Off-Axis Three-Mirror Optical System Designs: From Cooke’s Triplet to Remote Sensing and Surveying Instruments

Featured Application

Telescopes, remote sensing, and surveying instruments, incorporating highly specialized requirements, from a compact design to one with a wide field of view for remote sensing applications. Each may be designed starting from the paraxial optics (first principles), featuring a three-element on-axis system, and using only the paraxial optics as displayed in the Y-Ȳ diagram.

Abstract

The off-axis three-mirror optical system is derived from the classical Cooke triplet or a derivative of the inverse-telephoto lens. By properly arranging an internal reimaging mechanism or altering the location of the optical stop, one can create different versions of three-mirror optical systems. They include very compact configurations and wide field-of-view imagers. Insights into the optical design process, manufacturing, stray light management, and remote sensing applications are presented.

1. Off-Axis Optical Systems

The off-axis and decentered all-reflective optical systems (ODAOS), which were invented and developed in the 1970s, have been widely used in remote sensing, commercial, and military applications for 50 years (see the Acronyms). Among the current applications is monitoring the Earth’s vegetation, using a three-mirror system in an off-axis configuration and the James Webb Space Telescope [1,2]. The four main benefits of ODAOS over the traditional on-axis and rotational symmetrical refractive optical systems are the following. (1) They include no refractive or dispersive element; hence, they display no chromatic aberration. (2) The central optical system obscuration associated with the on-axis configuration is avoided, preserving the throughput. The absence of central obscuration is also favorable for the modulation transfer function (MTF). (3) The limitation on the size of the first optical element is eliminated because a single large piece of glass (blank) is not needed to fabricate this component. It is replaced with a metallic primary, featuring high strength for its weight; furthermore, it may be light-weighted with a honey-cone structure. (4) Modern fabrication technologies of diamond-turning coupled, with the recently developed free-form fabrication and testing techniques, are significantly more versatile and superior to the traditional manufacturing process of refractive optics and may be advantageously implemented. The off-axis configuration is also favored for stray light control [3,4].

The aim of this work is to briefly review and describe the optical designs of the off-axis three-mirror (3M) optical systems. In principle, 3M-optical systems are based on two conceptual classes of optical designs: the classical Cooke triplet and the inverse-telephoto derivative. These two classes of optical designs are described in several, easily accessible optical design textbooks [5,6,7,8,9,10]. In contrast to the presentation in these textbooks, here we focus on how to convert from either design class into a desired off-axis 3M optical system. In addition, we offer some key insights into optical design processes. For example, the positioning of the optical stop at specific locations along the optical axis may allow the optical system to form an intermediate image that results in the creation of a whole new format of reflective optical systems. This offers the possibility of inserting the Lyot stop at the intermediate focus for stray light control [11,12].

We start with the first-order optical layout of each optical system using the Y-Ȳ diagram [13,14,15]. This diagram in fact provides all important information about the parameters of the first-order design. We provide a complete conversion procedure only for the first design in each class. The members of the class share some common features among all the reflective optical systems; however, we do not wish to be repetitive to save time and space. Then, from the second design on, only the difference from the initial design is indicated.

In the next section, we deal with the reflective triplet (RT). The three-mirror anastigmat (TMA), Offner relay (OR), WALRUS, and Korsch triplet follow. Finally, optical design issues, manufacturing, stray light control, and applications are briefly discussed in Section 7.

2. Cooke Triplet

The classical Cooke triplet, presented in Figure 1, is one of the masterpieces in the field of optical design. Two positive lenses are placed—one in the front and one at the back of the middle lens—on a negative refractor, with the optical stop on top of it. Basically, this beautifully simple design corrects all the third-order Seidel aberrations. The optical power of the first two elements is 0.5 of the system optical power while the two lenses form an afocal system. The third lens focuses the incoming collimated beam onto the focal plane. Also, the central negative lens may be used to widen the system’s field of view. Its Y-Ȳ diagram is displayed in Figure 2. A short review of the Y-Ȳ diagram is presented in Appendix A.

The remote sensing instruments examine an object at infinity. In such a case, the marginal ray in the object space remains unchanged, or ever so slightly inclined with respect to the Ȳ-axis, while the slope of the chief ray increases from minus infinity. The object size is infinite; therefore, the graph only touches the Ȳ-axis there. The system forms a small image basically in the focal plane, at point I, where the generalized ray crosses the Ȳ-axis. The image size is equal to the distance OA-I.

The stop AS is located at the position where the generalized ray crosses the Y-axis. The stop height, the radius or one-half of the square side of the aperture, is equal to the distance OA-AS. The distance on both axes is measured in [m] or [mm], but we use a different scale on each axis. At the primary lens, which is convergent, the direction of the generalized ray deviates into the clockwise direction. At the secondary lens, which is divergent, the direction of the generalized ray deviates in a counterclockwise direction. The line segment between the points S and T is parallel to the Ȳ-axis.

The tertiary lens is convergent, deviating the generalized ray toward the Ȳ-axis. The image is formed at the position where the generalized ray crosses the Ȳ-axis. Between the secondary and tertiary mirrors, the generalized ray height remains unchanged for the preset selection of the power of the secondary component. It is equal to the stop height OA-AS. The power of the tertiary mirror is chosen to generate the desired image size, OA-I.

3. Reflective Triplet (RT)

Next, we describe how to convert the Cooke triplet into an off-axis, decentered, all-reflective system step by step. First, let us work on the first two elements. The corresponding reflective afocal optical system with the power of 0.5 is an afocal Mersenne design [16], illustrated in Figure 3a. Note that in this classical rotationally symmetrical system, the secondary mirror blocks some of the incoming beam bundle, introducing a so-called central obscuration.

The two designs with the power of 0.5, that is, the two afocal systems formed by the first two elements of the Cooke triplet and the afocal Mersenne design, are identical when presented in the framework of the Y-Ȳ diagram, shown in Figure 2.

The letter P for the primary denotes a positive primary element, while the letter S for the secondary denotes a negative one. In other words, element one in both the afocal portion of the Cooke triplet and the afocal Mersenne is positive, and the second one is negative. To mitigate the transmission loss caused by the central obscuration, the lower half of the afocal Mersenne is removed, as seen in Figure 3b, where the dashed line denotes the optical axis. This represents an off-axis optical system. The Y-Ȳ diagram of the afocal Mersenne design, with the optical stop located at the second mirror, is presented in Figure 4. The description follows the one that accompanies Figure 2, except that the final mirror of Figure 2, which projects the image on the focal plane, is missing in Figure 4.

If we rotate and/or decenter the second element slightly to avoid the beam obscuration and then add another positive mirror to form an image in the focal plane, we have now constructed a new conceptual optical design, that of an off-axis RT. Its Y-Ȳ diagram is the same as that of the classical Cooke triplet, presented in Figure 2.

When the first two elements of the RT do not form an afocal beam after leaving the second element, we obtain the layout shown in Figure 5, disclosed as the US Patent 4,240,707 [17]. The key features of the design are that all elements share a common optical axis (Item 34). Both the primary and the secondary mirror may be cut out from the same optical blank, sometimes referred to as the parent element. Therefore, RT is top–bottom optically symmetrical with respect to the optical axis. The Cooke triplet, on the other hand, is left–right symmetrical about the central negative refractor.

Next in Figure 5, the incoming beam, incident on the first mirror (Item 31), reflects off all three optical elements, without experiencing any transmission loss, and focuses on the focal plane (Item 36), except for small losses within the metal/AR coating on each optical component. Also, none of the reflectors are subject to chromatic aberration. We note that the incoming beam subtends an offset angle of about 22° with respect to the optical axis. We denote the beam axis with z and graphically with a broken line. An offset angle this large will, in general, create some distortion. It, however, does not appear in the left–right symmetrical Cooke triplet described here.

Additional interesting insights about RT are described next. (i) An improvement in the optical speed (related to the throughput per pixel) may be achieved [18]. When we take a casual look at Figure 5, we observe that the footprint of the triplet may be approximately enveloped with a square. The optical speed (the F/#) of the design in Figure 5 may be estimated to be about 3.5. This is too slow for space surveillance or a night-vision application. One simple remedy might be to increase the dimension along the x-coordinate, defined up through the plane of the paper, denoted as a dot in Figure 5 [19]. For example, if the ratio x:y is increased from about 3:1 to 4:1, then the effective optical speed may be improved to about F/1 or even smaller. Such a change often brings along additional undesirable consequences. Specifically, the sizes of the front protecting window and of the third element become prohibitively large.

However, if we move the optical stop to element one, then the size, and consequently, the cost, of the above three optical elements, together with the entire volume may be reduced. Unfortunately, the moving of the stop results in the introduction of some other higher-order aberrations. They, in turn, may be put under control by the addition of corrective high-order aspherical coefficients on each element.

(ii) The other benefit of increasing the ratio x:y in the aperture dimensions is that a huge field of view (FOV) may be achieved. Consider, for example, that the FOV of the nominal design in Figure 5 might be a square aperture with dimensions 10° × 10°. Then, a change of the square aperture into a rectangular one is possible to the degree that the FOV becomes, 2° × 35°. This is an ideal aperture configuration for a push-broom or scanning architecture [20,21]. We may perform the scan along the x-axis, that is, along the narrow field dimension, placing an inexpensive, linear detector array at the focal plane. By reconfiguring the aperture, we thus replace an expensive 2D focal plane array with an economical alternative. This architecture may be optimized for an inexpensive, high-performance, extremely wide-angle (effectively 65° × 35°) scanning sensor. Some would consider the 65-degree angle infinite in this configuration.

(iii) At the other extreme, we have learned that diffraction-limited performance may be achieved for an RT with an F/3-optical system and a FOV of 3° × 3° when the requirement on the image quality is very high.

(iv) Additionally, in this compact configuration, obviously the management of stray light emerges as a critical issue. For example, when the optical beam in Figure 5 strikes onto the last mirror, the light will scatter directly onto the focal plane unless an inner baffle (Item 40), is placed there to block it. In practical applications, a reasonable length of the front baffle coupled with an inner baffle (Item 40) is required to accomplish this goal.

(v) Finally, a flat image surface may be achieved when the sum of the powers (inverse radii of curvature) of the three mirrors approaches zero [5].

4. Three Mirror Anastigmat (TMA)

The traditional TMA [8,22,23,24] played an important role during the intercontinental star-war era. The term anastigmat literally means that most of the low-order Seidel aberrations, such as spherical aberration, coma, astigmatism, and field curvature, are corrected.

The conversion from the Cooke triplet to the TMA follows pretty much the same process as that of the RT case discussed above. Major differences between these conversions include only two aspects. First, the optical stop for the TMA is located at the first element, while for an RT it is placed at the central element. Consequently, and this is its second feature, the TMA forms an intermediate image between elements 2 and 3. Figure 6 depicts the situation where the optical stop is installed at lens 1 of the Cooke triplet. We note that an intermediate image is formed between lens 2 and lens 3.

The primary element P, a positive one (with the optical stop on it), and the second negative element, S, form an (intermediate) image at point II. These two elements taken together with the focal point represent an excellent example of a Cassegrain objective. Next, we slice off the lower half of the Cassegrain, as discussed above in the RT conversion process. There, we removed the lower half of the afocal Mersenne. Afterward, the remaining process is the same. The letter T represents the positive tertiary element that re-images the intermediate image onto the focal plane.

The equivalent Y-Ȳ diagram is displayed in Figure 7. The generalized ray is incident from the negative Ȳ-axis and, parallel to it on the primary element, similar to the case presented in Figure 2. In this configuration, a stop is placed on the primary element. The stop location at the primary then defines it as the location where the generalized ray crosses the Y-axis. The distance OA-AS is the aperture height, radius, or half of the square aperture side. The primary is convergent, so it deviates the direction of the generalized ray in a clockwise direction. Then, the generalized ray is incident on the secondary component S that deviates its direction of propagation in a counterclockwise direction. The secondary is a divergent element.

Soon after encountering the divergent component, the generalized ray crosses the Ȳ-axis, establishing an intermediate image II there. The size of the intermediate image is OA-II. The generalized ray continues unchanged until it encounters a tertiary component, T, a convergent lens, or a mirror. The point T is located on the negative half plane of the Y-axis; therefore, the convergent component T deviates the generalized ray in a counterclockwise direction, or toward the Ȳ-axis. The ray travels until it crosses the Ȳ-axis, where the image is located, and a focal array might be placed. The size of the final image is OA-I. This image is larger than the intermediate image. If a smaller image is desired, a strong tertiary component would redirect the generalized ray into the region OA-II.

One special feature of the TMA is the feasibility of the stray light control through the internal implementation of a Lyot stop [25,26,27]. One such practical implementation is illustrated in Figure 8. After adding a field stop at the intermediate image plane (Item 18), and the Lyot stop (Item 22) downstream, only the third-order diffraction and/or scattered radiation could reach the focal plane. The amount of such radiation is considered insignificant.

In general, similarly to the RT design case above, the three mirrors here also share the same common optical axis. Likewise, each element is simply a part of the same parent element. The primary and the tertiary mirrors are positioned very close to each other. Due to their proximity, a long front baffle is required for out-of-field stray-light control. A square aperture with a FOV of 2° × 2° is possible for the compact arrangement. A rectangular aperture with a FOV of 0.5° × 6° is also achievable. As suggested above, the optical stop may be placed at the front protection window to reduce cost. However, due to the creation of the internal, intermediate focal plane, the entire volume of the TMA is expected to be larger than that of the RT.

5. RT or TMA Coupling a Beam Splitter

If there is a need to spectrally separate the incoming beam into spectral channels, in either an RT or a TMA optical system, then a beam splitter is inserted between the tertiary mirror and the focal plane, provided that the back focal length is long enough. However, a non-parallel beam (i.e., an optical beam that has a field angle), passing through a plane parallel plate, introduces some lower-order Seidel aberrations, including lateral color [5,6,7,8,9,10].

We (YW) invented a dichroic beam splitter optical element [28] to mitigate the aberrations created by the newly inserted beam splitter. Its position within the optical system is illustrated in Figure 9. The beam splitter in the invention has one plane surface and a phase plate on the second surface. The plane surface, facing the incoming beam, has the required coating to spectrally separate the two spectral bands and reflects half of the incoming beam faithfully, without changing the image quality, into one channel. The phase plate, without the beam-splitting coating, with the basic antireflection coating, features a small wedge and power. Furthermore, some high-order aspheric coefficients are added on top. During the alignment phase, the technician slightly tilts and decenters the beam splitter to remove the newly added lower-order aberrations in the transmitting beam. Additionally, some higher-term aberration terms inherited from the parent RT are also reduced. In other words, the image quality of the transmitting channel is slightly better than that of the reflected one.

A version of an RT, developed for demonstration purposes, is shown in Figure 9. Item 109 is a flat plane mirror, which is used to fold the remaining portion of the optical train along the y-direction, avoiding extra intrusion into the x-direction. Item 111 is the new dichroic beam splitter; Item 121B is the reflected channel (carrying the RT original wavefront error), and Item 121A is the transmitted channel. Its beam quality features reduced wavefront error. By beam splitter design, the transmitted beam quality is better than that in the reflected channel.

6. Offner Relay (OR)

This is the final 3M conversion derived from the Cooke triplet described in this work. However, the OR [29] was initiated as a one-to-one relay in mind, i.e., it is a finite conjugate imaging system (point-to-point imaging) instead of infinite conjugate examples described here up to now.

Again, we start with the Cooke triplet of Figure 1. We assume that both the first and the third elements have the same power, equal to half of the second negative element. Next, the separation between elements 1 and 2 is identical to that between elements 2 and 3 and is equal to the radius of curvature of the second element, when dealing with a mirror system. This being a one-to-one conjugate system, the object distance is equal to the image distance, which in turn is equal to the radius of curvature of the element 1 or 3. Thus, the optical layout of the redefined Cooke triplet is displayed in Figure 10. Clearly, this ingenious finite conjugate imaging system exhibits perfect object- and image-space symmetry. It has a unit power, that is, no magnification.

The Y-Ȳ diagram, corresponding to the Offner relay of Figure 10, is presented in Figure 11. This diagram is similar to the Y-Ȳ diagram of Figure 2, except that the object in this case is at a finite distance. Furthermore, the components and generalized ray on the positive Ȳ-half-plane are symmetrical to those on the negative Ȳ-half-plane. This diagram has more similarity with the Y-Ȳ diagram in Appendix A than to the diagram in Figure 2. It represents a very special case of the diagram in Figure A3.

The imaging system produces an image with a unit magnification of the object in the image location. Furthermore, only a small object is being imaged at the edge of the object field. It is effectively reimaged to the other edge of the object.

The OR is an optical system that inverts the object and image onto itself. Thus, the chief ray Ȳ remains practically unchanged, until the narrow bundle of rays is incident on the (top outer) segment of the primary imaging component. The generalized ray remains practically parallel to the Y-axis because the ray bundle expands insignificantly. The convergent primary component deviates the generalized ray in the clockwise direction toward the small secondary lens or mirror that also functions as the aperture stop.

The secondary lens or mirror is divergent; therefore, it deviates the generalized ray in a counterclockwise direction. Afterward, the graph is symmetrical, because both the primary lens or mirror and the secondary one form a part of the same large component and the system features unit magnification. The object and the image are of the same size, O-OA is equal to OA-I. The power of the primary is equal to the power of the tertiary lens or mirror. OA-AS is the radius or half side of a square aperture.

A working conversion of the optical layout in Figure 10 or Figure 11 into the all-reflective configuration is shown in Figure 12. Both imaging component 1 and component 3 have the same power, the same separation from the second component and share the same optical axis. For this reason, they may be perceived and fabricated as a single component. This new single component and the second component are concentrically surrounding a shared and common focus. Examining either Figure 11 or Figure 12, we realize that OR is both object- and image-space telecentric. Additionally, the front lens CP (gray curved plate with a very weak power) is used to correct some small high-order aberrations.

7. Wide Angle Large Reflective Unobscured System (WALRUS)

None of the three designs (RT, TMA, OR) presented up to now has a large FOV. A 20° × 30° wide-angle unobscured design, for example, is attractive for many applications. One such system is described next. For convenience, we approach it conceptually from a three-lens optical system again, i.e., an inverse-telephoto derivative, shown in Figure 13. This is a negative-positive-positive lens combination. The aperture stop is placed between the first and the second positive lens (corresponding to the second and the third lens).

The pair composed of the first two elements performs as an inverse-telephoto lens. The matching two-mirror counterpart is the classical afocal Schwarzchild objective [30]. This similarity may best be appreciated by examining the Y-Ȳ diagram, except for the location of the optical stop.

The Y-Ȳ diagram, corresponding to the WALRUS of Figure 13, is displayed in Figure 14. The object is located at infinity, so the generalized ray is initially parallel to the Ȳ-axis. At the divergent primary component, the generalized ray is deviated counterclockwise, until it encounters a convergent, secondary one. The secondary imaging element converts the expanding beam into a collimated beam, causing the generalized ray to travel parallel to the Ȳ-axis.

The beam expander configuration/the inverse-telephoto lens transforms the narrow beam in the object space into a wide beam in the intermediary space, between two positive imaging components. After encountering the tertiary convergent element, the generalized ray is directed toward the Ȳ-axis, where the image is found when the generalized ray crosses the Ȳ-axis. This last part is similar to the Offner relay in Figure 11. The generalized ray is thus parallel to the Y-axis until it crosses the Ȳ-axis. A detector may be placed at the image location at Y = 0. Its size is OA-I. The aperture stop radius or half of the square aperture side is equal to OA-AS.

A slow optical system additionally featuring wide FOV results in a large-volume optical system. One easy way to mitigate this disadvantage is to insert a flat mirror [31] at the optical stop (Item 34), giving us the configuration presented in Figure 15. The secondary and the fourth mirror may be united into a single piece to simplify the manufacturing process and to further reduce the overall cost.

As mentioned already for the RT case, the offset angle may be as large as 45° or even larger, leading to a large amount of distortion [8,32]. One simple optical solution is the addition of a thin lens with a low power, placed near the focal plane. The location of a distortion corrector, Item 32, is illustrated in Figure 16 [32]. With modern artificial intelligence technology, a software solution might be implemented as well.

Just like in the development of the above OR case, we may improve the optical speed by increasing the dimension ratio x:y to about 3:1 or 4:1. This results in an effective optical speed of about F/1 or smaller. Currently, high-temperature 2D bolometer detector arrays promise to be soon fabricated by a molding process. With such technological advances in focal array fabrication, an inexpensive high performance extremely wide-angle staring sensor may soon become a reality.

8. Korsch Triplet

Finally, we introduce a creative and spectacular 3M optical design, the Korsch triplet [33], featured in Figure 17. The Korsch triplet is very similar to the TMA, from the geometrical optics point of view, including image quality, and especially its flat field. The difference between them lies in the location of the intermediate image plane. The TMA has the intermediate image between the secondary and the tertiary mirror, while the Korsch triplet places this image between the primary and the secondary mirrors. Also, the Korsch design requires a rotation of the primary from its on-the-parent orientation. This design is extraordinarily compact, with the F/# of each mirror still peculiarly moderate. While we admire the inventor’s creativity, the fabrication of the primary mirror, system alignment, and stray light control remain major concerns before the system may be employed widely.

9. Comments on Optical Design, Fabrication, and Potential Applications

Thorough and complete descriptions of all the designs mentioned here are outside the scope of this work. Only some key issues and concerns in optical design, fabrication, and applications are provided in this section.

First, and this is related to the design fundamentals, we believe that an excellent insight into the first-order geometrical optical layout is required for a good design. The Y-Ȳ diagram is an excellent tool for this purpose, especially when starting a new design completely from scratch. With that in mind, the key features of the Cooke triplet (or the inverse-telephoto derivative) as well as its derivatives ought to be fully understood. For example, symmetry with respect to the optical stop position can correct both coma and distortion simultaneously.

Next, distortion is the inherited nightmare of the off-axis and decentered 3M reflective systems. However, by inserting a weak lens or a phase plate near the focal plane, distortion may be substantially reduced. Lastly, during the final optimization, one can slightly tilt and/or decenter the last imaging element to yield a performance improvement by more than 20%.

One major drawback confronting the designs of the type RT, TMA, or WALRUS is the large volume envelope that contains the entire optical system. Space is always at a premium; the availability of such a large volume is very rare, especially in space and remote sensing applications.

Although OR comprises just two concentric spheres, photolithography, in general, is operated in the ultraviolet (UV) or shorter spectral region. The requirements in both surface figure and surface roughness control are extremely tight. Thus, special attention should be paid to polishing, optical testing, and bidirectional reflectance distribution function (BRDF) measurements to decrease scattering.

RT, TMA, and WALRUS have been applied mostly in the infrared (IR) spectral region where the requirement for both the surface figure and the surface roughness control are less stringent than those for the OR. Based on our experience, the misalignment in terms of the decenter and the tilt of the primary mirror in these three systems is extremely sensitive, resulting in performance degradation. A cascade of two-image forming systems in TMA and WALRUS results in their alignment being twice as challenging as for those of the other configurations.

Recently, we are witnessing significant advances in the fabrication and optical testing of both diffractive optics and emerging freeform optics [34,35]. The free-form optical shapes or optical surfaces are designed with little to no symmetry: they are free from any constraints of symmetry in their form and shape. In the technology advancement area, we are seeing improvements in the maturity of molding glass and non-glass materials.

The technological advances are very encouraging, bringing about more opportunities for the applications of all 3M optical systems and their derivatives. Their utilization will depend to a large degree on the tradeoff between cost and performance. For example, as mentioned above: when the molding technique is of sufficiently high quality and the high-temperature bolometer is commercially affordable, a new generation of inexpensive wide-angle night vision commercial cameras (equipped with either an RT or a WALRUS in a scanning or a staring sensor configuration) may be employed in a terrestrial vehicle (and/or in an unstaffed autonomous vehicle (UAV)), a ship, a building, a parking lot, and an airport during any weather conditions. Those include a clear, foggy, snowy, or stormy night, or for pandemic control.

Finally, for those who are interested in extremely wide-angle optical design, Ref. [36] is highly recommended.

10. Summary

The flexibility and clarity of the Y-Ȳ diagram make it an ideal first-order design tool for optical systems in remote sensing instruments and telescopes. This is especially indicated when stringent performance specifications make it impossible to base the design on an earlier layout.

Acronyms

3M	three mirror
F/#	f-number
FOV	field of view
ODAOS	off-axis and decentered all-reflective optical systems
OR	Offner relay
RT	reflective triplet
TMA	three-mirror anastigmat
WALRUS	wide-angle large reflective unobscured system