Abstract

We present new, stabilized shape-perturbation methods for calculations of scattering from rough surfaces. For practical purposes, we present new algorithms for both low- (first- and second-) and high-order implementations. The new schemes are designed with guidance from our previous results that uncovered the basic mechanism behind the instabilities that can arise in methods based on shape perturbations [D. P. Nicholls and F. Reitich, J. Opt. Soc. Am. A 21, 590 (2004)]. As was shown there, these instabilities stem from significant cancellations that are inevitably present in the recursions underlying these methods. This clear identification of the source of instabilities resulted also in a collection of guiding principles, which we now test and confirm. As predicted, improved low-order algorithms can be attained from an explicit consideration of the recurrence. At high orders, on the other hand, the complexity of the formulas precludes an explicit account of cancellations. In this case, however, the theory suggests a number of alternatives to implicitly mollify them. We show that two such alternatives, based on a change of independent variables and on Dirichlet-to-interior-derivative operators, respectively, successfully resolve the cancellations and thus allow for very-high-order calculations that can significantly expand the domain of applicability of shape-perturbation approaches.

© 2004 Optical Society of America

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    [CrossRef]
  2. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [CrossRef]
  3. 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]
  4. 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]
  5. E. Y. Harper, F. M. Labianca, “Perturbation theory for scattering 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]
  6. 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]
  7. 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]
  8. A. H. Nayfeh, O. R. Asfar, “Parallel-plate waveguide with sinusoidally perturbed boundaries,” J. Appl. Phys. 45, 4797–4800 (1974).
    [CrossRef]
  9. J. R. Wait, “Perturbation analysis for reflection from two-dimensional periodic sea waves,” Radio Sci. 6, 387–391 (1971).
    [CrossRef]
  10. 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]
  11. J. J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988).
    [CrossRef]
  12. 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]
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    [CrossRef]
  14. 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]
  15. 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]
  16. 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]
  17. A. A. Maradudin, “Iterative solutions for electromagnetic scattering by gratings,” J. Opt. Soc. Am. 73, 759–764 (1983).
    [CrossRef]
  18. 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]
  19. 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]
  20. 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]
  21. 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]
  22. D. P. Nicholls, F. Reitich, “Shape deformations in rough surface scattering: Cancellations, conditioning, and convergence,” J. Opt. Soc. Am. A 21, 590–605 (2004).
    [CrossRef]
  23. D. M. Milder, “The effects of truncation on surface-wave Hamiltonians,” J. Fluid Mech. 217, 249–262 (1990).
    [CrossRef]
  24. 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]
  25. D. M. Milder, “An improved formalism for wave scattering from rough surfaces,” J. Acoust. Soc. Am. 89, 529–541 (1991).
    [CrossRef]
  26. 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.
  27. 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]
  28. D. M. Milder, “Role of the admittance operator in rough-surface scattering,” J. Acoust. Soc. Am. 100, 759–768 (1996).
    [CrossRef]
  29. D. M. Milder, “An improved formalism for electromagnetic scattering from a perfectly conducting rough surface,” Radio Sci. 31, 1369–1376 (1996).
    [CrossRef]
  30. A. P. Calderón, “Cauchy integrals on Lipschitz curves and related operators,” Proc. Natl. Acad. Sci. USA 75, 1324–1327 (1977).
    [CrossRef]
  31. 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.
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    [CrossRef]
  36. 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]
  37. 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]
  38. 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]
  39. D. P. Nicholls, F. Reitich, “Analytic continuation of Dirichlet–Neumann operators,” Numer. Math. 94, 107–146 (2003).
    [CrossRef]

2004

2003

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

2001

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]

1996

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]

1995

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]

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]

1993

1992

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]

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]

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]

1991

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]

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

1990

1989

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

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

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

1978

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

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

E. Y. Harper, F. M. Labianca, “Perturbation theory for scattering 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

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

1971

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

1951

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

1907

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]

Baylard, C.

Bruno, O. P.

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).

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]

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.

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]

Gottlieb, D.

D. Gottlieb, S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).

Greffet, J. J.

Harper, E. Y.

E. Y. Harper, F. M. Labianca, “Perturbation theory for scattering 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]

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]

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]

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 for scattering 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]

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.

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, “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]

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.

Nayfeh, A. H.

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

Nicholls, D. P.

D. P. Nicholls, F. Reitich, “Shape deformations in rough surface scattering: Cancellations, conditioning, and convergence,” J. Opt. Soc. Am. A 21, 590–605 (2004).
[CrossRef]

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

Orszag, S. A.

D. Gottlieb, S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).

Quarteroni, A.

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

Rayleigh, Lord

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

Reitich, F.

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.

Tatarskii, V. I.

Versaevel, P.

Wait, J. R.

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

Commun. Pure Appl. Math.

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

J. Acoust. Soc. Am.

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 for scattering 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]

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]

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

J. Appl. Phys.

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

J. Comput. Phys.

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]

J. Fluid Mech.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Numer. Math.

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

Opt. Lett.

Phys. Rev. B

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

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]

Proc. Natl. Acad. Sci. USA

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

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. London Ser. A

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

Radio Sci.

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]

Other

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.

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

D. Gottlieb, S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).

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]

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.

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Figures (13)

Fig. 1
Fig. 1

Plots of relative error in absolute value of the Fourier coefficient (T2(g)[ξ]ˆ)p for two different incident waves, computed by using OE, Improved OE, and Slow OE implementations.

Fig. 2
Fig. 2

Plots of relative error in absolute value of the Fourier coefficient (v3ˆ)p for two different incident waves, computed by using PFE, Improved PFE, and Slow PFE implementations.

Fig. 3
Fig. 3

Plot of the rough surface (3.10) (d=1, h=0.1).

Fig. 4
Fig. 4

Energy defect (3.12) for a scattering configuration with the sinusoidal profile (3.9) using PFE, PTFE, and OE (λ/d=0.4368, h/d=0.3, d=1, a=2, Nx=64, Ny=48, n=64).

Fig. 5
Fig. 5

Energy defect (3.12) for a scattering configuration with the sinusoidal profile (3.9) using PFE, PTFE, and OE (λ/d=0.4368, h/d=0.5, d=1, a=2, Nx=64, Ny=48, n=64).

Fig. 6
Fig. 6

Energy defect (3.12) for a scattering configuration with the rough profile (3.10) using PFE, PTFE, and OE (λ/d=0.4368, h/d=0.25, d=1, a=2, Nx=128, Ny=48, n=64).

Fig. 7
Fig. 7

Energy defect (3.12) for a scattering configuration with the rough profile (3.10) using PFE, PTFE, and OE (λ/d=0.4368, h/d=0.35, d=1, a=2, Nx=128, Ny=48, n=64).

Fig. 8
Fig. 8

Energy defect (3.12) for a scattering configuration with the rough profile (3.10) using PFE, PTFE, and OE (λ/d=0.065, h/d=0.125, d=1, a=0.2, Nx=128, Ny=48, n=64).

Fig. 9
Fig. 9

Energy defect (3.12) for a scattering configuration with the rough profile (3.10) using PFE, PTFE, and OE (λ/d=0.065, h/d=0.175, d=1, a=0.2, Nx=128, Ny=48, n=64).

Fig. 10
Fig. 10

Convergence of specific efficiencies e-15 and e4 for the configuration of Fig. 9 using PFE, PTFE, and OE; comparison of diagonal Padé approximations ([n/2, n/2]) with an overresolved PTFE calculation ([32, 32] approximant).

Fig. 11
Fig. 11

Error (3.16) in the approximation of the current (using a diagonal Padé sum) for the configurations of (a) Fig. 7 and (b) Fig. 9 with incidence (3.13) and p=1; comparison with exact solution (3.15).

Fig. 12
Fig. 12

Error (3.16) in computation of the DNO by means of the OE and DIDO (b=0.05, 0.1, 0.15) algorithms for the sinusoidal profile (3.9) (λ/d=0.4368, h/d=0.1, Nx=128, n=0, , 60). The results in (a) were obtained with Taylor summation, while those in (b) were obtained with Padé approximation.

Fig. 13
Fig. 13

Error (3.16) in computation of the DNO by means of the OE and DIDO (b=0.05, 0.1, 0.15) algorithms for the rough profile (3.10) (λ/d=0.4368, h/d=0.1, Nx=128, n=0, , 60). The results in (a) were obtained with Taylor summation, while those in (b) were obtained with Padé approximation.

Tables (3)

Tables Icon

Table 1 Energy Defect for the Sinusoidal Profile (3.9) under Normal Incidence with a Wavelength-to-Period Ratio λ=0.4368, d=1, and a [32/32] Padé Approximant

Tables Icon

Table 2 Energy Defect for the Rough Surface (3.10) under Normal Incidence with a Wavelength-to-Period Ratio λ=0.4368, d=1, and a [32/32] Padé Approximant

Tables Icon

Table 3 Energy Defect for the Rough Surface (3.10) under Normal Incidence with a Wavelength-to-Period Ratio λ=0.065, d=1, and a [32/32] Padé Approximant

Equations (91)

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vinc(x, y)=exp(iαx-iβy)
v(x, y)=n=0vn(x, y),
vn=O(gn).
v(x, g(x))=-exp[iαx-iβg(x)].
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)
Pn(x)=-exp(iαx)[-iβg(x)]nn!-l=0n-1[g(x)]n-l(n-l)!yn-lvl(x, 0),
T(σ)[ξ](x)=Ξn(x)(x, σ(x)),
T(a)p=-ξˆp exp(iαpx)=p=-(-iβp)ξˆp exp(iαpx),
αp=α+2πdp,βp=k2-αp2.
v(x, y)=0d-Φn(x) (x-x, y-g(x))finc(x)+Φ(x-x, y-g(x))T(g)[finc](x)dx,
T(g)[ξ]=n=0Tn(g)[ξ]
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],
βD=iT(0)=-iy
βDpξˆp exp(iαpx)=pβpξˆp exp(iαpx).
vn(x, 0)=-δn,0 exp[iαx-iβg(x)]-l=0n-1[g(x)]n-l(n-l)!yn-lvl(x, 0).
βpi2πd|p|,|p|1,
T1(g)[ξ]=-(Dg)(Dξ)-Rg[βDξ],
T2(g)[ξ]=iβDSg[βDξ],
v2=-βDg22βDξ-Sg[βDξ],
v3=iβD2g36βDξ+igβDSg[βDξ].
Rf[η]=βD[fη]-fβD[η],
Sf[η]=-Rf2/2[η]+fRf[η],
Rf[η](x)=1d0dη(y)[f(y)-f(x)]Q(x-y)dy,
Sf[η](x)=-12d0dη(y)Q(x-y)[f(y)-f(x)]2 dy,
Q(x)p=-βp exp(iαpx)-i exp(iαx)2 sin2(πx/d) asx0
Rf[η]pˆ=q(βp-βp-q)fˆqηˆp-q,
Sf[η]pˆ=rqfˆrfˆqηˆp-r-q(βp-q-βp-r-q)-12βp-12βp-r-q=12rqfˆrfˆqηˆp-r-q[(βp-q-βp)+(βp-q-βp-r-q)]
Rf[η]pˆ=q[-(2π/d)q](αp+αp-q)βp+βp-qfˆqηˆp-q,
Sf[η]pˆ=rq(2π/d)2qrβp+βp-r-q×1+(αp+αp-q)(αp-q+αp-r-q)(βp+βp-q)(βp-q+βp-r-q)×fˆrfˆqηˆp-r-q,
(T1(g)[ξ]ˆ)p=-q[(2π/d)q]gˆqαp-qξˆp-q+q[(2π/d)q](αp+αp-q)βp+βp-qgˆqβp-qξˆp-q,
(T2(g)[ξ]ˆ)p=iβprq(2π/d)2qrβp+βp-r-q×1+(αp+αp-q)(αp-q+αp-r-q)(βp+βp-q)(βp-q+βp-r-q)×gˆrgˆqβp-r-qξˆp-r-q.
(v2ˆ)p=-12βprqgˆrgˆqβp-r-qξˆp-r-q-rq(2π/d)2qrβp+βp-r-q×1+(αp+αp-q)(αp-q+αp-r-q)(βp+βp-q)(βp-q+βp-r-q)×gˆrgˆqβp-r-qξˆp-r-q,
(v3ˆ)p=i6βp2r,q,sgˆrgˆqgˆsβp-r-q-sξˆp-r-q-s+ir,q,s(2π/d)2sqβp-r+βp-r-q-sβp-r×1+(αp-r+αp-r-s)(αp-r-s+αp-r-q-s)(βp-r+βp-r-s)(βp-r-s+βp-r-q-s)×gˆrgˆqgˆsβp-r-q-sξˆp-r-q-s.
ξˆp=p-2,gˆp=p-2,|p|F,
x=x,y=ay-g(x)a-g(x),
u(x, y)=n=0un(x, y),un=O(gn),
u(x, y)=vx, g(x)+a-g(x)ay
Δun+k2un=(1-δn,0)Fn(x, y),
un(x, 0)=-(-iβ)n exp(iαx)[g(x)]nn!,
yun(x, a)+T(a)[un(·, 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).
un(x, 0)=-δn,0 exp[iαx-iβg(x)].
Tn(g)[finc](x)=|p|<Nx/2τn,p exp(iαpx),
vn(x, y)=|p|<Nx/2dn,p exp(iαpx+iβpy)
u˜n(x, y)=|p|<Nx/2l=0Nyuˆn(p, l)exp(ipx)Tl2y-aa,
y=h2cos2πxd
y=h2cos2πxd+18sin6πxd+19cos6πxd+116cos8πxd;
ep=βpβ|Bp|2,
pUep=1.
U={p|βp>0}
ε=1-pUep,
vinc(x, g(x))=exp[iαpx+iβpg(x)]
v(x, y)=exp(iαpx+iβpy),
vn={[xg(x)]x-y}v|y=g(x)=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)|;
y=b(1-δ)+δg(x)=b-δ[b-g(x)],
T˜(δg)[ξ](x)=v(x, b(1-δ)+δg(x))·(δxg(x), -1),
T(g)[ξ]=T˜(δg)[ξ]|δ=1=n=0{T˜n(g)[ξ]}δnδ=1.
T˜(δg)[exp{iαpx+iβpδg(x)}]={[δxg(x)](iαp)-iβp}×exp(iαpx+iβp{b-δ[b-g(x)]}).
T˜n(g)[exp(iαpx)]=[xg(x)](iαp)exp(ibαp)(iβp)n-1[b-g(x)]n-1(n-1)!×exp(iαpx)-exp(ibαp)(iβp)n+1[b-g(x)]nn!×exp(iαpx)-l=0n-1T˜l(g)(iβp)n-l×{g(x)}n-l(n-l)!exp(iαpx).
exp(ibαD)pξˆp exp(iαpx)=pexp(ibαp)ξˆp exp(iαpx),
T˜n(g)[ξ]=x[b-g(x)]nn!x exp(ibαD)(iβD)n-1ξ+k2[b-g(x)]nn!exp(ibαD)(iβD)n+1ξ-l=0n-1T˜l(g){g(x)}n-1(n-l)!(iβD)n-lξ.
Rf[η](x)=pRˆp exp(iαpx)
Rˆp=q[-(2π/d)q](αp+αp-q)βp+βp-qfˆqηˆp-q.
αp2+βp2=αp-q2+βp-q2=k2,
αp-αp-q=(2π/d)q,
βp-βp-q=(αp+αp-q)[-(2π/d)q]βp+βp-q.
Rˆp=q(βp-βp-q)fˆqηˆp-q
Sf[η](x)=pSˆp exp(iαpx)
Sˆp=rq(2π/d)2qrβp+βp-r-q×1+(αp+αp-q)(αp-q+αp-r-q)(βp+βp-q)(βp-q+βp-r-q)ηˆp-r-qfˆrfˆq.
Sˆp=12rqfˆrfˆqηˆp-r-q[(βp-q+βp)+(βp-q-βp-r-q)]
Sˆp=12rqfˆrfˆqηˆp-r-q[(2π/d)q](αp+αp-q)βp+βp-q+[-(2π/d)r](αp-q+αp-r-q)βp-q+βp-r-q.
12rqfˆrfˆqηˆp-r-q(2π/d)22qαp2βp+-2rαp2βp=0.
A=βp+βp-q,B=βp-q+βp-r-q,
C=βp+βp-r-q,
a=[(2π/d)q](αp+αp-q),
b=[-(2π/d)r](αp-q+αp-r-q),
Sˆp=12rqfˆrfˆqηˆp-r-qaA+bB.
1A-1C=-bABC,1B-1C=-aABC,
Sˆp=12rqfˆrfˆqηˆp-r-qaC-abABC+bC-abABC=12rqfˆrfˆqηˆp-r-qa+bC-2abABC=12rqfˆrfˆqηˆp-r-q(2π/d)22qrC-2abABC,
a+b=(2π/d)2α(q-r)+(2π/d)2{q(2p-q)-r[2(p-q)-r]}by(2π/d)22qr.
2ab=-2(2π/d)2qr(αp+αp-q)(αp-q+αp-r-q),
Sˆp=rqfˆrfˆqηˆp-r-q(2π/d)2qrC×1+(αp+αp-q)(αp-q+αp-r-q)(βp+βp-q)(βp-q+βp-r-q),

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