## Abstract

In this paper, we propose a systematic approach to automatically retrieve the first-order designs of three-component zoom systems with fixed spacing between focal points based on Particle Swarm Optimization (PSO) algorithm. In this method, equations are derived for the first-order design of a three-component zoom lens system in the framework of geometrical optics to decide its basic optical parameters. To realize the design, we construct the mathematical model of the special zoom system with two fixed foci based on Gaussian reduction. In the optimization phase, we introduce a new merit function as a performance metric to optimize the first-order design, considering maximum zoom ratio, total optical length and aberration term. The optimization is performed by iteratively improving a candidate solution under the specific merit function in the multi-dimensional parametric space. The proposed method is demonstrated through several examples, which cover almost all the common application scenarios. The results show that this method is a practical and powerful tool for automatically retrieving the optimal first-order design for complex optical systems.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Zoom systems find applications in many areas, as the focal length or the transverse magnification for these systems can be adjustable by shifting some individual elements of the optical system. In many appliations, constant distance from the image plane to the object plane is a preferred requirement during the zooming process [1–11]. However, we no longer require stationary locations for the two optically conjugate planes in many other cases, where we are much more concerned to keep the two foci fixed when zooming. Therefore, the well-developed theory for designing common zoom systems cannot be applied to these occasions, and thus a new analytical method deserves to be explored to design these special zoom systems.

A zoom system with fixed spacing between focal points has much potential applications in the field of optical information processing [12–15]. In general, a spatial filtering has to be located in the Fourier spectrum plane of the lens system to work properly. Light beams with different spatial frequencies carry different spectrum information of targets, which means that the proposed special zoom system with fixed foci can help us obtain the information of different frequency range with the same spatial filter. The studied object is located at the front focal plane in the object space and the spatial filter is located at the rear focal plane in the image space. Another possible application is in the field of metrology and machine vision [15–20]. The proposed zoom system with two fixed foci is a crucial component for double-sided telecentric zoom systems, which can lead to better magnification constancy, better illumination uniformity at the detector and larger field depth. Compared with traditional telecentric systems with no zooming mechanism, zoom telecentric systems have unique superiority of the ability to provide a continuously varying focal length inside the whole specified range with superior imaging quality.

The designs of a special zoom lens with fixed spacing between focal points have been discussed in some papers [15,20–22]. The classical structure of a three-component zoom lens with fixed spacing between focal points is shown in Fig. 1(a). The front focal point $\textrm{F}$ and the rear focal point $\textrm{F}^{\prime}$ are located outside the optical system (${S_\textrm{F}} < 0$, $S_\textrm{F}^{\prime} > 0$). Here, ${S_\textrm{F}}$ is the front focal distance (FFD) that defines the distance of the front focal point from the first component of the system, and $S_\textrm{F}^{\prime}$ is the back focal distance (BFD) that defines the distance of the rear focal point from the last component of the system. Miks derives the equations for the paraxial design of first-order parameters based on Gaussian brackets and provides a 2X paraxial design. However, some parameters in the equations needs to be pre-determined and these parameters determine the zoom track of the zoom system as well as the maximum zoom ratio. We discussed the problem in our previous paper and presented a design method based on particle swarm optimization to find an optimal starting point for the design, which is especially important for the final design of zoom systems [23]. However, we find that the maximum zoom ratio of the system is still too small to meet some demanding requirements. We believe that the classical structure is not conducive for getting access to zoom configurations with relatively large zoom ratios. In addition, we have assumed that the Patzval sum has to be 0 for the correction of field curvature. However, the restriction seems too restrictive to acquire a first-order design with larger zoom ratio and probably is not a preferred choice.

In this paper, we propose an efficient and robust method to retrieve an optimal first-order design automatically based on a new optical structure. As shown in Fig. 1(b), object-space focal point $\textrm{F}$ locates inside the optical system (${S_\textrm{F}} > 0$, $S_\textrm{F}^{\prime} > 0$). We perform a straightforward derivation for the paraxial analysis of the described special zoom system without unnecessary restriction. Then the optical design work is transferred to retrieving an optimal solution in a specific search space by PSO algorithm. The maximum zoom ratio, compactness, and aberrations of the system are regarded as the performance metric for evaluation. A series of proper solutions can be automatically obtained as the candidate starting points for further optimization according to actual needs. We demonstrate that the proposed algorithm is efficient in designing zoom systems with two fixed foci as well as a given zoom ratio. Furthermore, we implement our method in the global optimum design for a specific evaluation performance metric. Combined with a fixed group, we make a minor change on the model, and succeed in the first-order design of a three-component 7X zoom system with two fixed foci. It is noted that our proposed method can be applied to finding an initial configuration for any other complex optical system. It indicates that PSO algorithm has great potential in searching an initial configuration for an optical system design, which is a significant step towards fully automated design with artificial intelligence.

## 2. Theory and design algorithm

#### 2.1 Paraxial analysis of a three-component zoom lens with two fixed foci

The constraints for our proposed zoom lens with two fixed foci are quite different from the traditional zoom systems. Theoretically, at least three moving components are necessary to construct a mechanically compensated zoom lens with two fixed foci [15]. In the case of a three-element optical system, all the locations of the three components have to be varied to satisfy all the constraints simultaneously. The proposed structure of a three-component zoom system with two fixed foci is schematically shown in Fig. 2. The optical power of the component *i*, which ranges from 1 to 3, is denoted as ${\phi _i}$. The relative locations of the optical components, the principal planes, and the two foci are given in Fig. 2. In order to derive the first-order properties of the zoom system, we apply the Gaussian reduction method to reduce the three-component system into a single Gaussian system represented by a group of principal points and a group of focal points.

Firstly, we combine the first two components into one single Gaussian optical system, whose principal planes of ${\textrm{P}}_{12}$ and $\textrm{P}_{12}^{\prime}$ are located at the positions illustrated in Fig. 2. Then the total power ${\phi _{12}}$ of the combination of the first two components, as well as its locations of the principal planes can be derived as follows:

^{st}component to the object-side principle plane of ${\phi _{12}}$, and $d_{12}^{\prime}$ defines the distance from the 2

^{nd}component to the image-side principle plane of ${\phi _{12}}$.

Now we combine the derived lens combination ${\phi _{12}}$ of the first two components with the 3^{rd} component into another single Gaussian optical system, whose principal planes of ${\textrm{P}_{123}}$ and $P{^{\prime}_{123}}$ are located at the positions illustrated in Fig. 2. Then the total power ${\phi _{123}}$ (also labelled as $\phi $) of the combination of the three components, as well as its locations of the principal planes can be derived as follows:

^{rd}component to the image-side principal plane of the combination ${\phi _{123}}$ of the three elements.

As one can see from Fig. 2, the distance *D* between the focal points and the focal length *f* ($f = 1/{\phi _{123}}$) of such a three-element optical system can be expressed as

^{st}element of the system (front focal length), $S_\textrm{F}^{\prime}$ defines the distance from the image-side focal point to the 3

^{rd}element of the system (back focal length). Combining these geometrical relationships in Eq. (3), we can obtain the equation as follows: Equations (1), (2) and (4) enable us to calculate the distances ${d_1}$ and ${d_2}$ for the given powers (${\phi _1}$, ${\phi _2}$, ${\phi _3}$), the distance

*D*and any focal length

*f*. As a result, we can figure out the axial displacement trajectories of all the three components during the zooming process by continuously varying the focal length

*f*from ${f_0}$ to $M{f_0}$. Here, ${f_0}$ is the minimum focal length of the zoom system during zooming and

*M*is the maximum zoom ratio of the zoom system. To guarantee the physical feasibility of the zoom system, ${d_1}$ and ${d_2}$ must satisfy the following conditions: To escape collisions between either of the focal planes and any one of the three components, we have to restrict the locations of the focal planes as follows: Equations (5) and (6) provide specific restrictions on paraxial parameters of the optical system, which results from the requirements of a physically feasible solution. While the paraxial analysis of the three-element zoom lens is completed, it is difficult to search for the optimal candidate solution in a multivariable space. In other words, the values of parameters ${\phi _1},\; {\phi _2},\; {\phi _3},\; D,\; f$ in Eqs. (1), (2) and (4), which determine the solvable interval of axial displacement equations as well as the maximum zoom ratio, need to be predetermined. There exists little information in published papers about the discussion of the possible optimal solutions for the first-order design of complex optical systems. In this paper, we employ the PSO algorithm to automatically search the globally optimal starting point design for the special zoom systems with fixed foci [24,25].

#### 2.2 PSO algorithm

PSO is a stochastic search method originally proposed by Dr. Eberhart and Dr. Kennedy, which utilizes a set of candidate solutions to find the global minimum of an objective function *T* (or merit function in optical design) [26,27]. In PSO, the position of a particle in the search space ${R^\textrm{S}}$ represents a candidate solution to the optimization problem at hand. Each particle determines its movement through the search space ${R^S}$ by combining aspects of its own history of current and best positions with those of other swarm members, with some random perturbations. Here ${R^S}$ is the domain of variables to be optimized and its dimension depends on the number of variables. As a result, PSO algorithm has been shown to be a robust method for global optimization in a high dimensional search space with multiple local optima [28–30].

In PSO, the position of the *i*^{th} particle of the swarm in the ${R^\textrm{S}}$ can be denoted as ${{\textbf X}_i} = ({{x_{i1}},{x_{i2}}, \ldots {x_{in}}} )$. The velocity of the particle is denoted as ${{\textbf V}_i} = ({{v_{i1}},{v_{i2}}, \ldots {v_{in}}} )$ and the best position of each particle in exploration histories is represented by the vector ${{\textbf P}_{\textrm{best}}} = ({{p_1},{p_2}, \ldots ,{p_i}, \ldots {p_N}} )$, where *N* is the population size. At each step, the merit function of the current position is evaluated and stored. If the position of *i*^{th} particle is better than any of its previous positions, it becomes the new best position ${p_i}$. The position ${p_i}$ with minimum merit function value in ${{\textbf P}_{\textrm{best}}}$ is set as ${{\textbf g}_{\textrm{best}}}$ and the best swarm's merit function value is reset to the current merit function value.

At first, each particle in the swarm is endowed with a random position and a random velocity inside the search space ${R^\textrm{S}}$. Then the ${{\textbf P}_{\textrm{best}}}$ and ${{\textbf g}_{\textrm{best}}}$ are identified through evaluation of merit function for all particles. The *i*^{th} particle will update its state in *k*-th iteration based on the following rules:

*w*is the inertia coefficient, ${w_1}$ and ${w_2}$ are acceleration coefficients, and

*ran*generates uniform random factors in the [1] interval.

*ran*is introduced for the purpose of maintaining the diversity of population and effectually preventing the optimization from prematurity. The term with ${w_1}$ represents the cognitive component which drives particles toward their own best position so far. The term with ${w_2}$ represents the social part which pulls the particles toward global best position that the swarm has been found. The inertia term

*w*is introduced into the velocity’s updating rule to keep the particle moving under the effect of those previous velocities, which essentially plays a trade-off role between the global search ability and local search ability. In general, ${w_1}$ and ${w_2}$ are respectively set a constant value in the range of [0.5, 2].

*w*is initialized in the range of [0.9, 1.2] and its value decreases linearly during the iterations [27,31–33].

#### 2.3 Design process by PSO algorithm

Based on the paraxial analysis in Section 2.1 and essentials of PSO algorithm in Section 2.2, the design problem can be transformed to searching the optimal solution with a nonlinear merit function *T* [31–33]. The merit function comprises basic first-order properties (e.g. total optical length), metric for zoom system (e.g. maximum zoom ratio), and aberrations to characterize the system. The total optical length of the zoom system can be defined as:

^{rd}component during the whole zooming process, and $\textrm{min}\{{{x_1}} \}$ denotes the minimum on-axis coordinate of the 1

^{st}component during the whole zooming process. The merit function can be defined as:

With the merit function, each candidate zoom system for optimization is regarded as a particle in the search space ${R^S}$ (*S *= 5). Unlike the fixed focal system, the initial state of the proposed zoom system consists of a group of multiple configurations with the focal length varying from ${f_0}$ to $M{f_0}$ (M is the maximum zoom ratio of the system). Therefore, the position of the *i*^{th} particle of the swarm is defined as a matrix ${{\textbf X}_i} = ({{{\boldsymbol x}_{i1}},{{\boldsymbol x}_{i2}}, \ldots {{\boldsymbol x}_{in}}} )$, which contains the information about the entire zoom progress. The vector ${{\textbf x}_{ij}}$ ($j = 1,2, \ldots n)$ represents the *j*-th configuration, and can be defined as ${{\textbf x}_{ij}} = [{{\phi_1},\; {\phi_2},\; {\phi_3},\; D,\; f} ]$. Similarly, the matrix of velocity ${{\textbf V}_i}$ have the same dimension. The initial ${{\textbf X}_i}$ and ${{\textbf V}_i}$ are generated by adding random values of the variable parameters relative to a starting point design that has been provided by the optical designer according to the actual situation. The boundary values for the velocity have to be selected carefully. If the velocities are too small, the particles will wander around the initial locations and fall into a locally optimal solution. On the other hand, excessive velocities may miss significant details and even fall outside of the search space.

In the next step, a new iteration starts. In our design of the proposed zoom system, these variable parameters should be constrained to conform to the physically realizable conditions. In other words, we have to guarantee that the neighboring components do not collide with each other during the motion process when zooming the system. As a result, we have to apply the conditions of physical feasibility expressed in Eqs. (5) and (6) to evaluate the current state during the update phase. Based on the simple derivation of formulas, ${d_1}$ and ${d_2}$ can be calculated for a given ${{\textbf x}_{ij}}$, as well as ${S_\textrm{F}}$ and $S_\textrm{F}^{\prime}$. If all the ${{\textbf x}_{ij}}$ in matrix ${{\textbf X}_i}$ satisfy the conditions, the configuration that *i*^{th} particle represents is evaluated as a reasonable solution and its merit function value will be computed. If current value is better than that of ${{\textbf P}_{\textrm{best}}}$, then ${{\textbf P}_{\textrm{best}}}$ is replaced by the particle’s current position. After all particles have been the evaluated, ${{\textbf g}_{\textrm{best}}}$ can be determined by the criterion. Of course, the solvable particles in the initial swarm should be enough in numbers. To guarantee that enough number of particles can start in a feasible region of the search space, the unrealizable particles will reinitialize until they can pass the feasibility check process. Then the velocities and positions of all checked particles are updated and optimized according to the rules expressed in Eq. (7).

After a maximum number of iterations, the optimization is terminated. Regardless of the optimal solution, we can also acquire a series of suboptimal solutions, which can as well be regarded as a starting point for the design work.

## 3. Design results and discussion.

#### 3.1 Example of zoom system with specified zoom ratio

Based on the mathematical models established in previous sections, we implement our method in Matlab to design a 4X zoom system with fixed spacing between focal points as a practical example. Getting back to Eq. (9), we can set ${w_M} = 0$ for the merit function because the zoom ratio is specified. The focal length of the special zoom system is constrained inside the range from 25 mm to 140 mm, which determines the probable search space of *D* and ${f_0}$. The type of the zoom system is classified according to the signs of optical power of each component. We choose the P-N-P (P represents positive optical power of the component, and N represents negative optical power of the component) type as a preferred design configuration, due to its symmetry and ability to compensate field curvature [34]. Considering the manufacturability and contribution of optical power to the whole system, the optical power of each component ${\phi _i}$ should be also constrained within a reasonable range. The search space can be adjusted according to the solution retrieval experience, to make the particles converge to the optimal position more quickly. The rough search space can be ascertained as what listed in Table 1. As the result of global optimization strongly depends on the parameters of proposed PSO algorithm, the magnitude of these parameters should be selected carefully. The detailed parameters for the PSO algorithm are also listed in Table 1, where $ite{r_{\textrm{max}}}$ defines the maximum number of iterations. The Petzval sum of the initial design is $6.98 \times {10^{ - 3}}$ mm^{−1}. The detailed first-order design result is shown in Table. 2, and the zoom track (loci of the three components) and locations of focal planes are depicted in Fig. 3.

In order to verify the validity of the first-order design result, we perform the optimization in optical design software. The working wavelength of the optical system ranges from 486nm to 656nm and the paraxial image height is set as 4 mm. After optimization, the distance between the two focal planes is 96.229 mm and the maximum optical length *L* is 113.05 mm. The F number of the system varies from F/6∼F/24. Figure 4(a) presents the shaded models of designed zoom system at sampled zoom ratio. The zoom trajectory is basically the same as the first-order design. Figures 4(b)–(d) show the MTF plots of the system at different zoom ratio. It illustrates that our proposed method can provide a satisfying first-order design for complex systems. Compared to our previous design [23], locating the front focal plane inside the optical system can help achieve a relatively higher zoom ratio.

#### 3.2 Example of zoom system with no specified zoom ratio

In this case, the maximum zoom ratio is not specified and is regarded as one of the very important performance metric for our designed zoom systems with fixed spacing between the two foci. Similarly, we can still apply the proposed PSO algorithm to resolve the complex design problem of retrieving global optimum based on a specific merit function. The coefficients in the merit function is decided to guarantee that all terms have comparable contribution to the final merit function value and as well satisfy actual requirements. In fact, the ratios ${w_M}/{w_L}$ and ${w_M}/{w_A}$ reflect the trade-off between zoom performance and optical performance. By repeating the experiment demonstrated in Section 3.1 under different maximum zoom ratios, we can roughly estimate the influence of the maximum zoom ratio on the optical total length and aberrations, which is essential for us to determine final trade-off between zoom performance and optical performance for our proposed special zoom system.

During the optimization process, we still adopt the P-N-P type and the minimum focal length ${f_0}$ is in the range of [25, 35]. The total optical length is restricted to 200 mm, where the coefficient ${w_L}$ will zoom to a much higher value once the *L* exceeds the specified boundary. The detailed parameters for the PSO algorithm and the coefficients in the merit function are listed in Table. 3. After 50 iterations of optimization by our proposed PSO algorithm, we can finally acquire a globally optimum initial design prescriptions for the special 5X zoom system. The final detailed first-order design result is shown in Table. 4, and the zoom track (loci of the three components) and locations of focal planes are illustrated in Fig. 5. The Petzval sum of the initial design is $1.76 \times {10^{ - 2}}$ mm^{−1}.

#### 3.3 Example of zoom system with an additional fixed group

The front focus of the examples above is located inside the optical system. The configuration with a virtual front focus is adopted because it can achieve a higher zoom ratio compared to the normal configuration with real front focus. However, it cannot be applied directly to design a double-sided zoom system because we cannot place a stop at the virtual focus. Otherwise, the vignetting effect due to virtual front focus will affect the telecentricity in image-side space. To overcome this issue, we locate another optical system with a fixed focal length in front of the zoom system to provide a conjugate real image point ${\textrm{F}_0}$ for the front focus $\textrm{F}$. As shown in Fig. 6, the real focus ${\textrm{F}_0}$ of the system is located outside the whole optical system. The locations of points $\textrm{F}$, $\textrm{F}^{\prime}$ and ${\textrm{F}_0}$ are stationary during the whole zooming process. Although the total length of the whole system becomes longer, the location of the front focus F for the zooming part becomes much more flexible. In fact, the front focal plane can be located at either in front of the 1^{st} component (i.e. ${S_{\textrm{F}}} < 0)$ or between the 1^{st} component and 2^{nd} component (i.e. ${S_{\textbf{F}}} > 0$ and ${S_{\textrm{F}}} < {d_1}$). Under such a condition, constraints described in Eqs. (5) and (6) can be simplified as

*w*between 0.2 and 1, and acceleration coefficients ${w_1} = {w_2}$ between 0.6 and 1.5. Actually the solvable region will become very narrow at high zoom ratio, and thus the proposed optimization algorithm probably cannot find the solutions at expected high zoom ratio. After several trial and error approaches, we finally acquire a 7X first-order design of the special zoom system with two fixed foci and the detailed design results for several sampled zoom ratios are presented in Table 5. The Petzval sum of the initial design is $2.71 \times {10^{ - 2}}$ mm

^{−1}.

## Conclusion

In this paper, we present an automated and globally optimized design method for a special type of zoom system with a constant distance between its front and rear focal points. Instead of traditional trial and error approach to find a proper starting point design, we apply the proposed PSO algorithm to automatically retrieve the globally optimal starting point design based on given constraints. The superior design results have shown that our proposed method is efficient and low-threshold. We have also demonstrated that the proposed zoom system design method can be able to deal with many different occasions by several examples. Besides, we have successfully found a 7X zoom system with two fixed foci with the help of our proposed PSO algorithm based on a new configuration. In addition, a large number of proper solutions for the specific zoom system can be automatically retrieved with the proposed method, which can provide more design freedoms for researchers to satisfy actual design requirements. Despite its simplicity in concept, our proposed PSO algorithm can be a very useful and promising tool during the design process to automatically retrieve a proper starting point for any other complex optical systems.

## Funding

National Natural Science Foundation of China (61805088, 61805089); Fundamental Research Funds for the Central Universities (2019kfyRCPY083, 2019kfyXKJC040).

## Disclosures

The authors declare no conflicts of interest.

## References

**1. **M. S. Yeh, S. G. Shiue, and M. H. Lu, “First-order analysis of a two-conjugate zoom system,” Opt. Eng. **35**(11), 3348–3360 (1996). [CrossRef]

**2. **M. S. Yeh, S. G. Shiue, and M. H. Lu, “Solution for first-order design of a two-conjugate zoom system,” Opt. Eng. **36**(8), 2261–2267 (1997). [CrossRef]

**3. **C. Chen, “Complete solutions of zoom curves of three-component zoom lenses with the second component fixed,” Appl. Opt. **53**(29), H58–66 (2014). [CrossRef]

**4. **Y. C. Fang, C. M. Tsai, and C. L. Chung, “A study of optical design and optimization of zoom optics with liquid lenses through modified genetic algorithm,” Opt. Express **19**(17), 16291–16302 (2011). [CrossRef]

**5. **J. Zhang, X. Chen, J. Xi, and Z. Wu, “Aberration design of zoom lens systems using thick lens modules,” Appl. Opt. **53**(36), 8424–8435 (2014). [CrossRef]

**6. **M. J. Kidger, “The importance of aberration theory in understanding lens design,” Proc. SPIE **3190**, 26–33 (1997). [CrossRef]

**7. **A. Miks, J. Novak, and P. Novak, “Method of zoom lens design,” Appl. Opt. **47**(32), 6088–6098 (2008). [CrossRef]

**8. **S. H. Jo and S. C. Park, “Design and analysis of an 8x four-group zoom system using focus tunable lenses,” Opt. Express **26**(10), 13370–13382 (2018). [CrossRef]

**9. **A. Miks and J. Novak, “Method of first-order analysis of a three-element two-conjugate zoom lens,” Appl. Opt. **56**(18), 5301–5306 (2017). [CrossRef]

**10. **M. Demenikov, E. Findlay, and A. R. Harvey, “Miniaturization of zoom lenses with a single moving element,” Opt. Express **17**(8), 6118–6127 (2009). [CrossRef]

**11. **K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component type,” Appl. Opt. **21**(12), 2174–2183 (1982). [CrossRef]

**12. **A. Miks and J. Novak, “Paraxial analysis of three-component zoom lens with fixed distance between object and image points and fixed position of image-space focal point,” Opt. Express **22**(13), 15571–15576 (2014). [CrossRef]

**13. **B. Javidi and C. Kuo, “Joint transform image correlation using a binary spatial light modulator at the Fourier plane,” Appl. Opt. **27**(4), 663–665 (1988). [CrossRef]

**14. **J. Choi, T. H. Kim, and H. J. Kong, “Zoom lens design for a novel imaging spectrometer that controls spatial and spectral resolution individually,” Appl. Opt. **45**(15), 3430–3441 (2006). [CrossRef]

**15. **A. Miks, J. Novak, and P. Novak, “Three-element zoom lens with fixed distance between focal points,” Opt. Lett. **37**(12), 2187–2189 (2012). [CrossRef]

**16. **S. Bloch and E. I. Betensky, “Telecentric zoom lens used in metrology applications,” Proc. SPIE **4487**, 42–52 (2001). [CrossRef]

**17. **G. Baldwin-Olguin, “Telecentric lens for precision machine vision,” Proc. SPIE **2730**, 440–443 (1996). [CrossRef]

**18. **M. Watanabe and S. K. Nayar, “Telecentric optics for computational vision,” in * Proceedings of European Conference on Computer Vision*, ed. (Springer, Berlin, Heidelberg, 1996) pp. 439–451.

**19. **M. Watanabe and S. K. Nayar, “Telecentric optics for constant magnification imaging,” IEEE Trans. Pattern Anal. Machine Intell. **19**(12), 1360–1365 (1997). [CrossRef]

**20. **A. Miks and J. Novak, “Design of a double-sided telecentric zoom lens,” Appl. Opt. **51**(24), 5928–5935 (2012). [CrossRef]

**21. **A. Miks and J. Novak, “Paraxial analysis of four-component zoom lens with fixed distance between focal points,” Appl. Opt. **51**(21), 5231–5235 (2012). [CrossRef]

**22. **J. Zhang, X. Chen, J. Xi, and Z. Wu, “Aberration correction of double-sided telecentric zoom lenses using lens modules,” Appl. Opt. **53**(27), 6123–6132 (2014). [CrossRef]

**23. **Z. Fan, S. Wei, Z. Zhu, Y. Mo, Y. Yan, and D. Ma, “Automatically retrieving an initial design of a double-sided telecentric zoom lens based on a particle swarm optimization,” Appl. Opt. **58**(27), 7379–7386 (2019). [CrossRef]

**24. **T. Yang, G. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light: Sci. Appl. **6**(10), e17081 (2017). [CrossRef]

**25. **A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. **9**(1), 1756 (2018). [CrossRef]

**26. **J. Kennedy and R. Eberhart, “Particle swarm optimization,” In Proceedings of IEEE Conference on Neural Networks (IEEE, 1995), pp. 69–73.

**27. **R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in Proceedings on Micro Machine and Human Science (IEEE, 1995), pp. 39–43.

**28. **W. Du, Y. Gao, C. Liu, Z. Zheng, and Z. Wang, “Adequate is better: particle swarm optimization with limited-information,” Appl. Math. Comput. **268**, 832–838 (2015). [CrossRef]

**29. **I. C. Trelea, “The particle swarm optimization algorithm: convergence analysis and parameter selection,” Inform. Process. Lett. **85**(6), 317–325 (2003). [CrossRef]

**30. **M. Clerc and J. Kennedy, “The particle swarm-explosion, stability, and convergence in a multidimensional complex space,” IEEE Trans. Evol. Computat. **6**(1), 58–73 (2002). [CrossRef]

**31. **H. Qin, “Particle swarm optimization applied to automatic lens design,” Opt. Commun. **284**(12), 2763–2766 (2011). [CrossRef]

**32. **H. Qin, “Aberration correction of a single aspheric lens with particle swarm algorithm,” Opt. Commun. **285**(13-14), 2996–3000 (2012). [CrossRef]

**33. **C. Menke, “Application of particle swarm optimization to the automatic design of optical systems,” Proc. SPIE **10690**, 39 (2018). [CrossRef]

**34. **J. Zhang, X. Chen, J. Xi, and Z. Wu, “Paraxial analysis of double-sided telecentric zoom lenses with three components,” Appl. Opt. **53**(22), 4957–4967 (2014). [CrossRef]