## Abstract

In this paper, a design method of surface contour for a freeform imaging lens with a wide linear field-of-view (FOV) is developed. During the calculation of the data points on the unknown freeform surfaces, the aperture size and different field angles of the system are both considered. Meanwhile, two special constraints are employed to find the appropriate points that can generate a smooth and accurate surface contour. The surfaces obtained can be taken as the starting point for further optimization. An f-θ single lens with a ± 60° linear FOV has been designed as an example of the proposed method. After optimization with optical design software, the MTF of the lens 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.

© 2013 Optical Society of America

## 1. Introduction

Compared with conventional rotationally symmetric surfaces, freeform optical surfaces have more degrees of freedom, therefore reduce the aberrations and simplify the structure of the system in optical design. In recent years, with the development of the advancing manufacture technologies, freeform surfaces have been successfully used in the imaging field, such as head-mounted-display [1–3], reflective systems [4–9], varifocal panoramic optical system [10] and microlens array [11].

Traditional freeform imaging system design uses spherical or aspherical system as the starting point. Then, some surfaces in the system are replaced by freeform surfaces to obtain satisfactory results [1, 2, 10]. However, with the increasing novelty and complexity of optical systems, this method is difficult to satisfy the design goal. A possible solution to this problem is to directly design freeform surfaces based on the object-image relations. One common method is to establish the partial differential equations based on incident and exit rays which determine the shape of the surfaces [12–14]. The points on the surfaces can be calculated next and the freeform surfaces are obtained after surface fitting. This method is simple and effective in imaging optics, especially for designing systems with a small field-of-view (FOV). Another ingenious method is the Simultaneous Multiple Surface (SMS) design method [15, 16]. Multiple freeform surfaces can be generated simultaneously, and several input and output tangential-ray bundles become fully coupled by the optical system.

For freeform imaging system design, a wide FOV is difficult to be achieved. Furthermore, the real size of aperture is expected to be considered while designing a wide FOV system. So, the light beams of different fields possibly have overlap area on the unknown surface during the design. The position and shape of the overlap area should meet the imaging requirement of different fields. It is both an interest and a challenge to directly design freeform surfaces under these conditions.

In this paper, a novel method to design a freeform imaging lens with a wide FOV has been developed. The proposed method has two key contents. Firstly, the aperture size and different field angles in a wide FOV system are both considered during the calculation of the data points on the freeform surfaces. Secondly, two special constraints are employed during the calculation. With these two key contents, a series of data points can be obtained, and a smooth and accurate surface contour can be generated after curve fitting with these data points. More importantly, the coordinates and normal vectors of these original data points can be approximately maintained, and the expected imaging relationship is ensured. The surfaces obtained can be taken as the starting point for further optimization in optical design software. Here, only the design method of the two-dimensional surface contour for tangential rays is covered in this paper. An f-θ single lens with a ± 60° linear FOV is designed as an example. A good starting point is obtained with the proposed method, and it is then optimized in optical design software to achieve good image quality and scanning linearity. The design method of three-dimensional freeform surfaces will be discussed in the future study.

## 2. Method

A freeform single lens is shown schematically in Fig. 1(a). Parallel light beams of different fields from the entrance pupil are refracted by the lens and focus on the image plane. The two-dimensional contour of the front surface in the tangential plane is firstly generated to control the tangential rays without the back surface, as shown in Fig. 1(b). This single surface is taken as the starting point for further optimization with optical design software. The back surface is then added in the next step. Assume that the center of the entrance pupil is located at the origin of an orthogonal coordinates system and the optical axis is along with the z-axis. The FOV of the system 2*ω* ( ± *ω*) is divided into 2*k* + 1 fields with equal interval *∆ω* between each two neighboring fields during the design process. So *∆ω* can be expressed as

#### 2.1 The feature rays for calculating the data points on the unknown surface

During the design of the freeform contour, each two neighboring fields are taken as a group. Several feature rays corresponding to different fields and pupil coordinates are defined in each group, and their intersections with the front surface can be calculated based on the relationships between the incident and outgoing rays. So, the data points on the unknown surface can be obtained group after group and the contour of the front surface can be then constructed with these data points. Consider an arbitrary group shown in Fig. 4. The two neighboring fields are labeled as field #1 and field #2 and the feature rays are respectively marked with a circled number and plotted in bold. Feature ray ① is the chief ray of field #1 and it will be refracted to its ideal image point ** P_{f}_{1}** by the front surface at data point 1. Feature ray ② is the bottom marginal ray of field #2 and it will be refracted to its ideal image point

**by the front surface at data point 2. When the beams of neighboring fields have overlap area (yellow area in Fig. 4(a)) on the unknown surface, the beams from two different fields have to be simultaneously controlled at this area on the surface. It means that two rays from field #1 and #2 which hit on the same point in the overlap area should be refracted to**

*P*_{f}_{2}**and**

*P*_{f}_{1}**respectively. This problem generally does not have an exact solution due to the over-determined problem. So in this paper, the calculation of the data points is taken as an optimization problem. As only the starting point of the system is concerned, an approximate solution obtained by a mathematical optimization process is adequate. Here in particular, at data point 2 (the bottom of the overlap area), feature ray ② from field #2 as well as another feature ray ③ (blue bold dotted ray) from field #1 will be refracted to their ideal image point**

*P*_{f}_{2}**and**

*P*_{f}_{2}**respectively. In short, two data points (point 1 and 2) on the surface are calculated in one field group using three feature rays (①②③) when the light beams have overlap area on the unknown surface. When the field angle is increasing and the light beams are separated, as shown in Fig. 4(b), there is no feature ray ③ and two data points (point 1 and 2) on the surface are calculated in one field group using only two feature rays (①②).**

*P*_{f}_{1}#### 2.2 Establishing the constraints to generate a smooth link line of the points

After the feature rays used in each group are defined, the data points on the front surface can be calculated based on the relationships between the incident and outgoing light rays. However, there are still some problems. As there are no geometric relationships between different groups, the two data points calculated in each field group are the optimum solution of a single problem specific to two fields. Therefore, the data points from different groups distribute irregularly in the tangential plane. A smooth and accurate surface contour is difficult to be obtained.

Another problem is related to the realization of the expected imaging relationship. The outgoing direction of a light ray which goes through a data point is determined by both the coordinates and the normal vector of this point. In this paper, the surface contour is obtained by curve fitting with the data points. In the ideal case, all the data points are on the fitted contour, and the original normal vector at each data point which determines ray direction is ensured as well. In this way, the light rays can be shifted into the expected direction by the surface. However, if no constraints are added when calculating the data points, the points obtained generally have considerable deviations from the surface contour after curve fitting, in other words, the fitting error is big. More seriously, the normal vector of the surface contour after curve fitting is not consistent with the normal at each original data point. As a consequence, the outgoing light rays are deviated from the expected direction and the expected imaging relationship will be not ensured.

To solve these problems, two special constraints are employed during the calculation of the data points. One constraint is used to establish the geometric relationships between neighboring field groups using the surface normal. The other one called stairs-distribution elimination constraint is used to improve the smoothness of the link line. With these constraints, the data points distribute regularly and form a smooth link line in the tangential plane. In this way, an accurate fitted contour is achieved and the deviation of the original data points from the fitted contour is very small. The way to maintain the consistency of the normal vectors after curve fitting is also involved in these constraints. Detailed analyses of these constraints are depicted in the following.

The data points calculated in the previous neighboring group are used during the calculation of the data points in each field group. The first constraint uses the surface normal vector at each data point to establish the geometric relationships between neighboring field groups. As shown in Fig. 5, ** P_{3}** and

**are the data points to be calculated in the current field group,**

*P*_{4}**and**

*P*_{1}**are the data points already calculated in the previous field group. The direction vector**

*P*_{2}**from**

*e*_{23}**to**

*P*_{2}**is constrained to be perpendicular to the unit normal**

*P*_{3}**at**

*N*_{3}**, and the direction vector**

*P*_{3}**from**

*e*_{34}**to**

*P*_{3}**is constrained to be perpendicular to the unit normal**

*P*_{4}**at**

*N*_{4}**. So, the constraints can be written as**

*P*_{4}Equations (2) and (3) establish the geometric relationships between neighboring field groups. The data points no longer distribute irregularly in the tangential plane after the constraint is added. Moreover, in this constraint, the original normal vector at each data point which determines the outgoing direction of light ray is perpendicular to the line connecting the neighboring point. As a consequence, the consistency of the normal vectors after curve fitting is approximately ensured, and the light rays can be shifted in the expected directions.

However, the link line of the data points may be not smooth enough. A typical case is shown in Fig. 6. The line connecting the two data points in each group is approximately parallel to the one in the neighboring group (Figs. 6(a) and 6(b)), which yields a distribution of points like stairs, as shown in Fig. 6(c). So the fitting accuracy of the contour is low and the data points have considerable deviations from the fitted contour. To solve this problem, the stairs-distribution elimination constraint is proposed. The intersection ** P_{i}** of the two lines which connect the two data points in each group is expected to be between

**and**

*P*_{2}**, as shown in Fig. 7. So the**

*P*_{3}*y*coordinate

*P*of

_{iy}**is constrained to be between the**

*P*_{i}*y*coordinates of

**and**

*P*_{2}**, and the**

*P*_{3}*z*coordinate

*P*of

_{iz}**is constrained to be between the**

*P*_{i}*z*coordinates of

**and**

*P*_{2}**. The constraint can be written as:**

*P*_{3}#### 2.3 Calculating the data points on the unknown freeform surface

With the defined feature rays and the special constraints depicted above, the next step is to calculate all the data points and obtain the freeform surface contour. In the ideal case, the feature rays used in each field group are refracted by the surface to their ideal image points respectively based on the Snell’s law. The vector form of the Snell’s law can be written as

where**= (**

*r**α*,

*β*,

*γ*),

**= (**

*r'**α'*,

*β'*,

*γ'*) are the unit vectors along the directions of the incident and exit ray.

*n*is the refractive index of the medium around the lens.

*n'*is the refractive index of the lens material.

**= (**

*N**i*,

*j*,

*k*) represents the unit normal vector at the data point. In the YOZ plane, Eq. (6) can be written in the scalar form [17]: where

*I*and

*I'*are the angles of incidence and refraction respectively. cos

*I*and cos

*I'*can be obtained by

*θ*respectively. So, for each feature ray, a tangent relation is required:

The components of ** r** and

**of each feature ray used in Eqs. (7)-(11) can be easily written out with the coordinates of its intersections with the entrance pupil, the unknown surface and the ideal image point. The calculation of the two data points in each field group is taken as a mathematical optimization problem. Two special constraints used in the optimization to obtain the data points that can generate a smooth link line have been depicted in section 2.2. As an exact solution may be not achievable to satisfy the Snell’s law for all the feature rays in each field group, Eqs. (7), (8) and (11) are also taken as constraints to control the direction of each feature ray in the optimization process. So, the constraints used to get the corresponding optimum solution in each group are Eqs. (2)-(5), (7), (8) and (11). Note that the constraints Eqs. (7), (8) and (11) will be used several times as there are more than one feature ray in a field group. The**

*r'**y*and

*z*coordinates (

*y*

_{1},

*z*

_{1}), (

*y*

_{2},

*z*

_{2}) as well as the y and z component (

*j*

_{1},

*k*

_{1}), (

*j*

_{2},

*k*

_{2}) of surface normal vector of the two data points in each field group are set as unknown variables. So, all the constraints can be expressed in terms of (

*y*

_{1},

*z*

_{1},

*y*

_{2},

*z*

_{2},

*j*

_{1},

*k*

_{1},

*j*

_{2},

*k*

_{2}). A merit function

*Φ*(

*y*

_{1},

*z*

_{1},

*y*

_{2},

*z*

_{2},

*j*

_{1},

*k*

_{1},

*j*

_{2},

*k*

_{2}) is formed by the sum of residual squares of the constraints. The optimization process is to minimum

*Φ*and to get the corresponding (

*y*

_{1},

*z*

_{1}) (

*y*

_{2},

*z*

_{2}). In this paper, the optimization is done by the commercial optimization software 1stOpt® [18]. Other commercial optimization software such as MATLAB® and Lingo® are also recommended.

The whole algorithm starts from the group of the first two fields containing the marginal field of the system. As shown in Fig. 8, when the coordinates (*y*_{1}, *z*_{1}), (*y*_{2}, *z*_{2}) of data point 1 and 2 in the first group are obtained, field #2 and the next neighboring field #3 are taken as the next group, and data point 3 and 4 can be then calculated with the same method. The above mentioned process is repeated until the all the fields are calculated. It should be emphasized that when calculating data point 1 and point 2, as no previous neighboring group exists, they can be obtained with the constraints Eqs. (3), (7), (8) and (11). The freeform contour in the tangential plane is finally obtained after curve fitting with all the data points, and it is taken as the starting point for subsequent optimization in optical design software, which will be depicted later in section 3.2.

## 3. Design example: A freeform f-θ single lens

#### 3.1 Designing the starting point of the system

As an example of the proposed method, an f-θ single lens with a wide linear FOV has been designed. The f-θ lens is used for a scanning range of ± 210mm in *y* direction. The system has a linear FOV of ± 60°, and it is divided equally into 61 fields with a 2° interval during the design process. As the scanning width *y* (mm) has a linear relationship with the scanning angle *θ* (°) for an f-θ lens, the f-θ property can be written as

Next, the starting point of the system was designed with the proposed method. Note that only half of the contour for 0° to 60° fields is needed to be generated because of the plane-symmetrical structure. If the data points are calculated only based on the equations to control the ray direction (Eqs. (7), (8) and (11)), these points are irregularly distributed in the tangential plane, as shown in Fig. 9. When the constraint to establish the geometric relationships between neighboring field groups using the surface normal is added (Eqs. (2) and (3)), the data points no longer distribute irregularly in the tangential plane, as shown in Fig. 10(a). However, the link line of the points is not smooth and the stairs-distribution is obvious. When the stairs-distribution elimination constraint (Eqs. (4) and (5)) is finally added, the stairs-distribution is removed and the data points obtained can generate a smooth link line, as shown in Fig. 10(b). A surface contour with high fitting accuracy is obtained after curve fitting. Then this starting point is entered into optical design software, in this paper, Code V® [19]. Figure 11(a) shows the layout of the system. Figure 11(b) shows the scanning error of each field. In this paper, the scanning error is defined as

where*h*is the ideal image height,

*h'*is the actual image height. For most of the sampling fields (0° to 50°), the error is within ± 0.4mm. For some larger field angles, the error is no more than ± 1mm. It shows that the light beams are well controlled by the starting point designed with the proposed method.

#### 3.2 Optimization of the starting point

The optimization of the starting point was conducted in Code V. In this paper, the surface type of the front surface chooses to be XY polynomials. XY polynomial surface is a commonly used nonrotationally symmetric freeform surface [1, 3, 6, 10, 14, 19]. The general expression for XY polynomials is shown in Eq. (14):

*c*is the curvature of the surface,

*k*is the conic constant, and

*A*is the coefficient of the

_{i}*x-y*terms. It has more parameters and thus offers more degrees of freedom to the optical design process. Generally, the XY polynomial surface can achieve a significantly lower error function and better image quality than other surface types such as anamorphic and even aspheric surface [1, 3]. Since the optical system is symmetric about the YOZ plane, only the even items of

*x*in XY polynomials are used. Moreover, the higher order polynomial terms only slightly improve the optimization result, and they lower the ray tracing speed and increase the difficulty in manufacture. Therefore, an 11 terms XY polynomial surface up to the 5th order is used in the design.

## 4. Conclusion

A freeform single lens design method to achieve a wide linear FOV is depicted in detail in this paper. The aperture size in a wide FOV system is considered during the calculation of the data points on the unknown freeform surfaces, and the calculation of the data points on the freeform surface is a mathematical optimization problem. Two special constraints are employed to find the appropriate data points which can generate a smooth link line. The constraint using surface normal vector at each data point establishes the geometric relationships between neighboring field groups. Moreover, the consistency of the normal vectors after curve fitting can be maintained. Then the smoothness of the link line is improved effectively by adding the stairs-distribution elimination constraint. With these constraints, a smooth and accurate surface contour can be finally obtained after curve fitting. The coordinates and normal vectors of the original data points can be approximately satisfied, and the expected imaging relationship can be ensured. The freeform surfaces are taken as the starting point for further optimization in Code V. A freeform f-θ single lens has been designed as an example of the proposed method. The overall system achieves a wide FOV of ± 60° with a scanning range of ± 210mm, and the MTF of each field is close to the diffraction limit. The proposed method to calculate the data points on the unknown surface is effective. Here, only the design of the two-dimensional surface contour for tangential rays is covered in this paper. The design method of the freeform contour can be extended to designing a three-dimensional freeform surface for imaging or illumination optics in the future study.

## Acknowledgment

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

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