We start this chapter with a brief discussion of the aberration-free PSF and OTF for a system with a square pupil, as, for example, a high-power laser beam with a square crosssection. We can obtain these results as a special case of the rectangular pupils discussed in the last chapter. Similarly, the square polynomials Sk can be obtained as a special case of the rectangular polynomials Rk discussed there, i.e., by letting c = 1 2 . However, we describe the procedure for obtaining them independently, and give expressions for the first 45 polynomials, i.e., up to and including the eighth order. 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 and then discuss the Strehl ratio as a function of it.
Orthogonal square polynomials were also obtained by Bray by orthogonalizing the circle polynomials, but he chose a circle inscribed inside a square instead of the other way around. Thus, his square with a full width of unity has regions that fall outside the unit circle. Defining a unit square as we have, where its semidiagonal is unity, has the advantage that the coefficient of a term in a certain polynomial represents its peak value. For example, since r has a maximum value of unity, the coefficients of astigmatism r2 cos2 q in S6 , or coma r3 cosq in S8 , or spherical aberration r4 in S11 represent their peak values.
As in the case of rectangular polynomials, products of the x- and y-Legendre polynomials, which are orthogonal over a square pupil, are not suitable for the analysis of square wavefronts, because they do not represent classical or balanced aberrations. For example, defocus is represented by a term in x2 + y2 . While it can be expanded in terms of a complete set of Legendre polynomials, it cannot be represented by a single 2D Legendre polynomial (i.e., as a product of x- and y-Legendre polynomials). The same difficulty holds for spherical aberration and coma, etc. However, products of Legendre polynomials are the correct polynomials for an anamorphic system, as discussed in Chapter 13.
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