An analytical, closed-form solution to the scattering problem from an infinite lossless or lossy elliptical cylinder coating a circular metal core is treated in this work. The problem is solved by expressing the electromagnetic field in both elliptical and circular wave functions, connected with one another by well-known expansion formulas. The procedure for solving the problem is cumbersome because of the nonexistence of orthogonality relations for Mathieu functions across the dielectric elliptical boundary. The solution obtained, which is free of Mathieu functions, is given in closed form, and it is valid for small values of the eccentricity h of the elliptical cylinder. Analytical expressions of the form S(h)=S(0)[1+g(2)h2+g(4)h4+O(h6] are obtained, permitting an immediate calculation for the scattering cross sections. The proposed method is an alternative one, for small h, to the standard exact numerical solution obtained after the truncation of the system matrices, composed after the satisfaction of the boundary conditions. Both polarizations are considered for normal incidence. The results are validated against the exact solution, and numerical results are given for various values of the parameters.
A new expansion of a function, inspired by the Taylor series expansion, is introduced, and various examples of functions admiting such new expansion, are given. In addition, it is shown that the new expansion, when applied to functions being completely monotonic, leads to some quite interesting results and applications. Finally, a new approach is suggested towards obtaining the solution of the difference equationand examples involving the gamma and psi functions are given.
The electron-density distribution is determined in a laser tube that is formed by a hyperboloid of revolution and spherical end surfaces. Calculations are made assuming different symmetrical optical resonator configurations ranging from near-planar to near-confocal. The tube geometry is specified in terms of the optical resonator parameters: resonator length L, mirror radius of curvature, R, and optical spot size at the mirror surface. The active medium is assumed to be a low-temperature, weakly ionized plasma. Schottky boundary conditions are assumed and the electron temperature is taken to be independent of position. The particle-diffusion equation is solved using oblate spheroidal coordinates. The oblate spheroidal angular dependence (which in the cylindrical limit corresponds to the variation perpendicular to the axis of the cylinder) of the density is found to be that of a zero-order Bessel function. The oblate spheroidal radial dependence (which in the cylindrical limit corresponds to the axial dependence) is obtained by means of numerical methods. Radial density profiles are calculated, assuming L=1.0 m, for various R/L ratios. The position Zp of the peak radial density is found to depend on the R/L ratio; as R/L decreases Zp is observed to move towards the end walls of the tube.
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