## Abstract

In this paper, a design method based on a construction and iteration process is proposed for designing freeform imaging systems with linear field-of-view (FOV). The surface contours of the desired freeform surfaces in the tangential plane are firstly designed to control the tangential rays of multiple field angles and different pupil coordinates. Then, the image quality is improved with an iterative process. The design result can be taken as a good starting point for further optimization. A freeform off-axis scanning system is designed as an example of the proposed method. The convergence ability of the construction and iteration process to design a freeform system from initial planes is validated. The MTF of the design result is close to the diffraction limit and the scanning error is less than 1μm. This result proves that good image quality and scanning linearity were achieved.

© 2014 Optical Society of America

## 1. Introduction

In comparison with conventional rotationally symmetric surfaces, freeform optical surfaces have more degrees of design freedom, and therefore improve system performance while decreasing the system size, mass, and number of elements in optical design. For non-imaging systems, the freeform surfaces is much more advanced due to the lower surface quality required. In recent years, with the development of the advancing manufacture technologies, freeform surfaces have been increasingly used in the imaging field, such as reflective systems [1–6], head-mounted-display (HMD) [7–10], varifocal panoramic optical system [11] and freeform microlens array [12, 13].

Traditional freeform systems design in imaging optics uses spherical or aspherical system as the starting point. Then, some surfaces in the system are gradually replaced by freeform surfaces to obtain satisfactory results [7, 9]. This design greatly depends on multiparameter optimization. The convergence may be slow for freeform surfaces with much more variables. If the starting point is far from the optimum point, designers may fail to find the useful solutions, or they have to spend long times improving the starting point. Furthermore, designing decenter and tilted systems is very difficult based on this design method as there are fewer starting points for choices. So, researchers turn to find ways to directly design the freeform surfaces based on the object-image relationships. One important method is to establish the partial differential equations (PDEs) based on incident and exit rays which determine the shape of the surfaces [14–18]. The points on the surface can be calculated, and then the freeform surfaces are generated by surface fitting. This design method is simple and effective in imaging optics. However, only a single field (or a small FOV) is considered while establishing the equations based on the object-image relationships. The other method called the Simultaneous Multiple Surface (SMS) design method is an ingenious way to generate several freeform surfaces simultaneously. After design, several input and output ray bundles can be fully coupled by the optical system [19, 20]. However, the number of ray-bundles considered in the design process depends on the number of surfaces in the system. So, there is a restriction on the number of fields considered in the design process.

An actual imaging system works for a certain object size and a certain width of light beam. Therefore, a basic requirement in imaging system design is to control the light rays of multiple fields and different pupil coordinates. Furthermore, as freeform surfaces have non-rotationally symmetric shapes and provide much more controlling freedom, they are often used in decentered and tilted non-symmetric systems. It is both an interest and a challenge to directly design freeform surfaces under these conditions.

In this paper, a novel method to design freeform imaging systems with linear FOV is developed. The unknown surface contours in the tangential plane are firstly designed to control the tangential rays of multiple fields and different pupil coordinates. This is a direct design process based on construction and iterations, and it may start from several simple initial surfaces such as planes. The number of fields considered in the design process is not restricted. The image quality for the tangential rays is then gradually improved with an iterative process and the desired freeform surface contours to control the tangential rays are obtained. This design result can be taken as a good starting point for further optimization. The freeform system with good image quality in both the tangential and the sagittal plane are obtained after optimization. The proposed method can be applied to designing decentered and tilted non-symmetric systems. A freeform off-axis scanning system has been designed as an example. The convergence ability of the method to design a freeform system from initial planes is validated. Good image quality and scanning linearity are obtained after optimization.

## 2. Method

In this paper, the method is proposed to design an actual system with a linear FOV in the tangential plane. So, the surface contours of the unknown surfaces in the tangential plane are firstly designed only to control the tangential rays. This design process is based on construction and iterations. Each unknown surface contour is constructed with the data points calculated in each iteration step based on the given object-image relationships. Then, the image quality of the system is improved with an iterative process. This design result can be taken as a good starting point for further optimization. The freeform surfaces with good image quality in both the tangential and the sagittal plane are obtained after optimization. In this paper, this method is employed to design double freeform surfaces in an optical system.

#### 2.1 Basic idea to design the freeform surface contour

An off-axis two-mirror system is taken as an example for illustration. As shown in Fig. 1, the light beams coming from the object space are expected to image on a given plane after two reflective surfaces. The design process of the 2D surface contours starts from an initial system set up with two planes, as shown in Fig. 1(a). After a construction and iteration process, a freeform system with much improved image quality for the tangential rays can be obtained, as shown in Fig. 1(b). The iterative loop of the method for designing freeform surface contours consists of four main steps. ①Establish the initial system with planes. The design of surface contours starts from this initial system. The initial surfaces can also be extended to other surfaces types such as spheres. ②To make the tangential light rays redirected to their ideal image points, the points on the new contour of surface #2 are calculated while the initial surface #1 is fixed. The new contour of surface #2 is then constructed with these points. It is used to replace the previous surface #2. ③Similarly, the points on the contour of new surface #1 are calculated based on the object-image relationships and the new surface #2 that has been calculated. The new contour of surface #1 is then constructed with these points. It is used to replace the previous surface #1. ④The new surface #1 and surface #2 are taken as the initial surfaces for the next iteration. Repeat the above steps and the image quality will be gradually improved. Detailed analysis and design process are depicted in Section 2.2 and 2.3.

#### 2.2 Construction of the freeform surface contours

### 2.2.1 The method to construct a single freeform contour

The method to construct the contour of an unknown freeform surface is depicted in the following section. For a system with M sampling fields in the design process, N different rays corresponding to N different pupil coordinates in the tangential plane are defined as the feature rays in each field. So, totally *K* = M × N feature rays in the tangential plane are used. The intersections of the feature rays with the unknown surface are taken as the data points on the surface. The surface contour is then constructed with these points.

To obtain all the data points ** P_{i}** (

*i*= 0,1,2…

*K*−1) on the contour of unknown surface

**, the intersections of the feature rays with surface**

*Ω*

*Ω**'*and

*Ω**”*, which are the two neighboring surfaces of

**, are employed, as shown in Fig. 2.The intersection of the ray with**

*Ω*

*Ω**'*is defined as the start point

**of a feature ray, and the intersection with**

*S*_{i}

*Ω**”*is defined as the end point

**. When the initial system and the feature rays have been decided, the start points**

*E*_{i}**(**

*S*_{i}*i*= 0,1,2…

*K*−1) of the feature rays and the directions of the rays after

*Ω**'*are known, but they are generally irregular. The end points

**(**

*E*_{i}*i*= 0,1,2…

*K*−1) are also determinate. The unit normal vector

**at each data point**

*N*_{i}**can be calculated based on the vector form of the Snell’s Law. For a refractive surface,**

*P*_{i}*n*and

*n'*are the refractive indices of the two media in Fig. 2. Similarly, for a reflective surface

During the calculation of the data points ** P_{i}**
(

*i*= 0,1,2…

*K*−1), an initial data point

**is firstly fixed which is the intersection of the marginal feature ray with the initial surface of the current iteration, as shown in Fig. 3. As the start point**

*P*_{0}**and end point**

*S*_{0}**of the initial feature ray can be easily obtained, the surface normal**

*E*_{0}**at**

*N*_{0}**can be calculated by Eq. (1) or Eq. (2). Then the tangent vector**

*P*_{0}**at**

*T*_{0}**can be obtained. To find the next data point on the unknown surface, we need to find the associated feature ray among the remaining**

*P*_{0}*K*−1 feature rays corresponding to different fields and different pupil coordinates. Here, the ray nearest to

**is taken as the feature ray corresponding to the next data point**

*P*_{i}**. As a realization of this principle,**

*P*_{i}_{+1}**is obtained by finding the point nearest to**

*P*_{1}**among the**

*P*_{0}*K*−1 intersections

**(**

*G*_{0}_{i}*i*= 1,2…

*K*−1) where the tangent vector

**intersects with the remaining**

*T*_{0}*K*−1 feature rays coming from

*Ω**'*, as shown in Fig. 3. Next, calculate the surface normal

**at**

*N*_{1}**with the start point**

*P*_{1}**and end point**

*S*_{1}**of its associated feature ray. Then find**

*E*_{1}**, which is nearest to**

*P*_{2}**among the**

*P*_{1}*K*−2 intersections of the tangent vector

**at**

*T*_{1}**with the remaining**

*P*_{1}*K*−2 feature rays. Repeat this process until all the

*K*data points on the unknown surface are obtained, as shown in Fig. 4.Finally, the contour of the surface is generated by curve fitting.

### 2.2.2 Constructing the contour of surface #2

Based on the method illustrated above, the two unknown freeform surfaces can be obtained from the initial planes (or other surface types). The contour of surface #2 which is neighbor to the image plane is designed first, as shown in Fig. 1. It is expected that all the feature rays can be redirected to their ideal image points when the new surface #2 is used while the initial plane #1 is fixed. The start point of each feature ray is its intersection with surface #1 and the end point is its ideal image point on the image plane. The contour of the new surface #2 is obtained by curve fitting with the data points on the unknown surface. Then, the previous surface #2 is replaced.

### 2.2.3 Constructing the contour of surface #1

Next, the contour of surface #1 shown in Fig. 1
is designed with the similar method. It is expected that all the feature rays can be
redirected to their ideal image points when the new surface #1 is used while the surface #2
that has been calculated is fixed. As shown in Fig. 5,
for a feature ray from the start point ** S_{i}** in the
object space,

**is redirected into**

*S*_{i}P_{i}**by surface #1. Then,**

*P*_{i}P_{i}'**is redirected to its ideal image point**

*P*_{i}P_{i}'**by surface #2. Here, Fermat's principle is used to calculate the unknown coordinates of**

*I*_{i}**. According to Fermat's principle,**

*P*_{i}'**is the point on surface #2 which minimizes the optical path length between**

*P*_{i}'

*P*_{i}*-*

*P*_{i}'*-*

**. The optical path length**

*I*_{i}*L*of

*P*_{i}*-*

*P*_{i}'*-*

**can be expressed as**

*I*_{i}*n*

_{1-2}and

*n*

_{2-image}are the refractive indices of the medium between surface #1 and surface #2 and the medium between surface #2 and the image plane respectively. So

**can be obtained by minimizing**

*P*_{i}'*L*and it is taken as the end point of a feature ray on surface #2. With

**,**

*S*_{i}**and**

*P*_{i}**, the incident and outgoing directions of a feature ray can be calculated and the normal**

*P*_{i}'**at each data point on surface #1 can be obtained using Eq. (1) or Eq. (2). All the data points on surface #1 can be calculated following the procedure depicted in Section 2.2.1. The new contour of surface #1 is finally obtained by curve fitting. Then, the previous surface #1 is replaced..**

*N*_{i}#### 2.3 Iterative process

With the method depicted in Section 2.2, two freeform surface contours can be obtained by the construction approach. The image quality for the tangential rays of the system will become better. To further improve the image quality, the contours of surface #1 and surface #2 constructed previously are taken as new initial surfaces for the next iteration, and then two new surfaces can be constructed using the same construction method. During the iteration process, the feature rays of the multiple fields and pupil coordinates are constrained to be redirected to their ideal image points in each iteration. After each time of construction, the data points are fitted into an analytical surface expression which is continuous. Therefore, the surface obtained with this method is always continuous. By repeating the above process, the image quality will be gradually improved. The feature rays can be redirected to their ideal image points approximately. The flow chart of the whole process is shown in Fig. 6. Note that the design result can also be output after constructing surface #2.

To increase the working efficiency, the whole design process is implemented into a program. The directions of the feature rays and their intersections with the surfaces are obtained with real ray trace using optical design software, in this paper, Code V®. MATLAB® is employed for data processing. As the Code V API (application programming interface) uses the Microsoft Windows standard Component Object Model (COM) interface, users can execute Code V commands using MATLAB which supports Windows COM architecture [21]. With the given the initial surfaces and the expected object-image relationships, the final MATLAB program enables automatic design of the freeform surfaces.

In this paper, as we focus on the design of an actual system with a linear vertical FOV, the freeform surface contours with good control of the tangential rays can be obtained with the method proposed in the above sections. The curvatures of the freeform surfaces in the sagittal plane are set to be zero first. This system can be taken as a good starting point for further optimization with optical design software. The freeform system with good image quality in both the tangential and the sagittal plane are obtained after optimization.

## 3. Design example

#### 3.1 System parameters and the iterative process

To validate the above design method, an actual freeform off-axis two-mirror imaging system for linear scanning has been designed. There are strict requirements on the exact position of the image on the image plane as well as the distance between the entrance pupil and the image plane. In addition, the final design requires excellent image quality and scanning linearity. Therefore, we use this design to validate the feasibility of the proposed construction and iteration process. The parameters of the system are given in Table 1.

An initial system was set up with two planes, as shown in Fig. 7. The two planes have a 45° and 47° tilt about the z-axis in the global coordinates respectively. The distance between the entrance pupil and surface #1, between surface #1 and surface #2, and between surface #2 and the image plane are 45mm, 70mm and 135mm, respectively. The two initial surfaces can also be extended to other surfaces such as spheres.

As the system has a linear 8° FOV in the vertical direction and 0° FOV in the horizontal direction, the design of the surface contours in the tangential plane which controls the tangential rays was firstly conducted. During the design process of the freeform scanning system, the fields over the full FOV have to be sampled. On the other hand, more sampling fields used in the construction process increases the computation time. Therefore, five fields over the 8° full FOV with equal interval which are respectively 0°, 2°, 4°, 6°, 8° are used in the construction process. Three different pupil coordinates of each field (the chief ray and two marginal rays) were employed. To analyze the effect of the iterative process, the iteration was conducted for 20 times. In fact, 10 to 15 times is enough for getting a good result in this design.

#### 3.2 Image quality analysis

In order to describe the effect of the iteration process more intuitively, the change of the
system layouts with iterations are given in Fig. 8.Due to the limited space, only the layouts after some representative iteration (1, 2, 4,
8 and 12 iterations) are plotted. It can be seen that the image quality is improved fast with
iterations. The systems after each iteration have not been optimized by optical design
software, which means that the direct design results after the construction and iteration
process are plotted in Fig. 8. The changes of the
freeform surface profiles with iterations are given in Table
2.The goal of the construction and iteration process is that the tangential rays of
multiple fields and different pupil coordinates can be redirected to their ideal image points.
As a consequence, the deviation of the actual image point of the feature ray from the ideal
image point should be evaluated. To be more specific, we use the spot diameter and absolute
distortion to evaluate the effect of the construction and iteration process. Figure 9(a) shows the convergence behavior of the maximum,
minimum and average spot diameter of the five sample fields (0°, 2°, 4°,
6°, 8°) versus the number of iteration steps. Figure 9(b) shows the convergence of the standard deviation σ_{spo} of
the spot diameters of these five fields. It can be seen that the spot sizes of different fields
and the difference between them reduce rapidly in few iterations, which means the convergence
of the design process is fast. Here, σ_{spo} of the initial system is not
plotted in Fig. 9(b) as it is generally zero due to the
structure of planes. After several iterations, the average 100% spot diameter converges to a
steady value around 240μm. The standard deviation of different field is less than
10μm, which indicates that the image qualities of different fields are significantly
improved simultaneously.

In this paper, the absolute distortion *∆h* for each field is defined as
the absolute value of the difference between the actual image height and the ideal image
height

*h*is the ideal image height of each field,

*h'*is the actual image height. Figure 10(a) shows the convergence behavior of the maximum absolute distortion of the five sample fields (0°, 2°, 4°, 6°, 8°) versus the number of iterations. Figure 10(b) shows the convergence of the standard deviation σ

_{dis}of the absolute distortion of these five fields. The distortion converges rapidly to a steady value in several iterations. Although some residual distortion exists, the standard deviation of the distortion of different fields is very small, which is around 100μm after several iterations, as shown in Fig. 10(b). It indicates that the distortion actually introduces a translation of the actual image plane.

As we can see above, with the construction and iteration process, a system by which the tangential rays are well controlled can be gradually approached from an initial system of planes. In the design example of a scanning system, the 100% spot diameter of tangential rays is around 240μm and the standard deviation of distortion is around 100μm after 12 iterations. This system can be taken as a good starting point for further optimization. The design and analysis for reflective systems in this design example can be also performed for refractive systems as well as for refractive-reflective systems.

#### 3.3 Further optimization with optical design software

The system after 12 iterations was taken as the starting point for further optimization with
optical design software, in this paper, Code V. XY polynomial surface was used to design the
system, which is a kind of commonly used nonrotationally symmetric freeform surface [1, 7, 10, 11, 16, 21]. Since the
optical system is symmetric about the YOZ plane, only the even items of *x* in
XY polynomials are kept. Moreover, the high order terms are not used as they lower the ray
tracing speed and increase the difficulty in manufacture. So, an eight terms XY polynomials up
to the 4th order is used:

*c*is the curvature of the surface,

*k*is the conic constant, and

*A*is the coefficient of the

_{i}*xy*terms. This kind of freeform surface is continuous and can be fabricated. In addition, the 4th order polynomials have enough design freedom for optimization. The default transverse ray aberration error function in Code V is used in this optimization. The scanning error (distortion) is controlled by constraining the imaging coordinate of the chief ray in each field using real ray trace data. The final system with good image quality in both the tangential plane and the sagittal plane using two 3D freeform surfaces was obtained quickly by optimization, as shown in Fig. 11.The primary mirror (surface #1) has a rectangular size of 6mm × 22.84mm. The secondary mirror (surface #2) has a rectangular size of 2.7mm × 26.18mm. The exact profiles of the two surfaces are given in Table 3.The final design has an F# of 47.7, which is normal for an f-theta scanning system. The spot diagram is shown in Fig. 12(a).The MTF of each field is close to the diffraction limit, which is shown in Fig. 12(b). Figure 13 shows the RMS spot diameter as a function of field in a curve. Figure 14 shows the scanning errors (i.e. the distortion values) of different fields. For all of the sampling fields, the error is not more than ± 1μm. Figure 15shows the relative distortion of the system, which is within 0.009% over the full FOV. This result proves that good scanning linearity was achieved.

The planar/spherical surfaces are generally far from the optimum point. Especially the planes which do not offer any power are generally not directly used in the freeform surfaces design. But experienced designers maybe achieve good design result using XY polynomial optimization starting from planar/spherical surfaces. However, with the construction and iteration process we proposed, the rays of multiple fields and different pupil coordinates are well controlled. Therefore, a better starting point using freeform surfaces can be obtained from the initial planes. It is easier to achieve an excellent final design and advanced optimization techniques are not required.

## 4. Conclusion

A design method of freeform systems with linear FOV is proposed in this paper. This method can be used to design freeform surfaces without complex derivations, and it can be applied to decentered and tilted non-symmetric structures. The design process starts from an initial system using only planes. The freeform surface contours in the tangential plane are firstly designed to control the tangential rays of the system. With an algorithm we proposed, each unknown surface contour is constructed with the feature rays of multiple fields and different pupil coordinates. Then the image quality of the tangential rays is gradually improved with an iterative process. The design result can be taken as a good starting point for further optimization. The freeform system with good image quality in both the tangential and the sagittal plane are obtained after optimization. A freeform off-axis scanning system has been designed as an example of the proposed method. The convergence ability of the construction and iteration process to design a freeform system from initial planes is validated. The construction and iteration process depicted in this paper provides an automatic optical design strategy which can be implemented into a program or a macro. In this paper, the construction and iteration process is restricted in the 2D tangential plane. Only the surface contour can be calculated with this process. Therefore, this method is suitable for designing the systems with a linear FOV. The construction and iteration process in full 3D geometry which leads to better solutions will be the future development of freeform surfaces design.

## Acknowledgment

This work is supported by the National Basic Research Program of China (973, No. 2011CB706701). The authors also thank the reviewers for their valuable comments and suggestions.

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