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Abstract
A novel, to the best of our knowledge, dual-state reflective optical relay system based on the Alvarez system is proposed, which can be used for remote sensing applications. By keeping the image and pupil positions constant, it can be combined with a telescope to achieve two different magnifications. As a compact structure with only two moving parts, freeform optical mirrors and a nearly diffraction limited performance for the infrared wavelength 8 µm make it an attractive subsystem for space applications. Different design tradeoffs and the preferred layout properties are discussed in detail.
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1. INTRODUCTION
This article presents a dual-state reflective optical system for space remote sensing, based on an adapted Alvarez concept first described in 1964 [1,2]. Instruments that image two different object fields within one optical system are a promising solution to extend the application possibilities of reflective systems. As opposed to classical zoom systems, mirror positions need only be accurate at two discrete locations, not over a range of zoom positions. In view of this, the entire optics must also be optimized only for these two discrete configurations. Fewer compromises are made on imaging quality than with continuous zoom optical systems.
An all reflective dual-channel foveated imaging system based on freeform optics is presented in [3]. This telescope setup consists of three mirrors that can image two different object fields onto two sensors simultaneously. This is accomplished by using two different mirror subareas on the primary mirror. There is a similar concept introduced in the form of a freeform optical system that incorporates multiple mirrors [4]. Two alternative options are described that image two different object fields. Both are based on the implementation of a moveable stop in a three-mirror arrangement. The secondary mirror consists of two independent freeform mirror surfaces with distinct mathematical descriptions. Depending on the stop position, the two different areas are illuminated and imaged on the sensor accordingly. A second variant uses the same concept but utilizes two image planes. It is also possible to combine a Ritchey-Chrétien (RC) telescope configuration with a four-mirror off-axis arrangement to represent two distinct fields of view with two image planes [5]. In this way, two different object fields can be imaged onto two sensors without displacement.
There are also solutions that utilize a three-mirror arrangement with different secondary mirror surfaces [6]. This system incorporates a rotatable secondary mirror with different surfaces on the front and back. In accordance with which mirror side is positioned in the beam path, one of the two fields of view will be imaged.
The disadvantage of these existing systems is that, in order to obtain an image of two fields, either two image sensors or different optical components for the configurations must be used. In this article, a dual field-of-view optical instrument with a single input channel and single image plane, which allows a discrete change of the total focal length through mechanical movement of the same optical elements, is presented.
In order to classify the system described in this article in general, Table 1 summarizes the current state of the art in variable optical systems (refractive and reflective), all of which aim to change focal length through the change of system parameters. They are classified according to their basic principles.
Table 1. State of the Art of Variable Component Systems (Refractive and Reflective)
Fig. 1. (A) Alvarez element in afocal configuration, (B) Alvarez element in focal configuration with displacement $\delta$, and (C) an adapted Alvarez concept for a reflective system (schematically).
Only systems where optical components are moved without leaving the ray path are considered. This means that different sections of the same optical components are used in every configuration. A system that changes its focal length by replacing entire optical components will not be considered because its basic principle is not comparable to the system being discussed. Also, multi-channel systems or systems with multiple image planes and segmented optical systems are excluded.
Four basic principles can be distinguished: The first is the axial displacement of optical components with fixed focal lengths and with predominant movement along the optical axis. This category includes classical mirror or lens zoom systems. The second principle is the lateral movement of components with a fixed focal length. As an example, the Alvarez lenses and the system described in this article belong to this category. The third and final basic principle of using fixed focal length optical components is the rotation of the component (e.g., Moiré lenses). The fourth category comprises solid state components with variable focal lengths. This category includes systems based on liquid lenses or deformable mirrors.
The four categories can be further divided into continuous and discrete modes of operation. Discrete systems have optical components that can be adjusted (e.g., their positions or focal lengths), but the overall system focal lengths are limited to two configurations. In contrast, continuous systems allow the focal length to be continuously adjusted within a defined range. References are only exemplary selections within each category.
Solid state zooms that utilize liquid lenses have major disadvantages for space applications (i.e., the sensitivity to temperature variations, the susceptibility to radiation damage, the potential for reliability issues, and the altered behavior under microgravity conditions). Consequently, they are not further considered.
The system discussed in this article refers to the category of lateral movement of fixed focal length components, which generally permits compact design. As a first step, the idea of transferring the Alvarez approach to a reflective solution is discussed; following that, the system concept and design considerations are discussed.
The original Alvarez concept describes a tandem lens system consisting of two refractive freeform elements shifted perpendicular to the optical axis by an opposite but equal amount [Figs. 1(A) and 1(B)]. Thus, a continuous change in focal length can be achieved, which is proportional to the displacement of the lenses and the sag of the surfaces. The advantage of this concept is that the change in focal length can be achieved with a small lateral shift of the elements. Furthermore, assuming ideal mechanics, the change in focal length can be highly reproducible and resistant to environmental influences [9,21]. Examples for imaging systems based on the Alvarez principle can be found in [21–23]. Within the scope of this paper, no further design methods, application examples, or detailed mathematical descriptions are discussed. Therefore, the references listed above and [8,24] should be consulted.
The approach has now been modified with two mirrors instead of refractive plates [Fig. 1(C)]. By doing so, one major disadvantages of refractive Alvarez systems is avoided: the dispersion of the materials with wavelength [25]. On the other hand, a larger finite distance between the Alvarez components causes a certain dependency between the mirror surfaces. Therefore, the correction is more complex, and freeform surfaces are required accordingly. Also, the reflective assembly cannot be as compact as a lens assembly.
There are two main differences between this realization and the classical Alvarez principle: First, there are only two discrete configurations rather than a continuous variation in focal length. Detailed information about the realization is provided in the following sections. Second, the mirror-based realization results in a significant increase in axial distance and lateral displacement. The original Alvarez principle is only valid for thin lens approximations, respectively, in paraxial approximations [2]. For small displacement ranges, the system behavior can still be described as linear in terms of lower order aberration difference. It is necessary to extend the classical description in order to correct higher order aberrations (e.g., suggested by [24,26]). In addition, a larger displacement range is required for correcting the system accurately, as discussed in the following sections.
As another minor difference, there is no longer a state in which the focal length is infinite as originally described [1,2]. Accordingly, this concept is intended to be used to image from a finite object plane to a finite image plane.
2. SYSTEM CONCEPT
The basic idea of building the reflective dual field-of-view optical system is to incorporate a mirror relay system into a three-mirror-anastigmat (TMA) as schematically shown in Fig. 2. It is intended to switch between the two systems states only within the relay group. The TMA is designed with an intermediate image plane, in which a double reflective freeform subsystem based on the just described concept is integrated. The small beam diameter around the intermediate image allows for a compact subsystem design.
In the following, particular emphasis is laid on the relay group. A correct paraxial framework must be used to design the overall system. This is the reason why a closer examination of the subsystem will be conducted. Based on the initial system design, it is possible to determine the magnitude of the parameter space for the subsystem (see Table 2). The performance of this subsystem has been specifically investigated and optimized, and the results will be incorporated into the overall design of the system. The overall system will be discussed in later publications.
Table 2. Preliminary Specifications of the Overall System
The relay group consists of four freeform mirrors and two plane fold mirrors as beam deflectors. An illustration of the paraxial layout of the system is illustrated in Fig. 3. The magnification can be changed from $m = - {1}$ to $m = - {0.25}$. An overview of the relay system in a simplified design and the light path is shown in Fig. 4.
Fig. 4. (A) Freeform subsystem schematic illustration. (B) Configuration 1. The arrows indicate the direction in which the mirror modules must be displaced in order to reach configuration 2. (C) Configuration 2.
Before going into the arrangement of the mirror surfaces in more detail, it is necessary to describe the basic mode of operation. The freeform surfaces were arranged so that two surfaces form a common mirror substrate (exemplified in Fig. 5). These two mirror substrates are then displaced in opposite directions by the same amount until they reach their final positions. By combining this movement with a position-constant entrance pupil, different areas on the mirror surfaces, later called functional areas, are illuminated. The optical system is optimized for two discrete configurations [Figs. 4(B) and 4(C)].
The first displacement position (configuration 1) arranges the first functional areas of displaceable mirror surfaces (M1-A and M4-A on mirror substrate 1, and correspondingly M2-A and M3-A on substrate 2). During the second displacement position (configuration 2), the second functional areas of the mirror surfaces are arranged in the beam path (M1-B and M4-B on mirror substrate 1, and correspondingly M2-B and M3-B on substrate 2). In each of the functional areas, the surface slope varies, resulting in different optical powers per surface and consequently in two different configurations with different magnifications. The arrangement of four movable mirror surfaces enables compensation of axial displacements and thus constant image positioning. This is an advantage with respect to the relay system, where the distance between the object and the image must be kept constant [27]. When used in conjunction with a telescope, as shown in Fig. 2, the two configurations correspond to two different fields of view. Therefore, this arrangement provides the advantage of maintaining a constant image plane for the entire system. This allows two different scenes to be captured with a single sensor.
As already described, the motion of the mirror surfaces is coupled by a common mirror substrate in each case. A detailed discussion of this topic can be found in [28]. Especially for freeform surfaces that do not have a common axis of rotation, manufacturing on a common mirror body is preferable. The fabrication of the two freeform surfaces can also be combined with a fabrication of mechanical references on the substrate (exemplified in [29]). Due to the machining in one setup, the relative positions of the mirror surfaces are already fixed. The remaining residual deviations resulting from ultra-precision machining are of a very small magnitude. A freeform surface usually requires adjustment of all six rigid body degrees of freedom. If two of these mirror surfaces are arranged on a common substrate, only six degrees of freedom have to be set instead of 12. This simplifies the overall system assembly.
Using this basic approach, a system application is developed in the next section.
3. OPTICAL DESIGN CONSIDERATIONS
In the first step, starting from a paraxial design, the behavior of the optical properties under different displacements of the axis point has been analyzed. Following this, further investigation has been conducted regarding the variation of the object field size and its corresponding numerical aperture. Below are outlined the key optical design considerations, as well as the general procedure applied here.
Systems that have to realize multiple configurations by common optical elements face the difficulty that one object point corresponds to multiple image points. This requires optical elements with non-uniform power distributions across their optical region [13]. As opposed to the classical Alvarez lens surface description, the reflective system uses a freeform description (Zernike-Fringe). A corresponding convergence of the system optimization has been achieved by optimizing and fixing the surface radius first, then increasing the freeform orders of the surfaces. The number of Zernike orders for the freeform surfaces is set to 25. The Zernike’s Z0 to Z3 are excluded as optimization variables. The Zernike term Z4 was limited in the merit function to a maximum value depending on the radius. This is due to the non-orthogonality of the conical basic shape surface description in relation to the Zernike polynomials. In both configurations, the mathematical description of the freeform surfaces is the same. Although this allows for more flexibility in the design, it is more difficult to generalize the principle.
To ensure that the surfaces could be manufactured, several surface parameters were analyzed during optimization, including the freeform part (with sag added to the rotationally symmetrical portion), radial slopes, azimuthal slopes, and machine axis accelerations.
As shown in Fig. 4, the freeform surfaces are arranged off-axis from the system optical axis. The lateral distance between the vertices of the freeform surfaces on the modules is set to 70 mm (see Fig. 5). It remains constant for all analyses described in the following sections. Furthermore, this results in an optical surface diameter of not more than 70 mm as a self-set mechanical boundary condition to prevent overlap between the surfaces. Moreover, this is required in order to maintain compactness.
Due to the simple mechanical implementation, integration, and testing of the modules, the tilt angle of the modules is 45°. When considering two-mirror-relay off-axis systems, it appears that the coma is linearly dependent on the field, as in rotational symmetric systems, but with an additional constant field part [30]. Additionally, astigmatism has a quadratic field dependence as well as a field constant component. The difficulty in correcting such a system is that the aforementioned aberrations, as well as defocus, have opposite dependencies on the tilt angle [30]. A similar behavior can also be seen for this system. For a more detailed discussion of the optical influence of the tilt angle, see [31].
Variable parameters during the optimization process were the mirror diameter (corresponding to the Zernike norm radius), the radius of curvature (whereby all mirrors should always have positive power), the freeform coefficients, and the $y$-distance to avoid obscuration. Additionally, the merit function requires that the axial beam is parallel to the optical axis throughout the system. This simplifies the overall adjustment of the two mirror modules.
For the overall system, it is important to clarify whether the object field or the image field is variable. Regardless of which variant is chosen, two basic relationships must be considered: First, from the relation ${H} = {f}\;{\cdot}\;{\rm \tan}(\omega)$, where $H$ represents the image height, $f$ the focal length, and ${2}\omega$ the angle of the full field of view, it emerges that a constant image height for different field of view is only possible with changing focal length [14]. Second, due to the definition of optical invariant (also referred to as etendue or invariance of throughput [15]) as a constant in paraxial geometrical optics, the numerical aperture must be changed. In this case, the field for the later narrow field-of-view configuration (configuration 1 in the subsystem) is smaller by definition, and the resulting numerical aperture is therefore larger. Thus, the resolution and signal-to-noise ratio is correspondingly better, which is also suitable for the intended application. In contrast, a larger numerical aperture makes configuration correction more difficult. The variable object field and constant image field have the consequence that the numerical aperture is constant at the image side and changes in relation to the focal length on the object side. In this case, a change in focal length on the order of 4 is aimed.
In a preliminary investigation, both options were evaluated. If only differences regarding to optical parameters are considered, it was shown that, if the image field is variable, the distortion of the image plane and the freeform part of the mirror surfaces are larger. From a system-level perspective, the constant image field option is preferable. The mirrors after the freeform optical assembly can be used more effectively to correct the remaining aberrations. For those reasons, it was decided to implement a constant image field with a variable object field.
In general, the concept of this system is not limited to a particular field shape, although the results presented refer to a circular object and image field. Since freeform surfaces are free of symmetries, a full circular field definition with 13 points is used (axis point, four maximum field axis points, and two zonal field points per diagonal).
As freeform surfaces are used, and therefore higher orders are expected, a pure paraxial beam consideration is not applicable. Obtaining reliable results in this case requires consideration of the real marginal rays. Moreover, the system arrangement is likely to cause a macroscopically effective astigmatism, which results in elliptical cross sections of the ray bundle. In order to achieve an isotropic resolution, the numerical aperture referred to the image plane is controlled by the merit function in both the $x$ and $y$ directions.
The output aperture is adapted to the corresponding input aperture, respectively, to the system presented in the following. Predictably, the system will exhibit a large amount of distortion. For this reason, the theoretically exact position of the field points in the image plane was required as a function of the focal length and the angle of incidence in the merit function. Since there is no intermediate image plane within the freeform system, a corresponding image rotation (in this case by $y$) must be considered.
4. DESIGN RESULTS AND PERFORMANCE ANALYSIS
In order to facilitate the comparison of different systems, the spot radius is employed as a reliable and consistent parameter, which is particularly useful when evaluating performance during the initial stages of optimization. An RMS value is calculated as a simplified comparison value for the spots.
A. Performance on Axis
In the first instance, general parameters of the subsystem, including the size of the mirror surfaces, the tilt angle to the optical axis, the displacement, and the resulting performance on the axis, are considered. For the application in space optical systems, an infrared wavelength of 8 µm is considered for the performance evaluation.
Figure 6 shows the dependency of the spot radius on the displacement between the mirror modules. Note that the figure shows eight different optical systems, optimized for the specified displacement. The stop position is not considered. A numerical aperture is first derived from the conceptual setup. The influence of apertures will be discussed in more detail in the following section.
The RMS spot radius decreases with increasing displacement values. The diffraction limit determines the minimum displacement of the mirror modules. Due to the requirement that both configurations have the same image-side numerical aperture, the diffraction limit for both configurations are 48.8 µm. As long as both configurations are within this limit, the displacement path is gradually reduced. The maximum displacement is defined as the footprints being completely separated on each mirror surface. In essence, this analysis focuses on finding the compromise between optical imaging quality and system compactness. The investigation of other systems beyond the 11 mm shift was therefore not conducted.
Figure 7 shows the relation between the displacement of the mirror modules and the footprints on the mirror. In addition, the minimum required mirror diameter is shown, which does not yet represent manufacturing-related enlarged edge areas. A larger diameter may be needed, for example, due to edge effects resulting from vibrating diamond turning tools or polishing processes.
Figure 7 shows that the separation of footprints on all mirror surfaces increases with increasing displacement $\delta$ of the mirror modules. Based on Fig. 7, it can be concluded that better separation is associated with lower RMS spot radii. This is because they can be corrected separately from each other.
The figure also shows that the non-circular shape of the footprints and their separation between the configurations increases with increasing displacement. This can be seen particularly well on the M3 and M4 mirrors, resulting in an increase in mirror size.
The results from performance on axis examinations are only initial indications, and the next step is to consider field points. Thus, the system with a displacement of 11 mm is used as the preferred variant, and the investigations on the image field are subsequently carried out on this system.
B. Performance with Field
To study the system, both the field angle at constant numerical aperture and the numerical aperture at constant field angle were increased. The parameter space for both variables is determined by the size of the intermediate image from the conceptual system design (see Table 2 and Fig. 2).
Further consideration should be given to the position of the stop in the system since the stop determines the diameter of the mirror surfaces and correction opportunities. In general, the telescope’s pupil and the relay’s pupil must match. A constant stop position is essential for the relay system. Considering that this requirement is more difficult to meet for the subsystem, the stop’s position is analyzed, optimized, and fixed for the subsystem. This results in a remote pupil from the telescope’s perspective, which is mostly not a severe problem. Due to the fact that the mirrors M1 and M2 as well as M3 and M4 are along the direction of the optical axis geometrically close to each other, the stop positions at M2 and M3 are not considered separately. Only the stop positions on M1, symmetrically in the system, and on M3 are discussed in detail.
The dependence of the RMS spot radius on the object field angle and the numerical aperture is shown in Figs. 8 and 9. The results are based on a displacement of 11 mm. Also shown is the Airy radius as representation for the diffraction limit of the application wavelength of 8 µm. Due to the relationship between numerical aperture and Airy radius, the Airy radius decreases with increasing numerical aperture.
Fig. 8. Comparison of RMS spot radii as a function of field angle and numerical aperture. The subsystem stop is located on M1. The displacement $\delta$ of the modules is 11 mm. The subsystem was re-optimized for each field or numerical aperture.
Fig. 9. Comparison of RMS spot radii as a function of field angle and numerical aperture. The stop is located symmetrically within the subsystem (between the fold mirrors). The displacement δ of the modules is 11 mm. The subsystem was re-optimized for each field or numerical aperture.
It is apparent from Fig. 8 that the RMS spot radius increases with an increase in the numerical aperture or field angle. A smaller numerical aperture will result in a smaller angle of entry into the system and consequently a smaller height of the chief ray. Both configurations perform about the same when manufacturing and position tolerances are not considered. It can be seen that the performance toward the zonal and maximum field points degrades with increasing maximum field angles. Configuration 2 shows the effect more clearly. For the application, this means that the resolution degrades toward the field edges. Figure 9 shows the results of the same examination for the stop position symmetrically in the system.
Compared to the stop position M1, the RMS spot radii for configuration 1 are slightly worse, and the azimuthal separation has increased. The values for configuration 2 are comparable in terms of the RMS spot radius and separation. But in the diagram, an area is marked for which no solution could be found in the specified range. This refers mainly to the geometric specification for the maximum mirror diameter mentioned at the beginning of this section. It can also be seen that the resulting spot radii and the degradation of performance toward the zonal and maximum field points scales are larger than for pupil position M1. For explanation, the footprints of the different pupil positions for a given image field at a given numerical aperture are shown in Fig. 10.
Fig. 10. Comparison of footprints for different mirror surfaces according to the location of the stop.
Coma, astigmatism, and distortion are chief ray aberrations and are therefore mainly influenced by the stop position. It is possible to correct constant field aberrations simultaneously if the stop is located on a mirror surface (such as M1 or M3). The other surfaces are only capable of correcting field-dependent aberrations. Regarding the fact that the coma part of the surfaces after the stop should be as low as possible due to the influence of coma correction on the image plane tilt, it seems more appropriate to locate the stop on a surface. If the stop is at the first mirror, the aperture in object space is constant; if the stop is located at the last mirror, the image space is constant [27]. Both variants are possible, but the definition of a constant image space should be preferred based on the previous definitions.
The footprints in Fig. 10 refer to a numerical aperture of 0.06 for configuration 1 and 0.015 for configuration 2. With correspondingly larger numerical apertures, the footprints between the configurations would be closer together.
In general, a stop positioned on mirrors M1 or M3 results in a low separation of footprints on the directly following surfaces as well. Considering the stop on M1, both configurations show relatively uniform footprints, but configuration 2 shows a greater spread of field points, which may allow a better field correction. The other case can be seen for the system in which the stop is located symmetrically. There is greater separation between the footprints of configuration 1, but the footprints are less uniform as well. In addition, it can be seen that a symmetrical or quasi-symmetrical system structure does not appear and therefore no compensation effect is achieved through equal chief ray heights. On the contrary, for the same numerical aperture with an assigned field, the chief ray heights are almost equal. Tolerancing-wise, a separation of the footprints is disadvantageous. There is a direct relationship between the degree of separation and the sensitivity of the optical system. Detailed discussions of the tolerancing of the system will be provided in future publications.
In the case of further increasing field sizes, a new adjustment of the displacement would therefore be necessary in order to improve imaging performance. It should also be noted that this is accompanied by an increase in the diameter of the mirror, which is already very close to the 70 mm limit.
5. SELECTION OF THE PREFERRED VARIANT
The preferred variant is the system with the stop position on M1 and the data summarized below. The selected variant represents a good balance between the quality of the optical features (related to the subsystem), the integration of the subsystem into the overall system, and the mechanical feasibility. Furthermore, when selecting the variant, care must be taken to ensure that the angle of entry into the system can be achieved through the primary TMA mirror. Therefore, it is not recommended to select a too small angle. Table 3 shows the key specifications of the preferred variant.
Table 3. Key Specifications of the Preferred Variant
Following the basic optimization and attainment of the basic requirements, the performance criterion can be changed to the RMS wavefront value. In contrast to spot radius, the system wavefront can provide a holistic assessment of aberrations in the complete system with the telescope parts and can be measured directly. In addition, the phase of the wavefront is additive to the overall system, enabling a more comprehensive evaluation of the system.
In Fig. 11, the RMS field map of the wavefront is shown for both configurations of the selected system. The black circles represent the zonal and maximum field points. The Maréchal criterion ($0.714 \cdot \lambda$), which serves as a general criterion for diffraction-limited systems [32], is used to evaluate the results. In Fig. 11, the Maréchal criterion is depicted as a contour line. First, the non-uniform field dependence on the RMS wavefront can be seen in both configurations. In configuration 1, the entire specified field is below the Maréchal criterion, making it diffraction limited. Since this configuration corresponds to a smaller field of view with a higher resolution in the later system application, it is advantageous in this regard. In contrast, configuration 2 exceeds the Maréchal criterion for the outer field points. Since this configuration 2 is designed for an overview of a scene with a larger field of view, lower resolution is acceptable.
The next step will be to integrate the subsystem with the introduced parameters into the overall system and to re-optimize the parameters. This will be covered in future publications.
6. CONCLUSION
A novel dual-state reflective optical relay system based on the Alvarez system was proposed, which can be used for remote sensing applications. It can be switched between two configurations with different magnifications ${m} = - {1}$ and ${m} = - {0.25}$, respectively, by moving two mirror modules consisting of two freeform surfaces each. It has been shown that for this type of system in particular, the challenge lies in balancing compactness (in terms of mirror diameters and travel distances) against achievable optical imaging quality. It is likely that a further increase in the Zernike orders (e.g., up to 36) can lead to a further improvement in imaging quality or compactness. The investigation refers to the general concept. Although there may be a loss of target tracking and information depending on the switching time, this system is a promising alternative to classic zoom systems for space systems. Advantages are given by compact simplified mechanics and optimized image quality. The design outcomes presented are based on ideal mechanical and optical conditions. Future publications will address tolerancing, as-built performance analysis, and manufacturing considerations. With the presented investigations of a reflective freeform system, based on the basic idea of the Alvarez lens system, a basis has been laid for successful implementation in an overall optical system. Further research and development of dual field-of-view systems has the potential to lead to improved optical instruments with various applications in science and engineering.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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