Manifold Learning for Knowledge Discovery and Intelligent Inverse Design of Photonic Nanostructures: Breaking the Geometric Complexity
Mohammadreza Zandehshahvar, Yashar Kiarashi, Muliang Zhu, Hossein Maleki, Tyler Brown, Ali Adibi
MManifold Learning for Knowledge Discovery andIntelligent Inverse Design of PhotonicNanostructures: Breaking the Geometric Complexity
Mohammadreza Zandehshahvar , Yashar Kiarashi , Muliang Zhu , Hossein Maleki ,Tyler Brown , and Ali Adibi
1, * School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA * [email protected] ABSTRACT
Here, we present a new approach based on manifold learning for knowledge discovery and inverse design withminimal complexity in photonic nanostructures. Our approach builds on studying sub-manifolds of responses of aclass of nanostructures with different design complexities in the latent space to obtain valuable insight about thephysics of device operation to guide a more intelligent design. In contrast to the current methods for inverse designof photonic nanostructures, which are limited to pre-selected and usually over-complex structures, we show thatour method allows evolution from an initial design towards the simplest structure while solving the inverse problem.
Keywords : manifold learning, knowledge discovery,inverse design, design complexity, photonic
Photonic nanostructures have been extensively employedfor different applications due to their unique features incontrolling the spatial, spectral, polarization, and eventemporal properties of an optical wavefront with sub-wavelength feature size. Thanks to recent advancesin optical materials and nanofabrication technologies,extensive design flexibility exists for photonic nanos-tructures through selection of constituent materials andgeometrical properties of individual nanostructures (alsoknown as nanoantenna or meta-atoms). This has enabledpractical photonic devices for a wide range of applica-tions in computing [1, 2], signal processing [3], imaging[4], planar lenses [5, 6], and wireless communication[7, 8], just to name a few. The large number of de-sign parameters (e.g., material selection and geometricalproperties of nanoantenna) requires new approaches forinverse design and optimization of these nanostructures.Unfortunately, there has been limited progress in thisdirection, and the continuous progress in nanofabrica-tion capabilities demands urgent progress in this newdirection. In addition, new techniques are needed to findthe best device architecture for a given response withminimum device complexity subject to constraints likeease of fabrication, sensitivity to fabrication errors, lowoptical losses, etc. Such techniques should allow the design algorithm to start from the under-defined archi-tecture and evolve to the most appropriate design beyondwhat initially envisioned by the designer.Considering the large number of design parametersand the ranges of their variations, traditional brute-forceinverse design approaches based on exhaustive search[9] of the design space or evolutionary approaches [10](e.g., genetic algorithms) cannot be used due to the com-putational complexity of state-of-the-art nanophotonicdesign problems. More recently, inverse design meth-ods based on artificial intelligence (AI) [11–34] haveshown promising performance in inverse design withconsiderably reduced computational requirements. How-ever, existing approaches are mainly focused on findingthe design parameters of a given nanostructure withoutchanging its structure. In addition, most of the reportedtechniques only consider finding the design parameterswithout providing insight about the underlying physicsof the device operation, e.g., by providing informationabout the hidden patterns in the data. More recently,there has been more interest in using AI to investigatethe design-response relation while optimizing the nanos-tructures [35–47]. New approaches for “knowledge dis-covery” in nanophotonics should utilize the “intelligent”aspects of AI rather than using AI primarily as an opti-mizer.The main focus of this paper is on addressing thesetwo major needs: 1) AI-based techniques to enable in-verse design while enabling the evolution of the nanos- a r X i v : . [ phy s i c s . op ti c s ] F e b e f l e c t i on Wavelength R e f l e c t i on Wavelength R e f l e c t i on Wavelength
Reflection responses
Latent Dimension 1 R e s pon s e s pa c e R e s pon s e s pa c e Latent space E n c ode r D e c o d e r Random sets of designs
ONE BLTL BLBRTHREE FOUR a Wavelength R e f l e c t i on R e s pon s e s pa c e Desired response
Desired response
FOURTHREEBLBRBLTLONE D e s i gn s pa c e R e s pon s e s pa c e BLBR b E n c ode r EM Simulation
Latent spaceOptimum design parameters La t en t D i m en s i on Latent Dimension 1 La t en t D i m en s i on Figure 1.
Workflow of the manifold-learning-based design approach . a Forming the feasible regions andlearning sub-manifolds in the latent space. Each sub-manifold corresponds to one of the five nanostructure classes,whose unit cells are shown. Random sets of design parameters are generated for each class, and the correspondingresponses are found using an electromagnetic (EM) solver. By training an AE, the dimensionality of the responsespace is reduced into 2 or 3 and each sub-manifold is modeled using a separate GMM. Each GMM covers therange of feasible responses from a given class of nanostructures. b For inverse design, the dimensionality of thedesired response is reduced using the trained AE to observe the feasibility of the response using different classes ofnanostructures in a . Using a trained neural network (NN) that relates the design space into the latent response space,we search for the optimum solution with minimal complexity. It is observed from simulations that the reflectionresponses have a single resonance peak with a Fano-like lineshape as shown in a and b .tructure architecture from the initial guess (by the de-signer) to the most appropriate architecture, and 2) en-abling knowledge discovery in photonic nanostructuresby uncovering the roles of different design parametersin the final response. For this purpose, we present anew approach based on manifold learning for breakingthe geometric complexity of nanophotonic structuresduring solving the inverse problem. We reduce the di-mensionality of the response space by training an autoen-coder (AE) [48] over a set of training data and modelthe sub-manifolds for different classes of nanostructureswith different degrees of design complexity in the la-tent space (i.e., low-dimensional space) using Gaussianmixture models (GMMs) [49]. This representation alsoresults in understanding the underlying patterns in thedata and provides valuable insight about the underlyingphysics of the nanostuctures. For solving the inversedesign, we will map the desired response into the latentspace and search over different sub-manifolds with re-sponse feasibility to evolve to design candidates withminimal geometric complexity. As a proof of concept and without loss of generality, we apply this methodto study dielectric metasurfaces (as a popular class ofphotonic nanostructures) formed by elliptical hafniumdioxide (HfO ) meta-atoms on a silicon dioxide (SiO )substrate. To show the capabilities of our approach, we study theinverse design of metasurfaces with reflection responsesof Fano-like lineshape [37] using the unit-cell structuresshown in Fig. 2. These structures are composed of a se-ries (one to four) ellipsoids of HfO on a SiO substrate.The design parameters are periodicity ( p ∈ [ , ] nm) and the radii of the ellipsoids ( R i ∈ [ , ] nm).The height of the ellipsoids (350 nm) is fixed due tofabrication limitations, and the substrate is assumed tobe infinite in thickness. The simplest design ( ONE inFig. 2) and the most complex one (
FOUR ) have 3 and 9
NE BLTL BLBR THREE FOUR pR R 𝑥 𝑦𝑧 Figure 2.
The metasurface unit cells with different geometric complexities . Each unit cell is one to four HfO ellipsoids on a SiO substrate. The design parameters are periodicity ( p ∈ [ , ] nm) and radii of the ellipsoids( R i ∈ [ , ] nm). The height of all ellipsoids is fixed at h el p =
350 nm while that of the substrate is assumed to beinfinite. The design complexity (i.e., the number of design parameters) of the structures are between 3 and 9 for thesimplest (
ONE ) and most complex (
FOUR ) cases, respectively. The resonance features of the selected ellipses resultin a single resonance peak in the reflection responses of the metasurface formed by a periodic array of each one ofthese unit cells (see Fig. 1).design parameters, respectively.For the AI analysis, a total of 8000 random sets ofdesign parameters for the five unit-cell structures aregenerated, and the corresponding reflection responses arecomputed using three-dimensional finite-difference time-domain (3D FDTD) simulations, implemented usingLumerical, in the 300 < λ < 850 nm range, where λ is thewavelength. The incident beam is a normally-incidentplane-wave from the top with linear polarization in the x direction in Figs. 1a and 2. Each reflection responseis sampled at 551 uniformly-placed wavelengths in theoperating range. This structure is capable of generatingFano-type reflection responses. Also, when co-polarizedresonances of different ellipsoids are strongly coupledin the polarization direction (i.e., x -direction in Fig. 2),overall resonant responses with relatively high qualityfactors (Qs) are observed. To form the feasible set of responses for each nanostruc-ture in Fig. 2, we train an AE to reduce the dimension-ality of the response space from 550 (i.e., the numberof samples in the spectral response) into two or three(i.e., two-dimensional (2D) and three-dimensional (3D)latent spaces, respectively), while minimizing the re-constructed mean-squared error (MSE). In addition, wetrain a convex-hull using the algorithm explained in Ref.[35] to encompass the range of feasible responses usingeach of the unit-cell structures in Fig. 2. Figures 3aand 3b show the representation of the responses and the convex-hulls of the feasible regions for each structurein a 2D latent space, respectively. The correspondingresults for the 3D latent space are provided in the Sup-plementary Information. As seen from Fig. 3, the rangeof feasible responses expands as we increase the designcomplexity (i.e., the number of ellipsoids in the unit cell).The feasible region of the structure with one ellipsoid(i.e.,
ONE ) is the smallest due to both the weak reso-nance in the nano-antenna and weak coupling betweenthe nano-antennas in the periodic metasurface. An in-teresting observation from Fig. 3b is the large disparitybetween the convex-hulls of the
BLTL and
BLBR struc-tures while both having two ellipsoids in their unit cell.Also, the convex-hulls of the
BLBR and
THREE struc-tures share similar regions despite having different levelsof complexity. These non-trivial observations need tobe understood using the physics of coupling betweendifferent ellipsoids (or meta-atoms).The latent-space representation of the responses (seeFig. 3a) provides priceless information about the under-lying patterns in the reflection responses. For example, itis observed that the clockwise movement around the fea-sible region results in a red shift, and the counter clock-wise movement results in a blue shift of the resonances.In addition, Fig. 3a shows an increase in the magnitudeand Q of the reflection as we move from the center ofthe latent space towards the edges. To better quantify theknowledge provided by the manifold-learning approach,we present the color-coded manifolds for the wavelengthand the Q of the Fano-type resonances of the reflectionresonances in Figs. 4a and 4b, respectively. Figure 4aclearly shows the red (blue) shifts of the resonance wave- lue ShiftRed ShiftLower Q
ONEBLTLBLBRTHREEFOUR a b 𝑥 𝑦𝑧
Latent Dimension 1 La t en t D i m en s i on Latent Dimension 1 La t en t D i m en s i on Figure 3.
Latent-space representation of the reflection responses . a Representation of the reflection responsesand b the corresponding convex-hulls of the feasible regions for unit-cell structures in Fig. 2 in the latent space.The smallest and largest convex-hulls in b correspond to the simplest ( ONE ) and most complex (
FOUR ) structures,respectively. The magnitude of the reflection peak and the Q of the resonance increase as we move from the centerof the latent space towards the edges in a . Clockwise (counterclockwise) movement in the latent space results in red(blue) shift in the resonance peaks in the reflection responses.length by clockwise (counterclockwise) movements inthe latent space, and Fig. 4b shows that higher Qs areachieved at the corners of the latent space.Comparing the feasible responses of structures withdifferent unit cells in Figs 3 and 4 suggests that: 1)the ONE and
BLTL structures cannot generate high-Qresponses, 2) the
BLBR structure is far more capablethan the
BLTL structure in forming a variety of differentresponses despite apparent similarity; 3) The
BLBR and
THREE structures have a similar capability in generatinghigh-Q responses despite their different levels of designcomplexity; and 4) the
FOUR structure provides thelargest range of responses thanks to its highest level ofcomplexity. While some of these conclusions (e.g., 4)might be trivial at the beginning, others (e.g., 2 and 3) arenot expected at the first glance. This clearly shows thepower of our manifold-learning approach in knowledgediscovery in nanophotonics.In addition to comparing different structures with dif-ferent levels of complexities, our manifold-learning ap-proach can provide valuable insight about the roles ofdifferent design parameters. To show this capability, westudy the effect of rotating one of the ellipsoids in the
BLBR structure (as the least complex structure with alarge range of high-Q responses) on the reflection re-sponse while keeping other design parameters fixed (seeFig. 4c). It is clear from Fig. 4c that by rotating oneof the ellipsoids (i.e., increasing θ from 0), both thepeak reflection magnitude and the Q decrease with a mi-nor resonance wavelength shift, however, the reflectionresponse outside the resonance range stays almost thesame. To see this in the latent space, the correspond-ing responses, after dimensionality reduction, have beenshown in Fig. 4b using triangles with the same colors asthose of the actual responses in Fig. 4c. The movementof these triangles towards the center of the latent spaceand the lack of considerable clockwise or counter clock-wise rotation by increasing θ in Fig. 4b confirms theability of the manifold-learning approach in uncoveringthe observed role of θ .The amount of visually observable information aboutdifferent classes of unit cells, seen from Figs. 3 and 4shows the efficacy of our manifold-learning approachin knowledge discovery, i.e., providing valuable observ-able insight about the physics of nanophotonic deviceoperation. a b c Latent Dimension 1 La t en t D i m en s i on Latent Dimension 1 La t en t D i m en s i on Wavelength (nm) R e f l e c t i on Figure 4.
Q factor and rotation analysis . a Wavelength distribution of resonances in the latent space. b High/low-Q distribution of the responses in the latent space. c Effect of rotation of the ellipsoid on the reflectionresponse.
To better quantify the effectiveness of each unit-cellstructure in Fig. 2 in forming a desired response, wemodel the sub-manifolds of the corresponding responsesin the latent space for each structure using GMMs(see Methods and Supplementary Information). TheseGMMs provide the levels of feasibility of achieving anygiven reflection response for metasurfaces with differentunit-cell structures in Fig. 2, which will be helpful in theinverse design.To find a structure that generates a desired reflectionresponse, the first step is to find the corresponding pointin the latent space by reducing the dimensionality of thedesired response using the trained AE (see the two exam-ples in Fig. 5a). Next, we find the log-likelihood of thefeasibility for the desired response using the five differ-ent unit cells in Fig. 2 by employing their correspondingGMMs. We select the unit-cell structure with the highestlog-likelihoods. Once the unit-cell of the metasurfaceis selected, we use exhaustive search with a separatelytrained neural network (see Methods) for the forwardproblem (i.e., connecting the design and response spaces)of that metasurface to find the optimum design parame-ters. Figure 5 shows the implementation of the inversedesign problem for two desired responses with high andmoderate Qs (see Figs. 5b and 5c, respectively). Tocompare the effectiveness of metasurfaces with differentunit cells and the importance of the GMMs, we imple-ment each design using all possible unit cells (regardlessof the design feasibility), and the corresponding resultsare shown in Figs 5b and 5c as well as Tables 1 and 2, respectively. This experiment mimics the conventionaldesign approaches focused on finding the design param-eters regardless of the feasibility of the response. Figure5a suggests that the desired response with Q = 52 canonly be generated using the
FOUR structure. This isconfirmed by comparing the actual optimal responses(see Fig. 5b) and the negative log-likelihood values( − log ( p ) in Table 1). Similarly, Fig. 5a suggests thatthe response with Q = 42 can be generated by BLBR , THREE , and
FOUR structures. This is confirmed bydifferent optimal responses and log-likelihoods from Fig.5c and Table 2, respectively. Tables 1 and 2 also providemeans for using a trade-off between design complexityand the response errors. The importance of the manifold-learning approach is that it enables the consideration ofthe feasibility before attempting to design a device usinga pre-selected structure.
Of the four insights and conclusions from Figs 3 and 4(as discussed in Section 2.2), the difference in the sizeof the convex-hulls of
BLTL and
BLBR structures andthe similarity of the convex-hulls of
BLBR and
THREE structures contradict the expected increase of the rangeof feasible responses and the design complexity. To ex-plain the reason for these important conclusions from themanifold-learning approach, we perform the near-fieldanalysis of these three structures using the 3D FDTDtechnique.Figure 6 shows the resonant nearfield enhancementby the
FOUR , BLBR , and
BLTL structures. Due to theresonance of the individual ellipsoids at the incident b c
Latent Dimension 1 La t en t D i m en s i on Wavelength (nm) Wavelength (nm) R e f l e c t i on R e f l e c t i on Figure 5.
Inverse design of Fano reflection responses . a Representation of two desired responses with an idealresonance lineshpe (zero reflection outside the resonant region) in the latent space. b-c
Desired (ideal) and thecorresponding optimized responses (found by our inverse design approach) for different unit-cell structures in Fig. 2.
Table 1.
The optimal design parameters (in nm), normalized MSE (NMSE), negative log-likelihood, for the Fanoreflection response in Fig. 5b Design ParametersStructure p R1BL R2BL R1BR R2BR R1TL R2TL R1TR R2TR NMSE − log ( p ) One 783 178 17 0 0 0 0 0 0 0.60 186.74BLTL 599 63 63 0 0 173 130 0 0 0.546 468.08BLBR 758 149 104 148 121 0 0 0 0 0.095 3.01THREE 684 151 135 170 132 160 86 0 0 0.084 3.24FOUR 769 123 74 148 99 142 80 128 148
Table 2.
The optimal design parameters (in nm), NMSE, negative log-likelihood, for the Fano reflection responsein Fig. 5c Design ParametersStructure p R1BL R2BL R1BR R2BR R1TL R2TL R1TR R2TR NMSE − log ( p ) One 672 78 180 0 0 0 0 0 0 0.36 276.8BLTL 777 76 64 0 0 168 157 0 0 0.31 214.26BLBR 851 143 134 163 132 0 0 0 0 x - and y - directions forall structures in Fig. 6. This contrast is caused by the dif-ferent levels of coupling between individual (resonant)ellipsoids in the directions parallel and perpendicular tothe incident polarization (i.e., x - and y - directions, re- spectively). Our analysis shows that the maximum fieldenhancement (or the strongest light-matter interaction)is obtained for ellipsoids with resonances co-polarizedwith the incident polarization with strong coupling inthe direction of incident polarization (i.e., x -directionin Fig. 6). It is clear that the coupled ellipsoids in thex-direction in both FOUR and
BLBR structures (Figs.
642 06420642 0642 𝑥𝑦 𝑥𝑦𝑥𝑦 𝑥𝑦
BLTL BLBRFOUR BLTL a bc d p = 764 nm 𝑅 ! = 95 nm 𝑅 " = 149 nm p = 764 nm 𝑅 ! = 95 nm 𝑅 " = 149 nmp = 764 nm 𝑅 ! = 95 nm 𝑅 " = 149 nm p = 764 nm 𝑅 ! = 149 nm 𝑅 " = 95 nm Figure 6.
Nearfield enhancement . The nearfield simulation of the a FOUR , b , BLBR , and c , d BLTL structuresunder normal illumination with a uniform planewave at λ = 576 nm, 567 nm, 390 nm, and 383 nm, respectively, withlinear polarization in the x -direction. The color code shows the electric-field amplitude divided by the amplitude ofthe incident filed (as a measure of field enhancement by the resonant structures).6a and 6b, respectively) are responsible for the strongfield enhancement while the coupled ellipsoids in the y -direction in the BLTL structure (Fig. 6c) cannot providesuch enhancement. As a result, the resonance strengthand Q of reflection in
FOUR and
BLBR structures aresomehow similar and they both are much stronger thanthose of the
BLTL structure as observed from Figs. 3 and4. It is important to note that the different distances be-tween adjacent ellipsoids in the x - and y -directions arenot the main contributors to the difference of the light-matter interaction strength in different structures. Toresolve this potential confusion, Fig. 6d shows a dif-ferent structure that includes two ellipsoids with smallspacing in the y -direction. It is clear from Fig. 6d thatdespite this closeness, the orthogonality of the couplingdirection and the incident polarization results in weakfield enhancement. A further evidence of this fact is seenfrom Fig. 4c, where by rotating one ellipsoids in the BLBR structure (and thus, weakening the coupling in thepolarization direction), the resonance strength is reducedand the responses move towards the center of the latentspace. The simulation results for the field enhancementin a variety of structures with different levels of coupling in x - and y -directions are provided in the SupplementaryInformation to further clarify this insight. Neverthe-less, this discussion clearly shows the capability of ourmanifold-learning approach in uncovering the physicsof device operation, which can be used to form moreeffective designs. For example, by using this insight, theunit-cell structures with coupling perpendicular to theincident field will be excluded from the design optionsbefore any design attempt. More importantly, this insightexcludes the option of using rotated ellipsoids for thedesign, which considerably reduces the computationalrequirements for any inverse-design approach.In addition to the valuable insight about the deviceoperation physics, the manifold-learning approach helpsforming more intelligent designs. First, it shows thelevel of feasibility of a response using a given class ofstructures. Secondly, it provides a series of options withdifferent levels of complexity (and potentially robustness,although not discussed in this paper) for the design. Asan example, our approach provides three options for thedesign problem in Fig. 5c. It allows moving from a morecomplex structure (e.g., FOUR in this case) to the leastcomplex one (
BLBR ) in our simple approach of usingfive options for the design. For a more complex design ool, we envision using a trained algorithm with consid-erably more options beyond what an average designercan consider. Such a design tool will allow the evolutionof an initial design structure by the user to a differentfinal design that might be considerably less complex,more robust, less power-hungry, etc. Considering thefact that in many cases, the initial designs are motivatedmore by using the utmost fabrication capabilities (andthus, most complex structures), such evolutionary de-sign approaches will be very helpful in forming practicalstructures with optimal resource management.
The 3D FDTD simulations are conducted with the com-mercial software Lumerical. The simulation domain islimited to one period (p) in the lateral directions (i.e., x and y in Fig. 2) and perfectly matched layers are usedon the top and bottom layers (in the z -direction in Fig.2) due to the periodicity of the structures. To form the latent space of the responses, an AE istrained on a total of 6000 reflection responses obtainedby 3D FDTD simulations for the random sets of designparameters of the structures in Fig. 2. Each reflectionresponse is calculated in the 350 nm < λ < 800 nm, andit is sampled uniformly with 550 samples in this range.The dimensionality of the reflection responses is reducedfrom 550 into 2 and 3 using the trained AE with 11 layers(550, 200, 100, 50, 20, d, 20, 50, 100, 200, 550 nodes ateach layer, respectively, where d is the dimension of thelatent space). The hidden layers have tangent-hyperbolic(tanh(.)) activation functions, and the input and outputlayers have linear activation functions. The MSE loss isminimized during the training using Adam optimizer inPython. The training is stopped after 500 epochs if therequired MSE is not reached.GMMs are used for modeling the sub-manifold of theresponses in the latent space for each design complexity.The distance metric is set to correlation, and the max-imum distance is 0.3, with a maximum of 5 Gaussiandistributions for each model. The GMMs are trained onthe training samples in 2D and 3D spaces. To find the optimum design parameters, a feed-forwardNN is trained from the design to the response space.The network has 8 layers with 9, 20, 50, 100, 100, 200, 400, 500 nodes at each layer. The activation functionsof the hidden layers is tanh(.). The design parametersare normalized to have zero mean and unit standarddeviation. The weights of the NN are trained using theAdam optimizer in Python to minimize the MSE.To perform the inverse design with a desired reflectionresponse and a given design complexity (i.e., a givenunit-cell structure in Fig. 2), we use the exhaustivesearch (with 10 random sets of design parameters) inthe design space using the trained feed-forward NN. Theoptimum solution with minimum mean-squared distanceto the desired reflection response for each design com-plexity is reported as the solution.Note that this is the simplest approach for the inversedesign using AI. In a more aggressive approach withless computation requirements, recent techniques liketraining a pseudo-encoder and combining an inverse AIdesign (from the response space to the reduced designspace) with a considerably smaller exhaustive search(from the design space to the reduced design space), asexplained in Ref. [36] can be used.All of the AI algorithms are implemented in Pythonand Keras on a system with Core i7 CPU, one RTX2080GPU, and 32 GB of RAM. Competing interests
The Authors declare no Competing Financial or Non-Financial Interests.
Acknowledgment
The work was funded by Defense Advanced ResearchProjects Agency (D19AC00001, Dr. R. Chandrasekar).
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