The field distribution in the focal plane of a paraboloidal reflector

The electromagnetic field in the focal plane of a finite axially-symmetrical paraboloidal reflector illuminated by a plane wave of arbitrary polarization incident nearly normally at the vertex, has been investigated. This paper describes the investigation, explains the method, and summarizes some of its results. The reflector is assumed to be of a focal length which is large compared with the wavelength of the incident wave, and the plane of the aperture is assumed to be about halfway between the focus and the vertex. The purpose is to view the diffraction pattern in the focal plane by treating the currents induced on the concave surface of the reflector so that the electromagnetic effect due to them on one part of the surface is taken into account at another by means of waves of current and charge on the paraboloid. The wave equation is derived by approximation from Maxwell's equations; this is feasible because the radii of curvature are large compared with the wavelength. The calculation proceeds by finding approximate solutions of the wave equation by the WKB methods and by noting the "turning point" of the waves at the same distance from the vertex as that of the geodesic corresponding to each pair of characteristic modes of the same harmonic order. The modes are determined and normalized, and thereon the coupling of the incident field is computed. From this representation of the current and charge distribution the electromagnetic field in the focal plane is derived. The integrations required are made by asymptotic approximation using the method of stationary phase. It is shown that in the vicinity of the focus the contribution from these secondary currents is very small compared with the field due to the primary currents
2(\overrightarrow{n} \times H^{inc}
). The primary field is represented by Fourier-Bessel series the number of terms in which is substantially [
m_{1} \sin \alpha +1/2
] where
m_{1}= (4\pi/\lambda) \times
focal length, and
\alpha
=angle of incidence at the vertex. The representation that approximates to geometrical optics when
m_{1}
is large enough does not emerge until the parameter
m_{1}^{1/2} \sin \alpha
is greater than unity and involves
m_{1}^{1/2}
times as many terms in the series. This does not occur until
\alpha
is many times the angular half beamwidth of the main lobe of the reflector.