Introduction

In many fields of contemporary science, from astrophysics to biology, scientists are required to find effective solutions for wave diffraction and scattering problems. This is, in particular, confirmed by the programs of regular conferences such as “Days on Diffraction,” “Mathematical Methods in Electromagnetic Theory,” “Electromagnetic and Light Scattering,” “PIERS,” etc.

The use of a priori information permits a significant increase in the efficiency of algorithms for solving such problems, and in several cases, the problem can principally only be solved by using such information. Here, we consider several examples to illustrate this.

1. Diffraction at thin screens. In problems of wave diffraction at thin screens, there is a complexity related to the fact that the current I(x) on the screen has a singularity near the screen edge. It follows from the Meixner condition [1] that the current component parallel to the screen edge has a singularity of the form −1/2, where ρ is the distance to the screen edge. Thus, for example, when solving the problem of diffraction at an infinitely thin band of width 2a (in the case of E-polarization), it is expedient to represent the desired current as

x=1a2−x2Jx,

where J(x) is now a smooth function.

If the boundary-value problem reduces to an integral equation, then the use of such a representation allows one to simplify the corresponding algorithm to solving this equation.

2. Diffraction at a periodic grating. We now consider the problem of diffraction of a plane wave at a periodic grating consisting of N elements. If there are infinitely many elements, then the problem reduces to a single period of the grating. If N is not large, then it is necessary to solve the problem of diffraction at a group of bodies. The situation is most difficult when N is finite but rather large. In this case, it is very difficult to solve the problem of diffraction at a group of N bodies because of the extremely large volume of calculations. Here it is appropriate to use a priori information stating that the distribution of fields or currents on <N central elements of the grating is approximately the same as the distribution on an infinite periodic grating [2]. Thus, the problem reduces to determining the field distribution on −M+1 elements of the grating, namely, on −M edge elements and one central element, which already allows the problem to be solved numerically.

3. Wave scattering at a body whose dimensions are much more than the wavelength. In this case, the use of the standard methods, for example, using integral equations, is not efficient because algebraization results in a system of large-sized algebraic equations. Here, it is expedient to use the hybrid method, where the current on a greater part of the scatterer is assumed to be equal to the current of geometric optics, i.e., the unknown current is equal to the current on the corresponding tangent plane on the illuminated part of the scatterer, and to zero on the shaded part [3]. The current remains unknown only near the “light-shadow” boundary, which permits a significant decrease in the dimensions of the corresponding algebraic system.

In the second and third examples, we discuss the use of a priori information of physical character, which was obtained by solving special problems. Already in the first example, we discuss the use of information about the analytic properties of the solution, which is directly contained in the boundary condition (the Meixner condition). This condition clearly contains the required information. From this standpoint, the problems of wave diffraction at smooth bodies, where the boundary conditions do not give any explicit information about the analytic properties of the solution, are much more complicated. We consider the diffraction problem for a monochromatic wave with the time-dependence t=eiωt (here ω is the cyclic frequency of oscillations) propagating in a homogeneous isotropic medium with relative dielectric permittivity ɛ and relative magnetic permeability μ. The mathematical model of such a problem is the system of Maxwell equations for the vectors of electromagnetic field strength with boundary conditions or, in the scalar approximation, the external boundary-value problem for the Helmholtz equation [1,4,5]. As is known in this case (e.g., see [5,6]), the solutions of these equations are real analytic functions that we call wave fields. The direct diffraction problem consists of determining the secondary wave field arising when the known primary field meets a certain obstacle, i.e., a scatterer. Throughout the monograph, we deal only with external problems of diffraction. The solutions of such problems are defined as everywhere outside the scatterer, i.e., outside a certain domain D with boundary S. Outside D, the wave fields vanish at infinity according to the Sommerfeld radiation condition [1,5]. This means that the wave field must have singularities in the domain D (or on its boundary), because otherwise (as will be shown below) the field would be zero.

We consider another example that illustrates the importance of such information.

Assume that we must solve the problem of diffraction of a primary wave field 0r→ at a compact scatterer bound by the surface S. When solving this problem, one can use the following representation for the field 1r→ scattered at the body [4,5]

1(r,θ,φ)=∑n=0∞∑m=−nnanm−in+1hn2krPnmcosθeimφ.

A similar representation in the form of a series in spherical harmonics also holds in the case of the vector [7]. Substituting this expansion for 1r→ in the boundary condition, we can determine the coefficients anm and hence solve the boundary-value problem. However, this can only be performed if the series written for 1r→ converges to the scatterer boundary S. This assumption is known in the literature as the Rayleigh hypothesis [8,9] and the bodies satisfying this hypothesis are called Rayleigh bodies. The series in spherical harmonics for the scattered field 1r→ is a power series in its infinite residuals (see below), and hence converges only outside the sphere containing all singularities of the function 1r→. Thus, only the bodies that are completely contained inside this sphere S are Rayleigh bodies. To know whether the body is Rayleigh or not is obviously of principal importance when solving the diffraction problem by the method described above. The fact that this information can be obtained a priori is also important [9,10].

What are these singularities of the function 1r→? We consider a simple example. Assume that a source of light is located in front of a plane mirror S. When looking at a mirror, we see the source of light at a point behind the mirror that is symmetric with respect to S (see Fig. 1). We see this imaginary source (image) as the straight continuation of the rays reflected from the mirror. The source image is precisely the result of analytic continuation of the field reflected from the mirror into the region behind the mirror.

We perceive the obtained image as an additional source of light on the other side of the mirror as if we can see “though the mirror.” Moreover, if we begin to move the source of light away from the mirror, then its image “behind the mirror” also moves symmetrically as far as possible.

The situation is quite different in the case of reflection from a nonplane surface. We now imagine that we begin to bend a plane mirror by lifting its “edges.” The space behind the mirror begins to “contract” forming a “fold.” A part of the image or even the entire image of the object may then disappear (this effect is familiar to everybody who has looked at their image in halls of mirrors, recall the laughter rooms). This effect arises due to formation of “folds” in the hypothetical media behind the mirror, i.e., of regions containing two (or even more) images simultaneously. In this case, the disappearing part of the image is “hidden in the fold”. A more detailed qualitative discussion of this effect is available (see [10]).

In the case of problems of diffraction at compact obstacles, the wave field (scattered at an obstacle) must also have singularities. Obviously, these singularities must lie outside the domain, where we seek the diffraction field (in the so-called nonphysical region). Thus, we speak of singularities of the analytic continuation of the wave field beyond the original domain of its...