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Abstract
Nanophotonics has joined the application areas of deep neural networks (DNNs) in recent years. Various network architectures and learning approaches have been employed to design and simulate nanophotonic structures and devices. Design and simulation of reconfigurable metasurfaces is another promising application area for neural network enabled nanophotonic design. The tunable optical response of these metasurfaces rely on the phase transitions of phase-change materials, which correspond to significant changes in their dielectric permittivity. Consequently, simulation and design of these metasurfaces requires the ability to model a diverse span of optical properties. In this work, to realize forward and inverse design of reconfigurable metasurfaces, we construct forward and inverse networks to model a wide range of optical characteristics covering from lossless dielectric to lossy plasmonic materials. As proof-of-concept demonstrations, we design a Ge2Sb2Te5 (GST) tunable resonator and a VO2 tunable absorber using our forward and inverse networks, respectively.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metasurfaces have been the focus of numerous studies in the past two decades due to their potential applications. In particular, they have great potential to replace conventional bulky optics as they enable control of amplitude, phase and polarization state of light within sub-wavelength dimensions. Many optical components, including lenses [1–4], polarizers [5], and holograms [6], have been realized by utilizing these nanophotonic structures. Current metasurface design approaches, employing either forward or inverse optimization, are heavily dependent on computationally expensive full electromagnetic simulations. Learning approaches, especially deep learning methods, stand out as an alternative approach with great potential to reduce high computational costs.
In recent years, nanophotonic structure design and simulation have joined the application areas of deep neural networks (DNNs) [7–19]. Optical response of a diverse range of nanophotonic structures have been modeled using various DNN architectures ranging from fully connected networks [7,9,10,12,17] to generative adversarial networks (GANs) [14–16]. Example models include transmission/reflection spectra of alternating layers [9,11,12], phase response and scattering parameters of cylindrical meta-atoms [7,17], and reflectance and circular dichroism of metal-insulator-metal structures [8,19]. Using the aforementioned networks, on-demand design of many nanophotonic devices such as metalenses [7,16,17], chiral matematerials [8], metagratings [14], and resonators [18] have been realized and proof of concept demonstrations have been reported.
Reconfigurable metasurfaces based on phase-change materials is another promising class of nanophotonic devices [20–29]. The tunable optical responses of this class of metasurfaces rely on the change of the optical properties with the phase transition of phase-change materials. Thus, DNN based design of such metasurfaces requires modelling of optical properties of these materials along the phase transition, which correspond to a wide range of dielectric permittivity values and even dielectric to plasmonic transitions [20–30].
Here, to design phase-change material based reconfigurable metasurfaces, we construct forward and inverse networks to model complex scattering parameters of reflected and transmitted light from arrays of cylindrical nanodisks. The amplitude and phase information are directly obtained from these parameters, and the materials of the nanodisks are represented by their complex optical constants (n + jk), which allow us to model both dielectric and plasmonic materials. Motivated by their capability of modelling complex-valued input-output relations, we employed complex-valued neural networks (CVNNs) for this task. As proof-of-concept demonstrations, we design a Ge2Sb2Te5 (GST) tunable resonator and a VO2 tunable absorber, utilizing our forward and inverse models, respectively.
2. Forward network and forward design of reconfigurable GST resonator
The structures investigated here consist of periodic arrays of low-aspect ratio nanodisks, made of either lossless dielectric or lossy plasmonic materials, on top of a low index substrate (${n_{subs}} \approx 1.72$) as shown at the lower left corner of Fig. 1(a). The wall-to-wall distance between adjacent nanodisks is kept constant (d=500 nm) and the corresponding design parameters are the radius (R), the height (H), and the complex refractive index (n + jk) of the nanodisk. The complex refractive index values used in the data generation are randomly chosen to cover both plasmonic and dielectric materials. The spectral range of operation is defined between λ=2 μm to λ=4 μm. The complex transmission (${S_{21}}$) and reflection coefficients (${S_{11}}$) are modeled independently as the optical response. (Details about data generation can be found in the Supplement 1).
Fig. 1. a Schematic of the complex valued neural network (CVNN). The network has 5 fully connected hidden layers. The input matrix includes optical constants (n-k), wavelength-normalized geometric parameters of cylindrical nanodisks (${\raise0.7ex\hbox{$R$} \!\mathord{/ {\vphantom {R \lambda }} }\!\lower0.7ex\hbox{$\lambda $}},$ ${\raise0.7ex\hbox{$H$} \!\mathord{/ {\vphantom {H \lambda }} }\!\lower0.7ex\hbox{$\lambda $}}$), and wavelength of operation (λ). The output is complex reflection(${S_{11}}$) or transmission(${S_{21}}$) coefficient. The unit cell structure of investigation is at the bottom left corner: cylindrical nanodisks with varying radius, height and optical constants on top of a lossless substrate $({n_{subs}} \approx 1.72$). The wall to wall distance between adjacent disks is 500 nm, both optical constants are randomly generated, $250 \le R \le 1500 $ nm, and $40 \le H \le 200 $ nm. b, c Probability density function (Pdf) of ${S_{21}}$ and ${S_{11}}$ networks’ mean square error (MSE) values, where 95% confidence borders of training and validation sets are indicated by blue and red dashed lines (2.0 × 10−4/ 4.0 × 10−4 and 3.1 × 10−4/6.3 × 10−4 for ${S_{21}}$ and ${S_{11}}$ networks respectively).
We use fully connected network structure that consist of 5 complex-valued hidden layers with 20, 80, 120, 80 and 20 neurons, as shown in Fig. 1(a). Two independent networks are trained to separately model reflection and transmission coefficients. The augmented data matrix ([X|y]) is prepared using our previously suggested wavelength normalization method7, where the geometric dimensions are normalized to the wavelength of operation and each of the spectral point corresponds to a row ([features|result]), as illustrated in Fig. 1(a). To feed the network, real and imaginary components of the input matrix are separated and horizontally stacked ([Real(X) Imaginary(X)]) resulting in a Nx8 input matrix, where N is the number of samples. Similarly, the result vector is converted to 2 columns of real and imaginary parts of the optical response. Note that complex-valued layers also have an imaginary component, doubling the number of output columns at each layer. As a performance metric, we use mean squared error (MSE) of the models. The overall training/test MSE’s are 7.0 × 10−5/1.2 × 10−4 and 1.2 × 10−4/1.6 × 10−4 for the transmission and reflection networks, respectively. Figure 1(b) and 1(c) show probability density function (Pdf) of error distributions over training and validation sets of both networks, where the dashed lines mark the 95% confidence borders (blue for training, red for validation). As indicated in Fig. 1(b),(c), the error distribution of training and test sets overlap for transmission and reflection networks, and 95% of the these sets have MSE $\le $ 2.0 × 10−4/4.x10−4 and MSE $\le $ 3.1 × 10−4/6.3 × 10−4, respectively. These low and consistent MSE values indicate that high prediction accuracy is achieved without overfitting. We also compare CVNNs with their acknowledged real-valued counterparts (RVNNs) for this particular task. We show that CVNNs have certain advantages as they preserve mathematical correlation between real and imaginary components of a complex-valued problem. The detailed discussion is provided in Supplement 1.
To visualize the performance of our model(s), we compare the predictions of our transmission model (similar analysis for reflection model can be found in Supplement 1) to FDTD simulations on two sample sets selected from unseen validation dataset, corresponding to a plasmonic and dielectric material, respectively. Figure 2(a),(c) depict the real and imaginary components of transmission coefficient (S21), and Fig. 2(b),(d) show the corresponding transmission amplitude (T) and phase (Δϕ) values. As seen in Fig. 2(b),(d), the transmission amplitude and phase responses of output light exhibit abrupt changes around resonance wavelengths, which is considered challenging to resolve for NN models [8]. The 0 to 2π jumps resulted from periodic nature of phase, as shown in Fig. 2(b),(d), is another source of singularity. The real and imaginary parts of complex transmission (reflection) coefficients are immune to such sudden jumps and singularities [17], as seen in Fig. 2(a),(c). The sample set indicated at the upper row (Fig. 2(a),(b)) employs a plasmonic material ($n\sim k$), while the next one (Fig. 2(c),(d)) uses a dielectric material (n constant, k=0). As indicated in Fig. 2(a),(c), the transmission coefficient is accurately predicted by our model in both cases. Furthermore, when its predictions are converted to amplitude and phase responses, the model maintains high prediction accuracy including the resonances and periodic jumps of the phase response for both the dielectric and plasmonic materials, as shown in Fig. 2(b),(d).
Fig. 2. a, c Comparison of predicted (real part: blue and imaginary part: red) and simulated transmission coefficients of exemplary samples from the validation set. a Plasmonic material ($n\sim k$), H=140 nm, and R=1500 nm. c Dielectric material (n constant, k=0), H=160 nm, and R=600 nm. b, d Corresponding transmission and phase values of unit cells that are indicated at a and c, respectively. Inset: Magnified image of optical response around resonant point (λ=2.88 μm to λ=2.96 μm).
As it is an inherent property of the wavelength normalization method [7] implemented here, we also briefly investigate spectral generalizability property of our model(s), where we improve over our previous results by replacing the wavelength of operation with a pseudo-feature. The detailed discussion is provided in Supplement 1.
The quick and reliable result obtained by our model suggests that a grid search method could serve as a feasible forward design method with accurate predictors such as our model(s). Regarding the grid search method, a grid is defined along the parameter space, on which the solution space is created. The optimal solution is determined with respect to the Figure of merit (FoM), which is defined according to the design objective. The grid search is straightforward, and with sufficient resolution (grid intervals), provides the optimum points along a continuous and differentiable solution space. However, creating a solution space over a fine grid is computationally expensive, especially with the commonly used simulation methods. Furthermore, the computational cost increases rapidly with the number of independent parameters that are needed to be optimized. These difficulties could be mitigated by utilizing predictive models, such as our transmission or reflection models, instead of full electromagnetic simulations, to obtain the optical response of the nanophotonic structures.
As a proof of this concept, we design a GST tunable resonator using the grid search method by utilizing our transmission model. GST is a well-known phase-change material which provides a large refractive index change upon phase transition and exhibits low losses [20,21]. The design objective is defined as obtaining the largest tuning interval (Δλ) between amorphous and crystalline phases of GST. The resonant dips should be sufficiently deep to provide zero transmission (or complete reflection, ignoring the absorption losses). Taking this criterion into account, the FoM is defined as (1-Tamorph)x(1-Tcrystal) x(${\raise0.7ex\hbox{${\Delta \lambda }$} \!\mathord{/ {\vphantom {{\Delta \lambda } {{\lambda_{range}}}}} }\!\lower0.7ex\hbox{${{\lambda _{range}}}$}}$.), where Tamorph and Tcrystal correspond to transmission at resonant wavelengths and, ${\lambda _{range}}$ is our spectral window of interest. ${\lambda _{range}}$ is introduced in order to convert the tuning range term to a unitless measure between 0 and 1. Within our parameter space, a grid with 8874 parameter sets (overall 4445874 data points with 501 wavelength points per set) is defined. Using an ordinary PC (i5-CPU), optical responses of all data points are calculated within 2 minutes by our transmission network. The search resulted in (R, H)optimum = (620 nm, 200 nm) corresponding to a tuning range of approximately Δλ=1 μm, and zero transmission (according to FDTD simulations) at the resonant wavelengths, as shown in Fig. 3. The resonant dip continuously red shifts as the GST phase shifts from amorphous (violet line) to 100% crystalline (red line).
Fig. 3. Comparison of predicted and simulated transmission response of GST tunable resonator designed using the transmission model (CVNN). a CVNN predicted spectra, b FDTD simulated spectra. The results are plotted for varying crystal fractions from amorphous, or 0%, (violet line) to crystal, or 100%, (red line). The tuned resonant deep occurs at 3012 nm and 3992 nm wavelengths for amorphous and crystal phases, respectively.
In order to verify the predictions of our transmission model for the GST tunable resonator, the spectral response of the designed resonator is also calculated using FDTD method. Figure 3(a) and 3(b) compare the predictions of our model to the FDTD simulations, showing very good agreement between the two tools validating successful forward design of reconfigurable GST resonator.
Note that, especially around smaller wavelengths, there are small magnitude differences between predicted and simulated results. We attribute these small errors to fixed simulation settings, which are used to accelerate the data generation process. Although they provide accurate simulation result for most of the samples, the fixed settings cause small deviations and oscillations around the correct solutions for a small part of the dataset creating discrepancies in the data. Such inconsistencies in training data lead small errors in predictions.
3. Inverse network and inverse design of the tunable VO2 absorber
We construct an inverse network to provide an R-H pair corresponding to the desired optical response (S11 and S21) of a specific material (n + jk) at a specific operation wavelength (λ). The inverse network consists of two main parts as a parameter retrieval part (Separate Sequential Network or SSN) and a forward network. The SSN consists of two independent networks (H and R networks), to predict the corresponding H and R values. The connection between these parameters is established as an input-output relationship. At the parameter retrieval part, the required geometric parameter set (R-H pair) is obtained from desired S11 and S21 pair, complex refractive index pair, and operation wavelength. Then, together with the given optical constants and wavelength of operation, they are combined as input parameters for the forward network, which consists of our pre-trained forward models, to predict the corresponding optical response, as shown schematically in Fig. 4(a). The weights of parameter retrieval part are tuned by comparing the predictions of the forward network to the desired S11 and S22 pairs.
Fig. 4. The general purpose inverse network. a The schematic. The R-network of SSN and the Forward network are pre-trained and fixed independently. b,c The comparison of requested and provided optical responses (b: S11, c: S21). Wavelength of operation is set to be 3 μm for all samples. The samples were chosen with varying refractive indices (n + jk) from the validation set. The corresponding predicted R-H pair and refractive indices are provided in Supplement 1.
The inverse network has to overcome the non-uniqueness in the dataset, that is, there are several R-H pairs yielding the same output. In such case of one-to-many mapping from input to output, the neural network experiences difficulties to converge. To address this problem, we use previously suggested tandem learning idea [7,9,31] where, R-network is pre-trained and its weights are kept constant during the training of the inverse network.
The inverse network achieved an overall MSE of 6 × 10−4. The network’s performance is demonstrated in Fig. 4(b),(c), where desired optical responses (complex reflection-transmission coefficient pairs) are compared with the ones corresponding to the network’s predictions (predicted R-H pairs). To provide a consistent and comprehensive demonstration, the operation wavelength is fixed to 3 μm, whereas the refractive indices of the material of interest are varied. As seen in the Fig. 4(b),(c), the network accurately recovers the desired optical responses.
The network is trained using the dataset of the transmission (reflection) model(s) covering a wide range of complex refractive indices so that the networks can operate with arbitrary materials. This network serves as a general-purpose inverse map for various optical response functions such as transmittance, reflectance, absorbance, phase responses, etc. as these properties and their combinations can be readily obtained from S11 and S21.
As a demonstration of this concept, we design a tunable infrared metasurface absorber employing the phase-change material VO2, a well-known material exhibiting insulator to metal transition over 68 °C [22–25,30]. The design objective is defined as determining geometric parameter set (R, H) that provides maximum tuning range at operation wavelength. The tuning range (ΔPabs) is defined as the absorbed power difference for two different refractive indices, which corresponds to the metallic and insulating phases of VO2. The operation wavelength (λ) is set to 3 μm.
This inverse design task contains an additional challenge of ambiguity of the solution. The ambiguity arises as the target response is not known but only relatively defined. For example, in our task, the target response is defined as the maximum tuning range, but its specific value is unknown. As a result, an inverse design network is not only responsible for finding an R-H pair, but must also verify that the tuning range provided by this pair is larger than all other possibilities without calculating them. To resolve this ambiguity, we redefine the target response as the best approximation to |ΔPabs|=1, which essentially yields the same result with the design problem.
For this task, the general-purpose inverse network architecture is modified to accept two complex refractive indices in the input. Two forward networks (one for each phase) are employed, and a calculation layer, which calculates ΔPabs from S11 and S21, is attached at the end of its output, as seen in the Fig. 5(a).
Fig. 5. Inverse Design of VO2 tunable absorbers. a The schematic of modified Separate Sequential Network (SSN). In the SSN structure the input matrix modified to include two refractive indices and desired optical response is converted to ΔPabs. In Forward path, two input matrices corresponding to the two different refractive indices are created and fed into our pre-trained and non-trainable forward networks.Then, the absorbed pover is calculated as Pabs_1(2)=1-|S21|-|S11|. Finally, the tuning range is calculated as the difference of absorbed power corresponding the two optical constat (n-k) sets. b Tuning range (ΔPabs=Pabs, hot- Pabs, cold) along the parameter space at λ=3 μm. Black stars corresponds to metasurfaces designed by SSN network and the grid search method (FDTD simulations).
Figure 5(b) indicates the predictions of the networks on top of results obtained by the grid search method (with intervals of 10 nm) in the solution space. As seen in Fig. 5(b), the prediction of the modified network is located at the white region indicating the largest available tuning ranges (ΔPabs). The network made a successful prediction with deviations ${\approx} $ 0.05 from the peak value calculated by FDTD solver.
4. Conclusion
In conclusion, we demonstrated DNN enabled forward and inverse design of reconfigurable metasurfaces, in which the generic geometry of cylindrical nanodisks on top of a transparent substrate is used as the building block. We simultaneously modeled a wide range of optical properties including that of lossy plasmonic and lossless dielectric materials. As proofs of the concept, we demonstrated the forward design of a GST reconfigurable resonator and inverse design of a VO2 tunable absorber. The forward and inverse models can also be employed to design and simulate other classes of metasurfaces with a large variety of materials.
Funding
Binational Science Foundation (2016388); Air Force Office of Scientific Research (FA9550-17-1-0348); Office of Naval Research (N00014-17-1-2425).
Disclosures
The authors declare no competing financial interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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