Optimum holographic elements recorded with nonspherical wave fronts

The problem of designing a flat, aspheric, holographic optical element that images a finite set of input wave fronts into a finite set of output wave fronts is rigorously analyzed. The optimum phase transfer function of the holographic optical element is analytically determined. The optimum phase transfer function is defined as the one for which the element has minimum mean-squared wave-front error averaged over the set of input wave fronts. It is also shown that in general it may not be possible to obtain a low value for this average mean-squared wave-front error by using a single holographic element. Furthermore, the performance trade-off of spherical aberration, coma, and astigmatism versus geometric distortion is clearly indicated by example.