In the configuration of long-focal-length reflective space cameras, it is necessary to consider the impact of parameters such as F-number, FOV, and focal length on the search space for distances between mirrors. To achieve a larger FOV, the distance between mirrors must be appropriately increased to balance the aberrations introduced by tilt (to eliminate obscuration), and it also varies greatly for space cameras with identical focal lengths but different F-numbers. Therefore, it is necessary to construct its search space, which adapts to various combinations of F-number, FOV, and focal length.

In this study, the search space for distances between mirrors is defined as, d max = τ 1 E D (1) d min = τ 2 E D (2) where ED is the diameter of the entrance pupil; τ ₁ and τ ₂ are functions of F-number, FOV, and focal length, τ 1 = τ 1 1 × f + τ 1 2 × F + τ 1 3 × x f o v + τ 1 4 × y f o v (3) τ 2 = τ 2 1 × f + τ 2 2 × F + τ 2 3 × x f o v + τ 2 4 × y f o v (4) where f is the focal length; F is the F-number; xfov and yfov represent the maximum FOV angles in the x and y directions as specified in the design requirements τ 2 3 = τ 1 3 = 0.05 , τ 2 4 = τ 1 4 = 0.1 . τ 2 2 can be expressed as, τ 2 2 = 1 / 10 ∗ N (5) where N is the total number of mirrors in the space camera.

For space cameras, longer focal lengths correspond to larger F-numbers. Therefore, to achieve the ideal distances between mirrors within the search space, longer focal lengths of space cameras require larger values of τ ₁ [23]. To obtain an upper limit for the search space that is more suitable for long-focal-length space cameras, τ 1 2 needs to dynamically vary with changes in focal length, τ 1 2 = f / 5000 / 10 ∗ N − 1 (6) τ 1 1 and τ 2 1 are defined as, τ 1 1 = τ c / 1000 ∗ N − 1 (7) τ 2 1 = τ c / 1000 ∗ N (8) where τ _c is a parameter that varies with focal length.

Due to τ _c varying with focal length, it is a function of f. To establish the mathematical relationship between τ _c and f, five different search spaces for distances between mirrors within the focal length range of 1,000–10,000 mm are designed in our study. The computed optimal τ _c values and the acceptable range of τ _c values are shown in Table 1, with trends illustrated in Figure 1. It can be observed that the data shows a power-law decay trend, and τ _c decreases more rapidly when the focal length is less than 5,000 mm and slows down as the focal length exceeds 5,000 mm. Therefore, the power function model a·x ^b + c is chosen as the fitting model, and a segmented fitting approach is employed. The least squares method is used for fitting. The relationship between τ _c and f is fitted as, τ c = 23712095.43 / f 2.19 + 1.05 , f < 5000 75507.73 / f 1.34 + 0.17 , f ≥ 5000 (9)

The fitting results are shown in Figure 2. The mean squared errors (MSE) of the fitting are 4.57 × 10⁻² and 3.59 × 10⁻⁹ for focal lengths less than and greater than 5,000 mm, respectively. The fitted τ _c values fall within the acceptable range, indicating that the fitting performance is satisfactory. There is a discontinuity at 5,000 mm in the fitting curve, which is attributed to the segmented fitting approach. Since the fitting aim is to establish a simple mapping relationship to calculate the corresponding τ _c value using a given focal length, rather than generating freeform surfaces or curves, this discontinuity is considered acceptable for real applications. The fitting results indicate that the fit provided by Figure 2B is superior for the focal length of 5,000 mm. Therefore, the fitting formula from Figure 2B ( τ c = 75507.73 / f 1.34 + 0.17 ) is used at 5,000 mm. Although the fitting in Figure 2A is somewhat less accurate than in Figure 2B, the fitting result in Figure 2A remains within an acceptable range at 5,000 mm, making the fitting curve in Figure 2A still useable for f < 5,000 mm.

Using the matrix method to establish the optical relationship between the mirror distance and curvature radii in a coaxial reflective system, B A D C = 1 − φ N 0 1 1 0 d N − 1 1 1 − φ N − 1 0 1 ⋯ 1 − φ 1 0 1 (10) where φ is the optical power, and the subscript is the serial number of the mirror.

Based on Equation 10, the objective function of curvature radius is obtained as, g c = argmin r n f + i N / A + d N − C / A + ∑ n = 1 N − 1 n + 1 1 / r n (11) where i N is the refractive index of the Nth mirror; n denotes the serial number of the mirror, 1 ≤ n ≤ N; r _n is the curvature radius of the nth mirror.

By substituting d _max and d _min into Equation 11, the corresponding combinations of curvature radii {r ^DMAX } and {r ^DMIN } can be obtained. Consequently, the curvature radius search space for each mirror is obtained as, r n max = ⁡ max r n D M A X , r n D M I N (12) r n min = ⁡ min r n D M A X , r n D M I N (13)

When the mirror rotates clockwise, the tilt angle is defined as positive; otherwise, it is negative. For reflective space cameras, the main objective of tilting the primary mirror is to eliminate the obscuration of the incident rays by the secondary mirror. The sign (positive or negative) of the tilt angle does not affect the optical path structure, as illustrated in Figure 3.

When the primary mirror’s optical power is negative, it results in a large secondary mirror area, thereby increasing the overall volume of the space camera. This is disadvantageous for creating a lightweight space camera. Therefore, this study limits the primary mirror’s optical power to positive values and constructs the search space for the tilt angle of the primary mirror based on this constraint. The search space for the tilt angle of the primary mirror can be represented in terms of its absolute value as, α 1 ∈ 0.5 ⁡ arcsin E D / 2 / d max , 0.5 ⁡ arcsin E D / d min (14) where α 1 represents the tilt angle of the main mirror, and the schematic diagram of its upper and lower limits is shown in Figure 4.

The optical path structure is affected by the tilt angle of the secondary mirror. According to the definition of optical path structure in Ref. [32], when the extension line of the incident ray on the secondary mirror (central ray of the center FOV) rotates anti-clockwise, aligning with the outgoing ray at an angle of less than 180°, the rotation vector of the secondary mirror is −1; otherwise, it is 1. Therefore, based on the optical path structure requirements, the search space for the tilt angle of the secondary mirror can be constructed, as shown in Figure 5.

It can be concluded that when α ₂ > 2α ₁, the rotation vector of the secondary mirror is −1; when α ₂ < 2α ₁, the rotation vector of the secondary mirror is 1. Simultaneously, to ensure structural compactness, the angle between the incident and outgoing ray is defined as acute angle. Therefore, the search space for the tilt angle of the secondary mirror is derived as, α 2 ∈ 2 α 1 − π / 4 ,   2 α 1 , V s = 1 α 2 ∈ 2 α 1 , 2 α 1 + π / 4 , V s = − 1 (15) where V s represents the rotation vector of the secondary mirror.

Then, the tilt angle (2 ( α 2 − α 1 )) of the tertiary mirror is computed when the incident ray on the tertiary mirror coincides with the outgoing ray, as depicted in Figure 6. Furthermore, the search space for the tilt angle of the tertiary mirror is determined as, α 3 ∈ 2 ( α 2 − α 1 ) − π / 4 , 2 ( α 2 − α 1 ) , V T = 1 α 3 ∈ 2 ( α 2 − α 1 ) , 2 ( α 2 − α 1 ) + π / 4 , V T = − 1 (16) where V T represents the rotation vector of the tertiary mirror.

To meet the lightweight requirements, it is necessary to consider the camera size. Based on Snell’s law and the reverse ray tracing method, ray tracing is performed on the central and edge rays of the FOVs of (0,0), (0, FOV _y ), (0, -FOV _y ), and (FOV _x ,0). Subsequently, the coordinates of the intersection points between each ray and the mirror as well as the image plane can be determined, along with the sets of x, y, and z coordinates of all intersection points, denoted as {X}, {Y}, and {Z}. Therefore, the size function of the space camera is given by, g s = max Y − ⁡ min Y × max Z − ⁡ min Z × 2 ⁡ max X (17)

The optical path structure is commonly a key design requirement, so it is essential to represent and assess the optical path of off-axis space cameras. Since the adaptive tilt angle search spaces have already been established based on the design requirements, it is only necessary to evaluate the intersections of the optical paths [32]. The cross function of the optical path is given as, g p = ∑ p = 1 N N − 1 / 2 V p R − V p C (18) where V p R is the pth cross vector from the design requirements; V p C is the pth cross vector of the current system.

To integrate the primary and tertiary mirrors, it is necessary to ensure that the primary mirror and the tertiary mirror are sufficiently close. Therefore, the primary/tertiary mirror distance discrimination function is defined as, g f t = 0 , d f t < Q 1 , d f t ≥ Q (19) where d f t is the minimum distance between the primary and tertiary mirrors; Q is a constant.

In systems with a z-shaped structure, setting the distance between the primary and secondary mirrors equal to the distance between the secondary and tertiary mirrors can increase the search speed, facilitating rapid convergence towards the goal of integrating the primary and tertiary mirrors. Therefore, when the design requirement of the optical path structure is z-shaped, an additional distances constraint condition needs to be added, that is, |d ₁| = |d ₂|.

To ensure that the rays within the system are not obscured by the mirrors or image plane, it is essential to control the off-axis degree. Furthermore, to prevent the risk of obscuration due to surface changes in the subsequent introduction of freeform surfaces, it is necessary to reserve a certain optical margin to address any potential optical path obscuration. And the off-axis degree function is given as, g o a = ∑ n = 1 N + 1 σ 1 O A n + σ 2 O A M n (20) where OA _n is the off-axis degree discrimination number, if the nth mirror does not obscure any rays, OA _n = 0, otherwise, OA _n equals to the shortest distance from the mirror to the obscured rays; OAM _n is the margin discrimination number, if the shortest distance between the nth mirror and any ray in the system is greater than 10 mm, then OAM _n = 0. Otherwise, OAM _n is the absolute value of the difference between the shortest distance from the mirror to the obscured ray and 10; σ 1 and σ 2 are weights.

To balance the imaging quality of the optical system, it is important to calculate the distances between the imaging positions of rays from different FOVs and the ideal image points. The tilt angle of the ideal image plane is given as, α I = 2 α 3 − α 2 + α 1 (21)

The coordinates of the ideal image point of the central ray in the central FOV is, x I = 0 y I = d 1 ⁡ sin − 2 α 1 − d 2 ⁡ sin − 2 α 1 + 2 α 2 + d 3 ⁡ sin − 2 α 1 + 2 α 2 − 2 α 3 z I = d 1 ⁡ cos − 2 α 1 + d 2 ⁡ cos − 2 α 1 + 2 α 2 + d 3 ⁡ cos − 2 α 1 + 2 α 2 − 2 α 3 (22)

Therefore, the ideal image point coordinates of other FOVs can be calculated [16]. Based on this, the image quality evaluation function is given as, g q = ∑ i = 1 K x c − x i 2 + y c − y i 2 + z c − z i 2 1 / 2 (23) where K is the total number of sampled rays; i is the sequence number of the sampled rays; (x _c ,y _c ,z _c ) is the coordinate of the ideal image point. It should be noted that since this study evaluates the imaging quality in the global coordinate system, the difference in the z-coordinate also needs to be calculated.

Subsequently, based on Equations 17–23 the structural parameter objective function is established as, G = argmin d n , α n , r n δ 1 · g s + δ 2 · g p + δ 3 · g f t + δ 4 · g o a + δ 5 · g q (24) where δ 1 to δ 5 are the weights of each function.

By searching for the minimum value of Equation 24, the structural parameters that best meet the design requirements are obtained. Then, the improved W-W method is used to obtain the discrete point combinations of the mirrors [27]. In this study, the Fringe Zernike polynomial is used for fitting, the expression is described as, Z = c ρ 2 / 1 + 1 − 1 + k c 2 ρ 2 + ∑ C j Z j (25) where c is the curvature; ρ is the radial height in the polar coordinate system; k is the conic coefficient; j is the serial number of the Zernike term. To avoid surface displacement due to the positional changes and tilts introduced by the first three Fringe Zernike terms, these terms are set to zero; C _j is the coefficient of the jth Zernike term; Z _j is the jth Zernike term; Z is the vector height of each point on the surface.

In summary, a designer can easily achieve a long-focal-length freeform off-axis reflective space camera with lightweight and integration of primary/tertiary mirror by only inputting the design requirements. The flow chart of the design method proposed in this study is shown in Figure 7.

In this experiment, our proposed method was used to design a z-shaped freeform off-axis reflective space camera, the specifications are presented in Table 2. First, τ ₁ and τ ₂ were obtained using Equations 3, 4, τ ₁ = 3.70, τ ₂ = 2.67. Then, the SA method was used to find out the search spaces for radii of curvature, and Equation 11 was used as the error function. The upper and lower limits of radii of curvature during the searching process, as well as the results, are shown in Table 3. The initial values of searching were randomly generated by MATLAB and Equations 12, 13 were used to set the search spaces for radii of curvature.

Subsequently, the search spaces for the tilt angle of the primary mirror was set as α ₁∈[-11.01, −3.88] according to Equation 14. According to the design requirements, V _S = −1 and V _T = 1 were obtained. Therefore, the search spaces for the tilt angles of the secondary and tertiary mirrors were calculated based on Equations 15, 16 as α ₂∈[2α ₁, α ₁+π/4] and α ₃∈[2 (α ₂-α ₁)-π/4, 2 (α ₂-α ₁)], respectively. Then, the SA method was utilized to search for the structural parameters of the z-shaped space camera. The search spaces and results related to the tilt angles, radii of curvature, and distances between mirrors are listed in the right column of Table 4. The initial temperature was set at 1,500, and Equation 24 was used as the error function, δ ₁ = 1 × 10⁻⁸, δ ₂ = δ ₃ = 50, δ ₄ = δ ₅ = 1, σ ₁ = 5, σ ₂ = 1, and Q = 100. Figure 8 illustrates the error function’s variation curve throughout the search process. Table 3 and Figure 8 demonstrate that a range of structural parameters can be searched within the search spaces obtained using the proposed method, which validates the rationality of the established search spaces. The error curve starts to stabilize and settles at a steady value of 17.28 after exceeding 90 iterations.

Then, the improved W-W method was employed to design the discrete points of the mirrors, and Equation 25 was used as the fitting function, the obtained freeform off-axis reflective space camera is outlined in Figure 9. It can be observed that the integration of the primary and tertiary mirrors in the designed space camera is effectively combined, with rays from all FOVs converging at the image plane. The RMS spot radius and wavefront of the central FOV are 13.32 μm and 0.10 λ, respectively. This suggests that the freeform off-axis reflective space camera created using the proposed method exhibits suitable structural parameters and optimal imaging performance. To validate the compactness of our designed system, the size of our space camera was calculated using Equation 17, and compared with off-axis three-mirror systems from Refs. [5, 11] that have the same design requirements. The calculation results are presented in Table 5. From Table 5, it is evident that, compared to the off-axis three-mirror system in Ref. [5], which does not consider volume constraints, the space camera designed in this study exhibits a significant reduction in size, which is 1.28 m³. Compared to the system in Ref. [11], which considers volume constraints but does not account for primary/tertiary mirror integration, our system still achieves a reduction in size. This is due to the primary/tertiary mirror integration search criteria used in this study, which reduces the dimensionality of the search spaces.

Finally, to further evaluate the optimization potential of the space camera designed in this study, the Zemax optical software was employed to optimize the system produced by our method. During the optimization process, the mirror positions and tilt angles were not altered; only the coefficients of higher-order terms above z ₃ for the freeform surfaces were optimized. And it was ensured that the minimum distance between the primary and tertiary mirrors did not exceed 120 mm and that the edge points of the mirrors did not obscure the rays in the space camera. The optimized space camera’s layout, modulation transfer function (MTF) curve, and spot diagram are illustrated in Figure 10. It can be seen that the layout of the optimized space camera remains unchanged from the space camera generated by our method. In the chosen FOVs, the MTF curves are all exceed 0.45 at 50 lp/mm, close to the diffraction limit. Most of the imaging spots are contained within the Airy disk, and its range is from 1.29 to 3.64 µm, which are smaller than the detector pixel size. These results demonstrate that our optimized space camera delivers excellent imaging performance, and the designed space camera has the potential for optimization. Table 6 presents a comparison of the system layout and performance across different stages of the design process, including search, improved W-W design, and final optimization results. Since the positions and tilt angles of the mirrors remained unchanged during both the improved W-W design and final optimization stages, only the surface shapes were modified. Consequently, the minimum distance between the primary and tertiary mirrors was utilized to assess subtle variations in the system layout. Additionally, the RMS spot radius of the central FOV was employed to evaluate system performance. It can be observed that using the search method produces a space camera with a high level of primary and tertiary mirrors integration, but with suboptimal imaging quality. This is because, at this stage, the space camera is an off-axis spherical system and has not yet employed freeform surfaces to correct aberrations. And it is evident that the use of the improved W-W method to generate freeform surfaces significantly enhanced the imaging performance of the system, while the minimum distance between the primary and tertiary mirrors remains relatively stable. Following the final optimization, further improvements in imaging performance are observed, alongside a change in the minimum distance between the primary and tertiary mirrors, yet the space camera maintains a robust integrated structure. This indicates that the structural parameters derived using the method outlined in this study effectively meet the design requirements of optical systems, exhibit high stability, and provide reliable input for subsequent stages, thereby reducing the complexity and difficulty of optimization. This further demonstrates that the method proposed in this study is capable of generating the freeform space camera with a high degree of integration and favorable imaging performance, providing a solid foundation for final optimization efforts.

To validate the applicability of the proposed method to different structures and focal lengths of space cameras, an off-axis three-mirror space camera with a focal length of 15,000 mm and a non-z-shaped structure was generated using our method. The system specifications are detailed in Table 7.

First, τ _c was calculated to be 0.36 using Equation 9, τ ₁ = 5.26 and τ ₂ = 2.61 were determined based on Equations 3 and 4. The search spaces for the radii of curvature were determined, and the bounds and final results of the search process are shown in Table 8. The primary mirror was set to rotate clockwise, and the search space for the tilt angle of the primary mirror was defined as α ₁∈[2.73, 11.28] according to Equation 14. According to the design requirements, V _S = −1 and V _T = 1, so the search spaces for the tilt angles of the secondary and tertiary mirrors were α ₂∈[2α ₁, α ₁+π/4] and α ₃∈[2 (α ₂-α ₁)-π/4, 2 (α ₂-α ₁)], respectively.

Since the system has a non-z-shaped structure, the condition |d ₁| = |d ₂| cannot be employed to reduce the dimensionality of the search spaces. Therefore, a two-stage SA method was utilized. The initial temperature was 1,500, Equation 24 was served as the error function, δ ₁ = 1 × 10⁻⁹, δ ₂ = δ ₃ = 50, δ ₄ = δ ₅ = 1, σ ₁ = 5, σ ₂ = 1, and Q = 100. In the first-stage search, when the value of the error function remained unchanged for 30 iterations, the corresponding structural parameters were automatically used as the initial values to begin the second-stage search. And the initial values were set to random values in the first-stage search. For the first and second stage searches, the error function curves are shown in Figures 11A, B, respectively. The search spaces and the results of the two-stage search are presented in Table 9. The following conclusions can be drawn from Figure 11 and Table 9.

1) In the first-stage search, when the number of iterations exceeds 70, the error value stabilizes at 146.64 and remains at this value for 30 iterations; In the second-stage search, when the number of iterations exceeds 60, the error value stabilizes at 35.82; After two-stage searching, suitable structural parameters can still be found within a total of 200 iterations.

Therefore, the method presented in this study can adaptively generate reasonable search spaces for structural parameters. In the methodology section of this study, multiple systems with a focal length of up to 10,000 mm (as shown in Figure 1) were used to determine the coefficient expressions for the search spaces for distances between mirrors (Equations 1–9). These expressions remain applicable to space cameras with a focal length of 15,000 mm (beyond the fitting range) in this example. Moreover, the radius of curvature and tilt angle search spaces obtained based on the search space are also suitable for this example, further validating the generalization and effectiveness of the proposed method.

After surface fitting with the improved W-W method, an off-axis reflective freeform space camera was designed and the resulting is illustrated in Figure 12. It is apparent that the integration degree of the primary and tertiary mirrors is high. The RMS spot radius and wavefront of the central FOV are 5.18μm and 0.08λ, respectively, quantitatively reflects that the imaging performance is satisfactory. Finally, the Zemax optical software was used to optimize the space camera, and the optimization constraints are consistent with the previous experiment. The layout, MTF curve, and spot diagram of the optimized space camera are illustrated in Figure 13. It can be observed that for rays from different FOVs, the majority of the imaging spots fall within the Airy disk, with radii ranging from 2.72 μm to 4.67μm, all of which are smaller than the pixel size; the MTF curves are all exceed 0.4 at 50 lp/mm. These experimental results prove that the designed space camera has the potential for optimization, and effectively ensures high imaging quality, lightweight, and ease of manufacturing. And Table 10 presents the comparison of the system layout and performance at different stages of the design process. Data from various design stages indicate that the minimum distance between the primary and tertiary mirrors exhibits limited variation, demonstrating a high level of integration in the space camera. For the non-z-shaped space camera in this experiment, the method proposed in this study still achieves a high degree of integration and satisfactory imaging performance. This proves that the design method presented in this study can provide a reasonable search space for structural parameters of long-focal-length space cameras tailored to different design requirements, and yield stable structural parameters. These parameters do not require further optimization in subsequent processes and provide a sound basis for the generation and optimization of freeform surfaces. Designers need only perform simple surface shape optimization on the freeform space camera generated using the method described in this study to achieve a space camera approaching the diffraction limit.