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Abstract
An efficient optimization strategy for liquid lens combining the uniform design and the deep learning is proposed to achieve improved dynamic optical performance and lowering driving force simultaneously. The membrane of the liquid lens is designed into a plano-convex cross-section, in which the contour function of the convex surface as well as the central membrane thickness is especially optimized. The uniform design method is initially utilized to select a part of uniformly distributed and representative parameter combinations from all possible parameter range, and their performance data is then obtained through simulation using MATLAB to control COMSOL and ZEMAX. After that, a deep learning framework is employed to build a four-layer neural network with its input and output layer representing the parameter combinations and the performance data, respectively. After 5 × 103 epochs, the deep neural network has undergone sufficient training, demonstrating effective performance prediction capability for all parameter combinations. Finally, a “globally” optimized design can be obtained by setting appropriate evaluation criteria which take the spherical aberration, the coma and the driving force into consideration. Compared with the conventional design using uniform membrane thickness of 100 µm and 150 µm as well as the previously reported “locally” optimized design, distinct improvements in the spherical and the coma aberrations across the entire focal length tuning range have been achieved, whilst the required driving force is largely reduced. In addition, the “globally” optimized design exhibits the best modulation transfer function (MTF) curves and provides the best image quality.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The liquid lens has garnered significant attention due to its ability to achieve optical zoom with only a single lens, and has been widely used in fields such as mobile phone cameras [1], continuous optical zoom microscopy [2], endoscopy [3], projector [4], holographic near-eye display [5], immersive head-mounted display [6], optical sectioning tomography [7], optical tactile sensor [8], depth sensor [9] and autofocus system [10,11], etc. With the widespread application of the liquid lenses, besides the focal length tuning capability, the requirements on their dynamic optical performance during the whole focal length tuning range are becoming more and more stringent. Different approaches have been proposed to improve performance. Among them, a strategy of introducing a non-uniform membrane with aspheric surface contour into the lens body has been widely studied due to its prominent dynamic aberration correction capability [12–17]. As for this case, a universal yet effective design method for the membrane structure is very important and much more desired. Recently, we developed a liquid lens design flow for dynamically correcting the spherical aberration over a user-defined focal length range [18]. Although good performance has been successfully achieved, there is still quite room for further performance improvement since the aspherical membrane structure is only optimized from limited parameter space in the design. However, finding a globally optimized design for the membrane structure remains a challenge due to the multivariate optimization problem caused by the various parameters including the surface contour function and the central membrane thickness that required to define the membrane structure. As a result, it is necessary to explore a suitable way to predict the performance data from all possible conditions efficiently. Besides, the requirement for driving force during liquid lens operation should also be seriously considered from an application perspective, particularly for the liquid lens with an integrated actuator configuration. For example, as for the most widely used voice coil type electromagnetic actuator, its driving force is dependent on the actuation current, which will in turn determine the power consumption as well as the cooling requirement.
The uniform design (UD) seeks design points that are uniformly scattered in the design domain [19], it has been widely adopted to reduce the number of the experiment by selecting a part of the representative and uniformly distributed combinations from the huge parameter database [20,21] In terms of the deep learning (DL), it can be used to reveal the key relationship between the original input and the final output from a given dataset by learning the inherent laws of the sample data, so as to provide accurate performance prediction. This distinct property of the DL has brought revolutionary developments in numerous applications, such as near-infrared (NIR) calibration [22], wind speed prediction [23], far-field radiation prediction [24], forward prediction, and inverse design for 3D chiral plasmonic metasurfaces [25], prediction of the modal frequency and the quality factor of the disk-shaped microscale resonators [26], and atmospheric visibility prediction [27], etc. Inspired by these characteristics, a method that combines UD and DL might provide a potential solution to address the above-mentioned structure optimization issue for the liquid lens.
In this paper, an efficient liquid lens structure optimization strategy based on uniform design and deep learning is proposed, targeting to improve its dynamic optical performance further whilst lowering the driving force requirement. Similar to the previous design, a membrane with aspherical plano-convex cross-section is adopted. To optimize its structure, the uniform design method is first adopted to select limited representative combinations from a certain range of structural parameters. With the aid of the MATLAB, the dynamic optical performance as well as the corresponding driving force of the liquid lenses that are composed of the membrane defined by the selected parameter combinations can be efficiently obtained through the mechanical and the optical analysis via the COMSOL and the ZEMAX, respectively. Moreover, a deep learning model with a four-layer neural network is especially constructed to reveal the intrinsic relation between the membrane structural parameters and the liquid lens performance. After iterative training with 105 epochs, the mean square error (MSE) will converge to below 8.0 × 10−6, validating its effective performance prediction capability. At last, specific evaluation criteria that take the spherical and the coma aberrations and the driving force into consideration are set for the “globally” optimized design selection. Compared with the conventional design using uniform membrane thickness of 100 µm and 150 µm, the spherical and coma aberrations over the whole focal length tuning range from ±1000 mm to ±15 mm have been distinctly decreased by 93% and 81%, respectively, but the required driving force is still quite similar. Moreover, in comparison to our previous aspherical membrane design, further decrease of the spherical and coma aberrations of about 21.6% and 39.8% can be obtained, whilst the driving force is reduced by half. In addition, the simulation results about the MTF curves and the resolution target imaging also demonstrate the improved performance.
2. Structure design
Figure 1(a) shows the schematic of the proposed liquid lens whose main body is derived from our previous work [28]. It consists of an elastic polydimethylsiloxane (PDMS) membrane, a glass backing plate and a cylindrical liquid chamber enclosed between them. The membrane is uniquely designed into a plano-convex cross-section with an aspheric surface contour for aberration correction. The convex end of the membrane is situated within the chamber, which is filled with a liquid whose refractive index is close to that of the membrane. With this configuration, the liquid lens has an initial focal length of infinity, enabling bidirectional wide-range focal length tuning capability, as shown in Fig. 1(b). Additionally, an annular ridge structure is incorporated onto the flat end of the membrane, allowing for a driving mechanism to be attached. Upon actuation, the driving force can be applied to the ridge in different directions and magnitudes, which is subsequently transferred to the liquid. Since the filling liquid is incompressible, the force causes deformation of the membrane, resulting in a positive or negative lens, as illustrated in Fig. 1(c) and (d), respectively.
It is no doubt that the optical performance of the liquid lens is crucial. In addition, considering its actual application conditions, the driving force required for zooming must also be taken into consideration so as to achieve efficient operation. For example, since the liquid lens proposed in this paper is driven by an electromagnetic actuator, its driving force is regulated by the coil current. The greater the driving force used to zoom the liquid lens, the larger the required coil current. Obviously, it will lead to increased power consumption and the coil heating due to the Ohm effect. Under the influence of which, the device temperature will increase and its optical properties might also be changed. More seriously, it will cause irreversible problems such as liquid volatilization and material ablation. Therefore, the driving force required for zooming must be considered during the design stage.
From the working principle shown in Fig. 1, the focus tuning is achieved by deforming the membrane. It’s obvious that not only the lens's optical performance but also its driving property are dominated by the membrane structure. As a result, the membrane structure optimization is the most critical task during the lens design, which includes the surface contour function of the convex end and the center thickness as shown in Fig. 2. Considering the fact that there involve four parameters (namely a, b, c and h) to define the membrane (it has been concluded in our previous research that only four parameters can be used to define the membrane well [18]), if assuming that each parameter has n values, the number of all possible parameter combinations will reach n4, that is to say even if n is only 1000, n4 will be as high as one trillion. Given such a huge amount of the membrane design, it is nearly impossible to directly analyze and extract the performance for all of them and subsequently find the “globally” optimized design. Therefore, it is much more desired to explore a reasonable method to solve this issue.
3. Membrane structure optimization
In order to find the “globally” optimized parameter combination easily and efficiently, a new optimization strategy for the liquid lenses combining the uniform design and the deep learning is proposed in this paper, which can be divided into 6 steps as shown in Fig. 3.
In the initial stage of this study, the uniform design method is employed to select a subset of n uniformly distributed and representative parameter combinations from all of the n4 possible parameter combinations of the liquid lens. Obviously, the amount of the performance data that need to be analyzed and extracted can be effectively reduced, thereby considerably shortening the time of the optimization procedure. In this paper, specific value ranges of each parameter are properly determined based on previous experience as illustrated in Table 1. To take both the data coverage and the time consumption into consideration, 1000 evenly distributed and representative parameter combinations are selected from all possible combinations using the uniform design method, ensuring that each parameter is covered only once at every value point within its designated range Table 2 shows these parameter combinations arranged according to the center membrane thickness in ascending order.
Table 1. The specific values of each parameter
Table 2. The selected parameter combinations
In the second step, the deformations of the membrane defined by the selected 1000 parameter combinations during the focal length tuning of the liquid lens are obtained. During this case, commercial finite element analysis software - COMSOL is adopted. This involved extracting deformation data from as many as 1000 liquid lens membrane models, which is a cumbersome process. To simplify this extraction process, a strategy of COMSOL controlled by MATLAB program is designed to realize automatic data input and output. The selected 1000 parameter combinations are input into COMSOL to build the corresponding liquid lens membrane model and the simulation results about the membrane deformation under different working conditions are exported into a data file.
In the third step, the obtained membrane deformation data are subsequently processed for the liquid lens optical performance analysis during the focal length tuning with the help of ZEMAX. The performance data under different parameter combinations include the optical aberration and the driving force required for full range focal length tuning. As for the driving force, its magnitude can be directly obtained from COMSOL. In comparison, the spherical aberration and the coma are extracted to represent the on-axis and off-axis optical performance of the liquid lens, respectively. In terms of the extraction of the spherical aberration, it is a bit more complicated and needed to be covered in detail. Figure 4(a) shows the schematic of the spherical aberration definition of a single lens with fixed focal length.
Fig. 4. Extraction and processing of spherical aberration: (a) schematic diagram of spherical aberration; (b) spherical aberration curve; (c) focal length-S1 curve.
The spherical aberration $\delta L_m^{\prime}$ can be calculated by:
where $L_m^{\prime}$ and $l^{\prime}$ are the image distance of the actual and the paraxial rays, respectively.It’s well-known that given a certain object point, lights with different incident heights will contribute to different sizes of spherical aberration, which can be described by a spherical aberration curve as shown in Fig. 4(b). Considering the fact that the liquid lens is a zoom device, an additional dimension of the focal length will be introduced. As a result, in order to characterize the spherical aberration performance of the liquid lens within its whole focal length tuning range (called dynamic spherical aberration), a special strategy is designed, in which the area enclosed by the spherical aberration curve and the vertical axis (green region in Fig. 4(b)) is calculated. This area value is defined as S1 and is used to represent the spherical aberration condition of the liquid lens at this corresponding focal length. Then, the respective S1 at each focal length point over the whole focal length tuning range can be calculated with the same method. Based on that, a curve revealing the change of the S1 with respect to the focal length can be obtained as shown in Fig. 4(c), and it is used for the liquid lens dynamic spherical aberration characterization. In order to quantitatively compare the dynamic spherical aberration performance of the liquid lens under different parameter combinations, the dimension of parameter combination is also introduced. Similar to the above processing method, for any parameter combination, the dynamic spherical aberration performance of the liquid lens under certain parameter combination can be represented by calculating the area enclosed by the S1 curve and the coordinate axis (blue region in Fig. 4(b)), which is defined as S2. Obviously, the smaller the S2 value, the better the dynamic spherical aberration performance. Moreover, in terms of the coma, since it varies with the field of view (FOV), for analysis simplification, the FOV of 20° is chosen and similar treatment can also be used to characterize the coma.
In the fourth step, a deep neural network is constructed with the help of the deep learning framework of TensorFlow. Figure 5 shows the structure of the deep neural network used to learn the relationship between the performance of liquid lenses and the parameter combinations. The input layer involves four neurons corresponding to the above-selected parameter combinations, namely h, a, b, and c. The output layer has three neurons to represent the performance data of the liquid lens, including the spherical aberration, the coma and the driving force. Since these parameters vary greatly in value, standardizing the parameter data is necessary to prevent the deep neural network from iterating solely in the direction of large value parameters during the training process. Z-score standardization is adopted in this paper, which can be expressed by [29]:
where ${z_i}$ is the data after normalization, ${x_i}$ is the raw data before normalization, $\mu $ and $\sigma $ are the mean value and the standard deviation of the data, respectively.After that, four hidden layers are set, and each hidden layer includes 16384 neurons. The activation functions of the first two hidden layers are set as ‘Hyperbolic tangent (tanh)’, while those of the last two hidden layers are set as ‘rectified linear unit (relu)’ instead. The loss function is chosen to be the mean-square error (MSE) loss since the training task is a regression problem. The MSE loss can be illustrated as follows [25].
where N is the number of data in the training dataset, ${y_i}$ is the actual value of the performance data obtained in the third step, and ${\bar{y}_i}$ is the predicted value of the performance data from the deep neural network.In order to mitigate the risk of overfitting during the model training process, a validation dataset consisting of 10% of the data is specifically selected, with the remaining data serving as the training dataset. From the results shown in Fig. 6, it is apparent that the MSE curves of both the training and the validation dataset significantly drop to a low level after 5 × 103 epochs, demonstrating that the deep neural network has undergone sufficient training.
Fig. 6. The MSE loss of the training dataset (red curve) and the validation dataset (blue curve) in every epoch, and the insets are local magnification.
It is not intuitive enough to judge the training degree of the deep neural network just by analyzing the MSE curve. For this reason, the comparison between the performance data obtained from the conventional analysis procedure and those predicted by the trained deep neural network for both of the training and validation dataset are shown in Fig. 7. The results indicate a high degree consistency between these two datasets, which demonstrates that the deep neural network has been effectively trained.
Fig. 7. Training effect of the deep neural network on performance data: (a) S2 of the spherical aberration; (b) S2 of the coma; (c) driving force.
However, the excellent training effect in Fig. 7 only shows that the deep neural network has been well trained for the data in the training set and the validation set. To ensure that the deep neural network can learn the complex relationship between the parameter combinations and the lens performance, it is necessary to evaluate the prediction accuracy of the deep neural network on data other than the training and the validation set as well. A random function is used in this paper to randomly generate 100 parameter combinations from all parameter combinations to form a test dataset. The actual performance data obtained using the conventional analysis procedure and corresponding ones predicted by the deep neural network for the test dataset are subsequently compared as shown in Fig. 8. It can be seen that the actual and the predicted values are also highly coincident, demonstrating that the deep neural network has been trained to learn the relationship between the performance data and the parameter combinations well so that it can be used to make accurate performance prediction.
Fig. 8. Prediction accuracy of the deep neural network on test dataset: (a) S2 of the spherical aberration; (b) S2 of the coma; (c) driving force.
In the fifth step, we utilize the well-trained deep neural network to predict the performance data of all possible parameter combinations. To achieve this, the entire set of parameter combinations are input into the trained deep neural network and their corresponding performance data can be obtained as shown in Fig. 9.
Fig. 9. Performance data under all parameter combinations predicted by the deep neural network: (a) S2 of the spherical aberration; (b) S2 of the coma; (c) driving force.
In the sixth step, appropriate evaluation criteria will be established by considering both of the dynamic optical performance, including the spherical aberration and coma, and the driving force. Since these indicators are quite different in magnitude, normalization is necessary, and their weights are as assigned in a ratio of 1:1:1 to calculate their weighted sum. After analyzing the predicted performance data under all parameter combinations, and taking their weighted sums as the goal, the parameter combination with the smallest weighted sum of indicators is chosen as the “globally” optimized parameter combination. The optimized parameter combination is presented in Table 3.
Table 3. The globally optimized parameter combination
4. Optimization result
To demonstrate the optimization effectiveness of the proposed method, three additional designs, including two conventional uniform membrane designs with different thicknesses and one non-uniform membrane design optimized with our previously reported method, are compared to the current design under the same operation statuses with respect to several key evaluation indicators.
Figure 10(a) and (b) depict the variations in the spherical aberration and the coma of different liquid lens designs during the focal length tuning from ±1000 mm to ±15 mm, respectively. In addition, the corresponding spherical aberration curves obtained at the focal length of 50 mm, 100 mm and 500 mm are especially provided in Fig. 10(c-e) for more straightforward comparison. Among them, the yellow line represents the conventional design of 100 µm uniform membrane thickness which is widely used in the liquid lenses, the green line represents the conventional design of 150 µm uniform membrane thickness which requires a comparable driving force to the current design for achieving the same tuning capability, the blue line represents the “locally” optimized design of non-uniform membrane thickness which is obtained using the optimization method in Ref. [18], and the red line represents the “globally” optimized design of non-uniform membrane thickness which is obtained using the current optimization method. In terms of the conventional design using the membrane with uniform thickness, the bottom and the top surfaces of the membrane are flat in the initial state, its structure is only characterized by the membrane thickness. The surface profile of the uniform thickness membrane after deformation is approximately a spherical profile, which will lead to distinct aberration. In contrast, the ‘locally’ optimized design is similar to the ‘globally’ optimized design, they both possess an aspheric membrane cross-section profile for the aberration correction. However, in our previous work, the aspherical membrane structure is only optimized from limited parameter space due to the time-consuming simulation procedure. As a result, the design with the best performance is more likely to be missed and the obtained design is actually treated as the ‘locally’ optimized design. In order for further performance improvement, a new optimization strategy combining the uniform design and the deep learning is developed to address the contradiction between the parameter space to be analyzed and the simulation efficiency. Therefore, distinctly enlarged parameter space can be used during the optimization procedure so as to facilitate the exploration of the ‘globally’ optimized design.
Fig. 10. Comparison of aberration curve: (a) S1 of spherical aberration; (b) coma; spherical aberration curve at (c) 50 mm, (d) 100 mm and (e) 500 mm focal length.
At the same time, for quantitative comparison, the detailed S2 values of both the spherical aberration and the coma as well as the maximum driving force required for the liquid lens operation are also listed in Table 4. The results clearly demonstrate the significant enhancement in spherical aberration and coma achieved with the “locally” and “globally” optimized designs as compared to conventional designs, owing to the non-uniform membrane strategy. Moreover, the “globally” optimized design outperforms the “locally” optimized one, with nearly half the required driving force and performance comparable to conventional designs, thereby validating the effectiveness of the current optimization method.
Table 4. Performance comparison
Moreover, the modulation transfer function (MTF) curves of different designs under the same imaging configuration are also provided. Considering the fact that the liquid lens cannot achieve the single-lens imaging in the state of negative focal length, in order to evaluate the imaging performance within the whole focal length tuning range, an additional solid lens together with the liquid lens is used to construct a zoom optical system in ZEMAX. In current case, a high-resolution microscopy objective lens (HPA50XAB, Thorlabs, USA) is selected as the solid lens to eliminate its potential impact on the imaging performance. Since the MTF curve is dependent on both the FOV and the object distance, to simplify the analysis, two fields of view (0° and 20°) are selected to demonstrate the on-axis and off-axis optical performance, respectively. Moreover, the object distances of 50 mm, 100 mm and 500 mm are chosen to demonstrate the optical performance of the lens working under short, medium and long focal lengths conditions. Figure 11 shows the simulation results obtained using ZEMAX, from which similar tendency to the above analysis can also be found.
Fig. 11. Simulation results about the MTF curves of the conventional and optimized designs working under different conditions.
Besides, its imaging performance is also studied using the ‘Image Simulation’ function in ZEMAX, in which a resolution test chart (USAF 1951) located at a distance of 500 mm is used as the target. From the results shown in Fig. 12, it can be seen that given the same imaging configuration, the currently proposed “globally” optimized design can provide the best image quality, demonstrating the effectiveness of the optimization method further.
Fig. 12. Imaging simulation results for the resolution test chart. (a) conventional design of 100 µm uniform membrane thickness; (b) conventional design of 150 µm uniform membrane thickness; (c) locally optimized design; (d) globally optimized design.yConclusions
5. Conclusions
In summary, a novel liquid lens optimization strategy combining the uniform design and the deep learning is developed for improving its dynamic optical performance and lowering the driving force simultaneously. Based on the uniform design, a representative set of parameter combinations is selected, and their performance data are obtained through a simulation process using MATLAB to control COMSOL and ZEMAX. Furthermore, a deep learning model of a four-layer neural network is built so as to explore the inherent relationship between the parameter combinations and the performance data of the liquid-filled lens quickly. After 105 epochs, the mean square error (MSE) of both the training dataset and the validation dataset dropped below 8.0 × 10−6, which means the deep neural network has been trained enough for subsequent prediction of the performance data under all possible parameter combinations. Finally, a “globally” optimized design can be obtained by setting appropriate evaluation criteria which take the spherical aberration, the coma and the driving force into consideration. Compared with the conventional design of 100 µm and 150 µm uniform membrane thickness as well as the “locally” optimized design, the S2 of the spherical aberration of the “globally” optimized design decreased by 93.25%, 93.04%, and 21.6%, respectively, and the S2 of coma decreased by 81.09%, 81.41%, and 39.8%, respectively. Furthermore, the driving force of the “globally” optimized design is close to that of the conventional designs of uniform membrane thickness, and only half of the “locally” optimized design. In addition, the “globally” optimized design exhibits the best MTF curves and image quality. All of the evaluation indicators reveal that the “globally” optimized design has exhibited the best performance which also demonstrate the effectiveness of the proposed optimization method in achieving superior performance for liquid lenses.
Funding
National Natural Science Foundation of China (12174137); National Key Research and Development Program of China (2020YFB2008800).
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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