Appendix A: Wavefront Reconstruction Based on Fourier Series

Fourier Wavefront Reconstruction Over a Rectangular Pupil The iterative and boundary methods developed here for circular and elliptical pupils are based on a method to reconstruct the wavefront over the rectangular pupil. To fully understand this development, it is necessary to first summarize the approach used by Freischlad and Koliopoulos to derive an inverse spatial filter to reconstruct a wavefront from its gradients within a square pupil. To perform this filtering operation, the Fourier spectra of xand y-gradients are multiplied with ju and jv (where j=√-1) respectively and then summed. According to the derivative theorem of Fourier theory, this combined spectrum corresponds to the Fourier transform of the wavefront Laplacian. Since the Laplacian operator in the spatial domain corresponds to a multiplication by a factor -2π(u+v) in the Fourier domain, dividing the combined spectrum by factor -2π(u+v) produces the spectrum of the wavefront itself. The inverse Fourier transformation is then applied to reconstruct the wavefront.