Higher-order phase integral

We emphasize that for the validity of (4.46) with the expressions (4.47a-c) for (2ra+1) the essential restriction is that d2Q2 z)/dz2 must not be too small at the top of the barrier, which means that close to its top the barrier is approximately parabolic, i.e., that the distance from the barrier to the transition points that are not associated with the barrier must be much larger than t" — t. However, when the energy is close to the top of the barrier, it is the slight deviation from parabolic shape close to the top that determines the values of the quantities K2n, n > 0, and one needs accurate values of these quantities for obtaining accurate values of in higher orders of the phase-integral approximation. 

The GD method is very useful for small phase distortions that are detected using the Z-scan technique, since only a few terms of the sum are needed. Also, this method can be easily extended to higher order nonlinearities. Spatially integrating Eg(r, t) up to the aperture radius results in the transmitted power through the aperture, giving ... 

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