Polarization Multiplexing Bifunctional Metalens Designed by Deep Neural Networks

As planar optical elements, metasurfaces confer an unprecedented potential to manipulate light, which benefits from the deep control of the interactions between nanostructures and light. In the past decade, considerable progress has been made in various metasurfaces for on‐demand functions, drawing great interest from the scientific community. However, it is a great challenge to integrate different functions into a single metasurface, due to the incapability of manipulating light at different dimensions and the lack of universal intelligent design strategy. Here, an intelligent design platform based on deep neural networks is proposed, which can map between structure parameters and optical response. The well‐trained network model can intelligently retrieve nanostructures to meet multidimensional optical requirements of metasurfaces. Four metalenses for chiral focusing are realized by the design platform and the simulation results are highly consistent with the design target. In addition, metalenses based on arbitrary polarization at various working wavelength are also demonstrated, showing that the method has powerful design ability. Various optical properties of nanostructures, such as phase shift and polarization, are manipulated by deep neural networks, which can greatly promote the development of multifunctional devices and further pave the way for optical display, communication, computing, sensing, and other applications.

Multiplexing metasurface can integrate different optical functions by the multidimensional manipulation of light. Therefore, the inverse design of nanostructures is more challenging due to the complex optical requirements. To solve the problem of inverse design, two typical approaches were implemented. Different functions are easily realized by different nanostructures via segmented, interleaved, few-layer designs, which results in the low efficiency and high background noises. [26][27][28][29] Different optical requirements can also be realized by the same nanostructures, which brings great challenges for the design of nanostructures. Obviously, based on traditional design strategy, like parameter sweeping or common optimization algorithms, it is a www.advancedsciencenews.com www.advphysicsres.com hard task to achieve the desired functions with limited computing resources. Recently, deep learning has been continuously introduced into interdisciplinary research, such as quantum optics, [30] protein structure analysis, [31,32] computational imaging, [33,34] and structure design. [35][36][37] Compared to rule-based or experimentbased strategies, deep learning can automatically discover useful information from a large amount of data, and then map between input and output. Due to its powerful data processing capability, deep learning has brought development opportunities to many photonics research fields, such as spectral analysis, [38] photonic crystal, [39] inverse design, [40][41][42][43] imaging system, [44] and integrated photonics. [45] In this work, we designed and demonstrated bifunctional reflective metalenses, which can realize independent focus functions under different polarized incidence. The relationship between geometric parameters and optical responses can be built by neural networks. Through the deep neural networks, we can further retrieve optimal structure parameters according to phase requirements of bifunctional focus. The bifunctional focusing metalenses at 1550 nm were designed based on deep learning, by which metalenses with focuses on arbitrary spots can be designed. The numerical simulation shows great consistency with design target. In addition, we further show other design examples in different polarization states and wavelengths, achieving expected focusing target, which proves that our method is flexible. Our method can fully explore the potential of the parameter space by deep learning, realize the independent manipulate of the bifunctional phase without segmented, interleaved, few-layer designs, and complete the multiplexing imaging under any two polarization states. Our method can be widely used in the design of multifunctional photonics devices.

Basic Principle of Chiral Focusing Metalenses
Two polarized components of light realized two independent focus functions. As shown in Figure 1, under normal incidence, the reflection is converted by metalens into the orthogonal polarization and converging into focus. Any two orthogonal polarization states on the Poincaré sphere can realize bifunctional metalens. Especially, the north pole of the Poincaré sphere represents left circular polarization (LCP) and south pole represents right circular polarization (RCP), which are used to demonstrate our bifunctional metalenses. The unit cell of metalenses is a V-shaped gold nanostructure with a 400 nm period. By strong plasmonic resonance between metallic nanostructures and incidence, properties of light are excellently controlled, such as polarization, amplitude, and phase shift. The LCP component of reflection focuses at F L (x L , y L , f L ) relative to the center of the metalens and the RCP component focus at On the plane of nanostructure array, refection has an abrupt phase shift relative to incidence, which originates from the interaction between light and nanostructures. Under RCP incidence, the phase shift of LCP reflection is L . Under LCP incidence, the phase shift of RCP reflection is R . According to the generalized Huygens principle, the reflection from each unit cell has same optical phase at the focus. Considering the reflection converging into the focus as a spherical wave, we can obtain the abrupt phase shits of nanostructures at position of (x, y) where refers to working wavelength.

Multiplexing Metalenses Design Workflow
In order to realize the bifunctional metalens without spatial multiplexing, bifunctional phase shifts must be realized by the same nanostructure, which brings great challenges to the design task. Therefore, we introduce a deep learning algorithm to help us retrieve nanostructures intelligently and efficiently. A dataset containing geometric parameters and optical responses is prepared, so that it can spontaneously find potential rules from the data and map between geometric parameters and optical responses. As a typical nanostructure to modulate phase shifts, V-shaped structure is composed of two aligned nanorods. The geometry of nanostructures is described by (l 1 , l 2 , a, b), where the length l 1 and l 2 represent two arms respectively. The angle a represents the azimuthal angel of their angular bisector while the angle b means the angle between two arms. These geometrical parameters can affect the interaction between V-shape nanostructures and incident light, and then modulate the phase shift. (More information about the phase modulation by V-shape nanostructures can be found in Section S1, Supporting Information.) As shown in Figure 2, the unit cell is a 30 nm thick V-shaped gold nanostructure, located on the top of a 100 nm thick glass (SiO 2 ) spacer and a bottom Au layer. The dataset is composed of 10 4 random nanostructures and their optical responses, which were obtained by FDTD (Finite Difference Time Domain) simulation. During the simulation, the optical responses at different wavelengths (530-1550 nm, step = 1 nm) have been recorded. The Jones matrix J( ) of one nanostructure is defined as: where R ij refers to the reflection in i-polarized component under j-polarized incidence at wavelength. Therefore, the output electronic field E out is: ) refers to target output. The phase shifts are obtained from E out . Significantly, the sine and cosine of phase shifts are added into dataset to avoid the training difficulty which is brought by the periodicity of phase shifts. The dataset is divided into 2 groups: 90% to train neural networks and 10% to test neural networks.
To realize the inverse design of multifunctional metalenses, a forward neural network and an inverse neural network were trained. Given geometric parameters, the forward network predicts the optical responses of nanostructures. Obviously, there is only one optical response for a certain nanostructure. The loss of forward network is the mean square error (MSE) between prediction of optical responses and the optical parameters from the dataset. By minimizing the loss, forward neural network was well trained. The inverse network recommends nanostructures according to optical responses as design target. However, different nanostructures may meet the same optical requirement, which brings great challenges to the training of inverse network. Therefore, the pretrained forward network is involved to guide the training of inverse network. During the training of inverse network, output geometry from inverse network is put into the forward network. The loss of inverse network measures the difference between the initial optical target and the prediction from forward network. The optimization algorithm Adam (adaptive moment estimation) is utilized to minimize the loss and learning rate is 0.001. After 360 epochs of training, the forward loss was reduced to 0.034 on training set and 0.040 on testing set. After 600 epochs, the inverse loss is reduced to 0.027 both on training set and testing set. These results show that our networks have convincing accuracy and great recommend ability. More information about the architectures of deep neural networks can be found in Section S2 and Figure S3 (Supporting Information).
The design workflow of bifunctional metalenses is shown in Figure 2. The forward and inverse networks were trained by dataset. The design target, which is the phase shifts of nanos- tructure array, is obtained by Equations (1) and (2) according to optical requirements. And for phase requirements at each point, the inverse network recommends the desired nanostructure and forward network predicts its optical responses. Due to the high computing efficiency of deep neural network, single nanostructure can be retrieved by inverse network within 1 s. The traditional design methods, such as parameter sweeping and elementary optimization, have great challenges in multifunctional metasurface design. These methods are incapable of recommending nanostructures outside the dataset. However, with the increasing dimensions of optical properties, required nanostructures increase exponentially. Obviously, it is a great challenge to complete the design of multifunctional metasurface with limited computing resources. Because of the weak generality, the other methods such as genetic algorithm and particle swarm optimization algorithm, consume lots of time to optimize numerous nanostructures. Compared with traditional design methods, our design method has overcome the mentioned challenges, and can design multifunctional metasurface efficiently with limited computing resources.

Design and Simulation Results of Bifunctional Chiral Metalenses
In order to verify our method, four different metalenses were demonstrated with a diameter of 32 m. The working wavelength is at 1550 nm. For the first sample (Figure 3a), the position of F L (x L , y L , f L ) is (12,0,15) m and F R (x R , y R , f R ) is (−12, 0, 30) m. According to the workflow in Figure 2, the nanostructure array was retrieved by inverse network within 1 ms, which is much more efficient than the traditional design method. The FDTD simulation of the whole array was performed. The reflective in-tensity in x-y plane and x-z plane is shown in Figure 3a, which is in great consistency with expectation. The clear focus spots show that the metalens can converge light to accurate focus spots excellently. The significant diffraction enhancement of output focus reduces the effect of background noise from the unconverted polarized component.
To further prove that our method has the ability to inverse design focus at any position, other three metalenses were demonstrated. For the second metalens (Figure 3a), the two reflective components converge to different off-axis spots with same focal length. For the third metalens (Figure 3c), both reflective components converge to optical axis with different focal lengths. For the fourth metalens (Figure 3d), both reflective components converge to the same spot, which means the reflection is converged to target position regardless of incident polarization. The simulation results in Figure 3 prove that our method can manipulate the two components independently, and further design a bifunctional metalens with arbitrary focal length. Essentially, our method provides an effective way to retrieve suitable nanostructures with more complex requirements of optical properties, so it can be extended to solve the design problem of other multifunctional photonic devices.

Expanding Design Cases of Bifunctional Metalenses
Besides circular polarization, other polarized components of reflection can also be designed by our platform. Therefore, three different bifunctional metasurfaces were demonstrated with orthogonal polarization states (labeled on the Poincare's sphere of Figure 4a). Under A polarized incidence, the B polarized component of reflection converges at focus (12,0,30) m. Under B polarized incidence, the A polarized component of reflection converges at another focus (−12, 0, 15) m. The bifunctional abrupt phase shifts of each nanostructure from the dataset were obtained according to Equation (4). The phase shifts as well as corresponding geometric parameters can construct an updated dataset. Similar with the workflow mentioned in Figure 2, the forward and inverse network were retrained with the updated dataset. Desired nanostructure arrays can be generated from retrained inverse network. The reflections from metalenses at xy plane and x-z plane were obtained by FDTD simulation (Figure 4a), which are in perfect consistency with the expectation. The powerful design ability to modulate light at any polarization were verified by the three metasurfaces. Therefore, our design platform can be effectively applied to optical encryption devices and multifunctional photonic devices.
As the working wavelength of dataset varies from 530 to 1550 nm, metalenses at other wavelength can be easily designed by our method. In order to obtain high efficiency, two metalenses at 850 and 1024 nm were designed based on circular polarization. As shown in Figure 4b, for metasurface at 850 nm, F L is (−12, −12, 30) m and F R is (12,12,30) m. For metasurface at 1024 nm, F L is (12, −12, 30) m and F R is (−12, 12, 30) m. According to the simulation results, the reflection converges at accurate position, which proves the powerful ability to design device at flexible polarization and working wavelength effectively. This indicates that our method may be combined with programmable metasurface in the future, exploring more powerful photonics devices that can dynamically manipulate light at flexible polarization and working wavelength. Besides metasurface, deep neural network may inspire various methods in other scientific fields.

Summary and Conclusion
In summary, we introduced deep neural networks to solve the inverse design problem of polarization multiplexing metalenses. Rather than different nanostructures realize different functions, like segmented, interleaved, few-layer designs, it is difficult to realize multifunction by single nanostructure conventionally. These nanostructures can be retrieved by our methods to manipulate the phase shifts in different polarization states independently. We have successfully implemented bifunctional focusing metalenses and the ability to converge any polarization component at any focus is shown. The simulation results by FDTD are in high consistency with the design targets. It should be noted that the multi-wavelength Jones matrix rather than the corresponding spectrum was recorded during the construction of the initial dataset. Therefore, our design method can be easily extended to devices with more complex requirement of polarization or working wavelength. Due to its high efficiency and great flexibility, our work inspires lots of research in multifunctional optical devices, which can significantly accelerate the applications of multifunctional devices such as optical encryption communication and multiplexing holography.

Experimental Section
Simulation Details: All simulation results in this work were accomplished by a commercial finite-difference time-domain methods solver (FDTD Solutions, Lumerical). The simulation domain consisted of structures, light source, and boundary conditions, where periodic boundary conditions and perfectly matched layers were used along the transverse and longitudinal directions with respect to the propagation of light. For each simulation, the far-field electric field was recorded, normalized by the reflection field of substrates. The polarization conversion efficiency and phase spectrum were obtained from the normalized electric field. The material properties of nanostructures and substrates used in the simulation were Au (gold)-Palik, and the SiO 2 was set as SiO 2 (glass)-Palik. When building the initial dataset, each simulation is implemented on one AMD Ryzen Threadripper 3970×32-Core Processor, which takes about 2−3 min. Thus, the whole dataset, including 10 000 nanostructures as well as corresponding multi-wavelength jones matrix, can be built in 400 h.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.