Читать книгу Optical Engineering Science - Stephen Rolt - Страница 74

Unlike spherical aberration and coma, there is less scope for correction of astigmatism and field curvature. In Eqs. (4.5c) and (4.5d), astigmatism is corrected at the aplanatic point and field curvature at the radial points. However, the convention used in Eq. (4.5c) to describe astigmatic correction corresponds to zero sagittal ray defocus. On the other hand, using the alternative convention set out in Chapter 3 we have:

(4.8a)

(4.8b)

From Eq. (4.8a), it is evident that at the aplanatic condition where u = −(n + 1)R, the astigmatism vanishes, as does the spherical aberration and coma. It is interesting to see what might happen to the field curvature where this condition is fulfilled:

(4.9)

This is related to the Petzval field curvature, which, by definition, is the field curvature that arises when the astigmatism in the system is zero. Relating this to Eq. 4.8b, then the field curvature may be expressed as:

(4.10)

Figure 4.4 Field curvature for single refraction.

It is clear that Eq. (4.9) represents, with its quadratic dependence upon the pupil location, r, a degree of defocus, Δf, or longitudinal aberration, that is quadratic in the field angle. This defocus is given by:

(4.11)

The systematic field dependent defocus can be represented as a spherical surface where the each field point is in focus. The curvature of this surface, C_PETZ and equivalent to 1/R_PETZ where R_PETZ is the Petzval Radius, is given by:

(4.12)

The sign is important, in that the Petzval curvature is in the opposite sense to that of the surface itself. This point is illustrated in Figure 4.4.

The most significant point about Petzval curvature is, in common with the underlying wavefront error, that it is additive through a system. To illustrate this, we might consider a system with N surfaces with radius of curvature R_i. The material that follows each surface has a refractive index of n_i. The Petzval curvature associated with the system is simply the sum of the individual curvatures and is referred to as the Petzval sum. This is given by:

(4.13)

The practical implication of Eq. (4.13) is that if a system consists of elements with entirely positive or entirely negative focal power, then that system will always exhibit field curvature. To achieve a flat field, or a zero Petzval sum, then any positive optical elements must be balanced by negative elements elsewhere in the system.

It must be emphasised that the condition for perfect image formation on the Petzval surface applies specifically to the scenario where astigmatism has been removed.