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General Third Order Aberration Theory

      Armed with a simple understanding of the basic concepts that lie behind the description of third order aberration, we can proceed to a more general and more powerful analysis. This analysis relies on a theoretical treatment of OPD as a measure of aberration. As pointed out earlier, although the lowest order aberration (beyond the paraxial approximation) has a fourth order dependence upon pupil function, this theory is still referred to as third order aberration theory. In the example we have hitherto considered, we have analysed an on axis object located at the infinite conjugate. For the more general treatment, we must consider off-axis objects with the chief ray having some non-zero field angle with respect to the optical axis. In addition, the object may have an arbitrary axial location and we must also consider the axial position of the pupil.

      This third order theory is referred to as Gauss-Seidel aberration theory and is of general applicability to optical systems of arbitrary complexity. There is, however, one important constraint. The theory assumes that the entire geometry, component surfaces and so on, is circularly symmetric about the optical axis. In formulating the theory, we assume that the object presents a non-zero field angle, θ, with respect to the optic axis which is assumed to be oriented along the z axis. The chief ray is tilted by rotation about the x axis, so the object is offset from the optical axis in the y direction. The third order aberrations are to be expressed in terms of the field angle, θ, and the normalised pupil function, p. However, in this instance, because of the non-zero field angle, the rotational symmetry of the pupil is removed, so that separate x and y components of the pupil function, p_x, p_y, must be introduced.

The assumption in the Gauss-Seidel theory is that the underlying third order aberrations in a symmetrical optical system are themselves symmetrical and proportional to the fourth power of the pupil function. However, the finite field angle will effectively introduce an offset in the effective y component of the pupil function, Δp_y, at some arbitrary optical component. This is illustrated in Figure 3.10 which shows generically how such an offset may be visualised.

      What is suggested by Figure 3.10 is that if a co-ordinate transformation is applied in y that is proportional to the field angle, θ, then the ray fan can be made symmetrical about this new optical axis. That is to say, in Figure 3.10b, any aberration generated would, in terms of OPD, simply be proportional to p⁴, with respect to the new axis. In arguing that the required offset is proportional to θ, rather than some other trigonometrical function, we are making an approximation based on linearization in θ. This is justified for third order analysis, since any error produced would only be visible in higher order aberration terms (than third order). In Figure (3.10), the pupil is shown at the optical surface under consideration. However, this is not a necessary condition; wherever the pupil is located a symmetrical ray fan may be produced by simple offset of the co-ordinate system in the Y axis.

Thus, by the argument presented here, any third order aberration may be represented by a pupil dependence of p⁴ augmented by a shift in the y component of the pupil function, Δp_y that is proportional to the field angle, θ. This is set out in Eq. (3.18), which describes the WFE, Φ, in terms of θ, p, and p_y. From this point, we use WFE, rather than OPD as the key descriptor, as we are describing OPDs across the entire pupil. The offset pupil is now denoted by p′

Figure 3.10 (a) Generic layout. (b) Layout with y co-ordinate transformation.

(3.18)

      c is a constant of proportionality for the pupil offset.

      Equation 3.18 may be expanded as follows:

(3.19)

Finally expanding Eq. (3.19) gives an expression for all third order aberrations:

(3.20)

Equation (3.20) contains six distinct terms describing the WFE across the pupil. However, the final term, c⁴θ⁴, for a given field position, simply describes a constant offset in the optical path or phase of the rays originating from a particular point. That is to say, for a specific ray bundle, no OPD or violation of Fermat's principle could be ascribed to this term, when the difference with respect to the chief ray is calculated. Therefore, the final term in Eq. (3.20) cannot describe an optical aberration. We are thus left with five distinct terms describing third order aberration, each with a different dependence with respect to pupil function and field angle. These are the so called five third order Gauss-Seidel aberrations. Of course, in terms of the WFE dependence, all terms show a fourth order dependence with respect to a combination of pupil function and field angle. That is to say, the sum of the exponents in p and in θ must always sum to 4.

3.5 Gauss-Seidel Aberrations

      3.5.1 Introduction

      In this section we will describe each of the fundamental third order aberrations in turn. Re-iterating Eq. (3.20) below, it is possible to highlight each of the aberration terms:

      We will now describe each of these five terms in turn.

      3.5.2 Spherical Aberration

The first term, spherical aberration, has a simple fourth order dependence upon pupil function and no dependence upon field. This is illustrated in Eq. (3.21):

(3.21)

      This aberration shows no dependence upon field angle and no dependence upon the orientation of the ray fan. Since, in the current analysis and for a non-zero field angle, the object is offset along the y axis, then the pupil orientation corresponding to p_y defines the tangential ray fan and the pupil orientation corresponding to p_x defines the sagittal ray fan. This is according to the nomenclature set out in Chapter 2. So, the aberration is entirely symmetric and independent of field angle. In fact, the opening discussion in this chapter was based upon an illustration of spherical aberration.

      Spherical

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