [ - i β p ( k ω ) - i k p ζ ( x 1 ) ] × exp [ - i β p ( k ω ) ζ ( x 1 ) + i k p x 1 ] B p } = [ - i α 0 ( k ω ) - i k ζ ( x 1 ) ] exp [ - i α 0 ( k ω ) ζ ( x 1 ) + i k x 1 ] .
p β r α p + k r k p β r - α p 1 a 0 a d x 1 × exp [ - i 2 π a ( r - p ) x 1 - i ( β r - α p ) ζ ( x 1 ) ] A p = β r α 0 - k r k β r + α 0 1 a 0 a d x 1 × exp [ - i 2 π r a x 1 - i ( β r + α 0 ) ζ ( x 1 ) ] .
1 a 0 a d x 1 exp [ - i 2 π p a x 1 - i γ ζ ( x 1 ) ] = λ = 0 ( - i ) λ λ ! γ λ ζ ˆ ( λ ) ( p ) .
ζ ˆ ( 0 ) ( p ) = δ p 0 ,
ζ ˆ ( λ ) ( p ) = r ζ ˆ ( λ - 1 ) ( p - r ) ζ ˆ ( r ) ,
A p = μ = 0 ( - i ) μ μ ! A p ( μ ) ,
A r ( ν ) = ( ω ) - 1 ( ω ) α r + β r [ N r ( β r + α 0 ) ν ζ ˆ ( ν ) ( r ) - p M r p λ = 1 ν ( ν λ ) ( β r - α p ) λ ζ ˆ ( λ ) ( r - p ) A p ( ν - λ ) ] ,
M r p β r α p + k r k p β r - α p ,             N r β r α 0 - k r k β r + α 0 .
A R ( 0 ) = δ r 0 ( ω ) α 0 - β 0 ( ω ) α 0 + β 0 ,
A r ( 1 ) = 2 α 0 [ ( ω ) - 1 ] ( ω ) α r + β r ζ ˆ ( r ) β r β 0 - ( ω ) k r k α 0 + β 0 ,
A r ( 2 ) = 2 α 0 [ ( ω ) - 1 ] [ ( ω ) α r + β r ] [ ( ω ) α 0 + β 0 p ζ ˆ ( r - p ) ζ ˆ ( p ) × { β 0 ( β r 2 - k r k ) + ( ω ) β r ( α 0 2 - k r k ) - 2 [ ( ω ) - 1 ] ( ω ) α p + β p ( β r α p + k r k p ) [ β p β 0 - ( ω ) k p k ] } .
k r p = ( ω ) - 1 ( ω ) α r + β r M r p 1 a 0 a d x 1 exp [ - i 2 π a ( r - p ) x 1 ] × { exp [ - i ( β r - α p ) ζ ( x 1 ) ] - 1 }
= [ ( ω ) - 1 ] β r α p - k r k p ( ω ) α r + β r × λ = 1 ( - i ) λ λ ! ( β r - α p ) λ - 1 ζ ˆ ( λ ) ( r - p ) ,
F r = ( ω ) - 1 ( ω ) α r + β r N r 1 a 0 a d x 1 exp ( - i 2 π r a x 1 ) × exp [ - i ( β r + α 0 ) ζ ( x 1 ) ]
= [ ( ω ) - 1 ] β r α 0 - k r k ( ω ) α r + β r λ = 0 ( - i ) λ λ ! ( β r + α 0 ) λ - 1 ζ ˆ ( λ ) ( r ) ,
A r + p k r p A p = F r .
A r + p k r p A p = 0.
A 0 + k 00 A 0 + p k 0 p A p = 0 ,
A r = - k r 0 A 0 - p k r p A p ,
A r = - k r 0 A 0 + p 1 k r p 1 k p 1 0 A 0 - p 1 p 2 k r p 1 k p 1 p 2 k p 2 0 A 0 + .
1 + k 00 - p 1 k 0 p 1 k p 1 0 + p 1 p 2 k 0 p 1 k p 1 p 2 k p 2 0 - p 1 p 2 p 3 k 0 p 1 k p 1 p 2 k p 2 p 3 k p 3 0 + = 0.
1 + k r r - p 1 k r p 1 k p 1 r + p 1 p 2 k r p 1 k p 1 p 2 k p 2 r - p 1 p 2 p 3 k r p 1 k p 1 p 2 k p 2 p 3 k p 3 r + = 0 ,
k r p ( k s ω ) = k r + s p + s ( k ω ) ,
A r = s G r s F s ,
G r s + p k r p G p s = δ r s .
G r s = δ r s - k r s + p 1 k r p 1 k p 1 s - p 1 p 2 k r p 1 k p 1 p 2 k p 2 s - .
G r r = 1 - k r r + p 1 k r p 1 k p 1 r - p 1 p 2 k r p 1 k p 1 p 2 k p 2 r + .
k r = k r r - p 1 k r p 1 k p 1 r + p 1 p 2 k r p 1 k p 1 p 2 k p 2 r - ,
G r r = 1 - k r + k r 2 - k r 3 + = 1 1 + k r .
l r s = k r s - p 1 k r p 1 k p 1 s + p 1 p 2 k r p 1 k p 1 p 2 k p 2 s - ,
G r s = - l r s + k r l r s - k r 2 l r s + = - l r s 1 + k r .
A r = 1 1 + k r s [ δ r s - ( 1 - δ r s ) l r s ] F s n r 1 + k r .
G r s = μ = 0 ( - i ) μ μ ! G r s ( μ ) ,
G r s ( 0 ) = δ r s
G r s ( μ ) = - [ ( ω ) - 1 ] p β r α p + k r k p ( ω ) α r + β r × λ = 0 μ - 1 ( μ λ ) ( β r - α p ) μ - λ - 1 × ζ ˆ ( μ - λ ) ( r - p ) G p s ( λ ) ,             μ 1.
1 + k r = μ = 0 ( - i ) μ μ ! k r ( μ ) ,             k r ( 0 ) = 1 ,
k r ( 0 ) = 1 ,
k r ( μ ) = - ν = 1 μ ( μ ν ) k r ( μ - ν ) G r r ( ν ) ,             μ 1.
n r = μ = 0 ( - i ) μ μ ! n r ( μ ) ,
n r ( μ ) = ν = 0 μ ( μ ν ) A r ( ν ) k r ( μ - ν ) .
p α r β p + k r k p α r - β p 1 a 0 a d x 1 exp [ - i 2 π a ( r - p ) x 1 + i ( α r - β p ) ζ ( x 1 ) ] B p = δ r 0 2 ( ω ) α 0 1 - ( ω ) .
B p = μ = 0 i μ μ ! B p ( μ ) ,
B r ( 0 ) = δ r 0 2 ( ω ) α 0 ( ω ) α 0 + β 0 ,
B r ( μ ) = ( ω ) - 1 ( ω ) α r + β r p ( α r β p + k r k p ) × λ = 1 μ ( μ λ ) ( α r - β p ) λ - 1 × ζ ˆ ( λ ) ( r - p ) B p ( μ - λ ) ,             μ 1.
B r = 2 ( ω ) α 0 ( ω ) α 0 + β 0 n ¯ r 1 + k ¯ r ,
1 + k ¯ r = μ = 0 i μ μ ! k ¯ r ( μ ) ,             k ¯ r ( 0 ) = 1 ,
k ¯ r ( μ ) = - λ = 1 μ ( μ λ ) k ¯ r ( μ - λ ) G ¯ r r ( λ ) ,             μ 1.
G ¯ r s ( 0 ) = δ r s ,
G ¯ r s ( μ ) = [ ( ω ) - 1 ] p α r β p + k r k p ( ω ) α r + β r λ = 0 μ - 1 ( μ λ ) × ( α r - β p ) μ - λ - 1 ζ ˆ ( μ - λ ) ( r - p ) G ¯ p s ( λ ) ,             μ 1.
n ¯ r = μ = 0 i μ μ ! n ¯ r ( μ ) ,
n ¯ r ( 0 ) = δ r 0 ,
n ¯ r ( μ ) = ( ω ) α 0 + β 0 2 ( ω ) α 0 ν = 0 μ ( μ ν ) k ¯ r ( μ - ν ) B r ( ν ) ,             μ 1.