Optical Engineering Science. Stephen Rolt

Figure 4.2, it is clear that the thickness of the hyperhemisphere is −R × (n + 1)/n, or −1.625 × R. To calculate the value of R, we set up a matrix for the system. The first matrix corresponds to refraction at the planar air/glass boundary, the second to translation to the spherical surface and the final matrix to the refraction at that surface. On this occasion, translation to the original reference is not included.

      From the above matrix, the focal length is −R/0.6 and hence R = −9.0 mm. The thickness, t, we know is −1.625 × R and is 14.625. In this sign convention, R is negative, as the sense of its sag is opposite to the direction of travel from object to image space.

      The (virtual) image is at (n + 1) × R from the sphere vertex or 2.6 × 9 = 23.4 mm.

      In summary:

      4.2.2 Astigmatism and Field Curvature

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.

4.3 Reflection from a Spherical Mirror

The third order analysis for a spherical mirror proceeds in very much the same way as the single refractive surface. That is to say, a ray is traced from the object location to the mirror and thence to the paraxial focus regardless as to whether the real ray actually terminates there. The general layout is shown in Figure 4.5. The sign convention used here is the same as applied to all previous analyses. That is to say, positive image distance is with the image to the right, and the image distance, as shown in Figure 4.5, is actually negative. However, it must be accepted that, as rays physically converge on this image point, then this image is actually real, despite v being negative. In addition, the same convention is applied to mirror curvature; the mirror depicted in Figure 4.5 has negative curvature.

Figure 4.5 Reflection at spherical mirror.

The analysis proceeds as previously. Firstly, we set out the object and image positions and the ray intercept at the stop.

      The optical path is given by: