1. Introduction

Optical metasurfaces allow for virtually arbitrary control of light fields, enabling applications in fields from optical imaging and sensing up to optoelectronics and quantum optics [1–6]. The optical resonances that are employed to tremendously alter properties such as the amplitude, phase and polarization, strongly depend on the structural parameters of each building block, i.e. metaatom [7–10]. Complex optical functionalities, for example the mutual control of amplitude and phase, typically come along with complex structures [11–14]. For instance, Jing et al. [11] reported on a gradient chiral metamirror with a maximum reflectivity of 94% at 10 GHz, where the metaatoms are described by not less than seven design parameters.

Metasurfaces with high reflectivity (metamirrors) and tailored phase and/or polarization properties are of particular interest for applications in high-precision optical interferometry, because they can outpace their conventional multilayer-based counterparts due to their performance in terms of thermal noise [15–18]. In the future, metasurfaces could be employed as cavity mirrors in these applications if tight requirements for the reflectivity, phase (and potentially polarization as well) are obeyed. For example, to reach a typical finesse of several hundreds of thousands that is used in ultra-stable laser cavities for optical atomic clocks, a reflectivity of $\gtrsim 99.997\%$ and typical radii of curvatures of $0.5\,$m to $1\,$m are necessary [19–24]. For future gravitational wave detectors the requirements will be even stricter [25]. Beside highly developed fabrication technology, a careful and accurate design of the metasurfaces will be the key tasks to reach these requirements. Deep Learning methods have proved to handle large parameter spaces with ease [26–30]. In this context, multiple approaches have been reported that implement inverse photonic design processes [31,32], ranging from generative adversarial networks (GANs) [33–35] to tandem neural networks [36]. GANs maximize the network’s creativity since the design space spans a multitude of degrees of freedom. While this helps finding novel structure types by circumventing human bias, generated structures are often extremely complex and non-intuitive. When it comes to high-performance nanophotonic devices, accuracy is a crucial part of the design process. Here, target values, e.g. reflectivities over $99.99\%$, are so sensitive that GANs fail to deliver satisfying accuracy. Especially, altering the structures, e.g. with Gaussian smoothing, to fit fabrication constraints might severely change their response. Hence, we choose a tandem neural network concept as the latter method insures that fabrication constraints for the metasurfaces can easily be considered. Despite the success of Deep Learning techniques, their accuracy did not yet meet the requirements for high-performance photonic devices. In this work we investigate Deep Learning for the design of a high-reflectivity focusing metamirror that optimises both, reflectivity and phase profile to a high level of accuracy. To this end, we utilize a tandem network consisting of three neural networks where one proposes metaatom designs based on a desired response to two (pretrained) networks that evaluate the design. After training, the first network is capable of generating designs to a wide range of requirements. Chapter 2 first describes the data acquisition. In chapter 3, this dataset forms the basis of two neural networks that predict phase and reflectivity of a given structure. These will be used in chapter 4 to implement a third neural network generating design parameters that satisfy the specific optical requirements in terms of reflectivity and phase. Finally, in chapter 5 we use the trained designing neural network to generate a metamirror with a focal length of 5 m and a mean reflectivity of 99.993 %.

2. Data acquisition

The building blocks of the metasurface, i.e. metaatoms, are aimed to be as simple as possible to allow a comparison to conventional design methods, yet effective for the desired mutual optimization of reflectivity and reflected phase. Although rectangular metaatoms with their two degrees of freedom are well investigated [10,37], a polarization independent metaatom is desirable. Hence, we choose symmetric cross-like structures made of amorphous silicon (refractive index of $n=3.445$ at a wavelength of $\lambda = 1550\,$nm) with only two degrees of freedom, i.e. lateral width Lx and height Ly of the cross’ arms as depicted in Fig. 1. The height of the structure is chosen to be 550 nm [7]. Since the goal is to design one metasurface, varying the lateral height of each metaatom is not feasible. As substrate we utilize fused silica with a refractive index of $n=1.444$ at a wavelength of $\lambda = 1550$ nm.

 

Fig. 1. Schematic of the cross-like metaatom with variable lateral height and width (Lx, Ly). Their structural height and period in both lateral directions are fixed to be 550 nm and 820 nm, respectively.

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First, two neural networks get trained to be able to map structure design pairs [Lx, Ly] to electromagnetic responses [phase, reflectivity]. For that, a dataset is necessary that contains enough information for the network to learn the complex non-linear relation between the input design pair and the output response. Here, we utilize Rigorous Coupled Wave Analysis (RCWA) [38] to generate a dataset of 184041 samples. To allow the network to predict with extremely high accuracy, hence being able to work with the tight requirements of our application, a larger sample size is necessary to improve the networks ability to learn the hidden connection between the metaatom design and the electromagnetic response. Afterall, even small deviations in the overall reflectivity has severe consequences for the metamirror’s performance as cavity mirror. The period of each metaatom in x- in y-direction (Px and Py) is fixed to 820 nm and Lx and Ly are described by the duty cycles in both lateral directions with respect to the period. Having the period of the metaatom as a free parameter would require complex stitching of all metaatoms to one macroscopic metasurface and is therefore not considered. For the data acquisition, both Lx and Ly are chosen to be in the interval [$0.3$ , $0.9$]. The impinging light has transverse electric polarization with a wavelength of $1550\,$nm.

3. Forward prediction

After splitting the ground truth data into a training set, a validation set and a test set of 75 %, 15 % and 10 % of the total amount, respectively, two networks are separately trained to predict the phase $\phi$ (P-NN) and the reflectivity $R$ (R-NN) of the metaatom. The network architecture is shown in Fig. 2. Each network contains six fully connected hidden layers where the number of nodes is halved with every consecutive layer, starting with 1024 nodes. A Leaky ReLU, Rectified linear unit with the slope $\alpha = 0.3$, serves as activation function. Adam with a learning rate of $1 \times 10^{-3}$ and a decay of $5 \times 10^{-6}$ is used as optimizer [39]. To prevent reflectivity values beyond 1, the last layers of both forward networks comprise sigmoid activation functions.

 

Fig. 2. Tandem Neural Network: First, the D-NN network predicts design parameters based on a desired input phase $\phi$. Both forward networks are then used to evaluate the given design so $\phi '$ matches the desired phase and $R$ is maximized.

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After 2000 epochs of training, the validation losses have decreased to $4 \times 10^{-5}$ for R-NN and $6 \times 10^{-4}$ for P-NN. Figure 3 shows the excellent prediction performance of the network and the ability to reproduce the given ground truth data. The mean absolute deviation of the phase achieved with the neural network compared to the RCWA simulation is 0.0323 rad, the respective absolute mean deviation of the reflectivity is 0.0029. Now, these forward predicting networks can be implemented in the framework of a tandem neural network in order to reverse-engineer metaatoms.

 

Fig. 3. Left: Ground truth training data from the RCWA simulation. Middle: Forward predictions of the trained neural networks P-NN and R-NN on the ground truth data. Right: Absolute deviation of both, phase and reflectivity predictions, showing excellent performance of the neural networks.

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4. Inverse design tandem network

Nonunique response-to-design mapping [36] hinders a single neural network predicting design parameter pairs based on desired responses to converge properly. It describes the ambiguity of two different nanophotonic designs exhibiting similar electromagnetic responses. To circumvent this problem, the forward neural networks trained a priori are utilized to form a tandem neural network as already introduced in the previous section 3 (see Fig. 2). The goal is to design a metaatom for a given phase response that simultaneously yields high reflectivity. For that, the inverse neural network (D-NN) takes a phase $\phi _\textrm {input}$ as input (the target reflectivity is always $1$) and proposes a design matching that specific phase (and reflectivity). This design is passed to the R-NN and P-NN network as input. The neural network D-NN then tries to minimize the following loss function for each batch of $n$ samples:

Here, $\phi _{\textrm {input}}$ is the desired phase for which the D-NN network generates the design $D_{\textrm {output}}$. It is important to emphasize that the D-NN network searches within the entire design space that was provided by the ground truth data to satisfy every possible request, i.e. a phase range of up to 2$\pi$. Since neural networks are smooth functions [40], an adiabatic phase change as input will consequently yield adiabatic changes in the output design parameters. This is beneficial for the metasurface containing metaatoms of different geometries, because otherwise, residual coupling between neighbouring metaatoms might noticeably distort the macroscopic response [41]. Figure 4 shows the result of the entire phase range between 0 and 2$\pi$. An $R^2$-value of 0.9994 illustrates that the network learned to provide design parameters that satisfy the requirements, i.e. maximizing the reflectivity ($R_{\textrm{mean}}=99.7\%$) for all possible phase values. The tandem neural network is especially advantageous when the inverse design process has to be repeated several times for example to determine the parameters of different metaatoms that shall form a metasurface. If the goal is to design a device that can be evaluated by one single simulation, other methods like genetic algorithms might be a good alternative, as Zhu et al. [42] have shown. Using the structural parameters determined with the tandem network, in the next chapter, we design a high-performance metamirror

 

Fig. 4. Inverse predictions of the D-NN for some part of the test set (normalized phase: $\phi /2\pi$) and the corresponding network predictions of phase (blue line) and reflectivity (orange line) for the proposed design should ideally lie one line (black line).

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5. Metamirror with parabolic phase profile

In general, the parabolic phase profile of a mirror can be describes by the following equation:

Here, $x$ and $y$ refer to Cartesian coordinates in lateral direction, $|\vec {k_0}|$ is the free space wave number, $f$ is the focal length and $\phi _0$ is the phase in the mirror center. The total phase difference $\phi _{\textrm {diff}}$ that is needed for fixed focal length $f$ and metasurface edge length $L$ can be retrieved by transposing Eq. (2) into $\phi _{\textrm {diff}} = \frac {|\vec {k_0}| \times L^2}{4 \times f}$. Given a wavelength of $\lambda _0$ = 1550 nm and a metasurface size of $L$ = 3 mm, a focal length of 5 m requires a phase range of $1.82\,$rad. The size of the metamirror is chosen to match a Gaussian beam with a beam waist of about $500\,\mu$m.

For the application of the metamirror in optical cavities, the most important demand is the integrated reflectivity to reach a high finesse. Aberrations of the mirror induced by phase mismatches don’t critically influence the finesse. Previous studies have shown that the fundamental cavity mode is robust against aberrations [43]. Nonetheless, to minimize the overall phase mismatch we discard in Fig. 4 regions, where the phase deviation is larger than 0.05 rad. Figure 5 then reveals an available phase change of $5.791$ rad that can be used to compose the focusing metamirror. The deviation at the outer edges (close to 0 and $2\pi$) are most likely caused by the cyclic behaviour of the phase. A neural networks struggles with learning the discontinuous behaviour of the given data. However, Fig. 5 shows that this effect does not matter with this application since the usable phase (or reflectivity) range is large enough. The network’s ability to predict the reflectivity of a given structure remains unaffected.

 

Fig. 5. Usable phase range where the error between desired phase and output phase is smaller than a threshold of 0.05$\,$rad covers 92% of a complete $2\pi$ roundtrip.

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The usable phase range can now be utilized to optimize the overall reflectivity of the focusing metamirror. For a Gaussian light beam, metaatoms in the center of the surface are exposed to higher light intensity than metaatoms at the outer edges, thus, adding more weight to the overall reflectivity of the metamirror. Hence, it is beneficial to shift the phase of the center metaatom $\phi _0$ to maximize the reflectivity at that specific position. After fixing the starting metaatom, the D-NN network can generate arbitrarily many metaatom designs although the initial ground truth step size is limited. In comparison to simple linear or cubic interpolation, a neural network approach accelerates the determination of the metamirror design parameters since the interpolation is only based on ground truth data. Figure 6 (top) shows the retrieved very smooth phase profile with a mean reflectivity (averaged over the area of the Gaussian beam) of $99.993\,\%$ and a mean phase mismatch of only $\Delta \phi = 0.016\,\%$. The inset shows an enhanced area of the radial phase profile of the metamirror exposing the staircase character due to the finite amount of metaatoms. Although finer design steps, thus finer phase values, can be generated by the neural network, the final minimum phase match will be limited by the fabrication resolution. Figure 6 (bottom) illustrates the absolute change of Lx and Ly, showing the adiabatic behaviour of both parameters. The results show that the neural network approach is suitable to achieve very high design reflectivities that reach the range, relevant for applications in high-performance optical components. It, thus, can be applied to design devices with an complex optical functionalities, ergo tight requirements, that may not readily accessible with conventional forward designing methods.

 

Fig. 6. Top: Radial phase profile of the metasurface with excellent phase agreement and high reflectivity. Bottom: Position dependent behaviour of both design parameters Lx and Ly, expressed in length units.

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6. Conclusion

We implemented two neural networks that can predict the electromagnetic response, i.e. phase and reflectivity, of a given nanostructure within the predefined design space with high accuracy. These networks were implemented into a tandem network that manages to search the design space, taking into account two figures of merit. That allowed us to utilize an inverse design network to compose a focusing metamirror with a reflectivity of 99.993$ \% $ and a minimal phase-mismatch of $\Delta \phi = 0.016\,\%$, consisting of over 13 million metaatoms. The application of a metamirror with large focal lengths requires a flat phase profile that is difficult to generate with the ground truth data only. The neural network allows the continuous generation of metatom design so even a flat phase profile can be rendered effortlessly. Here, interpolation relies only on the information within the ground truth data. A neural network only takes about 0.1 ms/prediction, making this method extremely fast (after a one-time investment of training). Our method only requires to define a design space where more parameters can either be fixed or left free, e.g. multiple wavelengths or polarization dependence. This makes it applicable for a variety of complex optical functionalities.