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Abstract
Metasurfaces provide a new approach for planar optics and thus have realized multifunctional meta-devices with different multiplexing strategies, among which polarization multiplexing has received much attention due to its convenience. At present, a variety of design methods of polarization multiplexed metasurfaces have been developed based on different meta-atoms. However, as the number of polarization states increases, the response space of meta-atoms becomes more and more complex, and it is difficult for these methods to explore the limit of polarization multiplexing. Deep learning is one of the important routes to solve this problem because it can realize the effective exploration of huge data space. In this work, a design scheme for polarization multiplexed metasurfaces based on deep learning is proposed. The scheme uses a conditional variational autoencoder as an inverse network to generate structural designs and combines a forward network that can predict meta-atoms’ responses to improve the accuracy of designs. The cross-shaped structure is used to establish a complicated response space containing different polarization state combinations of incident and outgoing light. The multiplexing effects of the combinations with different numbers of polarization states are tested by utilizing the proposed scheme to design nanoprinting and holographic images. The polarization multiplexing capability limit of four channels (a nanoprinting image and three holographic images) is determined. The proposed scheme lays the foundation for exploring the limits of metasurface polarization multiplexing capability.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Metasurfaces [1,2] are two-dimensional arrangements of substantial meta-atoms that can shape the electromagnetic wavefronts into almost arbitrary profiles. At present, metasurfaces have been used in many optical fields, such as holograms [3,4], meta-lens [5,6], beam deflectors [7,8], beam shapers [9,10], etc. However, most of the above devices can only perform specific functions, which cannot satisfy the growing demand for multifunctional devices. In recent years, realization of multiple functions utilizing a single metasurface has been widely studied. In general, multiplexing approaches of multifunctional metasurfaces can be divided into polarization multiplexing [11–21], wavelength multiplexing [22–24], angle multiplexing [25–27], orbital angular momentum multiplexing [28–30], etc. Polarization multiplexing takes the illumination polarization state as a degree of freedom, and the change of the function depends on the switching of the polarization state. It is common to realize the multiplexing for two beams of orthogonal linearly polarized light (OLPL) by manipulating the anisotropy of the meta-atom [11–14]. For example, the amplitude or phase of OLPL can be independently controlled by changing the lengths of the two arms of the cross-shaped structure, and thus polarizing beam splitter [11], dual-channel nanoprintings [12] and dual-channel holograms [13] can be realized. In fact, orthogonal circularly polarized light [15] (OCPL), any orthogonal polarized light [16] or even non-orthogonal polarized light [17,18] can achieve independent modulation by elaborately designing the dimensions and orientations of the meta-atoms. In addition, by designing different structures to manipulate different functions and then superimposing them into the same plane [19] or stacking them in different layers [20], polarization multiplexing can be achieved as well. Nevertheless, this method may result in strong coupling effect of adjacent structures and low efficiency. A supercell, a complex structure of several sub-atoms, is reported to deal with the dilemma. Based on a supercell with four nanopillars, the amplitudes and phases of three polarization states are arbitrarily regulated to realize the three-channel complex-amplitude holographic images [21]. Actually, the electromagnetic responses of the meta-atom under different polarized light illumination can be very complicated, and there may be coupling effects between the responses of different polarization states, even for simple shapes. Therefore, traditional design methods usually only aim at specific polarization states, and it is difficult to explore the polarization multiplexing capability limit of the meta-atom.
Recently, deep learning [31] has been introduced into the design of metasurfaces [32–37] due to its powerful ability to mine representations of data with multiple levels of abstraction. The design of multifunctional metasurfaces needs to explore the huge response space of meta-atoms, which can be effectively probed by deep learning. Currently, assisted by different meta-atoms with high degrees of freedom, four-channel holographic display of OLPL multiplexing at two wavelengths [38], and eight-channel holographic images of OLPL multiplexing at four wavelengths [39] have been realized by deep learning. However, the selection of illumination lights is limited to OLPL, and more attention is paid to the introduction of more wavelengths to achieve multiplexing of multiple channels. So far, although deep learning has shown great potential for designing multifunctional metasurfaces, it lacks the research of the exploration for the polarization multiplexing capability limits of the used meta-atoms.
In this work, a design scheme for polarization multiplexed metasurfaces based on deep learning is proposed. This scheme demonstrates an inverse design network (IDN) consisting of an inverse network (IN) and a forward network (FN). The cross-shaped structure is taken as an example to show the whole design scheme. We firstly establish the response space containing the co-polarization and cross-polarization responses with OLPL and OCPL illumination and then screen which polarization state combinations are most likely to be multiplexed in this response space. Subsequently, IDN is used to test the multiplexing effects of these polarization state combinations. The amplitude response is used to design nanoprinting and the complex amplitude (amplitude, phase) response is used to design holography. According to the imaging effects of the designed meta-devices, the four-channel multiplexing limit of the cross-shaped structure in the given response space is determined, and the multiplexing effect of a nanoprinting image and three holographic images is exhibited. The IDN considers the non-ideal responses in the design process, which can make full use of the response space and develop the full potential of polarization multiplexing for the cross-shaped meta-atom. Besides, the multiplexing of other polarization state combinations is also attained by proposed scheme.
2. Model and methodology
2.1 Establish the response space
To illustrate the design process of the proposed scheme, the meta-structure shown in Fig. 1(a) is used in this work. It is consisted of a silicon dioxide (SiO2) substrate and an upper titanium dioxide (TiO2) cross-shaped structure, where the values of height and period are set as 600 nm and 460 nm, respectively. The meta-structure contains five degrees of freedom: the length (Lx, Ly) and width (Wx, Wy) of the two arms and the rotation angle (θ). In term of polarization states, orthogonal linear polarization states and orthogonal circular polarization states are the most common polarization states. Therefore, the OLPL and OCPL at the wavelength of 532 nm are normally incident on the meta-atom from the substrate side and the transmitted co-polarization and cross-polarization responses are collected using CST Microwave Studio, as shown in the table of Fig. 1(a). Optical responses contain amplitude and phase, so the given response space includes 16-channel responses. Although the adopted structure still has other degrees of freedom and other polarization states, our design scheme concentrates on exploring the limits of polarization multiplexing capability in a given response space. Consequently, it is sufficient to use the current response space to illustrate the design process.
Fig. 1. IDN architecture and schematic diagram of the design process. (a) FN can predict the 16-channel responses of the meta-atom with 5-dimensional structural degrees of freedom. (b) IN architecture. (c) Final design obtained by IDN.
Generally speaking, a function can be realized by modulating phase and maintaining high amplitude. However, if the amplitude can also be arbitrarily controlled, the amplitude and phase responses of one polarization state can realize different functions [40,41]. Therefore, each of the 16 channels in the response space has the potential to implement a single function. Of course, it is difficult to realize the simultaneous regulation of 16 channels by only regulating 5 structural parameters (Lx, Ly, Wx, Wy, θ) of the used cross-shaped structure. This work is to find channels that can be independently regulated in the 16-channel response space and explore the limit of multiplexed channels. Herein, we make a simple provision: X-polarized light (XPL), Y-polarized light (YPL), left circular polarized light (LCPL), right circular polarized light (RCPL), amplitude and phase are referred to as X, Y, L, R, A and P, so the phase responses of transmitted RCPL with incident LCPL can be referred to as LRP.
2.2 Evaluation of composite correlation coefficient
If following the traditional design method, multiplexing is often achieved in some specific channels. In contrast, our method is able to consider all polarization states in the response space. Firstly, linear correlation coefficients between each channel are calculated. Nevertheless, linear correlation coefficients cannot be used to directly assess the multiplexing feasibility of more than two channels. In order to estimate the difficulties of various multifunctional design tasks, the composite correlation coefficients of all multichannel combinations under different numbers of channels are calculated based on linear correlation coefficients between each channel (the calculation method is shown in Section 1 of Supplement 1). Although multiplexing cannot be confirmed by the specific value of the composite correlation coefficient, the multichannel combination with lower composite correlation coefficient is more likely to achieve polarization multiplexing. LRA + LRP + LLP and LRA + LRP + LLP + RLP are selected because they achieve the lowest composite correlation coefficient in three-channel and four-channel combinations respectively (the composite correlation coefficients of them and other multichannel combinations are shown in Section 1 of Supplement 1). Their multiplexing feasibility will be further inspected through the IDN.
2.3 Evaluation of IDN
The proposed IDN can be divided into two parts: the first part is the FN that can predict the 16-channel responses, and the second part is the IN that generates designs based on the target responses, shown in Fig. 1(a) and (b). FN is a deep neural network with 12 hidden layers, which uses mean absolute error (MAE) as the loss function. It should be noted that for improving the effect of training, the amplitude and phase responses are converted into the complex transmission coefficients, because their real and imaginary parts remain smooth and nonsingular and their value ranges are limited to [-1,1] [42]. IN is a conditional variational auto-encoder [43] (cVAE) consisting of an encoder and decoder both with five hidden layers. The encoder compresses the input structure and the corresponding responses into a latent space, and then the decoder reconstructs the structure according to the latent variable sampled from the latent space conditioned on the target responses. Different from auto-encoder, the compressed latent space is not random but obeys a standard normal distribution. Therefore, the cVAE needs to be trained by minimizing following loss function:
where s is the input structure of the encoder, ŝ is the output structure of the decoder, r is the response and z is the latent variable. The first term is the reconstruction loss, which is calculated using the binary cross entropy. The second term is the Kullback-Leibler (KL) divergence between the true distribution q(z|s,r) of the latent variable and the standard normal distribution p(z). β is introduced to balance reconstruction loss and KL divergence. In addition, the 3-dimensional latent variables compressed by the encoder are the mean values and variances of distributions, so the decoder needs to reparameterize the latent variables [43].Similarly, the training of IDN is divided into two steps. The first step is the training of FN, whose accuracy has an important impact on the inverse design. An accuracy function is introduced to more intuitively and friendly observe FN’s training process:
where error_Adata and error_Pdata are amplitude and phase errors of a meta-atom’s 16-channel responses predicted by FN, respectively. It is considered to be a correct prediction when the amplitude errors are less than 0.1 and the phase errors are less than 15°. NUMdata means the number of correct predictions and the value of NUMdata/n is calculated during training, which can reflect the increasing process of accuracy.Secondly, after the FN training is completed, the training of IN needs to be carried out according to different design tasks. The loss function of cVAE cannot directly reflect the design error of each channel, so a training evaluation mechanism is introduced in combination with the well-trained FN. In detail, 500 candidate designs are generated according to the randomly generated target response, and then evaluated by FN to obtain the design with the smallest design error as the final design. The average design error of 200 random target responses is taken as the evaluation result during training, which can be used to roughly estimate the multiplexing feasibility. After the training is completed, the decoder is the final generative network (GN) of meta-atoms. Multiple candidate designs can be obtained through multiple sampling from the latent space. The candidate designs are evaluated and selected by FN to achieve the final meta-device. The entire design process is shown in Fig. 1. Additional details of IDN are presented in Section 2 of Supplement 1.
3. Exploration of polarization multiplexing capability limit
The FN is trained based on collected 16500 groups of data, among which 15000 are used for the training dataset and 1500 for the test dataset. For reducing the amount of used data to reduce the time of collecting the data, this work adopts the test dataset both for validation during training and testing after training. This is feasible because none of the samples in the test dataset participates in the training. The training process of FN is shown in Fig. 2(a)-(b). The test accuracy gradually increases as the training progresses, and finally maintains at 96% after 30,000 epochs. The histograms of each channel’s errors between the actual responses in the test dataset and FN-predicted values and each channel’s MAE marked in pink triangles are shown in Fig. 2(c). Most amplitude prediction errors are less than 0.01, and the mean absolute prediction error of each amplitude channel is less than 0.006. Most phase prediction errors are less than 5°, and the mean absolute prediction error of each phase channel is less than 5°. A few high prediction errors are distributed among multiple predictions, which results in a decrease in accuracy. However, the current precision has already ensured that FN can make very reliable predictions.
Fig. 2. (a) The decreasing process of FN’s training and test MAE. (b) The increasing process of FN’s training and test accuracy. (c) The histograms of each channel’s errors between actual responses in the test dataset and FN-predicted responses and their corresponding coordinate axes are numbers of amplitude and phase errors (below axes). The pink triangles represent the MAE (amplitude/phase) for each channel in the test dataset and their corresponding coordinate axes are amplitude and phase MAE (upper axes).
Subsequently, the multiplexing possibility of previously selected multichannel combinations will be inspected by IDN. The response data of LRA + LRP + LLP are extracted from the dataset to train IN. With the guidance of FN, the design error of each channel can be observed, as shown in Fig. 3(b). It can be found that the final design error of the amplitude channel is below 0.01, and the final design errors of two phase channels are around 2°. The extremely small errors indicate that IN can achieve an accurate design based on the target response, which is critical to achieving eventual multiplexing.
Fig. 3. Multiplexing effect evaluation of LRA + LRP + LLP. (a) The decline process of cVAE loss. (b) The decline process of the design error for each channel evaluated by FN. (c) The decline process of the cumulative error after design errors of all channels are normalized. (d) The design process of the meta-device. The simulation results of LRA channel’s nanoprinting image (e) and LRP and LLP channels’ holographic images (f)-(g).
For further demonstrating the validity of IN, we use IN to design a meta-device that displays a nanoprinting image of the LRA channel on the meta-device surface and two holographic images of the LRP and LLP channels in the meta-device's Fraunhofer region. It should be mentioned that the design error for each channel gradually plateaus after 2000 epochs of training, but there are still fluctuations. Therefore, the design errors of the three channels are normalized and accumulated to obtain the cumulative error in Fig. 3(c). The decoder with the smallest cumulative error is the final GN. The design process of the meta-device is shown in Fig. 3(d). The target phase distributions of the meta-device are generated by the Gerchberg-Saxton (GS) algorithm [40,44], which requires the meta-device surface’s amplitude distribution (MSAD) as the amplitude constraints on the object plane and the target holographic image as the amplitude constraints on the image plane. The target nanoprinting image in LRA channel is regard as the MSAD and the target holographic image in LRP channel is treated as the amplitude constraints on the image plane, and then the target phase distribution of the meta-device in LRP channel can be generated. Since LLA and LRA exhibit a highly negative linear correlation (their linear correlation coefficient is -0.91),
can be used to control LLA, and then the target phase distribution of the meta-device in LLP channel is generated.According to the final target responses, GN can give multiple candidate designs, which are selected with the smallest design error through FN as the final design. In fact, if we change the responses to observe the distribution of different designs in the latent space, it can be seen that these designs obey some specific distribution patterns (shown in Section 3 of Supplement 1), which reveals that the latent-space representation of the designs may help with viewing the underlying patterns between the cross-shaped structure and its response [45–48]. The number of the meta-device pixels is set as 300*300. Then the simulation results of the meta-device are calculated by finite difference time domain (FDTD) method. With LCPL incidence, the transmitted LCPL and RCPL are extracted from the FDTD results to display the nanoprinting image and two holographic images, as shown in Fig. 3(e)-(g). The coupling between meta-atoms is ignored in the design of each pixel of the meta-device, which leads to some background noise in the nanoprinting image. However, three images still show good visual effects. This indicates that the multiplexing of these three channels can be realized, which is consistent with the previous evaluation result of IN’s training process.
With increasing the number of multiplexed channels, the polarization multiplexing capability limit in the given response space can be further explored. The data of LRA + LRP + LLP + RLP are extracted for training, and the training process is shown in Fig. 4(a)-(c). The amplitude design error is about 0.03 and all phase design errors are around 15°. It is clear that the design error of each channel has increased significantly compared to LRA + LRP + LLP. Obviously, the response space of the meta-atom is gradually tapped when the number of multiplexed channels increases. Through the same design process, a meta-device that displays a nanoprinting image of the LRA channel on the meta-device surface and three holographic images of the LRP, LLP and RLP channels in the meta-device’s Fraunhofer region is designed. Notably, RLA (the MSAD corresponding to RLP channel) exhibits a highly positive linear correlation with LRA (their linear correlation coefficient is 0.99), so RLA can be regulated by controlling LRA, and then the target phase distribution of the meta-device in RLP channel is generated. The final simulation results of the designed four-channel meta-device calculated by FDTD method are shown in Fig. 4(d)-(g). The good display effects prove the feasibility of the four-channel polarization multiplexing using the cross-shaped meta-atom.
Fig. 4. Multiplexing effect evaluation of LRA + LRP + LLP + RLP. (a) The decline process of cVAE loss. (b) The decline process of the design error for each channel evaluated by FN. (c) The decline process of the cumulative error after design errors of all channels are normalized. The simulation results of LRA channel’s nanoprinting image (d) and LRP, LLP and RLP channels’ holographic images (e)-(g).
Furthermore, we verify the multiplexing effect of LRA + LRP + LLP + RLP + YXP due to its lowest composite correlation coefficient in five-channel combinations (shown in Section 4 of Supplement 1). The results show that this five-channel combination cannot achieve good multiplexing effect. This implies that the polarization multiplexing capability limit of the cross-shaped meta-atom in our given response space is four channels.
4. Multiplexing of other multichannel combinations
It is worth noting that the control of above meta-devices’ MSADs adopts the linear relationship between two channels. However, the linear relationship between amplitude channels doesn’t necessarily exist. LRA + LRP + YYP + RLP is taken as an example to explain this situation. YYA, the MSAD corresponding to YYP channel, has a linear correlation coefficient of -0.57 with LRA. These two channels do not have a highly linear relationship, and thus it is not appropriate to generate the target phase distribution of the meta-device in YYP channel by the previous method. Therefore, a MSAD optimization method for all phase channels is introduced, shown in Fig. 5(a).
Fig. 5. Meta-device design process and simulation results of LRA + LRP + YYP + RLP. (a) Meta-device design process containing the YYA’s optimization method. The simulation results of LRA channel’s nanoprinting image (b) and LRP, YYP and RLP channels’ holographic images (c)-(e).
LRA + LRP + YYP + RLP is used to illustrate the optimization process (the training process is shown in Section 5 of Supplement 1). The designed meta-device displays the nanoprinting image in the amplitude channel and the holographic images in the phase channels. The MSAD corresponding to RLP channel (RLA) is controlled using linear relationship, and then the target distributions of the meta-device in LRA, LRP and RLP channels can be determined. YYA (MSAD corresponding to YYP channel) is set to 1 at the beginning. According to the target holographic image (the letter ‘Q’), meta-device’s target phase distribution in YYP channel of the first iteration is calculated by the GS algorithm. In combination with other determined target values, the meta-device is designed by GN as previous meta-devices. FN can predict the actual responses of the designed meta-device, including the actual YYA values. The actual YYA values of the first iteration are used as the YYA values in the next iteration. The error between YYA’s given values and the designed meta-device’s actual YYA values predicted by FN can be calculated by:
where gi is the given value, ai is the actual value and n is the number of meta-device’s pixels. When YYA is set to 1, the error is 0.495. However, the error is reduced to 0.151 after 5 iterations. The error does not decrease significantly if continuing to increase the number of iterations. The YYA values after 5 iterations are regarded as the MSAD corresponding to YYP channel to generate the target phase distribution of the meta-device in YYP channel. The final simulation results of the optimized meta-device are shown in Fig. 5(b)-(e), and it can be seen that the YYP channel and other channels show the effects of highly matching the target images. Similar cases that do not contain linear relationships in a multichannel combination can be optimized in the same way. There are more choices for multiplexed channels combinations through this method. Realizations of other three-channel and four-channel combinations multiplexing are shown in Section 6 of Supplement 1.5. Conclusion
In summary, this work proposes a design scheme for polarization multiplexed metasurfaces based on deep learning. In the given response space of the cross-shaped structure, the four-channel multiplexing limit of a nanoprinting image and three holographic images is determined by the proposed scheme. Although only the cross-shaped structure is used to verify the feasibility of the scheme, the whole design process is carried on based on parameterized structural degrees of freedom. Thereby, it is expected to be applied in exploring the limit of polarization multiplexing capability based on other meta-atoms. Compared with the traditional design methods [11–21], the proposed scheme provides a richer range of choices for multiplexed polarization states, which is conducive to the designers’ customized design requirements. In addition, this scheme can become one of the potential routes to explore the limit of wavelength multiplexing capability. However, the increase in the dimension of the response space may cause the problem of difficult convergence of the network. Possible solutions are increasing the scale of the network and cascading a well-trained forward network after the decoder.
Funding
National Natural Science Foundation of China (61905031); National Key Research and Development Program of China (2020YFC1522900).
Acknowledgment
We thank the State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences for the software sponsorship.
Disclosures
The authors declare no conflicts of interest.
Data availability
All simulation data that support this work are available upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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