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In concentrating on third order aberrations, we shall, in the remainder of this chapter, seek to determine the impact of refractive surfaces, mirrors, and lenses on all the Gauss-Seidel aberrations. This analysis will proceed, initially, on the assumption that the surface in question lies at the pupil position. Subsequently, the impact of changing the position of the stop will be analysed. Manipulation of the stop position is an important variable in the optimisation of an optical design. The concept of the aplanatic geometry will be introduced where specific, simple optical geometries may be devised that are wholly free from either spherical aberration (SA) or coma (CO). These aplanatic building blocks feature in many practical designs and are significant because, in many instruments, such as telescopes and microscopes, there is a tendency for spherical aberration and coma to dominate the other aberrations. The elimination of spherical aberration and coma is thus a priority. Furthermore, by the same token, astigmatism (AS) and field curvature (FC) are more difficult to control. In particular, the control of field curvature is fundamentally limited by Petzval curvature, as alluded to in the previous chapter.
Figure 4.1 Calculation of OPD for refractive surface.
4.2 Aberration Due to a Single Refractive Surface
The analysis of the aberrations of a single refractive surface is based on the computation of the OPD of a generalised field point to the appropriate order (4th) in terms of field angle, θ and ray height, r, at the pupil. For this analysis, we will assume that the pupil is located at the lens surface. In calculating the OPD, we force all rays to go to the paraxial focus and compute the OPD with respect to the chief. Figure 4.1 shows an object with a field angle, θ, located at a distance, u from a spherical refractive surface of radius R. It must be emphasised, in this instance, that this analysis applies specifically to a spherical surface. In this geometry, it is assumed that the object is displaced from the optical axis in the y direction. The paraxial image is itself located at a distance v from the surface and the position of a ray at the surface (and stop) is described by its components in x and y – hx and hy.
The image in this case is the paraxial image and from the paraxial theory, the angle φ may be expressed in terms of θ as θ/n. To compute the optical path of a general ray as it passes from object to paraxial image, we need to define the ray co-ordinates at three points:
The z co-ordinate of the stop position is derived from the binomial expansion for the axial sag of a sphere including terms up to the fourth power. In making this approximation, it is assumed that h is significantly less than R. If we were to adopt the paraxial approximation we would only consider the first r2 term in the expansion. In the case of third order aberration, we need to consider the next term. It is then very straightforward to calculate the total optical path, Φ, for a general ray in passing from object to paraxial image:
The two square root terms represent the optical path of two ‘legs’ of the journey, with the path through the glass adding a multiplicative factor of n. The next stage of the process is an extension of the paraxial theory. It is assumed that rx, ry, and uθ are all significantly less than u. We can now approximate Φ from Eq. (4.2) using the binomial theorem. In the meantime collecting terms we get:
Before deriving the third order aberration terms, we examine the paraxial contribution which contain terms in h up to order r2.
As one would expect, in the paraxial approximation, the optical path length is identical for all rays. However, for third order aberration, terms of up to order h4 must be considered. Expanding Eq. (4.2) to consider all relevant terms, we get:
Four of the five Gauss-Seidel terms are present – spherical aberration, coma, astigmatism, and field curvature. However, clearly there is no distortion. In fact, as will be seen later, distortion can only occur where the stop is not at the surface as it is here. Of course, Eq. (4.4) can be simplified if one considers that u, v, and R are dependent variables, as related in Eq. (4.3). Substituting v for u, and R, we can express the OPD in terms of u and R alone. Furthermore, it is useful, at this stage to split the OPD contributions in Eq. (4.4) into Spherical Aberration (SA), Coma (CO), Astigmatism (AS), and Field Curvature (FC). With a little algebraic manipulation this gives:
(4.5b)
4.2.1 Aplanatic Points
It is worthwhile, at this juncture, to examine the four expressions in Eqs. (4.5a)–(4.5d) in some detail and, in particular, those for spherical aberration and coma. Before examining these expressions further, it is worthwhile to cast them in the form outlined in Chapter 3:
