1. INTRODUCTION

Chiral nanostructures are non-superimposable to their mirror image, and produce a different optical response for left circularly polarized (LCP) and right circularly polarized (RCP) light [13" target="_self" style="display: inline;">–3]. This difference can be quantified by measuring chiroptical responses such as circular dichroism (CD) and chiral anisotropy factor (gCD), providing metrics for the chiral asymmetry of these structures. Hindered by their small chiral asymmetry and electromagnetic interaction volume, naturally occurring chiral structures such as amino acids, DNA, and RNA exhibit low CD signals, which imposes limits on their detectability. This limitation can be resolved by coupling these naturally occurring structures to artificially engineered plasmonic nanostructures under resonant excitation [46" target="_self" style="display: inline;">–6]. Plasmonic metamaterials can strongly enhance a molecular CD signal [6] and even displace it to longer wavelengths due to the interaction of the strong resonant plasmonic near-field with the chiral biomolecules [5]. Chiral plasmonic metamaterials exhibit inherent CD signals whose amplitude depends on the degree of chiral asymmetry of their geometry and thus presents an avenue for designing and manipulating nanoscale chiroptical effects. By changing the geometry of these plasmonic metamaterials (chiral and nonchiral), one can tune a resonant spectrum and target specific applications using bio-assembled as well as nonbiological systems [7,8]. The interaction of biomolecular structures with superchiral fields from chiral plasmonic nanostructures can also induce asymmetric changes in the retardation phase effect of the modes in the nanostructures and provide fingerprints for enantiomer discrimination [9]. The range of applications for plasmonic chiral systems also includes, among others, improved spectroscopy techniques [10,11], photocurrent generation [3], and bolometry [12]. At the same time, while chiral plasmonic metastructures offer very strong CD signals, they also present design and optimization challenges due to the iterative and case-by-case simulations required to solve Maxwell’s equations for a given geometry, which can be a time- and resource-intensive process. Given the complex relationship between chiral plasmonic metamaterials and their chiroptical response, a data-driven approach to the design and optimization of the structures can improve its efficiency.

Deep-learning (DL) as a data-driven technique for analysis and prediction has permeated several disciplines such as natural language processing [13], image recognition [14], genetics, and biology [15,16]. Instead of generating simulation results by running through predefined systems of equations in a given geometry, a DL architecture can be trained to recognize patterns in a given dataset, identify attributes, and predict responses, thanks to its capability to reproduce arbitrarily complex functions. As a type of representation or feature learning, DL brings machine learning (ML) closer to artificial intelligence where human-like and exceedingly challenging tasks can be completed by trained systems. A typical example of the capabilities of this technique is the success of deep neural networks with reinforcement learning in playing games such as shogi, chess, and Go [1719" target="_self" style="display: inline;">–19]. In recent communications, neural network models have been utilized as a fast prototyping tool in the design of metamaterials [20–26" target="_self" style="display: inline;">–26]. The generative adversarial network (GAN) [27] and image processing [28] approaches have also been used to link the geometry of metamaterials with their optical response. Hybrid techniques consolidating compositional pattern-producing networks and cooperative coevolution [29] as well as those blending deep generative models with semisupervised learning [30] have also been proposed for the inverse design of metastructures. The prediction power of DL models even extends to the near- and far-field distributions of arbitrary 3D nanostructures [31]. However, DL-based prediction models for metamaterial design suffer from low accuracy owing to the huge mismatch between the dimension of input and output parameter, especially for inverse retrieval designs. Also, the generation of the training set via traditional simulation routes can be time and resource intensive for the study of nanoscale chirality, requiring an efficient use of the generated training set during the ML design and training. In addition, capturing the local optima (plasmonic/CD resonances) of chiroptical responses is challenging for most DL models and sometimes necessitates the setup of an auxiliary network, which adds to the complexity of the model [20].

In this communication, we propose an end-to-end multitask DL (MDL)-based model for the design and optimization of 3D chiral metamaterials. MDL models have gained root in the computational study of semantics [32] and transportation [33,34], as well as pose and action recognition [35]. Multitask learning draws on its implicit data augmentation, eavesdropping, attention focusing, representation bias, and regularization for effective and efficient generalization [36,37], eliminating the need for auxiliary networks or equivalent approaches to stabilize the model’s output of physically relevant information.

Our MDL model comprises a single bidirectional neural network solving two tasks: the accurate prediction of the full chiroptical response of a chiral metamaterial from a set of geometric parameters, via a forward prediction path, and the accurate retrieval of the geometric parameters that can produce a given input of a full chiroptical response, by solving the inverse problem via an inverse prediction path. To bridge the mismatch gap and enhance the prediction accuracy for both forward and inverse predictions, especially at CD and plasmonic resonances, a joint-learning feature is incorporated in the model training. This feature ensures the comparison of errors in the learning of tasks, allowing for a well-generalized system. Consequently, the MDL model ensures an efficient use of the training set to achieve faster convergence [38,39] and provides a practical guideline for implementing similar ML systems in a variety of design problems with nanotechnological applications. This work is organized in three major parts. The first concerns the description of the chiral metamaterial absorber modeled after the yin-yang symbol and its optical properties calculated by the finite element method (FEM). The second contains the discussion of the MDL model composed by the forward and inverse design paths (FDP and IDP, respectively). Finally, in the third, we apply a sample MDL-optimized structure in the application of sensing biomolecular enantiomers. We envisage that our work will inspire the use of ML as an effective and efficient data-driven metamaterial design and optimization tool, taking advantage of the multitask technique that we detail herein.

2. CHIRAL METAMATERIAL MODEL AND FORMALISM

Generally, the metamaterial absorber structure used in this study consists of top yin-yang-shaped Au nanoparticles (YNPs), a PMMA layer, a Au backreflector, and a bottom glass layer. Figure 1(a) shows the unit cell of the meta-absorber structure. The YNPs are defined by their radius R0 and thickness t1. The period, ρ, is set following 2R0+50nm. To eliminate spurious effects due to mathematically sharp edges, the top corners and the tip of YNP are rounded with a minimum radius of 10 nm, corresponding to an R0 of 100 nm, and scale up with larger values of R0 to preserve the YNP shape. The dielectric constants of glass and PMMA are 2.13 and 2.25, respectively. The dielectric constants of the Au YNP and backreflector are interpolated from the Johnson and Christy dataset [40]. The medium surrounding the metastructure is chosen to be air. The incident beam is normal to the metamaterial. These models were calculated using an FEM-based commercial software package, COMSOL Multiphysics.

Fig. 1. Schematic of (a) a single YNP chiral meta-absorber array with definition of incident circularly polarized lights and a unit cell with dimensions. (b) Three single YNP metastructure configurations: Au YNP/Glass (YG), Au YNP/PMMA/Glass (YPG), and Au YNP/PMMA/Au/Glass (YPAG). (c) Absorption and CD spectra of the three metastructure configurations (YG, YPG, and YPAG), showing their plasmonic resonances λp (λp=620nm, 625 nm, and 645 nm, respectively) and revealing a strong chiroptical response for the metamaterial absorber case (YPAG). Here, R0=100nm, t1=40nm, t2=50nm, t3=100nm, and t4=200nm.

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To illustrate the relevance of the multilayered structure on the intensity of the fields and chiroptical response under resonance, we consider single YNP under three structures of comparable dimensions: YNP/Glass (YG), YNP/PMMA/Glass (YPG), and YNP/PMMA/Au/Glass (YPAG). Figure 1(b) shows the three metastructure configurations with single YNPs. The YG setup comprises a 40 nm thick Au YNP and a glass substrate with thickness t4. For the YPG configuration, a PMMA polymer layer with thickness t2 is introduced on the glass substrate to create a YNP/PMMA interface. Fabrication of the homopolymer layer can be conveniently achieved through spin-coating. The third configuration (YPAG) adds an Au backreflector layer of thickness t3, between the PMMA and glass layers. Such structural combination with a backreflector largely increases the optical absorption of the system, by allowing the interaction between the nanoantenna and reflected modes [41–45" target="_self" style="display: inline;">–45]. Figure 1(c) compares the absorption and CD spectra of the YG, YPG, and YPAG metastructures. The YPAG structure shows a comparatively large and broad absorption peak and large differential absorption, both arising from the interaction between the Au YNP and the Au backreflector. Further comparison of the cross-sectional local field enhancement and surface charge density distributions for the three structures at their plasmonic resonances λp reveals sharp differences for LCP and RCP illumination at their respective plasmonic resonances (see Appendix A, Fig. 7). This further illustrates the strong chirality of the YNP-based structures especially in the meta-absorber case where the field is highly enhanced in the polymer spacer layer.

The enhanced chiral field and CD response in the metamaterial absorber case make it ideally competitive for chirality-related applications. The base YNP structure can be modified to generate other complex chiral designs, i.e., dimers and trimers (see Appendix A, Fig. 8). However, the optical response of these structures is highly sensitive to small variations of their geometric parameters. Thus, the optimization of a given design necessitates us to map out the response of many configurations across the design space. Each of these combinations requires solving the full electrodynamic problem under the illumination conditions, which is computationally costly. By combining a relatively small set of full wave simulations with our MDL approach, the design space can be explored in fine detail at a fraction of the computational cost. Experimentally and theoretically, a number of planar chiral geometries have been investigated and demonstrate huge CD signals including spirals [46–49" target="_self" style="display: inline;">–49], gammadions [50–54" target="_self" style="display: inline;">–54], L-shaped [55], Γ-shaped [12], and z-shaped [3] nanostructures. For the purpose of our demonstration of the MDL model, we adopt a variation of our YNP metastructure defined by double-top YNPs with radius, thickness, and gap distance R0, t1, and d, respectively (see inset in Fig. 2). The polymer thickness, t2, and the Au backreflector thickness, t3, affect the reflection of the beam and its interaction with the chiral resonance of the YNP, regulating the shift in the CD spectra. Due to the dominant effect of the Au backreflector, we have not explored the variation of thickness of the glass layer, t4, and assigned a fixed t4 value of 400 nm. The period of the structures is set following Px=3R0+50nm,Py=d+2(R0+25nm). Here, Px is the period along the x direction, and Py is the period along the y direction. By considering the double YNP structure, we introduce the gap between the nanoresonators as an additional parameter influencing the CD response, which is in a highly nontrivial relationship with the other geometric parameters. We have chosen this structure as a case study for our MDL model because of two sets of characteristics: (1) the continuous curved surfaces and the difference in scale between the whole structure and its finest features, such as its tip, increase the complexity of the numerical optical simulation of the metamaterial, thus representing a desirable candidate for alternative or supporting computational approaches; (2) the complex nonlinear relationship between the different design parameters and the structure’s optical response makes it a good example for the power of our MDL approach.

Fig. 2. Schematic of the bidirectional multitask deep-learning model for chiral metamaterial design consisting of forward design path (FDP) and inverse design path (IDP). Each path is composed by shared layers and task specific layers with joint optimization functionality. The model is set up in an end-to-end fashion where the geometric design parameters, CD, and LCP/RCP absorption spectra can be treated as input or output at specific ports. Here, the geometric design parameters are the YNPs thickness, PMMA thickness, YNPs radius, and YNPs (respectively represented as xi, i=1,2,…,6). x4 has been taken as a constant in the data shown in this study, but is nonetheless included in the model to represent the general parametrization of the system. The inset shows the metamaterial absorber geometry used to exemplify the use of the MDL.

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3. MULTITASK DEEP-LEARNING ARCHITECTURE

As depicted in Fig. 2, the model consists of an input layer, tensor layers, a normalization layer, optical and chiroptical task execution layers, and an optimization layer. The inset in the top right shows the schematic of the metamaterial absorber used to train the DL model. This model takes into account the plasmon resonant peaks with a regression learning objective. Our model has unique multitasking and joint learning capabilities that contribute to the efficient generalization of the parameter space [5658" target="_self" style="display: inline;">–58].

3.2 A. Forward Design

In the forward prediction path, the varying dataset scales across the input design parameters, absorption, and CD responses would make the model data-scale-dependent if trained directly, resulting in poor generalization. To eliminate the effect of the varying input length scales on the generalization of the model, we employ a normalization layer following the relation where a and b index the row and column, respectively, such that Xb is the bth column of the input parameter matrix. X¯b and σ(Xb) are the mean and standard deviation, respectively.

The normalization provides a well-conditioned dataset for optimization by ensuring that the training is less sensitive to the scale of features to be processed by the shared layer [59]. At the shared layer, we adopt a hard parameter sharing of four hidden layers, each with 1024 nodes. These hidden layers are shared between all the individual task-specific output layers of the network. The shared hidden layers hold computational weights from the task-specific layers. That is, the CD learning leverages the LCP/RCP spectra task-specific learning to enhance accuracy.

We regularize all hidden layers by applying penalties on layer activity during optimization with an L2 regularizer in order to learn sparse features and internal representations of raw observations [60]. The task-specific layer consists of two independent parts: the main task and auxiliary tasks. The auxiliary tasks are subtasks expected to assist in finding rigorous, rich, and robust representation of the input design parameters to benefit the main task. Learning auxiliary tasks restrict the parameter space during optimization and push for a faster convergence. The main task, which characterizes the desired output response, exploits and jointly learns from the auxiliary tasks via the shared layer. The main and auxiliary tasks correspondingly generate three single-task losses.

To optimize the MDL network, the joint multitask cost function comprising the three single-task losses is minimized. Here, the principal multitask cost function, subject to optimization, is expressed as [61] where i and j index the training set and the three learning tasks, respectively. yji refers to the simulated outputs from the three tasks (CD and LCP/RCP absorption signals). The model function f takes as input xi, which is the 1×5 design parameter matrix comprising the YNP radius R0, the gap distance d, the YNP thickness t1, the polymer thickness t2, and the Au backreflector thickness t3 with weight, w. The definition of d is illustrated by the inset in Fig. 3. See Appendix B for derivation.

Fig. 3. MDL model performance. (a) Numerical simulation and (b) MDL prediction CD results of the dimer structure at varying gap distance d (50–160 nm), across the visible and near-IR regime. R0=100nm, t1=40nm, t2=50nm, and t3=100nm. (c) Numerical simulation and (d) MDL prediction results of the dimer at varying YNP thickness (t1). The color legend has been truncated at ±0.2 for clarity. Inset, definition of the gap d. (e) Learning curve within 3000 epochs. (f) Discretized model performance at selected t1 = {60 nm, 80 nm, 100 nm} corresponding to the horizontal dots in (c) and (d). (g) Model performance at λ0=755nm across varying t1, corresponding to the vertical short dashes through (c) and (d). Here, R0=100nm, d=100nm, t2=50nm, and t3=100nm.

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We adopt the mean squared error (MSE)—the quadratic loss, which is the sum of squared distances between our target variable (simulated CD) and predicted CD values—to average the losses over the output. The Adam moment estimation stochastic optimization approach is used to compute an adaptive learning rate for each of the internal parameters of the model [62]. It is typical of ML models to miss the resonances of datasets with high volatility during prediction. This is because the probability distribution is centered at the off-resonance for each neuron in the output layer, neglecting the local optima. The joint loss optimization functionality enables collective error correction for the forward prediction task, allowing an accurate prediction at the local optima. The training set utilized ∼80% of the 640 collected samples with the remaining as the validation set. Each sample is constituted by the full 3×241 LCP absorption, RCP absorption, and CD spectra data points. These 241 data points were generated at 5 nm step intervals within the 400–1600 nm wavelengths. The MSE recorded is 0.000441, which is indicative of the model’s accuracy. After training, we use the validation dataset, which is unseen throughout the training, to evaluate the model. Within short prediction intervals, the MDL model exhibits prediction results comparable to simulation data. A comparison between simulated [Fig. 3(a)] and MDL-predicted [Fig. 3(b)] CD response at varying dimer gap length d shows a good agreement. The range of wavelength values corresponding to high and low chiroptical activity is essentially identical. Comparing Figs. 3(c) and Fig. 3(d) for varying YNP thickness also shows a good agreement even for ultrathin YNPs. The MDL prediction retains the same wavelength interval as the simulation, providing a full continuous prediction spectrum (241 prediction data points). In this case, essential information within short parameter intervals can be retrieved to enrich the analysis, design, and prototyping process. The learning curve in Fig. 3(e) illustrates the fast convergence of the MDL model. Specifically, the single end-to-end MDL model accurately captures both resonant and off-resonant CD signals of the metastructures for varying geometric parameters as depicted by Figs. 3(f) and 3(g). The training set emphasized larger gap dimensions, which are easier to fabricate consistently, rather than more experimentally challenging dimers with smaller gaps. This bias accounts for the small discrepancies observed in Fig. 3(b) with respect to the simulation values in Fig. 3(a), a deviation that could be further reduced by a more homogeneous sampling. The situation is similar, although less pronounced, for the data in Fig. 3(d).

Given the design parameter space of the structure, vast sets of CD responses from varied parameter configurations can be retrieved across the visible and near-IR regime (λ0=400–1600nm) via the trained MDL model. Figure 4 shows the interdependencies among the design parameters of the chiral metamaterial absorber and the evolution of CD generated by the MDL model with varying geometric parameter dimensions. For each plot, the unrepresented geometric parameters are held constant during the data generation. Figure  4(a) shows the CD values for varying YNP radius at t3={100nm,150nm,200nm}. We observe a slight shift and broadening of the CD peaks within the high chiroptically active wavelength interval (520–800 nm) for R0 within the range 100–250 nm at varying t3. Although the influence of the Au backreflector thickness, t3, on the CD response is significant, its effect is comparatively small. This is clear when we consider that the Au layer indeed serves as a reflector, and we should expect that changes in thickness, when it is already above the penetration depth of the radiation, will not be very impactful to the overall response of the metamaterial.

Fig. 4. MDL-predicted CD progressions. (a) CD evolution by varying YNP radius at t3 = {100 nm, 150 nm, 200 nm}. CD map plot by varying concurrently, (b) YNP radius and polymer thickness at t1 = {5 nm, 25 nm, 50 nm} for λ0=780nm, (c) YNP thickness and polymer thickness at R0 = {100 nm, 150 nm, 200 nm} and λ0=700nm, and (d) YNP radius and YNP thickness at t2 = {10 nm, 45 nm, 100 nm} and λ0=650nm. The color legend has been truncated at ±0.2 for clarity, but high-CD areas have been highlighted by adding the contour regions corresponding to CD values of 0.5.

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Figure 4(b) compares the CD response by concurrently varying the YNP radius and t2 at t1={5nm,25nm,50nm}. This plot is obtained at λ0=780nm. We observe the effect of the quasi-linear regions, where the CD remains constant irrespective of the coupling gap between the nanoresonators. Therefore, the YNP and PMMA thicknesses together with the YNP radius control the shift in the CD resonances. Varying these sets of parameters yields highly distinguishable CD map plots as illustrated in Figs. 4(b), 4(c), and 4(d). Figure 4(c) shows CD values at R0={100nm,150nm,200nm} generated by varying concurrently, t1 and t2 at λ0=700nm. The contour regions correspond to a CD magnitude of 0.5. Figure 4(d) illustrates the interaction between the YNP radius and YNP thickness at t2={10nm,45nm,100nm} for λ0=650nm. It can be deduced that the CD magnitude is generally lower for larger values of YNP radius. However, with thinner YNPs (t1<30nm), large CD signals can be realized by larger YNP radii (300nm<R0<400nm). For thicker YNPs (t1>45nm), large CD signals can be attained for both ultrathin (t2<20nm) and thicker (65nm<t2<100nm) PMMAs at larger YNP radius. Appendix D shows variations in CD across the studied wavelengths for varying PMMA thickness.

Having the trained network, we can easily extend its output to cover many more geometric parameter combinations, and for different wavelengths, although this sample should suffice for illustrating the method’s capabilities, as well as obtaining a qualitative understanding of the physical system.

It is relevant to underscore a point of particular interest of employing such an MDL approach, which is that it requires a much smaller amount of computational resources than producing an equivalently dense dataset through traditional simulation methods. Let us examine a quick estimate for the comparison of both approaches. For each combination of geometric parameters, it takes a typical i7-CPU PC (which, for simplicity, we will assume that consumes the total output of its 330 W power source regardless of the task performed) more than 3 h to compute a full spectrum when using COMSOL with precision levels adequate for our system. This order of magnitude for the duration of the computation is also representative of other simulation packages and frameworks. In our case, the total energy used for creating the model (combining the energy required to generate the training set with COMSOL and actually training the network) is approximately 0.63 MWh spanning 80 days using the above example PC. After training, using the MDL model requires less than a second to produce the response for a given geometric parameter set. On the other hand, for the five geometric parameters and our chosen sampling density, the total number of simulations required to obtain the sample result density that the trained MDL affords would be in the tens of thousands (∼28,106 samples), which would require an outrageous number of years (∼10,000 years) to produce, with a total energy consumption of approximately 29 GWh using the example PC. Therefore, in this (admittedly extreme) comparison, the MDL approach would outperform a naïve simulation-based approach by 7 orders of magnitude in terms of speed and energy expenditure.

3.3 B. Inverse Design

After obtaining a high prediction accuracy in the forward path, we proceed to design an IDP. It is important to realize that this task is very complex and sensitive, because of the large imbalance between the input and output dimensions of the model: five geometric parameters compared with the full CD, LCP, and RCP spectra (1×5 versus 3×241, respectively). While it is plausible to apply up- and down-sampling approaches to resolve this mismatch, essential features may be lost in the process, especially in the case of complex structures like the YNP. Such lost features introduce prediction errors that compound over a training loop, resulting in wide variations in the retrieved geometric parameters for comparatively small changes in the input spectra. Reverse-engineering the multitask forward design path makes use of the entire set of CD data points and maps each output geometric parameter to the desired full input spectra. The IDP takes a simulated CD spectrum as input, with the objective of retrieving the geometric parameters required to produce it. To achieve this objective, we connect the input spectrum to three isolated dense networks with two layers, responsible for mapping the chiroptical response to a shared latent-space representation with four layers, each composed by 2048 neurons. The shared latent space links to five task-specific layers with each layer managing the output of a designated geometric parameter (see Fig. 2).

Running this inverter network gives us a set of geometric parameters, which will be close to the ones generating the target CD spectrum. However, given the interdependency of these parameters and the high sensitivity of the CD response to geometric parameters [see, for instance, Fig. 4(b)], we should not assume that this is sufficient to obtain a satisfactory system prediction. Therefore, assessing the ML-predicted CD spectra will be central to characterizing the efficiency of the inverse retrieval model. For verification, the retrieved geometric parameters are re-fed into the forward prediction path, so that a comparison can be drawn between the target CD from the ground-truth geometric parameters and the ML-predicted CD from the ML-retrieved geometric parameters. Figure 5(a) shows a simulated CD serving as target spectrum (green line), with its corresponding geometric parameters in Fig. 5(c) (green bars). Upon feeding the simulated CD spectrum into the inverse path, the IDP retrieves the geometric parameters in Fig. 5(c) (red bars). Although it is clear that the inverter network found values that are very close to the ground-truth, we proceed to verify the retrieval by inputting these geometric parameters into the forward path and comparing its predicted CD response [red dots in Fig. 5(a)] with the simulated CD spectrum, thus confirming the success of the inverse path.

Fig. 5. Inverse design with the MDL model. (a), (b) Simulated (green solid lines) and predicted (red dots) CD spectra. (c), (d) Corresponding simulated (green bars) and retrieved (red bars) geometric parameters. Red dots in (a), (b) are predicted from the MDL model with geometric parameters retrieved [red bar in (c), (d)] for the target simulated CD spectra in (a), (b). [See Figs. 9(a) and 9(b) in Appendix C for absorption spectra comparison.]

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Figures 5(b) and 5(d) were generated following the same procedure. Both input spectra, in Figs. 5(a) and 5(b), were chosen to illustrate the model acting on two profiles with distinct properties (inflection points, negative and positive resonant CD peaks). Generally, there is a good agreement between simulation and ML-prediction responses, although we can observe slight differences in the spectra that cohere with the small differences between the sets of retrieved and ground-truth geometric parameters. Figure 9(c) in Appendix C compares different approaches to the training of the model such as training with and without the auxiliary tasks as well as separately training the LCP and RCP spectra. We observe comparatively better performance in the case of our MDL model. The MDL-optimized dimer structure can be adopted for several applications including photodetection of circularly polarized light (CPL), owing to its high consistency and accuracy [see Fig. 10 in Appendix D]. However, we will now explore the use of an example MDL-optimized chiral plasmonic structure in sensing adsorbed molecular enantiomers.

4. CHIRAL BIOSENSING

The strength of the chiroptical response from biomolecules is mostly limited by the small magnitude of their geometric chiral features, relative to the periodicity of CPL, in comparison with chiral plasmonic nanoantennas. Therefore, it is often necessary to use large molecular concentrations of nonracemic mixtures to provide detectable CD signals [5,63,64]. Moreover, chiral biomolecular structures have chiroptical activity in the UV and are thus difficult to detect with common instrumentation. Figure 6(a) shows the weak CD resonant peak of a molecular enantiomer pair in the UV. The optical activity of chiral molecules can be enhanced by a handedness-preserving Fabry–Perot cavity resonator [65] acting as a metamirror. Such metamirrors selectively reflect one CPL preserving its handedness while absorbing the other [66]. The molecular CD signal can also be enhanced by the interaction with the strong near-field of plasmonic structures, as well as duplicating its CD signal into the visible range, where they are easier to detect [5,67]. These advantages, which greatly enhance the detection capabilities of a chiral biomolecular sensor, are in principle independent from the chirality of the plasmonic metamaterial. Nonchiral nanoparticles can offer an effective and efficient sensing of molecular enantiomers [6,67] by coupling with the chiral molecules. However, local superchiral near-fields can offer additional possibilities for molecular CD detection with chiral plasmonic nanostructures on a metal-biomolecular platform [11]. Superchiral fields are those with a larger chirality than CPL, as characterized through their optical chirality, C [68], and can arise on chiral plasmonic structures illuminated with either linearly or CPL [9]. Such near-fields with enhanced optical chirality cause asymmetric phase differences between the chiral modes in the presence of a chiral dielectric, offering a path for sensing biomolecular chirality [69].

Fig. 6. Enantiomer detection. (a) The CD spectra of (red) left-handed medium (LHm), and (blue) right-handed medium (RHm) with molecular CD resonance (λm=380nm) in the UV. (b) CD spectra comparison of the right-handed chiral metamaterial absorber (RHcma) with (blue) and without (red) chiral medium (CM). Inset is the electric field at the plasmonic resonance, λp (λp=665nm), of the bare chiral metamaterial absorber for LCP and RCP light. (c) CD summation to remove metamaterial background CD signal to reveal the LH (blue solid line) and RH (green solid line) enantiomer pair CD signals. λm, λp, and λmp represent the resonant wavelengths for the CD of the bare molecules, the plasmonic chiral metamaterial absorber, and the metamaterial covered with chiral media, respectively. Inset, schematic representation of an enantiomeric protein molecular pair (L and D isomers). (d) Electric field, surface charge density, and optical chirality density distributions of the chiral metamaterial absorber covered by chiral media (CM) at λmp=720nm.

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In what follows, we present theoretical results illustrating the behavior of one of the ML-optimized nano-dimer metastructures in the presence of molecular enantiomers, using the excess CD method to quantify the chirality of the molecular sample [70]. The structure was optimized for value of CD of 0.35. The geometric parameters of the metamaterial are R0=100nm, t1=30nm, t2=50nm, t3=200nm, t4=200nm, and d=100nm. Modeling the chiral molecules, the dimer is covered by a 40 nm thick chiral medium with 1.6 refractive index.

The chiral dielectric medium is modeled following the constitutive equations: Here, ε0 and εr are the permittivity of free space and relative permittivity, respectively. Similarly, μ0 and μr are the permeability of free space and relative permeability, respectively. E and B are the complex electric field and magnetic flux density, respectively. D and H are the electric field displacement and the magnetic field, respectively. ξ is the chirality factor of the molecular sample, which shows very low values for low-density or near-racemic samples. Using a two-state model for the molecules, ξ can be expressed as a function of frequency as where βc controls the magnitude of the chiral asymmetry, ω is the angular frequency of the radiation, ω0=2πc0/λmol at the molecular excitation wavelength, λmol, and Γ12 defines the relaxation rate of the excited molecule, with its indices describing its quantum states. Here, λmol=380nm and Γ12=0.41eV. The expression and values for the chirality factor, ξ, are adopted from Govorov et al. [5,9] and follow from the quantum equation of motion for the electronic density matrix when assuming a dilute molecular sample.

Now, we proceed with the excess CD method to compute the chiral properties of the molecules, when in interaction with left-handed and right-handed chiral metamaterial (LHcma and RHcma, respectively). First, a baseline CD is calculated from the metamaterial absorber with racemate (ξ=0) molecular coverage. In the process, a distinguishable dielectric medium-induced CD redshift (∼55nm) is observed as illustrated in Fig. 6(b). This redshift arises from the refractive index of the molecular coverage, which is larger than that of the surrounding medium (air). The inset shows the local field enhancement of the bare metamaterial absorber without biomolecules, illustrating its strong and chiral near-field. Then, we can compute the sum of the CD signal of the two YNP metamaterial absorbers, with chiral and racemate molecular coverage, yielding a change in CD response associated with the biomolecules of opposite handedness. The coupling of the enhanced plasmonic near-field to the molecules is critical for biomolecular sensing. In Fig. 6(c), we can see how the CD signal of the chiral molecules shifts its peak to the visible range and increases its magnitude by approximately a factor of 2. The excess CD signal over that of the chiral metamaterial absorber, produced by the coverage of molecular enantiomers, is a detectable magnitude that allows us to characterize the presence and handedness of chiral biomolecules, as illustrated in Fig. 6(c).

Finishing this section, we present additional details on the chiral properties of the example metastructure, showing its near-field enhancement, surface charge density, and optical chirality parameter C. The latter is calculated, for harmonically oscillating fields, as [71] Figure 6(d) compares these three variables for the LHcma and RHcma systems, under the dielectric chiral medium described above. The first row is the top and cross-sectional view of the electric field enhancement distribution of the LHcma (red border) and RHcma (green border), when illuminated by LCP and RCP light. A y-z cross-sectional plane is taken to show the field enhancement inside the layered metamaterial. The surface charge density is illustrated in the second row, and the optical chirality maps in the third row. The electric field, surface charge density, and optical chirality are evaluated at the chiral media-modified plasmonic resonance, λmp=720nm. Importantly, the optical chirality maps reveal the superchiral near-field regions arising from the curvature and chiral geometry of the nanoantennas.

5. CONCLUSION

We propose a highly portable and functional MDL model to comprehensively study 3D, arbitrarily complex chiral metamaterials, and exemplify its usage with a chiral metamaterial designed after the yin-yang symbol. The model is composed by a single end-to-end bidirectional architecture, capable of performing optimization and inverse retrieval operations, and that takes advantage of the supporting role of two auxiliary tasks to facilitate the learning of the primary task, CD in our case. This feature in particular distinguishes it from other approaches discussed in the literature, and represents a reduction of the complexity of the DL framework while ensuring an efficient use of the training set toward a highly generalized system. As a data-driven approach, this MDL model requires a prior database of results created with methods such as FEM simulations or experimental data, but it helps in avoiding the huge computational cost that would be required to explore the vast design space of the physical system in fine detail. An additional advantage resides in the fact that a trained model is a fast, lightweight, highly transferrable tool that can drastically reduce the computational time used for subsequent studies of the system, both for the designers and for other research groups. Given a set of geometric parameters, the forward design path of the model predicts CD spectra with values virtually identical to the simulations we used as ground-truth. And, vice versa, for input CD spectra, the model retrieves the set of geometric parameters that would produce such input CD spectra by solving the inverse problem. As a result, the trained model can be used to explore the entire design space and thus render a complete account of the intricate relationship between the metamaterial’s geometric parameters and its chiroptical response. This is made possible by the joint-learning feature incorporated in the model. For nanophotonic applications, the design and prototyping process needs to be robust due to the complexity of light–matter interaction with nontrivial geometries. The multitasking DL-based prediction model presented herein can aid in engineering any potential fabrication of the nanophotonic structures for desired optical and chiroptical response toward a variety of applications. Illustrating a potential context for the utilization of this system, we have shown additional chiroptical properties of this chiral metamaterial absorber structure in the context of its interaction with molecular enantiomers. The high efficiency and accuracy of the end-to-end MDL model makes it a valuable tool for the study of complex physical phenomena, particularly for the design and prototyping of nanophotonic structures toward their application as biosensors, as photodetectors, or in polarization-resolved imaging and CD spectroscopy, among others.