Towards Automatic Design of Reflective Optical Systems

Traditional optical design processes can only obtain a few design solutions after a time‐consuming trial‐and‐error process. During this process, how to select the numbers and sequence of surfaces of different types (spherical, aspherical, and freeform) required by the system remains a difficult problem. Currently, no automatic design method is available for reflective optical systems that contain multiple surface types. Therefore, an automatic design framework for reflective systems is proposed that can output a series of design results with various combinations of surface types automatically after input specifications and constraints. In the first design example, 76 design results with imaging qualities near the diffraction limit are obtained that have 11 different surface type combinations. This framework realizes the automatic design of this type of system, paving the way toward general automatic optical design and significantly improving the efficiency of the optical design process. Using this framework, the numbers and sequence of the surfaces of different types required by the optical system can be determined reasonably, solving a difficult problem that has long puzzled designers. The proposed framework reduces the requirement for design experience and allows more people to participate in optical design processes.


Introduction
From the spectacles that have existed for hundreds of years to emerging head-mounted displays, from the microscopes that enable observation of the microscopic world to astronomical telescopes for observation of space, and from cameras that record images to spectrometers that are used to analyze spectral components, numerous optical instruments have been applied widely in various fields, including industry, agriculture, medicine, and scientific research. None of these optical instruments can be realized without the aid of optical design.
In recent decades, a series of studies of design methods for optical systems based on computer-aided design has been performed. These studies have promoted not only the development of these optical design methods, but also the development of technologies in the optical fabrication and optical inspection fields. However, traditional optical design methods are essentially computer-aided designs that rely heavily on the participation and experience of skilled designers, and there are still some well-known problems that hinder its further development. We believe that a truly automatic design method should have the following four features.
The design process should require almost no human involvement and be able to obtain a series of design results directly that has imaging qualities close to the diffraction limit. At present, almost every step in traditional optical design processes requires the participation of a designer because they are trial-and-error processes that have high design experience requirements. Generally, designers need to rely on their own experience to select the appropriate starting point, determine the next optimization strategy based on the current state of the designed optical system, and then obtain the design result through a process of progressive optimization. However, the starting point and the optimization strategy selected by the designer may not be optimal, and may even be wrong. Therefore, the designer may need to repeat the previous process multiple times to obtain a design result that meets the requirements.
Diversified design results should be output automatically by a single design process. The solutions obtained by traditional optical design methods lack diversity. Because of the tedious manual optimization processes, designers can only obtain small numbers of design results (and often only one result) at a time when using traditional design methods. Theoretically, there may be a series of solutions that meet the design requirements. If large numbers of design results that meet the design requirements can be obtained in a single design process, the design that best suits the requirements can then be selected by considering multiple factors synthetically.
The design method should be suitable for the design of optical systems that contain different surface types. As a result of advancements in fabrication technology, there are now more choices for surface types that can be used in the optical system. At present, spherical and aspherical surfaces are used widely in optical systems. In recent years, freeform surfaces have been used successfully in illumination systems, [1][2][3][4][5][6] spectrometers, [7][8][9] telescopes, [10][11][12] helmet display systems, [13][14][15][16] and lithography systems. [17,18] A general automatic design method should cover the design of systems that contain these different surface types.
After specifications are provided, the design method should have the ability to make a reasonable selection of the surface type for each reflective surface in the system. During the optical design process, how to make a reasonable selection of the type of each reflective surface in the system remains a difficult problem. Since the optical machining and test of freeform surfaces is challenging, [19] surfaces with lower numbers of degrees of freedom are generally preferred based on the premise of meeting design requirements. In this article, the surface types of all reflective surfaces in the system are described using the reflective surface type combination, which is abbreviated as RSTC here. There are no existing theories or methods that can provide an accurate estimate of the numbers and sequence of surfaces of different types required in a complex system. The RSTC of the system can be determined based on existing patents and entries in the literature. However, it may well be that a suitable starting point cannot be found. The designer can also list all possible RSTCs for the designed system by arrangement and combination method, select one RSTC according to their design experience, and then obtain a design result with this RSTC through optimization. If the design result can meet the design requirements, another RSTC that uses fewer freeform or aspherical surfaces is then selected, and a design result with this RSTC is then obtained through optimization. Otherwise, another RSTC that uses more freeform or aspherical surfaces is selected or another starting point is selected, and the tedious manual optimization process is then repeated. The current process for the design of a system with a reasonable RSTC is thus extremely complex and tedious. Moreover, it is difficult for designers to comprehensively consider various factors when selecting a suitable RSTC for the system, such as the number of freeform surfaces used, system volume, and tolerance.
In optical design research, large numbers of researchers have focused on design methods for the starting point, [20][21][22][23][24][25][26][27][28] such as the nodal aberration theory [29][30][31] and the partial differential equations method. [32] There are also researches on optimization methods. [33][34][35] However, none of these design methods can realize automatic optical system design. There are also automatic design methods for freeform off-axis optical systems that integrate the two functions of starting point calculation and optimization, [36,37] which can only be applied to the design of reflective systems of which each surface is a freeform surface. Compared to freeform systems, systems containing spherical and aspherical surfaces are easier to manufacture, test, and align, and thus have wider applications. Under the premise of meeting the design requirements, the designer will prioritize the use of spherical and aspherical surfaces as much as possible. Therefore, none of these methods can satisfy the four requirements listed above at the same time.
To obtain a design method that can meet all four requirements above simultaneously, a general automatic design framework for off-axis reflective systems is proposed that can obtain a series of design results with various RSTCs automatically. The proposed framework only requires the designers to input the specifications and constraints of the optical system, and it can then establish a series of alternative design results automatically that have imaging qualities close to the diffraction limit, thus freeing the designers from selecting a starting point, determining the RSTC, and optimizing the starting point. The design results obtained differ in terms of their RSTCs, system volumes, and optical power distributions. Within this framework, a planar system establishment method based on control of the geometric control parameters is proposed to obtain a series of design results. A selection principle for spherical systems is proposed that allows spherical systems with relatively reasonable optical power distributions and surface positions to be selected from larger numbers of spherical systems. New fitting methods for both aspherical surfaces and freeform surfaces are proposed to obtain systems that have imaging qualities close to the diffraction limit directly through the calculation and fitting of data points. An elimination principle for the RSTC categories is also proposed to improve design efficiency. The automatic design of off-axis three-mirror systems is used as an example to demonstrate the effectiveness of the proposed framework. In the first design example, 76 design results with imaging qualities near the diffraction limit are obtained that have 11 kinds of RSTCs. In the second design example, 56 design results with imaging qualities near the diffraction limit are obtained that have 3 kinds of RSTCs.

Experimental Section
An automatic design framework is proposed that can output a series of design results automatically by inputting only necessary parameters and can also help designers to reasonably determine the RSTC of the system. The automatic design process can be divided into the steps presented below. The flow chart for this design process is shown in Figure 1.
Using the proposed planar system establishment method, which is based on geometric control parameters, a series of planar systems with different surface positions are established.
A large number of spherical systems with different optical power distributions and surface positions are established, and potential spherical systems are then obtained by rough screening of their optical power distributions and surface positions. These systems will then serve as starting points for the next process.
The selected range of RSTCs is determined, and these RSTCs are sorted according to the proposed principles to obtain an arrangement of RSTCs: e.g., [RSTC 1 , RSTC 2 , RSTC 3 …]. The automatic design framework will then design the systems corresponding to each RSTC sequentially based on the order determined by the arrangement.
According to the given design sequence, the system with the tth type of RSTC is designed. A series of good starting points for the t-th type of RSTC is obtained by the proposed principle for the elimination of the search starting point. The design results for the t-th type of RSTC with good imaging quality are then obtained by searching the position parameters.
The proposed RSTC elimination principle is then used to judge whether it is necessary to eliminate some of the unreasonable RSTCs from the RSTC arrangement. Steps 4 and 5 are repeated until the system designs corresponding to all RSTCs in the RSTC arrangement are completed.

Establishment of Planar Systems
The positions of the surfaces in the planar system influence the imaging quality of the final design result significantly. To obtain diversified design results, a large number of planar systems with different surface positions need to be established automatically. The approach used to establish a planar system automatically according to the proposed method is then introduced, and is named the method to establish a planar system based on geometric control parameters.
It is assumed that the s-th reflective surface Ω s in the planar system is the aperture stop and that the planar system is symmetrical about the meridian plane. For the aperture stop, the upper endpoint is A, the lower endpoint is B, and ω 1 and ω 2 are the marginal fields in the meridian plane. The angle of incidence at the aperture stop for a ray in the field ω 1 is θ (s) and is smaller than that for ω 2 . As shown in Figure 2a, the intersection point of the lower rim ray that is incident on reflective surface Ω s and the upper rim ray that exits from reflective surface Ω s is C. When the value of θ (s) is given, the position of point C and the directions of ray incidence for reflective surface Ω s are determined. θ (s) is called the position parameters of the aperture stop.
Next, according to the requirement for the elimination of obscuration, the method used to determine the position and the directions of ray incidence for reflective surface Ω sÀ1 is introduced. As shown in Figure 2b, the lower endpoint of surface Ω sÀ1 , which is point D, should be located above the rays emerging from surface Ω s to eliminate obscuration. The extension of the line segment AB and the lower rim ray that is incident on surface Ω sÀ1 intersect at point E. To prevent obscuration, point E should be located above point A. The position of surface Ω sÀ1 and the directions of ray incidence on reflective surface Ω sÀ1 are determined after the selection of suitable lengths for the line segments CD and AE, which are l 1 (sÀ1) and l 2 (sÀ1) , respectively, as shown in Figure 2b. These two parameters are called the position parameters of surface Ω sÀ1 . As shown in Figure 2c, when using the same method, after the length l 1 (sþ1) ) of line segment CF and the length l 2 (sþ1) of line segment BG are selected, the position of reflective surface Ω sþ1 is determined. After the positions of all  www.advancedsciencenews.com www.adpr-journal.com reflective surfaces in front of the aperture stop are determined, the directions of ray incidence on the first reflective surface in the system can also be determined.
After the positions of all reflective surfaces in the planar system have been determined, the position of the image plane is then determined based on the principle of eliminating obscuration. As shown in Figure 3, the reflective surface adjacent to the image plane is Ω S and the upper endpoint of surface Ω S is K. For reflective surface Ω SÀ1 , the upper endpoint is H and the lower endpoint is J. A point L on the extension of the line segment HJ is then selected. The line segment KL intersects with the lower rim ray that exits from surface Ω SÀ1 at point M. The endpoint N of the image plane is located on the straight line on which line segment KL is located. After the length l 1 (Sþ1) of line segment JL and the length l 2 (Sþ1) of line segment MN are selected, the position of point N is determined. After the angle α (Sþ1) between the image plane and the vertical direction is selected, the position of the image plane can be determined. l 1 (Sþ1) , l 2 (Sþ1) , and α (Sþ1) are called the position parameters of the image plane. The angle of incidence of the chief ray on the image plane can be controlled by controlling α (Sþ1) .
These parameters, which are used to determine the positions of all reflective surfaces and the image plane of the planar system, are called the geometric control parameters of the planar system. Depending on the requirements for the compactness of the system, each geometric control parameter of the planar system is controlled within a specific range. After determining the value scopes of these parameters and letting each geometric control parameter of the planar system be a series of random numbers within its corresponding value scope, a series of planar systems with various surface positions are established using the method proposed in this section.

Construction of a Spherical System
This section describes the method used to obtain a series of spherical systems with relatively reasonable surface positions and optical power distributions. First, the section describes how a series of spherical systems that differ in terms of their optical power distributions or surface positions are established. Then, the way to use the selection principle for spherical systems proposed in this section to obtain a series of spherical systems with reasonable optical power distributions and surface positions is introduced.
Starting from a planar system P i , the process of establishing a series of spherical systems with different optical power distributions is as follows [38,39] : 1) Select a specific number of feature rays and calculate the position of the ideal image point that corresponds to each feature ray based on the object-image relationship. [40] 2) Construct the s1-th reflective surface Ω s1 as a spherical surface. A feature ray is selected, and the intersection of this ray with surface Ω s1 in the planar system is taken to be the first data point on Ω s1 . According to Fermat's principle and the object-image relationship, the ideal normal vector of surface Ω s1 at this data point is then calculated. Next, using the nearest ray algorithm, the positions of all data points remaining on the surface are determined in turn. These data points are then fitted to a spherical surface. 3) Using the same method, all remaining reflective surfaces are constructed as spherical surfaces in turn.
Through the above process, a spherical system SPH 1 is obtained. Next, its optical power is redistributed through the following process. For the spherical system SPH 1 , the spherical surface Ω s1 that is first constructed undertakes the main optical power. Next, the optical power of this surface needs to be distributed to other surfaces. Let the radius of Ω s1 be R s1 , and its optical power distribution parameter be ε s1 (ε s1 > 1). Let the radius of the surface Ω s1 be R s1 Â ε s1 , and reconstruct the s2-th surface Ω s2 . In this way, the s2-th surface undertakes more optical power. Let the radius of the surface Ω s2 be R s2 , and let its optical power distribution parameter be ε s2 . Next, let the radius of surface Ω s2 be R s2 Â ε s2 , and reconstruct Ω s3 . Repeat this step until the last reflective surface is reconstructed. In this way, a spherical system SPH 2 with an optical power distribution that is different from SPH 1 is obtained. By repeating this process, a series of spherical systems with different optical power distributions can be obtained.
By taking these spherical systems as starting points, a series of systems with selected RSTC can be obtained through an evolution process. [38] The positions of the reflective surfaces and the optical power distribution of the spherical system will both have important effects on the imaging quality of the system after evolution. A selection principle for spherical systems is proposed to determine whether the optical power distributions and the surface positions are reasonable. Using this principle, a series of potential starting points for the evolution process is obtained via the following process.
Starting from the series of planar systems that are established in Section 2.1, a series of spherical systems with differences in their optical power distributions or surface positions can be established using the method described above. Then, these spherical systems are evolved into systems of which each reflective surface is a freeform surface. Among them, some freeform systems have relatively poor imaging qualities. If the spherical systems corresponding to these freeform systems are evolved into systems with other kinds of RSTC, the imaging qualities of the systems obtained through evolution will be worse. Therefore, to improve design efficiency, such spherical systems are abandoned.

Design Process Sequence for Systems with Different RSTCs
To improve the design efficiency, it is necessary to arrange the design orders of systems with different RSTCs reasonably and www.advancedsciencenews.com www.adpr-journal.com eliminate unreasonable RSTCs from the RSTC category. Next, the method for the arrangement of the RSTCs and the elimination principle for the RSTC category proposed in this section are introduced. First, the category of RSTCs that can be used is determined. There are three possible types of reflective surfaces: spherical, aspherical, and freeform. Then, a system with S reflective surfaces has 3 S types of RSTC. If no empirical guidance is provided, the RSTCs that can be selected should include all 3 S types of RSTC. If the number of each type of reflective surface required by the system can be estimated roughly, the RSTC selection range can then be narrowed further.
Next, the RSTCs are sorted according to their ability to correct aberrations. Because a system that corresponds to an RSTC with stronger aberration correction ability is more likely to be designed successfully, the automatic design framework proposed in this paper prioritizes the design of systems with this type of RSTC. The following method is then used to judge whether the aberration correction ability of one RSTC is stronger than that of another RSTC. Freeform surfaces have stronger aberration correction ability than aspherical surfaces, and aspherical surfaces have stronger aberration correction ability than spherical surfaces. If the aberration correction ability of the type of each reflective surface in the t 1 -th type of RSTC is not weaker than (i.e., it is stronger than or equal to) that of the corresponding reflective surface of the t 2 -th type of RSTC, it is stated that the aberration correction ability of the t 1 -th type of RSTC is stronger than the t 2 -th type of RSTC. According to their aberration correction abilities, an arrangement of these RSTCs can be obtained, which is denoted by [RSTC 1 , RSTC 2 , RSTC 3 , …]. Using the sequence in the RSTC arrangement, a series of systems that corresponds to each RSTC in the RSTC arrangement is then designed successively.
Next, the proposed elimination principle for the RSTC category is introduced. To improve design efficiency, some unreasonable RSTCs will be eliminated from the arrangement of the RSTCs described above. If none of the design results for a specific RSTC obtained via the automatic design framework can meet the design requirements, then the RSTCs with weaker aberration correction abilities will be excluded from the RSTC arrangement.

Design of Systems with Good Imaging Qualities
This section describes how to obtain a series of design results for the same type of RSTC. Taking the potential spherical systems obtained in Section 2.2 as starting points and using the proposed principle for the elimination of starting points for the searching process, potential starting points for the searching process can be obtained through an evolutionary process. Next, using the proposed data point position calculation method based on the principle of equal optical paths and the proposed data point fitting method based on the best quadratic surface, the imaging qualities of the search process starting points are improved further through a single-degree-of-freedom search process for the surface position parameters. [36] When compared with freeform systems, systems that contain spherical and aspherical surfaces have fewer design freedoms and it is more difficult to find the solution for such systems. Therefore, the automatic design of systems containing spherical and aspherical surfaces has higher requirements in the following three aspects: 1) Higher requirements for selecting surface positions and optical power distributions; 2) Higher accuracy for calculation of data point positions; 3) Higher accuracy for fitting data points.
If the imaging quality of a system design that is derived by applying perturbations to the optical system position parameters is required to be near the diffraction limit, it is necessary to select a system with sufficiently good imaging quality to act as the starting point for the search process. Good starting points for the search process are obtained using a combination of the following two steps. The first step involves obtaining a series of potential spherical systems through rough screening of the optical power distributions and the surface positions. The second step involves the use of the elimination principle for the search starting point to further eliminate systems with unreasonable optical power distributions or surface positions, which obtains a series of good starting points for the search process. Next, the method used to obtain good starting points for the search process is introduced. Starting with a series of potential spherical systems obtained using the method proposed in Section 2.2, a series of systems with selected RSTC is established through an evolutionary process, [38] during which the fitting method proposed in this section is used. A specific number of systems with relatively higher imaging qualities are selected from these systems that will serve as starting points for the subsequent search process. Then, the imaging qualities of these systems are improved through a single-degree-of-freedom search process for the surface position parameters.
The data point calculation method based on the equal optical paths principle proposed in this section is introduced here. First, feature fields and feature rays are selected. As shown in Figure 4, R f (1) and R f (k) are the chief ray and the k-th feature ray of the f-th feature field, respectively, and D f (k) is the intersection of ray R f (k) with surface Ω s . For different feature rays from the same feature field, the optical distances from the object point to the intersection of the ray with the image plane are different. To improve the accuracy of the calculation of the data point positions, the chief www.advancedsciencenews.com www.adpr-journal.com ray is used as the reference ray, and the data point positions on surface Ω s are corrected based on the principle of equal optical paths. The distance OL f over which ray R f (1) travels from the object point to the ideal image point is OL f , which is taken to be the reference optical path of the f-th feature field. If the optical distance for ray R f (k) and the distance OL f are not the same, the position of the data point D f (k) located on ray R f (k) should be changed to let the optical path of the ray D f (k) be equal to the reference optical length OL f . Using the same method, the positions of all data points on surface Ω s are corrected. Next, the accuracy of the data point position calculations is improved further by searching for the reference optical distance for each feature field.
To solve for a system with good imaging quality, a highprecision data point fitting method is required. When fitting data points, both the coordinates and the normal vectors of the data points should be considered. [41] Next, the fitting method for aspherical and freeform surfaces proposed in this paper is introduced.
First, the method used to fit the data points to an aspherical surface when the position of the local coordinate system for the aspherical surface has been given is introduced. The expression for the aspherical surface is shown in Equation (1). Here, c is the radius of curvature, k is the conic constant, z c (x,y;c,k) is a quadratic term, z r (x,y;P a ) is the remaining term of the aspherical surface expression, and P a is a vector that consists of the (2o þ 1)-th order coefficients (where o = 0,1,2,3,….).
The proposed method finds the best base for the aspherical surface by first calculating the parameters of the quadratic term and then calculating the coefficients of the remaining terms. It is assumed that the coordinates of the d-th data point are (x d ,y d ,z d ), the ideal normal vector of the surface to be fitted at the d-th data point is (u d ,v d ,À1), and the total number of data points is NumD. The error function e a (c,k;P a ) is used to evaluate the fitting error for the aspherical surface, as shown in Equation (2), where W is the weight of the normal vector fitting errors.
First, the data points are fitted to a quadratic surface. Next, using this quadratic surface as the base and the function e a as the evaluation function, the data points are then fitted to an aspherical surface. The value of the function e a can then be calculated. Next, the error function e a is reduced further by searching for the parameters c and k. e a ðc, k; P a Þ ¼ If the position of the local coordinate system selected for the aspherical surface changes, then the minimum value of the error function e a (c,k;P a ) obtained using this method will also change. The best location for the local coordinate system of the aspherical surface is obtained via a search process. To determine the appropriate search range for the origin of the local coordinate system, the data points are fitted to a freeform surface, and the line of intersection of the freeform surface with the YOZ plane of the global coordinate system is then obtained. The search for the origin of the local coordinate system of the aspherical surface is conducted along the intersection line. During this process, it is also necessary to search for the angle between the global coordinate system and the local coordinate system used for the aspherical surface. The above method can be applied to the fitting of both coaxial and off-axis aspherical surfaces. For fitting of coaxial aspherical surfaces, there is no need to search for the position of the local coordinate system for the aspherical surface.
Next, the fitting method for the freeform surface is introduced. The freeform surface can be expressed in the form of Equation (3), where f c (x,y;c,k) is the conic term, f r (x,y;P f ) represents the remainder terms, and P f is a vector that consists of the coefficient A mn . Under the local coordinate system used for the freeform surface, assume that the coordinates of the d-th data point are (x d ,y d ,z d ), the ideal normal vector for the surface to be fitted at the d-th data point is (u d ,v d ,À1), and the total number of data points is NumD.
The function e f (c,k;P f ) is used to evaluate the freeform surface fitting error, as shown in Equation (4), where WP d is the weight of the fitting error for the d-th data point, and W is the weight of the fitting error for the normal vectors. First, the data points are fitted to a quadratic surface. Then, using the quadratic surface as the base, the data points are fitted to a freeform surface and the value of the error function e f is then calculated. Next, the value of e f is reduced further by searching for the parameters c and k.
When the image quality of the system is poor, the weight WP d , which is the weight of the fitting error for the d-th data point, is set at 1 to reduce the average value of the root mean square (RMS) wavefront error of the system. When the image quality of the system approaches the diffraction limit, the maximum wavefront error of the system can be reduced by controlling the weight WP d .

Design Results
This section demonstrates the effectiveness of the automatic design framework. The first example is an off-axis three-mirror system, working in visible band. Its field-of-view angle is 4°Â4°. The entrance pupil diameter of the system is 20 mm, and the F-number is 5. The automatic design framework obtains 280 design results, which have 14 kinds of RSTCs. Among these results, there are 76 design results with the maximum RMS values of the wavefront error (MAX RMS WFE) less than 0.085λ (where the wavelength λ = 587 nm), which have 11 kinds of RSTCs. The design is conducted on a computer of which the processor is Intel(R) Xeon(R) Platinum 9242 CPU @ 2.30 GHz. The design process took about 32.4 h. The optical pathway diagrams for the design results with the MAX RMS WFE less than 0.085λ are shown in Figure 5. The name of each system is marked on the top of that system. The RSTC of the system is marked above the system. "S" indicates that the surface is spherical, "A" indicates that the surface is aspherical, and "F" indicates that the surface is a freeform surface. For example, "ASF" indicates that the primary, secondary, and tertiary mirrors are aspherical, spherical, and freeform surfaces, respectively. The minimum, maximum, and average RMS wavefront error, the distortion, and the actual focal length of the design results with the MAX RMS WFE less than 0.085 λ are given in Table S1, Supporting Information.
The second example works in long-wavelength infrared band. Its field-of-view angle is 6°Â 8°. The entrance pupil diameter of the system is 50 mm, and the F-number is 1.7. The RSTC selection range is given to be ["FFF", "FFA", "FAF", "AFF"]. The automatic design framework obtains 384 design results, which have 4 kinds of RSTCs. Among these results, there are 56 design results with the MAX RMS WFE less than 0.085λ (where the wavelength λ = 10 μm), which have 3 kinds of RSTCs. The design process takes about 57.9 h. The optical pathway diagrams for the design results with the MAX RMS WFE less than 0.085 λ are shown in Figure 6. The RSTC of the system is marked above the system. The minimum, maximum, and average RMS wavefront error, the distortion, and the actual focal length of the design results with the MAX RMS WFE less than 0.085λ are given in Table  S2, Supporting Information.

Discussion
The design framework proposed in this paper is suitable for the automatic and multi-solution design of reflective optical systems; the proposed framework can determine the minimum number of freeform and aspherical surfaces required by the system and let the designer know which RSTCs corresponding design results can meet the design requirements. In traditional optical design methods, the determination of the RSTC has always been a very time-consuming and blind process. The framework proposed in this work can provide designers with a series of design results that have different RSTCs, which allows the designers to select the most suitable design results quickly according to their actual needs, and solves the long-standing and difficult problem of how to determine a reasonable RSTC for the system during the optical design process. In the first design example, design results with 14 types of RSTC were obtained using the proposed method. Among these results, there are design results with 11 types of RSTC that offer image qualities that are close to the diffraction limit. In this design example, the image qualities of the design results obtained when using three aspherical surfaces are already near the diffraction limit. In addition to the number of freeform surfaces and aspherical surfaces used to form the system, the sequence of the surfaces of different types in the system will also influence the image quality of the system. If the number of surfaces of different types used by the system has been given, the designer can then use the proposed method to select a better sequence. In the second design example, when compared with the RSTC of the "AFF" system, the RSTCs of the "FFA" and "FAF" systems are both beneficial in improving image quality.
After a single automatic design process has been completed, if additional design results with a specific RSTC are desired, then the design results obtained with this RSTC can be used as starting points, and finer searches for the power distributions and surface positions for these systems are then performed. More systems with this type of RSTC can also be obtained using the automatic design framework by expanding the search range for the geometric control parameters of the planar system and improving the search density.
Unlike traditional optical design methods that usually take the designer a long time to obtain a local optimal solution, the framework proposed in this paper can output large numbers of design results automatically after the specifications and constraints are input. In the first design examples, 280 design results were obtained using the proposed framework. Among these results, there were 76 design results with a MAX RMS WFE less than 0.085λ, 123 design results with a MAX RMS WFE between 0.085λ and 0.151λ, and 81 design results with a MAX RMS WFE between 0.3λ and 0.15λ. In the second design example, 384 design results were obtained using the proposed framework. Among these results, there were 56 design results with a MAX RMS WFE less than 0.085λ, 127 design results with a MAX RMS WFE between 0.3λ and 0.085λ, and 201 designs with a MAX RMS WFE more than 0.3λ. The solution that best meets the design requirements can then be selected according to the actual needs of designers. Unlike traditional optical design methods, through which only one or a few solutions can be obtained, the proposed framework can provide a large number of solutions. With the aid of the method proposed in this paper, the designer does not need to spend a great deal of time on improving the image quality of the optical system, and their main task is the evaluation and selection of the design results.
These design results provide us with important information that traditional methods cannot provide, such as: (a) the types of RSTCs that can meet design requirements; (b) the minimum number of aspherical and freeform surfaces required, and how to find a good location for them; (c) the range of volume, the range of tolerance, component sizes, and the number of freeform surfaces used of design results for each kind of RSTC. This allows designers to consider more factors comprehensively to select the final design result, which is advantageous for obtaining results that better meet design requirements.
Designers can pick out the design result that best meets actual needs through the following three methods. The first method is to compare these design results to obtain the most satisfactory design result, comprehensively considering various requirements such as system volume, number of freeform and aspherical surfaces used, component size, and tolerance. The second method is to gradually tighten the screening criteria until only one design result can meet all screening criteria. The proposed automatic framework can automatically output the volume range, the number range of freeform surfaces used, and the tolerance range of all design results whose imaging quality meets the requirements. Designers can comprehensively consider multiple factors based on these data, and gradually tighten the screening criteria, until there is only one design result that can meet the requirements remains. The third method is to select the design result that best meets actual needs by giving the evaluation function of the design result. If the evaluation function can be given, the most suitable design result can be quickly picked out by the computer.
If the framework proposed is applied to the design of systems with higher performance, it may obtain fewer solutions and takes longer time. Since the calculation of different systems can be  www.advancedsciencenews.com www.adpr-journal.com conducted at the same time, computers with more cores or computer cluster technology can be used to further improve design efficiency.
The proposed framework can also be applied to the design of off-axis reflective optical systems with other optical path configurations by changing the geometric control parameters of the planar system and the constraints.

Conclusion
An automatic design framework for reflective optical systems with surface types that include one or more of the spherical, aspherical, and freeform surface types is proposed that realizes the automatic design of this type of system for the first time.
Unlike traditional optical design methods, which rely heavily on the participation and the design experience of designers and can only obtain a few solutions, this framework only requires the designer to input design specifications and constraints, and it can then automatically output a series of solutions that offer good imaging qualities and various RSTCs. During the design process, the designer does not need to participate in the processes of establishing starting points, determining the RSTC, and optimizing the starting points. The framework can provide a reasonable determination of the numbers and sequence of the surfaces of different types required by the optical system, which resolves a long-standing problem that has plagued optical designers. This framework expands the number of people who can participate in the optical design process, which means that engineers with rich experience in the fabrication and inspection of optical systems  www.advancedsciencenews.com www.adpr-journal.com can also participate in the optical design process. This is conducive to obtaining design results that are more in line with the actual practical requirements. By changing the geometry control parameters of the planar system and constraints, this framework can also be applied to the automatic design of coaxial reflective optical systems and reflective off-axis optical systems with optical path configurations or number of mirrors that differ from the example. Using this framework, large numbers of off-axis reflective systems with imaging quality close to diffraction limit and different RSTC can be obtained, which can provide large amounts of data for deep learning frameworks. In the future, to further improve the design efficiency of the automatic design framework, nodal aberration theory can be used, which can guide the selection of optical path configurations, the selection of positions of surfaces, and the calculation of the shape of the reflective surface.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.