On plane-wave expansions of cylindrical waves

Wave transformations are often invoked in many waveguide discontinuity and electromagnetic scattering problems when different coordinate systems are employed to describe various waveguide cross-sections and scattering objects. There were a few formulas available in various books and papers to realize the transformation between plane wave and cylindrical wave. However, many of the formulas are limited to Bessel functions of the first kind or Hankel functions of order 0. This paper is mainly concerned with the plane-wave expansions of cylindrical waves. For the completeness, we first give the expansion of a plane wave in terms of cylindrical waves. After that, we derive the integral expression for the plane-wave expansion of Bessel function of the first kind. In the end, we present the plane-wave expansion of the Bessel function of the second kind.