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Chapter 6:
Transmittance Functions, Lenses, and Gratings
Abstract

The beam sources implemented in Chapter 5 are for the most part simple apertures illuminated by a plane wave. They are modeled with real functions and, in effect, have a zero phase component. In this chapter functions are presented that create a more complicated field by altering the magnitude and/or phase of the field. The functions can be used to apply "tilt" or "focus" to a field, model the effect of a periodic structure, or model a lens. In general, these transmittance functions can be thought of as multiplying an incident field to create a desired effect; however, some represent well-known optical components such as a diffraction grating or a lens.

The functions discussed in this chapter provide considerable utility in their own right, but like the basic functions they can be combined to create more elaborate fields. As a matter of convenience these functions are described as part of the source, or as applied in the source plane. However, they can be applied in other planes; for example, the pupil of an imaging system, which is coming up in Chapter 7.

6.1 Tilt

An optical beam can be steered in a propagation simulation by applying a "tilt" to the beam wavefront. Suppose a tilt of angle of α is applied to a wavefront, as indicated in Fig. 6.1, where the dashed line represents the tilted wavefront of the beam and the arrow indicates the intended direction of propagation.

An expression for the dashed line in Fig. 6.1 is z = −y tan α. The intent is to convert this line to a phase front in the xy plane at z = 0. This essentially requires replacing the position z with a phase quantity. As time progresses the wave moves in the positive z direction, but (as noted previously), the phase representation becomes more negative. This reverses the sign of the expression. The wavelength λ corresponds to 2π rad in the phase notation; so, using the wavenumber parameter k = 2π/λ, the phase function for producing the tilt is

(6.1)