To obtain accurate results, evaluating the Fresnel diffraction integral numerically requires some care. Therefore, this chapter rst deals with two simpler topics: diffraction with the Fraunhofer approximation and diffraction with lenses. This allows some optical examples of FTs to be demonstrated without the signicant algorithm development and sampling analysis required for simulating Fresnel diffraction. Vacuum propagation algorithms and sampling analysis for Fresnel propagation are the subjects Chs. 6-8. Computing diffracted fields in the Fraunhofer approximation or when a lens is present does not require quite so much analysis up front. Additionally, these simple computations involve only a single DFT for each pattern. Chapter 2 provides the requisite background. Consequently, readers may notice that the MATLAB code listings in this chapter are quite simple.

4.1 Fraunhofer Diffraction

When light propagates very far from its source aperture, the optical field in the observation plane is very closely approximated by the Fraunhofer diffraction integral, given in Ch. 1 and repeated here for convenience:

(4.1)

According to Goodman,⁵ "very far" is defined by the inequality

(4.2)

where Δz is the propagation distance, D is the diameter of the source aperture, and λ is the optical wavelength. This is a good approximation because the quadratic phase is nearly flat over the source.