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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Effect of pump depletion on degenerate four-wave mixing in saturable gain media

K.M. Hung, R. Gutiérrez-Castrejón, Trevor J. Hall

Published Jan 1, 1996• 0 citations
Publisher

Abstract

Co-polarised degenerate four-wave mixing (DFWM), in which all the four interacting beams have the same direction of polarisation, has been utilised to generate high phase conjugate reflectivities (~ 2500) recently [1]. In the co-polarised geometries six gratings are written in the material in which each beam forms a grating with another beam. Several theoretical models have been used to describe this co-polarised DFWM process in gain media [2]-[3]. However, all of them involve the use of small probe approximation and hence not all gratings are present in these models. Moreover, the effect of pump depletion has been neglected. A novel approach using the underlying symmetry is described recently to solve the co-polarised DFWM problem [4]. This approach allows us to take into account the effect of pump depletion, although the two standing gratings between the pumps and between the probe and the conjugate beams are ignored.

DOI
10.1364/cleo_europe.1996.ctuk34

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Co-polarised degenerate four-wave mixing (DFWM), in which all the four interacting beams have the same direction of polarisation, has been utilised to generate high phase conjugate reflectivities (~ 2500) recently [1]. In the co-polarised geometries six gratings are written in the material in which each beam forms a grating with another beam. Several theoretical models have been used to describe this co-polarised DFWM process in gain media [2]-[3]. However, all of them involve the use of small probe approximation and hence not all gratings are present in these models. Moreover, the effect of pump depletion has been neglected. A novel approach using the underlying symmetry is described recently to solve the co-polarised DFWM problem [4]. This approach allows us to take into account the effect of pump depletion, although the two standing gratings between the pumps and between the probe and the conjugate beams are ignored.
We model transient copolarized degenerate four-wave mixing in a saturable gain medium, using pulses of duration much shorter than the population lifetime with a finite probe beam. Initially we describe a model to include the simultaneous existence of both the reflection and transmission gratings for arbitrary field strengths. Then we develop a model to include all four possible gratings, including standing-wave gratings between counterpropagating pump beams and the probe and conjugate beams. By use of two expansion methods for the higher-order grating terms it is possible to model regimes in which either one grating is arbitrarily strong or all four gratings are strong. Comparisons are made among models including one, two, and four gratings. Good agreement is found between the theoretical four-grating case and pulsed copolarized degenerate four-wave mixing experiments in laser-pumped Ti:sapphire for a finite probe beam.
Phase distortion correction of an optical wave propagating in laser media is a crucial problem to be taken into account in most of laser systems. For this purpose, phase conjugation through stimulated Brillouin scattering using the master oscillator-power amplification configuration has shown a very good potential to obtain high beam quality in solid-state lasers that operate at variable repetition rates. Another attractive alternative to generate the phase-conjugate wavefronts is to use a four-wave mixing (DFWM) process in the laser amplifier. In this medium, the interference between the interacting beams leads to a spatial modulation of the complex susceptibility of the medium by gain saturation (amplitude gain grating). This all-solid-state solution presents the advantage of suppressing the external nonlinear medium and would realize the required functions in the laser medium itself. Moreover, due to the amplification experienced by all the interacting beams, associated to the resonant nonlinearity of the gain saturation, high reflectivities already have been achieved in Nd:YAG amplifiers using a multipass geometry.1,2
The effect of biasing illumination on the degenerate four-wave mixing (DFWM) in absorbing photorefractive (PR) media has been investigated. The input beams are considered to be spatially uniform. The analysis has been made both for transmission and reflection geometry. Considering pump beam depletion and the slowly varying amplitude approximation, nonlinear coupled wave equations are obtained and have been solved numerically by a shooting method. The effect of different parameters such as input beam intensities ratio, thickness of the interacting medium on the phase conjugate reflectively (PCR) has been investigated. It has been observed that PCR can be controlled to a large extent by controlling the biasing intensity.
The process of saturable-gain four-wave mixing (DFWM) in a diode-pumped Nd:YVO4 amplifier with interacting beams provided by a Nd:YAG oscillator is investigated. Multipass geometries are employed that arrange for the signal beam to experience an increased interaction length relative to that of the pump beams. DFWM reflectivities up to R = 170% and a diffraction efficiency η = 6% of the volume population hologram written in the Nd:YVO4 amplifier are obtained experimentally. A theoretical analysis of multipass DFWM is presented in the transient (pulsed) regime, including the laser amplification and the mutual interaction of all the interacting beams as well as the spectral mismatch between the beam frequency and the gain line-center frequency of the Nd:YVO4 amplifier. It is shown that because of the close spectral matchup of the Nd:YAG oscillator and the Nd:YVO4 amplifier, the amplitude (gain) grating contribution is predominant compared with that of the phase grating.
There is presently a considerable amount of interest in wave conjugation via four-wave mixing in saturable absorbing media. Typically, the four-wave process in volves the interaction, of two counter-propagating pump waves and an Input probe wave via a third-order nonlinearity in the polarization of the medium. Introduction of the probe wave produces a reflected signal wave in the retrodirection. Abrams and Lind have determined the magnitude of the reflected wave obtained from a two-level saturable absorber by solving a set of coupled equations for the probe and signal waves. To obtain this solution, they assumed that the probe and signal waves are weak compared with the pump waves and that the amplitudes of the pumps are constant. In this paper we shall consider the effect of pump attenuation caused by absorption and pump depletion caused by the transfer of energy to the probe and signal waves.
Degenerate four-wave mixing, arising from the thermal grating effect or other mechanisms such as the Kerr or photorefractive effect, can be described by a set of four coupled differential equations involving the four fields. Analytical solutions to these equations have been presented for cases where either the pump depletion or the effects of absorption were ignored.1 When both the pump depletion and absorption effects were included, a numerical result involving an iterative solution of the differential equations was presented.2 We present here a solution of the coupled-wave equations in which absorption as well as pump depletion effects are taken into account along with nonlinear refraction. The DFWM reflectivity is reduced to a quadrature using a method proposed by Kessel and Musin.3 For special cases of small absorption and no pump depletion, the solution reduces to the expressions presented earlier.4 The method of solution does not involve finding first integrals of the set of differential equations and can be extended to the description of DFWM using other physical mechanisms.