Solving the Axial Line Problem for Fast Computation of Mixed Potential Green's Functions for Cylindrically Stratified Media

Abstract

An improved approach is proposed for solving the axial line problem (|ρ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">→</sup> -ρ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">→</sup> | = 0) of mixed potential Green's functions (MPGFs) in spatial domain for cylindrically stratified media. Different from the previous work, to alleviate the convergence problem of the summation of cylindrical eigenmodes in the spectral domain, two asymptotic terms with respect to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> are extracted analytically. In this way, the higher order nonphysical singularities with respect to |ρ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">→</sup> - ρ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">→</sup> | are proven to vanish, and the lower order physical singularities can be transformed into the spatial domain completely using the Sommerfeld identity. Therefore, the remaining parts of the summation are free of singularities related to |ρ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">→</sup> - ρ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">→</sup> | and fast convergent, which can be easily transformed into closed-form spatial-domain expressions by applying the generalized pencil of function method. The present approach can be used to accurately calculate the spatial domain MPGFs over the entire cylindrical surface. Several numerical results are presented to verify the accuracy and efficiency of this approach.