Optical Engineering Science. Stephen Rolt

The Effect of Pupil Position on Element Aberration

In all previous analysis, it is assumed that the stop is located at the optical surface in question. This is a useful starting proposition. However, in practice, this is most usually not the case. With the stop located at a spherical surface, by definition, the chief ray will pass directly through the vertex of that surface. If, however, the surface is at some distance from the stop, then the chief ray will, in general, intersect the surface at some displacement from the surface vertex. This displacement is, in the first approximation, proportional to the field angle of the object in question. The general concept is illustrated in Figure 4.15.

      Instead of the stop being located at the surface in question, the stop is displaced by a distance, s, from the surface. The chief ray, passing through the centre of the stop defines the field angle, θ. In addition, the pupil co-ordinates defined at the stop are denoted by r_x and r_y. However, if the stop were located at the optical surface, then the field angle would be θ′, as opposed to θ. In addition, the pupil co-ordinates would be given by r_x′ and r_y′. Computing the revised third order aberrations proceeds upon the following lines. All the previous analysis, e.g. as per Eqs. (4.31a)–(4.31d), has enabled us to express all aberrations as an OPD in terms of θ′, r_x′, and r_y′. It is clear that to calculate the aberrations for the new stop locations, one must do so in terms of the new parameters θ, r_x, and r_y. This is done by effecting a simple linear transformation between the two sets of parameters. Referring to Figure 4.15, it is easy to see:

      (4.40a)

(4.40b)

Figure 4.15 Impact of stop movement.

      (4.40c)

The effective size of the pupil at the optic is magnified by a quantity M_p and the pupil offset set out in Eq. (4.40b) is directly related to the eccentricity parameter, E, described in Chapter 2. Indeed, the product of the eccentricity parameter and the Lagrange invariant, H is simply equal to the ratio of the marginal and chief ray height at the pupil. That is to say:

      (4.41)

      In this case, r₀ refers to the pupil radius at the stop and r₀′ to the effective pupil radius at the surface in question. As a consequence, we can re-cast all three equations in a more convenient form.

(4.42)

      The angle, θ₀ is representative of the maximum system field angle and helps to define the eccentricity parameter and the Lagrange invariant. We already know the OPD when cast in terms of r_x′, r_y′, and θ, as this is as per the analysis for the case where the stop is at the optic itself. That is to say, the expression for the OPD is as given in Eqs. and these aberrations defined in terms of K_SA′, K_CO′, K_AS′, K_FC′, and K_DI′. Therefore, the total OPD attributable to the five Gauss-Seidel aberrations is given by:

(4.43)

To determine the aberrations as expressed by the pupil co-ordinates for the new stop location, it is a simple matter of substituting Eq. (4.42) into Eq. (4.43). This results in the so-called stop shift equations:

(4.44a)

      (4.44b)

(4.44c)

(4.44d)

(4.44e)

What this set of equations reveals is that there exists a ‘hierarchy’ of aberrations. Spherical aberration may be transmuted into coma, astigmatism, field curvature, and distortion by shifting the stop position. Similarly, coma may be transformed into astigmatism, field curvature, and distortion and both astigmatism and field curvature may produce distortion. However, coma can never produce spherical aberration and neither astigmatism nor field curvature is capable of generating spherical aberration or coma. Equation (4.44e) reveals, for the first time, that it is possible to generate distortion by shifting the stop. Our previous idealised analysis clearly suggested that distortion is not produced where the lens or optical surface is located at the stop.

Another important conclusion relating to Eqs. (4.44a)–(4.44e) is the impact of stop shift on the astigmatism and field curvature. Inspection of Eqs. (4.44c) and (4.44d) reveals that the change in field curvature produced by stop shift is precisely double that of the change in astigmatism in all cases. Therefore, the Petzval curvature, which is given by K_FC−2K_AS remains unchanged by stop shift. This further serves to demonstrate the fact that the Petzval curvature is a fundamental system attribute and is unaffected by changes in stop location and, indeed component location. Petzval curvature only depends upon the system power. Thus, it is important to recognise that the quantity K_FC−2K_AS is preserved in any manipulation of existing components within a system. If we express the Petzval curvature in terms of the tangential and sagittal curvature we find:

(4.45)