On the Diffraction of an Electromagnetic Wave through a Plane Screen
Authors: Miles J. W. 
Published: January 2005

Following a discussion of several methods of attack and their application to various particular cases, the problem of the diffraction of an electromagnetic wave through a plane, infinitely thin, perfectly conducting, perforated screen is formulated (in generalized, cylindrical coordinates) in terms of the (generalized Fourier) transform of the tangential electric field in the aperture. The result is an integral equation for this transform, which may also be expressed as an integral equation for the aperture field. The power transferred through the aperture is calculated and cast in a variational form of the Schwinger type, the real and imaginary parts of the reciprocal of the complex power being stationary with respect to first-order variations about the real and imaginary parts of the exact aperture field. An aperture impedance is defined, whose real part (aperture resistance) is the ratio of the aperture scattering cross section to the geometrical cross section, and whose imaginary part is a measure of the standing waves in the neighborhood of the aperture. Moreover, the ratio of the aperture resistance to the impedance of the incident wave is equal to the ratio of the actual power transfer to that predicted by geometrical optics. The real (conductance) and imaginary (susceptance) parts of the aperture admittance are stationary with respect to first-order variations about the real and imaginary parts of the exact aperture field, the former being an absolute minimum in all cases, and the latter only in special cases. A complementary formulation, more suitable to the scattering due to a disk of finite area, is given in terms of the current flowing in the screen. An obstacle admittance, analogous to the aperture impedance, is developed. The two formulations are related by a rigorous form of Babinet's principle (due to Booker, Schwinger, and others). Two more alternative formulations, developed by Copson, are cited. Explicit formulations are given in two-dimensional Cartesian coordinates and in cylindrical polar coordinates. The results are applied to the diffraction of a plane wave through an infinite slit, where the magnetic field is parallel to the slit, and the diffraction of a normally incident plane wave through a circular aperture. The results for the slit compare favorably with the rigorous results computed by Morse and Rubenstein, while the results for the circular hole agree with those of Rayleigh and Bethe in the limit of large wave-length, and with geometrical optics in the limit of small wave-length. The Kirchhoff theory is developed in terms of the aperture conductance and compared with the more exact results. It is found to be very poor in the limit of large wave-length (where the ``static'' methods developed by Rayleigh are valid) and satisfactory in the limit of small wave-length (where geometrical optics give the transmission factor). The variational formulation provides a convenient continuation between the static and geometric limits and appears to be superior to the Kirchhoff theory for any presumed aperture field.