Photonic Inverse Design with Neural Networks: The Case of Invisibility in the Visible

Artificial intelligence is currently attracting unprecedented attention for its ability to tackle hard problems with huge datasets, which have been rendered tractable by the giant computational power and amount of training data available today. Photonic inverse design, in which one seeks to find objects of desired electromagnetic response, belongs to this class of complex problems that can greatly benefit from these ideas. In this work, artificial intelligence concepts are applied to advance the quest for invisible particles that do not perturb the background field; in particular, a fully connected neural network is proposed to address such a problem by learning the dynamics of visible-light interaction with low-scattering multilayered nanospheres. By swapping the roles between inputs and outputs, the same network can then be used for the inverse design of invisible nanoparticles, obtaining superior performance with respect to the best elements of the training set. The proposed approach can be generalized to approximate Maxwell interactions by simulating the electromagnetic response of more complicated optical configurations, and accomplish their inverse design directly, without successive iterations.

Figure 1

(a) A typical setup of a fully connected neural network with $D$ inputs and $R$ outputs, which constitute a plot $Q(\lambda )$ along variable $\lambda$ with $R$ samples and $L$ hidden layers comprising $U_l$ neurons each ($l=1,\dots ,L$). (b) Physical configuration of the nanoparticle subject to optimization: a five-layered sphere scatters a plane wave of wavelength $\lambda$ and is characterized by scattering efficiency $Q(\lambda )$.

Figure 2

(a) The mean relative error (MRE) of the neural network applied to the validation set, and (b) the average performance $\overline{Q}$ of optimal particles corresponding to different data partitions, normalized by the performance of the most weakly scattering particle of the dataset $Q_0\cong 0.08$, as a function of the number of neurons for each hidden layer $U$ for various learning rates $\alpha$. The plot parameters are $L=4$, $D=151$, $R=5$.

Figure 3

(a) Relative frequency $f$ of the samples belonging to three different training sets: totally random (green curve), targeted, comprised exclusively of low-scattering samples (red curve), and random around the average thicknesses $\mathbf{h}$ of the targeted set (blue curve) in proportion to their overall scattering efficiency $\overline{Q}$. (b) Mean relative error in decibels as a function of the number of epochs $E$ for the three aforementioned training sets (random, green; targeted, red; random targeted, blue). The testing set is the same in all cases and involves samples with suppressed $\overline{Q}$.

Figure 4

(a) The scattering efficiency metric $\overline{Q}$ of the proposed particle as a function of the desired flat output level $Q^{\ast }$. The dashed line corresponds to $\overline{Q}=Q^{\ast }$. (b) Optimal thicknesses of layers $h_{\rho }$ for $\rho =1,\dots ,R$ for the multilayered nanospheres of Fig. 4 (green curve) as a function of the selected $Q^{\ast }$. The dashed line corresponds to the design of minimal $\overline{Q}$. The plot parameters are $L=4$, $D=151$, $R=5$.

Figure 5

(a) The scattering efficiency $Q(\lambda )$ as a function of the operational wavelength $\lambda$ for a silver nanosphere equal in size to our optimal particle (silver), the best-cloaked particle from the training set (best), and the best prediction of our neural network [optimal design of Fig. 4]. Vertical dashed lines denote the visible part of the wavelength spectrum. (b) The performance metric $\overline{Q}$ (in decibels) of the optimal nanosphere [white cross, shown in Figs. 4 and 5] when the physical dimensions of its two external layers $(h_4,h_5)$ are perturbed.