1. Introduction

Optical zoom imaging systems play an important role in military and civilian applications such as aerial surveillance, frontier guarding, and microscopy [1–4]. In most optical payload systems, image motion compensation mechanisms such as fast steering mirrors (FSMs) installed on pupil planes are commonly used to improve the imaging quality and the image resolution due to the movement of the air vehicle relative to the targets, as well as owing to the vibration of the platform and the swing of the camera [5–7]. A special mid-wavelength infrared (MWIR) zoom system with a cooled detector is required to maintain stable focus on the fixed image planes of the two separate object planes, namely that of the object and that of the entrance pupil. Otherwise, wandering of the entrance pupil leads to the unstable height of a radiation beam at the FSM, which is unacceptable when it is essential to perform the image motion compensation in infrared zoom systems. And it is very clear, the well-developed theory on designing traditional zoom systems cannot be applied to such conditions that are defined as dual-conjugate zoom systems. Therefore, methods and reasonable solutions for the first-order parameters of MWIR dual-conjugate zoom systems should be discussed in detail in our research. As shown in Fig. 1, the varifocal dual-conjugate system which is connected to a cooled detector can be divided into two parts: objective with fixed focal length and zooming part with magnification varying from m to M. The detector cold stop is considered as the system aperture stop in order to realize 100% cold stop efficiency. The positions of exit pupil ($\overline {E^{\prime}} $) and image plane ($\overline {O^{\prime}} $) of the exit pupil are stable for that a fixed detector is preferred during the zooming process. Besides, the diameter of exit pupil is kept fixed due to the unchanged f-number (F^#) of the detector. Then the entrance pupil (E) is being relayed to the FSM via the objective with a diameter varying from Mf₀/F^# to mf₀/F^#, where f₀ is focal length of the objective.

 

Fig. 1. varifocal dual-conjugate system: (a) system configuration with magnification of M for Part II; (b) system configuration magnification of m for Part II.

Download Full Size | PDF

The first conception of dual-conjugate zoom systems, along with the solutions, was proposed by Hopkins [8,9]. Many years later, problems connected with the stabilization of the pupils of zoom systems reappeared in others’ works. An afocal zoom system used in the ophthalmoscope was described which kept both the images of the patient’s and clinician’s retina and pupils stable during the zoom [10]. Then, more attention was paid to the theories and first-order solutions than a practical zoom system with stable pupil positions. Multiple initial conditions with real or virtual images were discussed [11–13]. Besides Gaussian reduction, formulars based on matrix optics were introduced in which first order parameters were numerical solutions of the nonlinear equations [14]. Pal presented a system with only two separately moving elements to search for the solution of pupil stabilization, while, the positions of entrance and exit pupil were kept stabilized around some locations within an acceptable tolerance [15]. With the development of active optical elements, tunable focus lenses were also utilized to achieve required zooming properties without any mechanical movement of individual components [16–19].

In most of the circumstance, some parameters of the equations had to be predetermined and repeatedly calculated to find a reasonable first-order solution, which posed a serious challenge to inexperienced optical designers attempting to achieve the desired goals. Therefore, in order to reduce the reliance on experience, some optimization algorithms and automatic design methods were employed for the optical system design. Yang presented a point-by-point design process that can automatically obtain high-performance freeform systems [20]. A multi-objective evolutionary memetic optimization algorithm was presented and verified for the automatic design of optical systems, simultaneously addressing the image quality, tolerance, and complexity of the system [21]. Genetic algorithm was used both in lens design and zoom systems to search a point, which is the best fit with a set of prespecified requirements [22–24]. Neural network algorithm was demonstrated to realize bias locking for a Mach-Zehnder modulator used in optical communication [25]. Ant colony optimization was implemented and tested for optical design tasks [26]. Besides, global search algorithms on the basis of particle swarm optimization (PSO) algorithm were widely applied to lens designs [27–30] and retrieving the first-order designs of traditional zoom systems [31] or zoom systems with fixed foci [32,33]. Comparatively, PSO algorithm is easy to operate among the numerous global optimization method. Moreover, PSO algorithm have been preferred in recent years due to enjoying the advantages of great randomness, fast convergence, high solution quality, and good robustness in finding the optimal value in multidimensional constrained function space. Despite the well-developed theories on and the algorithms for optical design, there are barely enough reports on the dual-conjugate zoom design applied in practical MWIR systems.

In this paper, we present a method for the automatic design of a MWIR dual-conjugate zoom system. At least three moving lenses are required to stabilize the location of the paraxial pupils during the zooming process. The derivation of the paraxial imaging is performed with Gaussian optics. Combined with the zoom theory, an optimal first-order solution is derived based on the PSO algorithm. During the optimization process, the focal power of each component, the distance from the entrance pupil to the object, and the distance from the exit pupil to the image constitute the specific search space. The zoom ratio, the total length, and the Petzval field curvature of the zoom system are considered for the merit function, and the physical feasibility constitutes the boundary of the candidate solutions. Then, the design work is regarded as a mathematical optimization of the multidimensional constrained space. Combined with a fixed objective, a varifocal zoom system with fixed planes both for the image and for the entrance pupil and the exit pupil is designed. Furthermore, this method is implemented in a zoom system with a higher zoom ratio. We demonstrate that the proposed method can efficiently construct an initial configuration for the dual-conjugate zoom system. Meanwhile, the artificial intelligence applying the PSO algorithm has the great potential for searching the first-order solutions to the optical systems with which there is not enough experience.

2. Formulas and algorithm for the first-order solution

2.1 Paraxial analysis of a dual-conjugate zoom system with finite object distance

Different from a traditional zoom system which merely maintains the image plane stable, the dual-conjugate zoom system contains at least three moveable lenses as indicated in Hopkins’s papers [8]. A four-component varifocal system was introduced by Kryszczynski [12], in which the last lens was fixed. However, the front moving component in infrared zoom systems may exert pressure on the driving mechanism because of the larger aperture. Learning from the method in traditional zoom systems, we decomposed the desired varifocal system into a fixed objective and a subsystem with a finite distance between the object and the image and a finite distance between the entrance pupil and the exit pupil. The main task of the design work is transferred to the study on such subsystems. The paraxial optical parameters of the dual-conjugate zoom system are schematically presented in Fig. 2.

 

Fig. 2. Parameters determining the two optical conjugations of the zoom system.

Download Full Size | PDF

The convergence and the divergence of the beam determined by the optical lenses are shown in this figure as well. The object $\bar{O}$ is imaged at the image plane ($\overline {O^{\prime}} $) with a magnification of m, and the entrance pupil $\bar{E}$ is imaged at the exit pupil ($\overline {E^{\prime}} $) with a magnification of $\bar{m}$. The first and the second principal planes of the system are located at H and H’ respectively; also, ${\varphi _i}$ indicates the optical power of the ith component, where i ranges from one to three.

In Gaussian paraxial approximation, the distance from the image to the exit pupil, i.e. L’, and the distance from the object to the entrance pupil, i.e. L, remain unchanged during the zooming process and can be given by [11]:

From Eq. (1), a simple relationship between the focal length (f), m, L’, and L can be derived by eliminating the variable $\bar{m}$ as expressed in Eq. (2):

The optical power $\varphi $ and the distance $\Delta $ formed by three lenses with an optical power of ${\varphi _1}$,$\; {\varphi _2}$, ${\varphi _3}$ and intervals of d₁₂, d₂₃ can be expressed by:

The multiplication of d₁₂ and d₂₃ is obtained from the joint solution of Eq. (4):

By substituting Eq. (5) into Eq. (4), we can define d₂₃ as a linear function of d₁₂:

Then, d₂₃ can be calculated by Eq. (6). The distance from the first element to the object ${l_1}$ and distance from third element to image $l_3^{\prime}$ are still needed to confirm the position of each moving lens which can be calculated as

Therefore, we could calculate the distances ${l_1}$, $l_3^{\prime}$, d₁₂, and d₂₃ using Eqs. (2) to (10) for the given optical powers ${{\varphi }_1}$,$\; {{\varphi }_2}$, and$\; {{\varphi }_3}$; the given distances L, L’, and P; and any magnification. As we can infer from Eq. (2), the optical power varies with the magnification, so the locations of each component can be recalculated subsequently. As a result, we can calculate the kinematics of the three moving components in the zooming process by continuously varying the object–image magnification from m to M.

We form the varifocal system by locating an objective with a fixed focal length in front of the subsystem. A virtual or real object is provided by the front lens. As depicted in Fig. 3, the entrance pupil (E) is located in front of the whole optical system to provide an ample position for the FSM. The relative position of the entrance pupil ($\overline {{l_0}} $) can be obtained easily by Gaussian optics. E, $\overline {E^{\prime}} $, and $\overline {O^{\prime}} $ have a stationary location during the zooming process. Additional part III, which is called the reimage group with a magnification of ${m_4}$, can be applied if required. The plane E’ and O’ is considered as the aperture stop and image plane, then entrance pupil is relay to the FSM. The total focal length varies either from $m/{\varphi _0}$ to $M/{\varphi _0}$ or from $m{m_4}/{\varphi _0}$ to $M{m_4}/{\varphi _0}$ with a zoom ratio of R defined as R = M/m.

 

Fig. 3. Parameters determining the varifocal dual-conjugate system.

Download Full Size | PDF

The magnification of part II must vary from $- \sqrt R $ to $- 1/\sqrt R $ to keep symmetry of the dual-conjugate zoom system. Then ${\varphi _0}$ can be assigned roughly according to a desired range of focal length. A negative value for L’ is expected for the cooled infrared detector. To escape collisions between any two of the four components, as well as the exit pupil plane, the intervals between each component must satisfy the following conditions:

The specific restrictions expressed in Eq. (11), which result from the requirements of the physical feasibility, are added to the solutions for the first-order parameters of the dual-conjugate zoom system. In a previous study [11], the ranges of the solution to the system parameters were discussed at the mean position where the magnification of part II is either 1 or –1. Meanwhile, the ratio of the object to the image and the pupil magnification of the middle lens were considered to be either 1 and –1 or –1 and 1. Further, there was no merit function to evaluate the predetermined parameters. Thus, it is difficult for optical designers to search the optimal candidate solution in a multivariable space based on the traditional trial-and-error approach. In this paper, an intelligent algorithm based on the PSO is utilized to automatically search the globally optimal first-order solution for the dual-conjugate zoom system.

2.2 Particle swarm optimization algorithm

PSO is a type of group intelligent computation originally introduced by Kennedy and Eberhart on the basis of studying the bird swarm predation behavior [34]. The basic idea of the PSO algorithm is to find the optimal solution through the cooperation and information sharing among the individuals in a group. The flight process of the particle is the search process of the corresponding optimization problem. A particle contains two attributes, namely velocity and position, which describe how fast it moves and define the direction of the movement of the particle. Each particle is able to evaluate its current position by the merit function and to remember the optimal position. The moving velocity of the particle can be dynamically adjusted according to its historical optimal position and the optimal position of the whole population. After all the particles move, a new iteration starts to update the position and the velocity of the particle. Eventually, the swarm as a whole is likely to move close to an optimum of the merit function.

In the PSO, we randomly initialize the velocity and the position of N particles in the search space S^D at first and assume that the vector X_i= (x_i1, x_i2, …, x_iD) represents the position of the ith particle (i = 1, 2, …, N). The velocity of the ith particle is denoted by the vector V_i= (v_i1, v_i2, …, v_iD). The initial X_i and V_i are randomly generated within a limited range which is provided by the optical designer according to the actual situation. The particle updates its status after being evaluated by the merit function f(X_i). The ith particle in kth iteration updates its position and velocity as follows:

During the optimization process, the particles may escape from the search space S^D with an improper velocity. Therefore, methods should be used to correct for particles escaping the boundaries. In this paper, a candidate method in which an additional vector in the opposite direction is introduced to return the particles to the search space.

In Eq. (12), the first part ($w \cdot v_{id}^k$), known as the memory term, represents the impact of the previous velocity of the particle. The second part, i.e. c₁r₁(pbest_id-$x_{id}^k$), indicates the self-cognition component, which drives the particle from its current position to its best position. The third part, i.e. c₂r₂(gbest_id-$x_{id}^k$), provides a social term which points to the best position of the population, indicating the cooperation between the particles. The movement of a particle is determined by its own experience and the best experience of its peers.

We can locate the best position of a particle with fewer iterations if an appropriate inertia weight (w) is applied. A time-varying w is preferred to decrease the probability of falling into a local extreme value. A large inertia weight benefits the global search in the early iterations, and a small inertia weight strengthens the local search capability in the later stages. For simplification, a linear decreasing model of the inertia weight is applied as follows:

2.3 Design process by particle swarm optimization algorithm

The PSO algorithm is a suitable candidate for global optimization in the special zoom system with two fixed conjugate planes due to the easy implementation and moderate computation. The PSO algorithm can straightforwardly be utilized to search the optimal solution for the paraxial design parameters, namely ${{\varphi }_1}$,$\; {{\varphi }_2}$,$\; {{\varphi }_3}$, L, L’, P and R. The variable parameters are represented by particles with the position vector X_i= (x_i1, x_i2, …, x_iD) in the searching space which is determined by common sense, basic optics principles, and the physical feasibility. Generally, a multi-objective merit function which characterizes the optical system is used to select the optimal position. We tend to start with an initial structure which has the potential to provide a higher zoom ratio in a relatively compact space. Also, the field curvature aberration must be controlled for it cannot well be corrected only by the optical design software. As a result, the merit function used in the PSO optimization is defined as [32,33]:

To confirm the kinematics of the moving components, which do not collide with each other during the zooming process, we reconsider the physical restriction during the update phase of the optimization of the solutions. A few of the configurations with a magnification of part II of m = $- \sqrt R $ + $j\left( { - 1/\sqrt R + \sqrt R } \right)/Q$, where j = 0, 1, 2, …, Q, are chosen to verify the physical feasibility of the proposed paraxial design parameters. The intervals of each components should be changed continuously. There cannot exit discontinuity in the cam curve. The movements of three moving lenses are calculated between two zoom configurations and then limited in a range of δ, which is determined by the fabrication ability. The smoothness of the cam curve, as well as the constraints expressed in Eq. (11), is also evaluated. We can confidently believe that a continuous dual-conjugate zoom system can be realized if enough configurations satisfy the actual physical constraints. Thus, the position in the search space is a candidate solution to the paraxial parameters. Otherwise, the particles with an improper position are removed from the search space and the iteration.

The modified process of the optimal solution for the proposed PSO algorithm is depicted in Fig. 4. As discussed previously, the vectors X_i and V_i are firstly initialized by randomly adding values to the parameters, the constraints expressed in Eq. (11) is then used to confirm the physical feasibility for a given X_i, in case the impractical particles with the minimum merit function enter the optimization process; the values of the merit function are evaluated for the reasonable solutions using Eq. (15). Afterward, the best position of each individual (pbest_i) is replaced with a better value in its history, and the globally optimal position (g_best) is selected from pbest_i with the minimum merit function value. The inertia coefficient decreases with the iteration to ensure faster convergence. Next, all the checked particles update their velocities and positions following the rules expressed in Eq. (12). If the candidate particle runs out of the search space S^D, a reverse velocity vector is employed to return it, as expressed in Eq. (13). When either the predetermined accuracy or the maximum number of iterations is achieved, the optimization is terminated. Eventually, we obtain the optimal solution for the first-order design of the desired zoom system.

 

Fig. 4. Flow chart of the modified PSO algorithm.

Download Full Size | PDF

If few initial particles satisfy the constraints of the physical feasibility in the zoom system, the search space may be adjusted, and they will be reinitialized. Otherwise, if the found solution cannot possibly be used to develop a satisfying simulation with the optical design software, we can employ the suboptimal solutions for the other candidates to start with the design work.

3. Design results and discussions

3.1 Example of a dual-conjugate zoom system without reimage group

Firstly, we consider a simple system which only consists of fixed part I and zooming part II in Fig. 3. The equations and mathematical model in the previous section are programmed in MATLAB to calculate the paraxial parameters of the first-order design of an MWIR dual-conjugate zoom system. The distance from the exit pupil to the image plane is restricted to –19.1 mm to realize suppression of stray rays for the commercial detector. We find that the value of L makes little contributions to the optimization in our experiment. Thus, it can be specified a fixed value to obtain a reasonable location of the entrance pupil. Therefore, the search space of the optimization is reduced to (${{\varphi }_1}$,$\; {{\varphi }_2}$,$\; {{\varphi }_3}$, R, P) ∈S^D (D = 5).

In zooming part II, a negative lens is placed between the two positive lenses; the symmetry can help compensate for the aberrations. The optical power of each component should be within a reasonable range on account of the manufacturability and practicability, as well as its proper contribution to forming an optical system. We also tried other power combinations such as “+ - -”, “- + -”, “+ + -” but didn’t obtain reasonable solutions.

The value of ${{\varphi }_0}$ and L is set at 0.01 mm^-1 and -300mm, respectively. The location of the entrance pupil ensures ample space for the FSM. Afterward, a number of constraints expressed in Eq. (11) are added to select the reasonable solutions. We then add 200 particles into the search space to guarantee enough available particles in the optimization process. The positive value of P roughly represents the length of the system so that it should not be too large. The parameters w_R, w_L, w_A are set 50, 0.5 and 20 [33]. To summarize, the parameters that describe the particles and convergence in the PSO algorithm are listed in Table 1, and the detailed design results obtained from the optimization are presented in Table 2.

Table 1. Parameters of the PSO algorithm.

View Table | View all tables in this article

Table 2. Design data on the 4.6X dual-conjugate zoom system (Unit, mm).

View Table | View all tables in this article

As shown in Fig. 5, the merit function can converge quickly and begins to level off after 30 iterations. It is demonstrated that the PSO algorithm can be efficient for the optimal solutions. The focal length varies from 47 to 215 mm, corresponding to a zoom ratio of 4.6. As we can see in Fig. 6, the kinetics of the three moving components is obtained. Moreover, according to this figure, the locations of the two conjugate planes obviously remain stationary during the zooming process. The Petzval sum of the initial design is 3 × 10⁻³ mm⁻¹, and the total distance from the entrance pupil to the image plane is 227.1 mm.

 

Fig. 5. The convergence process of the merit function

Download Full Size | PDF

 

Fig. 6. The loci of the three elements and the locations of the image plane and the pupils

Download Full Size | PDF

We perform the optimization in the optical design software in order to validate the results of the first-order design. The initial structure is firstly constructed by the lens module of surface type in CODEV to confirm that none of the lights exceeds the optical system in the maximum field of view of each zoom configuration. The lens module is then replaced by the actual lens for aberration correction. Since the location of the entrance pupil relates strongly to the principal plane of each component, the actual intervals should be adjusted when the transformation is performed.

The working wavelength of the optical system ranges from 3.7 to 4.8 µm, and the field of view varies from 3.3° to 14.8°, corresponding to an image height of 12.2 mm. Furthermore, tractable aspheric surfaces are added to actual lens for the aberration correction, and none of diffraction surfaces is used. Also, the f-number is equal to four during the zoom process for suppression of stray rays. As displayed in Fig. 7, an eight-lens zoom system with a fixed entrance pupil and image plane has been successfully constructed. The zoom trajectory is basically the same as the first-order design, and the modulation transfer function (MTF) at three zoom positions is shown to evaluate the performance of the system. It is obvious that the zoom system can resolve 33 lp/mm or higher, which demonstrates that our proposed method can provide a satisfying first-order design for complex systems. The only weak point here is that the heights of the chief ray at the entrance pupil are not perfectly equal to zero. As shown in Fig. 8, the heights are distributed around zero and vary during the zooming process. Obviously, the chief ray is further away from the optical axis in a full field of view.

 

Fig. 7. The cross-section of the 4.6X MWIR dual-conjugate zoom system (left) and the MTF for three zoom configurations (right).

Download Full Size | PDF

 

Fig. 8. The height of the chief ray at the entrance pupil in different fields of view.

Download Full Size | PDF

3.2 Example of a dual-conjugate zoom system with reimage group

Although a higher zoom ratio is expected with the simple structure in Section 3.1, we found that the solvable region narrows rapidly at a high zoom ratio. As shown in Fig. 9, only 0.9% of the particles survived within the region at a zoom ratio of 16X. Although the proposed optimization method can lead to an advisable first-order solution with few particles, it performs much better at a low zoom ratio. By repeating the experiments using different L’ values, we found that more particles exist if the distance between $\overline {E^{\prime}} $ and $\overline {O^{\prime}} $ is liberalized.

 

Fig. 9. The variation in the percentage of the available particles with the zoom ratio at different L’.

Download Full Size | PDF

As discussed above, a higher zoom ratio can be achieved if the distance L’ is no longer restricted to –19.1 mm. However, such a condition cannot directly be applied to a zoom system with a cooled detector due to the vignetting effect. To overcome this issue, zooming part II must be followed by a reimage group. As shown in Fig. 3, the exit pupil and the image plane of part II are transferred by stationary part III to realize the suppression of the stray lights. Undoubtedly, the locations of the planes E, E’, and O’ are fixed when zooming the optical system. Although the total length of the whole system increases, the locations of $\overline {E^{\prime}} $ and $\overline {O^{\prime}} $ of part II become more flexible. With the existence of a reimage group, the plane $\overline {E^{\prime}} $ can be located either before or after the third moving component, that is either $\; l_3^{\prime} > {\; }0\; $or $l_3^{\prime} < {\; }0$. Under such conditions, the constraints described in Eq. (11) can be simplified to:

With the attachment of a reimage group, the merit function used in the PSO algorithm is modified as follows:

Since the merit function has become more complex due to the extension of the search space to (${{\varphi }_1}$,$\; {{\varphi }_2}$,$\; {{\varphi }_3}$, R, L’, P) ∈S^D (D = 6), we must make some minor modifications to the corresponding contents of the PSO algorithm. To this end, the range of L’ is determined by Eq. (18) for the reasonable location and magnification of the reimage group. After roughly placing the reimage group, the optical power φ₄ and the distance P’ in Eq. (17) can be calculated at different L’ values. During the optimization, the weights w_R and w_A are adjusted slightly to maintain the balance between the zoom performance and the optical performance. When the weight coefficient w_L is adequately reduced, the overall length tends to be large if a higher zoom ratio is desired. After performing a number of experiments, we finally created a 12.4X first-order design of the special zoom system. The modified parameters used in the PSO algorithm are tabulated in Table 3. Since the proportion of survival particles increases with the distance of the plane from $\overline {E^{\prime}} $ and $\overline {O^{\prime}} $, fewer particles are needed compared to the conditions in Section 3.1. The value of ${{\varphi }_0}$ and L is set as the same as that in Section 3.1.

Table 3. Parameters of the PSO algorithm.

View Table | View all tables in this article

After the optimization, we found the optimal solutions as listed in Table 4. The Petzval sum of the initial design is 3.3 × 10^-−mm⁻¹, and the focal length of the dual-conjugate zoom system ranges from 25.40 to 319.35 mm with the magnification of the reimage group being equal to –0.91. Figure 10 shows the loci of the three moving components, as well as the stationary lens, during the zooming process. The total distance from the entrance pupil to the image plane equals to 416.67 mm.

 

Fig. 10. The loci of the three moving elements of a dual-conjugate zoom system with the reimage group.

Download Full Size | PDF

Table 4. Design data on the 12.4X dual-conjugate zoom system (Unit, mm).

View Table | View all tables in this article

Similarly, the optimal solution is imported to CODEV for the simulation of an MWIR dual-conjugate zoom system. An elongated model with a mechanical length of 410 mm and a maximum aperture of 82.5 mm is developed, as shown in Fig. 11; also, the entrance pupil is located at a position of 215 mm before the front lens. Further, folded mirrors may be applied at intermediate image plane for further compactness. The fine imaging quality and the smoothness of the curves prove that our proposed method is a useful tool for automatically designing an MWIR dual-conjugate zoom system with a high zoom ratio. Actually, the zoom ratio of this special zoom system can be further enhanced if the total length of the system can be long enough.

 

Fig. 11. The cross-section of the 12.4X MWIR dual-conjugate zoom system (left) and the MTF for three zoom configurations (right).

Download Full Size | PDF

4. Conclusions

Although the solution areas and the design procedure were presented for different cases in previous works on dual-conjugate zoom systems, an optimal solution for an actual design problem cannot be found efficiently using the trial-and-error method. In this paper, the PSO was introduced as an efficient global optimization method for the solutions to the paraxial parameters of a first-order design. Thus, designers only need to know a general scope of particles in the search space. The two examples demonstrated that the PSO algorithm is able to manage a large number of proper solutions for the specific zoom system, which can give researchers the design freedom to satisfy the actual design requirements. Furthermore, the proposed optimization method helps construct a 12.4X dual-conjugate zoom system with an appended reimaging lens group; in addition, it is a very useful method for semiprofessional optical designers due to the easy implementation and conceptual simplicity. Finally, intelligent computation using the PSO algorithm has the potential to be applied to designing an initial construction of any other complex zoom systems.

We have investigated the first order solution of dual-conjugate zoom system and then will focus on the aberration design or thick lens design in the future. Moreover, other global optimization methods will be introduced to the study of this problem in next work.