A new multipole solution in plane-polar coordinates for the scattering of an arbitrary TE- or TM-polarized incident field by an infinitely extended slit in a plane screen is derived. To this end a classical three-domain problem with a complete multipole expansion in each of the domains is formulated. The unknown multipole amplitudes are found from the continuity conditions of the tangential electric and magnetic field components. Finally an infinite system of linear equations is derived which can be approximated by a numerically tractable finite one leading to a solution with an arbitrary accuracy. The results are numerically successfully compared to those ones obtained by classically solving the strip problem using Mathieu functions and by a subsequent application of the rigorous form of the Babinet principle. Numerical results include the comparison of the scattered fields obtained for an incident uniform complex-source beam and for an incident plane wave.

Because of its general theoretical and practical interest the scattering and diffraction of acoustic and electromagnetic waves by a slit has been very often discussed in the literature. An early contribution to that subject was given by

In the present work we introduce an alternative method to exactly solve the slit/strip problem

The numerical evaluation includes a successful validation of the method by comparing the results with those obtained by the aforementioned solution of the strip problem and application of the Babinet principle. Then we compare the scattered fields for an incident plane wave to those obtained for an incident uniform complex-source beam with different parameters. The goal of this investigation is to find out if and for which parameters a uniform CSB can replace an incident plane wave. This can be useful for many applications – particularly in the context of scattering from special sections as part of infinitely extended structures like the scattering by the tip of a circular or elliptic cone.

Consider the geometry in Fig.

Definition of the three-domains of the boundary value problem.

We split the entire space into three domains as sketched in Fig.

In each of the domains I, II, and III we introduce a complete plane-polar multipole expansion for the electric field intensity according to:

The

At the circular boundary around domain I the tangential components of the electric field intensity have to be continuous:

In each of the domains I, II, and III we introduce a complete plane-polar multipole expansion for the magnetic field intensity

The

In domain III, the scattered far-field can be obtained from Eq. (

Snapshot of a uniform CSB with

A complex-source beam (CSB) is obtained for a complex-valued point-source coordinate. As has been shown, in a paraxial approximation a CSB represents a Gaussian beam

Figure

Snapshots of the near fields obtained by the present approach using 60 eigenfunctions for the incident field, 40 eigenfunctions for the scattered field in regions II and III, and 20 eigenfunctions in region I.

Comparison between the normalized far fields obtained by the present approach using 40 eigenfunctions (solid) for the scattered field and by applying the rigorous form of Babinet principle to the solution found for the strip in elliptic coordinates using

Snapshots of the TM

Snapshots of the TE

Polar diagrams of the scattered far-fields for an incident plane wave and for incident uniform CSBs with different values of the focus length

Basically, all of the infinite series involved for solving the boundary value problem have to be truncated to come to a numerical solution. The maximal order of multipole functions needed for a desired accuracy of the scattered field (for instance,

We start with a comparison of the results of the current approach to those ones obtained by solving the complementary problem of a strip using Mathieu functions and a subsequent application of the rigorous form of the Babinet principle

Figures

Finally we remark that the code used for the calculation of the Mathieu functions

Polar diagrams of the scattered far-fields for an incident plane wave and for incident uniform CSBs with different values of the angle of incidence.

Relative maximum deviation between the normalized scattered far fields in case of an incident plane wave and an incident uniform CSB for different angles of incidence as a function of the normalized focus length

As described above the proposed solution can be easily extended to include the case of an incident uniform CSB. Of particular interest is the question whether a CSB can replace a plane wave to investigate the scattering by certain areas of a scattering object. As the best similarity of a uniform CSB to a localized plane wave is obviously in the waist, we choose its location almost at the center of the slit (

The relative maximum deviation for the TE

As expected, an increase of the focus length and corresponding beam width leads to a reduction of the maximum relative deviation. Moreover, for the TM

A new direct method to analytically solve the electromagnetic scattering and diffraction by a slit has been derived. The results are in agreement to those ones obtained by the classical solution in elliptic coordinates using Mathieu functions and a subsequent application of the rigorous form of the Babinet principle. It has been shown that an incident uniform complex-source beam can be used instead of an incident plane wave if the waist is chosen to be located nearby the slit and the beam width at the waist has been chosen to be sufficiently larger than the slit.

The data are available from the author upon request.

The author declares that there is no conflict of interest.

This article is part of the special issue “Kleinheubacher Berichte 2019”. It is a result of the Kleinheubacher Berichte 2019, Miltenberg, Germany, 23–25 September 2019.

This paper was edited by Thomas Eibert and reviewed by two anonymous referees.