Since the image plane is at a distance L from the secondary, the lateral shift is

in the opposite sense of the movement 5 for a Cassegrain telescope where m2 is negative. The angular lateral shift of the image related to the scale of the telescope in object space (pointing change) is

(AuP)dec = (m2 + 1)— rad = (m2 + 1)— 206 265 arcsec (3.423)

Such pointing changes are very large compared with angular decentering coma.

Rotation of the secondary through the angle upr2, using the notation of Eq. (3.375), produces an angular shift of the exit beam of 2upr2. The linear image shift is then

The corresponding angular shift for the scale of the telescope (pointing change) is

3.9 Zernike polynomials

The third order theory of telescope systems given in this chapter is based on the classical "Characteristic Function" of Hamilton discussed in § 3.2. Virtually all optical design programs provide analysis in these terms, but optimization of systems is usually performed using ray trace data which embraces all orders however they are defined. Nevertheless, most optical designers still think in terms of the "classical aberrations" because of the clear physical significance and the advantages of the simple third order terms. However, the Hamilton functions have the major disadvantage of being not only non-linear, but also of being (in general) non-orthogonal. The algorithms of optical design optimization routines have to take account of this.

In 1934 Zernike [3.119], in a classic paper, introduced the so-called Zernike circle polynomials. These have two very important advantages over the Hamilton terms. First, they are orthogonal functions and can be treated or corrected independently; second, they represent individually an optimum least square fit to the data within the accuracy of the degree of the polynomial.

The standard treatment of Zernike polynomials has been given by Wolf [3.120(a)] based on the derivation by Bhatia and Wolf [3.121] from the re-

quirement of orthogonality and invariance. We shall reproduce this here in a condensed form. They show that there exists an infinity of complete sets of polynomials in two real variables x, y which are orthogonal within the unit circle (in our case, the pupil with radius normalized to unity), i.e. satisfying the orthogonality condition


in which Va and Vß denote two typical polynomials of the set, V* being the complex conjugate, Saß is the Kronecker symbol11 and Aaß are normalization constants to be chosen. The circle polynomials of Zernike are distinguished from other sets by certain simple invariance properties, i.e. an invariance of form with respect to rotations of the axes about the origin. This has a resemblance to the Hamilton approach from the axial symmetry of a centered optical system. The function must then have the general type of solution

in which p and < have the same significance as in § 3.2, the radius and rotation angle in the pupil, and p is normalized to unity at the edge. In (3.427) R(p) = V(p, 0) is a function only of p. The function eil$ is now expanded in powers of cos < and sin <. If V is a polynomial of degree n in the variables x = p cos < and y = p sin <, it follows from (3.427) that R(p) is a polynomial in p of degree n and contains no 'power of p of degree lower than | 11. R(p) is an even or odd polynomial depending on whether l is even or odd. The set of Zernike circle polynomials is distinguished from all other possible sets by the property that it contains a polynomial for each pair of the permissible values of n (degree) and l (rotation angle in the pupil), i.e. for integral values of n and l, such that n > 0 ; l > 0 ; n > | 11 ; (n — | 11) is an even number. (3.428)

Wolfthen shows that the function may be normalized and expressed as

V„±m(p cos <,p sin <) = R^(p)e±i , (3.429)

in which Rm(p) are the radial polynomials and m = 111, a positive integer or zero.

The set of circle polynomials contains 1 (n + 1)(n + 2) linearly independent polynomials of degree < n. It follows that every monomial function xiyj where i and j are integers with i > 0, j > 0, and consequently every polynomial in x,y, may be expressed as a linear combination of a finite number of the circle polynomials Vnl .

To derive explicit expressions for the radial polynomials they are orthog-onalized by applying a special weighting factor related to the more general

Saß is 1 for a = ß and 0 for a = ß - see, for example, "Meyers Handbuch über die Mathematik", p. 251 and D 149 or Meschkowski, H. (1962) "Hilbertsche Raume mit Kernfunktionen".

one used in defining Jacobi polynomials. The Zernike normalization is then applied so that for all values of n and m

for the edge of the unit circle, the pupil. Wolf then derives the explicit form for the radial polynomials as

in which

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