Piezoelectric contributions to pulsed degenerate four-wave mixing

In solid-state materials lacking a center of symmetry, cascaded second-order nonlinear optical effects, especially the combination of optical rectification and the linear electro-optic effect, are an important contribution to degenerate four-wave mixing ~DFWM!, an experimental tool widely used to measure the third-order nonlinear optical susceptibility. This quantity determines the magnitude of several interesting phenomena, among which the optical Kerr effect, self-phase modulation, soliton formation, and the interaction of different optical waves, with applications like optical limiting and all-optical switching. In order to extract an accurate value of the third-order susceptibility of noncentrosymmetric materials from a DFWM experiment, it is necessary to take into account the peculiar geometry dependences introduced by second-order effects. They can both lead to misleading results, if ignored, or to alternative possibilities for calibrating DFWM experimental setups, if properly taken into account. Some predicted effects of the electro-optic and optical rectification contributions to DFWM have been observed in Refs. 4, 5, and 7. This letter now reports the experimental observation of another, associated effect: the nanosecond fast piezoelectric relaxation whose contributions to DFWM were calculated in Ref. 5. This is important because the possibility of piezoelectric relaxation introduces additional dependencies of the DFWM signal from the angle between the interacting beams and from the pulse length. This must be taken into account in order to be able to compare experimental results obtained with different DFWM setups. At the origin of the piezoelectric contributions discussed in this work are two space-modulated rectified polarizations induced by two pairs of interacting waves in DFWM. For the experimental configuration shown schematically in Fig. 1, the rectified polarizations can be described by two static ~on the time scale of the pulse length used in DFWM! plane waves with wave vectors ka and kb . 4,5 Approximating the interacting beams and the generated polarizations by plane waves with real amplitude (1/2)A(v ,k)exp@i(vt2k•x)# 1c.c. and complex amplitude A(v ,k), one can write the complex amplitude of the rectified polarization with wave vector ka as