Abstract

In optical systems, one kind of exceptional point (EP) is associated with the maximally unidirectional reflection. Here, we theoretically show that the intrinsic chirality of this kind of EP is only determined by the sign of the scattering rate difference, and that these EPs could only be on or within a fundamental scattering bound in an asymmetric resonant system. As a proof of our theoretical deviation, a bianisotropic metasurface is designed to exhibit an extreme EP with a definite chirality on the fundamental scattering bound. In addition, another EP with the opposite chirality is also available within this scattering bound in the same metasurface without any additional symmetry operation. Numerical results are in good agreement with our theoretical predictions based on the coupled mode theory. We believe that our results not only provide physical insights to explore EPs in resonant systems, but also have implications in designing unidirectional absorbers and thermal emitters.

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

1. Introduction

It is a common sense that Hermitian Hamiltonian lies at the heart of classical quantum mechanics. Unlike the Hermitian Hamiltonian, recently, the non-Hermitian system subjected to dissipation or decay loss has drawn a lot of research interests. The exceptional points (EPs) exclusive to non-Hermitian systems are mainly responsible for many counterintuitive phenomena [17]. Mathematically, an EP is a branch singularity where both eigenvalues and eigenvectors simultaneously coalesce. In optical system, EPs are linked to a lot of counter-intuitive phenomena such as loss-induced transparency [8], simultaneous lasing and absorption [9,10], and unidirectional light reflection [11,12], with potential applications for single-mode lasing [13] and enhanced sensor [1416]. The unprecedented phenomena have draw a lot of research interests to explore EPs in different optical systems [1725]. Lin et al. theoretically discussed the unidirectional invisibility coinciding with EP in a parity-time (PT) symmetric periodic structure, where the reflection is zero in one side but nonzero in the opposite side [11]. Without an exact PT-symmetric refractive index modulation, Feng et al. experimentally verified that the unidirectional reflection could occur at an EP in a fully passive optical waveguide structure [12]. However, delicate refractive index modulation is essential to explore this kind of EP, which causes stringent fabrication challenge for practical applications.

Among various artificial optical systems, metasurfaces are a special kind of interesting platform for exploring the physics of EPs because they possess an additional structural design freedom to effectively control the light-matter interaction [26]. Recently, the appearance of EPs and their associated optical properties have been explored in metasurfaces [2730]. For example, the unidirectional reflection due to the appearance of EP can be realized in a multilayer metasurface without a carefully modulated refractive index profile [27,31]. However, the physical mechanism is still unclear, especially the intrinsic chirality of EP [1] has not be identified up to now. Here, we map this kind of asymmetric metasurfaces into an asymmetric resonator,where the scattering asymmetry has been shown to be fundamentally bounded in a lossless system [3237]. When the so-called freedom of loss is added to generate an EP in such asymmetric system, a fundamental understanding on both the EP and its intrinsic chirality, especially the direct link between EP and the scattering bound, may be achieved.

In this article, we theoretically explore how EPs are realized in a designed metasurface possessing bianisotropic response [38,39], which could cause an asymmetric reflection. The optical response of this asymmetric resonant metasurface can be described by the coupled-mode-theory (CMT) formalism [3234], and bears fundamental scattering bounds [3537]. When the freedom of dissipation loss is taken into account, we discuss how EPs could occur on or within these scattering bounds, and demonstrate that the intrinsic chirality of EP in this scattering scenario is only determined by the sign of the scattering coupling rate difference. As a proof of our theoretical prediction, we numerically show that EPs on the scattering bound can be achieved in an actual metasurface design. Profited by the structural design freedom of the metasurface, another EP with opposite chirality is also numerically verified within the bounds in the same metasurface without any additional symmetry operation. Our results are verified by full-wave numerical simulations, which find good agreement with CMT. This investigation provides physical insight to explore EPs in various resonant physical systems not limited to metasurfaces and have implications in designing various practical devices such as unidirectional absorbers, thermal emitters and asymmetric gratings.

2. Theoretical description

As a first step, we apply CMT to explore the appearance of EPs in a resonant metasurface possessing only one resonant mode, as illustrated in the inset of Fig. 1. In CMT, $u$ denotes the complex oscillating amplitude of the resonant mode in the explored metasurface. The incident light described by $(s_{1,\mathrm {in}}, s_{2,\mathrm {in}})^{\mathrm {T}}$ interacts with the metasurface according to the coupled equation [3234]

$$\frac{du}{dt} = (if_{0}-\gamma_{t}-\gamma_{d})u+\left(\begin{matrix} d_{1}, d_{2}\end{matrix}\right)\left(\begin{array}{l}{s_{1,\mathrm{in}}}\\{s_{2,\mathrm{in}}}\end{array}\right),$$
where $f_{0}$ is the resonant frequency, $\gamma _{t}$ is the total scattering loss rate, $\gamma _{d}$ is the dissipation loss rate, and $d_{1} (d_{2})$ is the scattering coupling between the resonant metasurface and the incident light from the left (right) side. Moreover, the outgoing light described by $( s_{1,\mathrm {out}}, s_{2,\mathrm {out}})^{\mathrm {T}}$ shown in Fig. 1 is an interference of the direct transported component from the input and the resonance-assisted re-scattering one from the metasurface
$$\left(\begin{array}{l}{s_{1,\mathrm{out}}}\\{s_{2,\mathrm{out}}}\end{array}\right) = \left(\begin{array}{l}{d_{2}}\\{d_{1}}\end{array}\right)u + \mathbf{C}\,\left(\begin{array}{l}{s_{1,\mathrm{in}}}\\{s_{2,\mathrm{in}}}\end{array}\right).$$
This interference results in the so-called Fano resonance, which can be identified by examining the time dependence of the transmission or experimentally measuring the reflection/transmission spectra [3234,40].

 figure: Fig. 1.

Fig. 1. The scattering bounds and chirality diagram of EPs in the defined resonant system. These scatter points are EPs numerically obtained in our designed metasurface. Inset schematically illustrates an asymmetric resonator coupled by two ports.

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Here, for simplicity we assume that the background system without the resonant mode is a homogeneous lossless dielectric material, so the direct coupling matrix $\mathbf {C}$ is simply in the form of

$$ \mathbf{C} = e^{i\varphi_{0}}\left(\begin{matrix} t_{0} & ir_{0} \\ ir_{0} & t_{0} \end{matrix}\right), $$
where $t_{0}^2+r_{0}^2=1$, $\varphi _{0}$ denotes the accumulated phase, and $r_{0}$ ($t_{0}$) is the reflection (transmission) amplitude. Besides $\gamma _{t}=(\gamma _{1}+\gamma _{2})/2$, there exists a constraint from
$$ \mathbf{C}\left(\begin{array}{l}{d_{1}^*}\\{d_{2}^*}\end{array}\right) +\left(\begin{array}{l}{d_{2}}\\{d_{1}}\end{array}\right)=0, $$
where $d_{1,2}=\sqrt {\gamma _{1,2}}e^{i\varphi _{1,2}}$. This constraint implies that the scattering coupling ratio $\gamma _{1}/\gamma _{2}$ cannot be arbitrary [3537], that
$$\frac{\gamma_{1,2}}{\gamma_{2,1}}=\frac{1+r_{0}^2+2r_{0}\sin(\delta_{2,1})}{t_{0}^2},$$
where $\delta _{1,2}=2\varphi _{1,2}-\varphi _{0}$ .

Thus, according to Eqs. (1)–(4), the scattering response of the metasurface could be formulated by a scattering matrix $\mathbf {S}$ in the form of

$$ \left(\begin{array}{l}{s_{1,\mathrm{out}}}\\{s_{2,\mathrm{out}}}\end{array}\right)=\mathbf{S} \left(\begin{array}{l}{s_{1,\mathrm{in}}}\\{s_{2,\mathrm{in}}}\end{array}\right) =\left(\begin{matrix} t & r_{R} \\ r_{L} & t \end{matrix}\right)\left(\begin{array}{l}{s_{1,\mathrm{in}}}\\{s_{2,\mathrm{in}}}\end{array}\right), $$
with two sets of possible solutions
$$ \begin{aligned}r_{L}=\frac{r_{0}^2\gamma_{d}-\gamma_{\sigma}+i(f-f_{0})r_{0}^2\pm i\sqrt{t_{0}^2(r_{0}^2\gamma_{t}^2-\gamma_{\sigma}^2)}}{r_{0}[(f-f_{0})-i(\gamma_{t}+\gamma_{d})]},\\ r_{R}=\frac{r_{0}^2\gamma_{d}+\gamma_{\sigma}+i(f-f_{0})r_{0}^2\pm i\sqrt{t_{0}^2(r_{0}^2\gamma_{t}^2-\gamma_{\sigma}^2)}}{r_{0}[(f-f_{0})-i(\gamma_{t}+\gamma_{d})]},\\ t=\frac{it_{0}(f-f_{0})\pm i\sqrt{r_{0}^2\gamma_{t}^2-\gamma_{\sigma}^2}+t_{0}\gamma_{d}}{i(f-f_{0})+\gamma_{t}+\gamma_{d}}, \end{aligned}$$
where $\gamma _{\sigma }=(\gamma _{1}-\gamma _{2})/2$.

Here, we explore the existence of EPs in the scattering matrix $\mathbf {S}$, which is associated with the coalescence of both eigenvectors and eigenvalues of this scattering matrix $\mathbf {S}$. The eigenvalues of $\mathbf {S}$ are $\lambda =t\pm \sqrt {r_{L}r_{R}}$ with normalized eigenvectors $(1/\sqrt {|r_{L}|+|r_{R}|})$ $(\pm \sqrt {|r_{R}|}e^{0.5i(arg[r_{R}]-arg[r_{L}])},\sqrt {|r_{L}|})^{\mathrm {T}}$. Thus, the positions of EPs can be arrived by mathematically finding zeros of $r_{L}$ or $r_{R}$. In other words, an EP indicates a perfect unidirectional reflection from the metasurface characterized by an asymmetric factor of reflection $\Lambda =(|r_{R}|^2-|r_{L}^2|)/(|r_{R}|^2+|r_{L}^2|)=\pm 1$. Furthermore, an EP usually has a definite chirality $\Pi$ associated with the degenerated eigenvectors in the form of $( 1, i)^{\mathrm {T}}$ or $(1, -i)^{\mathrm {T}}$, where the investigated Hamiltonian is usually a symmetric matrix [1,41,42]. In our investigation, the scattering matrix $\mathbf {S}$ is not symmetric. We also note that an EP could correspond to maximally asymmetric transmission for incident circular polarization in the metasurface systems [28,30]. The investigated transmission matrix possessing EPs is symmetric in the linear polarization base, while asymmetric in the circular polarization base. So we assume the eigenvector base of our scattering matrix $\mathbf {S}$ is the circular polarization base, which can be similarly defined by the two incident (output) ports. The eigenvector chirality can be defined by the Stokes parameter $S_{3}=(|r_{R}|-|r_{L}|)/(|r_{R}|+|r_{L}|)$, where $\Pi =S_{3}=1$ indicates a left-handed polarization with $(1, 0)^{\mathrm {T}}$, and $\Pi =S_{3}=-1$ indicates a right-handed polarization with $(0, 1)^{\mathrm {T}}$. In this definition, the eigenvector of EP is directly associated with the C point [43]. The eigenvector evolution around each EP can be formulated by a scalar function $S_{1}+iS_{2}$, which could illustrate the topological structure of the EP [4]. The maximum $\Lambda =1$ ($r_{L}=0$ and $r_{R}\neq 0$) coincides with an EP with a chirality of $\Pi =1$, where the eigenvector is $(1, 0)^{\mathrm {T}}$. The minimum $\Lambda =-1$ ($r_{R}=0$ and $r_{L}\neq 0$) coincides with an EP with a chirality of $\Pi =-1$, where the eigenvector is $(0, 1)^{\mathrm {T}}$. Thus, the resulting chirality has a definite physical meaning and clearly indicates the asymmetric degree of reflection. A mirror symmetry operation along the propagation direction will flip the chirality of EP.

From the zeros of $r_{L}$ or $r_{R}$, we can find that EPs appear at the frequencies of

$$ f=f_{0}\mp\frac{t_{0}}{r_{0}}\sqrt{\gamma_{t}^2-\gamma_{d}^2r_{0}^2}, $$
and the chirality of EPs is determined by
$$ r_{0}^2\gamma_{d}=\Pi\gamma_{\sigma}. $$
We can see that the chirality $\Pi$ of EP is only determined by the sign of $\gamma _{\sigma }$. According to Eq. (5), EPs could only occur on or within the boundary $(1\pm r_{0})/(1\mp r_{0})$ when $\delta _{2}=\pm \pi /2+2n\pi$, where $n$ is an integer, as illustrated in Fig. 1. While different chiralities of EP can be classified by the phase boundary $\gamma _{\sigma }=0$ according to Eq. (9) due to $r_{0}>0$ and $\gamma _{d}>0$ in a lossy system. We should note that there is no asymmetric reflection due to the constraint of Eq. (5) when $r_{0}=0$. If the resonant frequency of EP is $f=f_{0}$, Eqs. (8) and (9) require $\gamma _{1}=r_{0}(1+\Pi r_{0})\gamma _{d}$ and $\gamma _{2}=r_{0}(1-\Pi r_{0})\gamma _{d}$, which coincide with the bounds in the asymmetric scattering [3537]. According to Eq. (7), the non-zero reflection amplitude at the resonant frequency is $2r_{0}(1+r_{0})^{-1}$ when $f=f_{0}$. To increase the non-zero reflection at EP, we need to increase the background reflection rate $r_{0}$. Especially, when $r_{0}=1$ both the transmission and reflection from one side are zero, while the reflection from the other side is $1$. The appearance of EPs on these scattering bounds provides a potential approach to explore these bounds in actual design where the dissipation loss can not be ignored. When deviating from the resonant frequency $f_{0}$, the position of EP could be predicted by Eq. (8), which still falls within these scattering bounds. More importantly, $\gamma _{\sigma }=0$ is the boundary separating EPs with opposite chiralities.

3. Metasurface and numerical verification

We propose a metasurface design to verify the above results, as illustrated in Fig. 2. To generate significant reflection difference, we adopt a metasurface possessing bianisotropic response, which has been used to control the asymmetric reflection or transmission in various synthetic metamaterials [38,39]. Units of the metasurface are arranged in a square lattice with a period $d = 10$ mm and the entire metasurface is free-standing in air (so that there is no high-order diffraction below $30$ GHz at normal incidence). The designed metasurface consists of three layers. The top and bottom layers are complementary metallic structures with geometric parameters $a=8$ mm, $b=5$ mm, $w=0.8$ mm, and thickness $t_{m}=0.5$ mm. The top layer is rotated by an angle of $45^{\mathrm {o}}$ with respect to the bottom layer. The metal is copper with an electric conductivity of $5.7\times 10^7$ $\mathrm {S}/\mathrm {m}$. The middle layer has a thickness of $h_{d}=0.8$ mm and the dielectric constant is $\varepsilon _{r}=4.3$. The total metasurface possesses a mirror symmetry along both the transversal $x$ and $y$ directions. Due to the existing mirror symmetries, the metasurface shows isotropic response for normal incidence, so we need only consider the scattering properties in a two ports system for a selected linear polarization state. The mirror symmetry is breaking along the $z$ direction, which causes asymmetric reflection . To explore the appearance of EPs in the scattering matrix $\mathbf {S}$, we select the dissipation factor $\delta _{d}=\varepsilon _{i}/\varepsilon _{r}$ and $f$ as the tuning parameters, where $\varepsilon _{i}$ is the imaginary part of the dielectric constant of the substrate. Using the finite-difference time-domain (FDTD) simulations, we numerically obtain the reflection and transmission spectra of the designed metasurface under normal incidence with a linear polarized light ($\mathbf {E}_{i}=E\mathbf {\hat {y}}$) in the frequency range from $9$ GHz to $17$ GHz.

 figure: Fig. 2.

Fig. 2. Schematic view of the scattering matrix we adopt and the unit cell design of the metasurface consisting of three layers.

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Figure 3 shows the numerically obtained asymmetric factor on reflection $\Lambda$. There are one pair of resonant peak and dip at different frequencies for each dissipation factor $\delta _{d}$. The distance between the dip and peak is around $3.4$ GHz. The resonant peak ($\Lambda >0$) represents a stronger reflection on the right side, while the resonant dip ($\Lambda <0$) implies a stronger reflection on the left side. At certain values of $f$ and $\delta _{d}$, the maximum of $\Lambda$ is at ($f=11.17$ GHz and $\delta _{d}=0.048$) with $\Lambda _{\mathrm {max}}=1$. The minimum of $\Lambda$ is at ($f=14.52$ GHz and $\delta _{d}=0.138$) with $\Lambda _{\mathrm {min}}=-1$. The presence of maximal and minimal asymmetric factors clearly proves that the designed metasurface possesses two EPs with opposite chiralities in the explored parameters space. The opposite chiralities of these two EPs are not linked by mirror symmetry operation, but only stem from the scattering coupling rate difference. To further illustrate the chirality of EP, the numerically obtained Stokes parameter $S_{3}$ is displayed in Fig. 3(b). The maximum (minimum) of $\Lambda$ coincides with the maximum (minimum) of $S_{3}$. The phase structures of eigenvector through $S_{1}+iS_{2}$, as illustrated in Figs. 3(c) and 3(d), clearly indicate that two EPs have opposite winding number. And the eigenvector will switch to the other eigenvector when the eigenvector continuously goes around the EP in a loop on the parameters plane.

 figure: Fig. 3.

Fig. 3. Numerical calculated asymmetric factor of reflection $\Lambda$ (a), the chirality $S_{3}$ (b), and phase structure of $S_{1}+iS_{2}$ around the 1st (c) and 2nd (d) EPs in the proposed metasurface design versus the frequency $f$ and the dissipation factor $\delta _{d}$.

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Besides the mode switch, crossing and anti-crossing behaviors for the eigenvalues also occur in our system, which are similar to the level repulsion and crossing for the Hamiltonian [3,42]. According to the eigenvalue of the scattering matrix $\mathbf {S}$, we make use of the eigenvalue part $\pm \sqrt {r_{L}r_{R}}$ without the transmission reference $t$ to indicate this crossing and anti-crossing behaviors around each EP, as illustrated in Fig. 4. Around the 1st EP, the eigenvalues show anti-crossing behavior for the real component [see Fig. 4(a)] and crossing behavior for the imaginary component [see Fig. 4(c)] when the dissipation factor $\delta _{d}=0.046$ is below the critical value of the 1st EP. While the eigenvalues show opposite behavior when the dissipation factor $\delta _{d}=0.050$ is above the critical value of the 1st EP. Around the 2nd EP, the real parts of the eigenvalue show crossing behavior, as shown in Fig. 4(b), while the imaginary parts of the eigenvalue show anti-crossing behavior when the dissipation factor $\delta _{d}=0.136$ is below the critical value of the 2nd EP, as illustrated in Fig. 4(d). Once the dissipation factor $\delta _{d}=0.140$ is above the critical value, we can find the opposite behavior with respect to the behavior below the EP. These crossing and anti-crossing behaviors around each EP consolidate the existence of EPs in our designed metasurface.

 figure: Fig. 4.

Fig. 4. Eigenvalue part $\pm \sqrt {r_{L}r_{R}}$ of $\mathbf {S}$ without transmission reference $t$ around each resonance below and above the critical dissipation loss factor.

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As shown in Fig. 3, the overlapping region of the two resonant modes (peak and dip) is small, we can separately fit their spectra according to CMT. Because the selected parameter is the dissipation factor $\gamma _{d}$ besides $f$, the microscopic parameters of $\gamma _{1,2}$ and $f_{0}$ are assumed to be constant around each resonance in the CMT fitting. For the fist resonant mode, the CMT fitting parameters are $\gamma _{1}=0.3004$ GHz, $\gamma _{2}=0.0045$ GHz, $f_{0}=11.17$ GHz. The background reflection $r_{0}$ could be fitted by the reflection of a homogeneous slab with an effective refractive index $n_{\textrm {eff}}=11.3399$ and an effective thickness $h_{\textrm {eff}}=0.8560$ mm. Once the fitted dissipation loss rate $\gamma _{d}$ is obtained for each $\delta _{d}$ according to CMT, the obtained dissipation loss rate $\gamma _{d}$ is fitted again in the form of $\gamma _{d}=0.0143+2.9720\delta _{d}$ in the unit of GHz. As shown in Fig. 5, the numerical asymmetric factor $\Lambda$ and CMT results find good agreement in the selected fitted frequency range from $10.5$ GHz to $11.5$ GHz. We also plot the asymmetric factor $\Lambda$ and the eigenvalue $|\sqrt {r_{L}r_{R}}|$ at the desired $\delta _{d}=0.048$ to indicate the degeneracy at EP in Figs. 5(c) and (d). At EP, these fitted parameters satisfy $r_{0}^2\gamma _{t}^2-\gamma _{\sigma }^2=0$ and $\gamma _{d}r_{0}^2=\gamma _{\sigma }$, which imply that EP in our design metasurface occurs at the asymmetric bound. Besides the zero amplitude of $r_{L}$, the non-zero amplitude of $|r_{R}|=2r_{0}(1+r_{0})^{-1}=0.9868$ is almost the same as the numerical prediction of $0.9751$. At the first EP position, the electric filed amplitude profile ($y=0$) at $11.17$ GHz for an incident light from the left and right sides are plotted in Figs. 5(e) and (f), respectively. We could find that the transmission is very weak (only around $0.0315$). When the incident light is from the left side as shown in Fig. 5(e), the reflection is very small because of the insignificant interference with the incident light, and the electric field in the metasurface region is very large due to the enhanced absorption. Physically, the zero reflection and enhanced absorption could be understood by the perfectly destructive interference from the background reflection and the re-radiation from the resonant mode in CMT. While the electric field in the metasurface region is small, the reflection is large due to obvious interference with the incident light when the incident light from the right side, as illustrated in the Fig. 5(f). Physically, the obvious reflection originates from near zero re-radiation of the resonant metasurface in this side.

 figure: Fig. 5.

Fig. 5. Variation of the asymmetric factor of reflection $\Lambda$ around the first EP from numerical calculation (a) and CMT (b). Evolution of $\Lambda$ (c) and $|\sqrt {r_{L}r_{R}}|$ (d) as the frequency $f$ is swept through the first EP at $f =11.17$ GHz, with $\delta _{d}=0.048$. Circles indicate numerical (FDTD) results and solid curves illustrate the theoretical predictions from the fitted CMT. The lower row shows the numerical electric field profile at $y=0$ when the light is incident from (e) the left and (f) the right side.

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For the second resonance around $f=14.5$ GHz, the CMT fitting parameters are $\gamma _{1}=0.1658$ GHz, $\gamma _{2}=1.5828$ GHz, $f_{0}=14.48$ GHz. The background refractive index is $n_{\textrm {eff}}=63.2614$ and the efficitive thickness is $h_{\textrm {eff}}=0.1419$ mm. The fitted dissipation loss rate $\gamma _{d}$ is in the form of $\gamma _{d}=0.0070+ 5.1142\delta _{d}$ in unit of GHz. As illustrated in Fig. 6, numerical and CMT $\Lambda$ find good agreements in the fitted frequency range from $14$ GHz to $15$ GHz. At the EP position with $\delta _{d}=0.138$, the minimum of $\Lambda$ coincides with the zero of $|\sqrt {r_{L}r_{R}}|$ at $f_{EP2}=14.52$ GHz, as shown in Figs. 6(c) and (d). The frequency position of EP $f_{\mathrm {EP}2}=14.52$ GHz at $\delta _{d}=0.138$ is the same as the predicted frequency from the fitted parameters according to Eq. (8), and these fitted parameters also satisfy $\gamma _{d}r_{0}^2=-\gamma _{\sigma }$ at EP. The scattering coupling ratio $\gamma _{1}/\gamma _{2}=0.1047$ for the second EP in our designed metasurface is slightly larger than the lower bound of $(1-r_{0})(1+r_{0})^{-1}=0.0016$. At the second EP position, the electric filed amplitude profile ($y=0$) at $14.52$ GHz for incident light from the left and right sides are plotted in Figs. 6(e) and (f). We could find that the transmission is on the finite level of $0.2288$. When the incident light is incident from the left side, as shown in Fig. 6(e), the reflection is large due to obvious interference with the incident light, and the electric field in the metasurface region is small. When the incident light comes from the right side, albeit the electric field in the metasurface region is large with increased absorption, the reflection is small because of an insignificant interference with the incident light, as illustrated in Fig. 6(f).

 figure: Fig. 6.

Fig. 6. Numerical (a) and CMT (b) variation of asymmetric factor of reflection $\Lambda$ around the second EP. Evolution of $\Lambda$ (c) and $|\sqrt {r_{L}r_{R}}|$ (d) as the frequency $f$ is swept through the second EP at $f =14.52$ GHz with $\delta _{d} = 0.138$. Circles show numerical (FDTD) results and solid curves show theoretical predictions from the fitted CMT. Numerical electric field profile at $y=0$ plane when the incident light from left (e) or right (f) side.

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4. Conclusion

We have theoretically shown that EPs could occur in a resonant asymmetric metasurface, and which factor determines the inherent chirality of the realized EP in the scattering approach. As a verified example, numerical results are in good agreement with the fitted CMT results in our designed metasurface. Our theoretical results could be useful for exploring the physics of the EP in other resonant systems not limited by the resonant metasurface system.

Funding

National Natural Science Foundation of China (11674244, 11874228, 11974259).

Disclosures

The authors declare no conflicts of interest.

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22. C. Zhang, R. Bai, X. Gu, X. R. Jin, Y. Q. Zhang, and Y. P. Lee, “Dual-band unidirectional reflectionless phenomena in an ultracompact non-Hermitian plasmonic waveguide system based on near-field coupling,” Opt. Express 25(20), 24281–24289 (2017). [CrossRef]  

23. K. Ding, G. Ma, Z. Q. Zhang, and C. T. Chan, “Experimental demonstration of an anisotropic exceptional point,” Phys. Rev. Lett. 121(8), 085702 (2018). [CrossRef]  

24. W. R. Sweeney, C. W. Hsu, S. Rotter, and A. D. Stone, “Perfectly absorbing exceptional points and chiral absorbers,” Phys. Rev. Lett. 122(9), 093901 (2019). [CrossRef]  

25. M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019). [CrossRef]  

26. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014). [CrossRef]  

27. M. Kang, H. X. Cui, T. F. Li, J. Chen, W. Zhu, and M. Premaratne, “Unidirectional phase singularity in ultrathin metamaterials at exceptional points,” Phys. Rev. A 89(6), 065801 (2014). [CrossRef]  

28. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with Terahertz metasurfaces,” Phys. Rev. Lett. 113(9), 093901 (2014). [CrossRef]  

29. D. L. Sounas, R. Fleury, and A. Alù, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. Appl. 4(1), 014005 (2015). [CrossRef]  

30. M. Kang, J. Chen, and Y. D. Chong, “Chiral exceptional points in metasurfaces,” Phys. Rev. A 94(3), 033834 (2016). [CrossRef]  

31. Y. Huang, Y. Shen, C. Min, S. Fan, and G. Veronis, “Unidirectional reflectionless light propagation at exceptional points,” Nanophotonics 6(5), 977–996 (2017). [CrossRef]  

32. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).

33. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004). [CrossRef]  

34. R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007). [CrossRef]  

35. K. X. Wang, Z. Yu, S. Sandhu, and S. Fan, “Fundamental bounds on decay rates in asymmetric single-mode optical resonators,” Opt. Lett. 38(2), 100–102 (2013). [CrossRef]  

36. H. Zhou, B. Zhen, C. W. Hsu, O. D. Miller, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Perfect single-sided radiation and absorption without mirrors,” Optica 3(10), 1079–1086 (2016). [CrossRef]  

37. D. L. Sounas and A. Alù, “Time-reversal symmetry bounds on the electromagnetic response of asymmetric structures,” Phys. Rev. Lett. 118(15), 154302 (2017). [CrossRef]  

38. R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B 91(11), 115119 (2015). [CrossRef]  

39. M. Odit, P. Kapitanova, P. Belov, R. Alaee, C. Rockstuhl, and Y. S. Kivshar, “Experimental realisation of all-dielectric bianisotropic metasurfaces,” Appl. Phys. Lett. 108(22), 221903 (2016). [CrossRef]  

40. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002). [CrossRef]  

41. S. Bhattacherjee, H. K. Gandhi, A. Laha, and S. Ghosh, “Higher-order topological degeneracies and progress towards unique successive state switching in a four-level open system,” Phys. Rev. A 100(6), 062124 (2019). [CrossRef]  

42. H. K. Gandhi, A. Laha, S. Dey, and S. Ghosh, “Chirality breakdown in the presence of multiple exceptional points and specific mode excitation,” Opt. Lett. 45(6), 1439–1442 (2020). [CrossRef]  

43. M. R. Dennis, “Polarization singularities in paraxial vector fields:morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002). [CrossRef]  

References

  • View by:

  1. W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17(2), 149–151 (2001).
    [Crossref]
  2. M. V. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54(10), 1039–1047 (2004).
    [Crossref]
  3. W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. A: Math. Gen. 37(6), 2455–2464 (2004).
    [Crossref]
  4. C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86(5), 787–790 (2001).
    [Crossref]
  5. T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Höfling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526(7574), 554–558 (2015).
    [Crossref]
  6. J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016).
    [Crossref]
  7. H. Xu, D. Mason, L. Jiang, and J. Harris, “Topological energy transfer in an optomechanical system with exceptional point,” Nature 537(7618), 80–83 (2016).
    [Crossref]
  8. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
    [Crossref]
  9. Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
    [Crossref]
  10. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
    [Crossref]
  11. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
    [Crossref]
  12. L. Feng, Y. Xu, W. S. Fegadolli, M. Lu, J. E. B. Oliveira, V. R. Almedia, Y. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
    [Crossref]
  13. Z. J. Wong, Y. L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10(12), 796–801 (2016).
    [Crossref]
  14. W. Chen, S. K. özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
    [Crossref]
  15. 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]
  16. W. Langbein, “No exceptional precision of exceptional-point sensors,” Phys. Rev. A 98(2), 023805 (2018).
    [Crossref]
  17. M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87(5), 053824 (2013).
    [Crossref]
  18. B. Peng, S. K. Ö zdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
    [Crossref]
  19. B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
    [Crossref]
  20. Y. Huang, G. Veronis, and C. Min, “Unidirectional reflectionless propagation in plasmonic waveguide-cavity systems at exceptional points,” Opt. Express 23(23), 29882–29895 (2015).
    [Crossref]
  21. K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6(2), 021007 (2016).
    [Crossref]
  22. C. Zhang, R. Bai, X. Gu, X. R. Jin, Y. Q. Zhang, and Y. P. Lee, “Dual-band unidirectional reflectionless phenomena in an ultracompact non-Hermitian plasmonic waveguide system based on near-field coupling,” Opt. Express 25(20), 24281–24289 (2017).
    [Crossref]
  23. K. Ding, G. Ma, Z. Q. Zhang, and C. T. Chan, “Experimental demonstration of an anisotropic exceptional point,” Phys. Rev. Lett. 121(8), 085702 (2018).
    [Crossref]
  24. W. R. Sweeney, C. W. Hsu, S. Rotter, and A. D. Stone, “Perfectly absorbing exceptional points and chiral absorbers,” Phys. Rev. Lett. 122(9), 093901 (2019).
    [Crossref]
  25. M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
    [Crossref]
  26. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
    [Crossref]
  27. M. Kang, H. X. Cui, T. F. Li, J. Chen, W. Zhu, and M. Premaratne, “Unidirectional phase singularity in ultrathin metamaterials at exceptional points,” Phys. Rev. A 89(6), 065801 (2014).
    [Crossref]
  28. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with Terahertz metasurfaces,” Phys. Rev. Lett. 113(9), 093901 (2014).
    [Crossref]
  29. D. L. Sounas, R. Fleury, and A. Alù, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. Appl. 4(1), 014005 (2015).
    [Crossref]
  30. M. Kang, J. Chen, and Y. D. Chong, “Chiral exceptional points in metasurfaces,” Phys. Rev. A 94(3), 033834 (2016).
    [Crossref]
  31. Y. Huang, Y. Shen, C. Min, S. Fan, and G. Veronis, “Unidirectional reflectionless light propagation at exceptional points,” Nanophotonics 6(5), 977–996 (2017).
    [Crossref]
  32. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).
  33. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
    [Crossref]
  34. R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
    [Crossref]
  35. K. X. Wang, Z. Yu, S. Sandhu, and S. Fan, “Fundamental bounds on decay rates in asymmetric single-mode optical resonators,” Opt. Lett. 38(2), 100–102 (2013).
    [Crossref]
  36. H. Zhou, B. Zhen, C. W. Hsu, O. D. Miller, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Perfect single-sided radiation and absorption without mirrors,” Optica 3(10), 1079–1086 (2016).
    [Crossref]
  37. D. L. Sounas and A. Alù, “Time-reversal symmetry bounds on the electromagnetic response of asymmetric structures,” Phys. Rev. Lett. 118(15), 154302 (2017).
    [Crossref]
  38. R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B 91(11), 115119 (2015).
    [Crossref]
  39. M. Odit, P. Kapitanova, P. Belov, R. Alaee, C. Rockstuhl, and Y. S. Kivshar, “Experimental realisation of all-dielectric bianisotropic metasurfaces,” Appl. Phys. Lett. 108(22), 221903 (2016).
    [Crossref]
  40. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
    [Crossref]
  41. S. Bhattacherjee, H. K. Gandhi, A. Laha, and S. Ghosh, “Higher-order topological degeneracies and progress towards unique successive state switching in a four-level open system,” Phys. Rev. A 100(6), 062124 (2019).
    [Crossref]
  42. H. K. Gandhi, A. Laha, S. Dey, and S. Ghosh, “Chirality breakdown in the presence of multiple exceptional points and specific mode excitation,” Opt. Lett. 45(6), 1439–1442 (2020).
    [Crossref]
  43. M. R. Dennis, “Polarization singularities in paraxial vector fields:morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002).
    [Crossref]

2020 (1)

2019 (3)

S. Bhattacherjee, H. K. Gandhi, A. Laha, and S. Ghosh, “Higher-order topological degeneracies and progress towards unique successive state switching in a four-level open system,” Phys. Rev. A 100(6), 062124 (2019).
[Crossref]

W. R. Sweeney, C. W. Hsu, S. Rotter, and A. D. Stone, “Perfectly absorbing exceptional points and chiral absorbers,” Phys. Rev. Lett. 122(9), 093901 (2019).
[Crossref]

M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
[Crossref]

2018 (2)

K. Ding, G. Ma, Z. Q. Zhang, and C. T. Chan, “Experimental demonstration of an anisotropic exceptional point,” Phys. Rev. Lett. 121(8), 085702 (2018).
[Crossref]

W. Langbein, “No exceptional precision of exceptional-point sensors,” Phys. Rev. A 98(2), 023805 (2018).
[Crossref]

2017 (5)

W. Chen, S. K. özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
[Crossref]

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]

Y. Huang, Y. Shen, C. Min, S. Fan, and G. Veronis, “Unidirectional reflectionless light propagation at exceptional points,” Nanophotonics 6(5), 977–996 (2017).
[Crossref]

C. Zhang, R. Bai, X. Gu, X. R. Jin, Y. Q. Zhang, and Y. P. Lee, “Dual-band unidirectional reflectionless phenomena in an ultracompact non-Hermitian plasmonic waveguide system based on near-field coupling,” Opt. Express 25(20), 24281–24289 (2017).
[Crossref]

D. L. Sounas and A. Alù, “Time-reversal symmetry bounds on the electromagnetic response of asymmetric structures,” Phys. Rev. Lett. 118(15), 154302 (2017).
[Crossref]

2016 (7)

M. Odit, P. Kapitanova, P. Belov, R. Alaee, C. Rockstuhl, and Y. S. Kivshar, “Experimental realisation of all-dielectric bianisotropic metasurfaces,” Appl. Phys. Lett. 108(22), 221903 (2016).
[Crossref]

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6(2), 021007 (2016).
[Crossref]

M. Kang, J. Chen, and Y. D. Chong, “Chiral exceptional points in metasurfaces,” Phys. Rev. A 94(3), 033834 (2016).
[Crossref]

H. Zhou, B. Zhen, C. W. Hsu, O. D. Miller, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Perfect single-sided radiation and absorption without mirrors,” Optica 3(10), 1079–1086 (2016).
[Crossref]

Z. J. Wong, Y. L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10(12), 796–801 (2016).
[Crossref]

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016).
[Crossref]

H. Xu, D. Mason, L. Jiang, and J. Harris, “Topological energy transfer in an optomechanical system with exceptional point,” Nature 537(7618), 80–83 (2016).
[Crossref]

2015 (5)

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Höfling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526(7574), 554–558 (2015).
[Crossref]

D. L. Sounas, R. Fleury, and A. Alù, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. Appl. 4(1), 014005 (2015).
[Crossref]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015).
[Crossref]

Y. Huang, G. Veronis, and C. Min, “Unidirectional reflectionless propagation in plasmonic waveguide-cavity systems at exceptional points,” Opt. Express 23(23), 29882–29895 (2015).
[Crossref]

R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B 91(11), 115119 (2015).
[Crossref]

2014 (4)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref]

M. Kang, H. X. Cui, T. F. Li, J. Chen, W. Zhu, and M. Premaratne, “Unidirectional phase singularity in ultrathin metamaterials at exceptional points,” Phys. Rev. A 89(6), 065801 (2014).
[Crossref]

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with Terahertz metasurfaces,” Phys. Rev. Lett. 113(9), 093901 (2014).
[Crossref]

B. Peng, S. K. Ö zdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

2013 (3)

M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87(5), 053824 (2013).
[Crossref]

L. Feng, Y. Xu, W. S. Fegadolli, M. Lu, J. E. B. Oliveira, V. R. Almedia, Y. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

K. X. Wang, Z. Yu, S. Sandhu, and S. Fan, “Fundamental bounds on decay rates in asymmetric single-mode optical resonators,” Opt. Lett. 38(2), 100–102 (2013).
[Crossref]

2011 (2)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref]

2010 (1)

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

2009 (1)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

2007 (1)

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
[Crossref]

2004 (3)

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
[Crossref]

M. V. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54(10), 1039–1047 (2004).
[Crossref]

W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. A: Math. Gen. 37(6), 2455–2464 (2004).
[Crossref]

2002 (2)

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

M. R. Dennis, “Polarization singularities in paraxial vector fields:morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002).
[Crossref]

2001 (2)

C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86(5), 787–790 (2001).
[Crossref]

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17(2), 149–151 (2001).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

Alaee, R.

M. Odit, P. Kapitanova, P. Belov, R. Alaee, C. Rockstuhl, and Y. S. Kivshar, “Experimental realisation of all-dielectric bianisotropic metasurfaces,” Appl. Phys. Lett. 108(22), 221903 (2016).
[Crossref]

R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B 91(11), 115119 (2015).
[Crossref]

Albooyeh, M.

R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B 91(11), 115119 (2015).
[Crossref]

Almedia, V. R.

L. Feng, Y. Xu, W. S. Fegadolli, M. Lu, J. E. B. Oliveira, V. R. Almedia, Y. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Alù, A.

M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
[Crossref]

D. L. Sounas and A. Alù, “Time-reversal symmetry bounds on the electromagnetic response of asymmetric structures,” Phys. Rev. Lett. 118(15), 154302 (2017).
[Crossref]

D. L. Sounas, R. Fleury, and A. Alù, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. Appl. 4(1), 014005 (2015).
[Crossref]

Bai, R.

Belov, P.

M. Odit, P. Kapitanova, P. Belov, R. Alaee, C. Rockstuhl, and Y. S. Kivshar, “Experimental realisation of all-dielectric bianisotropic metasurfaces,” Appl. Phys. Lett. 108(22), 221903 (2016).
[Crossref]

Bender, C. M.

B. Peng, S. K. Ö zdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Berry, M. V.

M. V. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54(10), 1039–1047 (2004).
[Crossref]

Bhattacherjee, S.

S. Bhattacherjee, H. K. Gandhi, A. Laha, and S. Ghosh, “Higher-order topological degeneracies and progress towards unique successive state switching in a four-level open system,” Phys. Rev. A 100(6), 062124 (2019).
[Crossref]

Bliokh, K. Y.

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M. Odit, P. Kapitanova, P. Belov, R. Alaee, C. Rockstuhl, and Y. S. Kivshar, “Experimental realisation of all-dielectric bianisotropic metasurfaces,” Appl. Phys. Lett. 108(22), 221903 (2016).
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Nat. Mater. (2)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
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Nat. Photonics (1)

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Nat. Phys. (1)

B. Peng, S. K. Ö zdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
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Nature (6)

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W. Chen, S. K. özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
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Figures (6)

Fig. 1.
Fig. 1. The scattering bounds and chirality diagram of EPs in the defined resonant system. These scatter points are EPs numerically obtained in our designed metasurface. Inset schematically illustrates an asymmetric resonator coupled by two ports.
Fig. 2.
Fig. 2. Schematic view of the scattering matrix we adopt and the unit cell design of the metasurface consisting of three layers.
Fig. 3.
Fig. 3. Numerical calculated asymmetric factor of reflection $\Lambda$ (a), the chirality $S_{3}$ (b), and phase structure of $S_{1}+iS_{2}$ around the 1st (c) and 2nd (d) EPs in the proposed metasurface design versus the frequency $f$ and the dissipation factor $\delta _{d}$.
Fig. 4.
Fig. 4. Eigenvalue part $\pm \sqrt {r_{L}r_{R}}$ of $\mathbf {S}$ without transmission reference $t$ around each resonance below and above the critical dissipation loss factor.
Fig. 5.
Fig. 5. Variation of the asymmetric factor of reflection $\Lambda$ around the first EP from numerical calculation (a) and CMT (b). Evolution of $\Lambda$ (c) and $|\sqrt {r_{L}r_{R}}|$ (d) as the frequency $f$ is swept through the first EP at $f =11.17$ GHz, with $\delta _{d}=0.048$. Circles indicate numerical (FDTD) results and solid curves illustrate the theoretical predictions from the fitted CMT. The lower row shows the numerical electric field profile at $y=0$ when the light is incident from (e) the left and (f) the right side.
Fig. 6.
Fig. 6. Numerical (a) and CMT (b) variation of asymmetric factor of reflection $\Lambda$ around the second EP. Evolution of $\Lambda$ (c) and $|\sqrt {r_{L}r_{R}}|$ (d) as the frequency $f$ is swept through the second EP at $f =14.52$ GHz with $\delta _{d} = 0.138$. Circles show numerical (FDTD) results and solid curves show theoretical predictions from the fitted CMT. Numerical electric field profile at $y=0$ plane when the incident light from left (e) or right (f) side.

Equations (9)

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d u d t = ( i f 0 γ t γ d ) u + ( d 1 , d 2 ) ( s 1 , i n s 2 , i n ) ,
( s 1 , o u t s 2 , o u t ) = ( d 2 d 1 ) u + C ( s 1 , i n s 2 , i n ) .
C = e i φ 0 ( t 0 i r 0 i r 0 t 0 ) ,
C ( d 1 d 2 ) + ( d 2 d 1 ) = 0 ,
γ 1 , 2 γ 2 , 1 = 1 + r 0 2 + 2 r 0 sin ( δ 2 , 1 ) t 0 2 ,
( s 1 , o u t s 2 , o u t ) = S ( s 1 , i n s 2 , i n ) = ( t r R r L t ) ( s 1 , i n s 2 , i n ) ,
r L = r 0 2 γ d γ σ + i ( f f 0 ) r 0 2 ± i t 0 2 ( r 0 2 γ t 2 γ σ 2 ) r 0 [ ( f f 0 ) i ( γ t + γ d ) ] , r R = r 0 2 γ d + γ σ + i ( f f 0 ) r 0 2 ± i t 0 2 ( r 0 2 γ t 2 γ σ 2 ) r 0 [ ( f f 0 ) i ( γ t + γ d ) ] , t = i t 0 ( f f 0 ) ± i r 0 2 γ t 2 γ σ 2 + t 0 γ d i ( f f 0 ) + γ t + γ d ,
f = f 0 t 0 r 0 γ t 2 γ d 2 r 0 2 ,
r 0 2 γ d = Π γ σ .

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