This is continuation from the previous tutorial - Microscopes

## 1. INTRODUCTION

During the initial stages of an optical design, many optical engineers take advantage of existing configurations that exhibit useful properties. This tutorial is a compilation of reflective and catadioptric objective designs that should help inform the reader of available choices and provide reasonable starting solutions.

This tutorial also includes a cursory introduction to some of the more important topics in system analysis, such as angular and linear blur size, image irradiance, scaling, and stray light control.

## 2. GLASS VARIETIES

Glasses used in the designs are represented in terms of index of refraction and Abbe number or $$V$$ number, below.

The $$V$$ number indicates glass dispersion. Most glasses can be obtained from a number of vendors.

## 3. INTRODUCTION TO CATADIOPTRIC AND REFLECTIVE OBJECTIVES

The variety of objectives presented in this chapter is large. Most of the intricate detail relating to each design is therefore presented with the design itself.

In the following paragraphs, analysis of the general features of the catadioptric and reflective objectives is undertaken.

### Conic Mirrors

It is apparent after a brief perusal of the designs that there are many surface types. Among these are the sphere, paraboloid, hyperboloid, prolate ellipsoid, and oblate ellipsoid. The oblate ellipsoid is a prolate ellipsoid turned on its side. The equation of a conic is given by the expression

$\tag{1}Z=\frac{ch^2}{1+\sqrt{1-(1+k)c^2h^2}}$

where $$Z$$ is the surface sag, $$k$$ is the conic constant, $$C$$ is the surface base curvature, and $$h$$ is the radial height on the surface. The relative shapes of these surfaces are illustrated in Fig. 1.

Conic mirrors give perfect geometric imagery when an axial point object is located at one conic focus and the axial point image is located at the other conic focus. Figure 2 illustrates these ray paths.

### General Aspheres

General aspheres are surfaces with fourth- and higher-order surface deformation on top of a flat or curved surface (see Schulz, 1988). The surface deformation of a rotationally symmetric general asphere is given by the relation

$\tag{2}Z=\frac{ch^2}{1+\sqrt{1-(1+k)c^2h^2}}+Ah^4+Bh^6+Ch^8+Dh^{10}$

where $$A$$, $$B$$, $$C$$, and $$D$$ are 4th-, 6th-, 8th-, and 10th-order coefficients that determine the sign and magnitude of the deformation produced by that order.

Although general aspheres allow correction of third - and higher-order aberrations and may reduce the number of elements in an optical system, general aspheres are more expensive than spheres or conics.

If aspheric deformation is required, conic surfaces should be tried first, especially since a conic offers higher-order correction (Smith,1992).

### Obscurations

Obscurations that block portions of the entering beam reduce image irradiance and image contrast (Smith, 1990, Everhart, 1959) in reflective and catadioptric systems. Several methods are used to reduce or eliminate completely the effects of an obscuration (see Fig. 3).

Figure 3a illustrates a commonly employed technique for reducing large-mirror

obscuration: a small secondary mirror close to the intermediate image moves the larger tertiary mirror out of the beam path.

Figure 3b is an illustration of an eccentric pupil system. All elements are symmetric about the same axis and the aperture stop is decentered for a clear light path.

Figure 3c is an example of an off-axis objective with the field of view biased to direct the center of view away from any intervening elements. All elements and apertures are symmetric about the optical axis

Figure 3d is an illustration of a tilted and decentered-component objective. Each element is rotationally symmetric about its own unique optical axis which may be tilted and / or decentered.

The imaging behavior of this system is more complicated to deal with than the imaging behavior of eccentric pupil and off-axis systems. Vector aberration theory (Thompson, 1980, Sasian, 1990a) has been developed to properly model the imaging behavior of these systems.

### Stray Light Suppression

Suppression of light diffracted from apertures and obscurations is facilitated with intermediate images and a real and accessible Lyot stop. Figure 4a illustrates a generic refractive configuration with an intermediate image and Lyot stop. Figure 4 b illustrates where the diffracted light (shaded region) originates and terminates (at one edge of each aperture, for clarity).

A field stop is placed at the focus of the first lens to block dif fracted light produced by the front light baffle. To block unwanted objects within the field of view, an occulting disc may be inserted at the focus of the first lens, as is done with a Lyot coronagraph in order to block the sun. By oversizing the field stop slightly, the light diffracted at the field stop falls just outside of the detector area.

Following the field stop is a second lens that reimages the intermediate image to the final image and the entrance pupil to the Lyot stop (the shaded region in Fig. 4a illustrates how the entrance pupil is imaged).

Undersizing the Lyot stop blocks the light diffracted at the entrance pupil. In this way the Lyot stop becomes the aperture stop of the system.

Another application of the Lyot stop in the infrared (assuming the Lyot stop is located

exterior to the objective optics) is as a cold stop (Fischer, 1992). The cold stop (Fig. 4a ) is a baffle that prevents stray infrared light, radiated from the housing, from impinging upon the detector from outside its intended field.

### Reflective and Catadioptric Objective Designs

The objectives to follow are listed according to focal ratio and design type. Objectives have a 20-cm diameter and catadioptric systems are optimized for a wavelength range from 480 to 680 nm unless otherwise stated.

The angles of surface and element tilts are with respect to the horizontal optical axis. Decenters are also with respect to the optical axis. Since many of the designs are aplanatic, anastigmatic, and free of chromatic aberrations, the position of the stop does not affect third-order aberration correction and may be chosen to minimize vignetting, distortion, element size, or stray light contamination.

All aberrations mentioned in this section are third-order unless otherwise stated.

Definitions of the abbreviated terminology used in the lens data are as follows:

$$\text{SUR}$$ Surface number.

$$\text{RDY}$$ Surface radius of curvature. A positive value means the center of curvature lies to the right of the surface; negative to the left.

$$\text{THI}$$ Thickness of element or distance to next element. The value is positive if the next surface lies to the right of the surface.

$$\text{GLA}$$ Glass-type or mirror surface, the latter referred to by the abbreviation $$\text{REFL}$$.

$$\text{CON}$$ Conic constant $$k$$.

$$\text{STO}$$ Stop surface.

$$\text{INF}$$ A surface with an infinite radius of curvature; that is, a flat surface.

$$\text A$$, $$\text B$$, $$\text C$$, $$\text D$$ The 4th-, 6th-, 8th-, and 10th-order aspheric deformation coefficients in Eq. (2).

A potential source of confusion is the terminology used to describe Mangin elements; that is, refractive elements with reflective back surfaces. This is illustrated in design 2 (F/4 Mangin): a ray enters the second refractive surface of the element (surface 2) and travels to the right where it intersects the mirror surface (surface 3). The thickness of surface 2 is therefore positive. The ray is reflected by the mirror surface (surface 3) and travels back through the glass element to surface 2; hence, the notation $$\text{F9/REFL}$$ and the negative surface 3 thickness. Since surface 2 and 4 represent the same surface, the radii are the same.

1. $$\text F$$/4 Paraboloid Objective

Comments: A single parabolic mirror objective can be arranged in a variety of forms, the best known being the Newtonian . Here a mirror flat diverts the image away from the light path.

A tipped-mirror configuration is the Herschelian; a modern version is untipped and employs an eccentric-pupil to give an accessible image. A ‘‘backwards’’ Newtonian, the Pfund has a large flat-mirror primary.

The Pfund has a smaller obscuration than the Newtonian and requires no diffraction-inducing support structure for the folding flat.

As has been mentioned, the on-axis performance of a paraboloid objective is perfect. Of f-axis, coma quickly degrades image quality.

For objectives slower than F/11, the easy-to-fabricate spherical mirror gives the same performance as a paraboloid when diffraction is also considered.

The paraboloid objective has image quality as good as a Cassegrain (design 3) of equivalent $$\text{FN}$$ and aperture diameter, and is easier to align. The Cassegrain has the advantage of being compact.

Comments: The Mangin (1876) was invented by a French engineer of the same name to replace dif ficult-to-fabricate paraboloids in light houses. The objective relies upon the overcorrected spherical aberration produced by the negative first surface to cancel the undercorrected spherical aberration produced by the focusing, reflective surface.

The chromatic aberration of the Mangin is reduced by achromatizing with two glasses of dif ferent dispersions. Secondary spectrum limits on-axis performance, and coma, one-half that of a spherical mirror, is the primary field-limiting aberration. Kingslake (1978) takes the reader through the design of a Mangin mirror.

Mangin mirrors are susceptible to ghost reflections from the refractive surfaces. Antireflection coatings are usually needed.

In some cases the overcorrected chromatic aberration of a Mangin is used to cancel undercorrected chromatic aberration produced by a refractive element. The Schupmann or medial objective (Schupmann, 1899, Olsen, 1986) has a positive refractive element with undercorrected chromatic aberration which is annulled by a Mangin element.

3. $$\text F$$/4 Cassegrain

Comments: The ubiquitous Cassegrain is predominant in situations where a small field of view, high resolution, compact size, long effective focal length, and accessible image are required. The classical Cassegrain is composed of a paraboloid primary and hyperboloid secondary, giving perfect imagery on-axis whenever the primary image coincides with the hyperboloidal focus. Coma and field curvature limit of f-axis performance.

Many books discuss the first- and third-order properties of Cassegrain objectives. The Rutten (1988), Schroeder (1987), Korsch (1991), and Smith (1990) texts are among these.

4. $$\text F$$/4 Ritchey-Chretien

Comments: The aplanatic Cassegrain or Ritchey-Chretien (Chretien, 1922) is also corrected for coma , leaving astigmatism and field curvature uncorrected. Both mirrors of the Ritchey-Chretien are hyperboloids.

Numerous modern telescope objectives are of Ritchey-Chretien form; among these are the Hubble space telescope and the infrared astronomical satellite $$\text{IRAS}$$.

5. $$\text F$$/9 Ritchey-Chretien Telescope with Two - Lens Corrector

Comments: This is a design by Wynne (1965) for the correction of the Cassegrain focus of a large (350-cm) Ritchey-Chretien. The corrector removes the inherent astigmatism and field curvature of the Ritchey-Chretien. Other Cassegrain focus correctors are discussed by Schulte (1966a), Rosin (1966), and Wilson (1968).

6. $$\text F$$/4 Dall - Kirkham

Comments: The Dall-Kirkham is another Cassegrain corrected for spherical aberration. The primary is an ellipsoid with 6th-order aspheric deformation and the secondary is spherical. An inverse Dall-Kirkham, or Carlisle, is just the reverse, with a spherical primary. There is zonal spherical aberration without the 6th-order deformation. Five times more coma is produced by the Dall-Kirkham than the classical Cassegrain, seriously limiting the field of view.

7. $$\text F$$/4 Cassegrain with Field Corrector and Spherical Secondary

Comments: By adding zero-power refractive correctors, the performance of a reflective objective is substantially enhanced. Zero power is maintained to prevent axial color. Such is the case with this objective similar to one designed by Rosin (1964). All third-order aberrations, with the exception of distortion, are corrected. The surfaces, with the exception of the primary, are spherical. One of the most attractive features of this design, in comparison to the Schmidt which will be discussed shortly, is the small size of the refractive elements. Add to this the capability of eliminating any remaining spherical aberration in an assembled objective by adjusting the axial positions of the lenses.

Zero Petzval sum and, hence, a flat image (in the absence of astigmatism) is ensured by giving the mirrors the same curvature and the lens elements equal and opposite power.

8.  $$\text F$$/15 Spherical - primary Cassegrain with Reflective Field Corrector

Comments: This well-corrected design from Korsch (1991) has an easily manufactured spherical primary and is intended for use as a large-aperture telescope objective. Another all-reflective corrector of Gregorian form has been developed for a fast (F/0.6) spherical primary (Meinel, 1984).

9. Afocal Cassegrain-Mersenne Telescope

Comments: The Mersenne contains two confocal paraboloids working at infinite conjugates. It is aplanatic, anastigmatic, and can be made distortion-free by choosing an appropriate stop location. The utility of confocal mirrors has been emphasized by Baker (1978) and Brueggeman (1968), and is illustrated in the following design.

10. Dual-magnification Cassegrain

Comments: This IR design is related to one introduced by $$\text T$$. Fjeidsted (1984). The system of fers two magnifications and fields of view. The high-magnification configuration is with the afocal Mersenne in the optical path. Removing the secondary lets light pass directly to the refractive assembly and a larger field of view is observed. The spectral range is from 3.3 to 4.2 $$\mu m$$.

11. $$\text F$$/3.2 Three-lens Prime Focus Corrector

Comments: This is a three-lens corrector for a 250-cm parabolic mirror. The corrector was developed by Wynne (1965, 1974) for the region of the spectrum extending from 365 to 1014 nm. It is used to extend the field of a parabolic mirror. Versions for a Ritchey-Chretien primary also exist. The corrector is able to correct spherical aberration, coma, astigmatism, and field curvature while keeping chromatic aberrations under control. The field of view can be extended considerably for smaller apertures.

The three-spherical lens corrector is one of the best large-optics prime-focus correctors to come along, both in terms of image quality and ease of fabrication. Other designs have either not performed as well or were heavily dependent on aspheric figuring.

This and other prime-focus correctors are surveyed in articles by Gascoigne (1973), Ross (1935), Meinel (1953), Schulte (1966b), Baker (1953), and Wynne (1972a).

12.  $$\text F$$/4 Gregorian

Comments: The classical Gregorian is aberration-free on-axis when the paraboloidal mirror image coincides with one of the ellipsoidal-mirror foci; the other focus coincides with the final image. Like the Cassegrain, off-axis image quality is limited by coma and field curvature. The ellipsoidal secondary reimages the entrance pupil to a location between the secondary and final image. Thus, there exists the possibility of unwanted-light suppression at the primary-mirror image and exit pupil

The Gregorian is longer than the Cassegrain and thus more expensive to support and house, but it produces an erect image and the concave secondary is easier to produce. In eccentric-pupil versions it has an accessible prime focus.

13. $$\text F$$/4 Aplanatic Gregorian

Comments: The aplanatic Gregorian is corrected for spherical aberration and coma. Both mirrors are ellipsoids. Astigmatism and field curvature limit of f-axis imagery.

14. $$\text F$$/1.25 Flat-medial-field Aplanatic Gregorian

Comments: The Gregorian’s field performance is enhanced if image accessibility is sacrificed. This version of the Gregorian (Korsch, 1991) is aplanatic. A flat medial image is achieved by balancing Petzval curvature with astigmatism, which remains uncorrected.

15. $$\text F$$/1.25 Flat-medial-field Aplanatic Gregorian with Spherical Primary

Comments: The field of this objective (Korsch, 1991) is larger than its cousins , the classical and aplanatic Gregorians, even with the spherical primary. Spherical aberration and coma are corrected, and the medial image is flat. The design has a real intermediate image and exit pupil. The obvious drawback is the size of the secondary in relation to the size of the entrance pupil, which is 15 cm in diameter.

Korsch (1991) analyzes two other designs that are loosely referred to as Gregorians.

16. Afocal Gregorian-Mersenne Telescope

Comments: The Gregorian Mersenne , also composed of confocal paraboloids, is aplanatic, anastigmatic, and can be corrected for distortion. The Gregorian-Mersenne has an intermediate image and an accessible exit pupil.

17. $$\text F$$/1.25 Couder

Comments: The Couder (1926), composed of two conic mirrors, is corrected for third-order spherical aberration, coma, and astigmatism. Unfortunately, the Couder is long for its focal length and the image is not readily accessible.

18. $$\text F$$/1.25 Aplanatic, Flat-medial-image Schwarzschild

Comments: The aplanatic , flat-medial-image Schwarzschild (1905) is similar in appearance to the Couder but the secondary mirror and image locations are different for identical secondary magnifications.

19. $$\text F$$/1.25 Aplanatic, Anastigmatic Schwarzschild

Comments: The spherical-mirror Schwarzschild (1905) is aplanatic, anastigmatic, and distortion-free (Abel, 1980). The Schwarzschild relies on the principle of symmetry for its high level of aberration correction and a large field of view. All surfaces have the same center of curvature at the aperture stop. Hence, there are no off-axis aberrations. Spherical aberration is produced but each mirror produces an equal and opposite amount, thus canceling the effect of the aberration. Some higher-order aberrations are also corrected (Abel, 1980). Eccentric portions of this design—above and below the optical axis in the picture—form well-corrected, unobscured designs. Zonal spherical aberration from the mix of third- and higher-order terms limits on- and off-axis performance.

An aspheric plate positioned at the center-of-curvature of the mirrors removes this aberration as illustrated in the next design.

Wetherell and Rimmer (1972), Korsch (1991), Schroeder (1987), Linfoot (1955), and Gascoigne (1973) offer a general third-order analysis of two-mirror systems. The closed-form solutions described provide insight into third-order aberration theory of reflective systems.

20. $$\text F$$/1 Aplanatic, Anastigmatic Schwarzschild with Aspheric Corrector Plate

Comments: With an aspheric plate at the aperture stop , spherical aberration is eliminated. The only aberrations still remaining are of higher order. To correct these, the mirrors must also be aspherized. Linfoot (1955) and Abel (1980) describe this design.

21. $$\text F$$/1.25 Anastigmatic, Flat-image Schwarzschild

Comments: With just two conics, this design type (Schwarzschild, 1905) achieves aplanatic and anastigmatic performance on a flat image surface. The flat field is attained by making the curvatures of the mirrors equal. Eccentric portions above or below the optical axis form unobscured versions; the design may alternatively be used off-axis. Sasian (1990a, 1990b) and Shafer (1988) have explored many of this design’s features.

22. $$\text F$$/1.25 Schmidt

Comments: The Schmidt (1932) also relies on the principle of symmetry; that is, the aperture stop is located at the center of curvature of the spherical mirror and hence the mirror produces no off-axis aberrations.

The Schmidt corrector is flat with aspheric deformation figured in to correct the spherical aberration produced by the mirror. It is positioned at the aperture stop because off-axis aberrations are independent of aspheric deformation when an aspheric surface coincides with the stop. Hence the Schmidt plate has no effect on of f-axis aberrations, and the benefits of concentricity are preserved.

The corrector introduces chromatic variation of spherical aberration (spherochromatism). A small amount of positive power in the corrector introduces undercorrected axial color to reduce the effects of this aberration. Further improvement is obtained by achromatizing the corrector with two glasses of different dispersions.

Higher-order aberrations degrade image quality at low focal ratios and large field angles. Kingslake (1978), Schroeder (1987), Maxwell (1972), and Linfoot (1955) provide additional details of this and other catadioptric objectives.

23. $$\text F$$/1.25 Field-flattened Schmidt

Comments: As is known from third-order aberration theory, a thin element will be nearly aberration-free, except for Petzval curvature, and distortion when it is placed in close proximity to an image. Therefore, by properly choosing the lens power and index to give a Petzval curvature of equal and opposite sign to the Petzval curvature introduced by the other optics, the image is flattened.

The image in the Schmidt above has been flattened with the lens near the image plane. The only aberrations introduced by the lens are spherochromatism and lateral color, lateral color being the most noticeable aberration; this can be removed by achromatizing the field-flattening lens. The close proximity of the lens to the image can cause problems with light scattered from areas on the lens surfaces contaminated by dirt and dust particles. Clean optics are a must under these circumstances.

The field-flattening lens provides two more positive results. First, the lens introduces a small amount of coma which is compensated by moving the Schmidt corrector towards the mirror somewhat, thus reducing the overall length of the objective. Second, oblique spherical aberration, one of the primary field-limiting, higher-order aberrations of the Schmidt, is substantially reduced.

Besides its usual function as a telescope or photographic objective, the field-flattened Schmidt has also been used as a spectrograph camera (Wynne, 1977).

24. $$\text F$$/1.25 Wright

Comments: The Wright (1935) is one-half the length of the Schmidt. It also relies on aspheric deformation of the corrector plate for the elimination of spherical aberration. Coma, introduced as the corrector is removed from the center of curvature of the mirror, is cancelled with conic deformation of the mirror; the surface figure is that of an oblate ellipsoid. The remaining astigmatism and Petzval curvature are balanced for a flat medial image. The only on-axis aberration, spherochromatism, is corrected by achromatizing the corrector.

25. $$\text F$$/4 Reflective Schmidt

Comments: Another way of defeating chromatic aberration is to eliminate it altogether with a reflective corrector (Abel, 1985). The elements are removed from the light path with a field bias $$(9^\circ)$$, and hence the objective is off-axis. Spherical aberration, coma, and astigmatism are all corrected. At large field angles, beyond about 12 8 half-field angle, oblique spherical aberration becomes evident, but otherwise this design provides excellent performance on a curved image over a $$24\^circ$$ field of view. In order to avoid severe obstruction of the light path, the full $$24^\circ$$ can be taken advantage of only in the plane that extends perpendicular to the picture of the design above. A considerably reduced field of view is allowed in the plane of the picture.

26. $$\text F$$/0.6 Solid Schmidt

Comments: All monochromatic aberrations, with the exception of distortion, are corrected by the appropriately-named solid Schmidt, a system used mostly as a spectrograph camera (Schulte, 1963). All chromatic aberrations, with the exception of lateral color, are corrected. The imaging theory behind the solid Schmidt is expounded by J. Baker (1940a). With a refractive index $$n$$, the solid Schmidt is $$n^2$$ times faster than the conventional Schmidt. Focal ratios of $$\text{F/0.3}$$ have been obtained. Schulte (1963) offers a design similar to the one given here.

27. $$\text F$$/1.25 Schmidt-Cassegrain

Comments: The Schmidt-Cassegrain (Baker, 1940b) represents a successful attempt to resolve the difficulties related to the curved image, considerable length, and awkwardly located image of the Schmidt objective, without destroying the positive attributes of the design.

The Schmidt-Cassegrain comes in a wide variety of forms—too many to go into here. Linfoot (1955) performs an extensive exploration of the design, with one and two aspheric plate correctors. Warmisham (1941) has gone as far as three. Wayman (1944) has analyzed a monocentric Schmidt-Cassegrain.

In this fast version of the Schmidt-Cassegrain, the corrector is close to the flat secondary. Usually one or both mirrors are aspherics. An achromatized image-flattening lens has been introduced.

An image-flattening lens is not usually required with a Schmidt-Cassegrain since enough degrees of freedom exist for aberration correction and a flat image. In this case, the secondary mirror is flat and one degree of freedom is lost. Additionally, the primary mirror power introduces a strong Petzval contribution which necessitates a field-flattening lens.

The first three digits of the six-digit code in the glass column identify the indices of the materials in the design, in this case plastics. These are 1.489 and 1. 583; the Abbe-numbers are given by the last three digits and they are 57.4 and 30.3, respectively.

The two plastics have nearly identical thermal coef ficients and are very light. Buchroeder (1971a) analyzes designs of this variety with two aspheric correctors. Shafer (1981) of fers a Schmidt- Cassegrain with an aspheric meniscus corrector. Only two elements are used since the secondary mirror surface is on the corrector. Rutten (1988) has examples of Schmidt- Cassegrains in a number of configurations.

28. $$\text F$$/3.4 Reflective Schmidt-Cassegrain

Comments: The reflective Schmidt-Cassegrain exhibits all the nice properties of the Schmidt-Cassegrain and, in addition, is achromatic. Schroeder (1978) points out that, because the corrector is tilted $$(9^\circ$$ here), adequate aberration correction requires a nonrotationally symmetric corrector plate. The nonaxially symmetric surface deformation in this design is given by

$\tag{3}Z=A[(1-AN)X^2+(1+AN)Y^2]^2$

where $$\text A$$ is the fourth-order symmetric coefficient and $$\text{AN}$$ is the fourth-order nonsymmetric coefficient. The $$\text y$$ dimension is in the plane of the picture; the $$\text x$$ dimension is perpendicular to it.

Since the corrector is tilted by $$9^\circ$$, the reflected rays are deviated by twice that amount. The element spacings $$\text{(THI)}$$ are no longer measured along the horizontal optical axis after reflection of f the corrector, but along the line of deviation. The secondary and tertiary are tilted by $$18^\circ$$.

29. $$\text F$$/2 Shafer Relayed Virtual Schmidt

Comments: Shafer (1978a) has introduced an eccentric-pupil (18-cm stop decenter), virtual Schmidt objective similar to this but with a decentered quaternary mirror. The center of curvature of the spherical primary is imaged by the concave secondary onto the flat Schmidt tertiary mirror. Now the Schmidt plate , which appears to be at the primary center of curvature, is aspherized to produce a well-corrected virtual image, hence the

name (see Fig. 5). In this configuration, the Schmidt plate is one-half the size of the primary.

The Schmidt plate and the spherical quaternary mirror form a finite conjugate Schmidt system. Thus, the spherical aberration of this mirror is also corrected.

Figure 5 shows a pictorial representation of the Shafer design with the last mirror decentered to provide a more accessible image. Since the primary and quaternary mirrors no longer share the same axis of symmetry, a two-axis Schmidt corrector is required to remove the aberrations of both mirrors.

The shape of this surface is described by Shafer, for an $$\text{F/1}$$, unobscured, wide-field design with an intermediate image and Lyot stop.

30. $$\text F$$/2.2 Spherical-primary Objective That Employs the Schmidt Principle of Correction

Comments: Baker (1978) reports on a system where the center of curvature of a large , spherical primary is imaged by a positive lens onto a much smaller mirror where aspheric correction of spherical aberration occurs. A small field offset $$(0.25^\circ)$$ is required so that the one-to-one relay doesn’t reimage the primary image back onto itself. To avoid overlap, this design is best used over a small or strip field of view.

Because of the geometry of the design, coma, astigmatism , image curvature , chromatic aberrations, and distortion are eliminated in addition to the spherical aberration correction from aspheric figuring of the tertiary mirror. Baker (1978) of fers several other interesting designs in his article , including an $$\text F/0.8$$, $$10.6$$-$$\mu m$$, $$180^\circ$$ field-of-view Roesch (1950), a design that incorporates a Schmidt with a strong negative lens or lens group before the aspheric corrector.

The strong divergence produced by this lens reduces the amount of light blocked by the image plane but increases the size of the spherical mirror.

31. $$\text F$$/2 Maksutov

Comments: The all-spherical Maksutov (1944) was intended as an inexpensive alternative to the Schmidt at slower speeds. In small sizes it is indeed less expensive. The meniscus corrector is ‘‘self-achromatic’’ when the following relationship is satisfied:

$\tag{4}t\frac{n^2}{n^2-1}(R_2-R_1)$

where $$R_1$$ and $$R_2$$ are the radii, $$t$$ is the thickness , and $$n$$ is the refractive index of the corrector.

Bouwers (1950) also developed a meniscus corrector. All elements of the Bouwers are concentric about the aperture stop. This ensures correction of third-order , off-axis aberrations over a nearly unlimited field of view. In exchange for the wide field, axial color is not well-corrected.

32. $$\text F$$/1.25 Solid Maksutov-Cassegrain

Comments: The solid Maksutov-Cassegrain shown here and the solid Schmidt- Cassegrains have been studied extensively by Wynne (1971, 1972b). Lateral color is the most consequential aberration left uncorrected.

33. $$\text F$$/1.2 Wide - field Objective e with Maksutov Correction

Comments: This very wide field imaging system similar to one in Courtes et al. (1971) is essentially a Maksutov focused on the virtual image of the object produced by the hyperboloidal mirror.

Both speed $$\text{(F/1)}$$ and a very wide field of view $$(80\times120$$ degrees) can be achieved with this design form on a flat image but only for small apertures—1.25 cm in this case.

Courtes (1983) describes similar systems with refractive and reflective Schmidt plates instead of a Maksutov corrector.

34. $$\text F$$/1 Gabor

Comments: Another meniscus design was invented by Gabor (1941). The Gabor is more compact than the Maksutov or Bouwers, and has a smaller focal ratio and field of view.

The design shown here began without the field lens. The lens was introduced into the design with the surface closest to the image being concentric about the chief ray and the other surface being aplanatic (Welford, 1986).

A surface concentric about the chief ray is free of coma, astigmatism, distortion, and lateral color. The aplanatic surface is free of spherical aberration, coma, and astigmatism with the result that the lens is coma-and astigmatism-free.

The spherical aberration produced by the lens is balanced against the spherical aberration produced by the two other elements. The chromatic aberrations were corrected by achromatizing the lens.

Shafer (1980a ,1980b) offers interesting suggestions for design with aplanatic and concentric surfaces. Several varieties of field-flattening lens are described. Kingslake (1978) runs through the design procedure for a Gabor.

35. $$\text F$$/4 Schmidt - meniscus Cassegrain

Comments: This system, originally by Bouwers, uses a slightly positive plate to compensate the overcorrected chromatic aberration produced by the meniscus. The Bouwers produces very good quality on a flat image, over a large field of view.

Fourth-and sixth-order deformation added to the plate eliminates any residual spherical aberration. Lateral color and oblique spherical aberration affect field performance, although both are small.

36. $$\text F$$/1.2 Baker Super - Schmidt

Comments: The Baker (1945) super-Schmidt, a design that incorporates both meniscus and Schmidt correction, achieves excellent performance over a wide field of view. The field-limiting aberration of a fast Schmidt, oblique spherical aberration, is controlled by adding a concentric meniscus lens which also introduces overcorrected spherical aberration, thus reducing the amount of overcorrection needed from the Schmidt plate. Since oblique spherical is proportional to the amount of overcorrection in the Schmidt plate, the effect of this aberration is reduced.

The most apparent aberration produced by the meniscus is axial color. This is minimized by achromatizing the Schmidt corrector. Spherochromatism is reduced since the magnitudes produced by the Schmidt corrector and meniscus are nearly equal and have opposite signs. Another meniscus element is added to further reduce aberrations.

37. $$\text F$$/1 Baker-Nunn

Comments: The Baker-Nunn (Baker, 1962) was born of work by Houghton (1944) during World War $$\text{II}$$. Houghton wished to find a less expensive alternative to the Schmidt. The result was a zero-power, three-lens combination with easy-to-make spherical surfaces. Spherical aberration and coma can be eliminated for any position of the corrector.

The surfaces have equal radii so they can be tested interferometrically against one another using the Newton ring method. Residual spherical aberration that remains after assembly is removed by altering the spacing between the lenses.

38. $$\text F$$/10 Houghton-Cassegrain

Comments: A two-lens, afocal corrector developed by Houghton and Sonnefeld (1938) is used here as a corrector for a Cassegrain system. Sigler (1978) has written on the subject of Cassegrains with Houghton, Schmidt, and Maksutov refractive correctors.

This Houghton- Cassegrain gives well-corrected imagery on a curved image surface. An afocal achromatized doublet corrector has also been tried (Hawkins and Linfoot, 1945).

39. $$\text F$$/3.6 Houghton-Cassegrain

Comments: Another Houghton corrector, with meniscus elements, is utilized in this design by D. Rao (1987). The spectral range is 550 to 850 nm. The design is similar to one introduced by Mandler (1951). Examples of other Houghton-Cassegrains of this form are studied by Gelles (1963).

40. $$\text F$$/1.25 Shenker

Comments: M. Shenker has studied a large number of variations on the theme of three-element correctors for a Cassegrain. This is related to one of the configurations developed by Shenker (1966). Note that the third corrector is also the secondary mirror. Zonal spherical aberration limits performance on-axis. This may be removed by aspherizing one or more surfaces.

All elements are of the same glass. Laiken (1991) has a similar version of this design as well as other catadioptric objectives. Maxwell (1972) has design examples and catadioptric imaging theory.

41. $$\text F$$/1.25 Mangin-Cassegrain with Correctors

Comments: Mangin mirrors are evident in this design by L. Canzek (1985) and two elements are used twice. The design has exceptionally good on-axis performance. Lateral color and higher-order aberrations limit the field.

42. $$\text F$$/1.25 Mangin - Cassegrain with Correctors

Comments: Another short and fast catadioptric by Max Amon (1973) is shown here. The second corrector is also the secondary mirror.

43. $$\text F$$/4 Eisenburg and Pearson Two - mirror, Three - reflection Objective

Comments: This aplanatic, two-mirror, three-reflection configuration was first introduced by Rumsey (1969). The design presented here comes from Eisenburg and Pearson (1987). The first and third surface represent the same surface.

44. $$\text F$$/4 Shafer Two-mirror, Three-reflection Objective

Comments: Shafer (1977) has documented numerous versions of multiple-reflection objectives. This is an aplanatic, anastigmatic design with field curvature. For optimum aberration correction, the primary is at the center of curvature of the secondary mirror. Shafer (1908b) suggests a ring field for a flat, accessible image on an annular surface, and a Lyot stop.

A simple ring field design is depicted in Fig. 6. Only one field angle q is required, easing the difficulties associated with off-axis aberration correction. The single viewing direction is rotated about the optical axis, forming, in this case, a ring image. In reality, less than half the ring image is used to avoid interference of the image with the entering beam.

45. $$\text F$$/15 Two-mirror, Three-reflection Objective

Comments: This is another aplanatic, anastigmatic, eccentric-pupil design which gives well-corrected imagery on a curved image. It has a 30-cm stop decenter.

46. $$\text F$$/15 Yolo

Comments: Arthur S. Leonard (1986a , 1986b) invented the Yolo (named after a scenic county in California) so that he could have an achromatic system without obscurations. The result is a tilted and decentered component objective that gives the high contrast of an unobscured refractive objective without the chromatic effects.

Spherical aberration is corrected by the hyperboloidal figuring of the first surface. The anamorphism introduced into the secondary (by a warping harness) corrects astigmatism; $$\text{RDX}$$ is the surface radius of curvature perpendicular to the picture. Coma is eliminated by adjusting the curvatures and tilting the secondary.

Relatives of the two-mirror Yolo are the Solano, an in-line three-mirror Yolo, or the three-dimensional, three-mirror Yolo (Mackintosh, 1986). As in design 28, thickness $$\text{(THI)}$$ is measured along the deviated ray paths. With the angle of reflection known, element decenter may be easily determined.

47. $$\text F$$/15 Schiefspiegler

Comments: The Schiefspiegler (‘‘oblique reflector’’ in German) was introduced about a century ago, and at the time was called a brachyt (or bent). The motivation behind the Schiefspiegler’s design is essentially the same as the Yolo’s. Like the Yolo, elements are tilted and decentered.

Coma and astigmatism are corrected by tilting the secondary and corrector lens. The lens is thin and slightly wedged to minimize chromatic effects. Spherical aberration is corrected with the aspheric deformation of the primary.

A three-mirror Schiefspiegler, or Trischiefspiegler, has been developed by Anton Kutter (1975). This design is all-reflective and completely achromatic. Like the Schiefspiegler, aspheric deformation of the primary corrects spherical aberration; coma and astigmatism are corrected with element tilts.

A four-mirror Schiefspiegler was recently introduced by Michael Brunn (1992). For more on unusual telescope objectives, see Manly (1991).

48. $$\text F$$/8 Catadioptric Herschelian Objective

Comments: Several centuries ago, Herschel tilted his large parabolic mirror to give him access to the image. A spherical mirror in this design by D. Shafer (Telescope Making 41) has been tilted for the same reason.

Element tilts in the Houghton corrector control the astigmatism introduced by tilting the mirror. The Houghton corrector also eliminates the spherical aberration of the mirror with lens bending. Note the smaller focal ratio of this design compared to either the Yolo or the Schiefspiegler.

Other catadioptric Herschelians, as well as Schiefspieglers and Yolos, have been studied by Buchroeder (1971b) and Leonard (1986b). Tilted, decentered, and unobscured Cassegrains are discussed by Gelles (1975).

49. $$\text F$$/4 SEAL

Comments: For an all-reflective objective, this flat-image design provides an exceptionally wide, unobscured field of view—greater than $$90^\circ$$ with a ring field. Referred to as the $$\text{SEAL}$$ (Owen, 1990), it is derived from its cousin the WALRUS (Hallam, 1986); a related design has been described by Shafer (1978d).

The $$\text{SEAL}$$ is another Mersenne-Schmidt hybrid: primary and secondary form an inverse-Mersenne; tertiary and quaternary (also the secondary) form a reflective Schmidt. Residual spherical aberration limits the performance , but by aspherizing the flat, this residual aberration is corrected as well.

Clearing all obscurations requires at least a $$22^\circ$$ field offset. The $$\text{SEAL}$$ shown here is optimized for a $$20^\circ$$ strip field although a square, rectangular, annular, or almost any shape field is possible.

50. $$\text F$$/4 Paul Three-mirror Objective

Comments: This design is based on work by Paul (1935) and later by Baker (1969), who was looking for an achromatic field corrector for a parabolic primary. Their efforts culminated in a design similar to this one, which combines the properties of an afocal Cassegrain-Mersenne in the first two elements with the properties of an all-reflective Schmidt in the secondary and tertiary elements.

Since both modules are corrected for spherical aberration, coma, and astigmatism to third order, the complete system is aplanatic and anastigmatic. Petzval curvatures are equal and opposite so a flat image is achieved. The conic deformation of the secondary is modified to give it an ellipsoidal shape. This gives the required Schmidt aspherization needed to correct the spherical aberration of the tertiary mirror.

Other all-reflective designs have been proposed by Meinel (1982, 1984) and Baker (1978). The Meinel-Shack objective (1966) exhibits similar performance and offers a more accessible image.

51. $$\text F$$/4 Alternative Paul Three - mirror Objective

Comments: This Paul objective has an aspheric tertiary mirror, instead of an aspheric secondary.

52. $$\text F$$/4 Off-axis, Eccentric-pupil Paul-Gregorian

Comments: This eccentric-pupil (22-cm), of f-axis $$(1^\circ)$$ design utilizes a Gregorian-Mersenne module in the primary and secondary mirrors. Spherical aberration produced by the spherical tertiary mirror is corrected by superimposing aspheric deformation on the paraboloid secondary, located at the tertiary mirror center of curvature.

With all concave surfaces, field curvature is uncorrected. A real and accessible exit pupil and intermediate image offer possibilities for excellent stray-light suppression.

As is the case with the virtual Schmidt system, the tertiary mirror may be decentered to provide a more convenient image location. This requires two-axis aspheric deformation of the secondary mirror (Shafer, 1978c).

53. $$\text F$$/4 Three-mirror Cassegrain

Comments: A design similar to the aplanatic, anastigmatic, flat-image design shown here was conceived by Korsch (1978) and is described by Williams (1979), Korsch (1977), and Abel (1985). The exit pupil is accessible and an intermediate image exists. A $$1^\circ$$  field offset is needed to displace the image from the folding flat. Residual coma limits field performance. Small element tilts and decenters will improve the performance of this design.

54. Three - mirror Afocal Telescope

Comments: This $$5\times$$ afocal design from Smith (1992) is an eccentric-pupil Cassegrain and a parabolic tertiary combined. The design is aplanatic and anastigmatic. The entrance pupil is decentered by 32 cm.

55. Three-mirror Afocal Telescope

Comments: A similar design by Korsch (1991) is also aplanatic and anastigmatic. The entrance pupil is decentered by 20 cm. Other afocal designs are described by Gelles (1974) and King (1974).

56. $$\text F$$/4 Three - mirror Cassegrain

Comments: Robb (1978) has introduced another aplanatic, anastigmatic, flat-image, three-mirror Cassegrain without an intermediate image.

57. $$\text F$$/6.7 Spherical Primary Three - mirror Objective

Comments: Making the largest element in an objective a spherical mirror reduces cost and may enhance performance. This aplanatic, anastigmatic, flat-image, eccentric-pupil design $$(-35$$ cm stop decenter) with an unobscured light path is similar to one described by Korsch (1991) and another developed for use as an astrometric camera by Richardson et al. (1986).

58. $$\text F$$/4 Spherical Primary Three-mirror Objective

Comments: Here is another aplanatic, anastigmatic, flat-field, eccentric-pupil design with a 17-cm stop decenter and large spherical primary. There is an intermediate image and an accessible exit pupil.

59. $$\text F$$/4 Three-mirror Korsch Objective

Comments: This of f-axis $$(5^\circ)$$ design by Korsch (1988) is aplanatic, anastigmatic, and has a flat image. The same configuration has been employed by Pollock (1987) as a collimator. Characteristics include a large field of view, low pupil magnification, accessible pupils, and an intermediate image.

The tertiary in this design is spherical. With reoptimization, the secondary may also be spherical.

60. $$\text F$$/4 Three-mirror Cook Objective

Comments: This objective was introduced by L. Cook (1981, 1979, 1992). The aplanatic, anastigmatic, flat-image design shown here has a larger pupil magnification and a smaller field than the previous design. The eccentric-pupil, of f-axis design has a $$-3.2$$-cm stop decenter and a $$5^\circ$$ field bias. A space-based surveillance objective in this configuration has been developed and built by Wang et al. (1991).

61. $$\text F$$/4 Three - mirror Wetherell and Womble Objective

Comments: Another aplanatic, anastigmatic, flat-image, off-axis $$(9^\circ$$ ) design has been introduced by Wetherell and Womble (1980). Figosky (1989) describes a variant of this form to be sent into orbit. The aperture stop is located at the secondary mirror; hence, this mirror is symmetric with respect to the optical axis.

62. $$\text F$$/10 Korsch Three-mirror, Four-reflection Objective

Comments: The three-mirror, four-reflection design shown here from Korsch (1974) is extremely compact for its 200-cm focal length, and the image is accessible.

63. $$\text F$$/1.25 McCarthy

Comments: McCarthy (1959) intended this design , which combines a Cassegrain- Mersenne primary and tertiary mirror with a quaternary and quintenary Schwarzschild arrangement, as a wide strip-field imager. Both the Mersenne and Schwarzschild groups are separately corrected for spherical aberration, coma, and astigmatism.

The Petzval curvature of the Mersenne is equal and opposite in sign to the Petzval curvature of the Schwarzschild and hence there is no net Petzval curvature. The quaternary mirror may be moved out from the entering beam with only a slight reduction in performance.

64. $$\text F$$/2.2 Cassegrain Objective with Schwarzschild Relay

Comments: S. Williams (1978) describes a technique for optimizing a high-resolution system similar to this one while maintaining proper clearances, obscuration sizes, and packaging requirements.

An all-reflective zoom system of the above configuration, developed by Johnson et al. (1990), gives a $$4\times$$ zoom range and a field-of-view range of 1.5 to 6.0$$^\circ$$. The Schwarzschild module and image position change with zoom position, while the front Cassegrain module remains fixed.

65. $$\text F$$/4 Altenhof Objective

Comments: This objective is similar to one designed by R. Altenhof (1975). Intended as a ring field system, the flat primary couples the light incident from a large azimuthal angle $$(60^\circ)$$ into the system where the spherical primary mirror focuses the light to a poorly corrected image.

The three-mirror Offner relay (Offner, 1975, Kingslake, 1978) , a unit magnification relay which is corrected for spherical aberration, coma , and astigmatism, improves the degraded image in the process of reimaging it to a flat focal surface in the form of an annulus.

A two-dimensional scene is imaged by rotating the flat mirror about an axis perpendicular to the picture so as to scan the other dimension. A two-dimensional mosaic image can also be produced by building up a series of one-dimensional annular strip images as the imaging system is moved along its trajectory.

66. $$\text F$$/4.5 Shafer Four-mirror, Unobscured Objective

Comments: This is a tilted and decentered-component infrared imaging system by David Shafer. Mirror tilts provide an unobscured path and help correct the aberrations. Thickness is measured along the deviated ray paths. With the reflection angle known, element decenter may be easily determined.

67. $$\text F$$/4.5 Shafer Five-mirror, Unobscured Objective

Comments: Another all-spherical, tilted, and decentered-component infrared imager by D. Shafer is presented here. The entrance pupil is accessible and there is an intermediate image. A number of variations on this arrangement are described by Shafer (1978b).

68. Korsch Two-and Three-mirror Objective

Comments: A new class of eccentric-pupil objectives has been introduced by Korsch (1991). Unlike most systems, which are conceived using third-order aberration theory, these systems are based upon the fulfillment of axial stigmatism, the Abbé sine condition, and the Herschel condition; meeting these three conditions guarantees a perfect axial point image, axially perpendicular image area, and axial line element, respectively.

Design examples are not given for two reasons. First, rays strike some mirror surfaces at angles greater than $$90^\circ$$, which can cause ray-trace errors. Second, some of the surface shapes are particularly complex and must be entered in design software as a user-defined surface.

Design (c) gives perfect imagery on-axis and less than one milliradian resolution at all other points over a $$6^\circ$$ field of view, for an aperture diameter equal to $$\text F$$/6.0 $$\text F$$ is focal length).

## 4.  FIELD-OF-VIEW PLOTS

The plots that follow give rms spot size and angular resolution as a function of half-field of view. The curves have been generated by calculating the resolution for a number of field angles and connecting them with smooth curves. The dashed horizontal line is the Airy disc diameter for 0.55-$$\mu m$$ radiation.

The numbers in the plots correspond to the designs presented in the previous section. The aperture of each design is 20 cm and the spectral range 480 to 680 nm, unless stated otherwise in the previous section.

It should be kept in mind that these are representative designs: they have usually been optimized for a specific purpose; they are meant to be a starting place for the design of a new system that may have entirely different requirements.

Flat-field designs show consistent performance out to the field angle for which the objective is optimized. Beyond this point, the graph leaps upward.

Reoptimization is needed if the field of view is to be extended further; a considerable increase in the average rms spot size may occur if this is attempted. The curved-image designs show a quadratic dependence with field angle.

Off-axis and eccentric-pupil designs have rectangular fields with most of the field of view in one dimension only. Data plotted for these designs are representative of the larger field.

In Figs. 8 and 10 plots for the curved image designs are provided. The curvature of the image is adjusted to give optimum performance. Figures 7 and 9 are for the flat image designs.