Analytical investigation of nonlinear phase shift via multistep χ(2) cascading in phase-matching configuration

The first expression of the nonlinear phase shift of the fundamental wave by the multistep second-order nonlinearity cascading is presented under the special phase-matching condition (Δk2+Δk3=0). The result obtained from our scheme can be split into two parts; two-step and three-step nonlinear phase shift of the fundamental wave. Comparing with two-step cascading, the sign of the multistep nonlinear phase shift of the fundamental wave can be controlled by the sign of wave vector mismatch,and the absolute value depends on the wave vector mismatch and input amplitude.