## Abstract

In this paper, we proposed a general direct design method for three-dimensional freeform surfaces and freeform imaging systems based on a construction-iteration process. In the preliminary surfaces-construction process, the coordinates as well as the surface normals of the data points on the multiple freeform surfaces can be calculated directly considering the rays of multiple fields and different pupil coordinates. Then, an iterative process is employed to significantly improve the image quality or achieve a better mapping relationship of the light rays. Three iteration types which are normal iteration, negative feedback and successive approximation are given. The proposed construction-iteration method is applied in the design of an easy aligned, low F-number off-axis three-mirror system. The primary and tertiary mirrors can be fabricated on a single substrate and form a single element in the final system. The secondary mirror is simply a plane mirror. With this configuration, the alignment difficulty of a freeform system can be greatly reduced. After the preliminary surfaces-construction stage, the freeform surfaces in the optical system can be generated directly from an initial planar system. Then, with the iterative process, the average RMS spot diameter decreased by 75.4% compared with the system before iterations, and the maximum absolute distortion decreased by 94.2%. After further optimization with optical design software, good image quality which is closed to diffraction-limited is achieved.

© 2015 Optical Society of America

## 1. Introduction

Freeform optical surfaces, which can be defined as the optical surfaces that are not rotationally symmetric (which means the sag varies around the periphery), have more degrees of design freedom than conventional rotationally symmetric surfaces in optical design. For non-imaging systems, freeform surfaces have been successfully used in the fields of LED uniform illumination [1], patterns illumination [2,3], vehicle lighting [4,5], road lighting [6,7], and laser beam shaping [8,9]. As the freeform surfaces can significantly improve system performance while decreasing the system size, mass, and number of elements in optical design, in the recent years, they have been increasingly used in the imaging field, such as off-axis reflective systems [10–14], head-mounted-display (HMD) [15–18], freeform microlens array [19], and panoramic optical system [20].

As freeform imaging systems generally employ off-axis nonsymmetric configurations and often require advanced system specifications, it is challenging to propose effective and easy design methods of freeform surfaces and systems for imaging optics. One traditional design method of freeform imaging systems is to find existing patents or other available systems as the starting point. Then, further optimization with optical design software is applied to obtain the final design result. However, as the freeform surfaces are generally used in the off-axis or unobscured forms, there are few existing patents or systems for choice. Moreover, the system configuration, the number of elements, and the system specifications (such as the field-of-view (FOV), the F-number and the focal length, etc.) of the existing systems are generally not consistent with the expect specifications of the current design. Therefore, designers may fail to find useful solutions, or they have to spend long times improving the starting point. Another common design approach is to firstly create a co-axial spherical or conical starting point based on the primary (or Seidel) aberrations theory. Then an off-axis or unobscured form is created by using an offset aperture and/or a biased input field, or tilting the surfaces until the light clears the mirrors. As the co-axial starting point is generally far from the optimum point, designers may also difficult to find useful solutions. Consequently, some direct design methods for freeform surfaces have been developed such as the partial differential equations (PDEs) method [21–26] and the Simultaneous Multiple Surface (SMS) design method [27,28]. However, both of these two methods have a restriction on the number of field points considered in the design process (or sometimes a wide field-of-view but with a very small aperture considering only chief rays, or unconstrained object-to-image mapping [25]), which limit the applications of these design methods.

In our previous work, we proposed a direct design method of the two-dimensional (2D) freeform surface contour for imaging optics [29]. However, as this method is restricted to the 2D design, the application is limited to designing systems with linear field-of-view and a small aperture. Later, a direct design method of three-dimensional freeform surface has been proposed [12]. In this method, the light rays from multiple fields and different pupil coordinates are considered. However, the image quality is generally low after the direct design process. For example, the average RMS spot diameter is above 1mm for the design example in Ref [12]. For some other design cases, especially for the systems with advanced specifications and high performance, the image quality and distortion after the direct construction process may be even worse and further optimization will be much more difficult (or even failure). Therefore, it is both an interest and a challenge to directly design three-dimensional freeform surfaces as well as the freeform systems with general applications and a much better image quality.

In this paper, we proposed a design method for three-dimensional freeform surfaces and the freeform imaging systems. This method is based on a three-dimensional point-by-point construction-iteration process, and this is, in some degree, a general direct design method of freeform surfaces which work for either a linear or rectangular field-of-view. In the preliminary surfaces-construction stage, the coordinates as well as the surface normals of the data points on the freeform surface can be calculated point-by-point directly based on the given object-image relationships (or in a more general concept, a given mapping relationship). Compared with other point-by-point design methods, the data points calculated in this method are used to control the light rays of multiple fields and different pupil coordinates. As a result, this method is suitable for designing general and actual imaging systems which work for a certain object size and a certain width of light beam. An iterative process is then employed to significantly improve the image quality or achieve a better mapping relationship of the light rays. Three iteration types which are normal iteration, negative feedback and successive approximation are given. After the preliminary surfaces-construction and iteration stages, multiple freeform surfaces with a base conic in the local coordinate system can be generated.

Another feature of the proposed construction-iteration method is that it can be used in the design of optical systems with special configurations and multiple structure constraints, which cannot be found among the existing patents or lenses database. With this method, systems with novel configurations can be obtained. In this paper, the proposed construction-iteration method is applied in the design of a low F-number, off-axis three-mirror system. The system has a special, easy aligned configuration. The primary and tertiary mirrors can be fabricated on a single substrate and form a single element in the final system. The secondary mirror is simply a plane mirror. With the above configuration, the alignment difficulty of a freeform system can be greatly reduced. With the preliminary surfaces-construction stage, the freeform surfaces in the optical system can be generated directly from an initial planar system. Then, with the iterative process, the average RMS spot diameter decreased by 75.4% compared with the system before iterations, and the maximum absolute distortion decreased by 94.2%, which proves the feasibility of the iterative process. After further optimization with optical design software, good image quality which is closed to diffraction-limited is achieved.

## 2. Design method of freeform surfaces with a construction-iteration process

The construction-iteration method for designing three-dimensional (3D) freeform surfaces consists of two stages. In the preliminary freeform surfaces-construction stage, the freeform surfaces in the system are designed successively with a direct construction process. Then, an iterative process is employed to further improve the image quality or achieve a better mapping relationship of the light rays. The details of the method are illustrated as follows.

#### 2.1 Preliminary freeform surfaces-construction process

Before the surface construction-iteration process, an initial system with planes or other surface types such as spheres is firstly established. The following surface design process starts from this initial system. Then, in this preliminary freeform surfaces-construction stage, each single freeform surface is designed with a direct construction process with the method depicted in Ref [12]. Here, we give a brief introduction of the point calculation process. The feature light rays of multiple fields and different pupil coordinates are used in this method. Based on the given object-image relationship (or in a more general concept, a given mapping relationship of the feature light rays), the ideal image points of feature light rays on the image plane (or the target points ** T_{i}** of the rays on a target surface) can be determined. The initial data point

**on the first feature ray**

*P*_{1}**is firstly fixed. It is taken as the intersection of**

*R*_{1}**with the unknown freeform surface**

*R*_{1}**. When the**

*Ω**i*th data point

**has been obtained, it is expected that the corresponding feature ray**

*P*_{i}**can be redirected to the corresponding target point**

*R*_{i}**on the target surface, as shown in Fig. 1. According to Fermat's principle, the variation of the optical path length**

*T*_{i}*S*between two fixed points is zero. Therefore, for the data point

**and the target point**

*P*_{i}**, we have**

*T*_{i}*ds*is the differential elements of path length along the ray,

*n*denotes the refractive index of the medium,

*δ*denotes the differential variation. With this principle, the outgoing direction of the ray

**after surface**

*R*_{i}**as well as the surface normal**

*Ω***at point**

*N*_{i}**can be calculated. Then, the next data point**

*P*_{i}**can be calculated based on the special “Nearest-Ray Algorithm” [12]. Repeat the above steps until all the**

*P*_{i}_{+1}*K*feature data points on the unknown surface

**are obtained. The detailed algorithm is given in [12]. The next step is to fit the data points into a freeform surface.**

*Ω*In this paper, the data points are fitted into a freeform surface with a base conic. In addition, the freeform surface terms are fitted into in the local coordinates system in consistent with the base sphere/conic. In this way, the freeform surface departure from the base sphere/conic can be reduced as much as possible, which benefits the fabrication process. Furthermore, the surface fitting process considers both the coordinate and surface normal of the data points, which is also important in obtaining the required and accurate freeform surfaces. The freeform surface can be written as

*c*is the base curvature of the surface at the vertex,

*k*is the conic constant, $\sum _{j=1}^{N}{A}_{j}{g}_{j}(x,y)$ are the freeform surface terms (such as XY polynomials and Zernike polynomials, etc.) with

*A*being the coefficients. Firstly, the feature data points are calculated in the global (initial) coordinates system with the proposed method, as shown in Fig. 2(a). The data point (

_{j}*x*,

_{o}*y*,

_{o}*z*) corresponding to the chief ray of the central field angle among the entire field-of-view (FOV) is taken as the vertex of the surface and the origin of the new local coordinate system. Then, the data points are fitted into a spherical surface with a least-squared fitting algorithm. Through this, the curvature

_{o}*c*of the surface at the vertex as well as its center of curvature (

*x*,

_{c}*y*,

_{c}*z*) can be obtained. The new

_{c}*z'*-axis passes through the center of curvature and the surface vertex, as shown in Fig. 2(b). Generally, the optical systems are symmetric about the YOZ plane. The tilt angle θ of the surface (as well as the rotation angle of the coordinates system) in the YOZ plane can be calculatedThen, the original coordinates and the surface normal of the data points

**can be transformed into the new local coordinates system. The relationship between the original coordinates (**

*P*_{i}*x*,

_{i}*y*,

_{i}*z*) and surface normal (𝛼

_{i}*, 𝛽*

_{i}*, 𝛾*

_{i}*) of each data point and the new coordinates (*

_{i}*x'*,

_{i}*y'*,

_{i}*z'*) and surface normal (𝛼

_{i}*'*, 𝛽

_{i}*'*, 𝛾

_{i}*'*) can be written as following.

_{i}*k*can be obtained, as shown in Fig. 2(c). If the coordinates and surface normal of the conic for

**(at**

*P*_{i}*x*=

*x'*and

_{i}*y*=

*y'*) are (

_{i}*x'*,

_{i}*y'*,

_{i}*z'*) and (𝛼

_{ic}*'*, 𝛽

_{ic}*'*, 𝛾

_{ic}*'*) respectively, we can get the residual coordinate and surface normal (with the

_{ic}*z*-component normalized to −1) of the data point when the impact of the base conic is excluded from (

*x'*,

_{i}*y'*,

_{i}*z'*) and (𝛼

_{i}*'*, 𝛽

_{i}*'*, 𝛾

_{i}*'*)

_{i}#### 2.2 The iteration steps

With the construction and fitting method depicted above, multiple freeform surfaces can be generated in order to achieve a given imaging or mapping relationship of the light rays. However, the actual intersection points of the feature light rays with the target surface may have a relatively large deviation from the expected target points. In this paper, an iterative method for the three-dimensional (3D) freeform surfaces is proposed to further decrease the deviation. After the freeform surfaces in the system have been generated in the preliminary surfaces-construction stage with the method depicted in Section 2.1, this system is taken as the initial system for the next iteration step. Each iteration step is decomposed into several sub-steps to re-generate each freeform surface individually. In each sub-step, the intersections where the feature light rays intersect with the initial surface are obtained first as the data points. It means the points on the surface generated in the last iteration step are preserved. Then the surface normal of each data point is calculated point-by-point with the method given in Section 2.1. In the normal iterative process, the target point ** T_{i}** for each feature ray at each sub-step is the corresponding ideal target point

**, as given in Eq. (8) and as shown in Fig. 3(a)**

*T*_{i,ideal}**for each ray at each sub-step is decided by a negative feedback method based on the ideal target point**

*T*_{i}**for each ray and the actual intersection**

*T*_{i,ideal}**with the target surface of the current initial system, as shown in Fig. 3(b). The feedback function can be written as:**

*T**_{i}*ε*(

*ε*>0) is the feedback coefficient. When ε = 0, no feedback is employed, which means the target

**is the ideal target point and it is the same case of the normal iterative process. The extent of feedback is increasing with the higher ε. However,**

*T*_{i}*ε*should not be too large to avoid instability of the iterative steps. Similarly, if the deviation between

**and**

*T**_{i}**is too large, there will be a dramatic change of**

*T*_{i,ideal}**and induce instability [31], So, a feedback threshold ∆(∆>0) is employed in the feedback function. Generally, the negative feedback will lead to a faster iterative process than the normal iterative process. However, the iterative process may be still not stable as improper**

*T*_{i}*ε*or ∆ may be used. The other type of iterative process is called the successive approximation. In this method, the target point

**is also decided by**

*T*_{i}**and**

*T*_{i,ideal}**where**

*T**_{i}*ρ*(

*ρ*>0) is the approximation coefficient. In this iterative type, the target point

**for each ray at each sub-step is step forward to the ideal target point**

*T*_{i}**from the current intersection**

*T*_{i,ideal}**, as shown in Fig. 3(c). This successive approximation approach is consistent with the general optical design and optimization strategy. This iteration type will lead to a steadier iterative process. However, the convergence will be slower than the normal iterative process as well as the negative feedback method. Generally, the normal or the negative feedback iteration types are preferred due to their faster convergence speed. However, if instability appears during the iterative process using these two types, the successive approximation has to be used. The above pros and cons of the iteration types are summarized in Table 1. With the above procedures, the data points and the surface normals can be calculated. Next, the new freeform surface is re-generated based on the fitting method depicted in Section 2.1. The previous surface is then replaced by this new surface and this new system is taken as the new initial system for the next sub-step to re-generate the next freeform surface. When all the freeform surfaces for design are re-generated, a single iteration step is completed. This iteration step can be repeated for several times to further decrease the deviation of the light rays. The flow diagram of the construction-iteration method is shown in Fig. 4. The system after the construction-iteration process can be taken as a good starting point for further optimization.**

*T**_{i}## 3. Freeform off-axis three-mirror system design

Based on the surface design method depicted in Sections 2, a freeform off-axis reflective imaging system with low F-number and rectangular field-of-view has been designed. Table 2 lists the specifications of the optical system. The system has a special, easy aligned configuration. The primary mirror (M1) and the tertiary mirror (M3) are expected to be fabricated on one single substrate and further obtain a single M1-M3 element in the final system. In addition, the secondary mirror (M2) is a plane mirror. With the above configuration, the alignment difficulty of a freeform system can be greatly reduced.

An initial planar system is firstly established, as shown in Fig. 5. To makes it easy to fabricate the two mirrors on one single substrate, the primary and tertiary mirrors are expected to be approximately continuous to each other in the final design [11,32]. So, to make the subsequent design easier, the difference between the *z* coordinates of the bottom edge point ** P_{a}** of M1 and the top edge point

**of M3 in the initial system is controlled. The secondary plane mirror is taken as the aperture stop. The system uses a biased 3° × 3° FOV in vertical direction with (0°, −12°) as the central field. Six sample fields over the half-full FOV were employed in the surface construction process which are (0°, −10.5°) (0°, −12°) (0°, −13.5°) (1.5°, −10.5°) (1.5°, −12°) (1.5°, −13.5°) respectively. The aperture of each field was divided into 16 angles with equal interval. Seven different pupil coordinates were sampled along the radial direction of each angle. So, totally**

*P*_{b}*K*= 672 feature rays were used in the surface construction process. In this design, the target point

**for each feature ray is its corresponding ideal image point**

*T*_{i}**on the image plane (target plane).**

*I*_{i}During the preliminary surfaces-construction process, a new freeform M3 would be firstly generated from the initial planar system with the method proposed in Section 2.1, as shown in Fig. 6(a). Then, a new freeform M1 is generated based on this new system with freeform M3 to further improve the image quality, as shown in Fig. 6(b). During the construction of M3 and M1, ** P_{b}** and

**are taken as the initial feature data points respectively to keep the space relationship of M1 and M3. Figure 7(a) compares the average RMS spot diameter of the two systems after generating M3 and after generating M1, which are 0.4822mm and 0.4033mm respectively. Figures 7(b) and 7(c) show the distortion grids of the two systems. The maximum absolute distortions of the two systems are 1.0009mm and 0.6mm respectively.**

*P*_{a}Then, the iterative process depicted in Section 2.2 is employed to further improve the image quality. Here, we use the RMS (root-mean-square) deviation σ* _{RMS}* of the actual image points from the ideal image points for the feature rays to evaluate the effect of the iteration process, as given in Eq. (11).

*K*is the total number of the feature light rays, σ

*is the distance of the actual image point from the ideal image point for the*

_{m}*m*th feature light ray. The iterations were conducted for 30 times. Figure 8 gives the change of the σ

*with normal iterations (red lines). The σ*

_{RMS}*decreased by 92.4% after 30 normal iteration steps. Figure 9 shows the system after iterations. The spot diagram of the system is shown in Fig. 10(a). The average RMS spot diameter is 99.13μm, which decreased by 75.4% compared with the system before iterations, as shown in Fig. 10(b). The distortion grid of the system after iterations is given in Fig. 11. Compared with Fig. 7(c), the distortion of the system is significantly reduced after iterations. The maximum absolute distortion is 0.0346mm, which decreased by 94.2% compared with the system before iterations. It can be seen that the iterative process is very important and effective for improving the image quality and decreasing the distortion of the system. Furthermore, we analyzed the changes of the σ*

_{RMS}*for the two other types of iterations (negative feedback (ε = 0.3) and successive approximation (ρ = 0.7)), as shown in Fig. 8 with pink and blue lines respectively. Table 3 compares the results of the three iteration types for the design example in this paper. From this table and Fig. 8, we can see that the preferred negative feedback and normal iteration types have an indeed faster convergence speed than successive approximation, and instability does not occur. So, these two iteration types are adequate for this system, and the successive approximation need not to be used.*

_{RMS}The final optimization process employed the system after 30 normal iterations as the starting point. The surface type for the primary mirror (M1) and tertiary mirror (M3) is XY polynomials up to the 4th order. As the optical system is symmetric about the YOZ plane, only the even items of *x* are used. A good starting point is of great importance for the optimization, especially when designing freeform optical systems which have much more variables and flexible structures. If we directly optimize the initial planar system with software, the optimization has little effect and no good design result can be obtained. However, with the construction-iteration method we proposed in this paper, a good starting point using freeform surfaces can be obtained from the initial planes based on the given object-image relationships. The system specifications and the requirements for the configuration and have been already considered during the design process of the starting point. The subsequent optimization with optical design software is still needed, but the final design can be easily obtained with optimization from the starting point in this design example, and less optimization skills are required. The final system after optimization is shown in Fig. 12. The modulation-transfer-function (MTF) plot of the final system is given in Fig. 13, which is above 0.5 at 16.7 lps/mm. The RMS wavefront error of the system is shown in Fig. 14(a), whose average value is below λ/30 at 10μm wavelength. Figure 14(b) shows the distortion grid. The maximum absolute distortion is 28.6μm, which is less than 1 pixel size. These results show that good image quality was achieved.

For a general off-axis three-mirror system, the degree of freedoms during the alignment process is 12. A system assembly method of a freeform off-axis three-mirror system using three individual elements has been proposed in Ref [33]. In this method, the residual field constant astigmatism that results from a figure error of the mirror surfaces can be removed. However, another method can be employed in which M1 and M3 are expected to be fabricated on a single substrate and reduce the alignment difficulty [11]. If M1 and M3 form a single M1-M3 element, the degree of freedoms can be reduced to 6. If M2 further employs a plane mirror, the freedom corresponding to the rotation (γ-tilt or clocking) of M2 perpendicular to the local z-axis is eliminated, which means the degree of freedoms can be reduced to 5. In addition, as the system is not sensitive to the decenters of the planar M2 in the local x and y directions compared to the general systems using a M2 with optical power and complex surface shape, the alignment difficulty of the system is further greatly reduced. If a fixed mechanical stop is used in this system with a separated plane mirror M2 mounted at the back of the stop, the degree of freedoms during the alignment process is reduced to 3.

## 4. Conclusion

In this paper, we proposed a general method for designing freeform surfaces and the freeform imaging systems based on a point-by-point three-dimensional construction-iteration process. In the preliminary surfaces-construction process, the coordinates as well as the surface normals of the data points on the multiple freeform surfaces can be calculated directly considering the rays of multiple field angles and different pupil coordinates. Then, an iterative process is employed to significantly improve the image quality. Three iteration types are given for the iterative process. The proposed construction-iteration method is applied in the design of an easy aligned off-axis three-mirror system with low F-number and rectangular FOV. The primary and tertiary mirrors can be fabricated on a single substrate and form a single element in the final system. The secondary mirror is simply a plane mirror. With the above configuration, the alignment difficulty of a freeform system can be greatly reduced. After the preliminary surfaces-construction stage, the freeform surfaces in the optical system can be generated directly from an initial planar system. Then, with the iterative process, the average RMS spot diameter and the maximum absolute distortion can be greatly decreased, which proves the feasibility of the iterative process. After further optimization with optical design software, good image quality which is closed to diffraction-limited is achieved.

## Acknowledgment

This work is supported by the National Basic Research Program of China (973, No. 2011CB706701).

## References and links

**1. **Z. Zhenrong, H. Xiang, and L. Xu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. **48**(35), 6627–6634 (2009). [CrossRef] [PubMed]

**2. **A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express **21**(9), 10563–10571 (2013). [CrossRef] [PubMed]

**3. **R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. **38**(2), 229–231 (2013). [CrossRef] [PubMed]

**4. **F. Chen, K. Wang, Z. Qin, D. Wu, X. Luo, and S. Liu, “Design method of high-efficient LED headlamp lens,” Opt. Express **18**(20), 20926–20938 (2010). [CrossRef] [PubMed]

**5. **C. C. Hsieh, Y. H. Li, and C. C. Hung, “Modular design of the LED vehicle projector headlamp system,” Appl. Opt. **52**(21), 5221–5229 (2013). [PubMed]

**6. **Z. Feng, Y. Luo, and Y. Han, “Design of LED freeform optical system for road lighting with high luminance/illuminance ratio,” Opt. Express **18**(21), 22020–22031 (2010). [CrossRef] [PubMed]

**7. **S. Wang, K. Wang, F. Chen, and S. Liu, “Design of primary optics for LED chip array in road lighting application,” Opt. Express **19**(S4Suppl 4), A716–A724 (2011). [CrossRef] [PubMed]

**8. **Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express **21**(12), 14728–14735 (2013). [CrossRef] [PubMed]

**9. **Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with Monge-Ampère equation method,” Opt. Commun. **331**, 297–305 (2014). [CrossRef]

**10. **K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express **19**(22), 21919–21928 (2011). [CrossRef] [PubMed]

**11. **J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. **17**(1), 015605 (2015). [CrossRef]

**12. **T. Yang, J. Zhu, W. Hou, and G. Jin, “Design method of freeform off-axis reflective imaging systems with a direct construction process,” Opt. Express **22**(8), 9193–9205 (2014). [CrossRef] [PubMed]

**13. **X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE **8486**, 848607 (2012). [CrossRef]

**14. **R. A. Hicks, “Controlling a ray bundle with a free-form reflector,” Opt. Lett. **33**(15), 1672–1674 (2008). [CrossRef] [PubMed]

**15. **D. Cheng, Y. Wang, H. Hua, and M. M. Talha, “Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism,” Appl. Opt. **48**(14), 2655–2668 (2009). [CrossRef] [PubMed]

**16. **D. Cheng, Y. Wang, C. Xu, W. Song, and G. Jin, “Design of an ultra-thin near-eye display with geometrical waveguide and freeform optics,” Opt. Express **22**(17), 20705–20719 (2014). [CrossRef] [PubMed]

**17. **H. Hua, X. Hu, and C. Gao, “A high-resolution optical see-through head-mounted display with eyetracking capability,” Opt. Express **21**(25), 30993–30998 (2013). [CrossRef] [PubMed]

**18. **A. Bauer and J. P. Rolland, “Visual space assessment of two all-reflective, freeform, optical see-through head-worn displays,” Opt. Express **22**(11), 13155–13163 (2014). [CrossRef] [PubMed]

**19. **L. Li and A. Y. Yi, “Design and fabrication of a freeform microlens array for a compact large-field-of-view compound-eye camera,” Appl. Opt. **51**(12), 1843–1852 (2012). [CrossRef] [PubMed]

**20. **T. Ma, J. Yu, P. Liang, and C. Wang, “Design of a freeform varifocal panoramic optical system with specified annular center of field of view,” Opt. Express **19**(5), 3843–3853 (2011). [CrossRef] [PubMed]

**21. **G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc. B **62**(1), 2–8 (1949). [CrossRef]

**22. **D. Knapp, “Conformal Optical Design,” Ph.D. Thesis, University of Arizona (2002).

**23. **D. Cheng, Y. Wang, and H. Hua, “Free form optical system design with differential equations,” Proc. SPIE **7849**, 78490Q (2010). [CrossRef]

**24. **J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. **8**(4), 281–283 (2001). [CrossRef]

**25. **J. Liu, P. Benítez, and J. C. Miñano, “Single freeform surface imaging design with unconstrained object to image mapping,” Opt. Express **22**(25), 30538–30546 (2014). [CrossRef] [PubMed]

**26. **J. Hou, H. Li, Z. Zheng, and X. Liu, “Distortion correction for imaging on non-planar surface using freeform lens,” Opt. Commun. **285**(6), 986–991 (2012). [CrossRef]

**27. **J. C. Miñano, P. Benítez, W. Lin, J. Infante, F. Muñoz, and A. Santamaría, “An application of the SMS method for imaging designs,” Opt. Express **17**(26), 24036–24044 (2009). [PubMed]

**28. **F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express **20**(5), 5576–5585 (2012). [CrossRef] [PubMed]

**29. **T. Yang, J. Zhu, and G. Jin, “Design of freeform imaging systems with linear field-of-view using a construction and iteration process,” Opt. Express **22**(3), 3362–3374 (2014). [CrossRef] [PubMed]

**30. **J. Zhu, X. Wu, T. Yang, and G. Jin, “Generating optical freeform surfaces considering both coordinates and normals of discrete data points,” J. Opt. Soc. Am. A **31**(11), 2401–2408 (2014). [CrossRef] [PubMed]

**31. **Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express **18**(9), 9055–9063 (2010). [CrossRef] [PubMed]

**32. **Q. Meng, W. Wang, H. Ma, and J. Dong, “Easy-aligned off-axis three-mirror system with wide field of view using freeform surface based on integration of primary and tertiary mirror,” Appl. Opt. **53**(14), 3028–3034 (2014). [CrossRef] [PubMed]

**33. **K. Fuerschbach, G. E. Davis, K. P. Thompson, and J. P. Rolland, “Assembly of a freeform off-axis optical system employing three φ-polynomial Zernike mirrors,” Opt. Lett. **39**(10), 2896–2899 (2014). [CrossRef] [PubMed]