Open Access
Add to BibSonomy
Abstract
We analyze the optical response of a parity-time ($\mathcal {PT}$) symmetric diatomic meta-molecular array. We show that the $\mathcal {PT}$ symmetry can transform an initial dark antisymmetric mode of the diatomic meta-molecule into bright, featured by a high-Q peak in the scattering spectrum. The Q factor is tunable by managing the magnitudes of the gain and loss. Furthermore, around this high-Q resonance an abnormal phase-shift is obtained, which persists even at exceptional point. We also show that the effective dielectric constant of the $\mathcal {PT}$-symmetric meta-molecular array can be modeled by an anomalous doubly-resonant Lorentz oscillator, which is different from that in electromagnetically induced transparency and Autler-Townes splitting. This study contributes to the advances of non-Hermitian optics and slow light.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the past decades we have witnessed the rapid advances in manipulating the dynamics of optical fields via the concept of artificial media, e.g. metamaterials and metasurfaces [1–8]. The key idea in almost all the schemes is to find a way in managing characteristics of artificial optical resonances so as to control the chirality, the propagating direction and the group velocity of an optical field. Among all the characteristics of an optical resonance, the $Q$ factor is of great importance because it not only determines the degree of field enhancement inside the artificial medium, but also provides a strong dispersion for various applications. A high-$Q$ factor also helps to maintain a fine detuning between paired resonances, which is desired in purposes such as these related to slow light [9–11]. To pursue a high-$Q$ resonance is just the reason of the surged study on bound-states-in-the-continuum (BICs) [12–17] in recent years.
Now, the advances of parity-time ($\mathcal {PT}$) symmetry provide us with another opportunity in manipulating artificial optical resonances [18–38]. A $\mathcal {PT}$ symmetric optical system is characterized by spatially distributed gain and loss. As a consequence of the non-Hermitian nature, eigenmodes in a $\mathcal {PT}$ symmetric system are no longer purely symmetric or anti-symmetric, and the detuning of two paired eigenmodes can be made extremely small, and even zero at the coalescent point named exceptional point (EP) [27–38]. Since the symmetry of an eigenmode determines its degree of darkness, e.g. a symmetry-protected BIC [12] is totally dark to the environment, it is then a very interesting question whether we can utilize this feature in controlling the $Q$ factor of the associated artificial resonance. Further more, because the detuning of paired eigenmodes and the abnormal dispersion in between are the keys to many attractive interference optical effects, e.g. electromagnetically induced transparency (EIT) and Autler-Townes splitting (ATS) [9–11,39], a study on the complex scattering spectra of paired $\mathcal {PT}$ resonances can reveal the physical mechanism behind and push forward potential applications in slow- or even fast-light. All these considerations motivate us in performing the study presented here.
In this article, we analyze the optical response of a $\mathcal {PT}$-symmetric diatomic meta-molecular array. We show that $\mathcal {PT}$ symmetry can transform an initial dark antisymmetric mode of the diatomic meta-molecule into bright, featured by a high-$Q$ peak in the scattering spectrum. The $Q$ factor is tunable by managing the amplitudes of the gain and loss. Further more, an abnormal phase-shift is achieved around this high-$Q$ resonance, which persists even at EP. Based on an effective-medium approach we deduce the effective dielectric constant of the meta-molecular array from the complex reflection coefficient. We find that although the abnormal phase-shift is similar to that in EIT and ATS, the effective dielectric constant follows an anomalous doubly-resonant Lorentzian function. This study highlights the novelty of $\mathcal {PT}$ symmetry, and contributes to the advances of non-Hermitian optics and slow-light science.
2. Theory and simulation
2.1 Structure and principle
First, let us explain the structure and the main principles of this investigation. Since a $\mathcal {PT}$ symmetry requires the interaction of gain and loss effects, and we are interested in the scenario that a dark mode exists before considering the $\mathcal {PT}$ symmetry, the simplest elemental structure is a diatomic meta-molecule as shown in Fig. 1. Each unit cell consists of two geometrically identical subwavelength particles with a radius of $d$. The refractive indices $n_R \pm jn_i$ of the two particles are conjugated with each other so that when one particle provides gain, the other provides loss effect. The two particles are placed close with each other, with a geometric distance $a$ much smaller than the period $p$. Since $p$ is smaller than the wavelength $\lambda$ in vacuum, a spatial array of these meta-molecules forms an effective medium that can be utilized in optical manipulation.
Fig. 1. Schematic of the configuration under investigation. Two geometrically identical particles (dielectric subwavelength spheres) with radius $d$ and distance $a$ form a diatomic meta-molecule, and are placed inside a cubic unit cell with a period of $p$. Refractive indices $n_R\pm jn_i$ of the two particles are conjugated with each other.
Resonances of the diatomic meta-molecule determine the response of the effective medium to an incident optical wave. Assuming each meta-molecule is placed horizontally in the $x-y$ plane, and considering an incident plane wave propagating in the $z$ direction, the resonance of each meta-molecule can be satisfactorily expressed by
The eigenfrequencies and eigenfunctions of Eq. (1) can be solved as documented in many literatures of optical $\mathcal {PT}$ symmetry [18–38]. Here our attention is paid to two characteristics of them. The first characteristic is about the resonant angular frequencies $\omega _{\pm }=\omega _0\pm (\alpha ^2-g^2)^{1/2}$, which are real in the conserved $\mathcal {PT}$ phase of $\alpha >g$ that discussed in this work. The detuning between them is then given by
We can see this pair of resonances approach each other when $g$ increases, and coalesce into a single one ($\Delta \omega =0$) at EP when $g=\alpha$. It is well known that a narrower detuning of paired resonances is preferred in EIT and ATS [9–11], and we can see the $\mathcal {PT}$ symmetric meta-molecule provides a similar opportunity here.
The second characteristic is about the brightness of the eigenmode because it determines the interaction strength of the meta-molecule with an external field and the associated $Q$ factor of the resonance. Usually two solutions can be found by solving Eq. (1), but here we are especially interested in the long-wavelength (low-frequency) one at $\omega _-$, as
On the contrary, the eigenmode of $\omega _+$ in the short-wavelength (high-frequency) side is always a bright mode because its eigenfunction is given by
The overlap integral of it with an external incident plane wave cannot be zero. The associated $Q$ factor is generally small.
To emphasize above discussed feature, we can propose a parameter termed the effective cross-section $S$ in characterizing the profile overlap of the eigenmodes with a plane-wave incidence in the $z$ direction, as
where $\Phi _\text {EM}$ is the normalized eigenfunction of the plane wave defined in the same parameter space as that of $E_g$ and $E_l$, $S_0$ is a constant obtained in the integral that has taken into account the plane-wave nature of the incident wave and the geometry of the particles. The variations of $S_\pm$ versus $g$ are shown in Fig. 2(a). When $g=0$, $S_-$ is null and the $\omega _-$ mode is totally dark. With increased $g$, $S_-$ slightly increases from 0. On the contrary, $S_+$ decreases monotonously from $S_0$ when $g$ increases. At EP of $\alpha =g$, $S_+$ and $S_-$ degenerate together because the eigenmodes coalesce here.Fig. 2. (a) Variation of the effective cross-section $S$ versus $g$, and (b) the reflection spectra $R=|s_{11}|^2$ at different $n_i$ values. We can see the $Q$ factor and the cross-section $S$ are correlated with each other.
2.2 High-$Q$ resonance
We perform three-dimensional full-field finite element optical simulations (COMSOL Multiphysics) on the scattering property of the $\mathcal {PT}$ symmetric diatomic meta-molecular array. Geometric parameters utilized here are $p=8$ $\mu$m, $d=1$ $\mu$m, and $a=10$ nm, respectively. The refraction index of the particles, before introducing the $\mathcal {PT}$ symmetry, is $n_R=5-j0.001$, where the small nonzero imaginary component is utilized to mimic the background absorption loss. The surrounding medium is air. Each meta-molecule is placed in the $x-y$ plane, with the inter-atomic axis along the $x$ direction. A plane wave with nonzero $H_y$ and $E_x$ components is launched from the input port, where the scattering parameter $s_{11}$ is recorded. The magnitude of the incident $E$ field is 1 V/m. Periodic boundary conditions are utilized in the $x$ and $y$ directions, and Perfectly Matched Layers (PMLs) are placed above and below the structure to absorb the reflected and transmitted waves. The gain and loss effects are introduced into the two particles via the additional non-zero imaginary components of $\pm n_i$, and we can approximately assume that $g$ is proportional to $n_i$ because $n_i$ is much smaller than $n_R$.
Figure 2(b) displays the reflection spectra $R=|s_{11}|^2$ when $n_i$ is increased from zero. We can see that in agreement with the analysis on the eigenfunction $\Psi$ and the cross-section $S$, only a broad reflection peak is obtained around $\lambda _+=10.152$ $\mu$m when $n_i=0$. This resonance is associated with the symmetric bright mode at $\omega _+$. The anti-symmetric mode $\omega _-$ at the long-wavelength side is invisible now. When $n_i$ increases slightly, the Hermitian nature of the coupled meta-molecule is broken, and this initial dark mode becomes bright [also see Fig. 3(a) at the wavelength of $\lambda _-=10.512$$\mu$m]. Because the weak nonzero $S_-$ helps to maintain an extreme confinement of field in the meta-molecule, this reflection peak possesses a high-$Q$ factor in comparison with the one of $\omega _+$. From the values of $\lambda _\pm$ we can find the values of $\omega _0$ and $\alpha$ by using $\omega _\pm =2\pi c/\lambda _\pm =\omega _0\pm \alpha$ (because now $g=0$), which gives $\omega _0=1.825\times 10^{14}$ $s^{-1}$ and $\alpha =3.18\times 10^{12}$ $s^{-1}$. With further increased $n_i$, the two peaks approach each other, which is in agreement with the standard theory of $\mathcal {PT}$ symmetry [18–38]. The $Q$ factor at $\omega _-$ also decreases substantially. At $n_i=0.11$ an EP is achieved, where not only the values of $\omega _\pm$ but also their $Q$ factors become the same.
Fig. 3. (a) Reflection coefficients $R$ versus wavelength at different $n_i$ values. (b) and (c) show the distribution of $H_y$ (background, unit $A/m$) and the vectorial direction of $D$ (arrows, no unit) at the two resonant peaks in (a).
To provide more evidences on the mechanism of these resonances, we also analyze the distributions of the electric displacement $D$ and magnetic field $H_y$. The electric displacement $D$ is well confined inside the particles, and can provide a clear and intuitionistic picture of the resonance. Figure 3 compares the difference between the reflection spectra of $n_i=0$ and $n_i=0.04$, and displays the distributions of $D$ and $H_y$ of the two resonances. We can see in each particle a circulating $D$ is excited. The $H_y$ component is enhanced inside the particles, reaches the maximum value at the central, and has no node. Consequently, the excited resonance in each particle is a magnetic dipole. As for the coupled resonance, at $\omega _-$ the circulating directions of $D$ in the two particles are opposite with each other, and the associated $H_y$ point toward opposite directions [see Fig. 3(c)]. These features confirm that it is a quasi-antisymmetric mode. As for the resonance at $\omega _+$, $H_y$ in the two particles are toward the same direction, and it is a quasi-symmetric mode.
2.3 Abnormal phase-shift
Above we have proved that the $\mathcal {PT}$ symmetry can induce a narrow reflection peak at the initial dark resonance $\omega _-$ by breaking the perfect anti-symmetry of it. The $Q$ factor of this peak is tunable, that with increased gain/loss rate $g$ it becomes broader. Now, paying attention to the other characteristic that the detuning $\Delta \omega$ between $\omega _+$ and $\omega _-$ approaches zero when $g$ increases toward EP, it is then an interesting question whether an abnormal dispersion, similar to that in EIT and ATS [9–11,39], can be induced in this $\mathcal {PT}$-symmetric system.
The features on dispersion can be found by analyzing the phases of the scattering parameters. Here we pay attention to the phase $\phi =\arg \{s_{11}\}$ that represents the phase delay of detected signal at the input port with respect to the input one. Some standard results are shown in Fig. 4. From Fig. 4(a) we can see where only a single resonance is excited ($n_i=0$), the phase-shift in $s_{11}$ is normal, that $\phi$ increases monotonously with the wavelength. However, when the high-$Q$ resonance associated with the quasi-antisymmetric mode at $\omega _-$ is excited, a sharp abnormal phase-shift can be observed between the two resonances. The abnormal phase-shift is characterized by a rapidly decreased $\phi$ versus wavelength. When $n_i$ increases and the two resonances approach each other, the position of the abnormal phase-shift also moves following the dip in the reflection spectra (see the light green shadows in Fig. 4). Note that this abnormal phase-shift persists even at EP, see Fig. 4(d). By comparing Fig. 4(a) for the Hermitian scenario with Fig. 4(d) for EP, we can see although in both cases only a single reflection peak exists, the phase-shift is obviously abnormal in the later scenario. The existence of this abnormal phase-shift can be utilized as an intrinsic feature of the non-Hermitian system, and might hint some other interesting physical mechanisms that deserve our further investigation.
Fig. 4. Reflection spectra $R$ (blue) and phase $\phi$ of $s_{11}$ (red) versus wavelength at (a) $n_i=0$, (b) $n_i=0.04$, (c) $n_i=0.08$, and (d) $n_i=0.11$(EP), respectively. The regions of abnormal phase-shift are labeled out by light green shadows.
To further reveal the origin of the abnormal phase-shift, we try to recover the effective dielectric constant $\epsilon _\text {eff}$ of the meta-molecular array by simply using the formula of reflection coefficient from a single interface, as
In this formula, the complex reflection coefficient $r$ is related to $s_{11}$ by $r=s_{11}\exp (j2kL_\text {eff})$, where $L_\text {eff}$ is the distance of the input port to an effective interface above the meta-molecule, $k=2\pi /\lambda$ is the wavevector in vacuum, and the factor 2 accounts for the forward and backward translation between the input port and the interface. By choosing a proper position of the effective interface and correcting this dynamic phase from $s_{11}$ we can briefly find the spectrum of $\epsilon _\text {eff}$. Two sets of results are shown in Fig. 5. We can see when $n_i=0$, only a single response around $\omega _+$ is obtained in $\epsilon _\text {eff}$. When $n_i$ increases, an additional sharp response emerges on the long-wavelength side. Once again, the analysis on $\epsilon _\text {eff}$ also agrees well with our above prediction on the tunable high-$Q$ resonance at $\omega _-$.
Fig. 5. Effective dielectric constant $\epsilon _\text {eff}$ deduced from $s_{11}$ when (a) $n_i=0$, and (b) $n_i=0.04$. (c) and (d) are the best fitted curves by using Eq. (7).
3. Discussion
Above brief fitting of the effective dielectric constant $\epsilon _\text {eff}$ confirms that the tunable high-$Q$ resonance and the abnormal phase-shift are all associated with the excitation of the quasi-antisymmetric mode at $\omega _-$. However, referring to the result shown in Fig. 5(b) we can see the spectrum of $\epsilon _\text {eff}$ is somehow different from that in ATS and EIT [9–11,39]. In ATS and EIT the effective response of the medium to an optical wave can be modeled by two Lorentz oscillators with resonant wavelength $\lambda _{1,2}$, so that $\epsilon _\text {eff}(\lambda )=\epsilon _\text {bak}+\sum _{i=1,2}\Lambda _i^2/(\lambda ^2-\lambda _i^2-j\delta _i\lambda )$, where $\epsilon _\text {bak}$ is the background, $\Lambda _{1,2}$ is the plasma wavelength, and $\delta _{1,2}$ is the decay length [9]. As for the $\mathcal {PT}$ symmetric meta-molecule discussed here, although each meta-molecule also supports a pair of resonances of wavelengths $\lambda _\pm =2\pi c/\omega _\pm$, the effective-medium approach gives a different curve from the doubly-resonant Lorentz-oscillator model [9]. The dispersion belongs to an anomalous doubly-resonant Lorentz oscillator [see Fig. 5(b)] with
The terminology of ‘anomalous’ refers to the last term of Eq. (7) that represents the contribution from the quasi-antisymmetric mode $\omega _-$. The negative sign before it implies that it is an effective oscillator with a gain effect, a scenario which is readily available in a population-inverted ($N_1<N_2$) 2-level atomic system because the denominator of the Lorentzian function is proportional to the net population of $N_1-N_2$ in the ground level. The positive sign before $\delta _-$ in the numerator implies that the amplitude of the oscillation grows with distance.
We fit the curves of Figs. 5(a) and 5(b) by using Eq. (7), and the best fitted curves are shown in Figs. 5(c) and 5(d). We can see they agree fairly well with each other. When $n_i=0$ where only a single peak exists, the best fitted parameters are $\epsilon _\text {bak}=0.9$, $\lambda _+=10.3$ $\mu$m, $\delta _+=0.08$ $\mu$m, $\Lambda _+=2.75$ $\mu$m, and $\Lambda _-$=0. When $n_i=0.04$, the best fitted parameters are $\epsilon _\text {bak}=0.9$, $\lambda _+=10.32$ $\mu$m, $\lambda _-=10.49$ $\mu$m, $\delta _+=0.08$ $\mu$m, $\delta _-=0.005$ $\mu$m, $\Lambda _+=2.75$ $\mu$m, and $\Lambda _-=1.1$ $\mu$m. Nevertheless, the deep-seated physics of the anomalous feature in Eq. (7) is beyond our ability now, and might be explained by the analogue of quantum physics and optics in the non-Hermitian frame [40,41]. Future studies on this anomaly would push forward many potential applications of $\mathcal {PT}$ symmetry especially these ones related to BIC and slow/fast light. Note that when $n_i$ further increases, e.g. $n_i>0.08$, the two resonances are not distinguishable in $\epsilon _\text {eff}$ because the wavelength difference between them is smaller than the width of the peak. The spectral features in $\epsilon _\text {eff}$ are smeared [42,43], and Eq. (7) cannot accurately describe the dispersion.
Besides the effort in explaining the anomalous doubly-resonant Lorentz-oscillator dispersion, in future work we can also try to overcome some other drawbacks of our study. For example, in this work we simply use the complex scattering parameter $s_{11}$ from a mono-layer meta-molecular array to find $\epsilon _\text {eff}$. In future work we can consider a bulk one containing multiple-layers of diatomic meta-molecular arrays. Secondly, although in our analysis on $\epsilon _\text {eff}$ we have subtracted the additional dynamic phase-shift between the input port and the meta-molecular array by assuming that the interface is flat and does not vary with frequency, it is hard to answer where the exact place of the effective flat interface is. In future work we can find ways in further reducing the dimension of the particles and burying them in a proper substrate so that the whole structure can be described as an effective medium in a much better way.
Before ending this article, we would like to briefly discuss potential experimental verification of our results. Because COMSOL is based on Maxwell’s equations, the analysis presented in this article is universal (not limited to 10 $\mu$m) and can be applied to other frequency regimes as well by simply resizing the geometric parameters. Future experiments can be performed in these frequency regimes where methods in realizing tunable gain and loss are available, e.g. external optical/electrical pumping of semiconductors and nonlinear optical processes utilized in surface plasmonics, meta-materials, and non-Hermitian optics [5–8,23,24,28,29,44–49].
4. Conclusion
In summary, here we analyze the optical response of a $\mathcal {PT}$-symmetric diatomic meta-molecular array. we show that $\mathcal {PT}$ symmetry can transform an initial dark antisymmetric mode of the diatomic meta-molecule into bright, featured by a high-$Q$ peak in the scattering spectrum. The $Q$ factor is tunable by managing the magnitudes of the gain and loss in the structure. Further more, an abnormal phase-shift is achieved around this high-$Q$ resonance, which persists even at EP. Based on an effective-medium approach we deduce $\epsilon _\text {eff}$ of the meta-molecular array from the complex reflection coefficient. We find that although the abnormal phase-shift is similar to that in EIT and ATS, $\epsilon _\text {eff}$ follows an anomalous doubly-resonant Lorentzian function here. This study highlights the novelty of $\mathcal {PT}$ symmetry, and contributes to the advances of non-Hermitian optics and slow-light science.
Funding
National Natural Science Foundation of China (12274241, 11874228, 11974259, 12104203, 12104227).
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.
References
1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef]
2. N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11(11), 917–924 (2012). [CrossRef]
3. F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007). [CrossRef]
4. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007). [CrossRef]
5. Y. M. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011). [CrossRef]
6. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). [CrossRef]
7. H. T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016). [CrossRef]
8. S. Sun, Q. He, J. Hao, S. Xiao, and L. Zhou, “Electromagnetic metasurfaces: physics and applications,” Adv. Opt. Photonics 11(2), 380–479 (2019). [CrossRef]
9. J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photonics 2(3), 287–318 (2010). [CrossRef]
10. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). [CrossRef]
11. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef]
12. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljacic, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016). [CrossRef]
13. A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kante, “Lasing action from photonic bound states in continuum,” Nature 541(7636), 196–199 (2017). [CrossRef]
14. H. M. Doeleman, F. Monticone, W. den Hollander, A. Alu, and A. F. Koenderink, “Experimental observation of a polarization vortex at an optical bound state in the continuum,” Nat. Photonics 12(7), 397–401 (2018). [CrossRef]
15. Y. Liang, K. Koshelev, F. Zhang, H. Lin, S. Lin, J. Wu, B. Jia, and Y. Kivshar, “Bound states in the continuum in anisotropic plasmonic metasurfaces,” Nano Lett. 20(9), 6351–6356 (2020). [CrossRef]
16. E. N. Bulgakov and D. N. Maksimov, “Topological bound states in the continuum in arrays of dielectric spheres,” Phys. Rev. Lett. 118(26), 267401 (2017). [CrossRef]
17. D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008). [CrossRef]
18. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007). [CrossRef]
19. S. Longhi, “$\mathcal {PT}$-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010). [CrossRef]
20. A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in $\mathcal {PT}$-symmetric optical systems,” Phys. Rev. A 89(3), 033808 (2014). [CrossRef]
21. J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91(3), 033825 (2015). [CrossRef]
22. S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546(7658), 387–390 (2017). [CrossRef]
23. L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11(12), 752–762 (2017). [CrossRef]
24. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and $\mathcal {PT}$ symmetry,” Nat. Phys. 14(1), 11–19 (2018). [CrossRef]
25. W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17(2), 149–151 (2001). [CrossRef]
26. H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87(1), 61–111 (2015). [CrossRef]
27. Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016). [CrossRef]
28. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017). [CrossRef]
29. W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhanced sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017). [CrossRef]
30. T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120(1), 013901 (2018). [CrossRef]
31. S. A. H. Gangaraj and F. Monticone, “Topological waveguiding near an exceptional point: defect-immune, slow-light, and loss-immune propagation,” Phys. Rev. Lett. 121(9), 093901 (2018). [CrossRef]
32. M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019). [CrossRef]
33. H. Z. Chen, T. Liu, H. Y. Luan, R. J. Liu, X. Y. Wang, X. F. Zhu, Y. B. Li, Z. M. Gu, S. J. Liang, H. Gao, L. Lu, L. Ge, S. Zhang, J. Zhu, and R. M. Ma, “Revealing the missing dimension at an exceptional point,” Nat. Phys. 16(5), 571–578 (2020). [CrossRef]
34. J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93(3), 033809 (2016). [CrossRef]
35. L. S. Sun, B. Zhao, J. Q. Yuan, and J. Chen, “Kerr-nonlinearity induced bistable-like parity-time phase transition in coupled waveguides,” Opt. Express 29(5), 7935–7947 (2021). [CrossRef]
36. J. B. Khurgin, Y. Sebbag, E. Edrei, R. Zektzer, K. Shastri, U. Levy, and F. Monticone, “Emulating exceptional-point encirclements using imperfect (leaky) photonic components: asymmetric mode-switching and omni-polarizer action,” Optica 8(4), 563–569 (2021). [CrossRef]
37. J. Q. Yuan, B. Zhao, L. S. Sun, L. T. Wu, T. J. Guo, M. Kang, and J. Chen, “Optical super-resonance in a customized $\mathcal {PT}$-symmetric system of hybrid interaction,” Opt. Express 29(16), 24663–24673 (2021). [CrossRef]
38. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of $\mathcal {PT}$ symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113(9), 093901 (2014). [CrossRef]
39. B. Peng, S. K Ozdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5(1), 5082–5092 (2014). [CrossRef]
40. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photonics Rev. 3(3), 243–261 (2009). [CrossRef]
41. N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge, 2011).
42. N. A. Mortensen, P. A. D. Goncalves, M. Khajavikhan, D. N. Christodoulides, C. Tserkezis, and C. Wolff, “Fluctuations and noise-limited sensing near the exceptional point of parity-time-symmetric resonator systems,” Optica 5(10), 1342–1346 (2018). [CrossRef]
43. W. Langbein, “No exceptional precision of exceptional-point sensors,” Phys. Rev. A 98(2), 023805 (2018). [CrossRef]
44. H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014). [CrossRef]
45. D. Kip, “Photorefractive waveguides in oxide crystals: fabrication, properties, and applications,” Appl. Phys. B 67(2), 131–150 (1998). [CrossRef]
46. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]
47. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009). [CrossRef]
48. J. B. Khurgin, “How to deal with the loss in plasmonics and metamaterials,” Nat. Nanotechnol. 10(1), 2–6 (2015). [CrossRef]
49. V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in $\mathcal {PT}$-symmetric systems,” Rev. Mod. Phys. 88(3), 035002 (2016). [CrossRef]
