Introduction

The ongoing trend towards miniaturization and increased integration of on-chip optical systems demand greater device versatility and data storage capacity1,2,3,4,5. Over the last decade, metasurfaces, a type of ultrathin planar optical component, have emerged as a promising solution for manipulating light and enabling a wide range of optical applications6,7,8,9. By designing the scattering properties of the individual meta-atom that make up the metasurface8,10,11, we are able to manipulate light wavefronts across multiple dimensions in physical space, including phase8,12, amplitude13,14, polarization15,16 frequency17,18, and so on, to enable a variety of fascinating functions such as focusing19, anomalous refraction20, polarization controlling21,22,23, polarization sensing18, and the generation of high-resolution holograms24,25,26. These functions are not available in conventional optics or require bulky optical components1,2,3,4. Yet, the current progress typically focuses on modulating one or two optical parameters, substantially limiting the potential for advancement. To enhance the performance and broaden the application scope of metasurfaces, it is essential to achieve independent and simultaneous control over multiple optical parameters on a single metasurface27. For instance, complex meta-atoms with intricate cross-sectional shapes can be employed to enable concurrent manipulation of phase and amplitude for superior holographic imaging quality28,29. Advanced functionalities such as vivid full-color holograms, dispersion-engineered devices, and achromatic metalens have also been demonstrated through the simultaneous control of phase and frequency30. Despite these achievements, integrated and on-chip metasurface design continues to face critical challenges such as a lack of multidimensional manipulation of optical waves, the absence of a universal design strategy, as well signal-to-noise ratio issues31. Therefore, exploring extra degrees of freedom (DOF) or versatile design strategies for manipulating optical fields is imperative.

Recent research has revealed that non-Hermitian systems, which involve resonators with loss and gain, provide an extra dimension to the manipulation of light propagation23,32,33,34. Typically, non-Hermitian phenomena are introduced as a small perturbation to Hermitian systems, resulting in global properties that closely resemble those of Hermitian systems35. However, investigations have demonstrated that non-Hermitian systems exhibit behaviors that diverge significantly from those of Hermitian systems. A most notable example of such divergence is the emergence of exceptional points (EPs)36,37,38,39,40, which occur as two or more eigenvalues and their corresponding eigenvectors coalesce simultaneously. Additionally, EPs are topologically robust and possess fascinating Riemann surfaces in their parameter space, distinguishing them from other types of topological singularities. The topological structure associated with the branchpoint singularity around an EP can be utilized for wavefront engineering. Specifically, the topologically protected 2π-phase accumulation by encircling an EP in parameter space, referred to as exceptional topological phase (ET phase)33, provides a DOF for phase-addressing. Similar concepts have been utilized to obtain the desired amplitude and frequency response32. This straightforward multi-channel approach creates additional research prospects for controlling electromagnetic fields with multi-optical parameters and developing highly compact integrated devices.

Though the EP exhibits a variety of unusual and counterintuitive physical phenomena, obtaining the EP is difficult due to its sensitivity to parameter changes. What’s more, to populate the meta-atom library around the EP, an efficient approach is required. Deep learning, an unconventional data-driven approach, has emerged as the most promising candidate for analyzing physical systems and solving engineering design problems41,42. It can be used to quickly discover scattering EP and observe the intriguing topological phenomena ignored by coarse scanning parameters or few data. Here, we propose a deep learning method to rapidly discover scattering EPs and fully utilize the scattering optical parameters distributed around them. Our approach efficiently summarizes the topological properties of scattering EPs and readily generalizes to other non-Hermitian systems by adjusting EP conditions. Furthermore, we demonstrate two multifunctional optical devices: an amplitude-multiphase multiplexing device and a wavelength division multiplexing device based on the meta-atom library at arbitrary wavelength. We anticipate this mechanism will pave the way for creating compact multifunctional optical devices in polarization optics, information encoding, optical data storage, and security.

Results

The unit of investigated metasurfaces is schematically shown in Fig. 1a, consisting of a metal-dielectric-metal sandwich structure. The metal layer is a planar chiral array structure formed by coupling L-shape nanostructures with straight nanorods with a spacing of g. Since the transmission is blocked by continuous back reflection, the chiral response of the metasurface is completely described by the reflected scattering matrix (or more precisely, for polarization conversion only, the Jones matrix):

$$S=\left(\begin{array}{cc}{S}_{11}&{S}_{12}\\ {S}_{21}&{S}_{22}\end{array}\right)$$
(1)

the cross-polarization term S21/S12, representing reflected circular polarization (CP) conversion from right/left circular polarization (RCP/LCP) to LCP/RCP, and the co-polarized term S22 is identical to S11 due to the optical reciprocity. The complex matrix element, i.e., \({S}_{mn}={{{{{{{\rm{real}}}}}}}}\left({S}_{mn}\right)+i* {{{{{{{\rm{imag}}}}}}}}\left({S}_{mn}\right)={A}_{mn}{e}^{i{\phi }_{mn}}(m,n=1,2)\), where Amn is the reflection coefficient and ϕmn represents the related phase. It has been demonstrated that the EP occurs under S12S21 = 0, which means that when the incident light is pure RCP or LCP, the output light is also pure RCP or LCP, namely the cross-polarization term A21 or A12 is blocked as 039. The further derivation leads to the eigenstates coalescing into CP light (RCP or LCP) gradually when the ellipticity angle gradually approaches ± 45∘ (See Supplementary Methods 1 for a detailed derivation of ellipticity angle). The appearance of EP gives rise to polarization conversion zero, which is a basic singularity in polarization conversion (Fig. 1b). This is invariably accompanied by a robust topological protect 2π phase accumulation and amplitude changes on the S21 channel around the EP in two-dimensional parameter spaces. This phenomenon, referred to as the ET phase, distinguishes itself from the 4π phase encircling EP in Riemann space. And the topological charge can be defined as:

$$w=\frac{1}{2\pi }\oint \,d\Phi$$
(2)

where Φ is the phase integrated along a path enclosing the EP counterclockwise. In comparison, the other phase and amplitude on the S12 channel remain constant throughout the parameter space33, thereby introducing a DOF for wavefront engineering. Different from the three primary mechanisms employed thus far - resonant phase, propagation phase, and Pancharatnam-Berry (PB) - ET phase-addressing mechanism possess inherent polarization decoupling properties and amplitude-modulated channels. Consequently, this straightforward method offers an opportunity for devices with multi-optical-parameters, as illustrated in Fig. 1d, showcasing a three-channel device. Moreover, the EP exhibits sensitivity to variations in external parameters, such as wavelength, which holds promise for applications in wavelength division multiplexing (WDM), as depicted in Fig. 1b. Here, we demonstrate a WDM device using two wavelengths as an example, and its schematic is presented in Fig. 1c.

Fig. 1: Structure of the chiral meta-atom and the design strategy of the multiple function devices.
figure 1

a The structure of chiral meta-atom, the dimensions are g = 70 nm, w = 50 nm, and P = 300 nm, t = 40 nm, h1 = 140 nm, h2 = 150 nm. b The amplitude and phase evolution from one exceptional point (EP) to a pair of EP on the S21 channel with wavelength in the parameter space. The appearance of an EP is accompanied by 2π phase accumulation and singular value amplitude change. c Schematic illustration of the wavelength division multiplexing metasurface based on exceptional topological (ET) phase to control S21 channel beams at two different wavelengths. The two wavelengths display holographic images A and B, respectively. d Schematic diagram of the amplitude-phase multiplexing metasurface that can grayscale imaging in amplitude channel and independent control of two cross circularly polarized channels in the far-field hologram. By combining the Pancharatnam-Berry (PB) and ET phases, it is possible to decouple S21 and S12 channels. This enables holographic imaging “D” and grayscale imaging “C” in S21 channel, while holographic imaging “E” in S12.

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Deep learning model construction

To fulfill the requirements of the aforementioned devices, it is essential to comprehensively cover the polar coordinate space with amplitude and phase values of the structural parameters. Figure S1 clearly illustrates that relying solely on the two parameters mentioned in reference by Song et al.33 is insufficient. Therefore, three parameters (L1, L2, and L3) (Fig. 2a) that play a primary role in the chiral harmonic oscillator coupling are taken into consideration. Subsequently, we construct a bidirectional neural network, on the one hand, to further fill the polar space efficiently, on the other hand, to assist in identifying EP at arbitrary wavelengths. The proposed deep learning model comprises a forward prediction neural network (PNN) that acts as an efficient simulator and an inverse neural network (INN) as a reverse retriever. The configuration of the forward-mapping PNN, which maps parameters to points, is illustrated in Fig. 2b. The parameters consist of the structural parameter R = (L1, L2, L3) and the wavelength parameter λi(i=1, 2, …, n), and the word ‘points’ here represents all the 8 values (real(Smn) and imag(Smn)) of the scattering matrix S at a specific wavelength λi. The reflectance spectrum of interest is set between 500 to 700 nm in the visible region and discretized into 100 data points with a resolution of 2 nm, as depicted in Fig. 2c, (the minimum resolution is dependent on the simulated spectrum utilized for training, here is 0.03 nm) which is equivalent to choosing 100 wavelength parameters (n = 100), meanwhile expanding the number of databases by a factor of 100. Consequently, the 3500 databases we simulated increased to 350,000. Then the entire database is fed into the network, of which 70% is assigned to the training set, while the remaining 30% is the test set to evaluate the network’s performance. During this training process, the network weights and bias are optimized by minimizing the loss function, which measures the mean square error (MSE) between predicted values generated from the network and simulation results. After training, the MSE of the overall test set is around 3.76 × 10−5 (Fig. S2a). We then input a set of structural parameters with EP in the S12 channel that the network has never seen before, demonstrating the accurate prediction ability of PNN. For example, fed (L1, L2, L3) = (30 nm, 100 nm, 250 nm) into our train PNN, the great consistency between the predicted value of the network (dots) and the simulated value (solid lines) exhibits the accuracy of the network. Subsequently, the ellipticity angle of the two eigenstates is derived, which degenerates as LCP at λ = 580 nm (Fig. 2d). The ellipticity angle is calculated by the prediction values of the forward PNN and obtained by the simulation values still has a good consistency, which once again confirms the accuracy of the forward PNN. Besides, the amplitude and phase of S matrix elements and the eigenvalues of S matrix results are plotted in Fig. S3. The derivation process is all written in the supplementary Methods 1. All these features indicate that an EP occurs at λ = 580 nm. All the above characteristics are similar when EP is present in the S21 channel. In this case, we fed the parameter (L1, L2, L3) = (50 nm, 160 nm, 110 nm) into the PNN, the results as shown in Fig. S4 also fit very well. The ellipticity angle of the eigenstate degenerated as RCP at λ = 570 nm where EP appears, so ellipticity is an intuitive characteristic quantity to characterize EPs and can be used as a convenient tool to reflect the existence of EPs in structures.

Fig. 2: Structure of the forward parameters-to-points prediction neural network (PNN) model.
figure 2

a The structure of chiral meta-atom, the structural parameters passed into PNN are L1 L2 and L3. b The forward PNN configuration of parameters-to-points. It deals with a regression problem between 1 × 4-dimensional parameters and 1 × 8-dimensional values. By superimposing the output values with different λi, 8 spectra can be obtained with a set of (L1, L2, L3) parameters. c The 8 spectra predicted by PNN are consistent with the finite element method (FEM) results. d The ellipticity angle of the eigenstates, which degenerate as left circular polarization at λ = 580 nm.

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The parameters-to-points forward-mapping PNN has advantages in several aspects. First, it addresses the challenges of insufficient resolution near resonance peaks in photonics and simultaneously solves the issue of dimensional mismatch fundamentally. Because increasing the number of n in the defined spectral range only increases the database size and does not affect the output dimension. Second, the PNN can achieve strong prediction capabilities even with a small training database. This capability reduces the amount of required data while maintaining reliable performance. Consequently, it proves suitable for scenarios that necessitate high-resolution spectra, such as those involving high-quality factors and sharp resonances. Lastly, the PNN exhibits a significantly shorter computational time for generating S-parameters compared to the finite element method (FEM) simulation. For example, when running on a laptop with Intel(R) Core (TM) i7-11700 @2.50 GHz, the deep learning model we proposed takes less than 30 s to predict more than 20,00,000 data, while it takes thousands of times longer to simulate 3000 data. Therefore, it can reveal interesting physics that was missed by the rough simulation parameter sweep. In contrast with the time-consuming traditional methods based on physics or rules, the reverse design method enables a direct and accurate search for the structural parameters with EP at desired frequencies. To achieve this, we constructed a tandem network (Fig. S5) that sequentially connects the INN and the pre-trained PNN (refer to Supplementary Note 1 for detailed construction and training procedures, the INN performance can be found in Fig. S6). Through these interconnected neural networks, the mapping relationship between structural parameters and eigenstate ellipticity is established directly or indirectly. Importantly, it should be noted that our inverse design methodology can be extended to other non-Hermitian systems by adjusting the EP conditions when exploring resonant or absorption EP systems.

EP evolution

Upon completion of the bidirectional neural network, the arduous and time-consuming trial-and-error numerical approaches become obsolete. Each inverse design task requires only a query of the INN that takes not more than a second. This means we can quickly and accurately obtain EP at an arbitrary wavelength. EP, as a sign of the emergence of a meta-atom library for multi-optical-parameter control. Subsequently, PNN enriches the parameter space around them, significantly enhancing the speed of our search for optimal meta-atom structures. This approach provides the necessary flexibility to choose the most suitable meta-atom structures and is crucial for investigating EP evolution (Fig. 3). Figure 3a shows the − 2π-phase accumulation (ET phase with a topological charge of −1) maintained for the reflected phase map irrespective of the closed path, as long as it encircles an EP in parameter space. Meanwhile, it is important to note that EPs are not static but vary with material parameters. It has been proved that EPs with a topological charge of ±1 appear in pairs in the Brillouin zone, and the theorem is also valid for compact two-dimensional parameter spaces34,43,44. Although the defined range of L1 and L3 is not closed in the system we studied, it still belongs to a part of a closed space, so we can consider adjusting material parameters and so on to gradually pull another EP into our field of view. As shown in Fig. 3a and b, a pair of EPs is pulled into our view by increasing L2 from 150 nm to 200 nm, the data showing reasonable consistency with the simulation results (Fig. S7). With the support of a powerful “simulator”, the evolution process of EP points is visible. Figure 3c and d demonstrate that a pair of EPs on the S21 channel can be gradually drawn closer by adjusting the length of L2 and λ. A pair of EPs is generated and gradually separated when fixing the L2 = 160 nm and varying λ from 500 nm to 658 nm. While fixing the λ = 500 nm and varying L2 from 80 nm to 160 nm, a pair of EPs gradually approach and eventually annihilate. Figure 3e, f summarizes the evolution of the two EPs in L1-L3 parameter space as a function of wavelength/L2 when the L2/wavelength is fixed. Therefore, by changing the structural parameters as well as the wavelength, we can control the position and number of EPs, which appear or disappear in pairs, thus maintaining the total topological charge conservation. In this process, the addition of a fast ‘simulator’ dramatically improves our efficiency in summarizing the topological charge evolution. Similar approaches could be considered in the future to study the topological charge evolution in the whole Brillouin zone.

Fig. 3: The evolution of exceptional points (EPs) in S21 channel predicted by parameters-to-points prediction neural network.
figure 3

EPs gradually get closer with the increasing of L2 from 150 nm (a) to 200 nm (b) when λ = 600 nm. c A pair of EPs generate and gradually separate with increasing λ. d A pair of EPs gradually approach and eventually annihilate with increasing L2. The color bar of (c) and (d) is consistent with (a) and (b). e, f Summarize the evolution of the two EPs in L1 − L3 parameter space as a function of wavelength/L2 in Fig (c, d) when the L2/wavelength is fixed. The red and blue dots represent the positions of two EPs in the parameter space, respectively.

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Multi-channel metasurfaces devices

The property of ET phase evolving with wavelength is promising in WDM applications, such as color holography, multi-channel communications, and storage24,25. To illustrate, we take the common wavelength 532 nm and 633 nm as an example. Through INN, EPs are acquired at the two wavelengths, and then PNN enriches the parameter space around them to fulfill the polar ordinate space. This enabled us with enough flexibility to choose meta-atom structures and is desirable for efficient wavefront modulation. Figure 4a exhibits the A21 and ϕ21 profiles around EPs in the three-dimensional (L1, L2 and L3) parameter space at λ = 532 nm and 633 nm. Structures marked 1 to 4 constitute a complete base covering 2π (2-phase gradient), thus, two holograms can be coded at 532 nm and 633 nm via the 2-bit coding method, see Fig. 4b for the device diagram. On the other hand, the amplitude of the cross-polarization term (A21) can also be manipulated around EP, as depicted in Fig. S8, which presents a powerful tool for multiple optical parameter modulation, such as controlling both the optical amplitude and phase at different polarization or frequency channels of input and output light. Figure 4c exhibits the A21 and ϕ21 profiles around EPs in the three-dimensional (L1, L2 and L3) parameter space at λ = 633 nm generated by the bidirectional neural network, similar with the Fig. 4a. The pink dots indicate that the ET phase can be continuously changed from -π to π when A21 varies continuously from the minimum value to the maximum value. On the premise that the amplitude and phase data on the S12 channel are basically unchanged (blue dots), we can screen out the 4-level data with an amplitude interval of 0.1 (the error is ± 0.02) and the 4-level data with a phase interval \(\Delta {\phi }_{21}^{ET}=9{0}^{\circ }\) (error range is ± 3∘) on S21 channel (red stars). Multiple structural candidates satisfied the above conditions at each location, with the scale accurate to 1 nm. Figure 4d and e exhibit the selected structures with amplitude range from 0 to 1 (uniformed) and phases from -π to π, and the S12 channel remains almost unchanged.

Fig. 4: Two multi-channel holographic devices designed by parameters-to-points prediction neural network.
figure 4

The first black box is the Wavelength Division Multiplex (WDM) metasurface: (a) Phase and amplitude profiles around exceptional points (EPs) in the three-dimensional (L1, L2 and L3) parameter space at λ = 532 nm and 633 nm. Structures marked 1 to 4 constitute a complete base covering 2π (2-phase gradient). b The WDM metasurfaces at 532 nm and 633 nm are realized by a 2-bit coding method. The second black box is the amplitude-phase multiplexing metasurface at λ = 633 nm: (c) The phase and amplitude data of S21 (pink point) fill the entire polar space, allowing us to regulate the two degrees of freedom independently and completely. 4-level amplitude and 4-levels phase data of S21 response are selected (red star) while satisfied S12 response remain basically constant (blue point). Each location has multiple candidates. 16 meta-atom designs are selected so that the amplitude and phase are decoupled in S21 (d) and the meta-atom library is created (e). f The phase in S21 and S12 decoupled without crosstalk through the combination of the Pancharatnam-Berry (PB) phase and exceptional topological (ET) phase.

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On this basis, we can combine it with PB phase to achieve independent control of the two cross-polarization channels without crosstalk, as depicted in Fig. 4f. PB phase, which has essentially opposite signs (\(\Delta {\phi }_{21}^{PB}=-\Delta {\phi }_{12}^{PB}=\Delta {\phi }_{0}\)) to the orthogonal CP light, resulting in locked and mirrored functionalities for RCP and LCP beams incidence45. In order to solve the crosstalk problem, we propose to directly overlay the ET phase addressing mechanism on the basis of the conjugate PB image, and convert it directly from the conjugate image into a different image. The specific design scheme is as follows: First, we implement PB phase (4-phase gradient) by rotating all the same structure so that S12 channel becomes image E and S21 channel is the conjugate image of E. Then, taking advantage of the characteristic that ET (4-phase gradient also) phase only S21 polarization channel has 2π phase accumulation, while S12 polarization channel phase is basically unchanged, replacing all the same structure with the encoded different structure profile so that the image is transformed from E to D and simultaneously achieve grayscale imaging C in the S21 channel. The phase profile of the encoded structure is equal to the difference between the phase profile of image D and image E (phase(D) - phase(E)). Therefore, the phase and amplitude presented by the whole structure should now be the sum of the PB phase and the ET phase of the structure itself. Since the phase and amplitude on the S12 channel are unchanged, the phase gradient remained: \(\frac{d{\phi }_{12}}{dx}=\frac{\Delta {\phi }_{12}^{PB}}{P}=-\frac{\Delta {\phi }_{0}}{P}\). while the phase gradient on the S21 channel changed to \(\frac{d{\phi }_{21}}{dx}=\frac{\Delta {\phi }_{21}^{ET}+\Delta {\phi }_{21}^{PB}}{P}=\frac{\Delta {\phi }_{21}^{ET}+\Delta {\phi }_{0}}{P}\) instead, manifesting the distinctly different spin-dependent linear phase gradients for orthogonal CP light incident. Thus, the completely independent control of the S12 and S21 channels can be realized without crosstalk.

As proof of concept, we design two multi-channel holographic devices. For the WDM metasurface, Roman numeral I and II are coded into one device, using the method outlined in Fig. 4b. The standard electron beam lithography technique was used to fabricate a multi-channel chiral metasurface (more details about the sample fabrication and optical setup for the measurement are presented in Figs. S9 and S10). As the experimental results show in Fig. 5a and b, when the light incident perpendicularly to the surface, the image of I is reconstructed in the far field at 532 nm, and the image of II is reconstructed at 633 nm. This strategy uses deep learning to quickly generate a complete base with simple planar structures for coding WDM holograms, avoiding time-consuming mass simulations in design and complex processing during manufacturing. For amplitude-phase multiplexing metasurface, two conjugated holograms (Maxwell 1st and 2nd equations) are encoded on the two CP channels by simple rotation through the meta-atom of the metasurface. The meta-atom is then assigned a specific structure that we select, which has encoded ET phase and ET amplitude on the S21 channel at 633 nm, while the phase and amplitude of the structure on the S12 channel are uniform. Hence, this will not affect the holographic imaging of the PB phase on the S12 channel, only convert the conjugate hologram of the PB phase to another hologram (Maxwell 3rd and 4th equation) on the S21 channel, and at the same time realize the amplitude grayscale imaging (Maxwell Avatar) at the object plane. The scanning electron microscopy (SEM) image of our designed chiral metasurface is shown in Fig. 5c and d. The device shows a grayscale distribution map of Maxwell’s portrait only under the LCP incident light when focused onto the object plane (Fig. 5e), while without any image under the RCP incident (Fig. 5g). At the same time, holograms without any crosstalk or clutter of light spots were displayed on the LCP and RCP channels respectively even though there is 4-level amplitude variation (Fig. 5f, h), which illustrates the advantages of deep learning and the decoupled strategy (PB phase + ET phase) we used. Our strategy can significantly improve information storage density and enhance information security and can be easily applied to other applications, such as information storage, anti-counterfeiting, encryption, optical communication, and many other related fields.

Fig. 5: Experimental results of the two multi-channel holographic devices.
figure 5

The first black box is the Wavelength Division Multiplex (WDM) metasurfaces: the image of I is reconstructed in the far field at 532 nm (a), and the image of II is reconstructed at 633 nm (b) in the S21 channel. The second box is the amplitude-phase multiplexing metasurfaces at λ = 633 nm: (c, d) Scanning electron micrograph (SEM) of the metasurface. When left circularly polarized light illuminates the metasurface, the Maxwell portrait and the Maxwell 1st and 2nd equations can appear on the S21 channel for grayscale imaging (e) and holographic imaging (f), respectively. Similarly, no image appears on the metasurface plane (g) under RCP incident light, while the PB phase encodes a holographic image (Maxwell 3rd and 4th equation) on the reflection plane (h). The scale bar of (e) and (g) is consistent with (c).

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Conclusion

The topological structure associated with the branch point singularity around a scattering EP can provide tools for controlling light propagation, which not only has the inherent polarization decoupling property but also provides the amplitude and wavelength modulation channel. This simple multi-channel approach creates research prospects for multi-optical-parameter electromagnetic field control and developing highly compact integrated devices. With the support of deep learning, we observe the conservation of topological charge under perturbation and fully use the optical parameter distribution of topological protection around the scattered EP. The observed ET phase evolving with wavelength provides a method for the design of WDM devices, and the amplitude and phase distribution around EP offer a method for wavefront engineering. Therefore, we propose a WDM metasurface for color hologram and a reflection chiral metasurface device that can simultaneously control both amplitude and phase. Combined with the PB phase, we completely decouple the RCP and LCP channels. It is important to emphasize that the CP conversion is naturally limited by the absorption loss of aluminum in our design system. Changing the material with lower loss is a way to improve CP conversion. The network architecture can be improved by convolution neural networks or residual neural networks to further solve the complex non-linear problems of multi-optical-parameter. There are three types of EPs extensively studied in physics: resonant EPs46,47,48, absorption EPs49 and scattering EPs50,51. Our approach can be extended to other non-Hermitian systems by adjusting the EP conditions, e.g., replacing the ellipticity with the ϵ = 0 when studying resonant EPs with PT-symmetry, where ϵ describes a resonance detuning52. Our work allows rapid and precise discovery of the topology of EPs, offers a powerful tool for digging related physics, and provides a paradigm for multi-optical parametric manipulation with high performance and less crosstalk, critical for imaging, encryption, and information storage applications.

Methods

Data collection

Figure 1 illustrates the general schematic of the plasmon structures under consideration. They consist of an aluminum meta-atom (upper layer), a SiO2 spacer (middle layer), and an aluminum mirror (bottom layer). During the modeling process, the meta-atoms are arranged in rectangular lattices. The parameters are created with (all lengths in nanometers) the following: L1 ∈ [0, 100], L2 ∈ [5, 250], L3 ∈ [0, 250] and then transferred to the commercial software package CST Microwave Studio for full-wave simulations. Since these parameter ranges include ample samples of phase and amplitude responses. Real part and imaginary part data of the reflection coefficient over the operating spectrum are calculated using CST frequency domain solver, with the unit cell boundary condition applied for all meta-atoms in both x and y directions. Opening boundaries are implemented in both the negative and positive z directions.

Details of deep neural network

Network Construction. A total of 3000 groups of meta-atom models and their corresponding complex reflection coefficients are collected and used for the training and testing of PNNs and INN. Detailed information on the PNNs and INN design network architectures, including hyper-parameters and learning curves, is listed in the Supporting Information. All models are constructed under the open-source machine-learning framework of TensorFlow.