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Abstract
The development of electronic industry demands miniaturization for zoom lens. Freeform surfaces and catadioptric lens system play important roles in the compactness of optical design. However, the initial structure of the catadioptric system is relatively rare and repeated optimizing in first-order lens design requires advanced knowledge and experiences. That brings challenges for establishing an initial structure for the freeform catadioptric zoom system, which is essential for optical designers. In this work, an automatic design method for the initial structure of a catadioptric system with freeform surface and zoom lens is proposed. As long as the focal length range and total length are determined, the initial structure of the catadioptric zoom lens with arbitrary zoom ratio in a certain range can be obtained for further analysis and optimization. The rapid design method reduces the dependence on experience for catadioptric system design, saves considerable time and can help to ease the difficulty in finding the initial structure of slim zoom lens.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Zoom lens has been more and more widely used for it combines the function of large field of view (FOV) and long shooting distance. Traditional zoom lens mostly adopts the coaxial zoom structure. Researchers need to search for the patent or use paraxial solution method to find the initial structure. However, the traditional design method of coaxial zoom lens has can't meet the requirements of miniaturization design, such as drone, AVR, mobile phones and other equipment with small volume. The catadioptric optical system can better meet the requirements of miniaturization. However, the initial structure of catadioptric zoom lens is relatively rare, which indicates that it is difficult to obtain through patent search. With the help of powerful global optimization from optical software CODE V, the initial structure of coaxial lens can be obtained. However, for complex off-axis zoom lens, even if all the boundary conditions are fully supplied or the global optimization is performed, the analysis results still require further exclusion and screening. The process is time-consuming and the feasible structures are usually limited. In addition, it is also difficult to use paraxial optics to analyze the catadioptric lens structure. Researchers need to spend a lot of time on repeated experiments to find the initial structure, and it is difficult to obtain initial structure parameters quickly.
To make zoom lenses as small as possible, the current solution is to fold ray path by using a right angle prism. In 2013, Park and Lee designed a compact zoom system using lens modules and aberration theory. However, the depth of this design is up to 12mm, which is difficult to meet the current requirements of mobile phones [1]. In 2018, Hou et al. introduced two Alvarez lenses to design a compact zoom lens with 3X zoom ratio [2]. The two Alvarez lenses adopted freeform surface and moved horizontally, which provided a new direction for the research of compact zoom system. Recently in 2021, a varifocal panoramic annular zoom lens for 360° rotating imaging was provided by Wang et al., which adopted a catadioptric structure [3]. In the field of off-axis reflection system design, Yang et al. realized three-mirror reflection system automatic design by point-by-point construction process [4]. Zhong and Gross extended initial system design method from rotationally symmetric systems to general non-rotationally symmetric systems by using the Nodal aberration theory and Gaussian brackets [5]. Chen et al. adopted deep learning to generate a good starting point for highly non rotationally symmetric freeform refraction, reflection and catadioptric systems by generating sufficient data sets in advance. However, for systems with high complexity, the effort required will increase significantly, and the results generated by deep learning may need to be further selected [6].
With the development of computer technology, the important value of matrix optics in ray tracing has been paid more and more attention. In 2022, Hong and Dunsby presented the coaxial ABCD matrix in dealing with automatic tube lens design process [7]. The ABCD matrix was applied to obtain the first-order properties of a pair of doublets. In 2021, Tang and Gross described a method to analysis the ray-tracing calculation and aberrations of symmetry-free system, and presented examples as well [8,9]. K. Dupraz et al. used ABCD matrix to simulate light propagation in reflection and refraction surfaces [10]. As we all know, comparing to off-axis reflection system, it is more difficult to represent initial structure of catadioptric system because the refractive index shifts at a surface needs to be considered by applying Snell’s law. In recent research, Snell's law has also been well applied in the design of new optical sensors, but this is not the focus of this paper. We mainly use Snell's law to realize the freeform surface design [11,12]. As far as we know, an easier way in obtaining initial structure of freeform catadioptric zoom lens automatically has not been investigated yet. Our work is to discuss about this problem. We are going to build the mathematic model of the catadioptric system and combine the math model with algorithm to achieve faster and more accurate initial structure design. As long as the focal length range and total length are given, the initial structure of the catadioptric zoom lens with arbitrary zoom ratio can be obtained for further analysis and optimization. The steps of this approach are mostly automated, which does not require the repeated experiments to find the initial structure or a lot of experience for designer. We also present two examples of initial structure design of slim catadioptric zoom system and compare the first-order results by MATLAB programming with those in CODE V.
2. Methods and mathematical fundamentals of catadioptric system design
2.1 Principle of initial lens construction
An idea of mixed structure based on freeform surface is proposed. A middle image is generated by compressing and folding ray path using a freeform prism. And the middle image is then magnified and focused by a conjugate zoom lens. The prism turns infinity objects into a middle image, while the function of the second lens group is to zoom the middle image in different magnification. The advantage of this structure is this: By changing the parameters of the second part, a series of catadioptric zoom lenses with various zoom ratios can be achieved, which makes it possible for catadioptric zoom system having automatic design process with a customized zoom ratio. In addition, this structure is symmetrical about the YOZ plane in the global coordinate system, which means that ray paths are symmetrical in the positive and negative direction on X axis. Moreover, the chief ray of the central field is located on the YOZ plane, so we only discuss the reflection and refraction in the YOZ plane without considering the influence of the X direction. And the mathematical model discussed below is based on this assumption as well.
In order to explore mathematical approach about a first-order quantitative analysis of the system, we introduce ABCD matrix and Gaussian brackets. The Gaussian brackets has been applied in optical design process for coaxial zoom system [13,14]. In this paper, we will combine Gaussian brackets with ABCD matrix to establish a mathematical model for initial structure of the catadioptric zoom lens, which is more convenient for computer calculation and storage. We present the expression of the system ABCD matrix ${M_Y}$ of the above catadioptric optical system as Eq. (1).
The dual-conjugate zoom system contains at least three moveable lenses [16–18]. Our conjugate zoom lens is the one in which the object and image position are fixed. Therefore, three moving groups can meet our needs. The finite conjugate zoom lens contains the following unknown parameters, group focal powers including ${\varphi _1}$, ${\varphi _2}$, ${\varphi _3}$, and air distances which are ${e_0}$, ${e_1}$, ${e_2}$, ${e_3}$. The finite conjugate zoom lens can be expressed by Gaussian brackets in Eq. (2), which can be expanded according to the transformation rules [15].
2.2 Off-axis ray tracing math model based on Gaussian brackets and ABCD matrix
Next, we will explore the mathematical foundation of automatic optical design of catadioptric systems. That is, off-axis refraction and reflection optical systems are represented by matrices for computer processing. Based on the laws of reflection and refraction, we obtain the following mathematical model. With the help of ABCD matrix, the first-order focal length characteristics of the off-axis imaging optical system can be obtained. Combining ABCD matrix and the above mathematical model, the initial structure formula of the refraction and reflection system can be obtained. Generally, the ABCD matrix has two input elements, namely, the ray height y and the aperture angle u. Among them, the description of aperture angle u is generally divided into two systems, U system and NU system. The input of aperture angle u of the NU system needs to be multiplied by the refractive index n of the medium. According to the properties of ABCD matrix, when light propagates in a uniform medium, the refractive index of the medium does not change suddenly, the U system is applicable. When light passes through different media, the change of refractive index should be taken into account, so it is more suitable to adopt NU system. In addition, the refractive index of air is 1, so the inputs of nu and u are equivalent to the ABCD matrix in air. Since the optical system in section 2.1 includes both glass and air, both systems are used in the mathematical model.
In the reflection system, in order to obtain the ABCD matrix of the off-axis optical system, we have the ray-tracing model in Fig. 1. The coordinate systems of the model are all local coordinate systems. Suppose the incident angle is named -θ, while the reflection angle is -θ’ for chief ray (the brown line). The incident angle and reflection angle of the next reflected chief ray are $\widetilde { + \theta }$ and $\widetilde { + \theta }^{\prime}$ respectively. For the upper reference ray (the purple line), the corresponding angles are $- {\theta _1}$, $- {\theta _1}^{\prime}$, $\widetilde { + {\theta _1}}$ and $\widetilde { + {\theta _1}^{\prime}}$ respectively. The angle between the upper incident reference ray and the chief ray, that is, the aperture angle, is expressed as + u, while the aperture angle of the reflection ray is + u’. The aperture angle of the next reflection is −u". In addition, we also specify the symbolic rules of the matrix. The right-hand Cartesian coordinate system is established based on the incident chief ray, which is the initial coordinate system. At present, the propagation direction of the ray is the positive direction of the Z coordinate, the Y coordinate component of the upper reference ray is positive, and the positive direction of the X coordinate is perpendicular to the paper. For the incident angle and reflection angle, we determine the sign of the angle according to the direction of the chief ray turning to the normal vector N. The sign is negative in clockwise direction and positive in counterclockwise direction. In particular, for the reflection angle, since the local coordinate system changes from the right-hand coordinate system to the left-hand coordinate system in ray tracing after the next reflection, the symbol of the reflection angle also needs to be changed. For the aperture angles u and u’ of the upper reference ray, when the chief ray turns clockwise to the upper reference ray, the symbol of the model is negative for the incident ray, and vice versa. Due to the change of the local coordinate system for reflection, the aperture angle of the reflected ray should also be changed. In addition, the Y coordinate of the incident ray is positive. When the rays travel along the direction from the vertex of the surface to the center of the curve, the radius of curvature R of the reflecting surface is positive, and vice versa.
According to geometric relationship, for surface 1 (S1), we have Eq. (3):
This simplifies to the following equation Eq. (4):
Among them, $\Delta {\alpha _1} = \frac{{{y_1}}}{{\cos \theta \cdot {r_1}}}$, where ${y_1}$ represents the vertical distance of the incident point of the upper reference ray relative to the chief ray. And ${r_1}$ represent the radius of S1. Here, we have ${y_1}$>0, ${r_1}$>0, so $\Delta {\alpha _1}$>0.
For surface 2 (S2), we have Eq. (5):
This simplifies to the following equation Eq. (6):
where, $\Delta {\alpha _2} = \frac{{{y_2}}}{{\cos \widetilde \theta \cdot {r_2}}}$, where ${y_2}$ represents the vertical distance of the incident point of the upper reference ray relative to the normal of the chief ray on S2. And ${r_2}$ represent the radius of S2. Here, we have ${y_2}$>0, ${r_2}$<0, so $\Delta {\alpha _2}$<0. The equations are in radians.In the reflection system, we notice that the calculation rules of the local coordinate y passing through the reflection surface are shown in Eq. (7) as follows. Where y and y’ stand for reference ray height related to chief ray in local coordinate system.
Therefore, the ABCD matrix corresponding to the mathematical model of the off-axis reflection system is indicated in Eq. (8), in which neither the refractive index n nor NU system needs to be considered.
The ABCD matrix of the off-axis refraction system is also analyzed in the local coordinate system, as shown in Fig. 2. For surface 1 (S1), we have Eq. (9) as follows.
According to the law of refraction and the above formula, Eq. (10) can be obtained.
where ${n_1}$ and ${n_2}$ are the refractive indices before and after surface S, $\Delta {\alpha _1} = \frac{{{y_1}}}{{\cos \theta \cdot {r_1}}}$. Here, ${y_1}$>0, ${r_1}$<0, so $\Delta {\alpha _1}$<0. The calculation method of the refractive surface ${r_2}$ is the same as that of ${r_1}$, and we have verified that the formula is valid even if the surface is plane. However, even we have Eq. (10), it is still difficult to obtain the relationship between the aperture angle u of incident ray and the aperture angle u’ of refracted ray directly. Without sacrificing the accuracy, we expand the sin(x) function by trigonometric function expansion and the arcsin(x) function by Taylor expansion to minimize the loss of accuracy. Taylor expansion can represent the original function, and it is more convenient for us to get the relationship between u and u’ quickly. In order to obtain more accurate solutions, we try Taylor expansions of different orders to ensure the best calculation accuracy with limited orders. After calculation, we find that the polynomial of the result is regular, which also facilitates our next matrix processing. Omitting the detailed Taylor expansion steps, we have the following expression in Eq. (11). Where y and y’ stand for reference ray height related to chief ray in local coordinate system.
From the above expression, we obtain the ABCD matrix corresponding to the refractive surface as follows. The ABCD matrix is based on U system.
Among them, the expansions corresponding to $\Delta {d_1}$, D and $\Delta {D_2}$ are listed as follows. The expansions are regular as the order increases. According to our experiment, the accuracy of Taylor's expansion can be good enough when it is expanded to 7 orders. However, if the expansion order exceeds 7, the cumulative error of equation simplification will increase slightly, as shown in Fig. 3. Here we expand to order 7 in Eq. (13). In this experiment, the focal power derived based on the mathematical model is represented by C from ABCD matrix [19].
In addition, for the incident angle θ at the intersection of the chief ray and the image plane, if the chief ray of the central field is not perpendicular to the image plane, the calculated focal length value needs to be divided by cos(θ) [20].
3. Construction of freeform prism and freeform lens
3.1 TIR Freeform surface prism design
The design inspiration of total reflection freeform prism comes from head-up display (HUD) [21,22]. In the paper, the prism images a finite object at infinity through a two-time reflection for human eyes to observe. The prism in our work is to project the object at infinity through a two-time reflection and refraction at the finite image plane. We also compared the single reflection and three-time reflection, and found that the structure with only secondary reflection is simple and compact. The imaging is only achieved by one prism, and the total reflection and refraction share one surface, which helps to reduce the number of surfaces in the catadioptric prism.
The conventional plastic material “OKP4-HT” with high refractive index (${n_d}$=1.633, and the corresponding total reflection angle ${\theta _{TIR}}$ of OKP4-HT is 37.31°) is chosen for the prism for plastic is more suitable for precise injection molding, as well as mass production, comparing to traditional glass. And it is found that the high refractive index of the material leads to a larger FOV for the prism based on the law of refraction.
In order to realize the total reflection and imaging of the prism, we take three planes at a certain angle to each other as the starting point, and then we optimize the prism. The surface S1 is a plane and also serves as the entrance pupil of the prism. Surface S2 and S4 are the same surface. During optimization, we control the angle of incident ray (AOI) on S2 to be greater than the total reflection angle which is 37.31°, and the incident angle on S4 surface to be less than the total reflection angle, so as to build the structure in which the ray is fully reflected on surface S2 and refracted at surface S4. In addition, the optimized boundary conditions are given to ensure that the ray is always transmitted within the boundaries of surfaces, so as to prevent the ray from leaking out. In addition, the telecentricity of refracted ray should be as small as possible to reduce the difficulty of subsequent work. For all fields in three-dimensional space, the optimized boundary conditions are shown in Eq. (14).
3.2 3D Freeform surface ray-tracing and fitting
After optimizing the freeform prism, we obtained the middle image of the object on the plane, as shown in Fig. 4. If we use the current middle image to design the rear coaxial zoom system, the zoom system will adopt the object telecentric structure, which will lead to large lens aperture and is not conducive to the miniaturization of the whole system. Therefore, in order connect the prism and the zoom system preferably, we consider using freeform lens to convergent chief rays from all fields. One of the lens surface is plane, setting close to S4 of the catadioptric prism. And the other side is a freeform surface. Considering the limitation of space, the freeform surface lens is placed obliquely.
There are three traditional methods for designing freeform surfaces. Simultaneous Multiple Surface (SMS) method originates from Illumination design. This method needs to consider that the number of surfaces should match the number of field points. In the design process, there are still restrictions on the number of field points [23–25]. Another important method is to establish partial differential equations (PDE), which gives numerical solutions by limiting boundary conditions. This method is often used in systems with one single field, or with large aperture and single ray, and is relatively complex [26,27]. In contrast, the construction-iteration (CI) method of point-by-point construction-iteration and fitting in three-dimensional space has relatively high design freedom, which can meet the three-dimensional imaging design and has high design efficiency [28]. The CI method we use is divided into two steps: The first step is to solve the three-dimensional discrete points, and the second step is to fit the discrete points into the freeform surface equation.
Next, we introduce the calculation process of three-dimensional discrete points. First, we insert a flat lens as the initial structure of the freeform surface lens into the CODE V file of the catadioptric prism, which tilts along the direction of the surface S4. In order to simplify the design, the flat lens uses the same material “OKP4-HT” as the catadioptric prism. One side of the flat lens close to the surface S4 is the surface S5, and the other side is S6. We use the macro command to read the coordinates and direction vectors of the ray on surface S5 for all fields. Since the optical system is symmetrical about YOZ plane, we select 11 points in the X direction and 21 points in the Y direction of the local coordinate system. We require all rays to converge at the stop center Q after passing through the surface S6.
Then, the surface S6 is divided into 231 discrete micro surfaces. Since the location and direction vector of the incident point are known, we can obtain the coordinates of the exit point of the surface S6. In this way, the central coordinates of all micro surface of S6 can be calculated in an iterative way. The specific steps are as follows:
- (1) Strat from the central field, we have z-coordinate of the point Q, global coordinates and the direction cosines of the incident chief ray, then the coordinate of refractive ray on S6 of the central field P00, incident ray vector ${{\boldsymbol r}_{\boldsymbol i}}$ and refracted ray vector ${{\boldsymbol r}_{\boldsymbol o}}$ can be obtained according to Eq. (15).
- (2) For the field points in YOZ plane, take the central field coordinate P00 as the base point, we build a micro surface $d{S_{00}}$ with the normal vector N00. Then we calculate the intersection P10 between the incident chief ray of the next field from surface S5 and micro surface $d{S_{00}}$, as shown in Fig. 5(a).
where, l, m and n are the directional cosine of the incident ray of the next field, t is the parameter of the incident ray equation of the next field in S6, and the intersection coordinates on the surface S6 can be updated according to Eq. (17). ${N_{0x}}$, ${N_{0y}}$ and ${N_{0z}}$ are the three directional cosines of the normal vector ${{\boldsymbol N}_{\boldsymbol 0}}$ of $d{S_{00}}$. According to Eq. (18), the coordinates Pi0 and normal vector Ni0 (- 10 ≤i≤ 10, i∈Z) of 21 fields in YOZ plane can be calculated, as shown in Fig. 6(a).
- (3) In the + X direction of surface S6, we take the central field as the starting point, using Eq. (17) in step (2) to calculate the coordinate P01 and corresponding normal vector N01 of the next field point in + X direction in S6, as shown in Fig. 5(b). and Fig. 6(b). And then we calculate the P11 point which has the same x coordinate as P01. In order to reduce the effect of error propagation, we adopt the “Nearest-Ray” Algorithm [28]. Our improvement is that the points array in our work is regular, so we only need to compare the distance in the X direction and Y direction for judgment, which is more efficient. We calculate the distance Δy of point P11 and P01 in micro surface $d{S_{01}}$ and the distance Δx between P11 and P10 in micro surface $d{S_{10}}$. After comparing Δx and Δy, we select the point corresponding to the smaller distance as P11, as indicated in Fig. 5(c). This cycle is calculated until the parameters of all points in + Y direction of S6 are obtained, as shown in Fig. 6(c). And then the points array in -Y direction of S6 is calculated in the same way. Finally, we have the coordinates and normal vectors of 231 discrete points corresponding to the surface S6, as shown in Fig. 6(d).
Fig. 5. Freeform surface ray-tracing from S5 to stop by CI method. (a) Calculations in YOZ plane. (b) Calculations in + X direction. (c) Calculations in first and fourth quadrants according to the “Nearest-ray” method.
Fig. 6. Calculation of points on surface S6 step by step. (a) Calculations of 21 fields in YOZ plane. (b) Calculations of 11 fields in + X direction. (c) Calculations in first quadrant according to the “Nearest-ray” method. (d) Coordinates and normal vectors of 231 discrete points on S6.
After obtaining the discrete points of freeform surface S6, we need to synthesize the points array into freeform surface equation for further lens design. However, because the freeform surface lens we designed is tilted and its discrete coordinate points are global coordinates, the freeform surface fitting cannot be proceeded correctly. So we need to convert the global coordinates into local coordinates to complete the fitting.
Firstly, after obtaining the coordinates of the points on S6, we use the nonlinear function optimization tool function in MATLAB to fit the discrete data points into a sphere and obtain the basic curvature and center coordinates of the sphere [29]. We also know that the coordinate corresponding to the central field is P00 (0, 0.7, 4.1). So the off-axis points array is rotated into on-axis points array along the Y-axis. The rotation angle θ can be calculated by the following formula, as shown in Eq. (19). Where, ${y_0}$ and ${z_0}$ are the central coordinates of S6, ${y_c}$ and ${z_c}$ are the coordinates of fitted sphere center.
Then, the global coordinate points shall take point P00 as the original point and translate according to Eq. (20).
Further, the shifted points array on S6 needs to rotate around the X-axis, as shown in Fig. 7. And the tangent plane of the rotated surface at point P00’ should be perpendicular to the Y axis. The matrix including shifting and rotating is as follows in Eq. (21).
In the second step, the freeform surface fitting based on the least square method is carried out by using MATLAB. We also use the commercial software 1STOPT to make a fitting comparison to verify the accuracy of the MATLAB program results. 1STOPT fitting adopts the global algorithm, while MATLAB fitting is fast. The XY polynomial coefficients of the two software are very close, as shown in Fig. 8. Root mean square error (RMSE) shows that the results of 1STOPT fitting has slightly higher accuracy, though calculation results are close to 0. Thus, we choose the fitting results from 1STOPT.
After inputting the fitted conic and XY polynomial coefficients into CODE V, we obtain the combination of catadioptric prism and freeform lens. By measuring each distance in which the central chief ray is transmitted as well as the refraction angle and reflection angle on surfaces, we establish the matrix of the combination in Eq. (22) and Eq. (23) according to Eq. (8), Eq. (12) and Eq. (13).
Besides, ${r_4}$ is −232.47mm, S5 is flat, so the radius of curvature is infinite, and ${r_6}$ is −12.18mm. It is worth noting that the curvature is not the curvature of the origin of the freeform surface, nor the curvature of the base sphere, but the curvature of the ray refraction point and reflection point, and this value must be accurately obtained. All freeform surfaces of this combination are expressed by XY polynomials. Due to the symmetry of YOZ plane, the coefficient of XY polynomials does not contain the odd term of X. The radius of curvature of a point on the freeform surface of XY polynomial is calculated according Eq. (24). Where y’ and y” are the first derivative and the second derivative evolved from XY polynomials, respectively.
4. Finite conjugate zoom lens design with user-defined zoom ratio
4.1 Establish of equations
After obtaining the catadioptric prism and freeform surface lens, the initial parameters of the coaxial zoom system are calculated. According to section 2, we assume that the finite conjugate zoom lens is composed of three moving groups, which contains three powers and four air spaces as variables. In the past, these values were obtained based on experience and repeated optimization, which was time-consuming. Here, for faster and more accurate calculation, we establish the ABCD matrix mathematical model of the zoom system in Eq. (25) derived from Eq. (1).
Further, the ABCD matrix of the coaxial zoom system is derived in Eq. (27).
In addition, for the optical system with finite conjugation, the element B of ABCD matrix is 0. Another equation for finite conjugate imaging can be obtained in Eq. (28) by simplifying Eq. (27).
Next, we get the equation for the total length in Eq. (29).
According to the finite conjugate zoom lens in Fig. 9 above, the paraxial object-image relation can be expressed by Eq. (30).
where, C is the focal power of the zoom lens. $(1 - D)/C$ represents the distance from the first group to the principal plane in object space, and $(A - 1)/C$ is the distance from the third group to the principal plane of the image. In order to reduce the variables in the equation, the image distance ${e_3}$ is replaced by ${e_0}$ and elements in Gaussian brackets, as shown in Eq. (31).
Now we have four equations, but the number of variables is still more than that of equations. We expand the number of equations by increasing the zoom positions to match the number of variables, which are wide, mid and tele. From the Eq. (26) to Eq. (30) in three different zoom positions, we can further obtain 12 equations with 12 variables, including ${\varphi _1},{\varphi _2},{\varphi _3}, {e_{01}},{e_{02}},{e_{03}},{e_{11}},{e_{12}},{e_{13}},{e_{21}},{e_{22}},{e_{23}}$.
4.2 Application of particle swarm optimization algorithm
In order to solve 12 equations, particle swarm optimization (PSO) algorithm is adopted. PSO is an intelligent algorithm that can quickly carry out global optimization, which has been well applied in solving the initial structure of coaxial zoom system [30,31]. Compared with the least square method, it can accommodate more equations and variables, and can avoid falling into local optimal solution. We use discrete particles to represent 12 unknown variables of the lens and initialize them with random number generation function. The velocity update and position update of each particle are determined according to the individual best pbest and global best gbest of the evaluation function in each iteration, and the best global solution is obtained after several cycles. The equations representing speed update and position update after ith iteration for particle j are listed in Eq. (32) and Eq. (33).
After identity transformation and simplification, the 12 equations form the final evaluation function in Eq. (35). Where, $valu{e_i}$ represents the predefined value of the system. It is not difficult to find out that the closer the merit function (MF) is to zero, the higher the accuracy of the solution.
We evaluate the output result according to the size of the final merit function. If it is less than 0.01, the calculation result is considered to be valid. In addition, we use CODE V to verify the accuracy of the MATLAB results. If the difference between results is slightly large, it is necessary to modify the lens parameters or increase the number of iterations.
5. Examples and results
After the combination of the prism, freeform lens and finite conjugate zoom lens, the initial structure of catadioptric zoom system is obtained, which can be further optimized. The following table shows the first-order design results of different zoom ratio. The search space for the power is [- 0.5,0.5] to ensure that the lens group has sufficient aperture. And the interval of the air space is between 0.5mm and TTL minus the total length of other minimum air spaces.
The calculation results of the initial structure of 3X zoom lens obtained by PSO algorithm are shown in Table 1. After 143.6 seconds, MF of the program decreases to 0.0049, which basically meets our requirements, as shown in Fig. 10(a). We input the parameters into CODE V and analyze the focal length by using lens model. Compared with the matrix calculation results in MATLAB, the average accuracy for all zoom positions of focal length is 99.76% under three zoom position, as indicated in Fig. 10(b).
Fig. 10. 3X catadioptric zoom system. (a) MF diagram of PSO calculation and initial structure. (b) Comparison of focal lengths between MATLAB program and CODE V results.
Table 1. Initial parameters of 3X catadioptric zoom system (Unit: mm)
After the same catadioptric prism and freeform surface lens are replaced with different finite conjugate zoom systems, the new initial structure can be obtained immediately. When the focal length range is set from −4 mm to −20 mm with the total length of 29.4 mm, we get an initial structure of 5X catadioptric zoom lens after the 429.3 second calculation in MATLAB, as shown in Table 2 and Fig. 11(a). The only change is to input four new parameters: three focal lengths for wide, mid, tele and TTL of finite conjugate zoom lens. It is worth noting that the convergent solution is easier to obtain when the total length of the zoom lens is greater than the longest focal length. Other relations of the four parameters will be discussed in our later work.
Fig. 11. 5X catadioptric zoom system. (a) MF diagram of PSO calculation and initial structure. (b) Comparison of focal lengths between MATLAB program and CODE V results.
Table 2. Initial parameters of 5X catadioptric zoom system (Unit: mm)
Similarly, the focal lengths of 5X catadioptric zoom lens are and compared in CODE V and MATLAB as follows in Fig. 11(b). The trajectory of the measured curve is accord with the MATLAB result perfectly, with the average approximation of 99.79%, which verifies the accuracy of the MATLAB program based on ABCD matrix.
This design not only verifies the correctness of the theoretical derivation, but also can be used as the initial structure of the actual zoom system. The initial structure discussed in this paper is a combination of three parts, but the design method of freeform catadioptric lens provided in this paper is not limited to this structure. We also apply the mathematical model and method to analyze the off-axis three mirror lens successfully. Thus, the validity and universality of the mathematical model are basically verified.
In this structure, not only can the finite conjugate zoom lens be designed automatically, but also the freeform prism is customizable. In 3X and 5X examples above, the distance d’ from the center of S6 to the stop center Q of the catadioptric prism system is 7.5 mm, which is slightly larger than its depth (6 mm). Because it is necessary to ensure that the freeform lens edge thickness is greater than 0.2 mm. Thus, the distance d’ cannot be too short. In addition, we found that a longer focal length requires a larger lens volume. The construction of mathematical model in X direction and the automatic optimization of high-order aberration will be the focus of our future work.
6. Conclusion
This paper presents an automatic design method to obtain the initial structure of a slim catadioptric freeform zoom system. With the aid of ABCD matrix and Gaussian brackets, this method combines the ABCD matrix of NU system and U system to construct the mathematical model of catadioptric system. At the same time, we propose a freeform catadioptric prism which is used to fold the ray path by 90°, reduce the depth of the zoom lens to 6mm, and realize the miniaturization of the zoom lens. CI construction method and least square fitting algorithm are adopted to generate high-precision freeform surface, and the calculation takes less than 20 seconds. The fitting accuracy of MATLAB program is verified by commercial software 1STOPT. Furthermore, the PSO algorithm is used to automatically solve the initial structural parameters of the finite conjugate zoom lens, which only needs the focal length range and total length. The 3X and 5X zoom lens are presented to demonstrate the feasibility and robustness of our method. The calculation of each lens by using MATLAB programming costs less than 8 minutes and the calculation results of the initial structure are also verified by CODE V software. The whole process is basically automatic, instead of providing parameters in the optical system in advance by guessing. It reduces the dependence on experience and does not need repeated time-consuming optimization with optical design software. We also provide the pseudo code of the core algorithm as a reference for readers.
Our work focuses on the first-order zoom automation design with customizable zoom ratio, which saves time for researchers in finding the initial structure. Compared with the previous work, which only presents the system with fixed zoom ratio [2,3], our method can produce a zoom lens with arbitrary zoom ratio. Also, the volume is smaller than the traditional design using right angle prism [1] or other coaxial zoom systems, which is more suitable for micro devices. Moreover, the catadioptric mathematical analysis framework we proposed is not limited to specific structures. In short, this method of quickly solving the initial structure of compact zoom lens is conducive to breaking the barrier of the initial structure design of slim catadioptric zoom lens. This method can provide some reference for the miniaturization research of zoom lens and the development of industry.
Funding
National Key Research and Development Program of China (2016YFB0402004).
Acknowledgments
The authors thanks Professor Yang and all the colleagues in Shanghai Key Laboratory of Modern Optical Systems of University of Shanghai for Science and Technology for their help.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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