Various integral transforms have been extensively used in the formulation of electromagnetic scattering, radiation, antennas, and electromagnetic interference-related problems. The integral transform technique [1,2] is an indispensable tool for representing the fields in the unbounded (open) region. This technique is often combined with the mode-matching method to solve electromagnetic boundary-value problems. The purpose of the present paper is to review the integral transforms that are applied in conjunction with the mode-matching method.
In the paper, we will limit our discussion to the canonical slotted conductors, which will enable us to use the technique of variable separation. The use of Fourier transform, Hankel transform, Mellin transform, Kontorovich-Lebedev transform, and Weber transform is discussed. We will show how these integral transforms can be incorporated into the pertinent electromagnetic boundary-value problems, which are formulated in terms of Helmholtz’s equation or Laplace’s equation. Depending on the problem geometries, a certain type of integral transforms must be chosen to facilitate the solutions to Helmholtz’s equation or Laplace’s equation for the potentials. From the integral transform definitions, we will show how the potentials in the unbounded region can be obtained. In the next section, we begin with Fourier transform. The time convention exp(-
The Fourier transform pair is as follows:
The Fourier transform technique has long been used for electromagnetic scattering, diffraction, and antenna applications. We will discuss the formulation of Fourier transform by considering radiation from slotted circular waveguides. Fig. 1 shows a circular waveguide with a narrow circumferential slot array. The circular waveguide is infinitely long in the
The
We will represent the scattered electric vector potential
We substitute Eq. (4) into Eq. (3) to obtain Bessel’s differential equation:
Since the field is finite at the origin, the Bessel function of the first kind,
A complete radiation analysis using the boundary conditions is available in [3].
The Hankel transform pair is as follows:
The Hankel transform is useful for the analysis of scattering from circular apertures. Consider electromagnetic scattering from a circular aperture in an infinitely extended, perfectly conducting plane, as shown in Fig. 2.
Assume that a uniform plane wave is incident on a circular aperture from below. The transmitted field in region I, which is above the slotted conducting plane, can be written in terms of the
Substituting Eq. (9) into Eq. (3) yields:
where Since the transmitted field must vanish when
The field representations and some computations are available in [4].
The Mellin transform pair is:
Consider the two-dimensional (
We will determine Փ(
Substituting Eq. (15) into Eq. (14) gives:
Since the electric field at
A complete solution to the potential problem is provided in [5].
The Kontorovich-Lebedev transform pair is as follows:
where is the Hankel function of the first kind. The incident wave impinges on the structure, as shown in Fig. 4. Assume that the scattering problem is two-dimensional (
We express
Substituting Eq. (21) into Eq. (20) yields:
Hence, the solution is:
Finally, we obtain:
A complete wedge-scattering analysis using the boundary conditions can be found in [6].
The Weber transform pair is [7]:
where
We wish to determine the electrostatic potential for the equivalent problem. The electrostatic potential Փ(
If the boundary condition for the equivalent problem requires Փ(
Substituting Eq. (28) into Eq. (27) gives:
If the boundary condition is such that the potential is zero when
A complete potential analysis is given in [8].
In this paper, the integral transform technique in electromagnetic boundary-value problems was shown. Fourier transform, Hankel transform, Mellin transform, Kontorovich-Lebedev transform, and Weber transform were introduced. Starting from Helmholtz’s equation or Laplace’s equation, pertinent potential expressions for the open regions were derived. The integral transform technique can be adequately applied to electromagnetic scattering and radiation problems, in particular when the scattering geometries have canonical cylindrical shapes.