Cutoff Wavelengths of Elliptical Metallic Waveguides

Abstract

The cutoff wavelengths lambda <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cmn</sub> of elliptical metallic waveguides with perfectly conducting walls are determined analytically. Two different methods are used for the evaluation. In the first, the electromagnetic field is expressed in terms of elliptical-cylindrical wave functions. In the second, a shape perturbation method, the field is expressed in terms of circular-cylindrical wave functions only, while the equation of the elliptical boundary is given in polar coordinates. Analytical expressions are obtained for the cutoff wavelengths, when the solution is specialized to small values of the eccentricity h=c/2a, (hLt1), with c the interfocal distance of the elliptical waveguide and 2a the length of its major axis. In this case, exact closed-form algebraic expressions, free of Mathieu as well as of Bessel functions, are obtained for the expansion coefficients g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mn</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2)</sup> and g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mn</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(4)</sup> in the resulting relation lambda <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cmn</sub> (h)=lambda <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cmn</sub> (0) [1+g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mn</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2)</sup> h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> +g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mn</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(4)</sup> h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> + O(h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sup> )] for the cutoff wavelengths. These expressions are valid for each m and n, namely, for the general mode. Numerical results for all types of modes and comparison with existing ones are also included.