1. Introduction

Off-axis reflective imaging systems have the advantages of fewer components, no central obscuration, a long focal length, a wide field of view (FOV), and high transmission. Therefore, they have been widely used in imaging applications such as remote sensing [1], adaptive optics scanning ophthalmoscopy [2], and infrared imaging systems [3,4]. In an off-axis reflective imaging system, the conventional spherical or aspherical surfaces can be replaced by freeform surfaces, which are defined as nonrotationally symmetric surfaces that offer more degrees of design freedom. The introduction of freeform surfaces ensures that when an optical system requires a large FOV and low F-number, the image quality will not be degraded by various aberrations [5].

Because of the asymmetric configuration of freeform surfaces, specialized design methods play an important role in realizing such optical systems. Among these methods, the multiparametric optimization method [5,6] is widely used in commercial optical design software to design various optical systems. However, direct design using commercial software requires extensive optical experience and a great deal of time because freeform surfaces require more parameters than conventional spherical or aspherical surfaces. Hence, a good starting point provided by a direct design method is essential for subsequent optimization by optical design software. Three major types of direct design method have been developed in recent years: the partial differential equation design method [7–9], the simultaneous multiple surface (SMS) method [10–12], and the three-dimensional construction-iteration (CI-3D) method [13]. These methods can generate freeform surfaces for high-performance optical systems, but the effect of manufacturing constraints on the system configuration and the shape of the freeform surfaces have not been taken into account.

If manufacturing constraints are not considered, systems designed only in terms of optical performance are likely to be unsuitable for practical production owing to poor machinability and huge time cost. Manufacturing constraints are notably useful for simplifying the machining process. Here, the concept of a reference surface is introduced to characterize the manufacturing constraints in various processing techniques. The more closely the shape of the workpiece surface resembles that of the reference surface, the less difficult it is to machine the workpiece. For the turning process, the reference surface is a revolving symmetric surface [14], and for milling, the reference should be an extruded surface [15]. The raster milling machining request a cylindrical reference surface to achieve high manufacture efficiency. In addition, the manufacturing perspective has been considered in order to reduce the need for alignment [16] caused by the asymmetry of freeform surfaces. In a previous study, a rotational tool-turning method was proposed to machine freeform prisms to avoid alignment by means of integrated manufacture [17]. Furthermore, we proposed an alignment-free manufacturing approach to machine a freeform off-axis multimirror system, which can achieve good optical performance after a customized machining process [18]. This raster milling process can be significantly simplified if all the freeform surfaces are approximated with respect to the same reference (cylindrical) surface; thus, the resemblance between the freeform surface and cylindrical surface needs to be considered in the optical design stage. Several design examples involving manufacturing and alignment have been reported, including the Artificial Accommodation System, in which the freeform surfaces of a micro-machine were optimized with respect to robustness tolerances [19], and an off-axis three-mirror system based on integration of the primary and tertiary mirrors [16,20]. However, there is currently no systematic freeform design method that concentrates on manufacturing constraints to regulate the shape of all the surfaces in an optical system.

In this paper, a freeform design method is proposed for a cylindrical freeform off-axis reflective imaging system considering the machining constraints for raster milling. Freeform surfaces are constructed based on the constant optical path length (OPL) condition to form the initial configuration of the optical system. Unlike the process in previous direct design methods, multiple reference points in different fields are fixed to control the distribution of the remaining feature data points on the freeform surface. Then an iterative process is employed to further improve the image quality and restrict the shape of the freeform surface to prevent excessive deformation by adjusting the coefficient of the surface expression using an optimization algorithm. The coefficients are adjusted after surface fitting for each freeform surface. Finally, a freeform off-axis three-mirror imaging system is designed as an example, where all the mirrors are approximated by a cylindrical surface for ultraprecise raster milling. The final system operates at F/2.0 with a 100 mm entrance pupil diameter and a 4° × 4° FOV. The freeform mirrors in the system are guaranteed to be distributed along a cylindrical surface with a radius of 150 mm.

2. Manufacturing demand and design strategy

The manufacturing constraints vary according to the practical application. Figure 1(a) demonstrates an integrated manufacturing configuration for an off-axis reflective imaging system based on raster milling. The curvature radii of all the freeform surfaces must not be smaller than the tool’s rotational radius. Hence, the position and shape of each freeform mirror need to be limited in the optical design according to this manufacturing constraint. It is therefore inferred that the freeform surface can be approximated by a certain reference surface, i.e., the cylindrical surface in the above case [Fig. 1(b)]. To characterize the deviation between the freeform surface and the reference surface in the design, an index called the degree of deviation from the reference surface (DDRS), σ_DDRS, is defined to evaluate the surface shape:

 

Fig. 1 (a) Machining configuration for freeform three-mirror system. (b) Geometric relationship between the freeform surfaces and the reference surface.

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Figure 2 illustrates the design strategy, which consists of the construction stage and the iteration stage. Here, we integrate the constraints of the machining configuration into the construction stage, in contrast to traditional direct design. The optimization combines both the optical function and the manufacturing constraints in the iteration stage. During these two stages, the system configuration and the shapes of the freeform surfaces are roughly determined in the construction stage, and the image quality is further improved in the iteration stage. Constraints are essential in both stages to decrease the deviation between the freeform and reference surfaces.

 

Fig. 2 Flow chart of the entire design process.

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The first step in the construction stage is to define the feature rays and feature points on the aperture stop. The essential task of the construction stage is to find the appropriate feature points on the feature rays to form freeform surfaces. In the traditional direct design method, one or a few fixed reference points are required as starting points to determine the coordinates of all the other feature points. Because there are only a handful of reference points, the coordinates of the remaining feature points are actually not well controlled by the designer. In contrast, in this study, a single reference point, which is on the reference surface, is fixed in each sampled field. In this way, the remaining feature points can be distributed extremely near the reference surface, leading to the desired freeform surface after surface fitting.

In tradition iterative design process, iteration focuses only on the image quality of the optical system; the shape of the freeform surface becomes unpredictable as iteration proceeds. Hence, in the iterative process, coefficient adjustment is performed after the freeform surface is refitted using the newly calculated feature points. The expression coefficients are taken as variables, and the image quality and manufacturing constraints are expressed in terms of a merit function; an optimization algorithm is then used to obtain the variables. Therefore, the method is considered to be a direct design method within a local multiparameter optimization to improve the image quality while restricting the position and shape of the freeform surface.

3. Construction and iteration of freeform surfaces

In imaging systems, light rays from multiple fields and different pupil coordinates should be redirected to their ideal image point according to the object–image relationship. Freeform surfaces can be generated from feature data points on the light rays. In this section, we propose a method to design freeform surfaces in a reflective optical system with shape and position constraints such that all the surfaces are expected to be in close contact with a cylindrical surface. The method consists of two main stages:

3.1 Construction of freeform shape based on the constant OPL condition

The entire design process is a manipulation of feature light rays and feature data points; hence, feature rays have to be defined before concrete calculation. For convenient ray tracing, feature rays and feature points are defined as in Ref [13]. Feature rays are normals to the wavefront that support optical design within the scope of geometric optics, whereas feature points lie on feature rays and constitute the freeform surface. Each feature point has a corresponding normal that indicates the reflection of a feature ray. As most optical systems use circular apertures, Fig. 3(a) shows the pupil coordinates on the aperture plane as an example. A total of K different pupil coordinates (data points) are sampled uniformly on the aperture, so K different feature rays are employed in each field. If M different fields are sampled in the design process, a total of M × K feature rays are emitted from the aperture, as shown in Fig. 3(b). For simpler computation and efficient design, only a configuration in which the aperture stop is located in front of the optical system is discussed in this paper.

 

Fig. 3 Data points and feature rays defined before calculation. (a) Sampled feature points on a circular aperture. (b) Feature rays emitted from data points.

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Ideally, all the feature rays should be redirected to their target points by the unknown freeform surfaces, as shown in Fig. 4. In an imaging field, target points are ideal image points on the image plane corresponding to feature rays. Feature points, incident feature rays, and target points can be used to calculate the surface normal vectors at feature points using the law of reflection. Then the unknown freeform surface is obtained by surface fitting. As incident feature rays and target points are determined in advance according to the given object–image relationship, the critical step of freeform surface construction is finding the feature data points on the unknown surface.

 

Fig. 4 Feature rays should be redirected to their corresponding target point after reflection by the unknown freeform surface.

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In the previous direct design methods that focus on data points, only one or two reference points are chosen, usually the intersections of the chief ray in the central field of view. with a plane placed at the approximate position of the unknown surface. The remaining feature data points on the feature rays are calculated according to the special Nearest-Ray Algorithm [13]. The distribution of feature points calculated by this method is uniform, and freeform surface fitting by these points shows high performance in ray redirection. However, the distribution of the remaining feature points is not controlled by the algorithm, because only one reference point is chosen by the designer.

Therefore, a novel method based on the constant OPL condition is applied to calculate the feature data points. According to Fermat’s principle, all the rays from a given wavefront share a constant OPL to their ideal image point. Feature data points on the feature rays can be calculated using the OPL determined by the chief ray in each field. The details of this method are described below.

The starting points of feature rays are very important, because the OPL is crucial to the calculation. In finite imaging optics, the starting points are simply the points on the object. However, in infinite imaging optics, the incident light rays are perpendicular to the plane wavefronts because the object is at a limitless distance. A dummy plane located at the aperture stop can be fixed to find the starting points, as shown in Fig. 5. For each sampled field, the dummy plane is obtained according to the center of the aperture stop S₀ and the direction of the chief ray R₀. The remaining feature rays R_i (i = 1, 2, …, K − 1) in this field intersect the wavefront plane; this fact can be used to obtain their starting points S_i (i = 1, 2, …, K − 1).

 

Fig. 5 Feature rays intersect the wavefront plane, determining their starting points. The optical path length in every field of view is calculated according to these starting points.

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In each sampled field, the chief ray R₀ emitted from S₀ is selected as the reference ray, on which a reference point P₀ is fixed according to the position of the unknown surface. P₀ is taken as the intersection of R₀ with the unknown freeform surface. The reference ray should be redirected toward the target point T (in this case, T is the ideal image point) after being reflected at P₀, as illustrated in Fig. 6. Because the OPL is constant, all the feature rays in this sampled field should have the same OPL, namely,

 

Fig. 6 Feature points P_i are calculated based on the constant OPL condition.

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Because a constant OPL is determined, feature point P_i (i = 1, 2, …, K − 1) can be found on every feature ray in this field, which satisfies the condition:

All the feature rays share the same target point T in this field, so the surface normal at each feature point P_i can be deduced using the law of reflection. The above steps are repeated for each sampled field to obtain the feature points and the corresponding normals. The freeform mirror is generated by fitting both the feature points and the normals. As the feature points in each field are essentially centered on the reference point, the selection of the reference point in each field strongly affects the distribution of feature points. Note that we can control the distribution of feature points by using well-chosen reference points, because multiple reference points are employed in this method. To restrict the shape of the freeform surface, the reference points should be at the intersections of the reference rays with the reference surface. Compared with traditional direct design method, this method with multiple reference points can generate freeform surfaces that are more similar to the reference surface, which will be verified in Section 3.

If there are other reflective or refractive surfaces between the unknown freeform surface and the image plane, the calculation process is more complicated. We assume that there is a reflective surface F₂ between the unknown surface F₁ and the image plane, as shown in Fig. 7. In this case, the optical path of the reference ray cannot be calculated directly. It is necessary to find a point M₀ on surface F₂ that satisfies Fermat’s principle, which means that the OPL between P₀, M₀, and T is an extremum (generally a minimum). Such process can be done by the least square method. After confirming M₀, the constant OPL in this field is determined as

 

Fig. 7 A search process is required to determine the feature points on the feature rays.

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For each remaining feature ray in the field, we search for a point P_i on the ray that satisfies the condition

The flow chart of the construction process is shown in Fig. 8. During the entire construction process, every freeform surface in the optical system is constructed in proper sequence, which is generally from the image plane to the object plane. This is because the construction of other surfaces can further enhance the image quality of the system if the last surface is constructed first. When one of the freeform surfaces is being constructed, the rest of the optical system must remain unchanged. All the reference points should be at the intersections of the reference rays with the desired reference surface so that all the freeform surfaces are well restricted according to the manufacturing constraints. The essential process in this method is to calculate the feature points on an unknown freeform surface with position constraints so as to control the approximate shape of the freeform surface after fitting. When all the surfaces in the system have been constructed as freeform surfaces, the optical system is built. The image quality may not be satisfactory; however, this is acceptable because the design must consider not only the image quality but also the manufacturing constraints. This result is taken as the initial structure for the next iterative optimization, in which the image quality will be further improved.

 

Fig. 8 Flow diagram of the construction process. U denotes the number of freeform surfaces in the system. M denotes the number of fields sampled in the design process.

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During the construction process, because the manufacturing constraints were considered, the reference surface can be employed directly as the initial surface to build the optical system instead of a planar or spherical surface. However, the feature rays may not converge on the next surface after being reflected by the reference surface, leading to oversizing of the next surface. Thus, an auxiliary point can be placed near the next surface as the target point for all feature rays. Then the reference surface is transformed into a freeform surface using the method described above so that the feature rays converge on the next surface, making the system configuration more reasonable.

3.2 Iterative optimization with coefficient adjustment

By using the construction method described above, multiple freeform surfaces can be constructed to obtain an initial optical system configuration. However, the actual intersection points of the feature rays with the image plane may deviate significantly from the ideal image points. Therefore, an iteration process is employed to eliminate the deviation. To evaluate the image quality of an optical system, here we use the root-mean-square (RMS) deviation σ_RMS of the actual image points from the ideal image points [13]:

After several iterations, the RMS deviation will be significantly reduced, improving the image quality remarkably. However, in the iteration process, the feature points on the surfaces are determined simply from ray tracing results without considering the manufacturing constraints. The distribution of feature points depends entirely on the initial configuration of the optical system and the target image points. Therefore, the freeform surface is very likely to deviate from the reference surface after iterative optimization, making our work in the construction process in vain. Additional constraints are indispensable during iteration to control the shape of the freeform surfaces.

To constrain the shape of the freeform surface, the coefficients of the surface expression are optimized. In the iteration process, the coefficient of the freeform surface expression can be modified directly, unlike the uncontrollable positions of feature points. Here, the optimization of a freeform surface is divided into three main steps: 1) Recalculation of the feature points and normals, 2) surface fitting according to these data, and 3) adjustment of the coefficients of this newly identified freeform surface. Coefficient adjustment reduces the RMS deviation well while minimizing the DDRS, where σ_RMS + w × σ_DDRS is set as the merit function, and the coefficients of the surface expression are set as variables to find the optimal solution that minimizes the merit function. In other words, after the feature points and the corresponding normals are determined, the surface expression coefficient will be optimized directly. An optical freeform surface is commonly expressed as a particular polynomial, for example, the XY polynomial, Zernike polynomial, or radial basis function. In this paper, the XY polynomial is chosen to describe the freeform surface. The XY polynomial surface is a commonly used nonrotationally symmetric freeform surface, the general expression of which is:

Because the shape of the surface is controlled mainly by the coefficients of the low-order x–y terms, only the first several coefficients of the XY polynomial are set as variable parameters in the optimization algorithm, which can also improve the computation speed. The specific number of adjusted coefficients needs to be determined by the designer on the basis of the system complexity. The simplex method is used to optimize the coefficients so that the merit function σ_RMS + w × σ_DDRS decreases continuously. Moreover, the weight w strongly affects the optimization result. A too-small value of w may reduce the constraint on the shape of the surface, whereas an excessively large w will result in a larger RMS deviation. The weight w should be properly chosen to balance the contributions of the RMS deviation and DDRS to the optimization, because manufacturing constraints may require that the image quality be sacrificed. The specific value of the weight w should be determined by the designer according to several experiments based on the practical design demand. Coefficient adjustment immediately follows surface fitting in the iteration process and not only controls the shape and position of the freeform surface, but also improves the image quality. It can further reduce the RMS deviation compared with that of an iteration process that focuses only on the feature data points and normals, which is equivalent to improving the performance of the iterative optimization. This will be shown in the following design case. The flow diagram of iterative optimization with coefficient adjustment is shown in Fig. 9.

 

Fig. 9 Flowchart of iterative optimization with coefficient adjustment. M denotes the number of fields sampled in the design.

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4. Case study and discussion

To verify the feasibility of the design method described above, an alignment-free freeform off-axis three-mirror imaging system with manufacturing constraints is designed. The specifications of the optical system are listed in Table 1. The system has a cylinder-like structure, as described in Section 2. All the freeform surfaces in the system should be in close contact with the surface of a cylindrical surface having a radius of 150 mm. This special configuration can significantly reduce the difficulty of integrated manufacturing.

Table 1. Specifications of freeform alignment-free off-axis three-mirror imaging system

View Table

4.1 Surface construction stage

The first step in the construction stage is to determine the sampled fields. Here, 81 fields over the entire filed of view were employed. In each field, 49 different pupil coordinates were sampled. Thus, a total of 3969 feature rays were applied to construct the freeform surfaces. The target point of each ray was its ideal image point on the image plane according to the object–image relationship. The expression for the cylinder used in the design process is:

 

Fig. 10 Layout of the alignment-free off-axis three-mirror imaging system. (a) Initial structure of primary mirror M1 and the auxiliary point used to construct M1. (b) Ray tracing result after M1 has been constructed. (c) Layout after all mirrors have been constructed.

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Here, an auxiliary point was set as the target point for all the feature rays; it was located close to the secondary mirror, as shown in Fig. 10(a). According to the auxiliary point and feature rays, the primary mirror was transformed into a freeform surface using the method described in Section 3.1. Thus, the reconstructed primary mirror converged all the feature rays, thereby reducing the size of the secondary mirror. The DDRS of the primary mirror constructed based on the constant OPL condition was 3.27619 mm, which is an acceptable value compared with the value of 3.43528 mm obtained by the design method with only one reference point. It is observed that the use of multiple reference points in this method is effective in constraining the shape of the freeform surface. In this case, the primary mirror is a freeform surface, the secondary mirror is a plane mirror, and the tertiary mirror is a cylindrical mirror, as shown in Fig. 10(b).

The tertiary mirror was constructed according to the ideal image points and reference points on the cylindrical surface. Then, the secondary and primary mirrors were constructed successively so that all the surfaces in the optical system were transformed into freeform surfaces, as shown in Fig. 10(c). The entire construction process followed three steps: 1) generating the tertiary mirror M3, 2) generating the secondary mirror M2, and 3) generating the primary mirror M1. The RMS deviation was used to evaluate the image quality, and the DDRS was used to evaluate the deviation between the freeform surface and the cylinder. Figure 11(a) shows the RMS deviation of the optical system after each step. Clearly, the image quality improved gradually as each surface was constructed as a freeform surface. Figure 11(b) shows the DDRS of each surface after the construction process. All of the freeform surfaces are sufficiently close to the cylindrical surface.

 

Fig. 11 (a) RMS deviation of the optical system after M3, M2, and M1 are generated. (b) Degree of deviation from the cylinder (DDRS) of freeform surfaces after three steps.

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4.2 Iterative optimization stage

The iterative process with coefficient adjustment was then employed to optimize the surfaces in the system to further improve the image quality. Because the optical system is symmetric about the YOZ plane, only the even-order terms of x in the XY polynomial are used. In this example, eight terms of the XY polynomial surface up to the fourth order were used in the design; the conic surface term was omitted, and the polynomial is expressed as:

In Eq. (10), A_i is the coefficient of the x–y terms. As described in Section 3, the simplex method was applied for coefficient adjustment, in which the first four coefficients (i.e., A₁, A₂, A₃, A₄) were chosen as variables, and σ_RMS + w × σ_DDRS was used as the merit function. The weight w was set to 0.1 to improve the image quality while maintaining the closeness of the freeform surface and the cylinder. For convenience, the iterative processes with and without coefficient adjustment are denoted as coefficient iteration and point iteration, respectively. Figure 12(a) shows the convergence behavior of the RMS deviation for coefficient and point iteration versus the number of iteration steps (red and blue lines, respectively). The RMS deviation decreased by 72.0% after 140 iteration steps in coefficient iteration. Note that coefficient iteration performs better at reducing the RMS deviation than point iteration, in which the RMS deviation converges rapidly after 20 iteration steps. The DDRS of each surface after iteration is shown in Fig. 12(b). In coefficient iteration, the DDRSs of the primary and tertiary mirrors are well constrained, although the DDRS of the secondary mirror increases slightly. For the entire optical system, the sum of the three surfaces’ DDRSs after coefficient iteration is 8.82358 mm, compared with 9.38861 mm after point iteration. Note that both iterative results share the same initial system structure, in which manufacturing constraints have been taken into account. Moreover, in practical ultraprecise machining, a DDRS reduction of 0.56503 mm can greatly enhance the processing efficiency or facilitate integrated manufacturing owing to the smaller lathe slider motion and larger tool rotation radius. Coefficient adjustment was proved to be effective in both constraining the shape of the freeform surface and improving the image quality. The optical system layout after coefficient iteration is shown in Fig. 13(a).

 

Fig. 12 (a) Convergence behavior of the RMS deviation for the two iteration types versus the number of iteration steps. (b) DDRSs of each surface after each type of iteration.

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Fig. 13 Layout of optical system after (a) coefficient iteration and (b) optimization using Zemax.

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The optical system after coefficient iteration is then imported into Zemax for further optimization. The DDRS should be considered in the operation using Zemax; otherwise, the constraints previously applied in the design process may disappear with multiparametric optimization. The layout of the optimized design result is shown in Fig. 13(b).

The imaging performance is evaluated in terms of the RMS spot radius in selected fields, as shown in Fig. 14. The RMS spot radius in various selected fields ranges from 4.626 to 10.417 μm, which is less than the side length of the detector pixels. The maximum field curvature is about 0.3 mm, and the maximum distortion is approximately −3%, as shown in Fig. 15. The average modulation transfer function (MTF) of this method is approximately 0.83503 at 30 lp/mm, as shown in Fig. 16. It can be deduced that the optimized freeform off-axis three-mirror imaging system exhibits the desired optical performance.

 

Fig. 14 RMS spot diagram in Zemax.

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Fig. 15 Field curvature and distortion of the optimized optical system.

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Fig. 16 MTF of the optical system.

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The DDRSs of the primary, secondary, and tertiary mirrors after optimization are 2.961, 0.129, and 5.350 mm, respectively. The sum of the three mirrors’ DDRSs is 8.44 mm, which indicates that the shape of the freeform surfaces was well constrained. The final design result balances the image quality and the shape requirement for the freeform surface.

This design example clearly demonstrates the feasibility of the design method using both a constant OPL and coefficient adjustment. By increasing the constraints on the shapes of the surfaces, the optical performance of the system was improved. The method is highly efficient and easily converges to a good design result in comparison to the use of optical design software starting directly from planes when high performance and manufacturing constraints must both be considered. Note that optimization by optical design software remains necessary, but the optimization process is much more effective because only a few parameters are set as variables using the proposed method.

5. Conclusion

Considering manufacturing constraints in design cannot only facilitate machining, but also have advantages such as eliminating optical alignment, which is of great value in expanding the application of freeform optics. In this paper, a direct design method for a cylindrical freeform off-axis reflective imaging system with manufacturing constraints is proposed, in which all the freeform surfaces are closely approximated by a reference surface to facilitate ultraprecise raster milling. The design method has two main steps:

The usefulness of this design method is demonstrated by designing a freeform alignment-free off-axis three-mirror imaging system in which all the freeform surfaces are in close contact with a cylindrical surface having a radius of 150 mm. The final optical system design guarantees acceptable image quality while simplifying practical manufacture. This design idea provides a feasible route to designing refractive or reflective systems with special requirements related to manufacturing constraints.