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Abstract
In this paper, we propose an off-axis reflective system design method based on a non-rotational symmetric conicoid-based freeform (CBF) surface description. The base description avoids complicated calculation of decenter and tilt when using the conventional conic expression, thus simplify the system modeling and optimization process, and it can reduce the number of coefficients that needed to represent mild freeform surfaces. A design method that includes the automatic initial system searching, preliminary optimization with rotationally symmetric surface deviation and fine-tuning with non-symmetric surface deviation is proposed. Two three-mirror systems have been designed to demonstrate the feasibility and conveniences of the proposed method.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Freeform optics plays an important role in improving the performance of off-axis optical systems. The non-symmetric property and extra degree of freeform surfaces enable new optical structures, reduce elements number, and improve the imaging performance [1–6]. Particularly, off-axis unobscured mirror systems designed with freeform surfaces generally outperform those with rotationally symmetrical surfaces, considering the available Ref. [7–13]. Freeform surfaces have also shown advantages in many other kinds of systems, such as near-eye displays [14–17], panoramic lenses [18], and illumination system [19–21].
Different design methods for freeform off-axis reflective systems have been put forward. Point-by-point surface contour calculation methods, e.g., partial equation methods [22–24], simultaneous multiple surface (SMS) method [25–27], construction-iteration (CI) method [28,29], are able to establish starting points with aspheric or freeform surfaces directly with certain initial values. F. Duerr et al. proposed a deterministic direct design method based on differential equations derived from Fermat’s principle and then solved using power series [30,31]. Yet to obtain a suitable result with best performance, an optimization process is necessary. Nodal aberration theory has become a promising guide in the choice of OPC for off-axis mirror systems [32–38], and researchers have combined the aberration theory with an automatic optimization process to find good initial systems [39,40]. Besides, the concept of solution space survey has been proposed in recent years, which aims to explore good candidates or best results with a pre-determined range of variable space [41,42]. These survey processes can be regarded as a super global optimization process; they are powerful but time-consuming for a certain design target. It remains to be found a design method that both releases the burden of designers and makes the optimization process efficient.
Since conic sections have special optical properties that are often exploited in lens design, it is conventional to express rotationally symmetric surface shapes in terms of their deviation from the sagittal representation. The classic description of freeform surface in the optical design community is described by the sum of the conic base and the power series polynomial deviation or other forms, it is most widely used but it is poorly suited for design purposes. Better ways to specify freeform shapes can facilitate optical design.
Many alternatives are presented for working with deviation from the conic base rather than the base itself. Rodgers studied different types of non-standard aspheric deviation as early as in 1984 [43]. Chase et al., used rotationally symmetric NURBS as the deviation [44]. In 2007, Forbes put forward the Q-type surface to describe aspheres, and later in 2012 the freeform Q-type surface was proposed [45,46]. By using Q-type surfaces, the fabrication and testing difficulty can be evaluated and controlled in the optical design process. In 2008, Cakmakci proposed the radial basis functions surface, where Gaussian functions are used to replace the traditional polynomial deviation [47]. Rogers proposed the keystone surface in which distortion transformation is made for variables x and y to improve the ability to correct aberrations [48]. The studies mentioned above combined the expression of additional deviation with optimization, fabrication and aberration characteristics. For the description methods mentioned above, the asymmetric deviation is only provided by the deviation items. In order to present the appropriate non-symmetrical profile for freeform surface, many asymmetric deviation coefficients are used as extra design degree of freedom. However, too many coefficients increase the difficulty in controlling the surface profile and slow down or even stagnate the optimization process.
Another issue often encountered in off-axial system design is the ray-tracing failures in optimization due to inappropriate surface displacements or tilts. For a rotationally symmetric or any other type of surface that has a surface normal parallel to its z-axis at its origin point, when the surface is only tilted, it is possible to obtain its folded OAR without any ray-tracing operation. But if the surface is both displaced and tilted, the folded OAR is also relevant to the incident position on the surface and can only be obtained via real ray-tracing accurately.
Hence, in the design process the maximum displacement mount is usually controlled for stability purpose and this means only the region near central area of the surface expression is used. This can be alleviated if the vertex (i.e., the origin) of the optical surface can be set as an arbitrary position on that surface. In this way, the center of aspheric or freeform terms can be on any position on one optical surface.
The origin position and tilt values of optical surfaces are determined ahead of the initial system design and maintain almost unchanged during the process. One reason for the shortage of initial system is that the non-symmetric deviations have important influence on the optical configuration. If the surface base provides the amount of non-symmetry and the deviation is rotationally symmetric, then the initial system can be obtained by dealing with the system described by surface bases that have fewer coefficients. Xu et al. and Schiesser et al. conducted research on the use of non-rotationally symmetric conicoid base [49–51].
In this paper, we discussed the motivations for proposing a general and non-rotationally symmetric conicoid base (CB) and a conicoid-based freeform (CBF) surface expression. It is described in a way that benefits the establishment of off-axis systems and speed up the optimization convergence by reduction of coefficients. A design method based on CBF surface upgrade is proposed to facilitate the design process of off-axis unobscured mirror systems.
2. Conicoid-based surface expression
In optical design software, conicoid is expressed as a conic surface with curvature c and conic factor k, satisfies the requirement (k < 0) [52,53]. This expression method cannot fully describe the conicoid surface. The drawback of current conic surface can only represent one-half of an ellipsoid (solid curve) with its center on one vertex of the major axis, as shown in Fig. 1. Points B and C cannot be represented, rays F1C and F1B cannot be traced. In this paper, the ellipsoid case is firstly analyzed.
In order to make use of the optical property of ellipsoids, it is often that two focal points are on different sides of surface normal of the incident point. And the optical region on the ellipsoid that is far from either end of the major axis may be used. In this case, the conventional conic expression is not suitable. Thus a general expression that can represent any section on the ellipsoid and hyperboloid will be very helpful.
Based on the two considerations mentioned above, the conicoid surface is described with three parameters, namely, lo, li, and φ with the assumption that Z-axis is the angular bisector of ∠F1OF2 and its foci F1 and F2 are both on YOZ plane. The diagram of the coordinate is shown in Fig. 2(a), where lo and li denote the distance between the surface origin O and two foci F1 and F2 respectively, and φ is the angle between OF1 and the normal of the surface at point O. For this ellipsoid surface in the coordinate system shown in Fig. 2(a), the following geometrical relations:
Fig. 2. Coordinate representation of the conicoid base. (a) The ellipsoid base and (b) the hyperboloid base.
Figure 3 shows the different coordinates in representing ellipsoid bases. In Fig. 3(a), the conventional expression using conic constant is utilized, in which the surface origin is placed on the major axis. Left section of the ellipsoid cannot be presented. It also involves complicated calculation to obtain the position of the two foci, and it requires surface decenter to model the system.
Fig. 3. Expression of the ellipsoid base. (a) The conventional conic expression, (b) CB expression with origin on the minor-axis, and (c) the CB expression with origin on neither major-axis nor minor-axis.
Moreover, the magnification of the surface cannot be calculated easily if the object plane is not perpendicular to the Z-axis. On the other hand, in CB expression, surface profiles in Fig. 3(b) and (c) can be easily represented and modelled with lo, li, and φ, in optical design software. The tilt angle equals φ when “decenter and bend” mode is used, and the distances from the focal points to the surface origin is also given in the surface expression. The method using lo and li to determine the magnification β is defined by
For the hyperboloid, its two foci are on different sides of the surface. According to the coordinate system shown in Fig. 2(b), the following geometrical relations can be obtained as
The sag of the hyperboloid, zh, can be expressed as
In the case of hyperboloid base, to obtain a concave shape like the ellipsoid, we assumed that lo ≤ li, which makes the origin of the surface stay on one of the two sheets of the hyperboloid.
The shapes of CB with respect to the surface coefficients for ellipsoid and hyperboloid are shown in Table 1 and Table 2, respectively. Since paraboloid surfaces can be regarded as extreme condition of ellipsoid and hyperboloid shapes where one focus is placed at infinity, they can be expressed in either Table 1 or Table 2. As indicated by the tables, the CB expression can represent any part of the conicoid, plane, or sphere, and by controlling its parameters, CB can be either rotationally or non-rotationally symmetric.
Table 1. Shape of the ellipsoid base
Table 2. Shape of the hyperboloid base
Figure 4 shows the shape and contour of different CB surfaces. For ellipsoid bases, an external plane is made to ensure all rays passing through XOY plane intersect with the ellipsoid base.
Fig. 4. Different shapes of the CB surface. (a) Planar symmetric ellipsoid base, (b) sphere base, (c) rotationally symmetric ellipsoid base with surface origin on major-axis, (d) planar symmetric ellipsoid base with surface origin on minor-axis, (e) planar symmetric paraboloid base (lo > li), (f) planar symmetric paraboloid base (lo > li), (g) rotationally symmetric paraboloid base, (h) rotationally symmetric hyperboloid base, and (i) planar symmetric hyperboloid base.
One advantage of the CBF base is the local properties, which means the origin of the surface can be any point on the ellipsoid or hyperboloid. For off-axis systems described by CB surfaces, this means the surface decenter/tilt can be replaced by changing the coefficients of the surface. No surface decenters relative to the optical axis ray (OAR) is needed when the “decenter and bend” type is used to set the off-axis system, which means the OAR passes the origin points of all mirrors. Thus, changing the position of the stop surface will not affect the OAR mentioned above, and thus the probability of ray tracing failure due to stop surface change might be avoided. This property may benefit the automatic optimization process, e.g., global searching using general-purpose algorithm, where the position of the stop varies.
The expression for CB has been derived, and since the base is non-rotationally symmetric, the deviation can be either rotationally symmetric or non-rotationally symmetric. For simplicity, the surface of CB base plus the rotationally symmetric deviation (e.g., aspheric or Q-type polynomials [45]) is abbreviated as CBR, and the surface of CB base plus non-rotationally symmetric deviation (e.g., XY-polynomials, Zernike polynomials, or freeform Q-type polynomials [46]) is abbreviated as CBN. When surface deviation of CBR and CBN takes the form of aspheric polynomials and XY-polynomials, respectively, the expressions for CBR and CBN are listed as
3. Design method based on CBF surface upgrade
A design strategy based on CBF surface upgrade can be utilized, as shown in Fig. 5. First a searching process using CB surfaces is adopted to establish an off-axis, unobscured initial system with certain optical path configuration, which can be automatically done with our previous work [54]. One benefit of using CB surfaces is that they can form a perfect image point for one object point when adjacent surfaces share a common focus, which is referred to as the coincided sequential foci (CSF) condition, regardless of the aperture. By sequentially coinciding the first focal point of the latter surface with the second focal point of the former surface, a perfect image point can be obtained. This property can be used to setup the initial system. This property benefits the automatic optical configuration variation process particularly, because the chance of ray tracing failure in that process can be reduced. Figure 6 shows a three-mirror system, where F1 is the common focus of M1 and M2, and F2 is the common focus of M2 and M3. In conventional conic surface expression, it requires complicated calculation to determine position of two foci, especially when the surface is decentered or tilted. Hence, the setup of the conicoid system can be greatly simplified when using CB surface as this info is contained in the surface expression. In the searching process, the error function contains three components for the target optical configuration, the elimination of obscuration, and correction of lateral aberration for edge field points.
Then a preliminary optimization is performed on the initial system after CB surfaces are upgraded to CBR surfaces with rotationally symmetrical deviation coefficient terms. In this process, a planar-symmetrical freeform system can be represented with much fewer variables needed compared with the conventional XY-polynomial (XYP) surface. Fewer surface coefficients offer fewer degree of freedom in optimization but prevents early stagnation in the optimization process. The result of the preliminary optimization process is an intermediate system with potential to be further optimized in the next fine-tuning stage.
CBR surfaces are upgraded to CBN surfaces in this fine-tuning stage to provide most degrees of freedom. This process aims to improve the imaging quality, decreasing distortion, and achieving the final design result.
After the fine-tuning, CBF surfaces can be converted to well-accepted freeform expressions, e.g., XYP surface, to verify the design result. When aspheric or XY-polynomial deviation is used for CBR or CBN surfaces, the deviation can be converted to XY-polynomials precisely. Thus the major task is to convert the CB into XY-polynomials and the conic base represent by paraxial curvature and k. This process cannot be achieved by formula convention but can be realized by surface approximation. Since conicoid surfaces are smooth without irregular or saltation changes in local curvature, the fitting precision can be generally guaranteed. We have found that 6th to 10th order of XY-polynomial can keep the conversion within acceptable precision, yet the residual error of the base can be further fitted.
4. Design examples
In this section, two planar-symmetric, off-axis examples are shown, namely a three-mirror zig-zag system and a three-mirror system with spherical package.
4.1 Three-mirror zig-zag system
The specifications of the design example are listed in Table 3. The system is designed with a positive-negative-positive structure and without intermediate image. In the initial system establishment stage where adjacent surfaces share a common focus, no intermediate images lead to two virtual foci of M2 and two real foci of M3. Thus, the ellipsoid base can present all the three mirrors in the system. Since the initial system meets the CSF condition, all aberration share the same nodal point, which corresponds to the center field. It can be discovered that the aberration increases with the field angle.
Table 3. Specifications of the design example
After an automatic searching process from the co-axis starting point [54], the off-axis, unobscured initial system is obtained, whose 2D layout is shown in Fig. 7(a). In this process, the obscuration can be addressed and eliminated, and target structure can be fixed. The aberration correction considering the actual FOV and aperture is involved. The curve of the synthesized error function, which includes components for optical path configuration, obscuration, and aberration, is illustrated in Fig. 7(b), and the structure parameters of the initial system are given in Table 4. αi denotes the tilt angle of the ith mirror from the optical axis, β2 denotes the magnification of M2, r is the ratio of distance from M1 to M2 over lo1, and δ is the distance between M2 and M3. In the automatic searching process using CB surfaces, the focal length of the zig-zag system can be expressed as the focal length of M1 times the magnification, β, of each successive mirror. The magnifications can be determined by Eq. (3), and the focal length of M1 is li1·cosφ1 since it is an off-axis paraboloid. The RMS spot sizes of the searching result is shown in Fig. 7(c). After the preliminary optimization with CBR surfaces, the RMS wavefront error is illustrated in Fig. 7(d), which indicates a drop of 91.9% in average RMS wavefront error as compared with the initial system. In the optimization process, system parameters, e.g., EFL and F-number, are evaluated and controlled by ABCD ray transfer matrix method. In CODE V, this can be realized by performing in-built function “fct_ABCD” [51].
Fig. 7. The initial system and the preliminary design using CBR surfaces. (a) 2-D layout of the initial system, (b) error function during the initial system searching process, (c) RMS spot size of the initial system, and (d) RMS wavefront error of the preliminary design using CBR surfaces.
Table 4. Solutions of the initial system
The 2D layout and RMS wavefront error of the CBN system are illustrated in Fig. 8(a) and 8(b). The surface deviation is described by10th order XY-polynomials, only even terms of x are used to maintain the planar-symmetry about YOZ plane. The maximum RMS wavefront error of the design result is about 1/40λ and the average RMS wavefront error is smaller than 1/60λ. To verify the design result, all the three optical surfaces were converted to standard XYP surfaces, and no optimization was performed after the surface conversion. The RMS wavefront error distribution after conversion is shown in Fig. 8(c). The average RMS wavefront error rise from 0.0158λ to 0.0164λ, showing a relative increase of only about 3.3%.
Fig. 8. The final design result after the fine-tuning. (a) 2-D layout, (b) RMS wavefront error of the CBN system, and (c) RMS wavefront error after convention from CBN to XYP.
The CBN and XYP coefficients for M1 are listed in Table 5 to show the surface conversion process. It can be noted that CBN offers 1 more independent coefficient as compared with XYP of the same order, since the degree-of-freedom for the off-axis angle is within the CB expression. In the conversion process, as discussed in the Section 3, the surface approximation is only performed on the CB to obtain a XY-polynomial expression for the base, and the original XY-polynomial terms of the CBN expression can be just added to the new XYP surface. In this way, the XYP surface has the same highest order as the original CBN surface.
Table 5. Surface coefficients for M1 before and after conversion from CBN to XYP
4.2 Three-mirror system with spherical package
The second example is a three-mirror system with spherical package. The specifications of the system designed are listed in Table 6. The structure of the second example is designed to be the “negative-positive-positive” form and have no intermediate images. Since M2 and M3 both have positive optical power and no intermediate image exists between M2 and M3, M2 or M3 will be a hyperboloid surface, and in this example, M3 is chosen to be the hyperboloid.
Table 6. Specifications of the design example
Similar to the first example, the initial system searching stage is performed to obtain an off-axis initial system. To benefit the searching of feasible folding structure, we set a smaller FOV of 8° (H) × 4° (V) and a greater F-number of 2 in this process, and the parameters in the initial system searching are listed in Table 7. After a progressive optimization with surface upgrade and system parameter adjustment, the intermediate design with CBR surfaces and the final design result with CBN surfaces are shown in Fig. 9(a) and 9(b), respectively. The RMS wavefront error of the final design result is illustrated in Fig. 9(c), indicating an average RMS wavefront error of 1/20λ. Additionally, the maximum distortion of the system is 1.34% in the tangential direction and 1.08% in the radial direction. The results indicate a good optical performance.
Fig. 9. The design result using CBF surfaces. (a) 2-D layout of the design with CBR, (b) 2-D layout of the design result with CBN surfaces, and (c) RMS wavefront error of the design result.
Table 7. Solutions of the initial system
5. Conclusions and discussions
We proposed a conicoid-based freeform (CBF) surface description and a design method based on CBF surface upgrade for planar-symmetric, off-axis reflective freeform systems. Its conicoid base facilitates the initial system establishment of off-axis mirror systems by parameterizing two foci as surface coefficients, and thus the initial system is obtained via an automatic searching process. In the preliminary optimization process, when aspheric deviation is used, the surface expression effectively reduces the number of freeform coefficients, which can alleviate the convergence problem commonly encountered when too many variables are used. The stability in optimization process can thus be improved. And by using the proposed CB, the lateral displacement of optical surfaces can be substituted by the changes in surface coefficients, which helps to improve the stability of ray-tracing. In the fine-tuning optimization process, aspheric deviation is upgraded to freeform deviation, offering more degrees of freedom to obtain better optical performance. Two three-mirror systems have been designed to demonstrate the feasibility of the proposed method.
Currently, our design method in this paper focuses on planar-symmetric off-axis systems since they are the most common off-axis systems. The CB itself is planar-symmetric because of its conicoid shape. But we believe the planar-symmetric CB base also has the potential to be used in systems without plane of symmetry to facilitate the initial system searching process, if tilt angle with respect to the y-axis and z-axis are allowed in off-axis setting. In addition, the ellipsoid in this paper only has two different axes, namely, the major axis and the minor axis, and its geometrical property enables a perfect imaging between its two foci. We can make use of this property and the surface base expression is proposed. But this good property is no longer valid for a three-axis ellipsoid. We think it will be interesting to study the three-axis ellipsoid surface in the design process in our future study.
Funding
National Key Research and Development Program of China (2017YFA0701200); National Natural Science Foundation of China (62005069); Young Elite Scientist Sponsorship Program by CAST (2019QNRC001).
Acknowledgments
We thank Synopsys for the educational license of CODE V.
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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