The lens equation

The parabolic wave-equation operator can be split into two parts, a complicated part called the diffraction or migration part, and an easy part called the lens part. The lens equation applies a time shift that is a function of x. The lens equation acquires its name because it acts just like a thin optical lens when a light beam enters on-axis (vertically). Corrections for nonvertical incidence are buried somehow in the diffraction part. The lens equation has an analytical solution, namely, $\exp [ i \omega t_0 (x)]$.It is better to use this analytical solution than to use a finite-difference solution because there are no approximations in it to go bad. The only reason the lens equation is mentioned at all in a chapter on finite differencing is that the companion diffraction equation must be marched forward along with the lens equation, so the analytic solutions are marched along in small steps.