Preamble

1967, Vol. III, K 517.946 : 9; 538.56 E. A. I. § 1. The problem of diffraction by a disk has been considered in [1], and for a sphere in [2, 3] and [4]. In this paper, we consider the case of a spheroid. In previous works [3, 5], solutions were obtained for various conditions (scalar, acoustic, and electromagnetic [6]), including the case of a perfectly conducting spheroid [6] and [7]. In [6], the solution (for the case of a dipole) was obtained (expressed in terms of spheroidal functions, for the case $a = 0$). However, it is found that in the center, the solution is not unique. From the condition at the boundary $\xi = \xi_0$, we obtain $\xi = \text{const}$. In [8], the problem is reduced to a system of equations for the coefficients $\Phi(\xi, \eta)$ as in [4], where $\eta = 1$ (at $\xi = \xi_0$), and the solution is sought in the form of a series [6] of functions $Y(\xi, \eta)$. We introduce a coordinate system related to the spheroid. Let $x = f[(\xi^2-1)(1 - \eta^2)]^{1/2} \cos\phi$, $y = f[(\xi^2-1)(1 - \eta^2)]^{1/2} \sin\phi$, $z = f\xi\eta$, where $1 < \xi < \infty$, $-1 < \eta < 1$, $0 < \phi < 2\pi$. The surface of the spheroid is defined by $\xi = \xi_0$. Let the incident field be a plane wave $\Pi^ = {1, 0, 0}$, where the potential is $R = (x^2 + y^2 + z^2)^{1/2}$. As $r \to \infty$, the field behaves as $\Pi = \Pi^ + \Pi^s$, where $\Pi^s$ is the scattered field. We represent the potentials as $\Phi = \Phi^0(\xi, \eta) + \Phi^1(\xi, \eta) \cos\phi$. The boundary conditions at $\xi = \xi_0$ are $\Pi_x = 0$ and $\Pi_y = 0$. The electromagnetic field components are given by $E = i k \mu \text{ rot } \Pi$ and $H = \text{ rot rot } \Pi = \text{ grad div } \Pi + k^2 \Pi$. Here $\Phi^0(\xi, \eta)$ corresponds to the primary field. The time dependence is assumed to be $e^{-i\omega t}$. The potential $u$ satisfies the wave equation $\Delta u + k^2 u = 0$. In spheroidal coordinates $(\xi, \eta, \phi)$, the solution can be expanded in terms of spheroidal wave functions.

From the boundary conditions (4), we find that at $\xi = \xi_0$:
$$\begin{aligned}
(A_n, \Pi_n) &= \dots \
\Phi \cos\phi + \Psi \cos\phi &= \dots
\end{aligned}$$
The coefficients are determined by the orthogonality of the functions $S_{mn}(c, \eta)$. Following the method of E. A. Ivanov [9], we obtain a system of algebraic equations for the unknown coefficients.

§ 3. Solution of the Problem

In this section, we expand the primary potential $\Phi^0$ as:
$$\Phi^0 = 2ik \sum_{n=0}^\infty \frac{S_{0n}(c, \eta)}{N_{0n}(c)} R_{0n}^{(1)}(c, \xi)$$
where $c = fk$, $S_{mn}(c, \eta)$ are the angular spheroidal functions, and $R_{mn}^{(3)}(c, \xi)$ are the radial spheroidal functions of the third kind. The normalization constants are $N_{mn}(c)$. The scattered field potential $\Phi^s(\xi, \eta)$ is represented as:
$$\Phi^s(\xi, \eta) = 2ik \sum_{n=0}^\infty a_n R_{0n}^{(3)}(c, \xi) S_{0n}(c, \eta)$$
For the case $m=0$, the boundary conditions (5) lead to a system of equations for the coefficients $a_n$. Similarly, for the components proportional to $\cos\phi$, we have:
$$W(\xi, \eta) = 2ik \sum_{n=1}^\infty b_n R_{1n}^{(3)}(c, \xi) S_{1n}(c, \eta)$$
Substituting these into the boundary conditions (9), we obtain a coupled system for $a_n$ and $b_n$. Using the properties of spheroidal functions:
$$\int_{-1}^1 S_{mn}(c, \eta) S_{mN}(c, \eta) d\eta = \delta_{nN} N_{mN}(c)$$
we can reduce the problem to an infinite system of linear algebraic equations. Let $C_{Nn}, D_{Nn}, V_{Nn}, W_{Nn}$ be the matrix elements derived from the boundary conditions. The system takes the form:
$$\sum_{n} (C_{Nn} a_n + D_{Nn} b_n) = B_N$$
where $B_N$ are coefficients related to the incident field.

7. $R_{mn}(c, \xi)$

Using the results from [6] and the expressions for the coefficients (14), we can analyze the convergence of the solution. For $N = 0, 1, \dots$, the terms with $n=N$ dominate the series. The system (14) can be solved numerically by truncation. For a perfectly conducting spheroid, the coefficients $a_n$ and $b_n$ are determined by the physical parameters of the problem.

The far-field radiation pattern is obtained by taking the asymptotic limit $r \to \infty$. In this limit, the spheroidal coordinates transition to spherical coordinates: $\xi \to r/f$ and $\eta \to \cos\theta$. The components of the electric field in the far zone are:
$$\begin{aligned}
E_\theta &= i k \Phi(\theta) \sin\phi \
E_\phi &= -i k \Psi(\theta) \cos\phi
\end{aligned}$$
where $\Phi(\theta)$ and $\Psi(\theta)$ are the scattering amplitudes. For the case of a small spheroid ($c \ll 1$), the results coincide with the known solutions for dipole scattering.

The case of a disk is recovered by taking the limit $\xi_0 \to 0$. In this limit, the spheroidal functions degenerate into Bessel functions and Legendre polynomials. The results obtained here are consistent with the classical solutions for diffraction by a circular aperture or disk [10, 11].

For a prolate spheroid, the coordinates are defined as:
$x = f[(\xi^2-1)(1-\eta^2)]^{1/2} \cos\phi$, $y = f[(\xi^2-1)(1-\eta^2)]^{1/2} \sin\phi$, $z = f\xi\eta$.
For an oblate spheroid, the transformation $\xi \to i\xi$ and $f \to -if$ is applied. The boundary conditions and the general form of the solution (9)-(23) remain valid under this transformation.

In conclusion, we have derived a rigorous solution for the diffraction of an electromagnetic wave by a spheroid using the method of separation of variables in spheroidal coordinates. The resulting infinite system of equations is suitable for numerical computation.

References

1. Meixner J., Andrejewski W. Ann. der Physik, 6 Folge, Band 7, 157–168, 1950.
2. Hönl H., Maue A. W., Westpfahl K. Theorie der Beugung (Theory of Diffraction). [Russian translation: Moscow, Mir, 1964].
3. Belkina M. G. Diffraction of Electromagnetic Waves by Certain Solids of Revolution. Moscow, Soviet Radio, 1957.
4. Ivanov E. A. Diffraction of Electromagnetic Waves by Two Spheres. [In: "Diffraction"], 1957.
5. Gliner E. B. Mathematical Physics Equations. Moscow, 1962.
6. Flammer C. Spheroidal Wave Functions. Stanford University Press, 1957. [Russian translation: Moscow, ILC, 1962].
7. Ritter E. K. Transactions IRE, Antennas and Propagation, AP-3, No. 4, 276, 1956.
8. Senior T. B. A. Canadian Journal of Physics, 44, No. 7, 1353–1359, 1966.
9. Ivanov E. A. Izvestiya AN BSSR, Ser. Fiz.-Mat. Nauk, No. 2, 1966.
10. Fock V. A. Journal of Experimental and Theoretical Physics (JETP), 19, No. 10, 916, 1949.
11. Weinstein L. A. Electromagnetic Waves. Moscow, Soviet Radio, 1957.

Submitted December 18, 1966.