If is the direction of propagation then without the nonlinearity the solution would be
(9.16)
where is the amplitude. is zero if this wave is not present at z=0. is the polarization and is related to the wave vector by the equation.
(9.17)
In the presence of nonlinear coupling the amplitude of the wave can vary with z. So, we write,
(9.18)
where or and Ai are the corresponding amplitudes.
We also assume that are slowly varying functions of z, i.e.
(9.19)
Substituting from eqs (9.18) in eq (9.5) and using eq (47) and approximation (9.19), we get an equations for . Then, taking scalar products of this equation with , respectively, we obtain.
(9.20)
where the constant K’ includes all factors independent of z. since we neglect pump depletion here.
is the direction of propagation then without the nonlinearity the solution would be 
is the amplitude. 
is zero if this wave is not present at z=0. 
is the polarization and is related to the wave vector by the equation. 
or 
and Ai are the corresponding amplitudes.
are slowly varying functions of z, i.e.
. Then, taking scalar products of this equation with 
, respectively, we obtain. 
since we neglect pump depletion here.