# Reflecting and Catadioptric Afocal Lenses

This is a continuation from the previous tutorial - stimulated transitions in the classical oscillator model.

Afocal lenses can be designed with powered mirrors or combinations of mirrors and refractors.  Several such designs have been developed in recent years for use in the photolithography of microcircuits. All-reflecting afocal lenses are classified here according to the number of powered mirrors they contain. They will be reviewed in order of increasing complexity, followed by a discussion of catadioptric afocal systems.

## Two-powered-mirror Afocal Lenses

The simplest reflecting afocal lenses are the variants of the galilean and keplerian telescopes shown in Fig. 19a and 19b. They may also be thought of as afocal cassegrainian and gregorian telescopes. The galilean/cassegrainian version is often called a Mersenne telescope. In fact, both galilean and keplerian versions were proposed by Mersenne in 1636, so his name should not be associated solely with the galilean variant.

Making both mirrors parabolic corrects all third order aberrations except field curvature. This property of confocal parabolas has led to their periodic rediscovery, and to subsequent discussions of their merits and shortcomings. The problem with both designs, in the forms shown in Fig. 19a and 19b , is that their eyepieces are buried so deeply inside the design that their usable field of view is negligible. The galilean form is used as a laser beam expander, where field of view and pupil location is not a factor, and where elimination of internal foci may be vital.

Eccentric pupil versions of the keplerian form of confocal parabolas, as shown in Fig. 19c, have proven useful as lens attachments. $$RO$$, $$RE$$, and the internal image are all accessible when $$RO$$ is set one focal length ahead of the primary, as shown. It is then possible to place a field stop at the image and pupil stops at $$RO$$ and $$RE$$, which very effectively blocks stray light from entering the following optics. Being all-reflecting, confocal parabolas can be used at any wavelength, and such attachments have seen use in infrared designs.

## Three-powered-mirror Afocal Lenses

The principle which results in third-order aberration correction for confocal parabolas also applies when one of the parabolas is replaced by a classical cassegrainian telescope (parabolic primary and hyperbolic secondary), as shown in Fig. 20, with two important added advantages.

First, with one negative and two positive mirrors, it is possible to reduce the Petzval sum to zero, or to leave a small residual of field curvature to balance higher-order astigmatism.

Second, because the cassegrainian is a telephoto lens with a remote front focal point, placing the stop at the cassegrainian primary puts the exit pupil in a more accessible location. This design configuration has been patented by Offner, and is more usefully set up as an eccentric pupil design, eliminating the central obstruction and increasing exit pupil accessibility.

## Four-powered-mirror Afocal Lenses

The confocal parabola principle can be extrapolated one step further by replacing both parabolas with classical cassegrainian telescopes, as shown in Fig. 21a.

Each cassegrainian is corrected for field curvature independently, and the image quality of such confocal cassegrainians can be quite good. The most useful versions are eccentric pupil.

Figure 21b shows an example from Wetherell. Since both objective and eyepiece are telephoto designs, the separation between entrance pupil $$RO$$ and exit pupil $$RE$$ can be quite large.

An afocal relay train made up of eccentric pupil confocal cassegrainians will have very long collimated paths. If the vertex curvatures of the primary and secondary mirrors within each cassegrainian are matched, the relay will have zero field curvature, as well. In general, such designs work best at or near unit magnification.

## Unit Power Finite Conjugate Afocal Lenses

The simplest catadioptric afocal lens is the cat’s-eye retroreflector shown in Fig. 22a, made up of a lens with a mirror in its focal plane.

Any ray entering the lens will exit parallel to the incoming ray but traveling in the opposite direction. If made with glass of index of refraction $$n=2.00$$, a sphere with one hemisphere reflectorized (Fig. 22b) will act as a perfect retroreflector for collimated light entering the transparent hemisphere.

Both designs are, in effect, unit power ($$M=-1.00$$) afocal lenses. Variations on this technique are used for many retroreflective devices.

Unit power relays are of much interest in photolithography, particularly for microcircuit manufacturing, which requires very high resolution, low focal ratio unit power lenses. In the Dyson lens, shown in Fig. 23a, the powered surfaces of the refractor and the reflector are concentric, with radii $$R$$ and $$r$$ given by

$\tag{30}\frac{R}{r}=\frac{n}{(n-1)}$

where $$n$$ is the index of refraction of the glass. At the center point, spherical aberration and coma are completely corrected. In the nominal design, object and image are on the surface intersecting the center of curvature, displaced laterally to separate object from image sensor (this arrangement is termed eccentric field, and is common to many multimirror lens systems).

In practice, performance of the system is limited by off-axis aberrations, and it is desirable to depart from the nominal design to balance aberrations across the field of view.

The unit power all-reflecting concentric design shown in Fig. 23b is patented by Offner. It was developed for use in manufacturing microcircuits, and is one of the most successful finite conjugate afocal lens designs in recent years.

The spheres are concentric and the plane containing object and image surfaces passes through the common center of curvature. It is an all-reflecting, unit power equivalent of the refracting design shown in Fig. 9 [refer to the keplerian afocal lenses tutorial].

Object and image points are eccentric field, and this is an example of the ring field design concept, where axial symmetry ensures good correction throughout a narrow annular region centered on the optical axis.

As with the Dyson lens, having an eccentric field means performance is limited by off-axis aberrations. Correction of the design can be improved at the off-axis point by departing from the ideal design to balance on-axis and off-axis aberrations.

The next tutorial introduces prism polarizers