Nonlinear Optical Theory in Solids

We use a simple rule giving the expectation value of a quantum operator for any perturbation order to calculate the general second-order conductivity tensor of a solid. The expression for the conductivity tensor amounts to a regrouping of terms in the conventional expressions and takes a considerably simplified form. The formula is applied to second harmonic generation in a free-electron gas and reduces to the classical equation given by Kronig and Boukema in the optical region. The second harmonic radiation generated in metals is shown to possess two resonances occuring at the plasma oscillation frequency for $p$ polarization and at half the plasma frequency for $p$ and $s$ polarization. The amplitude of the resonance is related to the imaginary part of the dielectric constant at the plasma frequency, ${\epsilon }_2\left({\omega }_p\right)$. Only metals with ${\epsilon }_2\left({\omega }_p\right)\ll 1$ (i.e., alkali metals, Ag and Al) will show resonant effects.