## Abstract

It is of urgent need to develop a point-by-point design method for off-axis aspheric systems in full field and full aperture. So a general full-field point-by-point method for off-axis aspheric systems is presented in this paper, in which light rays from different field angles and aperture coordinates are considered. Surface fitting is included during the point calculations, rather than after all the points are calculated. Data point calculations and aspheric surface fitting are repeated continuously to calculate an unknown aspheric surface. Both coordinate and surface normal deviations are considered. As an example, an aspheric off-axis three-mirror reflective system is designed to operate at F/2.4 with a 100-mm entrance pupil diameter and a 3° × 6° off-axis field of view. This method can also be used to design co-axial aspheric systems and novel systems with new structures.

© 2016 Optical Society of America

## 1. Introduction

Compared with spherical and quadric surfaces, aspheric surfaces [1–4] in off-axis optical systems have more degrees of design freedom and are better at eliminating many aberrations. However, considerable experience is needed to design off-axis aspheric optical systems, especially if they have novel structures. Thus an easier, effective design method is needed that does not require significant design experience.

The main idea when designing an off-axis aspheric system is to establish a proper initial system and then optimize it [5–7]. The optimization with optical software is important. However, a good starting point is also of great importance. There are three methods for establishing the initial system. The first uses the lens database, while the second uses paraxial optical theory [8–10]. The third one is point-by-point direct design method that calculates data points on the unknown surface and fits them. The latter is more versatile, reduces error, and requires less design experience. There are three common point-by-point design methods. The Wassermann-Wolf differential equations method [11–13] can be used to design two co-axial aspheric surfaces that are aplanatic at the on-axial field point. The Simultaneous Multiple Surface method [14, 15] can generate several surfaces simultaneously while considers as many field points as surfaces designed. The Construction-Iteration method [16, 17] considers light rays from full field and full aperture and can be used to build freeform surfaces. Here, we need a full-field, point-by-point, direct method for designing off-axis aspheric surfaces.

So in this paper, a full-field, point-by-point, direct design method is proposed where light rays from different field angles and aperture coordinates are considered. Surface fitting is included during the calculations and point calculations and aspheric surface fitting is repeated continuously to calculate an unknown aspheric surface. Deviations in both the coordinates and surface normal vectors of the data points are considered to obtain accurate fitting of the surface. The calculated off-axis aspheric system can be taken as a good starting point for further optimization. The final design can be quickly obtained with further optimization from the starting point and advanced optimization skills are not required. What’ more, the proposed method is a general point-by-point method for off-axis aspheric systems that requires less design experience and makes designing off-axis aspheric system easier. The benefit of this method is demonstrated by designing an aspheric, off-axis, three-mirror reflective system. It operates at F/2.4 with an 100-mm entrance pupil diameter and a 3° × 6° off-axis field of view, and approaches the diffraction limit after optimization.

## 2. Full-field point-by-point direct design method

Based on the given object-image relationships, data calculations and aspheric surface fitting are repeated to calculate an unknown surface. Rays from different field angles and aperture coordinates are focused on the image point after reflection from the calculated aspheric surface system. Both the coordinate and the normal deviations are used to reduce the fitting error. Below, the design method is discussed in two parts: calculation of data points and the fitting method.

#### 2.1. Calculation of the data points

Data point calculation is performed in three steps. (1) An initial system is established and *K* feature rays over different field angles and aperture coordinates are chosen. (2) Using the construction method [16], *m* feature data points are calculated, where *m*<*K*. (3) The *m* feature data points are fitted into an initial aspheric surface ** A_{m}**. The auxiliary point

**is introduced to calculate the (**

*G*_{m}*m*+ 1)th data point

**, and then the**

*P*_{m + 1}*m*+ 1 feature points are fitted to an aspheric surface

**. Next, auxiliary point calculations, feature point calculations, and aspheric surface fitting are repeated continuously to calculate**

*A*_{m + 1}**,**

*G*_{m + 1}**,**

*P*_{m + 2}**……**

*A*_{m + 2}**,**

*G*_{K-1}**and**

*P*_{K}**, where**

*A*_{K}**is the desired aspheric surface.**

*A*_{K}The first task is to construct an initial system with decentered and tilted planes for the subsequent calculation. Here, the obscuration needs to be eliminated firstly and the system needs to be with compact structure. An off-axis three-mirror system is shown as an example in Fig. 1(a).

*K* rays over different field angles and aperture coordinates are then sampled as feature rays that correspond to *K* feature data points, where *K* = *M* fields × *N* equal interval angles of the circular aperture in each field × *P* pupil coordinates of each angle. The feature data point ** P_{i}** is defined as the intersection of its corresponding feature ray

**with the unknown surface. The intersections of**

*R*_{i}**with two neighboring surfaces of the unknown surface are defined as the start point**

*R*_{i}**and the end point**

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

*E*_{i}**is the image point, as shown in Fig. 1(b).**

*I*_{i}Next, *m* feature data points are calculated using the Construction method, where *m*<*K*. During the Construction method, the feature data point ** P_{i}** is calculated as the intersection of its corresponding feature ray with the tangential plane of the nearest data point. The surface normal vector

**at**

*N*_{i}**can be calculated based on the Snell's Law. For a reflective surface:**

*P*_{i}The value of *m* is chosen to provide adequate number of known data points for an accurate fitted surface. The fitting error will be large if *m* is too small, and the subsequent feature data points are too little to further reduce the error if *m* is too big. Generally, *m* = *K* /2.

The calculated *m* feature data points are fitted to the smooth aspheric surface ** A_{m}**. The tangential plane

**at**

*T*_{m}**intersects with**

*P*_{m}**at one intersection line**

*A*_{m}**. As shown in Fig. 2, the data point that is on**

*L*_{m}**, and is nearest to**

*L*_{m}**, is defined as the auxiliary point**

*P*_{m}**.**

*G*_{m}The surface normal vector ** n_{m}** at

**can be calculated at the aspheric surface**

*G*_{m}**. The tangential plane at**

*A*_{m}**intersects with the remaining**

*G*_{m}*K−m*feature rays at

*K−m*intersections. At the intersection

**nearest to**

*P*_{m}^{'}**is a corresponding feature ray**

*G*_{m}**that is the feature ray of**

*R*_{m + 1}**, as shown in Fig. 3(a). Then,**

*P*_{m+}_{1}**is allowed to intersect with the tangent planes of the known**

*R*_{m + 1}*m +*1 data points (

**, 1≤i≤m and**

*P*_{i}**) at**

*G*_{m}*m +*1 intersections (

**, 1≤i≤m and**

*P*_{i}^{'}**). The pair of the data point and its corresponding intersection that has the shortest distance is defined as**

*G*_{m}^{'}**. As shown in Fig. 3(b), the intersection**

*P*_{j}–*P*_{j}^{'}**is the next feature data**

*P*_{j}^{'}**. The surface normal vector at**

*P*_{m+}_{1}**can be calculated from Snell's Law.**

*P*_{m+}_{1}After the coordinate and the surface normal of the (*m* + 1)th feature data point ** P_{m + 1}** are calculated,

*m*+ 1 feature data points are fitted into an off-axis aspheric surface

**. Step (3) is then repeated. The (**

*A*_{m + 1}*m*+ 1)th auxiliary point

**and the (**

*G*_{m+}_{1}*m*+ 2)th feature data point

**are then calculated. The**

*P*_{m + 2}*m*+ 2 feature data points are fitted to an off-axis aspheric surface

**, and so on. Auxiliary point calculations, feature data point calculations, and aspheric surface fitting are repeated continuously until**

*A*_{m + 2}**,**

*G*_{K-1}**, and**

*P*_{K}**are calculated. Finally, the coordinates and the surface normal values of**

*A*_{K}*K*feature data points

**(1≤i≤K) are calculated. The desired off-axis aspheric surface**

*P*_{i}**is obtained by fitting**

*A*_{K}*K*feature data points. A flow diagram of the entire calculation process is depicted in Fig. 4.

#### 2.2. Off-axis aspheric surface fitting

The coordinates and the surface normal values of *K* feature data points ** P_{i}** (1≤i≤K) are calculated under the same “global” coordinate system. These points are fitted to a rotationally symmetric aspheric surface by setting a local coordinate system with an origin at the aspheric symmetric center (0,

*a*,

*b*) and the

*Z*-axis at the rotational symmetric axis. The local coordinate system is described by parameters (

*a*,

*b*,

*θ*), where

*θ*is its rotation angle relative to the global coordinate system, as shown in Fig. 5(a). The feature data points in the global coordinate system can be transformed into the local system to be fitted to an aspheric surface by using the least-squares method.

By using different local coordinate systems, the same feature data points can be fitted into different aspheric surfaces that correspond to various fitting precisions. Here, the position parameters (*a*, *b*, *θ*) of the local coordinate system are determined within a certain range. The”local search algorithm”is used to find a set of optimal position parameters within the range that minimizes deviations of both the fitting coordinate and the normal.

The first step of data fitting is to validate (*a*, *b*, *θ*) and to establish the local coordinate system. A mesh is generated within a certain range [dotted line in Fig. 5(b)], which is around the feature data points and defines the grid nodes as (0, *a*, *b*), as shown in Fig. 5(b). The value of *θ* is generally [-π, π] for each set of (*a*, *b*). In the end, many sets of parameters (*a*, *b*, *θ*) can be derived that correspond to different positions of the local coordinate system.

From a set of (*a*, *b*, *θ*), the feature data points are transformed. The relationship between the global coordinate (*x*_{0}, *y*_{0}, *z*_{0}) and the surface normal (*u*_{0}, *v*_{0}, *w*_{0}), and that between the local coordinate (*x*, *y*, *z*) and the surface normal (*u*, *v*, *w*) can be written as:

The expression of an even aspheric surface defined under the local coordinate system can be written as:

The first term after the equal sign is the quadratic term. It can be transformed into linear XY polynomials by a Taylor series expansion to simplify the fitting process. Thus, the expression of the aspheric surface for further fitting calculations is:

The surface normal vector *N =* (*U*, *V*, *-*1) determines the direction of light rays. Therefore, we need to consider both the coordinate and normal deviations during the fitting to obtain an accurate aspheric surface [18]. The aim of the least-squares method is to find an optimization merit function *J* that can control deviations for both the coordinates and the normal. We propose *J* as:

*Z*is the fitting coordinate, (

*U*,

*V*, −1) is the fitting normal vector,

*z*is the real coordinate, (

*u*,

*v*, −1) is the real normal vector,

*ω*is the weight of the normal deviation,

*P*is the unknown coefficient matrix (

*A*,

*B*,

*C*,

*D*…),

*A*

_{1}is the matrix (

*r*

^{2},

*r*

^{4},

*r*

^{6},

*r*

^{8}…), and

*A*

_{2}and

*A*

_{3}are matrixes obtained by calculating the partial derivatives of

*A*

_{1}with respect to

*x*and

*y*separately.

We seek to minimize the optimization merit function *J*. Only the coefficient matrix *P* is unknown after the equal sign in Eq. (5). Thus, *P* that minimizes *J* is:

Thus coefficient matrix *P* for parameters (*a*, *b*, *θ*) can be obtained by Eq. (6), and its corresponding *J* is known by Eq. (5). Many sets of (*a*, *b*, *θ*) can be selected according to Fig. 5(b). The corresponding *P* and *J* are also calculated. We determine (*a*, *b*, *θ*) whose corresponding *J* is minimized and define it as the optimal local coordinate position. Its corresponding *P* is substituted into Eq. (4) to obtain the optimal fitting aspheric surface.

## 3. Off-axis aspheric three-mirror system

An aspheric off-axis three-mirror imaging system has been designed using the full-field point-by-point method. Specifications of the optical system are listed in Table 1. The goal of this design is to achieve diffraction-limited performance.

An initial system with three decentered and tilted planes is constructed and obscuration is eliminated in advance. The secondary mirror is the aperture stop as shown in Fig. 7(a). In total, there are *K* = 6 × 16 × 7 = 672 (where *K* = 6 fields × 16 equal interval angles of the circular aperture in each field × 7 pupil coordinates of each angle) light rays sampled as feature rays during the calculation. The six sampled fields are (0°, −16°), (0°, −13°), (0°, −10°), (1.5°, −16°), (1.5°, −13°), and (1.5°, −10°). The parameter *m* in the design method is 9K/16≈380, which is adequate for obtaining a good initial aspheric surface. The grid size of (0, *a*, *b*) is 0.2 mm × 0.2 mm. An aspheric expression up to the 6th order is used in this design.

Because the initial planar system is known, an aspheric tertiary mirror is calculated first based on the given object-image relationships. To calculate the feature data point ** P_{i}** (i = 1, 2…

*K*) on the tertiary mirror, the intersections of each feature ray with the secondary mirror and the image plane are the start point

**and the end point**

*S*_{i}**, respectively.**

*E*_{i}**is the ideal image point**

*E*_{i}**, as shown in Fig. 6(a). The aspheric coefficients (**

*I*_{i}*A*,

*B*,

*C*) and the positional parameters (

*a*,

*b*,

*θ*) are also obtained. In order to use Eq. (3) as the aspheric expression in the subsequent optimization, (

*A*,

*B*,

*C*) are transformed into corresponding (

*c*,

*k*,

*a*

_{1},

*a*

_{2}) by Eq. (4). There are three known numbers (

*A*,

*B*,

*C*) and four unknowns (

*c*,

*k*,

*a*

_{1},

*a*

_{2}). It is generally best not to use both a conic and an

*r*

^{4}surface, as they are mathematically quite similar [19]. So the value of

*k*is set to zero to simplify the calculation. The optical system after the calculation is shown in Fig. 7(b).

The aspheric secondary mirror is constructed next. The intersections of each feature ray with the primary mirror and the calculated aspheric tertiary mirror are the start point ** S_{i}** and the end point

**, respectively. According to the Fermat principle,**

*E*_{i}**is the point that minimizes the optical path length between**

*E*_{i}**and**

*P*_{i}**, as shown in Fig. 6(b). The optical system structure after the calculation is shown in Fig. 7(c).**

*I*_{i}The aspheric primary mirror is constructed last. The intersection of each feature ray with the calculated secondary mirror is the end point ** E_{i}**, which is the point that minimizes the optical path length between

**and**

*P*_{i}**as shown in Fig. 6(c). The optical system structure after the calculation is shown in Fig. 7(d).**

*I*,_{i}Thus, the whole off-axis aspheric three-mirror system is generated, with parameters listed in Table 2.

The root mean square (RMS) spot diameters and distortion grid of the system with three aspheric surfaces are shown in Fig. 8. The maximum relative distortion is 6%.

It can be seen from Fig. 7(d) that the rays from different field angles and aperture coordinates are basically focused on the image point. Therefore, this system can be taken as a good starting point for further optimization and indicates that the design method is effective. During the further optimization, many constraints such as: image height and keeping mirrors from obscuring each other are needed to be considered. The final design can be quickly obtained with further optimization from the starting point in Fig. 7(d). The optical system after optimization [Fig. 9(b)] approaches the diffraction limit.

The aspheric parameters after optimization are listed in Table 3.

The MTF, distortion grid, spot diagram, and RMS wavefront error of the system after optimization are shown in Fig. 10. Figure 10(a) is the MTF, which is almost diffraction-limited. Figure 10(b) is the distortion grid, where the maximum relative distortion is 2.5%. Figure 10(c) is the spot diagram of the system, where the maximum RMS spot diameter is less than 3 μm. Figure 10(d) is the RMS wavefront error of the system, with an average value of 0.04λ. In conclusion, the results reveal that good image quality is achieved in this optical system.

With this method, an aspheric off-axis three-mirror reflective optical system was designed from an initial plane system and was optimized to obtain near diffraction-limited resolution quickly. The ease of this method validates the feasibility.

## 4. Conclusions

Here, a general, full-field, point-by-point, direct design method of off-axis aspheric imaging systems was proposed in which light rays from different field angles and aperture coordinates were considered. Some of the feature data points were calculated first and fitted into an initial aspheric surface. Then, auxiliary point calculations, feature data point calculations, and aspheric surface fitting were repeated continuously to calculate remaining feature data points and the desired aspheric surface. A least-squares method with a local search algorithm was used for aspheric surface fitting and deviations in both the coordinates and normals were used to reduce error. In an example, an aspheric off-axis three-mirror system was designed directly and it can be taken as a good starting point for further optimization. It operated at F/2.4 with a 100-mm entrance pupil diameter and a 3° × 6° off-axis field of view, and was close to the diffraction limit after optimization. The aspheric design method does not require much design and optimization experience and can be used for novel systems with new structures.

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