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Chapter 9:
Systems with Rectangular Pupils
High-power laser beams have a rectangular cross-section; hence there is a need to discuss the diffraction characteristics of a rectangular pupil. We start this chapter with a brief discussion of the PSF and OTF of a system with such a pupil. Although high-power rectangular laser beams have been around for a long time [1], there is little in the literature on rectangular polynomials representing balanced aberrations for such beams. In this chapter we discuss such polynomials that are orthonormal over a unit rectangular pupil. These polynomials are not separable in the x and y coordinates of a point on the pupil. The expressions for only the first 15 orthonormal polynomials, i.e., up to and including the fourth order, are given for an arbitrary aspect ratio of the pupil becuase they become quite cumbersome as their order increases. However, expressions for the first 45 polynomials, i.e., up to and including the eighth order, are given for an aspect ratio of 0.75. The isometric, interferometric, and PSF plots of these polynomial aberrations with a sigma value of one wave are given along with their P-V numbers. The Strehl ratios for these polynomial aberrations for a sigma value of one-tenth of a wave are also given. Finally, we discuss how to obtain the standard deviation of a Seidel aberration with and without balancing. Products of Legendre polynomials (one for the x- and the other for the y-axis) which are also orthogonal over a rectangular pupil [4], are not suitable for the analysis of rectangular wavefronts of rotationally symmetric systems, since they do not represent classical or balanced aberrations for such systems. For example, the defocus aberration for such a system is represented by x2 + y2 . While it can be expanded in terms of a complete set of 2D Legendre polynomials, it cannot be represented by a single product of the x- and y-Legendre polynomials. The same difficulty holds for spherical aberration, coma, etc. However, products of such Legendre polynomials are suitable for anamorphic systems, as discussed in Chapter 13. Products of Chebyshev polynomials, one for the x- and the other for the y-axis, are also orthogonal over a rectangular pupil, but they are not suitable either for the rectangular pupils considered in this chapter for the same reasons as for the products of Legendre polynomials.
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