pass unwanted light, which is undesirable. The obstruction is not large enough to cause a serious degradation of the definition.

15.5.4 Three-Mirror System

An interesting modification of the classical Gregorian reflecting system has been suggested by Shafer,28 in which the marginal ray is reflected twice at the primary mirror, at equal distances on opposite sides of the axis, and once at the concave secondary mirror, the final image being formed at the center of the secondary mirror. The paraxial layout is shown in Figure 15.31. If the mirror radii are respectively 10 and 7.5, the separation is 7.5 and the focal length is 7.5. A few trials indicate that for the simultaneous correction of spherical aberration and OSC with a semiaperture of 1.0 (i.e., an f/3.75 system), the primary must be a concave ellipse with eccentricity 0.63782 and the secondary a concave hyperboloid with eccentricity 2.44. The aperture stop and both the pupils are at the primary mirror. Since the middle half of the entering beam is blocked out by the secondary mirror, the image receiver can cover the middle half of the secondary mirror without introducing any further obstruction. The angular field of our f/3.75 system is therefore1.9 . If the aperture is doubled, the angular field will also be doubled to 3.8 .

Figure 15.31 A three-reflection aplanatic system.

15.6 MULTIPLE-MIRROR ZOOM SYSTEMS

In the past several decades, there has been some interesting work on multiple mirror systems that often have a zooming capability. There are two general types, namely, obscured and unobscured pupils. In this section, both types will be discussed with some being fixed focused and other being zoom capable.

15.6.1Aberrations of Off-Centered Entrance Pupil Optical Systems

When the entrance pupil is decentered with respect to the optical axis of the optical system of an otherwise rotationally symmetric system, it breaks the normal symmetry and the system becomes known as a plane-symmetric system. The aberrations no longer appear as they do for a rotationally symmetric system and other new aberration coefficients appear in the aberration expansion equation. We will not discuss this expansion, but will now look at the general behavior of the aberrations. Both the terms decentered and off-centered are used interchangeably in the literature.

Figure 15.32 shows the distortion behavior for an off-centered optical system having zero, positive and negative aberration values for distortion, coma, astigmatism, and Petzval.29 When the pupil is centered, the coma, astigmatism, and Petzval do not affect the distortion of the image. Spherical aberration for an f/2 optical system with a centered pupil having a diameter of 5 is shown in the center of Figure 15.33. A close-up view of the focal region is shown at the top of the figure and five focus positions are included at the bottom. These are what we are accustomed to viewing. Now consider an off-centered entrance pupil system having a focal length of 10 and operating at f/5.

Figure 15.34 presents the spherical aberration for three pupil offset displacements of 2, 3, and 4 as well as for four defocus positions. In the zero defocus position, the spot diagram rather appears to have a comatic shape while the defocused positions appear to be a combination of astigmatism and coma;

Positive

Negative

Zero

Figure 15.32 Distortion for optical systems having an off-centered entrance pupil.

Figure 15.33 Spherical aberration for a centered-pupil optical system showing spot diagrams for several defocused positions.

Pupil offset = 4

Pupil offset = 3

Pupil offset = 2

Defocus

0.0500 Inch

Figure 15.34 Spherical aberration for an off-centered entrance pupil optical system showing spot diagrams for several defocused and entrance pupil offset positions.

however, they have their own unique shapes. Coma for the decentered pupil condition appears much like the centered pupil situation. Figure 15.35 illustrates the behavior for the f ¼ 10, f/5 optical system with the source 1 off-axis and the pupil decentered by 2. In the lower right corner of the figure, the coma for the same system with a centered pupil is shown and is scaled up by a factor of 2.

Up

Down

Centered pupil (image magnified by 2)

Figure 15.35 Coma spot diagrams for an off-centered entrance pupil optical system compared with the spot diagram for the same system having a centered entrance pupil.

In the following examples, the spot diagrams presented will be seen to contain what will appear as perhaps oddly shaped distributions but are consistent with the introduction just presented.

15.6.2 All-Reflective Zoom Optical Systems

Woehl published the first paper on all-reflective zoom systems.30 The principal purpose was to provide a means for beam shaping and image manipulation. The unobscured, off-axis six-mirror configuration had two fixed mirrors for input focusing and output reimaging, and two pairs of dual moving mirrors to affect zooming. The zoom range was 30:1 with the field-of-view limited by aberrations.

The first patent for an all-reflective zoom optical system was filed by Pinson in 1986.31 This system has a centered and obscured entrance pupil with no field-of-view offset. The patent covers a variety of configurations having two or more mirrors with examples for two to six mirrors. Practical designs require more than two mirrors due to aberration control, with four being a compromise. The generic form of the optics is back-to-back Cassegrain-type optics. The pupil shape is unusual and varies as a function of field angle and zoom ratio. Reasonably good resolution can be realized for f-numbers less than four. The system is uncompensated; that is, the images move as the optical system zooms. Two proof-of-concept four-mirror prototypes have been built.32

The first-generation unit was made from spherical and conic surfaces with a primary mirror diameter of about four inches. The zoom range was 4:1 with a field-of-view range of 1.5 to 6 and f/3.5. Remarkably, the overall length of the telescope varied from 13.5–15.5 inches while the focal length varied from 2.5 to 10 inches. The second-generation unit’s mirrors were all conics with higher-order aspherics on the secondary and tertiary mirrors. With an f/3.3 and a zoom ratio of 4:1, the primary mirror diameter was 4.9 inches and the overall length of the unit was about one inch less than the first generation. On-axis performance was diffraction limited and suffered some coma off-axis.

In 1989, Rah and Lee published a description of a four-mirror zoom telescope that maintained the aplanatic condition throughout the zoom range.33 This obscured-pupil, uncompensated design used spherical mirrors in a cascaded Cassegrain–Cassegrain configuration.34 By this is meant the order of surfaces is first primary mirror, first secondary mirror, second primary mirror, and second secondary mirror, in contrast to the Pinson configuration of first primary mirror, first secondary mirror, second secondary mirror, and second primary mirror. With the 2:1 zoom range, the f-number varied from 4 to 8 and the field of view (FOV) was maintained at a constant one degree throughout the zoom range. The FOV is limited by astigmatism; however, Rah and Lee observed that conic mirrors could be used to correct this aberration. It should be noted that the overall length of this structure changes quite dramatically with zoom.

In early 1991, Cook was awarded a patent for an all-reflective continuous zoom optical system.35 This obscured entrance pupil configuration comprises three mirrors arranged to form an anastigmat. The primary and secondary mirrors form a Cassegrain, with the tertiary mirror moving to affect the zooming function, basically serving as a relay mirror. The image is uncompensated and the line-of-sight changes with zoom in some realizations. This telescope was designed for a scanning system and has a narrow along-scan field-of-view and a wide cross-scan field-of-view. The image surface is flat and has a constant offset from the optical axis while the intermediate image has a varying offset. The basic design shown was a 2:1 zoom with the f-number varying from f/5.14 to f/10.2, focal length of 154.2 to 305.5, entrance pupil diameter of 30, and FOV offset of 3 to 1.5 . The structure of this system is as shown in

Table 15.8, where A6 and A8 are the sixthand eigth-order aspheric deformation coefficients.

The following year, Kebo received a patent for another all-reflective zoom optical system which taught afocal telescopes with object-space FOVs of up to a couple of degrees and zoom ranges of 2:1 (1.7 –3.6 ) and 4:1 (0.125 –0.5 ).36 The 2:1 design is a reasonably compact four-mirror unobs- cured-pupil configuration, having a common optical axis, that uses off-axis sections of rotationally symmetric mirrors for a FOV coverage of 0.95 to 2 . Unlike the prior systems by Cook, the FOV is centered on the optical axis in angular coordinates (but spatially translated). The basic mirror shapes are parabolic primary, hyperbolic secondary, and spherical tertiary and quaternary mirrors. To achieve the zoom function, it is required to move the final three mirrors. An intermediate image is formed by the primary and secondary mirrors at the location of the tertiary mirror. An interesting aspect of this design is that the exit pupil remains fixed with respect to the primary mirror and optical axis while the entrance pupil, lying prior to the primary mirror, utilizes different portions of the primary mirror with zoom. The on-axis 80% geometric blur diameters are 0.36 mrad (1.7 ) and 0.09 mrad (3.6 ).

The second of Kebo’s designs uses a three-mirror unobscured-pupil configuration with the primary mirror being the stationary mirror. It has a 4:1 (0.25 –1 object space) zoom range, and the remotely located entrance pupil in front of the primary mirror is fixed with respect to the tertiary mirror and the optical axis. The primary and secondary mirrors are ellipsoidal, and the tertiary mirror is hyperbolic, all having a common optical axis and being off-axis sections. In this case the output beam moves over the tertiary mirror with zoom. Also, this design is not compact and the tertiary mirror is much greater in size with respect to the primary mirror. The on-axis 80% geometric blur diameters are 0.084 mrad (0.125 ) and 0.15 mrad (0.5 ).

Also in 1992, Korsch investigated a dynamic three-mirror obscured-pupil zoom telescope for potential use in planetary observations.37 This design had a 4:1 (0.125 –0.5 ) zoom range operating at f/3.3 at a FOV of 0.5 . The image

and the tertiary mirror remain fixed with respect to one another during zoom. Both the primary and secondary mirrors move during zoom. An unusual aspect of Korsch’s design was the incorporation of a dynamically deformable primary mirror to correct for both aberrations and focus during zoom. The telescope has a flat image field, essentially no distortion, and excellent resolution.

An unobscured-pupil five-mirror all-spherical mirrors zoom telescope was designed by Shafer in 1993.38 The stop is located remotely in front of the primary mirror and remains fixed with respect to the primary mirror while the other mirrors and image move about during zoom. The FOV range is 8 to 3.2 (diameter) with the corresponding f-number range being 3.5 to 8.75. The geometric resolution over the entire FOV and zoom range is 100 mrad. Maintaining alignment during zoom can be challenging.

Many of these mirror systems are designed to have an accessible entrance pupil location so that they can mate with another optical system such as a camera or other sensor. The afocal type can be used to change the FOV of a sensor while the focal type may be used as a collimator or projector for coupling efficiently with a sensor under test or calibration.

15.6.3Off-Centered Entrance Pupil Reflective Optical Systems

In 1994, Johnson investigated a variety of three-mirror unobscured-pupil zoom telescopes for planetary science missions.39 Since the 1970s, a variety of fixed-focused, three-mirror, unobscured-pupil, anastigmatic optical systems have been developed and are often used in specialized space-based sensors and custom collimators for testing infrared sensors.40,41 Although the image is uncompensated, the configuration shown in Figure 15.36 required as small a volume as possible while providing a zoom range of 1.5 to 3 (square) and operating at f/3 at FOV ¼ 3 , and needed a flat image field. The aperture stop (entrance pupil) of diameter 152 mm is located at about a constant 1.5 m from the primary mirror. A decentered entrance pupil is used with all of the mirrors having a common optical axis. The mirrors are all segments of rotational symmetric forms, which are conic with aspheric deformations up to tenth order. The FOV center is offset from the optical axis by 5 , which means that all of the useful FOV is located at an off-axis portion of the image field of the telescope—that is, the actual image area is located no closer than 2 to the optical axis.

This technique allows the use of different portions of aspherized secondary and tertiary mirrors to be used in aberration control during zoom as shown in the Figure 15.36. The image is flat field, less than 1% distortion, and has little anamorphic error. This telescope also forms an intermediate image and a real

Secondary

mirror

Image

Wide

Narrow

Wide

Tertiary

Narrow mirror

Figure 15.36 Three-mirror unobscured-pupil zoom telescope with 2:1 zoom range.

exit pupil. It was found that the angular offset of the FOV is critical in achieving good optical performance over wide zoom ranges of up to 6:1 at relatively low f-numbers. Also, care must be given in selecting parameters since the size of the tertiary mirror and motion of the mirrors can become unacceptably large.

An example of an actual system used as an infrared collimator is shown in Figure 15.37. This configuration has a well-formed intermediate image formed

Figure 15.37 Three-mirror unobscured-pupil f/4 telescope having a common optical axis and remote entrance pupil.

by the primary and secondary mirrors which is relayed to the image plane by the tertiary mirror. A real and accessible exit pupil is also formed by the system. The total field-of-view of this system is 4.4 by 4.4 with a focal length of 600 mm operating at f/4, and the image field is flat. The central beam in the input FOV is þ4 with respect to the optical axis of the telescope.

In Figure 15.37, this beam is horizontal and it can be seen that the optical system is tilted by 4 clockwise (notice the image plane). The structure of the system is given by

and the sixththrough the twelfth-order aspheric deformation coefficients are shown in Table 15.9. The active object-space FOV is located at 1.8 to 6.2 in elevation (y-axis) and 2.2 in azimuth (x-axis). Defining the structure of such a non-rotationally symmetric optical system is more complicated than the typical rotationally symmetric optical system. This is also true of optimizing such systems.

Figure 15.38 presents the geometric spot diagram for this telescope and also shows the Airy disk diameter for flux at 10 mm. As should be expected, the shape of the images over the FOV varies significantly yet has symmetry about the meridional plane. Achieving the relatively large FOV was accomplished by using different portions of the secondary and tertiary mirrors in the image formation as the FOV angles change. The beam footprint on the secondary mirror is about 30% of the total active area of this mirror and, in a like manner, is about 40% for the tertiary mirror.

Table 15.9

Aspheric Coefficients for Optical System Shown in Figure 15.37

This design is actually for a 2:1 zoom telescope, but only the fixed-focus design for the wide FOV is given. For the infrared spectrum, this type of telescope can be manufactured using diamond turning technology with excellent results.

Another example of a compact three-mirror unobscured-pupil telescope with an accessible entrance pupil is illustrated in Figure 15.39. This configuration has an accessible intermediate image formed by the primary and secondary mirrors and folded upward by a flat mirror. The field stop is located at this image, which is relayed to the image plane by the tertiary mirror. A real exit pupil is also formed by the system, but is not accessible. Although very compact, this configuration can provide excellent stray-light suppression by the inclusion of baffles and the field stop at the intermediate image, which is tilted at about 20 with respect to the optical axis. The total field-of-view of this system is 1 by 1 with a focal length of 1000 mm operating at f/4. The central beam in the input FOV is þ1.5 with respect to the optical axis of the telescope, that is, the meridional FOV is þ1 to þ2 (positive slope angle).

Tertiary

mirror

Aperture

stop

Primary

mirror

Figure 15.39 Compact f/4 three-mirror unobscured-pupil telescope.

In Figure 15.39, this central beam is horizontal and the optical system is tilted by 1.5 degrees clockwise. The structure of the system is given by

and the sixththrough the fourteenth-order aspheric deformation coefficients are shown in Table 15.10.

The active object-space FOV is located at 1.0 to 2.0 in elevation (y-axis) and 0.5 in azimuth (x-axis). Defining the structure of such a folded nonrotationally symmetric optical system is more complicated than the typical rotationally symmetric optical system. As mentioned before, this is also true of optimizing such systems.

Figure 15.40 presents the geometric spot diagram for this telescope and also shows the Airy disk diameter for flux at 10 mm. As seen in the prior example, the shape of the images over the FOV varies significantly yet has symmetry about the meridional plane. Unlike the prior example, most of the area of each of the three powered mirrors is utilized in image formation as the FOV angles change.

Rodgers developed a folded four-mirror zoom collimator that can be either focal or afocal.42,43,44 Notice that this telescope utilizes the fold mirror in the

prior two optical systems as a powered element; however, these optical systems were independently developed. Two examples are presented in this discussion. The first is illustrated in Figure 15.41 on page 496 and provides a 2:1 zoom

Table 15.10

Aspheric Coefficients for Optical System Shown in Figure 15.39

capability with an FOV range of 1.5 to 3 (square) and a corresponding f-num- ber range of f/4.3 to f/8.6. The internal negative power mirror is used to control field curvature of the extended FOV. The specific novelty of this design is that the “folding mirror” has a weakly powered, highly aspheric, and highly tilted surface located near the internally formed image. This mirror serves as a field element and provides control of field aberrations. The tilt of this mirror is about 30 with respect to the beam incident upon it, which allows the fourth mirror to be located above the optical system and the image formed below it. The image is uncompensated with zoom.

The second optical system is presented in Figure 15.42. It follows the design methodology of the preceding telescope. Since this system is being used as a collimator, the exit pupil is located in front of the primary mirror, as illustrated in the figure. The zoom ratio is 2:1 with an FOV range of 1.5 to 3 (square) and a corresponding f-number range of f/4.3 to f/8.6. The exit pupil diameter is fixed at 100 mm with a pupil relief of more that 1200 mm. The folding mirror is

(a)

Figure 15.40 Eleven point images are shown in (a) with the blowup of each image shown in

(b). The scale on each side of the grid in (b) is 200 mm. Circles indicate the Airy disk diameter for a wavelength of 10 mm.

(b)

Figure 15.40 Continued

Entrance and exit pupils

Figure 15.41 Compact four-mirror unobscured-pupil zoom telescope.

Exit

Pupil

Entrance

Pupil

Image/Source

Figure 15.42 Compact four-mirror unobscured-pupil zoom telescope that has a very remote entrance pupil and accessible exit pupil.