Chapter 11

The Oblique Aberrations

In Chapter 4 we introduced the subject of the oblique aberrations of a lens, and in Chapter 8 discussed in detail the origin and computation of coma. In this chapter we continue the discussion, giving computing procedures for the remaining oblique aberrations, namely, astigmatism, field curvature, distortion, and lateral color.

11.1ASTIGMATISM AND THE CODDINGTON EQUATIONS

When a narrow beam of light is obliquely incident on a refracting surface, astigmatism is introduced, and the image of a point source formed by a small lens aperture becomes a pair of focal lines, a series of beam sections being indicated in Figure 11.1. One focal line (sagittal) is radial to the field and points toward the lens axis, while the other focal line (tangential) is tangential to the field. Both focal lines are perpendicular to the principal ray, and their locations can be calculated once the principal ray has been traced. The astigmatic images formed by the first surface become the objects for the second, and so on through the system. The locations of the focal lines are found by the two Coddington1,2 equations, which will now be derived.

11.1.1 The Tangential Image

In Figure 11.2, BP is an entering principal ray, B being the tangential object point distance t from the point of incidence P; the length t being measured along the principal ray, negative if the object point lies to the left of the surface as usual. The line BG represents a neighbor ray close to the principal ray, lying in the meridian plane, so close in fact that the short arc PG ¼ r dy can be regarded as tangent to the refracting surface itself.

The central angle y ¼ I U, and hence

DOI: 10.1016/B978-0-12-374301-5.00015-2

Figure 11.1 The astigmatic focal lines.

The short line PQ, perpendicular to the incident ray, is given by

PQ ¼ t dU ¼ PG cos I ¼ r cos I dy

The term on the right degenerates to the surface power (n0 – n)/r when the object point lies on the lens axis so that Ipr0 ¼ Ipr ¼ 0. It may be regarded as

the oblique power of the refracting surface for the principal ray. The oblique power is always slightly greater than the axial power, which provides a convenient check on the calculation.

11.1.2 The Sagittal Image

The other focal line is located at the sagittal image point Bs. This is a para- xial-type image formed by a pair of sagittal (skew) rays lying close to the principal ray. As explained in Section 8.1.1, the image of a point formed by a pair of sagittal rays always lies on the auxiliary axis joining the object point to the center of curvature of the surface. This property of sagittal rays enables us to derive the second Coddington equation locating the sagittal focal line.

In Figure 11.3 we show the principal ray, the sagittal object point B, and the sagittal image Bs, with the auxiliary axis joining B, C, and Bs. Now, the area of any triangle ABC is given by 12ab sin C; and since triangle BPBs ¼ triangle BPC plus triangle PCBs, we have

12 ss0 sinð180 I þ I 0Þ ¼ 12 sr sinð180 IÞ þ 12 s0r sin I 0

where

Figure 11.3 The sagittal focus.

Figure 11.4 (a) Opening equations. (b) Closing equations.

Example

As an example of the use of these formulas, we will trace a principal ray through the Section 2.5 cemented doublet used several times before. The principal ray will enter the front vertex at an angle of 3 . The tabular layout of the computation, as perhaps performed on a small pocket calculator, is given in Table 11.1. The closing equations give Zs0 ¼ 0.02674 and Zt0 ¼ 0.05641 recalling that l0 ¼ 11.28586 (Section 6.1). The tangential focal line is thus about twice as far from the paraxial focal plane as the radial focal line, and both are inside the focal plane.

11.1.4Graphical Determination of the Astigmatic Images

The location of the sagittal focus along a traced principal ray is easily found because the image lies on an auxiliary axis drawn from the object point through the center of curvature of the surface.

T. Smith4 credits Thomas Young with the discovery of a similar procedure for locating the tangential image point. Young’s method for the construction of the refracted ray itself involves drawing two auxiliary circles about the center of curvature of the refracting surface, one with radius rn/n0 and the other with radius rn0/n (Figure 11.5). The incident ray is extended to cross the second of these auxiliary circles at E, and then E is joined to the center of curvature C. This line crosses the first auxiliary circle at E0; then the refracted ray is drawn from P through E0.

Table 11.1

Calculation of Astigmatism Along a Principal Ray

Figure 11.5 Young’s construction for the sagittal and tangential foci (n ¼ 1, n0 ¼ 1.7).