Harmonic generation by an electromagnetic wave in an inhomogeneous isotropic plasma

The generation of harmonic waves by a plane electromagnetic wave normally incident on a collision-free plasma with a linear density profile is considered. The solution for the primary wave in the plasma region is first obtained, and nonlinear polarization currents for the higher harmonics are expressed in terms of the lower harmonic fields by a perturbation method. The analysis shows that the plasma excites a longitudinally oscillating second harmonic electric field which is totally confined inside the plasma and has a singularity. The third harmonic wave excited in the plasma, however, radiates back into free space, although the effect is negligibly small in a quasi-homogeneous limit. The amplitude of this radiated field oscillates as a function of
\omega^{3}
, and the electron density gradient has little effect on it.