## Abstract

In current off-axis system design process, optical path configuration (OPC) of the system often remains unchanged due to the explicit physical constraints added in the optimization process, and this prevents designers from obtaining potential better results with other OPCs. In this paper, we present a method to change the OPC automatically by ray-quadrangle-based optimization. In our method, a vector is utilized to represent the OPC and a penalty function based on the difference between the current vector and its target value is added into the optimization error function. During OPC variation, obscuration can be avoided without human interference. Examples are given as validation of the proposed method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical design is a classic branch in optics whose major part is to improve the performance of optical systems through optimization of parameters such as distances, optical surface shapes and lens materials. In order to obtain a physically feasible result, special constraints are imposed in the optimization process. Especially for off-axis systems, constraints are deliberately set to adjust the system to the previously-designed optical path configuration (OPC). Here, OPC refers to the rotation and cross condition of the optical axis ray (OAR) of an off-axis system. For demonstration, the OPC of the zig-zag system in Fig. 1(a) is obviously different from the compact looped system in Fig. 1(b). And the OPC often has a huge influence on the final design result, both on system compactness and imaging property. To date, the choice of OPC is often based on experience and existing systems. But the fact is that freeform optics is providing us with new configurations and more compact off-axis systems [1–6]. Configurations that were seldom used due to aberration issues (e.g., the systems in [2]) are now adopted, thanks to the aberration correction ability of freeform elements. And the optical path configuration, closely related to the space occupied by the system, are becoming more important in consideration than before. Hence, choosing by experience or referring to existing systems may not offer the best or most suitable results nowadays.

Starting points often refer to systems of specified OPCs and ready for further optimization. Nowadays, as a common practice, the OPC is determined in the establishment of starting point and is merely changed in the optimization afterward. Hence, many efforts have been devoted in studying the starting point establishment methods of specified OPCs. Point-by-point surface contour calculation methods, e.g., partial equation methods [7–9], simultaneous multiple surface (SMS) method [10–12], construction-iteration (C-I) method [13–15], and multi-field direct design method [4,16], are able to establish starting points with aspheric or freeform surfaces directly. But, like the conventional design process, the OPC of the system needs to be determined ahead of surface contour calculation, since the OPC is affected or even determined by the initial parameters such as optical path and the coordinates of initial points on surfaces. And in many surface contour calculation methods, the variation in initial parameters lead to total recalculation of all surface contours, which often takes a large amount of computation. Aberration theory can also help to guide designers to find suitable starting points [17–24]. Nodal aberration theory has become a promising guide in the choice of OPC for off-axis mirror systems [17–19]. But since the complexity of OPC increases dramatically with the number of mirrors, it is quite time-consuming to list all possible OPCs for a system with multiple mirrors manually, and to study the aberration property of every one of them is still much more difficult.

Besides starting point calculation, optimization also plays an important role in optical design. Studies in this field focus on optimization methods [25–28]. And in some researches, general-purpose optimization methods were used to minimize the cost-function based on aberration correction [26,27]. And escape functions [28] has been studied to achieve global synthesis. Although these studies provided designers with multiple ways or tools to achieve design results with better optical performance, seldom work is related to the variation of the OPC of off-axis systems during optimization.

OPC variation during optimization can be very helpful as it can not only decrease the importance of the initial OPC of the starting point but also lower the threshold of the experience of the designer involved. Typically, the configuration can be rotationally-symmetric at first and be then altered to different off-axis candidates to explore for the best or most suitable one, or the system can escape from local minimum in optimization by variation of its OPC.

Xu, et al. proposed a ray to handle obscuration problem automatically for off-axis mirror systems based on the relationship of ray-quadrangles in 2017 [29]. By optimizing an error function, the “shape” of the whole system can be adjusted to avoid obscuration. To fully take use of the ray-quadrangle-based optimization method to change the OPC, the key point is how to transform the current system to a desired OPC. And before that is the notation of different OPCs.

In this paper, we propose a method to notate the OPC of an off-axis mirror system by a vector, and a method to change the OPC of an off-axis mirror system via automatic optimization without human interference.

## 2. Notation of OPC

Lack of a common rotationally symmetric axis makes the configuration of off-axis systems more complicated than conventional on-axial ones. For off-axis mirror systems, a method to mark or notate different OPCs is needed, and a distance to measure the difference between OPCs is also needed if the difference in OPC is to be imposed into a quantified error function in optimization. In this paper, we differentiate the OPC in the following aspects. We think an off-axis, reflective system can be regarded to have a different OPCs according to the following conditions: (1) the rotation of the optical path, or (2) the number and position of the OAR interrupted differs. These two conditions are most relevant to the “shape” of the off-axis reflective system.

As shown in Fig. 2(a), the OAR turns clockwise after reflection on M1 and turns counter-clockwise after reflection on M2. Meanwhile in Fig. 2(b), the OAR turns clockwise after reflection on M2, which differs from the system in Fig. 2(a) in Condition (1). Systems in Fig. 2 (a) and (b) are also different in Condition (2), since OAR segment 2 intersects segment 4 as in Fig. 2(b) but not in Fig. 2(a).

To quantify the difference, a vector, *V*, is adopted to represent the OPC. The vector *V* has 2 parts and can be written as *V* = [*V*_{1}, *V*_{2}], where *V*_{1} and *V*_{2} are two vectors representing the Condition (1) and Condition (2) respectively.

For a system with *M* mirrors, it is obvious that the OAR will reflect on the *M* mirrors successively, and all the reflection will turn the OAR clockwise or counter-clockwise. Thus *V*_{1} can take the form of a vector with *M*-1 components to describe the OAR rotation. The *i*^{th} component is set 1 if the (*i* + 1)^{th} mirror turns the OAR into the same direction with the first mirror and −1 if the (*i* + 1)^{th} mirror turns the OAR into the opposite direction to the first mirror.

Figure 2(c) shows the process to obtain *V*_{2}. The function$Cross(i,j)$is introduced to represent whether the *i ^{th}* and

*j*segment intersect, as described in Eq. (1).

^{th}According to the pseudo-code in Fig. 2(c), *V*_{2} for the systems in Figs. 2(a) and 2(b) are [0, 0, 0] and [0, 0, 1] respectively. The number of components in *V*_{2} is$M(M-1)/2$. Thus, the number of components in *V* is$M(M+1)/2-1$. For a three-mirror system, *V* has 5 components in total.

## 3. Off-axis configuration variation

Although different OPCs can be obtained even from the same starting point by setting different limits for variables (distances and tilt angles) or constraints, this process is inefficient and hard to control. As in [19], the limits for variables were deliberately chosen to obtain an unobscured zig-zag configuration. To explore different configurations more efficiently, broader limits for variables are needed.

According to Section 2, the OPC of a certain system can now be notated by a vector V. If a target OPC exists, the varying degrees between current OPC and target OPC can be further expressed by the difference between them. Then the error function corresponding to the OPC mismatch,$Er{f}_{Str}$, can be expressed by the following Eq. (2).

where $|\cdot |$means the distance between the current OPC vector*V*and the target vector ${V}_{Tar}$, which can be fulfilled by norm or Hamming distance [30]. When$\left|V-{V}_{Tar}\right|>0$,

*bias*is a positive value whereas

*bias*will change its value to make$Er{f}_{Str}=0$, if $\left|V-{V}_{Tar}\right|=0$.

There is another condition where the aim of OPC variation is to escape from the current OPC. Hence, no target OPC is needed and the $Er{f}_{Str}$ can be expressed in another form, as described in Eq. (3).

The synthesized error function,$Er{f}_{Total}$, contains 3 components that address the OPC, obscuration, and aberration, respectively, as described in Eq. (4).

## 4. Applications and discussion

In this section, three typical applications of the proposed OPC variation method are introduced and discussed, and the success rate and robustness of the algorithm are demonstrated in experiments.

#### 4.1 Example 1: starting point searching

OPC variation can be utilized to help the designer to establish a starting point of certain OPC automatically. The first example is to search for an off-axis starting point with a desired OPC (*V* = [-1, −1, 0, 0, 1], corresponding to the OPC in Fig. 2(b)) from a set of initial parameters for an axial structure.

In the optimization, the optical power of M1, *φ*, the magnification of M2, *β*, the tilt angle of M1, M2, and M3 from the OAR, *α*_{1}, *α*_{2}, and *α*_{3}, and the distance from M1 to M2 and M2 to M3, *d*_{2} and *d*_{3} respectively, are utilized as the variables. The system has a full field-of-view (FOV) of 4° × 4°, F/# of 4.5, and a focal length of 450mm.

In order to investigate the effect of OPC variation and increase the stability of ray-tracing, conic surfaces (i.e. ellipsoid, hyperboloid, and paraboloid) are adopted and two adjacent surfaces share a common focus. In this way, one aberration-free image point can be obtained in the center of the image plane. To evaluate the aberration of the system, ray-tracing transverse aberrations of peripheral fields are used to establish the$Er{f}_{Abr}$. In the experiments, $Er{f}_{Abr}$can be directly evaluated by invoking CODE V when the system with current variables is modeled and the four peripheral field angles are set as the sampled fields. The combination of simulated-annealing method (SA) and down-hill simplex (DHS) method is used in the optimization of the error function. Initial temperatures for SA are 200 for all the variables, and the parameter *S _{i}*, which means the initial simplex for the variables in DHS, is [0.000001, 0.1, 0.1, 3, 3, 25, 25]. The initial values, upper and lower limits for the searching process and results for this example are listed in Table 1. Broader ranges of limits are set for the optimization instead of deliberately chosen ones, which allows the OPC to alter.

During the optimization, the system changes from the rotationally-symmetric starting point to the desired off-axis, unobscured system with *V* = [-1, −1, 0, 0, 1] without human interference (See Figs. 3(a) and (b)).

The error function during the starting point searching process is illustrated in Fig. 3(c). The exponential function is chosen as the *f*(·) in$Er{f}_{Str}$, and the expression of $Er{f}_{Str}$ is listed in Eq. (5).

In this example, we assign ${\omega}_{1}={10}^{6}$to$Er{f}_{Str}$, ${\omega}_{2}={10}^{2}$to$Er{f}_{Obs}$, and ${\omega}_{3}={10}^{-3}$to$Er{f}_{Abr}$. Considering$bias={10}^{6}$, it can be noted that the OPC meets the target after 19 iterations, and the obscuration is eliminated after the 20th iteration. Then the synthesized error function decreases slowly afterwards, which shows that the general-purpose optimization method used here is not efficient enough in correcting aberrations. Therefore, a commercial lens design software is suggested to perform the further optimization of aberrations after the OPC variation and obscuration elimination.

#### 4.2 Example 2: hybrid constraints in the design process

In some freeform reflective system designs, physical constraints are imposed into the design process for a more compact or efficient system by integrating two or more mirrors on one single element. In this circumstance, special physical constraints can be added to the error function. In this example, the starting point and the target OPC are the same as those in Example 1, while M2 and M3 are designed to be close enough to be fabricated on the same substrate. Hence, the upmost ray-intersecting point on M1 should be below but close to the lowest ray-intersecting point on M3, forming a small gap between these two mirrors. Referring to the ray-quadrangle defined in [29], these feature points are vertices of the previously-proposed ray-quadrangles. The angle between the two surfaces’ tangents in YOZ plane on the two feature points also needs to be controlled within a proper range, which is also approximated by operations on ray-quadrangle vertices. By adding the weighted squared distance between the two feature vertices and a weighted absolute value of the angle to the error function, the optimization process can handle user-defined physical constraints in the searching of starting points. Since the optimization has both general physical constraints and user-defined physical constraint, it is referred as the hybrid constraints condition.

In this example, a 101-iteration SA optimization was utilized to perform the OPC variation first and the rest 102-200 iterations were optimized with DHS. Parameters for SA and DHS and initial values and ranges for variables are the same as in Example 1. The modification is the added items in the error function. After the optimization with hybrid constraints, the result is shown in Fig. 4(a), and values for the variables are [*φ*, *β*, *α*_{1}, *α*_{2}, *α*_{3}, *d*_{1}, *d*_{2}] = [8.9527 × 10^{−4}, 3.4036, −24.579, −10.361, 6.5450, 396.99, 361.64]. The distance between the two nearest points on M1 and M3 is 5.79mm and the angle between the surface normal on the two nearest points is 5.2°, offering a proper gap between M1 and M3 for the following fine-tuning.

Its RMS spot size over the whole FOV is illustrated in Fig. 4(b), and the error function during optimization is shown in Fig. 4(c). Considering the weight and bias set for$Er{f}_{Str}$, the OPC reached the target after 17 iterations and rest of the optimization focused on the elimination of obscuration (18-19 iterations) and improvement of imaging performance and the user-defined weighted physical constraint (20-200 iterations), respectively.

#### 4.3 Example 3: OPC modification

OPC modification has the potential to be integrated to global synthesis of off-axis design to optimize beyond different OPCs for better results. The third example is about the modification of OPC of an off-axis three-mirror reflective system in the design phase. The original system is an unobscured, zig-zag system and the target OPC is designed to have a *V* = [1, −1, 1, 1, 0]. Unlike Example 1 and Example 2, in this example, the initial point itself is an unobscured zig-zag system, which means the OPC variation process will go through an unfeasible area with obscuration and reach the target OPC with the obscuration eliminated. The starting value, upper and lower bands for variables are listed in Table 2 and the result of the automatic OPC variation is shown in the rightmost column of Table 2.

Figure 5(a) shows the initial zig-zag type system and Fig. 5(c) shows the OPC modification result, respectively. Figure 5(b) illustrates an intermediate system occurred in the OPC variation process, which shows that the OPC modification process goes through a physically-unfeasible solution space before reaching the target OPC. Figure 5(d) illustrates the error function$Er{f}_{Total}$during the OPC variation process

#### 4.4 Algorithm analysis

Success rate is an important aspect of an algorithm. Based on the analysis of the OPC variation process and practical tests, several techniques are adopted in the proposed method to improve the efficiency and success rate.

(1) The SA method, whose global searching capacity has been tested by many engineering optimization problems, is adopted in the proposed method. The DHS method is adopted as a complement. In the very rare condition where SA is stagnated and fail to eliminate obscuration for a whole round, DHS will take place and its result will be adopted as the initial point in the next round. The benefit of using DHS is that the initial step length (e.g., 3° for the tilt angle) can be defined, while in SA only the initial temperatures can be defined. The other condition where DHS will take place is the latter part in hybrid constraints conditions. After OPC variation and obscuration elimination, the error function consists of $Er{f}_{Abr}$ and user-defined weighted constraints. In this case, DHS will be performed instead of SA to improve the efficiency and avoid the acceptance of worse results by SA.

(2) A multi-round optimization strategy is adopted instead of 1-round optimization with a larger iteration number. This strategy has two benefits. First, it can avoid the low efficiency of long round. Secondly, there’s a low probability that SA accepts a worse result and cannot get rid of that in the optimization. Then at the beginning of the next round, temperatures will be reset to a high value, increasing the probability to make the optimization continue properly. The reset of temperatures in our multi-round strategy corresponds to the “backfire” technique in SA.

(3) We change the probability distributions of variables, particularly for tilt angles. We find that obscuration is inclined to occur when the incident OAR coincides with the surface normal of the incident point. Thus, we change the distribution probability of tilt angles when target OPC is met but obscuration happens due to a small tilt angle. Suppose the threshold is ± ${a}_{Th}$ (${a}_{Th}$is positive), and if any ${a}_{k}$ in the variable set satisfies$\left|{a}_{k}\right|<{a}_{Th}$, then the Eq. (6) will be called.

(4) Another technique we use is to divide the range of tilt angles ${a}_{k}$ to sub-ranges. For the first round, all tilt angles are set with one half of its range, and the corresponding tilt angles in the results of the first round are denoted as${a}_{1k}$. If the second round is required, the new range for${a}_{k}$will be the intersection of a half range centered at ${a}_{1k}$and the original range for${a}_{k}$.

To test the success rate of the proposed method, 20 independent trials were performed for the Example 1 and Example 3 respectively. The maximum round allowable is 3, and the optimization process is regarded as unsuccessful if it fails to either obtain the target OPC or eliminate the obscuration in 3 rounds. And the optimization process is set to terminate at the end of the round in which the target OPC has been achieved and obscuration is eliminated. Initial temperatures set for all variables in SA are 200.

The error functions of the 20 trials for the first example during OPC variation optimization are illustrated in Figs. 6(a)-6(d), in which all the trials succeeded. The mean number of iterations to achieve an unobscured system with the target OPC is 34. It can be noted that 18 of the 20 trials succeeded in 1 round and the rest 2 trials succeeded in the second round. And all the 20 trials got the target OPC in 1 round, whereas 2 of them failed to eliminate the obscuration in the first round. It can be seen in Fig. 6(d) that the value of error function at iteration 0 for the second round in Trial 19 is different from the result of the first round. This is because the above-mentioned Technique (4) was called and the small tilt angle was moved toward its threshold.

For the third example, 16 of the 20 trials succeeded in 1 round and the rest 4 trials succeeded in the second round, which also indicates a 100% success rate. The mean number of iterations to achieve an unobscured system with the target OPC is 40. Two trials (i.e., Trial 16 and Trial 19) failed to obtain the target OPC in the first round, and another two trials (i.e., Trial 1 and Trial 8) obtained the target OPC but failed to eliminate the obscuration in the first round. An obvious sign of stagnation can be seen in the fours’ error function in the first round, as shown in Fig. 7. We think this shows the advantage of the multi-round strategy in dealing with stagnation.

Considering the above-mentioned techniques and the success rate in the experiments, the feasibility and robustness of the proposed method are demonstrated.

## 5. Conclusions

In this paper, we propose a method to change the optical path configuration for off-axis systems by optimization of OPC vector-based cost functions. Particularly, it can be utilized in the early stage of the design phase to obtain a suitable starting point with desired optical path configuration, or be used to alter the OPC in optimization to broaden the solution space. The effectiveness of the proposed method is demonstrated and validated by three examples corresponding to three typical applications and success rate is analyzed. This proposed method may help to achieve a more automatic way in the design of off-axis optical systems. In the future work, we will study the efficiency increase method in OPC variation by using adaptive parameters in cost functions.

## Funding

Zhejiang Provincial Natural Science Foundation of China (LQ19F050006); National Key Research and Development Program of China (2017YFA0701201); National Natural Science Foundation of China (NSFC) (61822502).

## Acknowledgment

We thank Synopsys for the educational license of CODE V.

## References

**1. **J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci. Appl. **6**(7), e17026 (2017). [CrossRef] [PubMed]

**2. **K. Fuerschbach, G. E. Davis, K. P. Thompson, and J. P. Rolland, “Assembly of a freeform off-axis optical system employing three φ-polynomial Zernike mirrors,” Opt. Lett. **39**(10), 2896–2899 (2014). [CrossRef] [PubMed]

**3. **D. Cheng, Y. Wang, H. Hua, and M. M. Talha, “Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism,” Appl. Opt. **48**(14), 2655–2668 (2009). [CrossRef] [PubMed]

**4. **Y. Nie, R. Mohedano, P. Benítez, J. Chaves, J. C. Miñano, H. Thienpont, and F. Duerr, “Multifield direct design method for ultrashort throw ratio projection optics with two tailored mirrors,” Appl. Opt. **55**(14), 3794–3800 (2016). [CrossRef] [PubMed]

**5. **Z. Zheng, X. Liu, H. Li, and L. Xu, “Design and fabrication of an off-axis see-through head-mounted display with an x-y polynomial surface,” Appl. Opt. **49**(19), 3661–3668 (2010). [CrossRef] [PubMed]

**6. **Q. Meng, W. Wang, H. Ma, and J. Dong, “Easy-aligned off-axis three-mirror system with wide field of view using freeform surface based on integration of primary and tertiary mirror,” Appl. Opt. **53**(14), 3028–3034 (2014). [CrossRef] [PubMed]

**7. **R. A. Hicks, “Controlling a ray bundle with a free-form reflector,” Opt. Lett. **33**(15), 1672–1674 (2008). [CrossRef] [PubMed]

**8. **D. Cheng, Y. Wang, and H. Hua, “Free form optical system design with differential equations,” Proc. SPIE **7849**, 78490Q (2010). [CrossRef]

**9. **J.-B. Volatier and G. Druart, “Differential method for freeform optics applied to two-mirror off-axis telescope design,” Opt. Lett. **44**(5), 1174–1177 (2019). [CrossRef] [PubMed]

**10. **J. C. Miñano, P. Benítez, W. Lin, J. Infante, F. Muñoz, and A. Santamaría, “An application of the SMS method for imaging designs,” Opt. Express **17**(26), 24036–24044 (2009). [CrossRef] [PubMed]

**11. **F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express **20**(5), 5576–5585 (2012). [CrossRef] [PubMed]

**12. **Z. Hou, M. Nikolic, P. Benítez, and F. Bociort, “SMS2D designs as starting points for lens optimization,” Opt. Express **26**(25), 32463–32474 (2018). [CrossRef] [PubMed]

**13. **T. Yang, G. F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light Sci. Appl. **6**(10), e17081 (2017). [CrossRef] [PubMed]

**14. **T. Yang, D. Cheng, and Y. Wang, “Freeform imaging spectrometer design using a point-by-point design method,” Appl. Opt. **57**(16), 4718–4727 (2018). [CrossRef] [PubMed]

**15. **R. Tang, B. Zhang, G. Jin, and J. Zhu, “Multiple surface expansion method for design of freeform imaging systems,” Opt. Express **26**(3), 2983–2994 (2018). [CrossRef] [PubMed]

**16. **Y. Nie, H. Thienpont, and F. Duerr, “Multi-fields direct design approach in 3D: calculating a two-surface freeform lens with an entrance pupil for line imaging systems,” Opt. Express **23**(26), 34042–34054 (2015). [CrossRef] [PubMed]

**17. **A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. **9**(1), 1756 (2018). [CrossRef] [PubMed]

**18. **J. C. Papa, J. M. Howard, and J. P. Rolland, “Starting point designs for freeform four-mirror systems,” Opt. Eng. **57**(10), 1 (2018). [CrossRef]

**19. **Y. Zhong and H. Gross, “Initial system design method for non-rotationally symmetric systems based on Gaussian brackets and Nodal aberration theory,” Opt. Express **25**(9), 10016–10030 (2017). [CrossRef] [PubMed]

**20. **J. W. Figoski, “Aberration characteristics of nonsymmetric systems,” Proc. SPIE **554**, 104–111 (1986). [CrossRef]

**21. **T. Schmid, K. P. Thompson, and J. P. Rolland, “A unique astigmatic nodal property in misaligned Ritchey-Chrétien telescopes with misalignment coma removed,” Opt. Express **18**(5), 5282–5288 (2010). [CrossRef] [PubMed]

**22. **J. Sebag, W. Gressler, T. Schmid, J. P. Rolland, and K. P. Thompson, “LSST telescope alignment plan based on nodal aberration theory,” Publ. Astron. Soc. Pac. **124**(914), 380–390 (2012). [CrossRef]

**23. **K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express **22**(22), 26585–26606 (2014). [CrossRef] [PubMed]

**24. **L. Yang, J. W. Qi, and Z. Bin, “Analytical expressions of the imaging and aberration coefficients of a general form surface,” J. Opt. Soc. Am. A **34**(12), 2077–2095 (2017). [CrossRef] [PubMed]

**25. **M. van Turnhout, P. van Grol, F. Bociort, and H. P. Urbach, “Obtaining new local minima in lens design by constructing saddle points,” Opt. Express **23**(5), 6679–6691 (2015). [CrossRef] [PubMed]

**26. **B. F. Carneiro de Albuquerque, F. Luis de Sousa, and A. S. Montes, “Multi-objective approach for the automatic design of optical systems,” Opt. Express **24**(6), 6619–6643 (2016). [CrossRef] [PubMed]

**27. **R. Wu, J. Sasián, and R. Liang, “Algorithm for designing free-form imaging optics with nonrational B-spline surfaces,” Appl. Opt. **56**(9), 2517–2522 (2017). [CrossRef] [PubMed]

**28. **M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, “Lens design: global optimization with escape function,” Opt. Rev. **2**(6), 463–470 (1995). [CrossRef]

**29. **C. Xu, D. Cheng, and Y. Wang, “Automatic obscuration elimination for off-axis mirror systems,” Appl. Opt. **56**(32), 9014–9022 (2017). [CrossRef] [PubMed]

**30. **D. J. S. Robinson, *An introduction to abstract algebra* (Walter de Gruyter, 2003).

**31. ***Code V manual book* (Synopsys, 2018).