The relative aperture size and the field-of-view (FOV) are two significant parameters for optical imaging systems. However, it is difficult to improve relative aperture size and FOV simultaneously. In this paper, a freeform design method is proposed that is particularly effective for high performance systems. In this step-by-step method, the FOV is enlarged from a small initial value in equal-length steps until it reaches the full FOV; in each step, part of the area of one system surface is constructed. A freeform off-axis three-mirror imaging system with large relative aperture size and a wide FOV is designed as an example. The system operates at F/2.5 with 150 mm effective focal length and a 60° × 1° FOV. The average root-mean-square wavefront error of the system is 0.089λ (working wavelength λ = 530.5 nm).
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Freeform optical surfaces are nonrotationally symmetric surfaces with multiple degrees of design freedom [1, 2]. Freeform surfaces have been used successfully in various fields in illumination, includeing beam shaping , road lighting  and special model illumination systems . In the imaging field, off-axis reflective imaging systems offer the advantages of high transmission, an absence of chromatic aberration, and elimination of obscuration  and are thus favored by optical designers. Because freeform surfaces have better capabilities for correction of aberrations than spherical and aspherical surfaces [7, 8], freeform surfaces are often used to obtain better imaging quality, particularly in off-axis systems. Additionally, use of freeform surfaces in imaging systems can significantly improve the system performance while reducing the system size [9–11]. They can also be used to produce unusual structures or special functional systems such as freeform prisms, which are used in see-through head-mounted displays [9, 12], compact reflective imagers [1, 13–15], and dual-focal-length imaging systems .
High-performance optical imaging systems, for example, large relative aperture size systems or wide field-of-view (FOV) systems, are commonly pursued by designers in fields including space detection, imaging spectrometers, and aerospace remote sensing. Systems with large relative aperture sizes can collect more incident light energy and thus improve each system’s signal-to-noise ratio (SNR) and resolution. A wide FOV can provide a large observation range and improve system detection efficiency. Some systems with large relative apertures [9, 17] or wide FOVs [15, 18–20] have previously been designed successfully using freeform surfaces. However, it is remains difficult to improve these two parameters simultaneously. It is both interesting and challenging work to improve the relative aperture size and the FOV simultaneously.
Traditional optical design methods begin by establishing an appropriate initial system with performance that is close to the target goals. The final design results are obtained by further optimization using optical design software [1, 21]. Generally, a good initial system and the further optimization by commercial software are two important key in optical design and it’s barely impossible to conduct a design without one another. However, there are few off-axis systems that can be used as starting points, particularly when designing high-performance systems with large relative apertures and wide FOVs. The design process can take a long time, and even may fail to reach the required solution when using co-axial starting points, because these points are generally far from the optimum point. In recent years, some more direct design methods for freeform surfaces have been developed, including the partial differential equations (PDEs) design method, the simultaneous multiple surface (SMS) method and the construction-iteration method (CI-3D). The PDEs is an effective method for imaging of only a single field [22, 23]. The SMS method can generate several freeform surfaces simultaneously. However, the number of field points considered in the process is same as the number of surfaces . The CI-3D method is efficient for control of the light rays of multiple fields and different pupil coordinates [25–28]. However, the CI-3D method is constructed face-by-face. The method constructs one surface considering the object-image relationships of full fields, which can cause the first constructed surface to deviate from the solution or even become impossible to solve when designing high-performance systems with both large relative apertures and wide FOVs. Determination of an effective direct design method for systems with both large relative aperture sizes and wide FOVs is therefore a significant task.
In this paper, a commonly used freeform design method is proposed for imaging system applications. This method is particularly effective for high-performance systems, for example, systems with large relative aperture sizes and wide FOVs. An initial planar system is first established. One surface of the initial system is constructed as freeform surface with point-by-point direct construction that only considers the rays of partial fields. Then, each surface is constructed one by one; in each construction process, the constructed fields are enlarged in steps of equal length. After all surfaces have been constructed, the surfaces are expanded via a surface expansion procedure to obtain wider fields. Multiple freeform surfaces are obtained until the full FOV is achieved and this can serve as a good starting point for further optimization. A high-image-quality freeform off-axis three-mirror imaging system with large relative aperture size and a wide FOV is designed as an example in this work. The final system operates at F/2.5 with effective focal length of 150 mm and a 60° × 1° FOV. The root-mean-square (RMS) wavefront error of the system at the working wavelength λ = 530.5 nm is 0.089λ.
2. Multiple surface expansion method
An actual optical imaging system operates with a certain entrance pupil size and a specific FOV. When a system operates with a smaller FOV, there are fewer aberrations to be corrected. One paraboloid surface can produce perfect imaging of a single field. However, as the number of fields increases, the imaging quality will deteriorate for all fields. Similarly, when only one surface is under construction, consideration of more fields simultaneously then leads to worse construction results being obtained. One potential approach to obtain better results in the construction process is described as follows: each constructed surface only considers partial fields of the full FOV. Then, the constructed fields are enlarged gradually until the full FOV is reached. Based on the above idea, a multiple surface expansion method is proposed. To describe the process involved in this method more clearly, some terminology must first be defined and introduced.
In our method, to control the light rays from multiple fields with different pupil coordinates, a finite number of the rays from the different fields are sampled to act as feature rays. Suppose that M fields from field φ1 to field φM are sampled equidistantly over the full FOV and N feature rays for each field are sampled according to specified rules . Ray Rm(n) is the nth feature of the field φm. This ray starts from point S and is reflected by the surface Ωk at point Pk,m(n) to E. Here, the subscript k indicates the serial number of the surface, the subscript m indicates the serial number of the field and the superscript n indicates the serial number of the ray. As shown in Fig. 1, the process is dependent on Snell's law:
As Eq. (1) indicates, the direction of a ray that is reflected/refracted after it reaches a surface is determined by both the coordinates and the normal vector at point Pk,m(n). We define Xk,m(n) as the solution for ray Rm(n) on the surface Ωk. This can be represented as follows:Fig. 2, Im(n) is the ideal point for ray Rm(n).
We define X as the solution for the system, and it can be represented by a set of solution of all surfaces as follows:
Next, we introduce the process of the multiple surface expansion method. To depict this process more clearly, we use two surfaces for the example shown in Fig. 3(a). Surface Ω1 is constructed first by considering only the feature rays of field φ1. The coordinates and the normal vectors of the feature points on surface Ω1 of field φ1 are calculated using a point-by-point method . X1,1 is then obtained. The freeform surface Ω1 is obtained after fitting of these discrete data points into the freeform . The rays of field φ1 can converge to its ideal point after being reflected by surface Ω1. However, the rays of field φ2 cannot converge to its ideal point after being reflected by surface Ω1 because the feature rays of field φ2 have not been considered during construction of surface Ω1. X1,2 is thus not calculated using the object-image relationship of field φ2, but is given by the extension area that is generated by X1,1 after the fitting process, as shown in Fig. 3(b). Then, surface Ω2 is constructed to perform field φ2 imaging. Because the changes in surface Ω2 also affect the imaging of field φ1, the feature rays of both field φ1 and field φ2 are used in this step. As shown in Fig. 3(c), X2,1 and X2,2 are obtained during this step. By enlarging the constructed fields gradually and constructing the surfaces alternately, X can gradually be optimized to provide an optimal solution.
The details of the design method can be broken down into three steps: (1) Establish an unobscured initial system with suitable planes for further design processing. Sample the feature fields equidistantly for the full FOV. (2) Construct one surface of the initial planar system using parts of the fields with a point-by-point direct construction process . Construct the surfaces alternately while gradually enlarging the constructed fields. (3) Expand the construction area of each surface to achieve the full FOV required. To depict this process more clearly, a system with K surfaces and a full FOV of Φ is taken as example. The flow diagram of the design method is as shown in Fig. 4.
2.1 Establish initial system and sample feature fields
In this step, an initial planar system is established for further design processing. Certain principles must be considered in this step, e.g., the system should have a compact structure without obscuration. While the initial planar system cannot be used to perform imaging, it can be used as the initial value of X. After the system is established, M fields ranging from field φ1 to field φM are sampled equidistantly over the full Φ FOV. The step length for these fields is small to produce a better effect. Here, a basic way to enlarge one field is introduced. In general, the system is symmetrical in the sagittal direction, and thus only half of the full field along the sagittal direction need be considered during the construction process. Suppose that the center of the FOV is (0, φy0). Center field is then taken to be the first constructed field φ1. In further construction processing, the constructed field is enlarged along the positive sagittal direction. In the meridian direction, the constructed field is enlarged in both the positive and negative directions. As shown in Fig. 5, field enlargement in the two directions will only stop when the full FOV has been achieved. Besides this basic way, other reasonable ways to enlarge field over full FOV can also be used.
2.2 Enlarge the constructed fields gradually while constructing surfaces alternately
In this step, a point-by-point direct construction process  is used to construct the surfaces. The construction process is performed in four steps. (1) Define the feature rays of the constructed field φ1. Two basic ways to define these feature rays and their start and end points are given in detail in . (2) Calculate the feature data points on surface Ω1 in the given object-image relationships. The intersection of the first feature ray with surface Ω1 is used as the initial data point. The next data point is calculated by obeying the nearest ray principle. The point is thus calculated as the intersection of the corresponding feature ray with the tangential plane of the nearest feature data point. Subsequently, the coordinates and the surface normal vectors of all data points of field φ1 can be calculated, and X1,1 is obtained. Then, the discrete data points are fitted into the freeform surface while considering both the coordinates and the normal . In this step, X2,1 to XK,1 are the initial values given by the initial planar system. (3) Enlarge one of the constructed fields, φ2, and construct the surface Ω2. Unlike the preceding step, both field φ1 and field φ2 are considered in this step. X2,1 and X2,2 are then obtained. X1,2 is the extension area that is generated by X1,1 after the fitting process. X3,1 to XK,1 and X3,2 to XK,2 are the initial values given by the initial planar system. (4) Enlarge the constructed fields gradually while constructing the surfaces alternately. When constructing surface Ωk, Xk,1 to Xk,k are obtained as shown in Fig. 6(a). After all surfaces have been constructed, XK,1 to XK,K are obtained as shown in Fig. 6(b). The calculation process denoted by Eq. (5) is shown in Fig. 7.
2.3 Expand the freeform area of each surface
In general, the number of sampled fields is much higher than the number of surfaces in the system. After all surfaces have been constructed, there are still some fields that have not been considered. In this step, these fields are computed using a surface expansion procedure. Unlike the construction process described in section 2.2, the intersections at which the feature rays intersect with the surfaces are used as the data points. This means that the points on the surface that was generated in the previous step are preserved and ensures that surface after expanded is continuous. The normal vector of each data point is then recalculated while following the object-image relationship. To enable this expansion process to be depicted clearly, suppose that three fields are used to construct the surface. The feature data points on the surface are inside their footprints, as shown in Fig. 8(a). The area has been constructed is an envelope formed by all three fields’ footprints. The area outside this envelope is the extension area that was generated during the surface fitting process. If one expansion field is added during the reconstruction process, the freeform area on the surface will expand after reconstruction as shown in Fig. 8(b).
The constructed field φK+1 is enlarged and surface Ω1 is reconstructed, and X1,1 to X1,K+1 are then obtained, as shown in Fig. 9(a). The constructed field is enlarged while reconstructing the surfaces alternately until all fields have been used. Suppose then that the full FOV was achieved when surface Ωk was reconstructed, as shown in Fig. 9(b). At this point, Xk,1 to Xk,M are then obtained. There are still some solution for fields on other surfaces that have not been solved, e.g., X1,K+2 to X1,M. Then, all surfaces should be reconstructed, except for surface Ωk, using the full FOV. All surfaces are expanded to a finish and the optimal solution for X is obtained, as shown in Fig. 9(c). The calculation process is illustrated in Fig. 10, after all surfaces have been expanded to a finish. This system can be used as a good starting point for further optimization processing. Further optimization processing is doing with optical design software.
3. Design of large relative aperture and wide FOV freeform imaging systems
In this section, a freeform off-axis three-mirror imaging system with high performance is designed based on the method proposed in Section 2. The optical system specifications are listed in Table 1. When compared with other systems operating over the same FOV, this system has a very large relative aperture.
To make the mathematical calculations more stable during the construction process, a 1/3 scale system is used in the design method. After construction, the system is triple-scaled to achieve the design objective. An initial system with specific planes is first established, in which the secondary mirror is used as a stop. The initial system layout is shown in Fig. 11(a). This system can be considered to be the initial value of Eq. (5). Here, the decentering and tilt of the planes are given based on the aim of eliminating obscuration. The system operates over 8° to 9° fields in the Y-direction, and −30° to 30° fields in the X-direction.
The system is symmetrical about the YOZ plane, and thus only half of the full FOV is considered during the construction process. We sample one field at intervals of Δφ = 0.5°. 61 fields along the X-direction and three fields along the Y-direction are sampled in total, as shown in Fig. 11(b). For each field, 98 feature rays are sampled by following the polar ray grids (14 polar angles with equal intervals and seven different pupil coordinates along each radial direction). Because the field along the Y-direction is narrow, three fields are considered along the Y-direction in each sub-step calculation process simultaneously. During the fitting process, a 4th order XY polynomial without odd components of x is used to describe the freeform design in this example.
where k is the conic constant, c is the surface curvature, and Ai is the coefficient of the x-y terms. The tertiary mirror is first constructed using the rays of field φ1 = 0.5°, as shown in Fig. 12(a). As the field is enlarged, the surfaces are constructed in a tertiary-secondary-primary order. When all surfaces have been constructed, system operates over −1.5° to 1.5° in the X-direction and 8° to 9° fields in the Y-direction. Then, the expansion process is used to obtain the wider fields required. The surface expansion order is the same as the construction order. The layout of the system with its −15° to 15° field along the X-direction is shown in Fig. 12(b). After all surfaces have been expanded, a freeform off-axis three-mirror system with a −30° to 30° FOV is obtained. The layout is shown in Fig. 12(c). The spot diagram of this system after triple-scaled is shown in Fig. vc 13. The RMS of spots of all fields are less than 1.5mm. The fields near the center of FOV have smaller spots which RMS are less than 0.5mm. Result shows that the light from each field can effectively converge near its ideal point and this system can be used as a good start point for further optimization.
Then, this system is used for further optimization. A system with good image quality is obtained after further optimization using optical design software. Some constraints are considered during the optimization process to prevent the light from being blocked. The final system can be obtained rapidly via optimization from the starting point. The primary mirror is rectangular with a size of 689.5 × 93.8 mm2. The diameter of the secondary mirror is 60.2 mm. The tertiary mirror is also rectangular with a size of 526.4 × 173.2 mm2. A cross-sectional view from the YOZ plane of the final system is shown in Fig. 14(a). A cross-sectional view from the XOZ plane of the final system is shown in Fig. 14(b). While the curvatures of the primary mirror and the secondary mirror are small, they still have important aberration correction effects. The modulation-transfer-function (MTF) plot of the final system is shown in Fig. 15(a), with modulation of more than 0.58 at 100 lps/mm. The average RMS wavefront error of the system is 0.089λ at a wavelength of 530.5 nm, which is sufficient for visible imaging. The field map of the RMS wavefront error is shown in Fig. 15(b). The maximum relative distortion of the system is 8.1%, which is much better than that of conventional wide FOV systems. The distortion grid is shown in Fig. 16. The results show that good image quality was achieved.
In this work, a multiple surface expansion method is proposed. The method is suitable for the design of optical systems with multiple freeform surfaces, particularly for systems with high performance measures such as large relative aperture sizes and wide FOVs. In this method, an initial planar system is first established. Each plane is the expanded to a freeform surface by enlarging the constructed fields. The construction result is then used as a starting point for further optimization using optical design software. The final design can be obtained rapidly via optimization from this starting point. The process of this method can be realized by a conventional software utilizing its programming features. A freeform off-axis three-mirror imaging system with large relative aperture size and a wide FOV was designed. The system operates at F/2.5 with an effective focal length of 150 mm and a 60° × 1° FOV. Good image quality was achieved in the visible bands. The RMS wavefront error of the designed system is 0.089λ (working wavelength λ = 530.5 nm). The system also has low distortion that is much better than that of conventional wide FOV systems.
National Natural Science Foundation of China (61775116).
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