## Abstract

Dielectric super-absorbing ($\gt\!{50}\%$) metasurfaces, born of necessity to break the 50% absorption limit of an ultrathin film, offer an efficient way to manipulate light. However, in previous works, super absorption in dielectric systems was predominately realized via making two modes reach the degenerate critical coupling condition, which restricted the two modes to be orthogonal. Here, we demonstrate that in nonorthogonal-mode systems, which represent a broader range of metasurfaces, super absorption can be achieved by breaking parity-time (PT) symmetry. As a proof of concept, super absorption (100% in simulation and 71% in experiment) at near-infrared frequencies is achieved in a Si-Ge-Si metasurface with two nonorthogonal modes. Engineering PT symmetry enriches the field of non-Hermitian flat photonics, opening-up new possibilities in optical sensing, thermal emission, photovoltaic, and photodetecting devices.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Dielectric super-absorbing ($\gt\!{50}\%$) metasurfaces have attracted growing attention in that they break the 50% absorption limit of a subwavelength thickness film [1], and exhibit advantages over metallic metasurfaces, such as low Ohmic dissipation, out-of-band transparency, and CMOS compatibility [2–9]. Losses in these metasurfaces enable them to exchange energy with the surrounding environment and exhibit complex eigenvalues, making them non-Hermitian [10–13]. Originating from quantum mechanics [14], non-Hermitian physics has been used to study the characteristics of optical systems, which led to the emergence of numerous novel phenomena and applications, such as loss-induced optical non-reciprocity [15], loss-induced optical transparency [16], selective thermal emitters [17,18], unidirectional invisibility [19,20], and exceptional points (EPs)-based sensing [21–23]. Parity-time (PT) symmetric systems is a particular family of non-Hermitian systems, which are invariant under the combined action of the $P$ $({i \to i,\hat x \to - \hat x,\hat p \to - \hat p})$ and $T$ $({i \to \!-\! i,\hat x \to \hat x,\hat p \to \!-\! \hat p})$ operators. The eigenfrequencies of such systems have distinct behavior in the PT symmetry and PT symmetry-breaking regime, and this characteristic opens up new possibilities of engineering spectral properties of photonic systems [24–27].

In dielectric non-Hermitian metasurfaces, previous efforts on realizing super absorption were making two spectrally overlapped modes reach the degenerate critical coupling condition, where the radiative and non-radiative decay rates are the same for each mode [28–41]. This method restricts the two modes to be orthogonal, i.e., the coupling between them is negligible compared to their losses. Consequently, the eigenfrequencies degenerate at the diabolic point [DP, see the green dashed line in Fig. 1(b)], where the eigenvectors are orthogonal (see Supplement 1). However, when the mode coupling is non-negligible, the strong interaction between different resonances can cause mode splitting, and the two-orthogonal-mode model is invalidated under this condition.

In this work, we demonstrate super absorption in dielectric metasurfaces by breaking PT symmetry. Our method is based on a two-nonorthogonal-mode model with EP degeneracies (both eigenvectors and complex eigenfrequencies coalesce), which can represent a broader range of optical systems. As a proof of concept, super absorption at near-infrared (NIR) frequencies is achieved in a Si-Ge-Si metasurface, with two nonorthogonal quasi bound states in the continuum (QBIC) modes. In this system, PT symmetry breaking is realized by engineering the loss difference between the two modes, and it successfully suppresses the mode splitting. Our work provides a clue for engineering light trapping in non-Hermitian flat photonics, thus having broad implications in optical sensing, photodetecting, thermal emission manipulation, and photovoltaic devices.

## 2. THEORETICAL MODEL

We start by considering a dual-port photonic system supporting two nonorthogonal modes ${\rm M}_1$ and ${\rm M}_2$ [Fig. 1(a)], whose resonant frequencies before coupling are ${f_1}$ and ${f_2}$, respectively. The two modes formed within a single resonator are connected by the near-field coupling coefficient $\kappa$. The radiative decay rate corresponding to mode $j$ ($j = 1,2$) can be expressed as ${\gamma _{j,{\rm R}}} = {\gamma _{je}} + \gamma _{je}^\prime$, where ${\gamma _{je}}$ and $\gamma _{je}^\prime$ are radiative decay rates of mode $j$ to Port 1 and Port 2, respectively. The total decay rate of mode $j$ can be given by ${\gamma _j} = {\gamma _{j,{\rm NR}}} + {\gamma _{j,{\rm R}}}$, where ${\gamma _{j,{\rm NR}}}$ corresponds to the non-radiative decay rate of mode $j $. The amplitude of incoming (outgoing) wave from Port 1 is expressed as ${S_{1 +}}$ (${S_{1 -}}$), whereas ${S_{2 -}}$ represents the amplitude of the outgoing wave from Port 2. This model can describe many dielectric systems such as dielectric meta-atoms on a transparent substrate illuminated from one side by incident light.

The far-field coupling induced by the radiation in two channels is mainly determined by the symmetric properties of the two modes. Here, we suppose that one mode decays symmetrically and the other decays anti-symmetrically into two ports, which leads to a negligible far-field coupling. Otherwise, if the two modes have the same symmetric properties, the total absorption cannot exceed 50% (see Supplement 1, Section 2). The effective Hamiltonian of the two-nonorthogonal-mode system in Fig. 1(a) is given by [42–44]

When the two resonant frequencies before coupling are the same (${f_1} = {f_2} = {f_0}$), the eigenfrequencies can be expressed as

The behavior of ${f_{\rm eigen}}$ is controlled by the coupling coefficient $\kappa$ and the loss difference ${\Delta}\gamma = | {{\gamma _1} - {\gamma _2}} |$. There are three cases: (1) $\kappa = {\Delta}\gamma /2$, which is the condition where both ${\rm Re}({{f_{\rm eigen}}})$ and ${\rm Im}({{f_{\rm eigen}}})$ of the two modes along with their associated eigenvectors coalesce, indicating the emergence of an EP [45]; (2) $\kappa \gt {\Delta}\gamma /2$, where, under this condition, the system is in passive PT-symmetric phase, and only ${\rm Im}({{f_{\rm eigen}}})$ of the two modes coincide; and (3) $\kappa \lt {\Delta}\gamma /2$, whereby the PT-symmetric phase is broken under this condition, and only ${\rm Re}({{f_{\rm eigen}}})$ of the two modes coincide. The coupling coefficient at the transition point of PT-symmetric phase is ${\kappa _{\rm EP}} = {\Delta}\gamma /2$ [see the vertical gray dashed line in Fig. 1(b)].

*Lineshape and Linewidth Control*—The eigenfrequencies are responsible for the lineshape features of absorption spectra. When $\kappa \gt {\kappa _{\rm EP}}$, two modes exchange energy strongly with each other, leading to the mode splitting. ${\rm Re}({{f_{\rm eigen}}})$ of the two modes differ from each other, and two peaks in the absorption spectrum can be seen consequently [see PT symmetry regime in Fig. 1(b)]. When $\kappa \lt {\kappa _{\rm EP}}$, ${\rm Re}({{f_{\rm eigen}}})$ of the two modes coincide and so do the two absorption peaks [see PT symmetry breaking regime in Fig. 1(b)]. Therefore, to obtain single-peak absorption spectra, we should either engineer $\kappa$ or ${\Delta}\gamma$ to make $\kappa \lt {\kappa _{\rm EP}}$, so that the coupling-induced mode splitting can be suppressed. In PT symmetry breaking regime, the quality-factors (Q-factors) are determined by the total loss (${\gamma _1} + {\gamma _2}$) according to the relationship ${Q} \propto {\omega}/({{\gamma _1} + {\gamma _2}})$, indicating that the total loss needs to be suppressed to obtain narrow band absorption spectra [see Fig. 1(b)].

*Amplitude Control*—In PT symmetry breaking regime, the amplitudes of absorption spectra ($ A $) can be optimized to be 100% by tuning the radiative and non-radiative decay rates. The calculated absorption in the parameter space $({\gamma _{1,{\rm NR}}}/{\gamma _{1,{\rm R}}}, {\gamma _{2,{\rm NR}}}/{\gamma _{2,{\rm R}}},\kappa /\sqrt {{\gamma _{1,{\rm R}}}{\gamma _{2,{\rm R}}}})$ is shown in Fig. 1(c), and unity absorption ($A = {100}\%$) is achieved when both Eqs. (3) and (4) are satisfied.

Here, we suppose ${\rm M}_1$ decays symmetrically, and ${\rm M}_2$ decays anti-symmetrically. The other case is that ${\rm M}_1$ decays anti-symmetrically, and ${\rm M}_2$ decays symmetrically, which requires $\frac{{{\gamma _{1,{\rm NR}}}}}{{{\gamma _{1,{\rm R}}}}} = 1 + \frac{\kappa}{{\sqrt {{\gamma _{1,{\rm R}}}{\gamma _{2,{\rm R}}}}}}$, and $\frac{{{\gamma _{2,{\rm NR}}}}}{{{\gamma _{2,{\rm R}}}}} = 1 - \frac{\kappa}{{\sqrt {{\gamma _{1,{\rm R}}}{\gamma _{2,{\rm R}}}}}}$ to realize unity absorption (more details are provided in Supplement 1, Section 3). At $\kappa = 0$, the two modes are orthogonal and unity absorption necessitates the radiative decay rate to be equal to the non-radiative decay rate for each mode, which is exactly the degenerate critical coupling condition studied before, see ${P_1}$ in Fig. 1(c). When the orthogonality of the two modes is perturbed by extra coupling ($\kappa \ne 0$), unity absorption can still be achieved as long as the radiative and non-radiative decay rates meet the condition described by Eqs. (3) and (4) [see ${P_2}$ and ${P_3}$ in Fig. 1(c)].

To sum up, in order to achieve super absorption for a two-nonorthogonal-mode system, we need to engineer the following three parameters: (1) lineshape: decrease the coupling coefficient $\kappa$ or increase the loss difference ${\Delta}\gamma$ to make $\kappa \lt {\kappa _{\rm EP}}$ so that PT symmetry is broken, and thereby the mode splitting is prevented; (2) linewidth: engineer the total loss to control the linewidth, and (3) amplitude: choose two modes with different symmetric properties, and tune the radiative and non-radiative decay rates of the two modes to satisfy Eqs. (3) and (4) so that the absorption can reach 100%.

## 3. EXAMPLES OF SUPER ABSORPTION VIA PT SYMMETRY BREAKING

To validate the theoretical model, a dielectric metasurface with two nonorthogonal modes is established [Fig. 2(a)]. The Si elliptical cylinders with an orientation angle $\theta$ on a ${\rm SiO}_2$ substrate support magnetic and electric QBIC (M-QBIC and E-QBIC) modes [3,46]. The mode coupling, which is comparable to the low radiative loss of QBIC modes, mainly arises from the substrate-induced interaction between electric and magnetic dipole resonances [47], and leads to the destruction of orthogonality. A thin layer of lossy Ge is inserted in the middle of lossless Si cylinders to introduce the non-radiative loss. In this Si-Ge-Si metasurface, the radiative decay rates ${\gamma _{{\rm E} - {\rm QBIC},{\rm R}}}$ and ${\gamma _{{\rm M} - {\rm QBIC},{\rm R}}}$ increase with the orientation angle $\theta$, and hardly change when Ge thickness ${h_2}$ varies (see Supplement 1, Section 4). Within the cylinders, the electric field of E-QBIC mainly concentrates in the lossy Ge layer, while that of M-QBIC is mainly in the lossless Si layer [Fig. 2(b)]. Therefore, the non-radiative decay rates ${\gamma _{{\rm E} - {\rm QBIC},{\rm NR}}}$ increases faster than ${\gamma _{{\rm M} - {\rm QBIC},{\rm NR}}}$ when Ge thickness ${h_2}$ increases, which causes the total loss difference ${\Delta}\gamma$ increases with Ge thickness ${h_2}$.

The PT-symmetric phase is tuned by changing Ge thickness ${h_2}$ [Figs. 2(c) and 2(d)]. To guarantee either ${\rm Re}({{f_{\rm eigen}}})$ or ${\rm Im}({{f_{\rm eigen}}})$ of the two modes coinciding at different ${h_2}$, ${P_x}$ is varied from 0.712 µm to 0.733 µm in the calculation (see Table 1 in Supplement 1, Section 4). At ${h_2} \lt 0.038\;\unicode{x00B5}{\rm m}$, the total loss difference ${\Delta}\gamma$ is not large enough to compensate for the coupling, indicating the system is in PT-symmetric phase. The field distributions are distorted due to strong mode coupling. At ${h_2} = 0.038\;{\unicode{x00B5}{\rm m}}$, both ${\rm Re}({{f_{\rm eigen}}})$ and ${\rm Im}({{f_{\rm eigen}}})$ of the two modes coalesce at EP. The electric field distributions of the two eigenmodes are the same, since two corresponding eigenvectors are parallel at EP. At ${h_2} \gt 0.038{\unicode{x00B5}{\rm m}}$, the system is in PT symmetry broken phase. Under this condition, ${\rm Re}({{f_{\rm eigen}}})$ of the two modes coincide. The weak coupling compared with the loss difference makes the two eigenmodes have distinct field distributions. In the simulation, eigenfrequencies are calculated using the complex refractive-index of the lossy Ge, and the eigenmodes distributions can be seen in the insets of Fig. 2(c) and Supplement 1, Section 5.

The dependence of eigenfrequencies on both Ge thickness ${h_2}$ and the cylinder thickness ${h_1}$ are calculated [Figs. 3(a) and 3(b)]. At ${h_2} = 0.010{\unicode{x00B5}{\rm m}}$, PT symmetry is not broken, indicating two modes have different ${\rm Re}({{f_{\rm eigen}}})$ and the same ${\rm Im}({{f_{\rm eigen}}})$ at the point marked with a rhombus in Fig. 3(d), which leads to an avoided crossing in the absorption spectra. Two peaks can be observed in the absorption spectra whose amplitudes are less than 0.5 [Fig. 3(e)]. At ${h_2} = 0.038\;{\unicode{x00B5}{\rm m}}$, both ${\rm Re}({{f_{\rm eigen}}})$ and ${\rm Im}({{f_{\rm eigen}}})$ of the two modes coincide at the EP [stars in Figs. 3(f) and 3(g)]. The superposition of the two modes makes the absorption peak larger than 0.5 [Fig. 3(h)]. At ${h_2} = 0.050\;{\unicode{x00B5}{\rm m}}$, PT symmetry is broken and two modes have the same ${\rm Re}({{f_{\rm eigen}}})$ and different ${\rm Im}({{f_{\rm eigen}}})$ at the point marked with a circle in Fig. 3(i). No mode splitting can be seen due to the coincidence of ${\rm Re}({{f_{\rm eigen}}})$, thus the absorption spectra corresponding to the two modes also show a crossing, and super absorption is obtained [Fig. 3(k)].

The super absorption resulting from breaking PT symmetry is experimentally validated by fabricating Si-Ge-Si metasurfaces with different periods ${P_x}$. Here, Ge thickness ${h_2} = 0.040\;{\unicode{x00B5}{\rm m}}$ corresponds to the case in which the absorption can reaches unity in simulation. The corresponding simulated eigenfrequencies and absorption spectra are plotted in Fig. 4(a) and Fig. 4(b), respectively. The PT symmetry breaking condition is marked with the gray dashed line. The measured absorption spectra of the fabricated Si-Ge-Si metasurfaces are provided in Fig. 4(c). At ${P_x} = 0.790\;{\unicode{x00B5}{\rm m}}$, the M-QBIC and E-QBIC modes are spectrally separated. As ${P_x}$ is gradually decreased, two absorption peaks cross due to the breaking of PT symmetry. When ${P_x}$ is decreased further, the two peaks gradually separate again. At ${P_x} = 0.710\;{\unicode{x00B5}{\rm m}}$, super absorption (71%) with Q-factor $\sim\! {41}$ is achieved [Fig. 4(d)]. The deviation of absorption spectra in the experiment from the simulated results arises due to the fabrication imperfection, such as reduced cylinder size after etching. More details regarding the fabrication process and spectra measurement are provided in Supplement 1, Section 6.

## 4. CONCLUSIONS

In conclusion, we demonstrate that super absorption in dielectric metasurfaces can be achieved by breaking PT symmetry. Utilizing two coupled modes, our work provides a novel method of breaking the 50% absorption limit and achieving super absorption in dielectric metasurfaces. In previous works, dielectric super absorption was predominately realized via making two orthogonal modes reach the degenerate critical coupling condition. To guarantee the orthogonality, the losses of two modes should be large enough to preclude the influence of mode coupling, which usually makes the Q-factors low ($\sim{10}$) [32–34,36]. Breaking PT symmetry provides an effective way to achieve super absorption in high-Q dielectric systems where the losses are comparable to the mode coupling. Besides, different from most previous experimental works where the PT-symmetric phase is broken by decreasing the coupling strength [21,48], the method of increasing the loss difference presented in this work enriches the field of PT-symmetric phase engineering. Moreover, the high-Q and out-of-band transparent properties make this dielectric absorber promising for applications in optical sensing and photodetecting devices. The working frequencies of these devices can be further extended to mid-infrared (MIR) and terahertz (THz) frequencies by replacing Si and Ge with other combination of materials: for MIR range, lossless Ge and lossy ${\rm Ge}_2{\rm Sb}_2{\rm Te}_5$ (GST) [49,50], ${\rm VO}_2$ [51]; and for THz range, lossless Si and lossy doped-Si [52]. Finally, we believe that the study of engineering PT-symmetric phase in low-loss dielectric systems can lead to the generation of non-Hermitian devices with novel properties, such as selective thermal emitters with high coherence, and highly sensitive spectrally tunable metasurfaces with multiple functionalities.

## Funding

National Key Research and Development Program of China (2017YFA0205700); National Natural Science Foundation of China (61775194, 61950410608, 61975181).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## Supplemental document

See Supplement 1 for supporting content.

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