Strong Electromagnetic Waves in Overdense Plasmas

We report two new results concerning the propagation of electromagnetic waves with a strength parameter $\nu =\frac{\mathrm{eE}}{m\omega c}$ sufficiently large ($\nu \gg 1$) to cause relativistic electron velocities. The first is an analytic solution of the nonlinear equations for linearly polarized waves in a uniform medium. The second is that propagation in a nonuniform medium increases the nonlinear penetration effect; the maximum nonrelativistic plasma frequency ${\omega }_p$ which allows transmission of a strong wave is ${{\omega }_p}^2\approx {\left(\frac{\omega L}{c}\right)}^{\frac{1}{2}}\frac{e{E}_i\omega }{\mathrm{mc}}$, where $L$ is the density-gradient scale length and $E_i$ the electric field in the absence of a plasma.