1. Introduction

Nanoplasmonics are indispensable in sensor and energy applications. Their electromagnetic (EM) responses are largely determined by their geometric parameters. One representative example of nanoplasmonics is the bowtie shaped nanoantenna formed by a pair of nanoprisms with extremely sharp tips and small gaps around 10 nm [1–4] as shown in Fig. 1(a). Such nanostructures can induce plasmonic resonance spectra in the far-field which is ultra-sensitive to any dielectric changes in the interfacial environment [5]. It can also concentrate energy at the hot spots formed around sharp tips and between small gaps to achieve enormous near-field enhancement for single molecule detection in surface-enhance Raman spectroscopy (SERS) [6–9]. Bowtie nanoantennas have also been explored in other applications like overtone spectroscopy [10], trapping atoms [11], integrated nanophotonics devices [12], and nanosensor switching devices [13]. Therefore, plasmonic nanoantenna is of great importance for obtaining high performance chemical and biological optical sensors [14,15]. For above mentioned applications, there is always a demand to tune the plasmonic resonance and maximize electric field intensity at very specific excitation wavelengths.

 

Fig. 1. The training dataset and DNN architecture. a) The bowtie nanoantenna geometry with variable parameters of width (W), length (L) of the prism and gap (G) between prism tips, with fixed thickness of 50 nm. b) The distribution of 3024 groups of training data collected by FDTD. c) Optimized DNN for predicting and designing the near-field EFIE (|E|²) with adaptive methods such as data preprocessing and cost function selection.

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However, there are great challenges to design such sophisticated, multi-parametered nanostructures that satisfy the performance requirements at target wavelengths. In conventional practices, the design starts based on intuition and experience followed by iterative optimization, with the hope to get desired resonances near the target wavelengths. Although different numerical optimization methods have been applied to plasmonic antenna design [16–19], It can be challenging for numerical optimization tool to search for the globally optimized parameter combination as the parameter space increases. Recently, by learning from adequate amount of training data, deep neural network (DNN) can accurately capture the physical relationship between nanostructure and their corresponding EM response, thus has been utilized for efficient and accurate inverse design with great success [20–26]. The simulation process can be expediated by orders of magnitude and direct inverse design that is long desired can be easily achieved. So far, a great number of nanophotonic devices has been explored, mostly including the scattering, transmission and reflection spectra of metasurfaces and plasmonics [27–29], or the structural color and wave filter applications derived from those spectra [30–32]. However, in nanophotonics, there still exists multiple degrees of freedom in the electromagnetic responses, such as near-field in the nanoantenna, phase manipulation in metasurfaces, and topological states in photonics etc. which have not been thoroughly studied with DNN training approach. Recent efforts have reported that 3D convolutional neural networks (CNN) can indirectly predict near-field and far-field properties of dielectric nanostructures through the learning of electric polarization inside the nanostructure and subsequent reconstruction of electric field intensity [33]. CNN has also been utilized to predict the two-dimensional distribution of electric field intensity of plasmonic nanoparticles [34]. However, it remains a question as to whether a simple multi-layer perceptron is effective enough to learn from nonlinear and complex near-field data structures to enable direct inverse designs. If a simple DNN also works, it can provide different machine learning approaches for those who wants to deal with complex, non-linear type of data.

Since bowtie plasmonic nanoantennas exhibit highly nonlinear, unpredictable relationship between microscopic geometric changes and their EM responses. In this work, we take such structure as a simple yet representative example, to discuss strategies to perform successful training of the DNN for learning the hidden relationship between nanoantenna geometry and near-/ far-field EM resonances. With the right data preprocessing and network training methods, accurate forward prediction and inverse design of transmission spectra and electric field enhancement are demonstrated simultaneously in fine nanoantenna structures. Previously, typical bowtie nanoantenna designed by conventional simulation approach only show electric field intensity enhancement (EFIE) on the order of ∼1000 [1,35,36] and rare cases of ∼10⁴ [37]. Our approach examines a much denser and larger parameter space, which results 1 to 2 orders of magnitude increase in EFIE peak values. Such finding provides useful strategy to deal with complex EM data features for multifunctional and high-performance photonic devices.

2. Results

In our approach, we collect sufficient training data of nanoantenna geometries and near-field EFIE at different wavelengths, which is a total of 3024 groups of sampled data in the whole G, W and L parameter space as shown in Fig. 1(b). Simulation details can be found in the materials and methods section. By using the ordinary DNN training method applied before [38–40] we end up with huge loss in the training process. Regular improvement strategies such as eliminating overfitting and altering network structure are not effective. We suspect there are two major reasons for the unsuccessful attempt. First, tiny changes of geometric parameters can cause significant shifts of the position and amplitude of both EFIE and transmission spectra as tested in similar designs [2–4]. Thus the data relationship between EM resonance and bowtie nanoantenna geometry is highly complex and nonlinear, which cannot easily be captured by ordinary DNN that worked for spectra data. Secondly, DNN recognizes the numerical values in the datasets and learn their hidden mathematical relationships, the large data range and unpredictable variations of EFIE significantly impact the training process. In previous DNN training [27–32,38–42] the common spectrum data applied to DNN are often limited in (0,1) or a small range less than 2 orders of magnitude. On the contrast, EFIE datasets have much larger difference exhibited at different wavelengths for a single antenna (horizontal difference) and among different antenna structures at the same wavelength (longitudinal difference), the ordinary DNN training process can hardly adapt to such EM data with abrupt and unpredictable changes over a wide range.

In order to address such challenge, we propose several modifications to the ordinary DNN structure and training process. As shown in Fig. 1(c), the main measures are summarized as two kernels: (1) for complex data structures such as EFIE, preprocess the output datasets with method of logarithm (Log), and the input with arithmetic operation (AO); (2) with regard to the inner sections of DNN, substitute the loss of mean squared error (MSE) for mean absolute error (MAE). Other settings of the hyper-parameters in the DNN include a Nadam optimizer [43], 5 hidden layers with 300 nodes in the first layer and 450 nodes in the rest. The input layer corresponds to the three geometric parameters of G, W and L, and the output layer includes EFIE of 101 wavelengths. More details of the DNN can refer to Table S1, Table S2 and Figure S1.

As shown in Fig. 2(a), among different data groups at the same wavelength of 880 nm, the maximum EFIE is over 60000 and the smallest value can be just a few single digits. This diverse longitudinal data difference can be further elaborated in the magnified inset and it determines the numerical orders of magnitude the network should cope with. Initial test of DNN training with direct feeding of unprocessed EFIE shows significant loss and total failure. Thus, a proper preprocess procedure and the final form fed into the DNN matter. Regular practice adopts the z-score normalization method ${\rm{y\; = \;}}\frac{{{\rm{x - \mathrm{\mu} }}}}{{\rm{\sigma }}}$ (Where µ is mean value and σ is standard deviation) which turns the data into a normal distribution. Such approach has demonstrated successful DNN training of small ranged circular dichroism signals in plasmonic metamaterials very recently [42]. However, although the z-score normalization can effectively eliminate horizontal difference in a single data group, the linear programming brings no substantial change to the significant longitudinal difference because of its feature-based processing. As can be seen in Fig. 2(b), the data is just linearly mapped to a narrowed range by z-score normalization with no changes in the distribution features, and the orders of magnitude remains the same. Other common preprocessing methods, the 0–1 normalization or min-max normalization for instance, are also excluded due to its feature-based attribute. Here, we propose to use logarithm method y = log₂x for such data processing and the effect is displayed in Fig. 2(c). Apparently, numerical values have been rearranged to (4,16) and extra dilution is put on as addition compared to the former distributions where small values were huddled in a narrow area. This makes the data more distinctive from each other and the feature differences can be more effectively distinguished by the DNN.

 

Fig. 2. The comparison between ordinary DNN and optimized DNN results for predicting the EFIE (|E|²). a) Different EFIE of training data at 880 nm. b) The z-score processing method for (a). c) The logarithm processing method for (a). d) An example of EFIE obtained by FDTD. e) The prediction result obtained by scale processing method for (d). f) The prediction result obtained by logarithm processing method for (d). g) The comparison of prediction results by method (e), method (f) with the simulation data. h) Error distribution of the test samples for predicting the EFIE by using MAE cost function (top) and MSE cost function (bottom). i) An example of predicted EFIE with an error value around 0.191.

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Another disadvantage of the z-score normalization methods exists. Comparing to the original EFIE of an antenna example in Fig. 2(d), we can observe that the spectrum processed by the z-score normalization shown in Fig. 2(e) has lost all the important characteristics like the peaks and tendencies. A mass of random peaks and burrs are generated, making the data unmeaningful. Such data preprocessing methods provides false data features to the DNN and the real electromagnetic response cannot be accurately learned. As a comparison, the result of logarithm method is displayed in Fig. 2(f), the features of the spectrum are mostly retained with only slight slowness of the trend, in the premise that the magnitude difference is effectively narrowed and values are confined to a small range of (0,8). By such preprocessing method, the sophisticated EM responses can be correctly captured by the DNN.

Another key factor in DNN training is the backpropagation and optimization procedure. Training loss tells DNN the similarity between predicted and true data, instructs the optimizer to reduce the loss continuously. The final loss value evaluates the effectiveness of training. In plenty of works associated to EM effects, MSE has been widely adopted as an effective evaluation indicator of loss with excellent performances, which can be expressed as Eq. (1) [28–31,38],

Where yi is the actual value and $y{(^)}$i is the predicted value. This function measures the accuracy of DNN by mean value of squared errors of all data. Our initial training results also demonstrate it is ineffective when applied to preprocessed EFIE datasets, no matter how effective the other hyper-parameters are optimized. In order to address such issues, we choose MAE as an alternative loss evaluation method, which proves to be a great improvement for the DNN training. The method can be described by Eq. (2),

Where yi is the actual value and $y{(^)}$i is the predicted value. Compared with the MSE method, MAE applied the absolute error as loss but not the quadratic term which brings a nonlinear magnification to the loss value. This would unreasonably magnify the influence of outliers and abnormal values during the judgement of loss and cause a compromise between the main rule which most data obey and the irregular distributions. The final results show minimized variance but larger deviation. In other words, the losses of outliers were decreased, but based on the worsening of accuracy of ordinary data, followed by degrading the overall model performance. Therefore, the MAE method is better at dealing with the case when outliers are detrimental to predicted results of all samples, such as the EFIF data here. And the actual results also prove the above deduction shown in Fig. 2(h). The combined effect of proper data preprocessing and loss evaluation methods can be seen in Fig. 2(g), which compares the predicted EFIE of the same nanoantenna structure before and after the modifications. The black solid line is the simulated result, the blue solid and red dash line represents the forward prediction by an ordinary DNN and the optimized DNN. The red line is accurate enough to predict most spectral characteristics and overlaps well with the simulated result among a large range of wavelengths. The blue line, in great contrast, exhibit a much inferior result and even output numerous negative values which is unacceptable, despite that the DNN has been laboriously trained with the aim to minimize loss configurations.

We test 100 groups of datasets which are not trained by the DNN and statistically evaluate the DNN performance. Figure 2(h) shows the relative errors generated by using MSE cost function (bottom) and MAE cost function (top), where the solid lines represent the mean value. The distribution clearly shows the improvement that the relative errors are narrowed into a smaller range and the mean value has decreased from 0.478 to 0.191. A prediction example by the optimized network with an error around 0.191 is shown in Fig. 2(i), which proves the effective training of DNN and it has greatly expanded the design space and density, that the impact of microscopic changes in geometry on EM near-field enhancement has been correctly captured. Therefore, the DNN can accurately predict EFIE with more results shown in Table S3 and Figure S2.

Our inverse iterative network can also address the inverse design problem by fixing the weights trained in the forward network but optimizing the input parameters which are randomly initialized, according to the target EFIE in the output layer. Several results may be obtained by different optimizations, and results with the minimum loss are chosen as the optimal designed parameters. Again, we use 100 groups of untrained data for network accuracy test and evaluate whether our DNN can produce maximized EFIE at the target wavelengths. The results are divided into three categories according to the differences between the actual and predicted peak EFIE wavelengths, i.e., a difference of less than 10 nm, between 10 nm and 30 nm and more than 30 nm, accounting for 42%, 30%, 28% of the above 100 data groups. The representative spectra of three groups are plotted in Fig. 3(a)-(c), with desired wavelengths at 843 nm, 758 nm and 801 nm respectively. For the first category, predicted EFIE nearly overlap with the actual spectrum and the peak EFIE can obtain a design value of 13501 compared to the target value of 14449. In the second category, wavelength of the designed EFIE peak only slightly differs from the target spectrum, but the designed EFIF still get a peak value of 7228 which is over half of the target value. The last category contains the rest situations. Although the two spectra have some differences, the main trends and features agrees with each other. Even in this case, the EFIE achieved a value around 854 which is slightly less than half of the target peak value at the wavelength of interest. More design results are shown in Table S4 and Figure S3. Among the 100 groups of design, the maximum designed EFIE can reach 89883 and almost half of the test groups exceed 10⁴ at desired wavelengths, which is about 1 to 2 orders of magnitude increase compared to those designed by conventional methods. Our approach can be adopted for high performance SERS and related applications to maximize device performance at any excitation wavelengths. The designed and original geometric parameters of G, W and L are plotted in Fig. 3(d)-(f). Results show good agreement between the two geometries in most test groups, and some results with certain discrepancy indicate that similar enhancement effect can be generated by different antenna structures.

 

Fig. 3. The inverse designed EFIE by DNN. The typical |E|² spectrum divided into 3 categories according to the differences between the actual and predicted peak EFIE wavelengths. a-c) The representative spectra. The top schematics show corresponding electric field intensity (logscale) at the peak wavelength of the xy plane and the xz plane. d-f) The comparison of designed gap (G), width (W), and length (L) with 100 groups of test data (the target value as the x-axis, design value as the y-axis).

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As previously investigated by conventional computational methods [44–46] and our quick examination shown in Figure S4, the near- and far-field resonances of optical nanoantenna are highly correlated. Our optimized DNN also provides an efficient approach to study the correlation between them. We propose a collaborative DNN that can simultaneously learn from the near-field EFIE and far-field transmission data at the same time. This collaborative DNN is innovative and convenient for designing multifunctional resonant nanostructure where chemical or biological molecules can be detected by both surface plasmonic resonance (SPR) spectrum shifts and SERS signal. As in real dual-functional sensor application scenario, maximum EFIE and SPR resonance at target working wavelengths are desired. Here, we directly train both data structures in the same network, by putting transmission and EFIE in the output layer together. Meanwhile, we increase nodes of 5 hidden layers from 300, 450, 450, 450, 450 to 300, 500, 500, 500, 500 for the requirement of raised network complexity. The forward and inverse network’s representative results are shown in Fig. 4(a)-(b) with additional results plotted in Table S5 and S6, Figure S5 and S6. The geometries of target nanoantenna are G = 7 nm, W = 210 nm, L = 120 nm. DNN with only EFIE design capability output geometries of G = 7.1 nm, W = 210.3 nm, L = 122.6 nm, DNN with only transmission spectra design capability output geometries of G = 7 nm, W = 210.1 nm, L = 120.2 nm while the collaborative DNN with both transmission and EFIE design capability output G = 7.5 nm, W = 210.0 nm, L = 120.0 nm. It approves that our single and collaborative DNN training has accurately captured the sensitive geometry-resonance relationship in bowtie nanoantenna for accurate inverse design. The designed geometry by collaborative DNN is plotted in Fig. 4(c)-(e) and these results show less discrepancy with actual values compared to the results generated by DNN with single data structure, which may suggest collaborative training of correlated EM data structures can enable the search of globally optimized designs.

 

Fig. 4. The results of collaborative DNN for simultaneously predicting and designing EFIE and transmission. a) The comparison between simulated, single DNN and collaborative DNN predicted EFIE and transmission spectrum. b) The comparison between simulated, single DNN and collaborative DNN designed EFIE and transmission spectrum. c-e) The comparison of designed gap (G), width (W), and length (L) with 100 groups of test data (the target value as the x-axis, design value as the y-axis).

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We design and plot nine examples by inverse collaborative DNNs in Fig. 5, covering a wide range of wavelengths. So far, advanced nanofabrication techniques such as bottom-up DNA origami assembly [14] and top-down electron beam lithography [15], can produce sub-5 nm bowtie nanoantenna gaps with high precision, together with powerful inverse design tools, better performing plasmonic sensors and SERS applications can be achievable [44–46] at target excitation wavelengths.

 

Fig. 5. The design tool application by collaborative DNN of both transmission and EFIE spectra over a broad wavelength range with details of designed nanoantenna geometry.

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3. Conclusion

In this work, we have reported the effective training of DNN for predicting and designing sensitive nanophotonics such as bowtie nanoantenna. By using a small group of training data with adaptive data preprocessing and training methods, our tool is capable to accurately predict and design both far-field transmission spectra and near-field electric field enhancement simultaneously in the same network. Our results demonstrate that DNN is powerful in learning multiple complex EM responses caused by microscopic geometric changes of nanostructures. In the meantime, the improvement of the DNN’s design capability and performance implies that our data training strategies have successfully overcome the problem of distinct EM data differences. Such strategies can be adopted for other EM data structures and help explore larger photonic design space for better device performance, for example, two orders of magnitude increment for near-field enhancement. Such implementation can inspire other complex EM response prediction and design, such as metalens phase manipulation [47] and topological photonic bandgap engineering etc., for next generation high-performance, multi-functional photonic devices [48,49].

4. Materials and methods

We test the sensitivity of spectrum to geometry structure in bowtie nanoantenna through simulation by using finite-different time-domain (FDTD) method (Lumerical Inc.), and then select gap (G, 5 - 40 nm), width (W, 20 - 250 nm) and length (L, 30–200 nm), as the input data, response spectra at wavelengths from 500 nm to 1000 nm, as the output data. Periodic boundary conditions with the period of 470nm for x, y directions and perfectly matched layer (PML) for z direction are adopted for the simulations. Meanwhile, a simulation area of ‘Mesh’ with a maximum mesh step of 1 nm were added given the sensitivity of the structures. Covering region and grid settings of ‘Mesh’ are finally determined by numbers of tests, to ensure accurate results and less time-cost in the meantime. Electric field intensity enhancement (EFIE) is calculated by