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Abstract
We present a physics-assisted artificial neural network (PhyANN) scheme to efficiently achieve focus shaping of high numerical aperture lens using a diffractive optical element (DOE) divided into a series of annular regions with fixed widths. Unlike the conventional ANN, the PhyANN does not require the training using labeled data, and instead output the transmission coefficients of each annular region of the DOE by fitting weights of networks to minimize the delicately designed loss function in term of focus profiles. Several focus shapes including sub-diffraction spot, flattop spot, optical needle, and multi-focus region are successfully obtained. For instance, we achieve an optical needle with 10λ depth of focus, 0.41λ lateral resolution beyond diffraction limit and high flatness of almost the same intensity distribution. Compared to typical particle swarm optimization algorithm, the PhyANN has an advantage in DOE design that generates three-dimensional focus profile. Further, the hyperparameters of the proposed PhyANN scheme are also discussed. It is expected that the obtained results benefit various applications including super-resolution imaging, optical trapping, optical lithography and so on.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Focus shaping of high numerical aperture (NA) lens has drawn considerable attention in many attractive applications, for example, super-resolution imaging [1,2], optical lithography [3], optical tweezers [4,5], optical magnetic recording [6] and so on. By virtue of vector diffraction theory (VDT) [7,8], the key problem to realize a specific focus shape is how to engineer the wavefront of incident beam at pupil plane of the optical lens that can yield the given targeted intensity profile. To this end, various approaches have been proposed to achieve full control over amplitude, phase and/or polarization of the light [9–26]. Among them, the forward design of focus shape requires physical perception, sweeping of parameters or the prior knowledge [9,10]. Nevertheless, the generated intensity profile is not always satisfactory or applicable to some special case only [11]. As a general method, the inverse design provides a better solution according to the desired intensity profile. Iterative algorithms are widely used in the inversion, such as the Gerchberg-Saxton (GS) algorithm [12–15] and evolutionary algorithm [16,17]. However, in the implementation of GS algorithm, one need to preset the total targeted intensity profile and pad zeros during FFT if one need to see the delicate structure in focal plane [13]. Evolutionary algorithm such as particle swarm optimization algorithm (PSO) may be trapped in the local optimal point, especially in the high dimensional variables problem [27]. As a result, several non-iteration inversion methods have been proposed. Chen et al. demonstrate an inverse design for the complete shaping of the focal field, including amplitude, phase and polarization [19]. Further, Zhang et al. extend this non-iterative method to be suitable for high NA systems and further engineered uniform-intensity focal fields [20]. However, to obtain perfect polarization vortices, the authors require a vector beam generator that guarantees the realization of the targeted complete shaping of focal field. Based on the VDT, an analytical procedure for the inversion of electrical and magnetic field at the focus to obtain incident light beam distribution has been presented, but focal field is limited to the on-axis position [21]. In addition, other non-iterative inverse designed methods are also developed, which include reversing of the radiation field of dipole arrays [22,23] or a uniform line source [24], Euler transformation [25] and cosine synthesized filter [26].
Recently, deep learning has shown great potential in design of photonic structures [28], computational imaging [29], biomedical imaging [30], holography [31] and inverse scattering problems [32]. Moreover, a method of inverse designing optical needles with central zero-intensity points by ANNs has been developed in [33]. In the data-driven frameworks, the trained deep neural networks (DNNs) give state-of-the-art performance in solving nonlinear inverse problems. However, the training data with labels is crucial to the well performance of such training-based DNNs. It is thus impossible to acquire the desired solutions due to mismatch between the test data set and the training data set [28–33]. To overcome this limitation, a scheme using untrained neural network (UNN) [34] has been demonstrated in quantitative phase microscopy [35] and phase imaging [36]. The significant advantage of this approach is that it does not need the training and iteratively optimizes weights of the neural network that generates the targeted phase profile as the input of the physical model.
In this paper, inspired by the UNN [34–36], we propose an iterative scheme named as PhyANN which combines a physical model with conventional ANN to flexibly and efficiently shape the focus of high NA lens. In the physical model based on VDT, the focus intensity can be expressed by weighted sums of focal field contribution from a series of fixed annular region of DOE. In PhyANN, the used ANN is a generator transforming arbitrary inputs, maybe a constant, to transmission coefficients of the DOE, from which, we use the physical model to generate focus intensity profiles. The defined loss function with respect to the focus intensity is then employed to optimize weights and biases via gradient descent, eventually resulting in a desired solution that satisfies targeted focus profile.
The main merits of the proposed PhyANN are summarized as follows. First, compared to the iterative algorithm based on gradient descent, which is only suitable for derivable targeted function and could be trapped into bad local minima [37], the PhyANN has higher possibility to arrive the global minima due to the use of ANN [38–40]. Secondly, the input of ANN in the PhyANN is an arbitrary constant rather than the focus intensity profile, which is different from that in Ref. [35] and [36]. The focus profile can be explicitly implemented by incorporating corresponding restriction term in the loss function, such as sidelobe and peak intensity. In this regard, the sub-diffraction spot with 0.41λ resolution, flattop spot with tunable radius, optical needles with 0.41λ resolution and high flatness, and multi-focus region with 0.45λ resolution are easily obtained. Thirdly, in contrast to direct deep learning method that directly regresses targeted parameters from focus intensity profiles [33], the PhyANN learns how to modulate the contribution of different annular zone of the DOE rather than how to approximately describe the well-known VDT.
This paper is organized as follows. In Section 2, the physical model based on VDT and the optimization problem are introduced. In Section 3, the pipeline of the proposed PhyANN is explicitly presented. In Section 4, the focus shaping examples including sub-diffraction spot, flattop spot, optical needle and multi-focus region are obtained using the PhyANN and the comparison with PSO algorithm is also provided. In Section 5, the hyperparameters of the PhyANN scheme are analyzed and discussed. Finally, conclusions are drawn in Section 6.
2. Physical model of the focal field generation
The schematic diagram of focus shaping is shown in Fig. 1. The radially polarized beam with uniform distribution is chosen as the incidence, which passes through a DOE located at the pupil plane and then focused by an aplanatic lens of high NA. Based on VDT [7,8], the radial and longitudinal electric field components at cylindrical coordinate (r, z) around the focus can be expressed by
Fig. 1. Schematic illustration of the focusing geometry of high NA lens. (a) The radially polarized beam is modulated by a DOE and then focused by high NA lens. The inset is zoom in focus intensity profile. (b) The DOE with transmission coefficients, Tn, of each annular zone.
Here, we set the DOE to be circular symmetry and θmax is divided into N equal parts, which corresponds to annular regions of the DOE in Fig. 1(b). The boundaries of the various region are given by θn, n = 1,2,…N, and thus the transmission function T(θ) is given by
Fig. 2. The focal field distribution when the NA = 0.95 and the total number of annular zones is N = 25. All the amplitudes are normalized by their maximums. (a) Radial component Er and (b) Longitudinal component Ez of electrical field at focal plane (z = 0) with respect to annular zone number n and radial coordinate r. (c) Radial component Er of electric field along longitudinal direction is zero due to the zero contribution from annular zones. (d) Amplitude and (e) phase of electric field component Ez contributed by each annular zone along longitudinal direction.
Based on the different contribution of each annular zone to the focal field, the purpose of focus shaping is to acquire the proper transmission coefficient T1∼TN of each annular zone of the DOE, which can modulate the amplitude and phase of the incident beam to fulfill the targeted intensity I(r,z) in Eq.(4). Such process is cast into an inverse problem, which is to estimate the unknown transmission coefficient T1∼TN by minimizing the discrepancy between the computed and targeted intensity distribution:
where f (Tn) is a loss function, we need to define. In fact, some other optimization targets should be also considered, such as side lobe and Strehl ratio (Central intensity of focal field), all of which are critical parameters for the application, such as superresolution imaging and optical trapping. Therefore, we define the loss function asIn the loss function, ‖ · ‖2 represents Euclidean distance and thus the first term is a fidelity term measuring the closeness of an estimated intensity Ig to the targeted intensity. The second term restricts relative intensity of the maximum side lode to ISL. The third term uses Strehl Ratio (SR) as the loss function, which can maximize the intensity of the focal point, I(r = 0, z = 0). To be more specific, SR is defined as the ratio of the central intensity of the focal field with DOE to that of the focal field without DOE
Usually the value of α and β (α>0, β > 0) is chosen empirically to balance the contribution of corresponding term to the total loss function. One typical method is to solve the minimization problem in Eq.(5) using direction optimization algorithm, for example, gradient decent method. In this work, a PhyANN scheme that combines the physical model with ANN is proposed in the following. We make our implementation of PhyANN as open software in Python, which can be found at https://github.com/shepherd-cc/FocusShaping.
3. ANN-based optimization method – PhyANN
As is known to all, a typical ANN method is to optimize its weight and bias parameters using a sufficiently large set of training data so that the ANN can represent a universal function that maps the transmission coefficient Tn to the targeted intensity distribution It, but it suffers from the limited generalization ability due to the mismatch between the test data set and the training data set [33]. Such learning-based ANN method is usually considered as a black box and thus tends to be more obscure [32]. Instead, a strategy to incorporate the knowledge of underlying physics as well as insight originated from objective function approaches into the ANN structure are the desired solution.
In this work, the pipeline of PhyANN is proposed in Fig. 3 by combining the resulting physical model of Eq.(4) with an ANN. Accordingly, the retrieval of transmission coefficients in Eq.(5) is converted to the retrieval of weights of ANN:
Compared to the conventional ANN, PhyANN does not need training data set. Instead, it uses the physical model of Eq.(4) to generate the intensity distribution Ig from the transmission coefficients Tn, and then employs the defined loss function of Eq.(6) to optimize the weights and biases via gradient descent. This forces the generated intensity Ig to converge to the focus profile with designated parameters as the iterative process proceeds.
Fig. 3. The architecture of PhyANN scheme. An arbitrary constant (a unity constant here) rather than focus intensity profile is the input to the neural network. The output of the neural network is the transmission coefficients of DOE, which modulates the incident beam of the physical model to generate the focus intensity Ig. The loss function is defined as summation of the mean square error between normalized Ig and targeted intensity It and other restriction term, which is employed to iteratively optimize the weights and biases via gradient descent.
The ANN used here is a generator transforming arbitrary input to transmission coefficients, Tn. Without loss of generality, a constant of c = 1 is adopted in this work. Specifically, the ANN consists of an input layer with single neural node followed by rectified linear unit (ReLU), three hidden layers with 300 neural nodes followed by ReLU in each hidden layer and an output layer with N neural nodes followed by activation function tanh. The number of weights can be easily tuned by changing the number of layers and the neural nodes of each layer. The weights of ANN will be renewed with error backpropagation and gradient descent according to Eq.(8). Once the optimal weights W are obtained when the iteration is terminated according to the targeted loss function, the output Tn of ANN are the optimal transmission coefficients of annular regions.
In the simulation, a normal personal computer with the configuration of CPU: Intel Core i5-9400F 3.20G and RAM:16G is used for all the computation. We use TensorFlow 2.3 and Python 3.7.7 to construct such a pipeline of PhyANN. The optimizer we used is adaptive moment estimation (Adam) [41], a stochastic gradient descent method that is based on adaptive estimation of first-order and second-order moments. We use 0.001 as the learning rate, which is a good default setting for the tested machine learning problem [41].
4. Result
In the physical model, the radially polarized beam at operating wavelength 532 nm is passing through the DOE and focused by an aplanatic lens of NA = 0.95. Six DOEs that generate four different kinds of focus intensity profiles, including sub-diffraction spot, flattop spot, optical needle, and multi-focus region are inversely designed using the proposed PhyANN scheme. For comparison, typical PSO algorithm results are also provided [42]. It should be noted that the convergence speed of two approaches is dependent on their hyperparameters: for example, the number of hidden layers and node number of each layer for the PhyANN and particle population for PSO.
For PSO algorithm, the particle population indeed has an important impact on the performance when other characteristic parameters are the default values in [42]. More specifically, the optimization performance can be improved by increasing the particle population. However, the bigger particle population PSO algorithm chooses, the more time it takes per iteration. To fairly compare the proposed PhyANN with PSO algorithm, we choose the proper number of the hidden layer (three layers) and node number of each layer (300 nodes for each layer) for PhyANN, and obtain the DOE design results. Then, we delicately modify the particle population of PSO algorithm to ensure that the iterations and computation time are almost same as that in the proposed PhyANN. Finally, we compare the loss function value and the SR of the focus profiles. The optimization parameters and the comparisons are provided in Table 1. It is found that for DOE designs that generate two-dimensional focus profiles (subdiffraction spots and flattop spots), the loss functions and the optimized SR are comparable for both methods. For DOE designs that generate three-dimensional focus profiles (optical needles and multi-focus region), the PhyANN provides smaller loss function value and better SR than PSO algorithm (Table 1).
Table 1. The characteristic parameters of the focus shaping examples using PhyANN and PSO algorithm
4.1 Inverse design of sub-diffraction spots
Generally, the resolution of optical microscopy is limited to 0.5λ/NA due to diffraction limit, which is described as the full width at half maximum (FWHM) of the focal spot. The sub-diffraction spot is of great importance to the resolution enhancement of optical microscopy. To achieve the sub-diffraction spot using the proposed PhyANN scheme, two targeted intensity points at r = [0, FWHM/2] along radial direction with corresponding normalized intensity It = [1, 0.5] is chosen in the loss function.
For the first case, the targeted FWHM of sub-diffraction spot is set as 0.41λ without sidelobe constraint and the optimized transmission coefficients of Tn are obtained as shown in Fig. 4(a). The generated intensity distribution in the focal plane (calculated from the generated Tn) is presented in Fig. 4(b). To be clear, the cross section along radial direction is also provided in Fig. 4(c). It is found that two targeted intensity points (red dots in Fig. 4(c)) exactly fall on the line, proving the effectiveness and accurateness of the proposed approach. But we should note the rising sidelobe due to the narrowing of main spot.
Fig. 4. Inverse design results of sub-diffraction focal spot for targeted FWHM = 0.41λ. The first row (a-c) is presented without sidelobe restriction (α = 0) while the second row (d-f) with sidelobe loss function (α = 0.5). (a) and (d) are the resulting transmission coefficient distributions of DOE, corresponding to which, the normalized intensity distributions of focal plane are plotted in (b) and (e). Their cross section (blue lines) along radial direction are shown in (c) and (f), respectively, where the target intensity points (the red dots) exactly fall on the lines.
To alleviate the rising sidelobes, we include the sidelobe constraint in the loss function by setting α = 0.5 and ISL = 0.2. As expected, the relative intensity of maximum sidelobe has been restricted to 0.2 at the expense of reduce of peak intensity of focal field, which is reflected in Fig. 4(e) and Fig. 4(f). Consequently, the transmission coefficients Tn of annular regions in Fig. 4(d) have a complex combination to fulfill the restriction of the sidelobe, compared to the transmission coefficients in Fig. 4(a).
4.2 Inverse design of flattop spot
The flattop spot is attractive to many applications due to it relatively well-defined size, shape and uniform intensity distribution, which finds application in material processing, lithography and so on. Using the PhyANN scheme, we can easily achieve flattop spots with high flatness and tunable radius by selecting targeted points with uniform intensity. As examples, two flattop spots with their flattop radii of 0.17λ and 0.55λ are shown in Fig. 5(a)- Fig. 5(c) and Fig. 5(d)- Fig. 5(f), respectively. The transmission coefficients of Tn in Fig. 5(a) and Fig. 5(d) are [0.79, 0.97, 0.97, 0.94, 0.97, 1, 0.98, 0.98, 1, 1, 1, 1, 1, 1, 1, 1, 0.98, 1, 0.97, 1, 1, 0.53, −1, −1, −1] and [0.62, 0.96, 1, 1, 0.96, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0.76, −0.3, −0.98, −1, −0.98, −0.8, 0.57, 1, 1], respectively.
Fig. 5. Inverse design results of flattop spots for targeted radii: (a-c) 0.17λ and (d-f) 0.55λ. (a) and (d) are the resulting transmission coefficient distributions of the DOE, corresponding to which, the normalized intensity distributions of focal plane are plotted in (b) and (e). Their cross section (blue lines) along radial direction are shown in (c) and (f), respectively, where the target intensity points (the red dots) exactly fall on the lines.
4.3 Inverse design of sub-diffraction optical needle and multi-focus regions
In the previous two sub-sections, the focal field is shaped only in the focal plane (z = 0). However, focus shaping in three-dimensional (3D) volume is imperative for 3D microscopy imaging, optical lithography and Raman spectroscopy. Here, we demonstrate the proposed PhyANN scheme capable of shaping the focal spot in 3D volume by achieving sub-diffraction optical needles and multi-focus regions. More specifically, a data set of targeted points and the corresponding intensity are properly chosen according to characteristic parameters such as the length of optical needle, FWHM at a certain plane and so on.
As examples, an optical needle with 10λ depth of focus and a sub-diffraction resolution of 0.41λ and a multi-focus region with a sub-diffraction resolution of 0.45λ are introduced using the PhyANN scheme. The data sets of targeted points and the transmission coefficients of the designed DOE are shown in Table 2. The design results are plotted in Fig. 6(a)-Fig. 6(f). One can see that the resulting transmission coefficients of optical needle and multi-focus region in Fig. 6(a) and Fig. 6(d) have much more changes than that of focus shaping (Fig. 4 and Fig. 5) in the focal plane only, which attributes to the complexity of the 3D focal field shaping. In addition, it should be emphasized that the optical needle (Fig. 6(b)) designed by the PhyANN has quite high flatness of almost same intensity value, which is much better than that in [15,33]. This is because the generated intensity at the targeted position along z-axis is exactly same as the targeted unity intensity as shown in Fig. 6(c). Similarly, all of focus in the multi-focus region have almost same profile as shown in Fig. 6(e) and Fig. 6(f).
Fig. 6. Inverse design results of (a-c) optical needle with targeted needle length of 10λ and (d-f) multi-focus region. (a) and (d) are the resulting transmission coefficient distributions of the DOE, corresponding to which, the normalized intensity distributions at the meridian plane are plotted in (b) and (e). Their cross sections (blue lines) along radial direction are shown in (c) and (f), respectively, where the red dots represent the target intensity points.
Table 2. The data set of targeted points and transmission coefficients Tn for focus shaping
5. Discussion
In the previous section, the capability and efficiency of the proposed PhyANN approach have been demonstrated by designing different DOEs that generate subdiffraction spot, flattop spot, optical needle and multi-focus region, respectively. Compared to the typical PSO algorithm, the PhyANN has advantage in three-dimensional focus profile designs. In this section, we focus on the PhyANN and discuss the effect of various hyperparameters on the optimization process. For the learning rate, we use the default setting of 0.001 in the optimizer of Adam [41]. In the following, we discuss α and β, the number of hidden layers and number of nodes per layer in details.
The hyperparameters, α and β, play an important role in the optimization. By adjusting α, a compromise is achieved to suppress the sidelobes and satisfy the targeted characteristic parameters of the focus profile. The appropriate compromise highly depends on the choice of α. If α is too large, the generated focus profile does not fit the targeted characteristic parameters, while if α is too small, the sidelobe cannot be suppressed as expected. The role of β is similar to that of α, but a compromise is achieved to increase the SR and satisfy the targeted characteristic parameters of focus profile. In this regard, the strategy to choose α and β is as follows. By setting α = 0, we first initialize β as a large value (normally β = 1), and then decrease it until the generated focus profile fits the targeted characteristic parameters. This value is the right value of β we need. When β is determined, the value of α can be obtained according to the same strategy.
As an example, the optimization results of sub-diffraction focal spot with α = 0 is shown in Fig. 7(a) when different values of β are used. Among them, β = 1×10−3 is the value determined by the proposed strategy. It is found that the targeted intensity points match well with the focus intensity profile and the SR is maximized to 0.34 as shown in Table 1. The bigger value of β (blue line β = 1×10−1) leads to the unfit of the targeted points with the focus profile. The case for the smaller value (yellow line β = 1×10−5) has a smaller SR of 0.26, although the targeted intensity points match well with the focus intensity profile. For the subdiffraction spot design with sidelobe constraint, the best value of β = 1×10−3 is adopted. Figure 7(b) shows the different optimization results when α=0.01, α=0.5 and α=1. Similar phenomena can be found and α = 0.5 is determined by the proposed strategy. Therefore, the proposed strategy of how to choose the hyperparameters, α and β, is effective and important. Some other methods to determine such regularization parameters in a nonlinear optimization problem can be also found in [43,44].
Fig. 7. The performance of the PhyANN with different α and β, hidden layers or different nodes per layer. (a) The optimization results of subdiffraction spot (Fig. 4(a) to Fig. 4(c)) with different β. (b) The optimization results of subdiffraction spot with sidelobe constraint (Fig. 4(d) to Fig. 4(f)) with different α when β = 1e-3. The loss curve of (c) subdiffraction spot (Fig. 4(a) to Fig. 4(c)) and (d) optical needle (Fig. 6(a) to Fig. 6(c)) with different number of hidden layers when the number of nodes per layer is 300. The loss curve of (e) subdiffraction spot (Fig. 4(a) to Fig. 4(c)) and (f) optical needle (Fig. 6(a) to Fig. 6(c)) with different number of nodes per layer when the number of hidden layers is 3.
Regarding the hyperparameters of the number of hidden layers and number of nodes per layer, we take the sub-diffraction spot (Fig. 4(a-c)) and optical needle (Fig. 6(a-c)) for examples to evaluate the performance of the PhyANN. The hyperparameters, α and β, are the best value determined by the proposed strategy. To avoid the loss of generality, we train 10 times for every situation and calculate the average value. When the number of nodes per layer is set as 300, the base-10 logarithm of loss function of PhyANN with different hidden layers is shown in Fig. 7 (c)-Fig. 7(d). As seen from Fig. 7 (c) and Fig. 7 (d), when the iteration process proceeds, the loss function for both cases promptly drop to a much smaller value and then gradually decrease to a stabilized value, which is attributed to the presence of the third term of the loss function, SR.
Further, with the increase of the numbers of hidden layers, the optimization process converges faster and has a smaller loss fucntion value. When the number of hidden layers is set as 3, the base-10 logarithm of loss function of PhyANN with different number of nodes per layer is shown in Fig. 7(e)-Fig. 7(f). It is also found that the optimization process converges fast at beginning and then go to a steady state. However, there is not much improvement when we increase numbers of nodes per layer. To sum up, more hidden layers or nodes per layers indicate better performance in the optimization. However, in practical, we should note that the more layers or nodes means the increasing computation cost. Therefore, we should make a tradeoff between computational cost and optimization performance according to the complexity of the problem we need to resolve.
6. Conclusion
In this work, we present an ANN-based iterative scheme, named as PhyANN, for DOE design to shape focus of a high NA objective lens. In contrast to the conventional ANN that requires a training process, the PhyANN reconstructs the DOE by combining a conventional ANN with a physical model of VDT that generates focus intensity profiles from the transmission function of the DOE. The defined loss function with respect to the focus intensity profile is employed to optimize weights and biases via gradient descent, eventually resulting in a desired solution that satisfies targeted focus profiles.
By delicately designing the loss function and then feedback to the ANNs, we achieve a sub-diffraction focal spot of FWHM = 0.41λ with the intensity of maximum sidelobe less than one fifth of the peak intensity. Flattop spots with different size are easily obtained by increasing the target points with unit intensity. Meanwhile, the proposed scheme shows the high freedom to shape the focus in 3D volume by achieving a sub-diffraction optical needle with a 10λ depth of focus and 0.41λ beyond diffraction limit resolution. A multi-focus region along longitudinal axis with lateral resolution of 0.45λ is also obtained. It should be mentioned that we can efficiently achieve the focus profile with the targeted characteristic parameters from second to tens of seconds. This is attributed to the architecture of the proposed PhyANN that incorporates knowledge of the underlying physics and insight originated from the designed loss function.
In the physical model based on VDT, the amplitude and phase hybrid modulation DOE is adopted and is divided into a series of annular regions with fixed widths, which allows us to optimize the transmission coefficients instead of the annular widths that is normally used in bibliography [11–17]. As a result, the vector diffraction integral of Eq. (3) is computed for one time only and then can be saved as a data base. Further, the amplitude and phase hybrid modulation DOE means the extension of the degree of the freedom for generation of focus profiles. However, we should also note the implementation of the amplitude and phase hybrid modulation DOE is a little bit complicated than common phase-only case. Fortunately, some amplitude and phase hybrid modulation DOEs have been developed [45] and the emerging metasurface devices can also provide both amplitude and phase modulation of light beam in subwavelength scale using the prescribed nanostructures [16,46]. Therefore, it will not be a problem for the experimental implementation.
In addition, we have compared the proposed PhyANN with the typical PSO Algorithm. It is found that for DOE designs that generate the two-dimensional focus profiles (subdiffraction spots and flattop spots), the loss functions and the optimized SR are comparable for both methods when the almost the same computation time and iterations are set in the optimization process. For DOE designs that generate three-dimensional focus profiles (optical needle and multi-focus region), the PhyANN provides slightly better loss function value and better SR than PSO algorithm. It can be concluded that the proposed PhyANN can offer an alternative method to effectively design the annular pupil filter that generates the targeted focus profile. Such study provides not only an interesting opportunity for many applications such as super-resolution microscopy, optical tweezers, optical lithography and optical storage, but also a promising direction that combining underlying physics with ANN to solve nonlinear inverse problems.
Funding
National Natural Science Foundation of China (61905291, 61805288); Guangdong Basic and Applied Basic Research Foundation (2020A1515010626); The Open Project Program of Wuhan National Laboratory for Optoelectronics (2019WNLOKF020); Fundamental Research Funds for the Central Universities (19lgpy271).
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.
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