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Abstract
In this paper, the adaptive liquid (AL) lenses are introduced into the double-sided telecentric zoom system, which could greatly decrease the mechanical motion group compared with the traditional zoom system, and only one movable aperture stop (STO) is retained. Firstly, this paper derived the Gaussian brackets used in this system, and we found the appropriate screening method to get the suitable initial structure parameters from the solution space. Then we used the lens module design method to create the initial system. Finally, we used CODEV to further optimize the system, and we got an excellent design result, which controlled the telecentricity of both sides within 0.1°, the distortion was controlled within 0.5%, and the MTF of each zoom configuration above 0.3. This optical system has high application potential and value in the field of precision machine vision. The design method proposed in this article can effectively solve the problem that the zoom system using adaptive liquid lenses lacks the initial structure.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The emergence of adaptive liquid lenses provides a fully new way of thinking in zoom system study [1–8]. Compared to the traditional zoom optical system [9–14], which realizes the changes of magnification and the compensation of the image plane by the motion of the lens group, the new optical zoom system used adaptive liquid lenses is more advanced in mechanical vibration control, the system stability and the response speed [7]. For now, there already exists a lot of relative research [7,15–24], among them, Miks et al. derived the calculation method of the paraxial imaging properties and design parameters of a double conjugate zoom lens composed of three tunable-focus lenses with a fixed position [15]; Li et al. designed an ultrathin zoom telescopic objective used three electrowetting liquid lenses, which largely decrease the total length [16]; Lenk et al. propose to combine the linear lens movements with tunable lenses to achieve the change of focal length, and they did experiments to prove this idea [20]; Miks et al. present a methodology of calculation of the inner structure of two- and three-component hybrid liquid membrane lenses with variable focal lengths that have corrected spherical aberration and coma [23].
With the continuous development of current machining technology, the demand for precision machine vision in the market is increasing. The remarkable feature of the double-sided telecentric optical system is the chief rays enter the system parallel to the optical axis and eventually exit the system parallel to the optical axis. So, whether the object plane or the image plane happens a tiny shift, the magnification can always keep constant [25]. For a double-sided telecentric system, when an object point deviates from its optimal object plane, the corresponding image spot will expand, while the position of its centroid can remain unchanged due to the characteristics of its chief ray entering the system in parallel, which enables the recognition algorithm to recognize the exact position of the object point, which means when measuring high-thickness objects, the double-sided telecentric system can achieve a larger accurate measurement range compared to the conventional industrial cameras. The above advantages show the reason why the double-sided telecentric system is active in the field of industrial monitoring. However, the manufacturing cost of the double-sided telecentric optical system is far more than the conventional optical system. If the zoom system can be combined with the double-sided telecentric system, the manufacturing cost performance will be effectively increased and the applicability scenarios can be greatly extended, which will undoubtedly be more competitive in the market.
At present, studies on the double-sided telecentric zoom optical system are rare, and most of them rely on mechanical group motion to achieve zooming [26–31]. As mentioned above, such an optical system has large mechanical vibration and a certain delay response time, which is difficult to meet the working environment requirements of high-precision device monitoring. If the adaptive liquid lenses can be used to replace the mechanical movement group to realize zooming, the demand for mechanical movement can be greatly reduced and this problem can be effectively improved.
In this paper, the double-sided telecentric system is combined with the zoom system using adaptive liquid lenses. Firstly, we derived Gaussian brackets applied in four components double-sided telecentric zoom AL system and then established an effective screening method to find the suitable initial structure parameters from the solution space obtained in MATLAB [32]. We also use ZEMAX [33] to build the ideal lens models to prove the effectiveness of our screening method. After that, the lens module design method is used to design the four components, and the module combination method applied in this system is deduced. As a result, we built the initial structure of the system with sufficient potential. Finally, CODEV [34] was used to further optimize this initial structure, and the final design result has an excellent performance in telecentricity control, distortion control, and aberration correction. The research of this paper fills the gap in the current field and has enough potential and value in industrial application.
2. Theoretical analysis of double-sided telecentric zoom AL system
2.1 Gaussian brackets used in the double-sided telecentric zoom AL system
Firstly, let’s show the derivation of the Gaussian brackets used in a four-component zoom system based on adaptive liquid lenses, which can be written as follows [24]:
And we can further expand Eqs. (1) to Eqs. (2):
And the calculation rules are as follows:
And we can get more first-order parameters according to the following Eqs. (4).
Now we should summarize the characteristics of the double-sided telecentric system and combine them with the above Gaussian brackets, then we can obtain the equations suitable for the double-sided telecentric zoom AL system. And the structure of the four-component double-sided telecentric system is shown in Fig. 2.
From this figure, we can find that the position of the STO is at both the rear focal point of the front lens group (GROUP1) and the front focal point of the back lens group (GROUP2), so the chief ray from the object space can enter the system parallel to the optical axis and eventually exit the system parallel to the optical axis. And the geometric relationship of this system can be expressed as Eqs. (5):
where d2 is the distance between Lens2 and Lens3, BFL1 is the back focal length of GROUP1, FFL2 is the front focal length of GROUP2, L1 and L2 are used to indicate the STO position. So, we can construct a double-sided telecentric system by reasonably controlling the position of the STO and the first-order parameters of the system.Now, let’s derive the solving method for the double-sided telecentric zoom AL system. The double-sided telecentric zoom AL system is a finite conjugate system, so we need to use magnification m to present the features in different zoom configurations, as shown in EQ1. The second equation EQ2 is used to describe the position of the STO. And EQ3 and EQ4 are used to represent the object distance and the image distance, respectively. The optical system model used in this algorithm has been shown in Fig. 2. And the above equations are shown in Eqs. (6).
where f12’ is the focal length of GROUP1, f34’ is the focal length of GROUP2. And we should note that the sign of the BFL1 is according to the plane of Lens2, while the sign of the FFL2 is according to the plane of Lens3. This means there are other hidden conditions in EQ2, which are BFL1 > 0 and FFL2 < 0 should satisfy simultaneously.
2.2 Screening method used in the double-sided telecentric zoom AL system
After deriving the Gaussian brackets used in the double-sided telecentric zoom AL system, we can put it into MATLAB to get the solution space. And now we need to screen the suitable solution system from the solution space as the initial structure parameters for the optical design. The screening process and results are shown in Fig. 3.
According to Eqs. (6), the input parameters should include the distance ${d_j}$, the magnification m, the object distance $S,$ and the image distance $S^{\prime}$. And for the selection of the ranges of these parameters, we set as the following Table 1.
Table 1. The range of the double-sided telecentric zoom AL system parameters.
In order to simplify this solving question, we set the distance ${d_j}$ between the different components to the same value. For the selection of the range of object distance S, it is set to a range that varies with the distance d, which is set to $- 1.4d < S < - 0.5d$. The value of the image distance $S^{\prime}$ should be reasonably chosen according to the value of the distance d, which is set to $0.5d < S^{\prime} < 1.4d$ in this paper. After solving the Eqs. (6) based on Table 1, we can acquire a big solution space. It’s hard to select a suitable solution system directly, so we need to find the appropriate screening method.
In this paper, we used a three-step screening method, as shown in Fig. 3 before. For the first screening, we select the solutions based on the optical power range of each group. The zoom range of the AL must be kept within a reasonable range to meet the practical application requirements. And a lower focal length span between different zoom configurations in each component is more beneficial in controlling the difficulty of the design, which could be seen more clearly in the subsequent design process. For the double-sided telecentric zoom AL system, we set the focal length span of each group as shown in Fig. 3(b).
In the next screening step, we selected only those solution systems that could meet the requirements of the span of magnification. And the select results are shown in Fig. 3(c). And each of these solution systems contains only one set of solutions, so for the third screening step, we determine the superiority of each set of solutions based on the self-defined evaluation functions, which are defined as:
In this step, the value of the self-defined function of each zoom configuration can reflect the optimization potential of the configuration in this solution system to some extent. For this design, we believe that every configuration is equally important, so we choose the set of solutions with a uniform distribution of the self-defined evaluation functions for each zoom configuration and the lowest curve, which is $\; d = 0.3,S ={-} 0.27,S^{\prime} = 0.3$.
The specific parameters of this set of solutions and the ideal optical models in ZEMAX are shown in Fig. 4(a)_1 to (a)_3, and we can see from these pictures that the motion of the STO just showed a monotonic linear motion, which is beneficial in the subsequent optimization.
Fig. 4. (a) The screening results of the double-sided telecentric zoom AL system and (b) the examples of the bad solving results.
We also selected one solution system for each zoom configuration from these discarded solutions to establish an ideal optical model for comparison, as shown in Fig. 4(b)_1 to (b)_3. And it’s obvious that there even exists more than one focus in these models, which may greatly increase the difficulty of design. So, we can say the screening method we used during the design process is effective and superior.
3. Method of building the initial structure of the optical system
In the previous part, we used the algorithm to obtain the focal length of each lens group of the four-component zoom AL system in different zoom configurations, and we need to obtain the initial structure of the system. In this paper, we decide to use the lens module design method to build the initial structure. In this section, we first introduce the module combination method used in the double-sided telecentric zoom AL system, then we introduce the design principles and results of each lens module, and at the end of this section, we show the initial structure and evaluate it.
3.1 Combination method used in the double-sided telecentric zoom AL system
The modules of the double-sided telecentric zoom AL system are combined based on the principal plane positions of each group, as shown in Fig. 5.
Figure 5 shows the geometric distance relationships for each lens group. First, we specify the concept, ${z_{i,j}}$, where $i = 1,2,3$ correspond to different zoom configurations, and $j = 1,2,3,4$ correspond to different lens groups.
And there are several data we should calculate when synthesizing the system, the distances between the adjacent vertices of the groups, which are marked with the black two-way arrows; the distance between the rear vertices of the second group and the STO, and the distance between the front vertices of the third group and the STO, which are marked with the green two-way arrows. And we can obtain the Eqs. (8) according to Fig. 5.
It should be noted that the positive and negative values of ${z_{i,j}}$ and ${z_{i,j}}^{\prime}$ are relative to their local coordinates corresponding to the vertex of the group, while the value of ${d_j}$ satisfies the conventional sign rule for optical systems, with the front end of the system being the global coordinate starting point.
In particular, for the convenience of subsequent calculations, we can list Eqs. (9) to describe the distances relevant to the STO position:
We can also list Eqs. (10) to describe the work distance and the image distance used in this finite conjugate system:
After driving the Eqs. (8) to (10), now we can easily combine these four lens modules into a whole system, and we can optimize this synthesis system in optical design software.
3.2 Lens module design method used in the zoom AL system
We first need to select suitable ALs from the current market to ensure the practicability of the design. And after a well-consideration, we choose the series ML-20-35 products from Optotune. The optical specifications are given in Table 2.
Table 2. The optical specifications of the ALs.
For the zoom system, the material combination is particularly important, and we can determine the material combination of the four ALs to reduce the design complexity. According to the optical design experience, we have decided to adopt three high refractive index ALs with low dispersion, ML-20-35-VIS-HR, and one low refractive index AL with high dispersion, ML-20-35-VIS-LD. And we plan to put the one ML-20-35-VIS-LD into the second or the third group to help reduce the chromatic aberration.
Then we will use lens group i as an example to illustrate the initial structural parameters and material combination chosen method of a lens group containing an adaptive liquid lens. Figure 6(a) shows the basic structure and geometric relationships of lens group i. We can use Eqs. (11) to reflect the relationship between the focal length of the normal lens group ${f_1}^{\prime}$, the focal length of the adaptive liquid lens ${f_2}^{\prime}$, and the focal length of the whole lens group $f^{\prime}$:
After deriving Eqs. (11) with respect to ${f_2}^{\prime}$ :
Whether $f^{\prime}$ takes positive or negative, for a relatively compact lens group, $f^{\prime}({d - f^{\prime}} )$ is generally less than 0, which means ${f_1}^{\prime}$ changes with ${f_2}^{\prime}$ is monotonic. For the zoom module using an adaptive liquid lens, the value of ${f_2}^{\prime}$ is always unique if ${f_1}^{\prime}$ is determined, so we only need to reasonably allocate the optical power of the normal lens group and the adaptive liquid lens according to experience, and then calculate the relevant parameters according to Eqs. (11).
In this example, we take the focal lengths of the three zoom configurations of the lens group i as 60 mm, 70 mm, and 80 mm, respectively. In order to make the lens group i have a good optimization potential, we select an air-spaced achromatic doublets as the normal lens group, and we set the focal length of ${f_1}$‘ as 100 mm and d as 15 mm. According to Eqs. (11), ${f_2}^{\prime}$ of the three zoom configurations is 127.5 mm, 198.3 mm, and 340.0 mm, respectively, which is in a reasonable range according to Table 2. And we can calculate the radius of curvature through $f^{\prime} = \frac{{{R_1}{R_2}}}{{({n - 1} )({{R_2} - {R_1}} )}}$. And we choose ML-20-35-VIS-HR as the adaptive liquid lens used in this example, the initial structure of the lens group i is shown in Fig. 6(b).
Next, we need to preliminarily optimize this lens group, so that the synthesis system formed by all lens groups could have a nice initial aberration correction characteristic, which is beneficial in reducing the design difficulty and improving the design efficiency. In this step, we mainly focus on the correction of chromatic aberration and spherical aberration. The material matching is automatically completed by the optical design software. The spherical and chromatic aberration coefficients are given in Fig. 6 (b) and (c). After this optimization, it can be seen that the accuracy of the focal length becomes worse, and the maximum differences of the principal plane position in each zoom configuration are expanded from 3.5 mm to 6.8 mm, while the spherical and chromatic aberration coefficients are significantly reduced, which is worth the cost. In addition, the red circle in Fig. 6(b) shows the irrational thickness of the adaptive liquid lens in each zoom configuration, we also solve this problem in this step.
Through this optimization process, we can also find that there is a contradiction between the aberrations correction and the differences in the position of the principal plane in different zoom configurations. Here, we can answer the question we mentioned above, why should the span of the focal length of each group not be too large? When designing in these groups, we’d like to eliminate aberrations as much as possible and improve the optimization potential of the initial synthetic structure, so we do not strictly restrict the principal plane position in different zoom configurations but allow a certain gap exists. If the focal length span of the component is too large, it will greatly increase the contradiction of the position difference of the principal plane and the aberration correction, as a result, the initial structure may hard to be optimized.
Based on these above design principles, we have designed the four groups, as shown in Fig. 7(a). And the maximum variation in the position of the principal planes in different zoom configurations of all lens group is kept within 6.6 mm, and the total length differences caused by this reason is not serious, as shown in Fig. 7(b). In this design, the maximum difference is 6.6 mm. For a double-sided telecentric optical system, we should maintain the symmetry of the system as much as possible, which is helpful to correct the aberrations. So we adopt a mirror symmetry design method when designing each module, which means the initial structures of GROUP2 and GROUP3 are mirror symmetry, and the initial structures of GROUP1 and GROUP4 are mirror symmetry, as shown in Fig. 7(a).
Now we can combine these four groups into an overall synthesis initial structure based on Eqs. (8) to (10). Figure 7(b) shows the synthesis system, and we can find that this initial structure has a paraxial working F/# of 10 and a half-image plane of 3 mm, and this system has an excellent performance in terms of magnification and object-side telecentricity and image-side telecentricity, which means we have got a good initial structure. After further optimizing in aberrations correction, the distance differences correction, and the other optical specifications, such as the object distance, and the image plane size, this optical design can be completed.
4. Design results of the double-sided telecentric zoom AL system
Finally, we show the design result of the double-sided telecentric zoom AL system using CODEV, as shown in Fig. 8. The paraxial working F/# is 4.0, and the total length of the system is 195 mm, which has an increase of about 45 mm compared to the initial synthesis system. The reference application background of this system is the field of precision machine vision. Therefore, it is necessary to eliminate aberrations as much as possible and improve the imaging quality. Aspheric surfaces have greater advantages than ordinary spherical surfaces in correcting spherical aberration, coma, and astigmatism [36–38]. In addition, one aspheric surface can replace multiple spherical surfaces to reduce the length of the system. What’s more, the aspheric surfaces have more optimization variables that can be released, which makes the optical system have higher optimization potential and can obtain better optical performance. Based on the above considerations, We added two aspheric surfaces in GROUP2 and GROUP3, respectively, and we gradually released their variables to the 8th-order parameters. We also added three ordinary lenses to help correct the residual aberrations in GROUP4. In Fig. 8, the positions of these aspheric surfaces are indicated by blue surfaces. And the AL of each group is marked in the diagram with a red surface, similarly. The size of the image plane is 8 mm.
The reference CCD image element size chosen for this system is 4.65 µm, so the Nyquist frequency at 108lp/mm needs to be examined according to $Nyquist = \frac{{1000lp}}{{2 \times pixels/mm}}$. It can be seen that the MTFs of all zoom configurations are above 0.3 for each field of view, as shown in Fig. 9.
And we also offered the spot diagrams of all zoom configurations, as shown in Fig. 10. And we can find that the RMS is controlled well during zooming, which is enough to be compatible with the CCD sensor according to our experience.
For this double-sided telecentric zoom AL system, we need to examine both the object-side telecentricity and image-side telecentricity, as shown in Table 3 and Table 4. And the object-side telecentricity and the image-side telecentricity of the central wavelength (587.5618 nm) are both controlled to within 0.10° for all zoom configurations. The system has excellent telecentricity control and is fully capable of meeting the requirements of practical engineering applications.
Table 3. The object-side telecentricity/° of the double-sided telecentric zoom AL system of the central wavelength.
Table 4. The image-side telecentricity/° of the double-sided telecentric zoom AL system of the central wavelength.
Figure 11(a) shows the distortion control of the system, which is within 0.5% in all zoom configurations for all wavelengths.
Finally, we need to evaluate the STO movement of the system, as shown in Fig. 11(b), and it can be found that the STO movement pattern of the system is also monotonically linear, which is corresponding to the initial structure parameters and is conducive to the practical application requirements. The specific values of the AL focal lengths and STO position for each zoom configuration are given in Table 5. Combined with the layout Fig. 8, we can find that the variation of the focal length of each AL is kept within a reasonable range. In addition, the double-sided telecentric optical system has a high requirement for the correction of stray light. Therefore, when manufacturing this system, the stray light can be eliminated by applying matting coating on the optical non-working surfaces and the mechanical structure surfaces, designing extinction threads on the surface of the mechanical structure, and strictly controlling the aperture of the lens [39]. The content mentioned above is important for the system to meet the needs of practical applications.
Table 5. The STO positions/mm and the focal length/mm of each AL.
In summary, the final design results have an excellent performance in object-side/image-side telecentricity control, distortion control, and aberration correction.
5. Conclusion
In this paper, the adaptive liquid lens and double-sided telecentric zoom system are combined, and we use four adaptive liquid lenses and one movable STO to realize zooming, which could greatly reduce the bad influence produced by the mechanical vibration during zooming. This system has stronger adaptability in industrial monitoring than the conventional double-sided telecentric system.
Firstly, we improved the Gaussian brackets according to the characteristics of the double-sided telecentric system, and we used the appropriate screening method to select the suitable system parameters from the solution space solved by MATLAB, and we put it with the other filtered solutions together for comparison to prove the effectiveness of our screening method. Then we use the lens module design method to establish the initial structure, and then expand the module combination method to apply in the finite conjugate system. After building the initial structure, we use CODEV to get a final design result, which controlled the telecentricity of both sides in the central wavelength within 0.1°, the distortion was controlled within 0.5% for all wavelengths, and the MTF of each zoom configuration is above 0.3. What’s more, the moving mode of the STO is basically linear, which is beneficial in decreasing the complexity of the system.
The double-sided telecentric zoom system using adaptive liquid lenses proposed in this paper has a high industrial potential and application value, the proposed design method is simple and efficient in solving the current problem that it is hard to find a reasonable initial structure in this field. This study will promote the further development of this field.
Funding
The Instrument Development of Chinese Academy of Sciences (No. YJKYYQ20200060); Sichuan Province Science and Technology Support Program (No. 2021JDRC0084, No. 2022103, No. 2022YFG0249); Chengyu Reginal Center project of large Instrument (No.E2R3010010); Cutting-edge Distribution Program of Institute of Optics and Electronics Chinese Academy Sciences (No. C21K003).
Disclosures
The authors declare no conflicts of interest.
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.
Supplemental document
See Supplement 1 for supporting content.
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