An Integral Equation for the Surface Current in the Problem of the Natural Modes of a Waveguide Containing a Resistive Film

Technical Summary: An Integral Equation for the Surface Current in the Problem of the Natural Modes of a Waveguide Containing a Resistive Film

Problem Statement
This paper addresses the eigenmode problem for a uniform waveguide containing an infinitely thin resistive film characterized by a surface impedance $W$. The objective is to determine the electromagnetic field distribution of the eigenmodes, defined by the propagation constant $\gamma$ and transverse wavenumber $\kappa$. The physical configuration involves a waveguide with a perfectly conducting boundary and a resistive film located within the cross-section. The film introduces two-sided boundary conditions where the tangential electric field is proportional to the surface current density, and the magnetic field exhibits a discontinuity proportional to the current. The challenge lies in formulating a rigorous mathematical framework to solve for the surface current $h(v)$ on the film, which satisfies both the wave equations and specific edge conditions at the film's boundaries.

Methodology
The authors formulate the problem by deriving a vector integral equation for the surface current density $h(v)$. The methodology proceeds through the following steps:

1. Green's Function Formulation: The electromagnetic field components ($E_z, H_z, E_v$) are expressed in terms of the surface current using two-dimensional Green's functions ($G_e$ and $G_h$) that satisfy the Helmholtz equation and the appropriate Dirichlet or Neumann boundary conditions on the waveguide walls.
2. Boundary Condition Application: By applying the two-sided boundary conditions on the resistive film and utilizing Green's theorem, the field components are represented as integrals over the film's contour $L$.
3. Derivation of the Integral Equation: The tangential electric field components on the film are related to the surface current. By taking the limit as the observation point approaches the film surface, the authors derive a vector integral equation of the form:
   $$\int_L dv_1 \overleftrightarrow{R}(v, v_1) \cdot h(v_1) + W h(v) = 0$$
   Here, $\overleftrightarrow{R}(v, v_1)$ is a tensor kernel dependent on the Green's functions and their derivatives.
4. Handling the Kernel Singularity: A critical technical hurdle identified is the evaluation of the term $D(v, v_1)$, which involves the limit of a second derivative of the magnetic Green's function ($G_h$) normal to the film surface. The authors note that standard methods involving term-by-term differentiation of eigenfunction series (requiring summation in the Abel–Poisson sense) are difficult to implement for non-Cartesian geometries. Consequently, they develop alternative methods to evaluate this limit explicitly without relying on Abel–Poisson summation.

Key Contributions and Results
The primary contribution of the paper is the derivation of explicit expressions for the kernel tensor $\overleftrightarrow{R}$ for two specific geometries: rectangular and circular waveguides.

• Rectangular Waveguide:
  For a rectangular waveguide, the authors introduce four Green's functions ($\Gamma_{pq}$) satisfying various boundary conditions. They derive an identity relating the mixed derivatives of the Green's functions to the Laplacian of these auxiliary functions. This allows the evaluation of the limit $D(v, v_1)$ explicitly. The resulting expression for the kernel involves the Green's functions $\Gamma_{he}$ and $\Gamma_{eh}$ and their derivatives, valid for films at arbitrary angles $\theta(v)$, including special cases like the E-plane and diagonal plane.

• Circular Waveguide:
  For a circular waveguide, the Green's functions are initially expressed as a double series of Bessel functions. To avoid the difficulties of differentiating this series directly, the authors utilize an alternative representation involving a sum over the angular mode number $m$ with radial functions $g_{pm}$.

  • General Case: For films not lying in the diametral plane, the authors express the mixed derivative $\partial^2 G_h / \partial u \partial u_1$ in terms of radial and tangential derivatives. They derive an explicit formula for the limit by differentiating the radial functions $g_{hm}$, carefully handling the jump discontinuity at $\rho = \rho_1$.
  • Diametral Plane Case: For the specific case where the film lies in the diametral plane ($\phi = 0, \pi$), the authors derive a distinct expression for $D(v, v_1)$ using the differential equation satisfied by $G_h$, relating the normal derivative to the radial derivative and the function itself.

Significance and Claims
The paper claims to provide a complete formulation for the integral equation governing the surface current in waveguides with resistive films. Its significance lies in the explicit derivation of the kernel $D(v, v_1)$ for both rectangular and circular cross-sections. The authors emphasize that their method for calculating the limit (16) is distinct from previous approaches that relied on Abel–Poisson summation of eigenfunction series. By avoiding this summation, the proposed method offers a more direct and potentially robust procedure for evaluating the kernel, particularly for geometries where eigenfunctions depend on polar coordinates. The derived expressions are intended to facilitate the numerical solution of the integral equation for finding the natural modes of such waveguide systems.