Theory of Optical Harmonic Generation at a Metal Surface

It is shown that a light wave of the high intensity obtainable from lasers produces a sufficiently strong nonlinear polarization on a reflecting metal surface to result in an observable amount of second harmonic generation. The analysis is based upon a self-consistent set of Maxwell's equations and the classical Boltzmann equation, respectively, for the electromagnetic fields and the distribution function of the conduction electrons. The conduction electrons are considered to be completely free except for a potential barrier at the metal surface, and the equations are solved for the fields varying with the frequency $\omega$ of the incident wave, and also for the fields varying with the frequency $2\omega$ in the approximation where the surface barrier can be taken as a step potential. The effect of the incident light wave is treated as a perturbation to the motion of the electrons and the frequency $\omega$ is assumed to be less than half the plasma frequency ${\omega }_p$ so that neither the fundamental nor the second harmonic wave can lead to plasma resonance. The part of the polarization varying as $e^{-2i\omega t}$ which is quadratic in the incident field is found to have the form ${\mathrm{P}}_2\left(\mathrm{NL}\right)=\alpha ({\mathrm{E}}_1\times {\mathrm{H}}_1)+\beta {\mathrm{E}}_1\mathrm{div}{\mathrm{E}}_1,$ where ${\mathbf{E}}_1$ and ${\mathbf{H}}_1$ are, respectively, the electric and magnetic fields varying as $e^{-i\omega t}$ and where the magnitudes of the coefficients $\alpha$ and $\beta$ have been determined. Since div ${\mathbf{E}}_1$ differs from zero only near the surface of the metal, the second term in ${\mathbf{P}}_2$ (NL) can be considered as a surface contribution in contrast to the volume contribution of the first term. It is shown that these two terms give rise to comparable effects of second harmonic generation. The ratio of the average energy flux reflected with frequency $2\omega$ from the surface to the incident flux is found to be of the order of magnitude ${\left(\frac{e\left|E_{\mathrm{inc}}\right|}{\mathrm{mc}{\omega }_p}\right)}^2$, where $E_{\mathrm{inc}}$ is the amplitude of the incident electric vector.