Total Transmission from Deep Learning Designs

Bei Wu | Zhan-Lei Hao | Jin-Hui Chen | Qiao-Liang Bao | Yi-Neng Liu |Huan-Yang Chen

Abstract—Total transmission plays an important role in efficiency improvement and wavefront control,and has made great progress in many applications,such as the optical film and signal transmission.Therefore,many traditional physical methods represented by transformation optics have been studied to achieve total transmission.However,these methods have strict limitations on the size of the photonic structure,and the calculation is complex.Here,we exploit deep learning to achieve this goal.In deep learning,the data-driven prediction and design are carried out by artificial neural networks (ANNs),which provide a convenient architecture for large dataset problems.By taking the transmission characteristic of the multi-layer stacks as an example,we demonstrate how optical materials can be designed by using ANNs.The trained network directly establishes the mapping from optical materials to transmission spectra,and enables the forward spectral prediction and inverse material design of total transmission in the given parameter space.Our work paves the way for the optical material design with special properties based on deep learning.

Index Terms—Artificial neural networks (ANNs),deep learning,forward spectral prediction,inverse material design,total transmission.

1.lntroduction

In the past decades,optics and photonics have developed rapidly,showing a strong capability in tailoring light-matter interactions.For example,photonic crystals have the ability to achieve complete photonic band gaps[1].Metamaterials demonstrate special properties,such as the negative refractive index[2],[3].Surface plasmon polaritons can break the traditional diffraction limit and regulate electromagnetic waves at the subwavelength scale[4].Meanwhile,the structure design plays a central role in optics and photonics,ranging from individual plasmonic nanoparticles[5]-[7]to metamaterials composed of meta-atoms[8],and to integrated photonic devices[9].At present,there are mainly two traditional photonic structure design methods.First,we can rely on the physics-based method,which is mainly proceeded from scientific intuition and prior knowledge.For example,dielectric and metallic nanoparticles with simple geometries can be accurately calculated by the Mie theory[10].However,when the geometric structure or material property of the photonic structure is complex,it will be difficult to design accurately.Second,we can resort to electromagnetic modeling based on numerical methods,which mainly include the method of moment (MOM),finite element method (FEM),and finite difference time domain (FDTD).By setting up sufficient mesh grids and iterative steps,the optical response of a specific structure can be calculated accurately.However,it is often necessary to fine tune the geometry and simulate repeatedly in order to gradually approach the optimal structure,which is a waste of computing resources.Moreover,only limited design parameters can be obtained in the optimization process.

In this work,we realize total transmission through a reasonable photonic structure design.In the past,the realization of total transmission often required the simultaneous analysis of homogenization and transformation optics[11]-[14].The transmission characteristics of a multi-layer stack can be analyzed by replacing it with an effective medium[15].In electromagnetics,the concept of homogenization is to replace a subwavelength-sized complex structure with a uniform effective medium,which goes back to the Lorentz-Lorenz and Maxwell-Garnet effective medium theories[1],[16].Nowadays,the modern method of homogenization can be used to calculate the effective parameters more accurately,and can also be applied to calculate various geometric structures.Although the effective medium theory is available and accurate in some scenarios,in the presence of gain or the strong surface wave interaction,the transmission characteristics of metal dielectric multi-layer stacks will be very sensitive to the detail structures,such as the cases related to hyperbolic metamaterials[17].Moreover,if the incident wavelength is reduced to about the size of the photonic structure,homogenization will be no more effective in periodic metal-based structures[15],[18],[19].

Hence,a method that could solve the above problems is highly anticipated.And deep learning may be the most promising one.Although it draws inspiration from the biological mechanisms of life,the concept of deep learning goes far beyond the analogies of biological nerves.The unique advantage of deep learning lies in its data-driven ability,which enables a model to quickly design the optimal structure that matches the target optical response,and avoids the tedious iterative procedure.In recent years,deep learning allows the ondemand design for many applications,such as the plasmonic nanostructure design[7],three-dimensional (3D)vectorial holography[9],and self-adaptive microwave cloak[20].It can commendably solve the time-consuming and low-efficiency problems in traditional design methods.In fact,there are many groups applying deep learning to inversely design the photonic structure according to the given spectra.For example,Peurifoyet al.trained artificial neural networks (ANNs) to design the thickness per shell of a nanophotonic particle according to the given scattering cross section[6].Liuet al.exploited deep learning to design the layer thickness of a thin film according to the given transmission spectrum[21].However,these studies are based on ordinary spectra and their applications are limited.The photonic structures with special properties,such as total transmission,total reflection,and total absorption,have not been inversely designed with deep learning.Therefore,the inverse-design algorithm based on deep learning might show a strong ability in dealing with complex systems for total transmission or other similar effects with a large degree of freedom.

In this paper,we exploit deep learning to design the optical materials with the target optical properties,specifically,total transmission at any incident angle for a broadband frequency range.Our design approach involves designing an appropriate equivalent description about the transfer matrix method (TMM),training the network model for the forward spectral prediction,and achieving the inverse-design model based on the pre-trained tandem network.Finally,the material parameters can be obtained by applying the total transmission spectrum on the inverse-design network,which is also sketched inFig.1.

2.Methods

2.1.Generating Training Data with TMM

We exploit TMM to generate the training data,which is a powerful tool to analyze the propagation of light through layered dielectric media[22].We consider the dielectric multi-layer stack displayed inFig.1(a),and put it in the air.Here,we take the p-polarized incident wave as an example,and the magnetic fields on both sides of the interface are polarized along theydirection and can be written as the forms ofH1,yandH2,yin (1):

wherekl,xandkl,z(l=1,2) are the components of the wave vector along thexandzdirections,respectively;alandblare the field coefficients,which can be solved with the transfer matrix of the interface.On the right side of the equation,the first and second terms in parentheses represent the waves propagating in the directions ofzand -z,respectively.For the p-polarized wave,by applying the boundary conditions,the transmission matrix for the p-polarized wave can be obtained as

whereεl(l=1,2) is the relative dielectric constant.The transmission matrix connects the fields across the interface of different medium layers.When the fields propagate in a homogeneous medium,it can be shown that the electric or magnetic fields atz+Δzcan be related to those at thezposition by a 2×2 propagation matrix[23]:

ForNlayers shown inFig.1(a),the total transfer matrix can be obtained from the transmission matrices on different interfaces and the propagation matrices in different homogeneous media.We can relate field coefficients of incident and outgoing waves by az+Δztransfer matrix M,namely,

Eventually,we can calculate the transmittance for the p-polarized wave as

whereM11is the element in the first row and the first column of the transfer matrix M.

Here,we focus on the material properties of the multi-layer stacks.In order to prepare data for deep learning,we construct a series of input-output pairs as follows.For the convenience of expression and analysis,we introduce the state vector d=(ε1,ε2,···,εI)Tas the input,which is composed of the material properties per layer,such as the dielectric constant,andIrepresents the number of parameters.Then we exploit TMM to calculate the corresponding transmittance and introduce the label vector β=(T1,T2,···,TN)Tas the output,which is obtained by the transmittance for different incident angles and wavelengths.Specifically,we select a point every 2° in the range of the incident angle from 0° to 88°,and a point every 22 mm in the range of the incident wavelength from 70 mm to 158 mm.It is worth noting that the light transmittance is very low at normal incidence.However,in the inverse design,to achieve total transmission,the label vector needs to be an all-one vector,and the low transmittance at normal incidence will make the convergence slow or diverge.Therefore,when we generate the training data,we limit the maximum incident angle to 88°,which will accelerate the convergence speed of ANNs without decreasing the accuracy.

From the above description,we generate a 5×104-sample dataset (80% for the training set,10% for the validation set,and 10% for the test set).Here,the training set is the data sample for fitting the network;the validation set is for adjusting the hyper-parameters,such as the learning rate,loss function,and weight initialization and activation function,and for the preliminary evaluation of the ability of the neural network;the test set is used for evaluating the ultimate generalization ability of the network model.Next,ANNs will be trained and evaluated with these samples.

2.2.Building and Training the Deep Learning Network

Deep learning has a strong generalization ability in the given design space.Furthermore,it can be used to design the optical materials that conform to our target quickly and accurately without time-consuming numerical calculations.In order to implement two functions of the forward spectral prediction and inverse material design,we build a tandem network model,in which an inverse-design network G cascades with a forward-modeling network F,as shown inFig.2(a).Here,the forward-modeling network predicts the transmittance F(d) based on the input state vector d.It has 6 hidden layers with 225 neurons per layer,and the training target is to minimize the cost function designed as

where βiis the label vector of the input state vector,F (d) is the predicted transmittance,andLis the number of neurons in each batch.Then we exploit the AdaDelta optimizer to train the forward-modeling network,which is an adaptive learning rate optimization algorithm.The results show that it converges quickly (nearly after 800 epochs),because an array of material parameters will definitely correspond to one transmission characteristic:One-to-one relationship.

Fig.2.Schematic representation of the neural network model,state vector,and label vector:(a) tandem network in which an inverse-design network G cascades with a forward-modeling network F,(b) state vector d composed of the material properties per layer,and (c) label vector β obtained by the transmittance for different incident angles and wavelengths.

Similarly,the inverse-design network has 6 hidden layers with 225 neurons per layer.It designs the material parameters G(β) based on the input label vector β.The trained inverse-design network can accurately and rapidly design the material parameters satisfying the target transmission characteristics.However,the same training approach for the inverse-design network would make it difficult to converge due to non-uniqueness mapping:The transmission spectrum does not correspond to only one combination of materials.The tandem network model can effectively solve this problem.It means that instead of directly training the inverse-design network with the training datasets β and d,we replace the datasets with β and F(d) to train the whole tandem network model.In this way,the designed material parameters G(β) do not need to be the same as the state vector d.On the contrary,as long as the predicted transmittance of the designed material F(G(β)) is consistent with the label vector β,the designed material parameters are in line with the requirements.The training target of the tandem network model is to minimize the cost function designed as the mean-square error (MSE) of each point on the spectrum,which indicates the difference between the label vectors and the predicted ones.

During the tandem network model training,the weights of the trained forward-modeling network are frozen,and only the inverse-design network is updated.Similarly,the AdaDelta optimizer is exploited to train it,and the tandem network will converge after nearly 1200 epochs.Table 1presentsMSE for different sizes of multi-layer stacks.Results show that the errors are close,indicating no over-fitting results caused.

Table 1:Accuracy of the neural network for different sizes of multi-layer stacks

3.Results and Discussion

3.1.Deep Learning Can Realize Spectral Prediction

For the spectral prediction,we expect that when we input any material parameters into the network model,it can quickly calculate the corresponding transmission spectra.The results show that deep learning can effectively realize the spectral prediction.

We test the spectral prediction on an instance—a six-layer stack with the thickness of each layer fixed at 1 mm.The relative permeability per layer is fixed at 1 to ensure that the materials are easily available,whereas the relative dielectric constant can be any value that matches the condition (named the random model),as shown inFig.2(b).We set the state vector to be d=(ε1,ε2,ε3,ε4,ε5,ε6)T=(2.1,4.8,3.3,4.3,2.5,7.0)T,and then calculate the corresponding transmission spectra with TMM,deep learning,and COMSOL Multiphysics,respectively,as shown inFig.3(a).It is worth noting that,unlike other ordinary spectra,we select a point every 22 mm in the range of the incident wavelength from 70 mm to 158 mm to meet the broadband characteristics.

It can be clearly found that the results of the three methods are all very accurate and consistent to each other.However,under the same hardware conditions,it only took 0.07 s to calculate with deep learning,but 70 s to calculate with COMSOL Multiphysics.Deep learning is obviously 1000 times faster than numerical calculations.

3.2.Deep Learning Can Achieve Total Transmission

To achieve total transmission,the label vector is set to be an all-one vector and applied on the inversedesign network,then the optical materials will be designed to obtain the transmission spectrum that satisfies the condition.It is essentially an inverse-design problem,and the results show that deep learning can effectively realize this function.

We test this inverse design on two examples.The first one is an alternative six-layer structure (named the ABAB model),and the state vector is d=(εa,εb,μa,μb)T,as shown inFig.2(b).Similarly,the thickness per layer is fixed at 1 mm,and the incident wavelength ranges from 70 mm to 158 mm.We have the network model learn what optical materials could produce the label vector β=(1,1,···,1)T,and the result shows that d=(2,0.35,0.15,3.2)T.The transmission spectra corresponding to the designed material parameters are shown inFig.3(b).It can be seen that total transmission is realized.

As explained in Section 1,the effective medium theory can also be exploited to analyze the material properties to achieve total transmission.According to the theory,the ABAB-type multi-layer structure can behave as an anisotropic medium if the thickness of each layer is the same and much less than the operative wavelength ofλ.Taking the above ABAB model as an example,we can calculate the equivalent dielectric constants (ε) and permeability (μ)[24],[25]:

whereμaandμbare the relative permeability of A and B layers,respectively;εaandεbare the relative dielectric constants of A and B layers,respectively;ε‖andε⊥are the equivalent relative dielectric constants in parallel and perpendicular to the optical axis,respectively.

According to transformation optics,total transmission can be achieved if the following (9) is satisfied[26]:

It can be found that in the above ABAB model,the material designed by the neural network does not exactly satisfy transformation optics,but as shown inFig.3(b),it can also realize total transmission at most incident angles (except for angles around 90°) for a broadband frequency range.In fact,in addition to the material parameters we showed above,the neural network has also designed many other material parameters,some of which meet the requirements of transformation optics.Nevertheless,we choose the above optical material as a demonstration,just to show that the material parameters designed by deep learning are not restricted by the traditional physical methods.However,such irregular material parameters might have similar properties.Moreover,if the wavelength is reduced to about the thickness,the performance of the effective medium theory will become invalid,but our method does not have this limitation,even if the wavelength and thickness are almost the same in the realm of photonic crystals,the calculation results will still be accurate.Based on the discussion,we believe that deep learning goes beyond traditional physical methods in a sense.

In the second example,the model is consistent with that in the spectral prediction (the random model).Specifically,the relative dielectric constant of each layer is not correlated and able to be any value that matches the condition,whereas the relative permeability is fixed at 1.In this case,the state vector is d=(ε1,ε2,ε3,ε4,ε5,ε6)T,similarly,the designed optical material needs to satisfy the label vector β=(1,1,···,1)T,as shown inFig.2(c).The result shows that d=(15.02,0.32,0.42,0.39,-0.63,-0.74)T,and the corresponding transmission spectra are shown inFig.3(c).It can be seen that if the wavelength is greater than 100 mm and the incident angle is in the range from 0° to 85°,the transmittance is close to 1,i.e.,total transmission.We can verify this effect from the electric field distributions simulated by COMSOL Multiphysics,as shown inFig.4(a).In the simulation,we chose 80 mm as the incident wavelength,and the transmission spectra change little with the incident wavelength,as shown inFig.4(b).It is worth noting that this instance does not strictly satisfy the broadband condition.When the incident wavelength is too small,the multi-layer stack will no longer achieve total transmission,because the material designed by the neural network cannot be equivalent to the air with the effective medium theory and the light will be reflected in this case.

Fig.4.Validation results of the total transmission case:(a) electric field distributions simulated by COMSOL Multiphysics and (b) transmission spectra varying with the incident wavelength.

It will be very complicated if we solve the second instance with transformation optics,and the designed materials will be gradient,which are difficult to manufacture.By introducing deep learning,the timeconsuming numerical calculations are avoided.Moreover,the material designed by the neural network is not limited by transformation optics,but could be the non-gradient materials (maybe with some kind of randomness).It is worth mentioning that in order to test the generalization performance of the network model,our given training data do not include total transmission samples.Under this condition,deep learning can still design the materials satisfying total transmission,and the generated parameters in the random model are beyond the value range of the state vector d from 2 to 8.This shows that deep learning is not simply fitting the data,but further learning the underlying physical laws of the data[27].

4.Conclusions

In this article,we proposed to exploit the deep learning approach for predicting the transmission spectra based solely on the optical material parameters.We have designed,trained,and tested the proposed scheme,showing a very accurate prediction of the transmission characteristics for given materials and significantly speeding up the spectral prediction of the multi-layer stacks.Although we focused on the optimization of material parameters,this approach can be easy to be extended to design the thickness when the material is specified,and to other research fields,such as 3D vectorial holography[9],topological photonics[28],and optical logic operation[29].It can also effectively solve the inverse-design problem.Specifically,we exploit deep learning to achieve total transmission at any incident angle for a broadband frequency range,which might even go beyond traditional physical methods,such as transformation optics.

Disclosures

The authors declare no conflicts of interest.