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Abstract
For off-axis reflective systems, there is no effective method that can obtain a small-volume starting point automatically. Reducing system volume using the optimization method is usually a difficult and cumbersome process, which usually takes the designer a long time. This paper proposes a method of designing small-volume starting points of freeform off-axis reflective imaging systems, which does not require human involvement after inputting a planar system, specifications and constraints. In the design example presented in this paper, it took only about ten minutes to obtain a small-volume starting point. The starting point obtained by this method makes the optimization process easier, takes less time of designers and more likely to succeed. In the design example, it is also demonstrated that the method has the ability of establishing small-volume starting points with different specifications, which can be used to quickly estimate the minimum volumes that systems with different specifications can achieve and provide guidance for determination of specifications and optimization.
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1. Introduction
Off-axis reflective optical systems have the advantages of no chromatic aberration, low weight, no central obscuration, and less absorption of light energy and are thus of interest to many researchers of optical design [1–7]. Under the premise that the system specifications are the same, the volumes of off-axis systems are generally larger than those of coaxial systems. It is therefore important to study methods of designing small-volume off-axis reflective systems. Applying the freeform surface to the off-axis optical system not only corrects the asymmetric aberrations existing in the off-axis system but also improves the imaging quality of the optical system, reduces the number of elements required by the system, and reduces the system volume [8–10]. Freeform surfaces have been applied to illumination systems [11–14] and beam shaping [15,16] in non-imaging fields and to spectrometers [17–19], telescopes [20–22], helmet-mounted displays [23–25], and mobile phone lenses in the imaging field.
Generally, optical designers prefer off-axis reflective optical systems with smaller volumes, which have compact structures, smaller component sizes, and low weight. In traditional design process, the designer usually first find a starting point and then reduce its volume through optimization process with the help of the optical design software. The starting point can be obtained by searching the patent library, solving the coaxial system and creating the off-axis form, adopting aberration theory [26], and using direct design methods. Direct design methods include the partial differential equations method [27–29], the simultaneous multiple-surface method [30–32], and the construction and iteration method [33–35]. However, there is no design method for starting points of off-axis reflective systems that address compressing the volume of system automatically. Although the system volume can be reduced using the traditional optimization method including the volume constraints, it is a trial-and-error process which requires the participation of designers and usually takes designers a long time for the following reasons. (1) Optimization methods usually use linear approximation to deal with complex nonlinear optimization problems, and thus the optimization step size cannot be too large. Otherwise, the optimization algorithm may not be able to find a solution that satisfies the current constraints. During the optimization process, to reduce the system volume, the designer needs to adjust specific constraints step by step according to the current situation of the system and design experience. (2) When reducing the system volume using the optimization method, the current system may fall into local optimal solution. Jumping over the trap of local optimal solution not only demands high for the design experience and optimization skills of designers but also take up a long time of designers. If the volume of the starting point is much larger than the expectation, the system may fall into local optimal solution many times, and the designer may fail to find a satisfying design result. (3) Since the existing patents of reflective off-axis optical systems are limited, it is difficult for designers to estimate the minimum volume for the given system specifications. If the expected system volume is too small, a useful solution may cannot be found. Therefore, the design of a small-volume system is a blind process, and the designer may need to make multiple attempts to obtain a small-volume design result.
The starting point can have a significant influence on the final design results. Obtaining a good starting point is beneficial for reducing the time that the optimization process takes and increasing the possibility of obtaining a more satisfying design result, which is a very important step. Therefore, this paper proposes a method of designing the starting points of small-volume off-axis reflective imaging systems automatically. After inputting specifications, constraints and a planar system without light obscuration, a starting point of small volume is obtained without the participation and guidance of the designer. Unlike the traditional design method that uses volume constraints and search for a large number of variable parameters, the method proposed neither uses volume constraints nor search variable parameters. Instead, the positions and shapes of reflective surfaces of the system are obtained by solution. The designs of off-axis three-mirror systems are taken as examples to demonstrate the effectiveness of the method proposed. In the Section 3.1, starting from a planar system with a relatively large volume, a small-volume starting point is automatically obtained, which takes about 10 minutes. Since the volume and focal length of the starting point basically meet the requirements, its optimization is easier and takes less time of the designer. In Section 3.2, the proposed method is used to the design of small-volume starting points with three kinds of system specifications.
Compared with other design methods of starting points, the proposed method has the following advantages. (1) After a planar system, specifications and constraints are inputted, this method can establish small-volume starting points of off-axis reflective systems automatically, which does not require the participation of designers and has versatility. It is beneficial for reducing the time of the designer taken up by the optical design process, reducing requirements of design experience of designers, reducing the difficulty of the optimization process, and improving the possibility of obtaining a satisfying result. (2) This method allows the designer to have a rough estimate of the minimum system volume that can be achieved with the given specifications, and provides guidance for optimization. Since the patents of off-axis reflective systems are limited, it is difficult for designers to estimate the minimum volume that can be achieved with the given specifications through traditional design methods. This will lead to blindness in the optimization process. The method can be used to roughly estimate the minimum volume that the given specifications can reach, which is beneficial for reducing blindness of the optimization process. (3) This method can provide designers with a series of small-volume starting points with different specifications. It can be used to quickly and roughly estimate the minimum volumes that systems with different specifications can achieve and provide guidance for the determination of specifications at the beginning of optical design, which is very important and meaningful for optical design.
2. Methods of designing small-volume starting points
The design of an off-axis three-mirror reflective system without obscuration is taken as an example to introduce the method of designing a small-volume starting point. As shown in Fig. 1, the distances between the mirror ends and their closest marginal rays, which are referred to as constraint distances L1–L5 in the paper, will affect the volume of the off-axis reflective system. The beam overlap areas of mirrors, which is introduced in Section 2.1, will also affect the volume of the off-axis reflective system. Therefore, in the method proposed, the system volume is compressed by the cooperation of the following two steps. The first step is to reduce constraint distances L1–L5 of the initial planar system to the minimum values that can be accepted by the designer, which achieves a preliminary compression of the volume. In the second step, the system volume is further compressed by keeping constraint distances L1–L5 unchanged and reducing the beam overlap areas of mirrors. During the design process, keeping constraint distances L1–L5 unchanged and reducing the beam overlap areas of mirrors is a key step, which is introduced in Section 2.1. The entire design process is introduced in Section 2.2.
2.1 Method of reducing beam overlap areas of mirrors
As shown in Fig. 2, the beam overlap area of a mirror is defined as the overlap area of the reflected rays and incident rays of a mirror in the meridian plane, which is a two-dimensional area. As shown in Fig. 2, in the meridian plane, the lower rim-ray striking the mirror is R1, and the upper-rim ray emitted from the mirror is R2. The beam overlap area of the mirror is an irregular figure surrounded by ray R1, ray R2 and the contour of the mirror in the meridian plane. This section describes how to compress the system volume by keeping constraint distances L1–L5 unchanged and reducing the beam overlap areas of mirrors.
It is introduced how to reduce beam overlap area, taking freeform system C1 as starting point. During this process, the position of the secondary mirror remains unchanged, and the system remains symmetric about the meridian plane. The system C1 is symmetric about the meridian plane and has no light obscuration, and the secondary mirror is the aperture stop, as shown in Fig. 3(a). It has a relatively compact structure. For system C1, the distances between the endpoints of mirrors and their closest marginal rays are the shortest distances that can be accepted by the designer. Let the minimum acceptable value of constraint distance Li be li (i = 1, 2, …, 5). The establishment of such a system will be introduced in Section 2.2. Taking system C1 as the starting point, a system with smaller volume is obtained through reducing beam overlap areas of the secondary, primary and tertiary mirrors in turn. The specific process is as follows.
Fig. 3. Process of reducing the system volume using the proposed method of reducing beam overlap areas of mirrors: (a).initial freeform system, (b) reduction of beam overlap area of the secondary mirror, (c) reduction of beam overlap area of the primary mirror, and (d) reduction of beam overlap area of the tertiary mirror.
The center of the secondary mirror is the geometric center of the secondary mirror, which locates at the midpoint of the line connecting the upper and lower endpoints of the secondary mirror. To reduce the beam overlap area of the secondary mirror, the secondary mirror is rotated clockwise by α1 around its center. The tertiary mirror and image plane are rotated clockwise by 2α1 around the center of the secondary mirror. The obtained system is C1’, as shown in Fig. 3(b).
The beam overlap area of the primary mirror is then reduced according to the requirement of eliminating light obscuration and compressing system volume. The position of the primary mirror and its incident ray directions are changed to make the distance between the low end of the primary mirror and the upper-rim ray exiting the secondary mirror is l1 and the distance between the upper end of the secondary mirror and the lower-rim ray striking the primary mirror is l2. Adopting this method, system C1'’ is obtained, as shown in Fig. 3(c).
Next, it is introduced how to reduce the beam overlap area of the tertiary mirror. The position of point A is solved, which is located on the upper-rim ray emitted from the secondary mirror and is at a distance l3 from the lower-rim ray emitted from the primary mirror. The point A is expected to be the upper end point of the new tertiary mirror. According to the requirement of eliminating light obscuration and compressing system volume, the position of the upper end point of the new image plane, which is point B, is calculated. The position of the image plane is determined after selecting the tilt of the image plane. After recalculating the shape of the tertiary mirror, system C2 is obtained, as shown in Fig. 3(d).
Using the proposed method of reducing beam overlap areas of mirrors, a system C2 having smaller volume is obtained, which takes the system C1 as the starting point. For system C2, the distances between the endpoints of mirrors and their closest marginal rays are also the shortest distances that can be accepted by the designer (Li = li, i = 1, 2, …, 5), which is same as the system C1.
If the constraint distances of the system C1 are greater than the minimum acceptable value (Li>li), a more compact system C2, of which the constraint distances L1–L5 are the minimum acceptable values (Li = li, i = 1, 2, …, 5), can also be obtained using the method of calculating the surface positions proposed in this section. That is to say, the method of calculating surface positions proposed in this section can also be used to reduce the constraint distances L1–L5 of a system and establish an off-axis reflective system with smaller volume. It will be used in the process of compressing the volume of the planar system in Section 2.2. However, in this case, since the constraint distances L1–L5 change, the beam overlap areas of mirrors are not necessarily reduced from the process of C1 to C2.
2.2 Design process of the small-volume starting point
This section describes how to obtain a small-volume starting point, which takes an initial planar system PPP0 established by the designer, specifications and constraints as input.
To facilitate understanding, the combination of the surface types of the primary mirror, secondary mirror, and tertiary mirror of the system is used in naming the systems established in the process. As an example, “F” indicates that the reflective surface is a freeform surface whereas “P” indicates that the reflective surface is a plane. “FPF” means that the secondary mirror of the system is a plane and the remaining mirrors are freeform surfaces. In this process, the root mean square σRMS of the distances between the ideal and actual intersections of the sample rays and the image plane is used to evaluate the imaging quality of the system, as expressed in Eq. (1), where δm is the distance between the ideal and actual intersections of the m-th sample ray and the image plane.
The first step is to reduce the system volume by reducing distances between the mirror ends and their closest marginal rays, which is L1–L5 in Fig. 4, of the initial planar system PPP0. During the process, light obscuration need to be avoided. For off-axis reflective systems, light obscuration can be avoided by controlling the distances between the mirror ends and their closest marginal rays [35–37]. These distances can also influence the system volume. Using the method of calculating positions of surfaces proposed in Section 2.1, a planar system PPP1, of which the constraint distances L1–L5 are the minimum values, is obtained. The aim of this step is to compress the system volume by reducing constraint distances Li to its minimum value li (i = 1,…,5), instead of reducing beam overlap areas of mirrors. Taking specifications, constraints and a planar system without light obscuration as the input, a compact planar system is established automatically and quickly using this method. Therefore, our method has no requirement for the compactness of the initial planar system PPP0. There is no need for designers to establish an initial planar system with compact structure through multiple attempts.
After reducing distances L1–L5 to the minimum values, the volume of the system can still be further compressed. The second step of process of compressing the system volume is then introduced. The system PPP1 is then taken as the starting point and a freeform system FPF1 is obtained using the construction and iteration method [34]. Taking the system FPFi as the starting point, system FPFi+1 with smaller volume is obtained using the method of reducing beam overlap areas of mirrors (i = 1, 2, 3,…) proposed in Section 2.1. The process from FPFi to FPFi+1 can be regarded as once iteration. As the number of iterations increases, the volume of the system gradually decreases, and the σRMS of the system generally increases gradually. If the σRMS of a system is too large, its imaging quality will be very poor, and such a system cannot serve as a good small-volume starting point. Therefore, if the σRMS of the system FPFi+1 exceeds the maximum value σMAX given by the designer, the iterative process stops. Since the σRMS of the system FPFi+1 does not meet the requirement for imaging quality of the starting point, the system FPFi+1 is abandoned, and system FPFi obtained in the last iteration is taken as the design result of the system compression process.
Using the method in Section 2.1, the surface positions of the system FPFi can be calculated. The shapes of the tertiary and primary mirrors are recalculated using the following strategy proposed. First, the positions and shapes of the primary and secondary mirrors are kept unchanged, and the tertiary mirror is regenerated using the construction and iteration method. The primary mirror and tertiary mirror are then regenerated for several times using the construction and iteration method. This strategy is used to avoid the aberrations that need to be corrected by the primary mirror are too large. The recalculation of the shape of the primary mirror will influence the directions of incident rays on the secondary mirror and the tertiary mirror, while the recalculation of shape of the tertiary mirror does not affect the rays striking the primary and secondary mirrors. If the aberrations that need to be corrected by the primary mirror are too large, the fitting accuracy of the primary mirror may be very low, which will cause the actual intersections of the rays and the image plane to deviate greatly from the corresponding ideal image points. This strategy is conducive to improving the tolerance of the method to a single reduction of beam overlap area of the primary mirror.
The flowchart of establishing a small-volume starting point is shown in Fig. 5. After inputting initial planar system PPP0, specifications and constraints (σMAX and l1–l5), a small-volume starting point is automatically obtained.
In traditional optical design process, the system volume is reduced during the optimization process. The method proposed aims to find a small-volume starting point before the optimization begins. The differences between the traditional system volume reduction method that includes volume constraints in a ray trace optimization routine and the design method proposed in this paper are as follows. (1) The traditional optimization method requires the participation of designers, demands high for their design experience, and usually takes them a long time. The proposed method almost does not require the participation of the designer, reduces the time that the designer need to invest in the optical design process, reduces the requirements of the design experience and the difficulty of optimization, and improves the possibility of successful optimization. (2)Traditional optimization methods gradually compress the system volume by establishing evaluation functions and constraints, and searching for variable parameters. This process is a complex nonlinear optimization process, and it is also a trial and error process. In the proposed method, the compression of the system volume is achieved by the proposed method of reducing beam overlap areas of mirrors. The position and shape of each surface of the system are obtained by solving. During the process, there is no need to search for variable parameters, nor does it need to calculate the gradient of the evaluation function. (3) Traditional optimization methods require designers to establish complex constraints, and adjust these constraints step by step according to the current state of the system and design experience. If the constraints are changed too drastically, the system is likely to crash. This results in the traditional optimization process generally taking up a long time of the designer. The proposed method has relatively simple constraints. In the process of designing a small-volume system, these constraints remain unchanged, and do not need the adjustment of the designer.
3. Design results
This section takes the design of off-axis three-mirror reflective systems as examples to demonstrate the efficiency of the proposed design method and its ability of establishing small-volume starting points with different system specifications. In the design example, the type of each freeform surface is the XY polynomial.
3.1 Design of a small-volume system
The design of a starting point of an off-axis three-mirror reflective system is taken as an example to show the process of designing a small-volume system. The system specifications are shown in Table 1. After inputting specifications, constraints and a planar system without light obscuration, a small-volume starting point is obtained without the participation and guidance of the designer, which takes about 10 minutes. The computer we used has 256GB of RAM, operating at 2.10 GHz. During the design process, 324 sample rays are used to calculate the σRMS of the system, and the value of σMAX is chosen as 0.7 millimeters.
Table 1. System specifications
A planar system PPP0 without light obscuration is established. Its optical path diagram is shown in Fig. 6. Using the method described in Section 2.1, the planar system PPP1 having smaller volume is established. As shown in Fig. 6, the structure of system PPP1 is more compact than that of system PPP0. The first step of the system volume compression process is then completed.
Taking planar system PPP1 as the starting point and using the method described in Section 2.2, system FPF1, of which the primary and tertiary mirrors are freeform surfaces and the secondary mirror is a plane, is established. To reduce the beam overlap area of the secondary mirror, the secondary mirror of system FPF1 is rotated clockwise around the center of the secondary mirror by 3° and the tertiary mirror and image plane are rotated clockwise by 6° around the center of the secondary mirror. The beam overlap areas of the primary and tertiary mirrors are then reduced using the method described in Section 2.1. Using the method in Section 2.2, a system FPF2 is then obtained, of which the primary mirror and the tertiary mirrors are freeform surfaces and the secondary mirror is a plane. The volume of system FPF2 is 67% of system FPF1. Then, taking system FPFm as a starting point, system FPFm+1 is obtained using the same method. After seven iterations, the σRMS of the system FPF7 exceeds σMAX. Then, the iteration process stops, and system FPF7 is abandoned. The second step of the system volume compression process is then completed.
The optical path diagrams of systems FPF1–FPF6 are shown in Fig. 6. In the figure, for the convenience of comparison, all surfaces and the rays of each system are rotated by a given angle along the selected direction so that the secondary mirror of the system is perpendicular to the horizontal direction.
For the convenience of calculation, approximation processing is used when calculating the beam overlap area of mirrors. The area enclosed by the line connecting the upper and lower endpoints of the reflective surface and the contour of the reflective surface in the meridian plane is ignored. The beam overlap areas of the mirrors of systems FPF1–FPF6 are shown in Fig. 7(a). The abscissas give the system numbers, and the ordinates give the beam overlap areas of the primary, secondary and tertiary mirror of system FPFi (i = 1, 2, …, 6). As the system number increases, the beam overlap areas of the primary, secondary and tertiary mirror are all decreases. The system sizes in x direction of systems FPF1–FPF6 are shown in Fig. 7(b). The dimension in the x-direction first decreases and then and then increases slightly. The volume of the smallest cuboid that can contain reflective surfaces and image planes is taken as the volume of the system. The volumes of systems FPF1–FPF6 are shown in Fig. 7(c). The abscissas give the system numbers, and the ordinates give the system volumes. As the system number increases, the volume of the system decreases. The volume of system FPF6 is 14% of system FPF1. The maximum relative distortion of systems FPF1–FPF6 are 2.7%, 6.2%, 4.8%, 5.8%, 6.5% and 7.3%, respectively.
Fig. 7. Changes in beam overlap areas of mirrors, system size in x direction and system volume during the process of designing the small-volume starting point: (a) beam overlap areas of the mirrors of systems FPF1–FPF6 that are obtained in the design process, (b) system size in x direction of systems FPF1–FPF6, and (c) volumes of systems FPF1–FPF6.
In this way, taking the specification, constraints, and initial planar system PPP0 with a volume of 3.871 liters as input and using the method proposed, the freeform system FPF6 with a volume of 0.237 liters is automatically obtained. The volume of system FPF6 is 6.1% of the input planar system PPP0. As long as the designer invests a lot of time in reducing the size of the system during the optimization process, it is also possible to obtain a system with the same volume as system FPF6. However, the traditional optimization method requires the designer to participate in tedious optimization processes to gradually compress the system volume, which may takes several hours or even several days. Therefore, the method proposed is helpful for effectively reducing the time that the optical design process takes.
System FPF6 is a good starting point of the optimization process. The spot diagram of system FPF6 is shown in Fig. 8(a), and the RMS wavefront error of system FPF6 is shown in Fig. 8(b). The entrance pupil diameter and field of view of system FPF6 are constant with the expectations. Its focal length and volume basically meet the design requirements.
Fig. 8. Imaging quality of the system FPF6 automatically obtained using the proposed method: (a) Spot diagram of system FPF6, and (b) RMS wavefront error of system FPF6.
System FPF6 is optimized using the optical design software Code V. During the optimization process, the secondary mirror evolves from a plane to a freeform surface. Volume constraints are used to limit the volume of the optimized system, and obscuration is avoided by controlling the distances between the mirror ends and their closest marginal rays. At the beginning of optimization process, the constraints are relatively looser. As the optimization processes proceeds, the constraints are gradually tighten. The optical path diagram, RMS wavefront error, and Spot diagram of the optimized result FFF6(opt) is shown in Fig. 9. For this system, its volume is 0.232 liters, and the average values of the RMS wavefront error of the system is 0.0447λ (where λ = 10µm).
Since there was no need to spend a long time on reducing the system volume through the progressive optimization process, the optimization process was easier and took less time. Besides, this method enables the designer to have a rough estimate of the minimum volume that the system with given system specifications can achieve, provide guidance for optimization, and reduce blindness in the optimization process. Therefore, taking the starting point obtained by the method proposed, a small-volume design result with good imaging quality is obtained efficiently.
Although designers generally prefer design results with small volumes, compressing the volume of the system may increase the manufacturing difficulty and decrease the imaging quality. Using this method, a series of compact design results with different volumes can be obtained. The most suitable one can be selected from them and taken as the starting point for optimization.
Next, it is discussed that how to choose appropriate values for α1, σMAX, li (i = 1,…,5). If α1 is too large, the imaging quality is likely to be poor after rotating the secondary mirror and recalculating the positions and shapes of the primary and tertiary mirrors. If α1 is too small, the process of establishing a small-volume starting point will take a long time. According to our design experience, it is better to pick a value for α1 between 2°–5°. σMAX is given by the designer based on the expectations of the starting point. If the σMAX selected by the designer is larger, then the small-volume starting point obtained by the proposed method generally has smaller volume and poorer imaging quality. If the σMAX selected by the designer is smaller, then the small-volume starting point obtained generally has larger volume and better imaging quality. The designer can balance the imaging quality and volume of the starting point by choosing a reasonable σMAX. Since the method is proposed for starting point, there is no need to evaluate the imaging quality of the system accurately. It is only need to roughly select an appropriate value for σMAX. li is the acceptable minimum value of the constraint distance Li (i = 1,…,5), which is mainly determined by the manufacturing technique. According to the design requirements and manufacturing technique, a set of suitable values for distances li (i = 1,…,5) can be selected.
3.2 Discussion of system volumes with different specifications
The method proposed in this paper can also be used to roughly estimate minimum volumes that systems with different specifications can achieve, and help the designer to determine the system specifications at the beginning of the optical design, which is useful and significant.
Next, systems with three kinds of system specifications are designed using the method proposed. These three kinds of system specifications are shown in Table 2. For ease of understanding, “FPFi(s)” is used to denote the systems established in this section. Among them, the superscript “s” denotes the number of the system specifications in Table 2, and the subscript “i” represents that the system is the i-th freeform system obtained during the design process of the small-volume starting points with the s-th kind of system specifications through the proposed method.
Table 2. Three kinds of system specifications
First, a small-volume starting point with the first kind of system specifications is designed. The initial planar system PPP0(1) has the same surface position and incident light direction as system PPP0. The optical path diagram of system PPP0 is shown in Fig. 6. The entrance pupil diameter of system PPP0(1) is 50 mm. Taking this planar system as a starting point, a series of freeform systems are established using the same method in Section 3.1, which are FPF1(1)–FPF6(1). The small-volume starting point with the first kind of system specifications is system FPF6(1).
A small-volume starting point with the second kind of system specifications is then designed. The initial planar system PPP0(2) has the same surface positions and incident light direction as system PPP0, and its entrance pupil diameter is 60 mm. Using the same method in Section 3.1, freeform systems FPF1(2)–FPF6(2) are established. The small-volume starting point with the second kind of system specifications is the system FPF6(2).
Then, a small-volume starting point with the third kind of system specifications is designed. The initial planar system PPP0(3) is the same as the system PPP0(1) except for the field of view. Adopting the same method, freeform systems FPF1(3)–FPF6(3) are established. The small-volume starting point with the third kind of system specifications is the system FPF6(3).
The volumes of these freeform systems FPFi(s) (s = 1,2,3; i = 1,2,…,6) are shown in Fig. 10. Among them, “s” represents the number of system specifications in Table 2, “D” represents the entrance pupil diameter, and “FOV” represents the field of view, “f” represents the focal length. The entrance pupil diameter of the system FPF6(2) is 1.2 times that of the system FPF6(1), and the volume of the system FPF6(2) is 1.76 times that of the system FPF6(1). The field of view in the meridian plane of the system FPF6(3) is 1.5 times that of the system FPF6(1), and the volume of the system FPF6(3) is 1.38 times that of the system FPF6(1).
4. Conclusions
Traditional methods of designing the starting points of off-axis reflective optical systems have not considered how to compress the system volume automatically. The system volume is generally reduced in the optimization process. However, if the volume of the starting point is too large, the designer may cannot find a design result meeting design requirements. Even if a good design result is obtained through the optimization process, it usually takes the designer a long time since the optical system may fall into the local optimal solution many times or even crash. These problems need to be handled by designers according to their design experience, which demands high for their optimization skills. Therefore, obtaining a good starting point is significant. This paper proposes a method of designing starting points of small-volume off-axis reflective imaging systems. Unlike the traditional optimization method that uses volume constraints and searching for a large number of variable parameters, this method almost does not require the participation and guidance of the designer, nor does it need to search for variable parameters, and thus saves the time of designers and improves the design efficiency. The method only needs the designer to input system specifications, constraints and a planar system without light obscuration, and a small-volume starting point is then automatically obtained. Since the starting point is closer to the desired design result, its optimization is more likely to be successful and takes up less time of the designer. In addition to the ability of obtaining a small-volume starting point automatically, this method also can be used to quickly and roughly estimate the minimum volume that systems with different specifications can reach, which is an important function that cannot be achieved by other design methods. The designs of off-axis three-mirror reflective systems, of which the secondary mirror is the aperture stop, are taken as examples to show the effectiveness of the proposed method. The proposed method is still valid if the primary mirror is the aperture stop. The aim of this paper is to propose a method for designing starting points of freeform off-axis reflective imaging systems of small volume. When searching for the small-volume starting point, freeform surfaces are introduced to improve the imaging quality of the starting point. If the fitting method of the data points is changed, the method proposed in this paper can also be used to the design of small-volume starting points of off-axis reflective systems containing spherical and aspherical surfaces. The design method can be applied to the design of systems that are symmetric about the meridian plane and have no vignetting. The research on how to further reduce the system volume by using configuration that is asymmetric about the meridian plane or considering vignetting can be conducted in the future.
Funding
National Natural Science Foundation of China (62175123).
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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