Modulation instability of electromagnetic radiation in narrow-gap semiconductors V. I. Berezhiani Laboratoire POMA, EP 130 CNRS, Universite ´ d’Angers, 2, Boulevard Lavoisier, 49045 Angers, Cedex 1, France and Institute of Physics, The Georgian Academy of Science, Tbilisi 380077, Georgia V. Skarka Laboratoire POMA, EP 130 CNRS, Universite ´ d’Angers, 2, Boulevard Lavoisier, 49045 Angers, Cedex 1, France R. Miklaszewski Institute of Plasma Physics and Laser Microfusion, P.O. Box 49, street Hery 23, 00-908 Warsaw, Poland Received 28 July 1997 The modulation instability of electromagnetic radiation propagating in narrow-gap semiconductor plasma is considered. In a dense n -doped InSb semiconductor plasma the instability can develop in a picosecond time scale for the field intensities E 1.610 5 V/cm. The numerical simulations confirm the analytical predictions that a modulated localized long pulse splits into a chain of solitonlike structures. Possible applications of this effect to generate short picosecond pulses and the dynamical light grating of the semiconductor at midinfrared wavelengths are suggested. S0163-18299801411-8 The recent progress in femtosecond laser techniques has stimulated investigations on the interaction of ultrashort elec- tromagnetic EM pulses with matter. The nonlinear self- interaction of EM waves in different kinds of optical media is a subject of considerable interest. The magnitude and re- sponse time of nonlinear refraction that leads to EM wave self-interaction depend on the mechanism of nonlinearity. Different types of optical nonlinearities with fast response time have been studied in semiconductors. 1 In n -doped semi- conductors moderately large free-carrier concentration may occur via thermal excitation of impurities. Early research on the free-carrier optical nonlinearities in the semiconductors concentrated on the effects of nonparabolicity of conduction- band electrons. In the pioneering work of Patel, Slusher, and Fleury 2 large optical nonlinearities arising from conduction electrons have been observed in n -doped narrow-gap semi- conductors NGS. This work stimulated the research de- voted to the nonlinear modulation interactions of EM radia- tion propagating through semiconductor plasma. Semiconductor plasma is a reasonably high density conduction-band electron plasma in a fixed neutralizing ionic background. In the two-band approximation of Kane’s model the Hamiltonian of the conduction-band electrons is analo- gous to the relativistic one H = m * c * 2 =( m * 2 c * 4 +c * 2 p 2 ) 1/2 . 3 Here c * =( E g /2m * ) 1/2 plays the part of the speed of light, m * is the effective mass of the electrons at the bottom of the conduction band, E g is the width of the gap, and p is the electron quasimomentum. The characteristic velocity c * in the Kane’s dispersion law is much less than the speed of light e.g., c * 3 10 -3 c for InSb. Therefore, the nonlinear effects appear for a considerably lower EM field intensity than that required for the normal gaseous plasma. As a con- sequence, the methodology of relativistic gaseous plasma has been used to show that due to the velocity-dependent effec- tive mass of the conduction electrons it is possible to have: a self-focusing of laser light in NGS, 4 a parametric amplifica- tion of EM waves and different kind of resonant excitations of density waves. 5 Since these investigations have been car- ried out two decades ago, the dynamical properties of the nonlinear interaction of EM waves in NGS were studied mainly on a nanosecond or even slower time scales. In order to avoid the breakdown of the semiconductor these consid- erations have been limited to the field intensities much lower than 10 7 W/cm 2 the surface ionization intensity for InSb is 3 10 7 W/cm 2 Ref. 4. Consequently, ‘‘relativistic’’ non- parabolicity factor p 2 / m * 2 c * 2 e 2 E 2 / m * 2 c * 2  o 2 was much less than unity ( E and  o are electric field and frequency of laser radiation, respectively. Current technology, however, has made it possible to pro- duce picosecond intense laser pulses with wavelengths rang- ing from the ultraviolet to the midinfrared. 6 Since such pulses are much shorter than the breakdown time which typically develops in 10 -10 sec the breakdown does not oc- cur for higher intensities of laser radiation allowing us to consider effects with finite strength of nonparabolicity factor p 2 / m * 2 c * 2 ( 1). Notice however, that the pump fluence should be smaller than 0.5 J/cm 2 otherwise it will melt the semiconductor even on a subpicosecond time scale. 7 In our recent publication it has been shown that an intense short laser pulse propagating through the semiconductor plasma will generate longitudinal Langmuir waves in its wake. 8 Also, an alternative approach for strong longitudinal wave generation by two near frequency laser beams has been suggested. 9 In these studies we focused basically on the case of transparent plasma with characteristic frequency of the laser radiation  0 much larger than the plasma frequency  * e . Consequently, the laser field distortion due to the modulation instability has been neglected. However, the modulation instability may have fast growth rate for the case of dense plasma. Such an instability leads in general to the spontaneous breakup of the EM beam into a periodic pulse train as a result of an interplay between the nonlinear and dispersive effects. In this paper we consider the modulation instability of PHYSICAL REVIEW B 15 MARCH 1998-I VOLUME 57, NUMBER 11 57 0163-1829/98/5711/62514/$15.00 6251 © 1998 The American Physical Society