The goal of this chapter is to develop methods for modeling near-field optical-wave propagation with high delity and some flexibility, which is considerably more challenging than for far-field propagation. This chapter uses the same coordinate convention as in Fig. 1.2. It begins with a discussion of different forms of the Fresnel diffraction integral. These different forms can be numerically evaluated in different ways, each with benefits and drawbacks. Then, to emphasize the different mathematical operations in the notation, operators are introduced that are used throughout Chs. 6-8. The rest of this chapter develops basic algorithms for wave propagation in vacuum and other simulation details.

The quadratic phase factor inside the Fresnel diffraction integral is not bandlimited, so it poses some challenges related to sampling. There are two different ways to evaluate the integral: as a single FT or as a convolution. This chapter develops both basic methods as well as more sophisticated versions that provide some flexibility. There are different types of flexibility that one might need. For example, Delen and Hooker present a method that is particularly useful for simulating propagation in integrated optical components. Because the interfaces in these components are often slanted or offset and the angles are not always paraxial, they developed a Rayleigh-Summerfeld propagation method that can handle propagation between arbitrarily oriented planes with good accuracy.^28,29

In contrast, the applications discussed in this book involve parallel source and observation planes, and the paraxial approximation is a very good one. When long propagation distances are involved, beams can spread to be much larger than their original size. Accordingly, some algorithms discussed in this chapter provide the user with the flexibility to choose the scaling between the observation- and source-plane grid spacings. Many authors have presented algorithms with this ability including Tyler and Fried,³⁰ Roberts,³¹ Coles,³² Rubio,³³ Deng et al.,³⁴ Coy,³⁵ Rydberg and Bengtsson,³⁶ and Voelz and Roggemann.³⁷ Most of these methods are mathematically equivalent to each other. However, one unique algorithm was presented by Coles³² and later augmented by Rubio³³ in which a diverging spherical coordinate system was used by an angular grid with constant angular grid spacing. This was done specifically because the source was a point source, which naturally diverges spherically.