## Shape deformations in rough-surface scattering: cancellations, conditioning, and convergence

JOSA A, Vol. 21, Issue 4, pp. 590-605 (2004)

http://dx.doi.org/10.1364/JOSAA.21.000590

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### 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

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(050.2770) Diffraction and gratings : Gratings

(290.0290) Scattering : Scattering

**Citation**

David P. Nicholls and Fernando Reitich, "Shape deformations in rough-surface scattering: cancellations, conditioning, and convergence," J. Opt. Soc. Am. A **21**, 590-605 (2004)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-4-590

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### References

- Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
- S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
- G. V. Anand and M. K. George, “Normal mode sound propagation in an ocean with sinusoidal surface waves,” J. Acoust. Soc. Am. 80, 238–243 (1986).
- G. V. Anand and M. K. George, “Normal mode sound propagation in an ocean with random narrow-band surface waves,” J. Acoust. Soc. Am. 94, 279–292 (1993).
- E. Y. Harper and 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).
- E. Y. Harper and 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).
- W. A. Kuperman and 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).
- A. H. Nayfeh and O. R. Asfar, “Parallel-plate waveguide with sinusoidally perturbed boundaries,” J. Appl. Phys. 45, 4797–4800 (1974).
- J. R. Wait, “Perturbation analysis for reflection from two-dimensional periodic sea waves,” Radio Sci. 6, 387–391 (1971).
- W. C. Meecham, “On the use of the Kirchoff approximation for the solution of reflection problems,” J. Rat. Mech. Anal. 5, 323–334 (1956).
- J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
- J. M. Chesneaux and A. A. Wirgin, “Response to comments on ‘Reflection from a corrugated surface revisited’,” J. Acoust. Soc. Am. 98, 1815–1816 (1995).
- J. J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988).
- J. J. Greffet, C. Baylard, and P. Versaevel, “Diffraction of electromagnetic waves by crossed gratings: a series solution,” Opt. Lett. 17, 1740–1742 (1992).
- J. J. Greffet and Z. Maassarani, “Scattering of electromagnetic waves by a grating: a numerical evaluation of the iterative-series solution,” J. Opt. Soc. Am. A 7, 1483–1493 (1990).
- D. R. Jackson, D. P. Winebrenner, and A. Ishimaru, “Comparison of perturbation theories for rough-surface scattering,” J. Acoust. Soc. Am. 83, 961–969 (1988).
- 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).
- C. Lopez, F. J. Yndurain, and N. Garcia, “Iterative series for calculating the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 970–972 (1978).
- A. A. Maradudin, “Iterative solutions for electromagnetic scattering by gratings,” J. Opt. Soc. Am. 73, 759–764 (1983).
- 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).
- 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).
- 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).
- O. P. Bruno and 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).
- O. P. Bruno and F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
- O. P. Bruno and 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).
- O. P. Bruno and 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).
- O. P. Bruno and F. Reitich, “Calculation of electromagnetic scattering via boundary variations and analytic continuation,” Appl. Comput. Electromagn. Soc. J. 11, 17–31 (1996).
- O. P. Bruno and F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am. 104, 2579–2583 (1998).
- O. P. Bruno and F. Reitich, “High-order boundary perturbation methods,” in Mathematical Modeling in Optical Sci-ence, 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. SPIE 1558, 213–221 (1991).
- D. M. Milder, “An improved formalism for wave scattering from rough surfaces,” J. Acoust. Soc. Am. 89, 529–541 (1991).
- D. M. Milder and 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 and H. T. Sharp, “An improved formalism for rough surface scattering. II: Numerical trials in three dimensions,” J. Acoust. Soc. Am. 91, 2620–2626 (1992).
- P. J. Kaczkowski and 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).
- D. M. Milder, “Role of the admittance operator in rough-surface scattering,” J. Acoust. Soc. Am. 100, 759–768 (1996).
- D. M. Milder, “An improved formalism for electromagnetic scattering from a perfectly conducting rough surface,” Radio Sci. 31, 1369–1376 (1996).
- R. A. Smith, “The operator expansion formalism for electromagnetic scattering from rough dielectric surfaces,” Radio Sci. 31, 1377–1385 (1996).
- A. G. Voronovich, Wave Scattering from Rough Surfaces, 2nd ed. (Springer-Verlag, Berlin, 1999).
- K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
- H. Hasimoto and H. Ono, “Nonlinear modulation of gravity waves,” J. Phys. Soc. Jpn. 33, 805–811 (1972).
- K. M. Watson and B. J. West, “A transport-equation description of nonlinear ocean surface wave interactions,” J. Fluid Mech. 70, 815–826 (1975).
- D. M. Milder, “The effects of truncation on surface-wave Hamiltonians,” J. Fluid Mech. 217, 249–262 (1990).
- W. Craig and C. Sulem, “Numerical simulation of gravity waves,” J. Comput. Phys. 108, 73–83 (1993).
- R. A. Smith, “An operator expansion formalism for nonlinear surface waves over variable depth,” J. Fluid Mech. 363, 333–347 (1998).
- D. P. Nicholls, “Traveling gravity water waves in two and three dimensions,” Ph.D. thesis (Brown University, Providence, R.I., 1998).
- D. P. Nicholls, “Traveling water waves: spectral continuation methods with parallel implementation,” J. Comput. Phys. 143, 224–240 (1998).
- P. Dinesen and J. S. Hesthaven, “Fast and accurate model- ing of waveguide grating couplers,” J. Opt. Soc. Am. A 17, 1565–1572 (2000).
- P. Dinesen and 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).
- G. Bao and 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.
- O. P. Bruno and 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).
- D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, Berlin, 1998).
- D. P. Nicholls and F. Reitich, “Analytic continuation of Dirichlet–Neumann operators,” Numer. Math. 94, 107–146 (2003).
- A. P. Calderón, “Cauchy integrals on Lipschitz curves and related operators,” Proc. Natl. Acad. Sci. USA 75, 1324–1327 (1977).
- R. Coifman and 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 and F. Reitich, “A new approach to analyticity of Dirichlet–Neumann operators,” Proc. R. Soc. Edinburgh, Sect. A 131, 1411–1433 (2001).
- D. P. Nicholls and F. Reitich, “Stability of high-order perturbative methods for the computation of Dirichlet–Neumann operators,” J. Comput. Phys. 170, 276–298 (2001).
- C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1988).
- A. Friedman and 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).
- J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Mod. Opt. 11, 235–241 (1980).
- L. Li, J. Chandezon, G. Granet, and J. P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
- O. A. Ladyzhenskaya and 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 and 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).

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