Abstract

The temporal instability of counterpropagating beams in a Kerr nonlinear medium due to relaxation is well exploited.¹ We demonstrate that another even more fundamental physical parameter, linear dispersion, can lead to instability. We investigate instability in the simplest eigenarrangement² with the polarization of two counterpropagating beams being linear and parallel to each other. The dynamic equations for forward and backward propagating envelopes E₁(z, t) and E₂(z, t) are $i\left[{\left(-1\right)}^{j-1}\partial E_j/\partial z+v_g^{-1}\partial E_j/\partial t\right]-\mu {\partial }^2{E}_j/2\partial t^2=-{\chi }_2{k}_0\left(2I_{3-j}+I_j\right)E_j/2$, where j = 1, 2 and I_j = |E_j|² is the intensity of the respective wave, μ = ∂²k/∂ω² is a linear dispersion parameter, χ₂ is third-order susceptibility giving rise to nonlinear refractive index, and v_g is the (linear) group velocity. For large dispersion, unstable modes with lowest frequencies are excited first. The threshold intensity I required to excite these unstable modes diminishes as dispersion increases. In the case of I₁ = I₂= I and sufficiently large parameter p = χ₂k₀/L/2, this threshold intensity I is determined by $p\simeq \log \left(\mu v_g^2/2L\right)$, where L is the total length of the nonlinear medium. More modes are excited when the intensity exceeds this threshold, and their combinations may yield chaotic behavior. The mechanism of the resulting instability can be explained in terms of positive distributed feedback provided by the nonlinear index grating in the situation that the phase speed of small perturbations with slightly different frequencies is altered by linear dispersion.

© 1988 Optical Society of America