Stationary Degenerate Four-Wave Mixing

Similarly, the wave E4 is due to a nonlinear polarization P4 of the form P4 -EiEiEi, with an associated wave vector mismatch AA 4 = k4 (Iki - ki) (see Fig. 11.5). 

To illustrate the physics, we assume that all waves are polarized in the x direction. In plane wave form the total electric field E and the nonlinear polarization P are given by 

Furthermore, we impose a simplifying condition, namely, that the intensity of is much higher than the intensity of E2 in this case the generated wave E is much more intense than the wave E. Under this simplifying assumption, the terms contributing to the generation of waves in the E direction, for example, are (E E ) E, (E2EI)E, ( i ) 3, E eI)Ex, ( 3 i ) ,and E eI)E2- 

The effects associated with these static and grating terms become clearer when we write down the corresponding coupled Maxwell wave equations. From Equation (11.47), ignoring the transverse Laplacian term, the equations for the amplitudes A of these three waves can be derived to give 

For simplicity, we have let all the nonlinear coefficients associated with the various contributing terms [(EjEI)E2, E E )Ex, etc.] in these equations be equal to A typical nonlinear polarization term (for P ) associated with E E E in terms of the amplitude A, is given by 

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