## Abstract

In this paper, mixed-surface-type optical system is defined as the optical system comprised of various types of surfaces, including spherical, aspheric, and freeform surfaces. A general point-by-point design method for mixed-surface-type systems is proposed. In detail, methods for spherical system design and optical power assignment are proposed during the point-by-point design process. Additionally, three surface evolutions are proposed: from spherical to aspheric surfaces and freeform surfaces, and from aspheric surfaces to freeform surfaces. A mixed-surface-type off-axis three-mirror system is designed as an example and as a starting point for further optimization. The sphere receives most of the system optical power while the freeform surface has little power, which can reduce system fabrication difficulty.

© 2017 Optical Society of America

## 1. Introduction

Mixed-surface-type optical system is defined as the optical system that is comprised of various types of surfaces (spherical, aspheric, and freeform surfaces). Following the rapid development of precision machining and testing technologies, spherical and aspheric surfaces have been more widely used in optical systems [1–4]. More recently, freeform surfaces, which are not rotationally symmetrical and have more degrees of freedom, have been used in optical systems to improve the system performance [5–9]. Mixed-surface-type systems comprising spherical, aspheric, and freeform surfaces can be expected to be used widely in the future because they can be designed more flexibly to satisfy optical performance requirements.

Off-axis reflective systems that can change the decentering and tilting of their surfaces to eliminate aberrations offer advantages that include compact structures and the absence of chromatic aberrations and obscuration, and are thus widely used in space-based cameras, telescopes, and infrared/ultraviolet systems [1,3,4].

An optical system is usually designed by establishing an appropriate initial system and then optimizing it [1–3,10] and the mixed-surface-type system design process is similar. The optimization with optical software is important. However, a good initial system is also of great importance. Two methods have been used to establish an initial system during the design of mixed-surface-type systems: one method uses the lens database, while the other uses paraxial optical theory [11–16]. When using paraxial optical theory, the co-axial spherical system is taken as the starting point for further optimization. The paraxial optical theory is calculated under the paraxial condition and is better at designing co-axis systems, while the error may be large if it is used to design the off-axis system, especially when the off-axis magnitude of the system is very large.

Some point-by-point design methods that calculate and fit data points on unknown surfaces have been used to establish initial systems for the design of aspheric and freeform optical systems, which are calculated under the off-axis directly and avoid the error occurred during the conversion from co-axis to off-axis, including the Wassermann-Wolf (W-W) differential equations design method [17–19] that considers on-axial field points, the Simultaneous Multiple Surface (SMS) design method [20,21] that considers the same numbers of field points and surfaces to be designed, and the Construction-Iteration (CI) method [22–24] that considers the light rays from full fields and full apertures.

The desired mixed-surface-type system design method must meet the following requirements. First, it is necessary to control the light rays from the full field and from a full aperture because an actual imaging system usually works for a specific object size and a specific light beam width. Second, a point-by-point design method is needed because it is more generally applicable, more accurate and requires less design experience. Third, surfaces with fewer degrees of freedom are used preferentially to satisfy the optical performance requirements, which is the design principle used in this paper.

In this paper, a full-field full-aperture point-by-point design method is proposed for mixed-surface-type optical systems. Each surface in a mixed-surface-type system can be a different type of surface (including spherical, aspheric, and freeform surfaces). The proposed method for mixed-surface-type system design does not simply require each surface to be calculated directly: instead, the method involves evolution of the surface from a surface that has fewer degrees of freedom to one that has more degrees of freedom. The following three main points in the design method are proposed: a point-by-point design method for spherical systems; a method to assign optical power during the point-by-point design process; and surface evolution methods, including those from spherical to aspheric surfaces, from spherical to freeform surfaces, and from aspheric surfaces to freeform surfaces, which are essentially the calculations from fewer degrees of freedom to more degrees of freedom. In detail, the data point calculations and sphere fitting processes that consider both the coordinates and the normal deviation are repeated continuously to calculate a best-normal-fit sphere. Surfaces with more degrees of freedom can be evolved from the surfaces with fewer degrees of freedom by fitting the coordinates of the data points on the fewer-freedom surfaces and the recalculated normal vectors.

An off-axis three-mirror mixed-surface-type system is designed as an example that can serve as a good starting point for further optimization. In the example, a spherical system is first calculated on a point-by-point basis and the optical power is assigned. Then, spheres with lower optical power are gradually evolved into aspheric or freeform surfaces to correct aberrations, because sometimes freeform surfaces are preferably used in the mirror with lower optical power while spheres that are more easily fabricated contain most of the optical power and thus reduce the system fabrication difficulty. The resulting system operates at F/2.2 with an entrance pupil diameter of 100 mm and a 3° × 6° field-of-view (FOV), and is close to the diffraction limit after optimization in the long wave infrared (LWIR). The mixed-surface-type optical systems have many combinations from which designers can choose according to their requirements. This paper only shows the calculations for one combination, while other combinations are similar.

## 2. Design method for mixed-surface-type systems using a point-by-point construction process

In this section, a point-by-point method is proposed for mixed-surface-type optical systems design that is calculated based on given object-image relationships. A general design idea is proposed: the spherical system is designed first and the optical power is assigned approximately, and then some of the spheres are gradually evolved into aspheric or freeform surfaces for further aberration correction, which is similar to the general optimization process. It is preferable to use surfaces with fewer degrees of freedom to satisfy the optical performance requirements, and this is the design principle used in this paper. A flow diagram that illustrates the entire calculation process is depicted in Fig. 1, and involves surface evolution processes from surfaces with fewer degrees of freedom to surfaces with more degrees of freedom rather than simple direct calculations of each surface.

The calculation process can be broken down into the following five steps.

- Step 1: an initial planar system is established without obscuration, and this system needs to have a compact structure.
- Step 2: the spherical system is then calculated point-by-point.
- Step 3: the optical power of the spherical system is assigned approximately. This step is repeated until the optical power of each mirror in the system meets the design requirements; then, the spherical system after power assignment is used to perform any further calculations or optimization. The design process ends if the spherical system obtained after optimization satisfies the optical performance requirements. If not, the process continues to Step 4.
- Step 4: mixed system that contains both spherical and aspheric surfaces is then calculated point-by-point. The aspheric surfaces in the mixed system can be evolved from spheres. The design process ends if the mixed system composed of spherical and aspheric surfaces satisfies the optical performance requirements after optimization. If not, Step 5 is performed.
- Step 5: if necessary, mixed system that contains spherical, aspheric and freeform surfaces is then calculated point-by-point. The freeform surfaces in the mixed system can be evolved from either spherical or aspheric surfaces for further correction of system aberrations. Then, the resulting mixed system, which comprises spherical, aspheric and freeform surfaces, is used for further design optimization.

Based on the five steps above, a mixed-surface-type optical system containing spherical, aspheric, and freeform surfaces can be generated point-by-point. The details of Step 2 are shown in Section 2.1; Step 3 is shown in Section 2.2; Steps 4 and 5 are shown in Section 2.3.

#### 2.1 Design method for spheres with full-field full-aperture point-by-point process

To calculate a sphere using a point-by-point process, sphere fitting process that considers both the coordinate and normal deviations is deduced and is defined as the best-normal sphere fitting method. Sphere that is obtained by this method is defined as the best-normal-fit sphere. The data points on the unknown surface are calculated by repeating the data point calculations and the best-normal sphere fitting process continuously and the resulting points are fitted into a best-normal-fit sphere. Below, the sphere design method is discussed in the following two parts: the calculation of the feature data points and the best-normal sphere fitting method.

### 2.1.1 Calculation of feature data points

The calculation process for the data points is based on given object-image relationships and rays from different field angles and aperture coordinates are focused on the image point after calculation in the ideal case. Here, we present an introduction to the point calculation process.

(1) An initial planar system without obscuration is established. *M* fields are sampled and the circular aperture of each field is then divided into *N* polar angles with *P* sampled pupil coordinates for each angle. Therefore, *K* = *M* × *N* × *P* light rays over the different field angles and aperture coordinates are selected to be used as the feature rays that correspond to the *K* feature data points on the unknown surface.

(2) The intersections of the feature ray ** R_{i}** with two neighboring surfaces of the unknown surface (i.e., the initial plane in this step) are regarded as the start point

**and the end point**

*S*_{i}**of**

*E*_{i}**.**

*R*_{i}**is the image point. As shown in Fig. 2(c), the normal vector**

*I*_{i}**for each feature data point**

*N*_{i}**is calculated using Snell's law:**

*P*_{i}**and**

*r*_{i}=*S*_{i}*P*_{i}/|*S*_{i}*P*_{i}|**are vectors in the directions of the incident ray and the exiting ray, respectively. The tangential plane at**

*r*_{i}' =*E*_{i}*P*_{i}/|*E*_{i}*P*_{i}|**can then be obtained.**

*P*_{i}The feature data point ** P_{i}** (where 2≤

*i*≤

*m*) is calculated as the intersection of the corresponding feature ray

**with the tangential plane of the nearest feature data point**

*R*_{i}**(where 1≤**

*P*_{q}*q*<

*i*), as shown in Fig. 2(a), where

**is calculated as the intersection of the corresponding feature ray**

*P*_{1}**with the initial plane.**

*R*_{i}Subsequently, *m* feature data points and their corresponding normal vectors are calculated and fitted to an initial best-normal-fit sphere *A _{m}*

_{.}In general,

*m*=

*K*/2, which is sufficient to obtain an accurate fitting sphere. The fitting error will be high if

*m*is too small, but the number of the remaining feature data points will be too low to reduce the fitting error further if

*m*is too large.

(3) The remaining *K*-*m* feature data points are calculated by repeating the best-normal sphere fitting and feature data point calculations continuously. Essentially, *j* (where *m*≤ *j*≤ *K*−1) feature data points (from ** P_{1}** to

**) are fitted to a best-normal-fit sphere**

*P*_{j}**. The feature data point**

*A*_{j}**is calculated as the intersection of the corresponding feature ray**

*P*_{j + 1}**with the best-normal-fit sphere**

*R*_{j + 1}**, as shown in Fig. 2(b). Subsequently,**

*A*_{j}*j*+ 1 feature data points (ranging from

**to**

*P*_{1}**) are fitted into a best-normal-fit sphere**

*P*_{j + 1}**, and the feature data point**

*A*_{j + 1}**is then calculated based on sphere**

*P*_{j + 2}**. The best-normal sphere fitting and feature data point calculation are then repeated continuously until the remaining**

*A*_{j + 1}*K*-

*m*feature data points have all been calculated.

The normal vector ** N_{j}** for each data point

**is calculated using Snell's law, where the unknown surface in this step is the fitting sphere**

*P*_{j}**(where**

*A*_{j}*m*≤

*j*≤

*K*−1).

Finally, the coordinates and normal vectors of all *K* feature data points ** P_{i}** (where 1≤

*i*≤

*K*) are calculated. The best-normal-fit sphere

**is obtained by fitting all**

*A*_{K}*K*feature data points and their corresponding normal vectors. The best-normal sphere fitting method that is used during these calculations is introduced in the next section.

### 2.1.2 Best-normal sphere fitting method

The surface normal vector *N =* (*u*, *v*, −1) determines the propagation direction of the light rays. Therefore, both the coordinates and the normal deviation must be minimized during the fitting process to obtain the best-normal-fit sphere [25].

During these calculations, the calculated feature data points and the corresponding normal vectors will be fitted to a best-normal-fit sphere using the least squares method. The coordinates of the calculated feature data point ** P_{i}** (where 1≤

*i*≤

*K*) are (

*x*

_{i},

*y*

_{i},

*z*

_{i}), the corresponding normal vector is (

*u*

_{i},

*v*

_{i}, −1), the center of the sphere is (

*A*,

*B*,

*C*), and the radius is

*r*. The expression for the sphere is thus:

This expression can be written in matrix form as:

The form of Eq. (3) is the same as **Ax = b**. Therefore, both sides of Eq. (3) can be multiplied by **A ^{T}** to solve for the parameter

**x**[26]:

*D*=

*A*

^{2}+

*B*

^{2}+

*C*

^{2}−

*r*

^{2}. Equation (4) can be converted via a matrix transformation to remove the unknown

*r*. The resulting expression for calculation of the center of sphere using the coordinates of the feature data points is thus:

*x*

_{i}and the other parameters have similar definitions.

The expressions for the normals *u* and *v* are obtained by calculating partial derivatives of the spherical expression with respect to *x* and *y* separately, as shown in Eq. (6) and Eq. (7).

Similar to Eq. (2), Eqs. (6) and (7) can be written in matrix form to enable calculation of the center of the sphere based on the normal vectors of the feature data points, as shown in Eq. (8) and Eq. (9), where *U*_{i} = (1 + 1/*u*_{i}^{2})*x*_{i} and *V*_{i} = (1 + 1/*v*_{i}^{2})*y*_{i}.

The sphere center (*A*, *B*, *C*) is contained in Eqs. (5), (8) and (9), where Eq. (5) is obtained by the coordinates of the feature data points and Eqs. (8) and (9) are obtained by the normals of the feature data points. As shown in Eq. (10), the center of the best-normal-fit sphere (*A*, *B*, *C*) can be calculated using the following weighting procedure among Eqs. (5), (8) and (9), where *ω* is the weight of the calculation based on the normal vectors.

The radius *r* is contained in Eqs. (2), (6) and (7), where Eq. (2) is obtained by the coordinates of the feature data points and Eqs. (6) and (7) are obtained by the normals of the feature data points. As shown in Eq. (11), when the center of the best-normal-fit sphere is known, the radius *r* can be obtained using a weighting procedure among Eqs. (2), (6) and (7), where *ω* is again the weight of the calculation based on the normal vectors.

As shown above, Eqs. (10) and (11) are both equalities composed of matrices, and the unknown parameters of the best-normal-fit sphere can be calculated by solving them using the matrix manipulation.

Based on the fitting method presented above, the best-normal-fit sphere is obtained and its own optical power, which is determined based on the radius, is also obtained. Because the design freedom is limited with only one sphere, the above sphere design method can be used multiple times to generate multiple spheres in a given system, where the different surfaces in that system are connected according to Fermat's principle [22] that the optical path length between these surfaces is an extremum. After the complete spherical system is obtained, a method to assign the system’s optical power is then proposed.

#### 2.2 Method to assign optical power during the point-by-point design process

One major concern in allocating optical power to each surface is the correction of field curvature. Allocating optical power can also be for avoiding one mirror receives too much system optical power, which can reduce system fabrication and assembly difficulty.

Using the off-axis three-mirror system as an example, an initial planar system is established where all obscuration has been eliminated in advance, as shown in Fig. 3(a).

Using the proposed design method, a spherical system with three mirrors has been calculated point-by-point. As shown in Fig. 3(b), the mirror (sphere C) that was calculated first obtains most of the system optical power because rays incident from different field angles and aperture coordinates are basically focused on the image point after it has been calculated. The other two mirrors (spheres B and A) calculated next receive very little optical power so that they look like plane [Fig. 3(c)]. In fact, they are spheres with small curvatures. Although they offer less power, they are important in aberration correction. The relationship among the optical powers of the three mirrors is shown in Eq. (12)

Sometimes, depending on the design requirements, the distribution of the optical power on a specific mirror may not be appropriate (too little or too much) after the calculations and must be reassigned. Therefore, a method to assign optical power in a spherical system is proposed. After one sphere is calculated, the optical power of this sphere as determined by its radius is also obtained. If the optical power of the sphere is too high or too low, its radius *r* can be changed into *ε* × *r* to change its optical power, where the number *ε* is determined by the designers based on the sphere’s fabrication and alignment requirements. Then, the other spherical mirrors must be recalculated to compensate for the change in optical power.

Using Fig. 4 as an example, sphere C is calculated first [Fig. 4(a)]. *r*_{C}* _{'}* =

*ε*

_{C}×

*r*

_{C}is used to change the optical power of the tertiary mirror [Fig. 4(b)], where

*ε*

_{C}= 1.25. Then, to compensate for the change in the optical power of C

**, sphere B is calculated under the same optical performance conditions [Fig. 4(c)].**

^{'}*r*

_{B}

*=*

_{'}*ε*

_{B}×

*r*

_{B}is used to change the power of the secondary mirror [Fig. 4(d)], where

*ε*

_{B}= 0.6. Finally, sphere A

**is calculated to compensate for the changed optical power of sphere B**

^{'}**[Fig. 4(e)]. After these calculations, the power is distributed more evenly over the three spherical mirrors.**

^{'}In this way, the optical power of the spherical system can be reassigned over the three mirrors. The sphere after assignment can be with more or less optical power, or with negative power for a system with positive power as a whole to correct the field curvature, which is decided according to the design requirements. This method can also be used in the conventional CI method to assign the optical powers of the optical system.

The spherical system after the power assignment process can then be used to perform further calculations or optimization. During the optimization, the surface parameters and decenters and tilts of the mirrors and the image plane are adjusted with some constraints added to get the best image quality. The optimization process ends if the RMS wavefront error and MTF meet the requirements. The following optimization processes of mixed systems are similar.

#### 2.3 Surface evolution processes: from sphere to aspheric surface; from sphere to freeform surface; and from aspheric surface to freeform surface

A spherical system generally cannot satisfy optical performance requirements because it has fewer degrees of freedom, and thus the use of aspheric and freeform surfaces to correct the higher order aberrations of such a system is a significant necessity. In the proposed design method, aspheric and freeform surfaces are calculated point-by-point from the spheres that were generated in the previous design step, which can produce a better starting point for further optimization processes.

Three different surface evolution processes are proposed here: from spherical to aspheric surfaces (Step 4); and from spherical or aspheric surfaces to freeform surfaces (Step 5), and these are essentially calculations that allow the surfaces to evolve from having fewer degrees of freedom to having more degrees of freedom.

In Step 4, the evolution of the aspheric surface consists of three main points. First, the data points on the sphere are sampled and their coordinates are preserved; second, the normal vector of each sampled data point is recalculated point-by-point using Snell's law; third, the sampled coordinates and recalculated normal vectors of the data points are fitted to an aspheric surface using the least squares method while considering both the coordinates and the normal deviation.

In Step 5, the evolution of the freeform surface is similar to the process in Step 4 in terms of the first two main points, but differs during the fitting process because the freeform surface is not rotationally symmetrical.

- (1) The data points on the best-normal-fit sphere are sampled to record the information of the sphere. In Section 2.1,
*K*feature data points are calculated and are then fitted to a best-normal-fit sphere. The sampled data points*A*_{K}(where 1≤*D*_{i}*i*≤*K*) are considered to be the intersections of the*K*feature rays with the sphere. The coordinates*A*_{K}of the sampled data points*Z*_{i}are then preserved for use in further calculations.*D*_{i} - (2) The normal vectors are then recalculated for each sampled data point. Both the coordinates and surface normals decide the propagation direction of the light rays and the impact of surface normals is lager. Therefore, to focus light rays that are incident from different field angles and aperture coordinates on the image point, the normal
of each data point*N*_{i}must be recalculated based on the given object-image relationship using Snell's law, as shown in Section 2.1.1, where the unknown surface in this step is the best-normal-fit sphere*D*_{i}.*A*_{K} - (3) The preserved coordinates
and the recalculated normals*Z*_{i}are then fitted to an aspheric surface. The aspheric surface is rotationally symmetrical, so the main idea during the fitting process is to find an optimal aspheric vertex among the grid nodes that minimizes both the coordinate and normal deviations using a local search algorithm around the sampled data points [24], as shown in Fig. 5. The aspheric parameters are then calculated using the least squares method under a local coordinate system with its origin at the aspheric vertex (0,*N*_{i}*a*,*b*) and its*Z*-axis along the rotational symmetry axis.

As the freeform surface is not rotationally symmetrical, the data point that corresponds to the main ray of the central field angle throughout the entire field-of-view (FOV) is taken to be the vertex of the surface and is used as the origin of the local coordinate system. The sampled coordinates and the recalculated normal vectors are then fitted to a freeform surface with a base conic in the local coordinate system using the least squares method.

A point-to-point imaging system is used as an example to compare between the direct surface calculations and the surface evolutions. The corresponding optical structure is shown in Fig. 6.

The unknown surface in Fig. 6 can be a spherical, aspheric or freeform surface. The surface (spherical, aspheric or freeform) can be calculated directly from the plane based on fitting of the calculated feature data points and the corresponding normals, while aspheric or freeform surfaces can also be evolved from fewer-freedom surfaces by fitting of the sampled data points and the corresponding normals. The root mean square (RMS) spot diameter is used as the evaluation index here.

As shown in Fig. 7, a freeform surface that is evolved from an aspheric surface using the data points on the aspheric surface [Fig. 7(f)] produces the best image quality. Aspheric and freeform surfaces that are evolved from spheres using the data points on the sphere [Fig. 7(c), (e)] are better than those that were calculated directly from the plane [Fig. 7(b), (d)]. The above analyses prove that surface evolution processes can produce better image quality than direct surface calculations.

In a mixed optical system, based on the three key points described above, aspheric surfaces can be evolved from spheres and freeform surfaces can be evolved from both spherical and aspheric surfaces. In this way, mixed systems with any combinations of surfaces can be designed using the proposed design method and used for further optimization.

## 3. Example design of a mixed-surface-type system

It is preferable to use surfaces that have fewer degrees of freedom to satisfy the optical performance requirements, and this is the principle that is used during the point-by-point design process. The surface with strong curvature receives more optical power of the system, and it is more difficult to manufacture and assemble, especially when it has more degrees of freedom. Therefore, if a freeform surface must be used in the mixed-surface-type system, it is preferred to be in the form of a mirror with lower optical power to reduce the fabrication difficulty, while the more easily fabricated spheres can handle most of the optical power.

According to the analysis above, an off-axis three-mirror mixed system with freeform primary, aspheric secondary and spherical tertiary mirrors is designed as an example using the point-by-point design method proposed in Section 2. The specifications of this optical system are listed in Table 1, where the FOV ranges from −16° to −10° in the meridian direction.

An initial planar system is constructed and all obscuration is eliminated from this system in advance. The secondary mirror is used as the aperture stop, as shown in Fig. 8(a). In total, *K* = 14 × 16 × 7 = 1568 light rays are sampled as feature rays during the calculation, in which 14 fields are sampled and the circular aperture of each field is then divided into 16 polar angles with 7 sampled pupil coordinates for each angle. An aspheric expression up to the 6th order is used as shown in Eq. (13) and a 12-term XY polynomial surface up to the 6th order is used as the freeform expression, which is shown here as Eq. (14).

After the initial planar system is determined, a system composed of three spheres is calculated, with the tertiary, secondary, and primary mirrors being calculated in order. The sphere is calculated by repetition of data point calculations and best-normal sphere fitting processes. The resulting spherical system after the calculation is shown in Fig. 8(b). When the optical power is assigned, the radius of the tertiary mirror is *ε* times of the calculated value, where *ε* = 1.25 in this example. The corresponding secondary and primary mirrors are then calculated, as shown in Fig. 8(c). When compared with the distribution in Fig. 8(b), the optical power of the spherical system is distributed more evenly over the three spherical mirrors in Fig. 8(c). After the optical power has been assigned, the resulting spherical system is then used for further calculations, as shown in Fig. 9(a), and the corresponding RMS spot diameter is shown in Fig. 9(b).

Next, the spherical system is used as the initial system from which the aspheric secondary and primary mirrors are calculated in order, as shown in Fig. 9(c). The aspheric surfaces are evolved from the spheres by fitting of the sampled spherical coordinates and the recalculated normal vectors using the least squares method, where a local search algorithm is used to find the optimal axis of symmetry. The corresponding RMS spot diameter is shown in Fig. 9(d).

Finally, the desired freeform primary mirror is evolved from the corresponding aspheric surface. The freeform surface is calculated by fitting of sampled coordinates and the recalculated normal vectors using the least squares method. The final calculated mixed system is shown in Fig. 9(e), and the corresponding RMS spot diameter is shown in Fig. 9(f).

Thus, an entire mixed-surface-type off-axis three-mirror system is generated, in which the spherical tertiary mirror receives most of the system’s optical power. Figure 10(a) shows that the light rays from different field angles and different aperture coordinates are all basically focused on the image point. Therefore, this system can be used as a good starting point for further optimization processes.

The conventional optimization method is used by adjusting decenters and tilts of the mirrors and the image plane, as well as the surface parameters, and the distances between each mirrors. During these optimization processes, many constraints must be considered, including image height, ensuring that the tertiary mirror receives most of the optical power, and preventing the mirrors from obscuring each other. The optimization process ends if the maximum of the RMS wavefront error is less than the diffraction limit (0.07λ). The final design [Fig. 10(b)] can then be obtained quickly by further optimization from this starting point. The optical system after optimization approaches the diffraction limit at LWIR.

The modulation transfer function (MTF) and the RMS wavefront error of the mixed system after optimization are shown in Fig. 11. Additionally, the maximum relative distortion of the system is 7% in the sagittal direction and 3% in the meridian direction. These results show that good image quality can be achieved using this optical system.

By the proposed full-field full-aperture point-by-point design method, an off-axis three-mirror mixed system containing freeform primary, aspheric secondary and spherical tertiary mirrors was designed based on an initial planar system and the optical power was assigned during the design calculations. The designed system was rapidly optimized to obtain near diffraction-limited resolution in the LWIR, and advanced optimization skills were not needed.

## 4. Conclusions

The aim of this paper is to propose a point-by-point design method for mixed-surface-type optical systems. Three main aspects of the design method are proposed: a point-by-point design method for sphere systems; a method to assign the optical power of the system during the point-by-point design process; and surface evolution processes including those from spherical to aspheric surfaces, from spherical to freeform surfaces, and from aspheric surfaces to freeform surfaces, which are essentially calculations that allow the surfaces to evolve from having fewer degrees of freedom to having more degrees of freedom. The sphere in the mixed system is calculated through repeating data point calculations and best-normal sphere fitting, where both the coordinate and normal deviations are considered during the fitting process. Surfaces with more degrees of freedom can be evolved from surfaces with fewer degrees of freedom by fitting the coordinates of the data points on the fewer-freedom surfaces and the recalculated normal vectors. As an example, a mixed-surface-type off-axis three-mirror system with freeform primary, aspheric secondary and spherical tertiary mirrors is designed that operates at F/2.2 with an entrance pupil diameter of 100 mm and a 3° × 6° FOV. In the design process, the spherical system is initially calculated point-by-point and the corresponding optical power is assigned simultaneously. Then, some spheres with lower powers are gradually evolved into aspheric or freeform surfaces to further correct system aberrations, and this ensures that the sphere receives most of the optical power. Light rays that are incident from different field angles and aperture coordinates are then basically focused on the image point after completion of the calculations. This system can then be used as a good starting point for further design optimization. The final design can be obtained rapidly based on optimization from this starting point and the designed system operates close to the LWIR diffraction limit, while advanced optimization skills were not needed. Optical designers can use the proposed method to design mixed optical systems with various surface combinations and can then select the most reasonable combination according to their specific design requirements.

## References and links

**1. **D. Korsch, “Anastigmatic three-mirror telescope,” Appl. Opt. **16**(8), 2074–2077 (1977). [CrossRef] [PubMed]

**2. **P. N. Robb, “Three-mirror telescopes: design and optimization,” Appl. Opt. **17**(17), 2677–2685 (1978). [CrossRef] [PubMed]

**3. **L. G. Cook, “three-mirror anastigmatic used off-axis in aperture and field,” Proc. SPIE **183**, 207–211 (1979). [CrossRef]

**4. **J. M. Rodgers, “Unobscured mirror designs,” Proc. SPIE **4832**, 33–60 (2002). [CrossRef]

**5. **O. Cakmakci and J. Rolland, “Design and fabrication of a dual-element off-axis near-eye optical magnifier,” Opt. Lett. **32**(11), 1363–1365 (2007). [CrossRef] [PubMed]

**6. **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]

**7. **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]

**8. **R. Wu, H. Li, Z. Zheng, and X. Liu, “Freeform lens arrays for off-axis illumination in an optical lithography system,” Appl. Opt. **50**(5), 725–732 (2011). [CrossRef] [PubMed]

**9. **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]

**10. **G. Xie, J. Chang, J. Zhou, K. Zhang, and X. Wang, “Research on all movable reflective zoom system with three mirrors,” Proc. SPIE **9618**, 96180O (2015). [CrossRef]

**11. **J. H. Pan, “The Design, Manufacture and Test of the Aspheric Optical Surface,” Science (1994) (in Chinese).

**12. **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]

**13. **Q. Meng, H. Wang, K. Wang, Y. Wang, Z. Ji, and D. Wang, “Off-axis three-mirror freeform telescope with a large linear field of view based on an integration mirror,” Appl. Opt. **55**(32), 8962–8970 (2016). [CrossRef] [PubMed]

**14. **Y. Zhong, H. Gross, A. Broemel, S. Kirschstein, P. Petruck, and A. Tuennermann, “Investigation of TMA systems with different freeform surfaces,” Proc. SPIE **9626**, 96260X (2015). [CrossRef]

**15. **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]

**16. **C. Huang and X. Liu, “Design of off-axis four-mirror optical system without obscuration based on free-form surface,” Proc. SPIE **97952**, 97952D (2015).

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

**18. **Y. Bian, H. Li, Y. Wang, Z. Zheng, and X. Liu, “Method to design two aspheric surfaces for a wide field of view imaging system with low distortion,” Appl. Opt. **54**(27), 8241–8247 (2015). [CrossRef] [PubMed]

**19. **T. W. Tukker, “Beam-shaping lenses in illumination optics,” Proc. SPIE **6338**, 63380A (2006). [CrossRef]

**20. **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). [CrossRef] [PubMed]

**21. **J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamaría, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE **7429**, 74290C (2009). [CrossRef]

**22. **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]

**23. **T. Yang, J. Zhu, X. Wu, and G. Jin, “Direct design of freeform surfaces and freeform imaging systems with a point-by-point three-dimensional construction-iteration method,” Opt. Express **23**(8), 10233–10246 (2015). [CrossRef] [PubMed]

**24. **T. Gong, G. Jin, and J. Zhu, “Full-field point-by-point direct design method of off-axis aspheric imaging systems,” Opt. Express **24**(26), 29417–29426 (2016). [CrossRef] [PubMed]

**25. **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]

**26. **Z. Guan and J. F. Lu, “The fundamental of numerical analysis,” High. Educ. (1998) (in Chinese).