Abstract

We analyze the conditioning properties of classical shape-perturbation methods for the prediction of scattering returns from rough surfaces. A central observation relates to the identification of significant cancellations that are present in the recurrence relations satisfied by successive terms in a perturbation series. We show that these cancellations are precisely responsible for the observed performance of shape-deformation methods, which typically deteriorates with decreasing regularity of the scattering surfaces. We further demonstrate that the cancellations preclude a straightforward recursive estimation of the size of the terms in the perturbation series, which, in turn, has historically prevented the derivation of a direct proof of its convergence. On the other hand, we also show that such a direct proof can be attained if a simple change of independent variables is effected in advance of the derivation of the perturbation series. Finally, we show that the relevance of these observations goes beyond the theoretical, as we explain how they provide definite guiding principles for the design of new, stabilized implementations of methods based on shape deformations.

© 2004 Optical Society of America

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  60. L. Li, J. Chandezon, G. Granet, J. P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
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2003 (1)

D. P. Nicholls, F. Reitich, “Analytic continuation of Dirichlet–Neumann operators,” Numer. Math. 94, 107–146 (2003).
[CrossRef]

2001 (6)

D. P. Nicholls, F. Reitich, “A new approach to analyticity of Dirichlet–Neumann operators,” Proc. R. Soc. Edinburgh, Sect. A 131, 1411–1433 (2001).
[CrossRef]

D. P. Nicholls, F. Reitich, “Stability of high-order perturbative methods for the computation of Dirichlet–Neumann operators,” J. Comput. Phys. 170, 276–298 (2001).
[CrossRef]

A. Friedman, F. Reitich, “Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth,” Trans. Am. Math. Soc. 353, 1587–1634 (2001).
[CrossRef]

P. Dinesen, J. S. Hesthaven, “Fast and accurate modeling of waveguide grating couplers. II. The three-dimensional vectorial case,” J. Opt. Soc. Am. A 18, 2876–2885 (2001).
[CrossRef]

O. P. Bruno, L. A. Kunyansky, “A fast, high-order algorithm for the solution of surface scattering problems. I: Basic implementation, tests and applications,” J. Comput. Phys. 169, 80–110 (2001).
[CrossRef]

K. F. Warnick, W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (3)

R. A. Smith, “An operator expansion formalism for nonlinear surface waves over variable depth,” J. Fluid Mech. 363, 333–347 (1998).
[CrossRef]

D. P. Nicholls, “Traveling water waves: spectral continuation methods with parallel implementation,” J. Comput. Phys. 143, 224–240 (1998).
[CrossRef]

O. P. Bruno, F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am. 104, 2579–2583 (1998).
[CrossRef]

1996 (4)

D. M. Milder, “Role of the admittance operator in rough-surface scattering,” J. Acoust. Soc. Am. 100, 759–768 (1996).
[CrossRef]

D. M. Milder, “An improved formalism for electromagnetic scattering from a perfectly conducting rough surface,” Radio Sci. 31, 1369–1376 (1996).
[CrossRef]

R. A. Smith, “The operator expansion formalism for electromagnetic scattering from rough dielectric surfaces,” Radio Sci. 31, 1377–1385 (1996).
[CrossRef]

O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations and analytic continuation,” Appl. Comput. Electromagn. Soc. J. 11, 17–31 (1996).

1995 (2)

V. I. Tatarskii, “Relation between the Rayleigh equation in diffraction theory and the equation based on Green’s formula,” J. Opt. Soc. Am. A 12, 1254–1260 (1995).
[CrossRef]

J. M. Chesneaux, A. A. Wirgin, “Response to comments on ‘Reflection from a corrugated surface revisited’,” J. Acoust. Soc. Am. 98, 1815–1816 (1995).
[CrossRef]

1994 (1)

P. J. Kaczkowski, E. I. Thorsos, “Application of the operator expansion method to scattering from one-dimensional moderately rough Dirichlet random surfaces,” J. Acoust. Soc. Am. 96, 957–972 (1994).
[CrossRef]

1993 (5)

1992 (4)

J. J. Greffet, C. Baylard, P. Versaevel, “Diffraction of electromagnetic waves by crossed gratings: a series solution,” Opt. Lett. 17, 1740–1742 (1992).
[CrossRef] [PubMed]

L. Kazandjian, “Comparison of the Rayleigh–Fourier and extinction theorem methods applied to scattering and transmission at a rough solid–solid interface,” J. Acoust. Soc. Am. 92, 1679–1691 (1992).
[CrossRef]

O. P. Bruno, F. Reitich, “Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh Sect. A 122, 317–340 (1992).
[CrossRef]

D. M. Milder, H. T. Sharp, “An improved formalism for rough surface scattering. II: Numerical trials in three dimensions,” J. Acoust. Soc. Am. 91, 2620–2626 (1992).
[CrossRef]

1991 (2)

D. M. Milder, “An improved formalism for wave scattering from rough surfaces,” J. Acoust. Soc. Am. 89, 529–541 (1991).
[CrossRef]

J. Roginsky, “Derivation of closed-form expressions for the T matrices of Rayleigh–Rice and extinction-theorem perturbation theories,” J. Acoust. Soc. Am. 90, 1130–1137 (1991).
[CrossRef]

1990 (2)

1989 (1)

A. Wirgin, “Scattering from hard and soft corrugated surfaces: iterative corrections to the Kirchhoff approximation through the extinction theorem,” J. Acoust. Soc. Am. 85, 670–679 (1989).
[CrossRef]

1988 (2)

D. R. Jackson, D. P. Winebrenner, A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988).
[CrossRef]

J. J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988).
[CrossRef]

1986 (1)

G. V. Anand, M. K. George, “Normal mode sound propagation in an ocean with sinusoidal surface waves,” J. Acoust. Soc. Am. 80, 238–243 (1986).
[CrossRef]

1983 (1)

1980 (1)

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

1978 (1)

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

1977 (2)

W. A. Kuperman, F. F. Ingenito, “Attenuation of the coherent component of sound propagating in shallow water with rough boundaries,” J. Acoust. Soc. Am. 61, 1178–1187 (1977).
[CrossRef]

A. P. Calderón, “Cauchy integrals on Lipschitz curves and related operators,” Proc. Natl. Acad. Sci. USA 75, 1324–1327 (1977).
[CrossRef]

1975 (3)

K. M. Watson, B. J. West, “A transport-equation description of nonlinear ocean surface wave interactions,” J. Fluid Mech. 70, 815–826 (1975).
[CrossRef]

E. Y. Harper, F. M. Labianca, “Perturbation theory forscattering of sound from a point source by a moving rough surface in the presence of refraction,” J. Acoust. Soc. Am. 57, 1044–1051 (1975).
[CrossRef]

E. Y. Harper, F. M. Labianca, “Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean,” J. Acoust. Soc. Am. 58, 349–364 (1975).
[CrossRef]

1974 (1)

A. H. Nayfeh, O. R. Asfar, “Parallel-plate waveguide with sinusoidally perturbed boundaries,” J. Appl. Phys. 45, 4797–4800 (1974).
[CrossRef]

1972 (1)

H. Hasimoto, H. Ono, “Nonlinear modulation of gravity waves,” J. Phys. Soc. Jpn. 33, 805–811 (1972).
[CrossRef]

1971 (1)

J. R. Wait, “Perturbation analysis for reflection from two-dimensional periodic sea waves,” Radio Sci. 6, 387–391 (1971).
[CrossRef]

1965 (1)

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[CrossRef]

1956 (1)

W. C. Meecham, “On the use of the Kirchoff approximation for the solution of reflection problems,” J. Rat. Mech. Anal. 5, 323–334 (1956).

1951 (1)

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

1907 (1)

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

Anand, G. V.

G. V. Anand, M. K. George, “Normal mode sound propagation in an ocean with random narrow-band surface waves,” J. Acoust. Soc. Am. 94, 279–292 (1993).
[CrossRef]

G. V. Anand, M. K. George, “Normal mode sound propagation in an ocean with sinusoidal surface waves,” J. Acoust. Soc. Am. 80, 238–243 (1986).
[CrossRef]

Asfar, O. R.

A. H. Nayfeh, O. R. Asfar, “Parallel-plate waveguide with sinusoidally perturbed boundaries,” J. Appl. Phys. 45, 4797–4800 (1974).
[CrossRef]

Bao, G.

G. Bao, D. Dobson, “Variational methods for diffractive optics modeling,” in Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001), pp. 37–70.

Baylard, C.

Bruno, O. P.

O. P. Bruno, L. A. Kunyansky, “A fast, high-order algorithm for the solution of surface scattering problems. I: Basic implementation, tests and applications,” J. Comput. Phys. 169, 80–110 (2001).
[CrossRef]

O. P. Bruno, F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am. 104, 2579–2583 (1998).
[CrossRef]

O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations and analytic continuation,” Appl. Comput. Electromagn. Soc. J. 11, 17–31 (1996).

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
[CrossRef]

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities,” J. Opt. Soc. Am. A 10, 2307–2316 (1993).
[CrossRef]

O. P. Bruno, F. Reitich, “Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh Sect. A 122, 317–340 (1992).
[CrossRef]

O. P. Bruno, F. Reitich, “High-order boundary perturbation methods,” in Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001), pp. 71–110.

Butzer, P. L.

P. L. Butzer, R. J. Nessel, Fourier Analysis and Approximation (Academic, New York, 1971).

Calderón, A. P.

A. P. Calderón, “Cauchy integrals on Lipschitz curves and related operators,” Proc. Natl. Acad. Sci. USA 75, 1324–1327 (1977).
[CrossRef]

Canuto, C.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1988).

Chandezon, J.

L. Li, J. Chandezon, G. Granet, J. P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
[CrossRef]

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

Chesneaux, J. M.

J. M. Chesneaux, A. A. Wirgin, “Response to comments on ‘Reflection from a corrugated surface revisited’,” J. Acoust. Soc. Am. 98, 1815–1816 (1995).
[CrossRef]

Chew, W. C.

K. F. Warnick, W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

Coifman, R.

R. Coifman, Y. Meyer, “Nonlinear harmonic analysis and analytic dependence,” in Pseudodifferential Operators and Applications (American Mathematical Society, Providence, R.I., 1985), pp. 71–78.

Colton, D.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, Berlin, 1998).

Craig, W.

W. Craig, C. Sulem, “Numerical simulation of gravity waves,” J. Comput. Phys. 108, 73–83 (1993).
[CrossRef]

Dinesen, P.

Dobson, D.

G. Bao, D. Dobson, “Variational methods for diffractive optics modeling,” in Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001), pp. 37–70.

Evans, L. C.

L. C. Evans, Partial Differential Equations (American Mathematical Society, Providence, R.I., 1998).

Friedman, A.

A. Friedman, F. Reitich, “Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth,” Trans. Am. Math. Soc. 353, 1587–1634 (2001).
[CrossRef]

Garcia, N.

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

George, M. K.

G. V. Anand, M. K. George, “Normal mode sound propagation in an ocean with random narrow-band surface waves,” J. Acoust. Soc. Am. 94, 279–292 (1993).
[CrossRef]

G. V. Anand, M. K. George, “Normal mode sound propagation in an ocean with sinusoidal surface waves,” J. Acoust. Soc. Am. 80, 238–243 (1986).
[CrossRef]

Granet, G.

Greffet, J. J.

Harper, E. Y.

E. Y. Harper, F. M. Labianca, “Perturbation theory forscattering of sound from a point source by a moving rough surface in the presence of refraction,” J. Acoust. Soc. Am. 57, 1044–1051 (1975).
[CrossRef]

E. Y. Harper, F. M. Labianca, “Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean,” J. Acoust. Soc. Am. 58, 349–364 (1975).
[CrossRef]

Hasimoto, H.

H. Hasimoto, H. Ono, “Nonlinear modulation of gravity waves,” J. Phys. Soc. Jpn. 33, 805–811 (1972).
[CrossRef]

Hesthaven, J. S.

Hussaini, M. Y.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1988).

Ingenito, F. F.

W. A. Kuperman, F. F. Ingenito, “Attenuation of the coherent component of sound propagating in shallow water with rough boundaries,” J. Acoust. Soc. Am. 61, 1178–1187 (1977).
[CrossRef]

Ishimaru, A.

D. R. Jackson, D. P. Winebrenner, A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988).
[CrossRef]

Jackson, D. R.

D. R. Jackson, D. P. Winebrenner, A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988).
[CrossRef]

Kaczkowski, P. J.

P. J. Kaczkowski, E. I. Thorsos, “Application of the operator expansion method to scattering from one-dimensional moderately rough Dirichlet random surfaces,” J. Acoust. Soc. Am. 96, 957–972 (1994).
[CrossRef]

Kazandjian, L.

L. Kazandjian, “Comparison of the Rayleigh–Fourier and extinction theorem methods applied to scattering and transmission at a rough solid–solid interface,” J. Acoust. Soc. Am. 92, 1679–1691 (1992).
[CrossRef]

Kress, R.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, Berlin, 1998).

Kunyansky, L. A.

O. P. Bruno, L. A. Kunyansky, “A fast, high-order algorithm for the solution of surface scattering problems. I: Basic implementation, tests and applications,” J. Comput. Phys. 169, 80–110 (2001).
[CrossRef]

Kuperman, W. A.

W. A. Kuperman, F. F. Ingenito, “Attenuation of the coherent component of sound propagating in shallow water with rough boundaries,” J. Acoust. Soc. Am. 61, 1178–1187 (1977).
[CrossRef]

Labianca, F. M.

E. Y. Harper, F. M. Labianca, “Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean,” J. Acoust. Soc. Am. 58, 349–364 (1975).
[CrossRef]

E. Y. Harper, F. M. Labianca, “Perturbation theory forscattering of sound from a point source by a moving rough surface in the presence of refraction,” J. Acoust. Soc. Am. 57, 1044–1051 (1975).
[CrossRef]

Ladyzhenskaya, O. A.

O. A. Ladyzhenskaya, N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations (Academic, New York, 1968).

Li, L.

Lopez, C.

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

Maassarani, Z.

Maradudin, A. A.

Maystre, D.

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

Meecham, W. C.

W. C. Meecham, “On the use of the Kirchoff approximation for the solution of reflection problems,” J. Rat. Mech. Anal. 5, 323–334 (1956).

Meyer, Y.

R. Coifman, Y. Meyer, “Nonlinear harmonic analysis and analytic dependence,” in Pseudodifferential Operators and Applications (American Mathematical Society, Providence, R.I., 1985), pp. 71–78.

Milder, D. M.

D. M. Milder, “Role of the admittance operator in rough-surface scattering,” J. Acoust. Soc. Am. 100, 759–768 (1996).
[CrossRef]

D. M. Milder, “An improved formalism for electromagnetic scattering from a perfectly conducting rough surface,” Radio Sci. 31, 1369–1376 (1996).
[CrossRef]

D. M. Milder, H. T. Sharp, “An improved formalism for rough surface scattering. II: Numerical trials in three dimensions,” J. Acoust. Soc. Am. 91, 2620–2626 (1992).
[CrossRef]

D. M. Milder, “An improved formalism for wave scattering from rough surfaces,” J. Acoust. Soc. Am. 89, 529–541 (1991).
[CrossRef]

D. M. Milder, “The effects of truncation on surface-wave Hamiltonians,” J. Fluid Mech. 217, 249–262 (1990).
[CrossRef]

D. M. Milder, H. T. Sharp, “Efficient computation of rough surface scattering,” in Mathematical and Numerical Aspects of Wave Propagation Phenomena (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1991), pp. 314–322.

D. M. Milder, “An improved formalism for rough-surface scattering of acoustic and electromagnetic waves,” in Wave Propagation and Scattering in Varied Media II, V. Varadan, ed., Proc. SPIE1558, 213–221 (1991).
[CrossRef]

Nayfeh, A. H.

A. H. Nayfeh, O. R. Asfar, “Parallel-plate waveguide with sinusoidally perturbed boundaries,” J. Appl. Phys. 45, 4797–4800 (1974).
[CrossRef]

Nessel, R. J.

P. L. Butzer, R. J. Nessel, Fourier Analysis and Approximation (Academic, New York, 1971).

Nicholls, D. P.

D. P. Nicholls, F. Reitich, “Analytic continuation of Dirichlet–Neumann operators,” Numer. Math. 94, 107–146 (2003).
[CrossRef]

D. P. Nicholls, F. Reitich, “A new approach to analyticity of Dirichlet–Neumann operators,” Proc. R. Soc. Edinburgh, Sect. A 131, 1411–1433 (2001).
[CrossRef]

D. P. Nicholls, F. Reitich, “Stability of high-order perturbative methods for the computation of Dirichlet–Neumann operators,” J. Comput. Phys. 170, 276–298 (2001).
[CrossRef]

D. P. Nicholls, “Traveling water waves: spectral continuation methods with parallel implementation,” J. Comput. Phys. 143, 224–240 (1998).
[CrossRef]

D. P. Nicholls, “Traveling gravity water waves in two and three dimensions,” Ph.D. thesis (Brown University, Providence, R.I., 1998).

Ono, H.

H. Hasimoto, H. Ono, “Nonlinear modulation of gravity waves,” J. Phys. Soc. Jpn. 33, 805–811 (1972).
[CrossRef]

Plumey, J. P.

Quarteroni, A.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1988).

Raoult, G.

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

Rayleigh, Lord

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

Reitich, F.

D. P. Nicholls, F. Reitich, “Analytic continuation of Dirichlet–Neumann operators,” Numer. Math. 94, 107–146 (2003).
[CrossRef]

A. Friedman, F. Reitich, “Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth,” Trans. Am. Math. Soc. 353, 1587–1634 (2001).
[CrossRef]

D. P. Nicholls, F. Reitich, “Stability of high-order perturbative methods for the computation of Dirichlet–Neumann operators,” J. Comput. Phys. 170, 276–298 (2001).
[CrossRef]

D. P. Nicholls, F. Reitich, “A new approach to analyticity of Dirichlet–Neumann operators,” Proc. R. Soc. Edinburgh, Sect. A 131, 1411–1433 (2001).
[CrossRef]

O. P. Bruno, F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am. 104, 2579–2583 (1998).
[CrossRef]

O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations and analytic continuation,” Appl. Comput. Electromagn. Soc. J. 11, 17–31 (1996).

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities,” J. Opt. Soc. Am. A 10, 2307–2316 (1993).
[CrossRef]

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
[CrossRef]

O. P. Bruno, F. Reitich, “Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh Sect. A 122, 317–340 (1992).
[CrossRef]

O. P. Bruno, F. Reitich, “High-order boundary perturbation methods,” in Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001), pp. 71–110.

Rice, S. O.

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

Roginsky, J.

J. Roginsky, “Derivation of closed-form expressions for the T matrices of Rayleigh–Rice and extinction-theorem perturbation theories,” J. Acoust. Soc. Am. 90, 1130–1137 (1991).
[CrossRef]

Sharp, H. T.

D. M. Milder, H. T. Sharp, “An improved formalism for rough surface scattering. II: Numerical trials in three dimensions,” J. Acoust. Soc. Am. 91, 2620–2626 (1992).
[CrossRef]

D. M. Milder, H. T. Sharp, “Efficient computation of rough surface scattering,” in Mathematical and Numerical Aspects of Wave Propagation Phenomena (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1991), pp. 314–322.

Smith, R. A.

R. A. Smith, “An operator expansion formalism for nonlinear surface waves over variable depth,” J. Fluid Mech. 363, 333–347 (1998).
[CrossRef]

R. A. Smith, “The operator expansion formalism for electromagnetic scattering from rough dielectric surfaces,” Radio Sci. 31, 1377–1385 (1996).
[CrossRef]

Stein, E. M.

E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton U. Press, Princeton, N.J., 1970).

Sulem, C.

W. Craig, C. Sulem, “Numerical simulation of gravity waves,” J. Comput. Phys. 108, 73–83 (1993).
[CrossRef]

Tatarskii, V. I.

Thorsos, E. I.

P. J. Kaczkowski, E. I. Thorsos, “Application of the operator expansion method to scattering from one-dimensional moderately rough Dirichlet random surfaces,” J. Acoust. Soc. Am. 96, 957–972 (1994).
[CrossRef]

Ural’tseva, N. N.

O. A. Ladyzhenskaya, N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations (Academic, New York, 1968).

Uretsky, J. L.

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[CrossRef]

Versaevel, P.

Voronovich, A. G.

A. G. Voronovich, Wave Scattering from Rough Surfaces, 2nd ed. (Springer-Verlag, Berlin, 1999).

Wait, J. R.

J. R. Wait, “Perturbation analysis for reflection from two-dimensional periodic sea waves,” Radio Sci. 6, 387–391 (1971).
[CrossRef]

Warnick, K. F.

K. F. Warnick, W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

Watson, K. M.

K. M. Watson, B. J. West, “A transport-equation description of nonlinear ocean surface wave interactions,” J. Fluid Mech. 70, 815–826 (1975).
[CrossRef]

West, B. J.

K. M. Watson, B. J. West, “A transport-equation description of nonlinear ocean surface wave interactions,” J. Fluid Mech. 70, 815–826 (1975).
[CrossRef]

Winebrenner, D. P.

D. R. Jackson, D. P. Winebrenner, A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988).
[CrossRef]

Wirgin, A.

A. Wirgin, “Scattering from hard and soft corrugated surfaces: iterative corrections to the Kirchhoff approximation through the extinction theorem,” J. Acoust. Soc. Am. 85, 670–679 (1989).
[CrossRef]

Wirgin, A. A.

J. M. Chesneaux, A. A. Wirgin, “Response to comments on ‘Reflection from a corrugated surface revisited’,” J. Acoust. Soc. Am. 98, 1815–1816 (1995).
[CrossRef]

Yndurain, F. J.

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

Zang, T. A.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1988).

Ann. Phys. (1)

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[CrossRef]

Appl. Comput. Electromagn. Soc. J. (1)

O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations and analytic continuation,” Appl. Comput. Electromagn. Soc. J. 11, 17–31 (1996).

Appl. Opt. (1)

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]

J. Acoust. Soc. Am. (15)

G. V. Anand, M. K. George, “Normal mode sound propagation in an ocean with sinusoidal surface waves,” J. Acoust. Soc. Am. 80, 238–243 (1986).
[CrossRef]

G. V. Anand, M. K. George, “Normal mode sound propagation in an ocean with random narrow-band surface waves,” J. Acoust. Soc. Am. 94, 279–292 (1993).
[CrossRef]

E. Y. Harper, F. M. Labianca, “Perturbation theory forscattering of sound from a point source by a moving rough surface in the presence of refraction,” J. Acoust. Soc. Am. 57, 1044–1051 (1975).
[CrossRef]

E. Y. Harper, F. M. Labianca, “Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean,” J. Acoust. Soc. Am. 58, 349–364 (1975).
[CrossRef]

W. A. Kuperman, F. F. Ingenito, “Attenuation of the coherent component of sound propagating in shallow water with rough boundaries,” J. Acoust. Soc. Am. 61, 1178–1187 (1977).
[CrossRef]

J. M. Chesneaux, A. A. Wirgin, “Response to comments on ‘Reflection from a corrugated surface revisited’,” J. Acoust. Soc. Am. 98, 1815–1816 (1995).
[CrossRef]

D. R. Jackson, D. P. Winebrenner, A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988).
[CrossRef]

L. Kazandjian, “Comparison of the Rayleigh–Fourier and extinction theorem methods applied to scattering and transmission at a rough solid–solid interface,” J. Acoust. Soc. Am. 92, 1679–1691 (1992).
[CrossRef]

O. P. Bruno, F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am. 104, 2579–2583 (1998).
[CrossRef]

J. Roginsky, “Derivation of closed-form expressions for the T matrices of Rayleigh–Rice and extinction-theorem perturbation theories,” J. Acoust. Soc. Am. 90, 1130–1137 (1991).
[CrossRef]

A. Wirgin, “Scattering from hard and soft corrugated surfaces: iterative corrections to the Kirchhoff approximation through the extinction theorem,” J. Acoust. Soc. Am. 85, 670–679 (1989).
[CrossRef]

D. M. Milder, “An improved formalism for wave scattering from rough surfaces,” J. Acoust. Soc. Am. 89, 529–541 (1991).
[CrossRef]

D. M. Milder, H. T. Sharp, “An improved formalism for rough surface scattering. II: Numerical trials in three dimensions,” J. Acoust. Soc. Am. 91, 2620–2626 (1992).
[CrossRef]

P. J. Kaczkowski, E. I. Thorsos, “Application of the operator expansion method to scattering from one-dimensional moderately rough Dirichlet random surfaces,” J. Acoust. Soc. Am. 96, 957–972 (1994).
[CrossRef]

D. M. Milder, “Role of the admittance operator in rough-surface scattering,” J. Acoust. Soc. Am. 100, 759–768 (1996).
[CrossRef]

J. Appl. Phys. (1)

A. H. Nayfeh, O. R. Asfar, “Parallel-plate waveguide with sinusoidally perturbed boundaries,” J. Appl. Phys. 45, 4797–4800 (1974).
[CrossRef]

J. Comput. Phys. (4)

D. P. Nicholls, F. Reitich, “Stability of high-order perturbative methods for the computation of Dirichlet–Neumann operators,” J. Comput. Phys. 170, 276–298 (2001).
[CrossRef]

D. P. Nicholls, “Traveling water waves: spectral continuation methods with parallel implementation,” J. Comput. Phys. 143, 224–240 (1998).
[CrossRef]

W. Craig, C. Sulem, “Numerical simulation of gravity waves,” J. Comput. Phys. 108, 73–83 (1993).
[CrossRef]

O. P. Bruno, L. A. Kunyansky, “A fast, high-order algorithm for the solution of surface scattering problems. I: Basic implementation, tests and applications,” J. Comput. Phys. 169, 80–110 (2001).
[CrossRef]

J. Fluid Mech. (3)

R. A. Smith, “An operator expansion formalism for nonlinear surface waves over variable depth,” J. Fluid Mech. 363, 333–347 (1998).
[CrossRef]

K. M. Watson, B. J. West, “A transport-equation description of nonlinear ocean surface wave interactions,” J. Fluid Mech. 70, 815–826 (1975).
[CrossRef]

D. M. Milder, “The effects of truncation on surface-wave Hamiltonians,” J. Fluid Mech. 217, 249–262 (1990).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

J. Phys. Soc. Jpn. (1)

H. Hasimoto, H. Ono, “Nonlinear modulation of gravity waves,” J. Phys. Soc. Jpn. 33, 805–811 (1972).
[CrossRef]

J. Rat. Mech. Anal. (1)

W. C. Meecham, “On the use of the Kirchoff approximation for the solution of reflection problems,” J. Rat. Mech. Anal. 5, 323–334 (1956).

Numer. Math. (1)

D. P. Nicholls, F. Reitich, “Analytic continuation of Dirichlet–Neumann operators,” Numer. Math. 94, 107–146 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (2)

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

J. J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

A. P. Calderón, “Cauchy integrals on Lipschitz curves and related operators,” Proc. Natl. Acad. Sci. USA 75, 1324–1327 (1977).
[CrossRef]

Proc. R. Soc. Edinburgh Sect. A (1)

O. P. Bruno, F. Reitich, “Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh Sect. A 122, 317–340 (1992).
[CrossRef]

Proc. R. Soc. Edinburgh, Sect. A (1)

D. P. Nicholls, F. Reitich, “A new approach to analyticity of Dirichlet–Neumann operators,” Proc. R. Soc. Edinburgh, Sect. A 131, 1411–1433 (2001).
[CrossRef]

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

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

Radio Sci. (3)

J. R. Wait, “Perturbation analysis for reflection from two-dimensional periodic sea waves,” Radio Sci. 6, 387–391 (1971).
[CrossRef]

D. M. Milder, “An improved formalism for electromagnetic scattering from a perfectly conducting rough surface,” Radio Sci. 31, 1369–1376 (1996).
[CrossRef]

R. A. Smith, “The operator expansion formalism for electromagnetic scattering from rough dielectric surfaces,” Radio Sci. 31, 1377–1385 (1996).
[CrossRef]

Trans. Am. Math. Soc. (1)

A. Friedman, F. Reitich, “Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth,” Trans. Am. Math. Soc. 353, 1587–1634 (2001).
[CrossRef]

Waves Random Media (1)

K. F. Warnick, W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

Other (13)

G. Bao, D. Dobson, “Variational methods for diffractive optics modeling,” in Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001), pp. 37–70.

R. Coifman, Y. Meyer, “Nonlinear harmonic analysis and analytic dependence,” in Pseudodifferential Operators and Applications (American Mathematical Society, Providence, R.I., 1985), pp. 71–78.

D. P. Nicholls, “Traveling gravity water waves in two and three dimensions,” Ph.D. thesis (Brown University, Providence, R.I., 1998).

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, Berlin, 1998).

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1988).

O. A. Ladyzhenskaya, N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations (Academic, New York, 1968).

L. C. Evans, Partial Differential Equations (American Mathematical Society, Providence, R.I., 1998).

P. L. Butzer, R. J. Nessel, Fourier Analysis and Approximation (Academic, New York, 1971).

E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton U. Press, Princeton, N.J., 1970).

A. G. Voronovich, Wave Scattering from Rough Surfaces, 2nd ed. (Springer-Verlag, Berlin, 1999).

D. M. Milder, H. T. Sharp, “Efficient computation of rough surface scattering,” in Mathematical and Numerical Aspects of Wave Propagation Phenomena (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1991), pp. 314–322.

O. P. Bruno, F. Reitich, “High-order boundary perturbation methods,” in Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001), pp. 71–110.

D. M. Milder, “An improved formalism for rough-surface scattering of acoustic and electromagnetic waves,” in Wave Propagation and Scattering in Varied Media II, V. Varadan, ed., Proc. SPIE1558, 213–221 (1991).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Plot of “energy defect” (error in conservation of energy) versus perturbation order n for a sinusoidal profile under normal incidence; the height-to-period ratio is 0.05, and the wavelength-to-period ratio is 0.065. A comparison is given of FE and PFE implementations.

Fig. 2
Fig. 2

Plots of relative error in (absolute value of) the Fourier coefficients (T2(g)[ξ]ˆ)p and (vˆ3)p for different values of the wave number (and oblique incidence) as computed from the original OE and PFE recursions, respectively [cf. Eqs. (3.1c) and (3.16)]; the results are for rough profile g and incidence ξ with gˆp=ξˆp=p-2 for |p|F. To avoid aliasing effects and isolate the instabilities, here F was chosen as F=Nx/3 for T2 and F=Nx/4 for v3, where Nx=2048 is the number of discretization points.

Fig. 3
Fig. 3

Maximum error [cf. Eq. (3.23)] in the evaluation of the DNO with the OE, PFE, and PTFE algorithms. A comparison is given with exact solution (3.22) and p=1 for normal incidence with a wavelength-to-period ratio of 0.4368. (a) Sinusoidal profile (3.21a) (δ=0.09, Nx=64), (b) “rough” profile (3.21b) (δ=0.03, Nx=256), (c) Lipschitz profile (3.21c) (δ=0.03, Nx=256).

Fig. 4
Fig. 4

Maximum error [cf. Eq. (3.23)] in the evaluation of the DNO with the OE, PFE, and PTFE algorithms. A comparison is given with exact solution (3.22) and p=1 for normal incidence with a wavelength-to-period ratio of 0.4368. The effect of different discretizations for Lipschitz profile (3.21c) is given, showing the existence of an optimal number of discretization points (Nx=512), a compromise between aliasing and conditioning errors: (a) effect on OE scheme, (b) effect on PFE algorithm.

Fig. 5
Fig. 5

(a) Results of the implementation of the recursion (4.1) in exact and finite-precision arithmetic, (b) root test for the sequences Dn and Δn in Eqs. (4.1) and (4.2), respectively.

Equations (144)

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

v˜inc(x, y, t)=exp(-iωt)vinc(x, y)=exp(-iωt)exp(iαx-iβy)
y=g(x),g(x+d)=g(x),
v˜scat(x, y, t)=exp(-iωt)vscat(x, y).
Δvscat+k2vscat=0
vscat(x+d, y)=exp(iαd)vscat(x, y).
vscat(x, g(x))=-vinc(x, g(x))=-exp[iαx-iβg(x)].
vscat(x, y)=p=-Bp exp(iαpx+iβpy)
αp=α+2πdp,βp=k2-αp2.
T(σ)[ξ](x)=vscatn(x)(x, σ(x)),
vscat(x, σ(x))=ξ(x).
ξ(x)=p=-ξˆp exp(iαpx),
vscat(x, y)=p=-ξˆp exp[iαpx+iβp(y-a)],
T(σ)[ξ](x)=T(a)[ξ](x)=-vscaty(x, a)=p=-(-iβp)ξˆp exp(iαpx).
vscaty(x, a)+T(a)[vscat(·, a)](x)=0.
Δv+k2v=0,
v(x, g(x))=-exp[iαx-iβg(x)],
yv(x, a)+T(a)[v(·, a)](x)=0,
v(x+d, y)=exp(iαd)v(x, y)
v=v0(x, y)=-exp(iαx+iβy),
v(x, y)=dp exp(iαpx+iβpy).
V(x, y;δ)=n=0vn(x, y)δn.
Δvn+k2vn=0,
vn(x, 0)=Pn(x),
yvn(x, a)+T(a)[vn(·, a)](x)=0,
vn(x+d, y)=exp(iαd)vn(x, y).
V(x, δg(x);δ)=-exp[iαx-iβδ(x)].
Pn(x)=-exp(iαx)[-iβg(x)]nn!-l=0n-1[g(x)]n-l(n-l)!yn-lvl(x, 0),
v(x, y)=V(x, y;δ=1)=n=0vn(x, y).
vn(x, y)=p=-dn,p exp(iαpx+iβpy)
dn,p=-(-iβ)nCn,p-l=0n-1q=-Cn-l,p-q(iβq)n-ldl,q,
[g(x)]l/l!=p=-Cl,p exp(ipx).
Bp=n=0dn,p,
v(x, y)=0dΦn(x)(x-x, y-g(x))v(x, g(x))-vn(x)(x, g(x))×Φ(x-x, y-g(x))dx,
Φ(x, y)=i4n=-exp(-iαnd)H0(1)(k[(x+nd)2+y2]1/2)=i2dp=-exp(iβp|y|)βpexp(iαpx)
v(x, y)=0d-Φn(x)(x-x, y-g(x))finc(x)+Φ(x-x, y-g(x))T(g)[finc](x)dx,
finc(x)=vinc(x, g(x))=exp[iαx-iβg(x)].
v(x, y)=p=-Bp exp(iαpx+iβpy),
Bp=i2dβp0dfp(x){[iαpxg(x)-iβp]+T(g)}[finc](x)dx,
fp(x)=exp[-iαpx-iβpg(x)].
T(0)p=-ξˆp exp(iαpx)=p=-(-iβp)ξˆp exp(iαpx).
T(g)=T(δg)|δ=1=n=0Tn(g)δnδ=1.
wp(x, y)=exp(iαpx+iβpy)
T(δg)[exp{iαpx+iβpδg(x)}]=[iαpδxg(x)-iβp]exp[iαpx+iβpδg(x)].
Tn(g)[exp(iαpx)]=[xg(x)](iαp)(iβp)n-1[g(x)]n-1(n-1)!exp(iαpx)-(iβp)n+1[g(x)]nn!exp(iαpx)-l=0n-1Tl(g)(iβp)n-l{g(x)}n-l(n-l)!exp(iαpx).
ξ(x)=p=-ξˆp exp(iαpx),
Tn(g)[ξ]=[xg(x)][g(x)]n-1(n-1)!p=-(iαp)×(iβp)n-1ξˆp exp(iαpx)-[g(x)]nn!p=-(iβp)n+1ξˆp exp(iαpx)-l=0n-1Tl(g){g(x)}n-l(n-l)!×p=-(iβp)n-lξˆp exp(iαpx)
Tn(g)[ξ]=x[g(x)]nn!p=-(iαp)(iβp)n-1ξˆp exp(iαpx)+k2[g(x)]nn!p=-(iβp)n-1ξˆp exp(iαpx)-l=0n-1Tl(g){g(x)}n-l(n-l)!×p=-(iβp)n-lξˆp exp(iαpx).
D=-ix
βD[ξ]=p=-βpξˆp exp(iαpx),
Tn(g)[ξ]=x[g(x)]nn!(iβD)n-1xξ+k2[g(x)]nn!(iβD)n-1ξ-l=0n-1Tl(g){g(x)}n-l(n-l)!(iβD)n-1ξ.
Tn(g)[ξ]=(iβD)n-1x[g(x)]nn!xξ+k2[g(x)]nn!ξ-l=0n-1(iβD)n-l[g(x)]n-l(n-l)!Tl(g)[ξ].
V(x, δg(x);δ)=-exp[iαx-iβg(x)],
v0(x, 0)=-exp[iαx-iβg(x)].
Pn(x)=-l=0n-1[g(x)]n-l(n-l)!yn-lvl(x, 0).
T0[ξ]=-iβDξ,
T1(g)[ξ]=-DgDξ+k2gξ-iβDgT0[ξ],
T2(g)[ξ]=iβD{-D(g2/2)Dξ+k2(g2/2)ξ-iβD(g2/2)T0[ξ]-gT1(g)[ξ]},
βD[fη]-fβD[η]=Rf[η],
most singular part ofβD[fη]=fβD[η].
T1(g)[ξ]=-(Dg)(Dξ)-g(D2ξ)+k2gξ-gβD2ξ-Rg[βDξ],
βD2=k2-D2,
T1(g)[ξ]=-(Dg)(Dξ)-Rg[βDξ].
T2(g)[ξ]=iβD{-(g2/2)D2ξ-g(Dg)(Dξ)+k2(g2/2)ξ-(g2/2)βD2ξ-Rg2/2[βDξ]-g((Dg)(Dξ)-Rg[βDξ])}
T2(g)[ξ]=iβDSg[βDξ],
Sf[η]=-Rf2/2[η]+fRf[η].
y=iβD.
v0=ξ,
v1=-igβDv0,
v2=g22βD2v0-igβDv1.
v2=g22(k2-D2)ξ-gβDgβDξ,
v2=g22(k2-D2)ξ-g2βD2ξ-gRg[βDξ]=g22(k2-D2)ξ-g2(k2-D2)ξ-gRg[βDξ]=-g22(k2-D2)ξ-gRg[βDξ].
βDg22βDξ=g22(k2-D2)ξ+Rg2/2[βDξ],
v2=-βDg22βDξ+Rg2/2[βDξ]-gRg[βDξ]
v2=-βDg22βDξ-Sg[βDξ].
βDξ(x)=r=-br exp(iαpx),
[g(x)]2/2=r=-C2,r exp[i(2π/d)px].
[g(x)]22βD2ξ(x)=r=-p=-C2,r-pβpbpexp(iαrx),
βDg22βDξ(x)=r=-βrp=-C2,r-pbpexp(iαrx).
p=-FFC2,r-pβpbp,
βrp=-FFC2,r-pbp,
v3=ig36βD3v0+g22βD2v1-igβDv2
v3(x)=iβD2g36βDξ+igβDSg[βDξ],
mostsingularpartofvn=(-i)nβDn-1gnn!βDξ.
ip=-FFC3,r-pβp2bp
iβr2p=-FFC3,r-pbp.
gs(x)=cos(x),
gr(x)=(2×10-4)x4(2π-x)4-c0,
gL(x)=-2πx+1,0xπ2πx-3,πx2π,
gr(x)=k=196(2k2π2-21)125k8cos(kx),
gL(x)=k=18π2(2k-1)2cos[(2k-1)x],
gr,P(x)=k=1P96(2k2π2-21)125k8cos(kx),
gL,P(x)=k=1P/28π2(2k-1)2cos[(2k-1)].
T(g)[exp{iαpx+iβpg(x)}]=[iαpxg(x)-iβp]exp[iαpx+iβpg(x)].
Error=Error(n, Nx)=max1jNx|T(g)[ξ](xj)-Tn,Nxapprox(xj)|.
g(x)=p=-MMgˆp exp(ipx),
[g(x)]ll!=p=-lMlMCl,p exp(ipx).
Dn=-(-iβ)nCn,nM-l=0n-1Cn-l,(n-l)M(iβlM)n-lDl,
Dl=dl,lM.
g(x)=2 cos(x)=exp(ix)+exp(-ix)(M=1);
Δn=|β|n|Cn,nM|+l=0n-1|Cn-l,(n-l)M||βlM|n-lΔl.
|Dn|Δn,
x=x,y=ay-g(x)a-g(x),
Δu+k2u=F(x, y),
u(x, 0)=-exp[iαx-iβg(x)],
yu(x, a)+T(a)[u(·, a)](x)=R(x),
u(x+d, y)=exp(iαd)u(x, y)
u(x, y)=v(x, {[a-g(x)]/a}y+g(x)),
F(x, y)=divx2g(x)axu-[g(x)]2a2xu+(a-y)xg(x)ayu-(a-y)g(x)xg(x)a2yu+y(a-y)xg(x)a·xu-(a-y)g(x)xg(x)a2·xu-(a-y)2|xg(x)|2a2yu-xg(x)a·xu+g(x)xg(x)a2·xu+(a-y)|xg(x)|2a2yu+2g(x)k2au-[g(x)]2k2a2u,
R(x)=g(x)aT(a)[u(·, a)](x).
u(x, y)=n=0un(x, y),
Δun+k2un=(1-δn,0)Fn(x, y),
un(x, 0)=-(-iβ)n exp(iαx)[g(x)]nn!,
yun(x, a)+T(a)[u(·, a)](x)=Rn(x),
un(x+d, y)=exp(iαd)un(x, y)
Fn(x, y)=divx[Fn(1)(x, y)]+yFn(2)(x, y)+Fn(3)(x, y),
Fn(1)(x, y)=2g(x)axun-1-[g(x)]2a2xun-2+(a-y)xg(x)ayun-1-(a-y)g(x)xg(x)a2yun-2,
Fn(2)(x, y)=(a-y)xg(x)a·xun-1-(a-y)g(x)xg(x)a2·xun-2-(a-y)2|xg(x)|2a2yun-2,
Fn(3)(x, y)=-xg(x)a·xun-1+g(x)xg(x)a2·xun-2+(a-y)|xg(x)|2a2yun-2+2g(x)k2aun-1-[g(x)]2k2a2un-2,
Rn(x)=g(x)aT(a)[un-1(·, a)](x).
un2K1Bn
us2=l=0sp=-(1+|p|2)s-l0a|yluˆp(y)|2 dy.
ϕ0=p=-0a|ϕˆp(y)|2 dy,
μ1/2=p=-(1+|p|2)1/2|μˆp|2,
η3/2=p=-(1+|p|2)3/2|ηˆp|2
Δw(x, y)+k2w(x, y)=ϕ(x, y),
w(x, 0)=η(x),
yw(x, a)+T(a)[w(·, a)](x)=μ(x),
w(x+d, y)=exp(iαd)w(x, y)
w2Ke(ϕ0+η3/2+μ1/2)
FN0K1K0(CgBN-1+Cg2BN-2),
RN1/2K1K0CgBN-1,
uN2KeFN0+-(iβ)N exp(iαx)[g(x)]NN!3/2+RN1/2KeK1K0(CgBN-1+Cg2BN-2)+βNCgNN!+K1K0CgBN-1K1BN
xlyml!m!un2C2BnDm(m+1)2Al(l+1)2,
Rf[η]=βD[fη]-fβD[η].
Q(x)=p=-βp exp(iαpx),
βp=i|p|+iα sgn(p)-ik22|p|+O(p-2)asp,
Q(x)=exp(iαx)-i21sin2(πx/d)-αcos(πx/d)sin(πx/d)+ik22ln[sin2(πx/d)]+P(x),
η(x)=p=-ηˆp exp(iαpx)
βD[η](x)=p=-βpηˆp exp(iαpx)=p=-βp exp(iαpx)1d0dη(y)exp(-iαpy)dy=1d0dη(y)Q(x-y)dy,
Rf[η](x)=βD[fη]-fβD[η]=1d0dη(y)[f(y)-f(x)]Q(x-y)dy.
[f(y)-f(x)]Q(x-y)
Sf[η]=-Rf2/2[η]+fRf[η].
Sf[η](x)=1d0dη(y)Q(x-y)f(x)[f(y)-f(x)]-[f(y)]22-[f(x)]22dy,
Sf[η](x)=-12d0dη(y)Q(x-y)[f(y)-f(x)]2 dy.

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