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Abstract
Traditional optical design methods require designer intervention in the system’s evolution from the starting point to the final design. Trial-and-error during design optimization improves system performance step by step but requires much time and effort. A new optical design framework, end-to-end fast automatic design, is proposed and achieved for the freeform reflective optics in this paper, which promotes a new optical design mode. Compared with the traditional mode through improving performance after each trial, an optical system with good image quality can be directly obtained in the end-to-end design process with simple input and no human involvement within a short time. If there is still the possibility for performance improvement of the obtained system, the designer can vary the input parameters repeatedly to obtain multiple systems with good image quality. Finally, the desired system is selected from these systems. Compared with the step-by-step trials in traditional optimization, this new optical design mode involves high-speed trials of the end-to-end automatic design process, reducing the dependence on experience and skill. In this paper, an end-to-end fast automatic design method for freeform imaging systems is developed based on a new design route. Using an initial plane system as an input, a freeform system with excellent image quality can be designed automatically within 1–2 min. After several trials of the end-to-end fast design process, three high-performance freeform systems are designed successfully that consider volume control, beam obscuration, and mirror interference.
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1. Introduction
Optical design has been playing an important role in scientific development and engineering progress over the past century, and the optical design mode has changed along with the advances of multi-discipline. In the early stages of optical design, ray tracing and aberration calculations are often performed by the designers themselves. Subsequently, the relevant numerical calculations are performed by computers. To date, optical design has been developed to optimize the aberrations at the starting point automatically using commercial software. However, the current approach to optical design necessitates the designer's intervention in the system's development step-by-step, which takes a great deal of time and effort. The strategy chosen by each designer determines the speed and effectiveness of the optimization process. The development of optical design requires experienced and skilled designers. In the development of optical design, the freeform surface, i.e., a continuous surface without rotational symmetry, has undoubtedly triggered a design revolution [1–3]. Various types of polynomials, including XY and Zernike polynomials, can be used to describe these surfaces. The freeform surface has a high number of degrees of freedom [4–8], which means that freeform optical systems are often expected to achieve high performance [9–14] or novel functions [15–21]. This pursuit of high performance means that the optical design of freeform systems faces major challenges. Starting points of the freeform imaging system with high performance are relatively rare in the lens database, and the increased numbers of degrees of freedom make the optimization process more complex. The traditional mode, which consists of creating the starting point and the optimization process, often requires a lot of effort and time from the designer. If the requirement for human involvement in optical design can be reduced and automatic design can be realized, the designers will then be freed from this more tedious work. We believe that this automatic design method should have the following two characteristics. (1) In addition to the optical specifications and the necessary constraints, only a simple initial system should be required as an input; alternatively, no other input may even be required. (2) After the initial input, an optical system with good image quality can be obtained directly without the need for the designer to participate in any intermediate process.
In recent years, research into automatic design methods for freeform imaging systems has emerged, which can be divided into two routes. The first route is to automatically generate a starting point with a small aberration that is close to the design goal, and then an automatic optimization process without human involvement is performed to obtain the final optical system. The contribution of Zernike terms of an optical surface to the aberration can be predicted through the Nodal Aberration theory [6,22–26]. Based on the Nodal Aberration theory, design methods for the freeform system have been proposed by some researchers [24,25]. The starting point with the best potential for aberration correction can be created through the method and the corresponding optimization strategy can be given to guide the designers to use optical software for system optimization. In J. C. Papa’s work, a design method to automatically explore the solution space of freeform imagers is described [27]. A grid search process is performed for the starting points of the freeform system and then an automatic optimization process without human involvement is used to improve the image quality of all grid points. The first-order, second-order, and third-order aberrations are used to reduce the number of degrees of freedom in the grid search, thus allowing computing resources to be saved. In the “first time right” method [28,29], multi-dimensional nonlinear equations based on Fermat’s principle and the aberrations are solved to establish a starting point of the freeform system. Because the high-order aberrations are considered in the calculation of the starting point, the image quality of the starting point is excellent. The final system can then be obtained by an automatic optimization process with the position of surfaces as variables [28]. In Y. Nie’s work, the design examples based on the “first time right” method and automatic optimization process require five minutes of calculation time [29].
The second route is to use the direct design method to get the final system and the automatic optimization process for improving the image quality is not performed. System design through the direct design method [30–35] is both effective and direct, which does not depend on the selection of the input system. Using the direct design method, only an input system with a simple configuration, such as a planar system [34,35], is required to obtain a freeform system with good image quality. In T. Yang’s work, by combining the direct design method with the single-freedom-search strategy, an automatic optical design method for freeform imaging systems has been realized with only a planar system as input [36]. Furthermore, the automatic design framework, which involves a grid search over the system parameters, is combined with the direct design method to obtain the solution space under only certain optical specifications as the input [37]. The direct design method is used to construct the freeform system for each grid point, which requires large computing resources, and it takes half an hour on average to obtain a good image quality system. In order to further improve the speed of automatic design, a fast automatic design method based on system construction and system correction is proposed, where an automatic design process of the examples takes six minutes [38]. With the initial plane system as the input, an intermediate system with fundamental optical parameters is generated through the system construction. In the subsequent system correction process, the image quality is improved to a high level through the iterative process of image plane correction and surface correction. The speed of the method is fast but not effective enough for imaging systems with high performance.
At present, the current automatic design methods require several minutes, or even hours, to produce a freeform system with satisfactory image quality. If the system does not meet the design requirements for volume, fabrication processing, assembly, or other aspects, it is necessary to revise and wait for an additional few minutes to a few hours again. Such extended delays may disrupt the designer's working rhythm and interrupt their design ideas. In our opinion, the typical behavior of optical designers is to wait for around one minute, which is consistent with the general time taken by a single local optimization process when using traditional optical software. The waiting time of about one minute would not disturb the production of the designer’s ideas and the established work rhythm. Thus, during the trial-and-error process, the optimization strategy can be adjusted repeatedly by the designer according to the optimization results until the system satisfies the final requirements.
If the speed of a single automatic design process for a high-performance system can be increased several times, a new optical design framework, referred to as end-to-end fast automatic design, can be achieved. It is possible to directly obtain an optical system with good image quality using simple input and no human involvement within the time of a single local optimization using the optical software. “End-to-end” refers to a system with excellent image quality, from a simple input, and with no human involvement in optical design from the input end to the output end. The new design framework is proposed in this paper and promotes a new optical design mode. The basic idea of the traditional design mode is to perform a step-by-step optimization process from a starting point to the final system. A single local optimization in the traditional method results in only an optical system whose performance is likely to approach the design target, while in the end-to-end design process, an optical system with good image quality can be automatically and directly obtained within the time of a single traditional optimization. If the system output by the end-to-end automatic design process does not meet some relevant requirements or if there is still room for performance enhancement, the input parameters can be varied continuously and the end-to-end design process can then be repeated until the system successfully meets the requirements. The end-to-end automatic design is performed in a short time, which enables consecutive and fluent thinking for the designer during the design process. During the end-to-end automatic design trial period, the designer's creative production remains uninterrupted. After many systems with good image quality are obtained through the end-to-end automatic design, the desired system can be selected based on actual requirements from the array of designed systems.
However, achieving end-to-end automatic design remains elusive through the two routes of current automatic design methods. In the first route, the automatic optimization process will take a long time if the starting point deviates far from the design target since the computation time depends on the quality of the starting point. In the second route, the computational cost is relatively high using the direct design method although the input can be a concise planar system. In this paper, a new route of automatic design for the freeform system is proposed. In the new route, a simple input system with the potential of small aberration is created first, and then the freeform imaging system is constructed employing the direct design method. Using this route, it is not necessary to consider the quality of the starting point. A freeform system with good image quality can be obtained in a short time, enabling the realization of end-to-end fast automatic design.
In the proposed end-to-end fast automatic design method based on the new design route, a freeform system with excellent image quality can be obtained after 1–2 min of fast automatic design processing, which is several times faster than the current methods. Based on the immediacy of this method, when the system obtained does not meet the requirements for volume, fabrication processing, assembly, or other aspects, the designer can change the input parameters constantly and rapidly obtain a series of freeform systems with good image quality. Finally, the system desired by the designer is selected from these systems. Using this method, high-performance freeform imaging systems can often be realized after several trials of the end-to-end design process. The end-to-end fast automatic design method is divided into two parts that combine the advantages of the direct design method with those of the aberration analysis method. With a planer system as input, a first-order geometry (FOG) is initially established, in which the positions and optical power of the surfaces are determined by some of the low-order aberrations. The direct design method is then used in the construction and correction of the freeform system to perform further correction of both low-order and high-order aberrations.
During the trials of the end-to-end fast automatic design process, volume control, beam obscuration, and mirror interference aspects are all considered. A new design can be obtained rapidly by simply adjusting the initial plane system. After several high-speed trials of the end-to-end design process, three different types of high-performance freeform imaging systems are designed successfully. These systems are an off-axis three-mirror system working in the visible band, a three-mirror imaging system with a small F-number working in the long-wave infrared band, and a four-mirror system with a long focal length working in the visible band. In the design of these three systems, the average calculation times for their end-to-end design processes were 49 s, 1 min 12 s, and 1 min 51 s, respectively. The effectiveness of both the proposed end-to-end fast automatic design method and the new optical design mode is verified by the designs of these three high-performance freeform imaging systems. The method and the optical design mode involved are attractive and valuable not only for optical engineers but also for practitioners working in other fields of optics.
2. End-to-end fast automatic design method
The proposed end-to-end fast automatic design method is introduced in this section. Expressions for the system parameters and the description of the optical power are presented first. Then, the two aberration correction equations used in this method are introduced, which comprise the flat field equation and the field-linear astigmatism elimination equation. Finally, the two parts of the design method are described separately. In this method, the FOG from which some low-order aberrations have been eliminated is established based on the aberration correction equations, and the direct design method is then used in the construction and correction process to obtain a high-performance freeform imaging system.
2.1 Description of the optical power
The parameters used to describe the surface positioning of the FOG are defined first. The number of optical surfaces in the system to be designed is N. These optical surfaces are denoted by S1, S2, …, Sk, …, SN. In this method, the surface position is described using the surface distance d and the angle of incidence θ. If the main ray in the center field is defined as the optical axis ray (OAR), dk is then the distance between the corresponding points of the OAR on surfaces Sk and Sk + 1, and θk is the angle of incidence of the OAR on Sk. When a three-mirror system is used as an example, the positions of the mirrors and the image plane are described by the parameters d1, d2, d3, θ1, θ2, and θ3, as shown in Fig. 1.
An important step toward the determination of the FOG is the determination of the optical power distribution based on the field curvature and astigmatism correction equations. In the off-axis system, it is inappropriate to use the radius of curvature to describe the optical power. The image distance and the object distance are used to express the optical power φ of the surface, as shown in Eq. (1), where n′ and n are the refractive indices in front of and behind the surface, respectively. li and lo are the image distance and the object distance of the surface, respectively. The object distance of a surface is the distance between the object point and the corresponding point of the OAR on the surface. The image distance of a surface is the distance between the image point and the corresponding point of the OAR on the surface. Using the matrix approach for the FOG, the transfer matrix of the surface Sk and the transfer matrix of a light ray propagating from surface Sk to Sk + 1 are shown in Eq. (2), where φk is the optical power of surface Sk, and nk is the refractive index behind surface Sk. The transfer matrix for the entire optical system is given in Eq. (3). To satisfy the given effective focal length f requirement, the variables must ensure that Eq. (4) holds.
Subsequently, it is then necessary to convert the optical power into specific surface parameters to obtain the FOG. When the FOG of the off-axis system is established using a sphere, which is a rotationally symmetrical surface, the same radius of curvature will correspond to different optical powers on both the meridian plane and the sagittal plane. For systems with a wide field of view (FOV) or a large aperture, the off-axis angles of the surfaces must be large enough to eliminate obscuration, and thus the difference between the meridian optical power and the sagittal optical power will be greater at the same radius. Therefore, a toroidal surface is applied to the determination of the FOG in this method. In the off-axis system, a toroidal surface can have different radii of curvature on the meridian plane and the sagittal plane [39]. The use of the toroidal surface can ensure that the focal power determined for each surface is consistent on the meridian plane and the sagittal plane. According to the Coddington equations [40,41], the radii of the toroidal surface on the meridian plane and the sagittal plane can be determined using the object and image distances of a surface, as shown in Eq. (5), where θ and θ’ are the angle of incidence and the exit angle, respectively, and RT and RS are the radii of curvature on the meridian plane and the sagittal plane, respectively.
2.2 Flat field equation and field-linear astigmatism elimination equation
The flat field condition and the field-linear astigmatism elimination equation will be used in the proposed method to determine the optical power and position for each surface.
The field curvature is an important aberration. In the absence of astigmatism, the Petzval curvature, which is the sum of the optical powers of the mirrors, must be equal to zero to achieve a flat image plane, as shown in Eq. (6). This condition is referred to as the “flat field”. By substituting Eq. (1) into Eq. (6), an equation using the object and image distances of the surfaces as variables can be obtained.
In the off-axis system, the field-asymmetric field-linear astigmatism often dominates an important part of the aberration [24,42]. In this method, the equation proposed by Chang [43] is used to eliminate the field-asymmetric field-linear astigmatism. In Eq. (7), mk is the ratio of the image distance to the object distance for each surface, and θk is the angle of incidence of the OAR on each surface. When γ is zero, the field-linear astigmatism is eliminated. If the object surface is oriented perpendicular to the OAR, the image plane will then be perpendicular to the OAR [43].
The characteristic of the required equation is that it can be regarded as a scalar product of two vectors. For example, consider the case where the number of surfaces is three, as shown in Eq. (8). Vector A is related only to the object and image distances of each surface, i.e., it is closely related to the optical power of each surface. Vector B is only related to the angle of incidence. When vector A lies orthogonal to vector B, the field-asymmetric field-linear astigmatism will then be eliminated.
The astigmatism correction transformation based on Eq. (7) will be illustrated below and will be used to correct the surface position in this method. When the optical power is determined for each surface, vector A is also determined. If and only if vector B lies on the plane Ω perpendicular to vector A, γ can then be zero, and field-linear astigmatism can be eliminated. Therefore, vector B can be projected onto the plane Ω using a projection transformation to obtain vector B’, as shown in Fig. 2.
After the projection transformation shown in Fig. 2, vector B becomes the vector B’ that lies orthogonal to vector A. With vector A remaining unchanged, the scalar product γ of vectors B’ and A becomes zero, and vector B’ can then minimize the modulus length of the difference between vectors B and B’. Understanding this projection transformation from the perspective of the system structure allows field-linear astigmatism to be eliminated through minimal adjustment of incidence angles when the optical power distribution is determined. This transformation is later called the astigmatism correction transformation (ACT).
2.3 Fast automatic design strategy for the FOG
The input required for the end-to-end fast automatic design method is the system specifications, which include the focal length, FOV, and F-number, along with an initial planar system as conceived by the designer. There are some basic requirements for an appropriately established initial planar system. There is no beam obscuration or interference between the mirrors. The structure of the planar system should be compact in order to avoid a large volume. If the final system deviates too far from the initial planar system in each end-to-end design, the designer cannot judge the general structure of the final system based on the given planar system. The trials of the end-to-end design process will become random and out of the designer's control. To ensure that the final system does not deviate too far from the initial planar system during the end-to-end design process, the distance d for each surface in this method remains unchanged during the calculation of the FOG in most cases, and the angle of incidence θ of the OAR on each surface only changes when the ACT is used. According to a simple geometric relationship, the difference between the image distance of surface Sk and the object distance of surface Sk + 1 is dk. Therefore, if the image distances of S1, S2, …, SN−1 are known, then the object distances of S2, S3, …, SN can be determined. Next, the object distance of S1 is the given object distance for the optical system, which is usually infinity, and the image distance of SN is dN, i.e., the distance between SN and the image plane. In short, if the image distances of S1, S2, …, SN−1 are known, then the object and image distances can be obtained for each surface, and the optical power of each surface can be solved using the object and image distances for each surface based on Eq. (1). Therefore, the image distances of S1 to SN−1 can be set as variables to solve the aberration equations.
The design strategy used in this method is to ensure that the FOG satisfies the following three conditions as far as possible. The focal length of the system must be equal to the given value, as shown in Eq. (6). The sum of the focal powers of the surfaces ψ must be equal to zero, as shown in Eq. (7). The parameter γ related to field-linear astigmatism must be equal to zero, as shown in Eq. (8). The three conditions above can be used to form three equations with respect to the optical power distribution, which are called the focal length equation, the flat field equation, and the astigmatism equation in this paper. For the case where multiple solutions are realized when solving these equations, the corresponding first-order geometries are calculated separately, and the system with the best image quality is then selected automatically for subsequent construction of the freeform system.
Different design strategies can be selected based on the number of reflectors included in the end-to-end fast automatic design method proposed in this paper, which can be applied to a four-mirror system at most. As mentioned above, there are (N−1) variables in a system with N surfaces. If N is greater than three, then the number of variables will not be less than the number of equations. The design strategy is to solve for the optical power of the FOG by solving the three equations simultaneously. If N is not greater than three, then the number of variables will be lower than the number of equations. The strategy is to combine the three equations and the ACT to determine the FOG.
2.3.1. Design strategy for the two-mirror and three-mirror system
The design strategy for the FOG of the three-mirror system is introduced first here. Different design strategies can be performed automatically based on the number of reflectors. There are (N−1) variables in a system with N mirrors. The three-mirror system has two variables, which are the image distances of the primary and secondary mirrors. The number of variables is thus lower than the number of equations in this case. To satisfy the three conditions as far as possible and obtain the optimal result, the proposed design strategy for the three-mirror system is divided into two steps.
The first step is to attempt to find a FOG that can satisfy all three equations simultaneously. The flat field equation and the focal length equation are solved simultaneously to determine the optical power distribution, and the ACT is then performed to correct the position for each surface. Simultaneous the flat field equation and the focal length equation have analytical solutions. The optical power obtained is converted into parameters of the toroidal surface. At this time, the first two equations in the system are satisfied, but the astigmatism equation is not satisfied, i.e., γ is not equal to zero. The ACT is then performed on the system to adjust the angle of incidence for each surface at the optical powers determined. As a result of the first step, a FOG that satisfies all three equations at the same time is obtained.
If the absolute value of γ before the ACT is small, this means that the position of the surfaces lies close to the position that satisfies the astigmatism equation. The adjustment to the angle of incidence caused by the ACT will be relatively small, which means that beam obscuration may not occur. The corresponding FOG can be used directly in the construction of the freeform system, and it is not necessary to enter the second step. This case corresponds to optical structures with small astigmatism, e.g., the three-mirror-anastigmat (TMA) structure. When the absolute value of γ is large, a large movement in the surface will be caused by the ACT such that a large-scale beam obscuration may occur. This actually reflects the fact that it may be difficult to find a solution that can satisfy all three equations around the initial plane system. Therefore, a fast and automatic search process for the optical power distribution must be used as the second step for the three-mirror system design strategy.
The second step involves searching for the FOG with the best potential for small aberrations. Multiple FOGs with different distributions of optical power are first established. Multiple FOGs have the same d and incident angle symbols as the initial plane system, which makes the multiple FOGs close to the given initial plane system. The different distributions of optical power are obtained by simultaneously solving the focal length equation and the astigmatism equation using different values of γ that give analytical solutions, where γ is selected to have values ranging from zero to positive and negative values with a specific step length. The ACT is then performed on these geometries and a series of first-order geometries with the required potential for small aberrations can be obtained. The average root mean square spot radius for each field is used to evaluate the image quality. The FOG with the best image quality will then be selected for use in the freeform surface construction process.
Under the determined optical power and with minimal adjustment of the angle of incidence, the ACT is actually used to obtain a FOG with a small aberration potential. The approach of solving the astigmatism equation using different values of γ is not only intended to consider the astigmatism aberration; the essence of the approach is to obtain a variety of different optical power distributions. Searching for γ from zero means that the positional adjustments of the surfaces caused by the ACT range from small to large. When beam obscuration occurs, the search process will then stop.
The design strategy for the FOG of the two-mirror system is introduced next. For the two-mirror system, if only the image distance of the primary mirror is selected as the variable based on the rules for the selection of variables, it will not be conducive to actually solving the equations. Therefore, the distance between the secondary mirror and the image plane is also set as a variable when dealing with the two-mirror system to enable a strategy similar to that used for the three-mirror system to be implemented for the two-mirror system.
2.3.2. Design strategy for the four-mirror system
In a four-mirror system, after the surface distance and the angle of incidence are determined according to the given initial plane system, the four-mirror system has three variables, which comprise the image distances for the first three mirrors. The number of variables is equal to the number of the equations, i.e., three. The optical power distribution can be obtained by solving the focal length equation, the flat field equation, and the astigmatism equation simultaneously. To solve these nonlinear equations, numerical solutions can be calculated using an optimization algorithm. The Universal Global Algorithm in the First Optimization (1stOpt) software is used in this method to solve these equations. The algorithm is fast, stable, and does not rely on the initial values. Correct solutions can be obtained in most cases.
2.4. Construction and correction process for the freeform system
The direct design method is used for the system construction and correction process. Because the FOG has been subjected to optical power distribution and surface position corrections based on a combination of astigmatism and field curvature, many of the low-order aberrations have been corrected. At this time, it is possible to obtain excellent image quality by applying the direct design method to correct for higher-order aberrations.
The direct design method used in the system construction process is the normal correction method [44], which can realize the construction of the freeform imaging system rapidly. In this direct design method, each surface is subjected to multiple fitting processes. Using the fitting method that considers both the coordinates and the normal [45], a freeform surface can be obtained by fitting the preserved coordinates and the corrected normals of the data points. These data points are obtained by ray tracing of the feature rays that correspond to multiple fields with different apertures. The coordinates of the data points are preserved, while the normals of the data points are corrected. These corrected normals are determined based on the incident rays and the exit rays according to the law of reflection. The incident rays remain in their original directions and the exit rays point toward the ideal image points. The freeform system is built by fitting each freeform surface in the appropriate order. Then, the fitting process for all freeform surfaces in the system is performed several times until the image quality no longer improves. The freeform imaging system construction process is then finally completed. In this direct design method, the focal length of the imaging system can be controlled by the coordinates of the ideal image points. The mirrors only move in a small range in the method so that the obtained freeform system will not deviate from the designer's expectations.
The correction process for the freeform system is introduced next. The image plane lies perpendicular to the OAR in the FOG, which only takes the field-asymmetric field-linear astigmatism into account. To improve the image quality further, an iterative process that alternates between image plane correction and correction of the surfaces is performed for system correction [38]. During the iterative process, the aberration terms associated with the image plane are considered for image plane correction, and the normal correction method is used for the correction of the surfaces. Because the system’s image quality is already good before the system correction process, the correction process required for the freeform system can often be completed within several iterations. Different from the method based on aberration analysis [24,25], the shape parameters of the mirrors are determined by using the direct design method. Freeform systems with good image quality can be obtained directly so that the optimization process can be not employed in the automatic design.
In summary, the entire design process for the end-to-end fast automatic design method proposed in this paper is as follows. After the designer provides the system specifications and the initial plane system, the FOG with the determined optical power distribution and the surface positions is established based on the flat field condition and the field-linear astigmatism elimination equation. Then, the normal correction method is used to construct the freeform imaging system. Finally, the iterative process that alternates between image plane correction and surface corrections is performed to complete the system correction process. As a result, a freeform imaging system with excellent image quality can be designed automatically without involving the designer in any intermediate process.
3. Results and discussion
The design method proposed in this paper realizes the end-to-end fast automatic design of a freeform imaging system, which will drive change in the optical design mode. Because of the immediate and efficient features of the method, when the system obtained does not meet the requirements in terms of the designed volume, fabrication, assembly, or other aspects, the designer can constantly vary the input planar system and rapidly obtain a series of good systems with good image quality. Finally, the desired system is selected by the designer from these systems. An excellent design can usually be obtained after only a few high-speed trials of the end-to-end design. For designers who are familiar with the various optical structures of freeform systems, an imaging system with high performance can be obtained in only one or two attempts. For beginners in optical design or for practitioners in other fields of optics, the desired design can usually also be obtained after a few trials. Even if there is no clear idea of the final result in the beginning, the designers can also gradually clarify the idea in the trials of the end-to-end design process.
The input required for the end-to-end fast automatic design method proposed in this paper is an initial plane system. During the automatic design process, the positions of the surfaces only move within a small range. This feature ensures that the optical structure of the system will follow the designer's original concept. As long as the initial plane system is established appropriately by the designer, the possibilities of beam obscuration or mirror interference will be reduced. Even if these phenomena do occur, beam obscuration and mirror interference can both be avoided by adjusting the initial plane system only and then performing another end-to-end design process.
If a compact optical system is to be designed, a loose initial plane system can be presented in the first end-to-end design process. Based on the output system obtained from the first end-to-end design process, the designer predicts the shape and size of each surface in the output system in the subsequent end-to-end design processes. The initial plane system can then be adjusted continuously to reduce the system volume and achieve a compact optical structure.
Through the high-speed trials of the end-to-end automatic design, three freeform imaging systems with excellent performance are designed successfully. The first design is an off-axis three-mirror reflective system. The system works in the visible band and has an F-number (FNO) of 3, an effective focal length (EFL) of 540 mm, and an offset FOV of 3°×3° with a center field of (0, −10°). After several rapid trials of the end-to-end design process, a freeform imaging system that can meet the various design requirements is obtained. The results obtained from the last end-to-end fast design are shown in Fig. 3. An initial plane system was established first in which the aperture stop was the secondary mirror, as shown in Fig. 3(a). The FOG with the determined optical power distribution and the surface positions is established based on the flat field condition and the field-linear astigmatism elimination equation. The image distances of S1 and S2 are set as variables. The ACT is then performed using the first step described in Section 2.3.1, the FOG obtained s as shown in Fig. 3(b). No beam obscuration occurs in the FOG and the normal correction method can be used in the construction and correction of the freeform surfaces. XY polynomials ranging up to the sixth order are used to describe these freeform surfaces. After three iterations between image plane correction and surface correction, the final system design is obtained. The optical layout is shown in Fig. 3(c). A field map of the average root mean square wavefront error (RMSWFE) is presented in Fig. 3(d), and the average error value obtained is 0.0655λ (where λ=587.6 nm). The distortion grid is shown in Fig. 3(e). The results obtained here indicate that the designed system provides good image quality that is close to the diffraction limit. The method in this work runs on a personal computer with a single CPU (Intel i9-10920) and not on a workstation. The related program runs on MATLAB, and the ray tracing data are read from CODE V. A single end-to-end design process takes an average time of 49 s. After the first trial of end-to-end design, the designer can predict the optical power distribution, mirror size of the system, and other aspects of the subsequent design. After several trials resulted in several obtained designs, three of these designs are shown in Fig. 4, which illustrates the evolution of the freeform system. During the trials of the end-to-end design process, the system volume was controlled and reduced from 117 L to 62 L.
Fig. 3. Process and results for the last end-to-end fast design iteration for example 1. (a) Layout of the initial plane system. (b) Layout of the FOG. (c) Layout of the final system. (d) RMSWFE and (e) distortion grid for the final system.
The second design is an off-axis three-mirror imaging system with a low F-number. The system works in the long-wave infrared (LWIR) band and has an F-number of 1.8, a focal length of 180 mm, and an FOV of 6°×6°. After several rapid trials of the end-to-end design process, a freeform imaging system that can meet the various design requirements is obtained. The results obtained from the last end-to-end fast design are shown in Fig. 5. The initial plane system is shown in Fig. 5(a), in which the aperture stop is the secondary mirror. After the distribution of optical power and the ACT in the first step in Section 2.3.2, beam obscuration occurs. Upon that, a search process of multiple optical power distributions is performed through the second step in Section 2.3.2, where γ starts from 0 and the step size is 0.01. The beam obscuration occurs when γ is −0.03 and 0.24 respectively. When the γ is −0.02, the FOG with the best image quality is obtained, as shown in Fig. 5(b). With the FOG with the most potential for small aberrations as input, the construction and correction of the freeform surfaces are performed through the normal correction method. XY polynomials ranging up to the sixth order are used to describe the freeform surfaces. After five iterations between image plane correction and surface correction, the final designed system is obtained. The optical layout is shown in Fig. 4(c). The field map of the average RMSWFE is presented in Fig. 5(d), and the average error value is 0.0588λ (where λ=10 µm). The distortion grid is shown in Fig. 5(e). It also indicates good image quality that is close to the diffraction limit. A single end-to-end design process takes an average time of 1 min 12 s. After several trials resulted in several obtained designs, three of these designs are shown in Fig. 6, which illustrates the evolution of the freeform system. During the trial process, the beam obscuration is eliminated, and the interference between the primary mirror and the tertiary mirror is prevented.
Fig. 5. Process and results for the last end-to-end fast design iteration for example 2. (a) Layout of the initial plane system. (b) Layout of the FOG. (c) Layout of the final system. (d) RMSWFE and (e) distortion grid for the final system.
The third design is an off-axis four-mirror imaging system with a long focal length. The system works in the visible band and has an F-number of 8, a focal length of 3000 mm, and an FOV of 1.5°×1.5° with a center field of (0, −10°). After several rapid trials, a good freeform imaging system is finally obtained. The results obtained from the last end-to-end fast design are shown in Fig. 7. The initial plane system is shown in Fig. 7(a), in which the aperture stop is the secondary mirror. Simultaneous focal length equation, field curvature equation, and astigmatism equation are used to solve the optical power, and the corresponding FOG is shown in Fig. 7(b). After the construction and correction of the freeform surfaces, seven iterations of image plane correction and surfaces correction are performed to develop the image quality. XY polynomials ranging up to the sixth order are used to describe the freeform surfaces. Finally, the optical layout of the designed system is shown in Fig. 7(c). The system is compact with a length of less than 800 mm in the z-direction although a focal length of up to 3000 mm. The field map of the average RMSWFE is presented in Fig. 7(d), and the average error value is 0.0558λ (where λ = 587.6 nm). The distortion grid is shown in Fig. 7(e). After several trials resulted in several obtained designs, three of these designs are shown in Fig. 8, which illustrates the evolution of the freeform system. In order to avoid mirror interference in Fig. 8, the initial plane system is adjusted, and another end-to-end design process is then performed. During the trial process, the beam obscuration is eliminated, and the system volume is also decreasing.
Fig. 7. Process and results for the last end-to-end fast design iteration for example 3. (a) Layout of the initial plane system. (b) Layout of the FOG. (c) Layout of the final system. (d) RMSWFE and (e) distortion grid for the final system.
Several representative automatic design methods and the corresponding design examples for the freeform system are listed in Table 1. There are mainly three-mirror systems and four-mirror systems in these design examples. For the three-mirror system, examples 1 and 2 in this paper work in the visible and long-wave infrared bands respectively. The performance of example 1 is close to that of the three-mirror examples working in the visible band in the work of Refs. [24,28], and the computational speed is improved by at least several times compared with the current methods. The computational time is reduced from several minutes to less than one minute. Under similar parameters of F-number and FOV, the focal length of example 2 is nearly two times higher than that of other design examples working in the long-wave infrared band in the work of Refs. [36–38]. The four-mirror system has diverse structural forms and substantial attention has been paid to it. The performance and structural forms of the four mirror design examples in Table 1 have different characteristics, and it is difficult to compare which is better from the perspective of optical performance. Example 3 in this paper is a long focal length four-mirror optical system working in the visible band. Compared to the other four mirror systems in Table 1, design example 3 in this paper has the longest focal length. Furthermore, example 3 is compact and the calculation time is lower than 2 minutes. From the design examples of various methods given in Table 1, it can be seen that the end-to-end fast design method can be applied to high-performance imaging systems.
Table 1. Design examples of automatic design methods for freeform systems
A modular design form for the end-to-end automatic design of freeform systems is also provided by the method proposed in this paper. This design method can be divided into two modules. The first module is based on the use of aberration theory to solve for the system parameters of the FOG, including the surface position and the optical power distribution. The second module involves the use of the direct design method to calculate the relevant freeform surface parameters. In the method proposed in this paper, aberration correction equations based on astigmatism and field curvature are used in the first module to establish the FOG. This process is fast and stable and can produce an output design that is close to the initial plane system proposed by the designer. Naturally, a variety of other aberration theories can also be used to construct the FOG, including the Nodal Aberration theory [6,22–24,26] and the “first-time right” method [28,29]. In module two of the proposed method, the direct design method is both efficient and universal. Faster and more efficient methods can also be used in this case, which is worthy of further research.
4. Conclusion
To achieve the end-to-end fast automatic optical design and to promote change in the optical design mode, an end-to-end fast automatic design method is proposed in this work for freeform imaging systems. Only the system specifications and an initial plane system are used as the input, and a freeform imaging system with good image quality can be obtained after approximately 1 min without involvement by the designer. The method is suitable for the design of high-performance imaging systems. To the author’s knowledge, it is the fastest automatic design method. The advantages of aberration analysis and a direct design method are combined, which provides an important concept for research of automatic design methods.
Because of the immediacy of the proposed method, if the obtained system does not meet the design requirements for volume, fabrication, assembly, or other aspects, the designer can vary the input parameters constantly and then obtain a series of systems with good image quality. Finally, the desired system can be selected from these output systems. During the trials of the end-to-end fast automatic design process, aspects of volume control, beam obscuration, and mirror interference are all considered. After several high-speed trials of the end-to-end fast automatic design process, three freeform systems with high performance were designed successfully, covering the range from the visible band to the long-wave infrared band and for both three-mirror and four-mirror imaging system designs. The effectiveness and applicability of the proposed method and the optical design mode have been verified by these three design examples. This method and the optical design mode involved are attractive and valuable not only for optical engineers but also for practitioners working in other fields of optics.
In addition, the end-to-end fast automatic design method described still has room for further improvement. (1) The method is currently run in MATLAB. Using some other underlying programming languages will further improve the calculation speed further. (2) Since only the light ray of the center field is used to determine the FOG to calculate the optical power currently, the method may not work well for systems with a very large FOV. For systems with larger numbers of mirrors or very large FOV, other aberration theories can be used to determine the surface position and the optical power distribution, e.g., Nodal Aberration theory.
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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