Parametric instabilities of phonons: Nonlinear infrared absorption

Nonlinear infrared absorption by parametric phonon processes is shown to be negligible in the low-absorption region of exponential frequency dependence of the optical-absorption coefficient $\beta$, but observable at the reststrahl resonance and in Raman scattering. At low intensity, the transmission $T$ is independent of intensity $I$ as usual, but at high intensity the $T\left(\omega \right)$ curve broadens and the transmission at resonance increases. This behavior results from the parametric instability in the process in which an intermediate-state reststrahl phonon is annihilated and a pair of phonons is created. An effective relaxation frequency of the reststrahl phonon, which is greater than the low-intensity value as a result of the increase in the amplitudes of the pair phonons above their thermal equilibrium values, is quite useful in understanding absorption and Raman-scattering results. The time constant for the approach to the steady state is important since the steady state is not attained in short laser pulses in important cases in which long-lived phonons give rise to low steady-state threshold intensities for anomalous absorption. The threshold for the parametric instability is quite sharp when considered as a function of the amplitude of the fundamental phonon, but the deviation from linear absorption with increasing intensity is quite smooth. Contrary to previously accepted results, even crystals such as NaCl having a center of inversion could have anomalously low thresholds since the threshold is controlled by the phonon (in the pair) having the longer lifetime. Chain instabilities and enhanced relaxation from mutual interaction of excited pair phonons are negligible for the phonon instabilities, in contrast to previous results for plasmas and parallel pumping in ferromagnetic resonance, respectively. The method of calculation, using boson occupation numbers rather than mode amplitudes, has the simplicity and power to yield more information about parametric instabilities, including effects above the threshold, than has been possible previously.