## Non Axisymmetric Surfaces and Zernike Polynomials

Non-axisymmetric optical surfaces are used for non-centered systems. Such surfaces may also represent a wavefront shape including some of the optical aberrations. Taking the origin of the azimuth angle 9 in the z, x plane by setting x = r cos 9, y = r sin 9, the general shape may be represented by

Z = XZn,m = X (An,mrn cosmQ + Bn,mrn sinmQ), (1.40a)

n,m with n, m positive integers, n + m even, m < n and where An,m, Bn,m are coefficients. A particular class of non-centered systems presents a symmetry plane: setting this symmetry in the z, x plane allows studying optical systems with Bn,m = 0. The surface Z(r, Q) is generated by successive optical surface modes zn,m as

Z =Aoo + A20 r2 + A40 r4 + A60 r6 + + A11 r cosQ + A31 r3cosQ + A51 r5cos Q +

where the coefficients An,m are denoted Anm for simplicity. The general representation of a surface - having a continuous tangent plane - is obtained by co-adding the Bnmrn sin mQ terms.

In the polynomial representation introduced by Zernike [178, 179] of an optical surface, or a wavefront, each polynomial has certain simple properties of invariance. Let us consider a dimensionless aperture radius p = r/rmax, and represent a particular surface by

Z = X Rn,o{p} + X Rn,m{p} cos mQ + £ Rn,m{p} cos mQ, (1.40c)

where the dimensioning coefficients in front of each term Rn,m{p} have been assumed equal to unit length (they do not appear here). The terms of this expansion are called Zernike circle polynomials. The normalization and determination of the radial components Rn,m{p} are such as (cf. Born and Wolf [17])

A cnm dimensioned representation of a surface with the first Zernike circle polynomials is

Z = c00 + c20(2p2-1) + c40(6p4 — 6p2 +1) + c60(20p6-30p4 +12p2-1) + + cnp cosQ + c3i(3p2-2)p cosQ + c5i(10p4-12p2+3)p cosQ + + c22p2cos2Q + c42 (4p2-3)p2cos2Q +

+ c33p3cos3Q +

where a similar series including coefficients c'nm for the sinus terms must be added.

1.8 Seidel Representation of Third-Order Aberrations

Geometrical aberrations introduced by centered optical systems have been investigated for improving the performance of lenses used in photography, an invention discovered by Nicephore Niepce in 1816 and improved by J. Daguerre in 1839. These lenses required large apertures and large fields of view, and thus needed a better correction of on- and off-axis aberrations. The early investigations on optical aberrations were by ray tracing, but J. Petzval obtained considerable results by an analytical study (Sect. 1.1). Although his manuscript was destroyed in a fire by thieves, he demonstrated the importance of his calculations by constructing his four-lens objective in 1840 well-known as "Petzval portrait lens" [17, 29].

### 1.8.1 The Seidel Theory

Considering centered systems, the first analysis on geometrical aberrations was published by W. Rowan Hamilton, in 1833 [71] (cf. Wilson [170]), where he introduced its Characteristic function. He derived the general form of the primary aberrations by using three fundamental parameters: the aperture height of the ray, its azimuth angle, and the field height. L. Seidel published a famous analysis on opical aberrations in 1856 [144]. His theory provides a powerful evaluation of the amount of primary aberration types introduced by each surface of an optical system, which then allows one to calculate their sums for the whole system. In 1895, an analytical study on aberrations led Bruns to consider a function related to the aberration function that he called eikonal function. Historical notes on this work are in the book by Herzberger [75]. Schwarzschild introduced the Schwarzschild eikonal in 1905, which is closely related to its perturbation function in the motion of planets, and called by him the Seidel eikonal (cf. Born and Wolf [17]).

In general the rays from a point source in the object field and emerging from a centered system do not lean against a spherical wavefront. The general shape of the aberrated wavefront or aberration function can be defined from three variables which characterize all the aberration terms. We follow Welford ([167] Sect. 8.2) and denote these dimensionless variables as:

p: aperture radius at exit pupil plane x, y, normalized from 0 to 1, 9: azimuth angle at exit pupil plane, polar coordinate with p,

7): image height at image focal plane X, Y, normalized from 0 to 1, where x = p cos 9, y = p sin 9, and the normalized image height n = n'/V^ax is here taken in the X-direction which is parallel to the x-axis. In the Gaussian approximation, the normalized field angle (p = ('/(^ax is equivalent to the normalized image height ((p = n); to simplify the writing, the prime on n and (p have been omitted.

With respect to the Gaussian reference sphere, the wavefront only including primary aberrations is represented at the exit pupil level by the aberration wavefront function

W[4](p,e,n) =1 Sip4 + 1 Sn fi p3cos 9 + 1SM f)2p2cos2d

where SI to SV are the five 3rd-order Seidel coefficients which have the dimension of length and represent respectively the amount of primary spherical aberration, coma, astigmatism, field curvature and distortion. These terms are called third-order aberrations. To distinguish them from those of the same family, which could also include higher-order coefficients, we will refer to them as

Explicit relations have been derived to calculate the transverse image size of all third-order aberrations for a single surface, thus providing each of its Seidel coefficients Si as functions of p, 9, fj parameters. These relations take also into account the case of an aspheric surface. Since Seidel, many authors have re-formulated the explicit expressions for Si coefficients. One may refer, for instance, to the books by Conrady [36], Chretien [29], Kingslake [86], Welford ([167], Appendix A), and Wilson [170] who explicitly gives the Seidel sums for two-mirror telescopes. Of universal practice in raytracing analysis, each of the Si coefficients of a surface efficiently allows one to calculate, by simple co-addition, the Seidel sum X Si of a whole system.

Given the field height of the source point and its associate principal ray which is normal to the wavefront W[4], (1.41) allows deriving the lateral shift at the local image plane for any aperture ray. The two components of this shift, which define the amount of transverse aberration (or lateral aberration), are rAX = -1 [Sip3 sin 9 + Sn fj p2 sin29 + (Sm+Sv) f\2p sin 9 ], AY — -1 [ Si p3cos 9 + Sn f) p2(2+cos29) (1.43)

This allows one to draw the geometrical shape of the aberrated image for Sphe 3, Coma 3, Astm 3, and its global radial shift Dist 3. Given an aberration term, its image pattern may be traced for all rays emerging at a fixed height fj = constant and then varying p and 9 .

Several important properties may be immediately derived from the aberration function (1.41).

If Sphe 3, Coma 3, and Astm3 vanish, i.e. Si — Sii — Sm — 0, we see that the value of SV is a measure of the field curvature; this curvature is also called the Petzval curvature, after Petzval who discovered, in 1840, that the Petzval condition SV — 0 is a necessary condition for obtaining a flat image field.

If Astm 3 is not corrected, Sjji = 0, we may regroup the third and fourth terms of (1.41) in the form

which, in the image space, are respectively the x, z and y, z sections of the tangential focal surface and sagittal focal surface of curvature 1/Rt and 1/Rs. Denoting 1/Rp the curvature of the Petzval focal surface, from (1.44), these surfaces are related by

0 0