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**Scattering theory for diffraction gratings.**
*(English)*
Zbl 0541.76001

Applied Mathematical Sciences, Vol. 46. New York etc.: Springer-Verlag. IX, 163 p. DM 48.00; $ 17.90 (1984).

This very interesting and nicely written monograph develops a theory of the scattering of transient electromagnetic and acoustic fields by diffraction gratings. The theory is based on an eigenfunction expansion for gratings in which the eigenfunctions are Rayleigh-Bloch waves. The eigenfunction expansions are generalizations of T. Ikebe’s theory of distorted plane wave expansions first developed for quantum mechanical potential scattering [Arch. Ration. Mech. Anal. 5, 1-34 (1960; Zbl 0145.369)]. Central to the development of the theory is the study of a linear operator A, called the grating propagator, which is a self-adjoint realization of the negative of the Laplace operator acting in a Hilbert space of square integrable acoustic fields. A fundamental result of this analysis is a representation of the spectral family of A by means of Rayleigh-Bloch waves.

The theory of scattering by gratings developed in the monograph is restricted to two-dimensional wave propagation. In particular waves are assumed to be solutions of the wave equation in a two-dimensional grating domain and to satisfy Dirichlet or Neumann boundary conditions on the grating profile. These problems provide models for the scattering of sound waves by acoustically soft or rigid gratings and of TE or TM electromagnetic waves by perfectly conducting gratings.

The methods employed are applicable to the scattering of scalar waves by three-dimensional gratings and to systems such as Maxwell’s equations and the equation of elasticity.

In order to make the monograph accessible to a variety of users it is divided into two parts. Part I can be interpreted both as a complete statement, without proofs of the physical concepts and results of the theory and also as a summary and introduction to the complete mathematical theory developed in part 2.

A key step in the development of the theory is the introduction of the reduced grating propagator Ap which depends on the wave momentum. The Hilbert space theory of such operator waves is initiated by H. D. Alber [Proc. R. Soc. Edinb. Sect. A 82, 251-272 (1979; Zbl 0402.35033)]. Indeed Alber’s powerful methods influence much of theory and in particular the analytic continuation of the resolvent of Ap.

The theory of scattering by gratings developed in the monograph is restricted to two-dimensional wave propagation. In particular waves are assumed to be solutions of the wave equation in a two-dimensional grating domain and to satisfy Dirichlet or Neumann boundary conditions on the grating profile. These problems provide models for the scattering of sound waves by acoustically soft or rigid gratings and of TE or TM electromagnetic waves by perfectly conducting gratings.

The methods employed are applicable to the scattering of scalar waves by three-dimensional gratings and to systems such as Maxwell’s equations and the equation of elasticity.

In order to make the monograph accessible to a variety of users it is divided into two parts. Part I can be interpreted both as a complete statement, without proofs of the physical concepts and results of the theory and also as a summary and introduction to the complete mathematical theory developed in part 2.

A key step in the development of the theory is the introduction of the reduced grating propagator Ap which depends on the wave momentum. The Hilbert space theory of such operator waves is initiated by H. D. Alber [Proc. R. Soc. Edinb. Sect. A 82, 251-272 (1979; Zbl 0402.35033)]. Indeed Alber’s powerful methods influence much of theory and in particular the analytic continuation of the resolvent of Ap.

Reviewer: B.D.Sleeman

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76Q05 | Hydro- and aero-acoustics |

74J20 | Wave scattering in solid mechanics |

78A45 | Diffraction, scattering |