A Closed-Form Solution for Electromagnetic Wave Propagation in Spatially Unbounded, Linear Time-Varying Plasmas

We present a closed-form solution for the electromagnetic wave equation that governs propagation in a spatially unbounded, linear time-varying gaseous plasma. Using a proper change of variable, the wave equation is converted into the Lommel's transformed version of the Bessel differential equation. The analytical wave functions in time are then expressed in terms of Bessel functions of the first kind with complex orders, which appears to be a very rare case in physical problems. Unlike previous analytical treatments of the problem, such as the use of adiabatic and Laplace transform approximations, which deal with slow and fast rise times of the plasma, respectively, the solution presented here imposes no constraints on the rise time relative to the time period of the source wave.