+2n = 6)

Each aberration type, defined by a column with a given value of n, represents an infinite series in ascending powers of p. The order of an aberration term is defined from Eq. (3.16) as

The two aberrations in the first group are first order aberrations, i.e. errors in the region of Gaussian optics. The term i knap cos ^ denotes a lateral shift of each image point from its nominal Gaussian position, the error being proportional to the nominal image height, i.e. a scaling error. In wavefront terms, this error is of great practical importance in telescope optics as it represents pointing, tracking or guiding error. It is best understood as a tilt of the imaging wavefront by a shift of its centre of curvature from the nominal position. The other Gaussian term 0k20p2 is also of great practical importance as it represents defocus from the correct nominal image plane. Defocus therefore gives a parabolic departure from the perfect reference sphere, which, in the Gaussian region, simply means a change of radius of the wavefront. This Gaussian concept of defocus can, of course, be extended at once to cover all orders from Eq. (3.11), as a change of radius of a spherical wavefront.

The five aberrations in the second group are the monochromatic third order aberrations or Seidel aberrations [3.4] and are, with the first order effects above, the dominant errors of telescope images and optical systems in general.

The column with n = 0 represents a family with no dependence on so the effect is symmetrical to the axis. The terms

represent spherical aberration, a symmetrical fourth power phase error of the wavefront in the third order case. The term

is a defocus error which also depends on the square of the field: it is therefore field curvature.

The column with n =1 has two third order terms. The term 1k31ap3 cos ^ is one of the most important in practical telescope optics: third order coma. The other term 3k11a3p cos ^ has the same aperture function m + n = 1, n = 1 as the equivalent first order (tilt) term and is therefore also a lateral shift of the image from its nominal position. But it is no longer simply an error of scale since it varies with the cube of the field size. It is the aberration known as distortion. Distortion does not reduce the quality of a point image, but displaces it from its nominal position in a non-linear manner. In most basic telescope systems, distortion is small and of little significance. However, as soon as field correctors or other elements are introduced fairly near the image, distortion can become important.

The last third order term is in the column with n = 2. This is 2k22a2p2 cos2 <p and represents astigmatism, again extremely important in practical telescope optics.

The fifth order terms can be interpreted in a similar way as higher order effects, combining the basic types of aberration. The column n = 0 gives all axisymmetric combinations of spherical aberration and field curvature. The nomenclature of the other columns is somewhat arbitrary. There is general agreement that the columns with even values of n are "astigmatic" types, in combination with various orders of spherical aberration and field curvature. For example, 2k42a2p4 cos2 ^ is normal fifth order astigmatism.

Some authors also call the types with odd values of n greater than 1 astigmatic. Personally, I prefer the term "comatic" for all odd values of n. Certainly, n = 1 is a comatic family, the term 1k51ap5 cos ^ being normal fifth order coma. But 3k31a3p3 cos ^ is also a form of fifth order coma. 5k11a5 p cos ^ is fifth order distortion. The fifth order term with n = 3, 3k33a3p3 cos3 is also a term with considerable importance in practical telescope optics, as we shall see in later chapters. We shall call it "triangular coma".

The field dependence in the aberrations, the power of a, is of no consequence for the degrading effect on the image. Nevertheless, it is of great importance in telescope optics to know that, for third order aberrations, spherical aberration is independent of the field, coma linearly dependent, astigmatism dependent on the square, and distortion on the cube.

We shall return to these formulations later in connection with telescope testing and active control of telescope optics. It will also be necessary to compare the properties of the above Hamilton formulation with those of Zernike polynomials and "natural modes". But optical design procedures are still largely based on the classical Hamilton/Seidel formulation which has many virtues of simplicity and clear physical interpretation. For this reason we shall use it in the treatment of telescope systems in this chapter.

3.2.2 The Seidel approximation:

third order aberration coefficients

The Hamilton formulation of Table 3.1 is of little direct use in optical design because the coefficients k are not known in terms of the constructional parameters of the optical system. This was the problem first solved by Seidel [3.4], and possibly, to some extent, earlier by PetzvaLi An excellent summary is given by Bahner [3.5] which we take over here in an expanded form. However, the reader familiar with Bahner's admirable book should note that there are differences of sign arising from our use of the strict Cartesian sign convention of Welford [3.6]. The formulation is instructive, and nothing reveals better how a telescope form achieves its correction, and what its limitations are, than an analysis of the third order aberrations. However, such calculations are rarely performed by hand in an age when powerful optical design programs, often working with PC's, can calculate these aberrations reliably and

1 The extent to which Petzval possessed a practical version of third order theory, as so admirably formulated by Seidel, has been hotly debated for over a hundred years, even before Petzval's death in 1891. His first, and only seriously scientific paper in 1843 [3.151], reveals a clear understanding of the Hamilton (Characteristic Function) basis, almost certainly independently developed, but gives no explicit formulae. Subsequent publications of Petzval are semi-popular and scientifically trivial in comparison. The best classical historical analysis of Petzval's work was given by von Rohr in 1899 [3.152]. He points out the excellent quality of Petzval's famous portrait objective, including balance of higher order spherical aberration at the remarkable relative aperture for that time of f/3.4. He also states that Petzval did not use, or indeed believe in, iterative trigonometrical ray tracing, although this had been first published in complete form in 1778 in the (then) well-known book by Klugel [3.153]. This is an astonishing and important piece of information: for if Petzval used no ray tracing, the only way he could have calculated such an excellent objective would have been by calculation of the aberration coefficients to the third order and, at least for spherical aberration, probably to the fifth order as well! Without explicit formulae such as those of Seidel, this would have been impossible. But von Rohr confirms the general view that Petzval never published such formulae or his means of deriving them, a tragedy for his historical reputation.

New light has recently been thrown on the subject through the admirable thesis of Rakich (see § 3.6.5.1 and § 3.7.2.4). Rakich quotes a paper given in 1900 [3.164] by the notable designer of photographic objectives, H.L. Aldis, which has remained unmentioned in the classical literature. Aldis gives equations for third order aberrations which he attributes directly to Petzval, without any mention of Seidel! However, he gives no reference to written work by Petzval, nor to any meeting with him with verbal information - Petzval had died nine years earlier in Vienna. After further study, I hope to publish an analysis of this Aldis paper, regarding Petzval's work, soon.

effortlessly. As in all calculations in geometrical optics, great care with the sign convention is always required.

A basic aspect of third order calculations must be emphasized from the start: the linear superposition of the effects of successive surfaces in the optical system. In other words, the Seidel surface coefficients (Sq)v are calculated from the formulae below to give by algebraic addition the Seidel sum for each aberration:

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