Abstract

We propose a scheme to achieve a controllable unidirectional reflectionless propagation at exceptional point (EP) in a non-ideal parity-time metasurface consisting of two silver ring resonators. The unidirectional reflectionless propagation can be manipulated by simply adjusting the angle of incident wave and the distance s between two silver rings based on the far field coupling. In addition, the angle of incident wave in a wide range of ∼25° is available to achieve the unidirectional reflectionless propagation. Moreover, the unidirectional reflectionless propagation at EP is insensitive to the polarization of incident wave due to the two-ring structure.

© 2017 Optical Society of America

1. Introduction

It is well known that Hermitian Hamiltonians have real spectrums in quantum mechanics. However, Bender et al. [1] proposed non-Hermitian Hamiltonians that can still exhibit entirely real spectrums if they have parity-time (PT) symmetry. Thereafter, PT symmetry has been extensively studied in various fields, such as quantum field theories [2], open quantum systems [3], non-Hermitian Anderson models [4], Lie algebras [5], electronics [6], acoustics [7–11] and so on. Particularly, researches on PT symmetry have been perfectly extended to optics domain from quantum mechanics in the last decade, such as optical lattices [12–16], photonic structure [17–20], plasmonics [21–23], waveguide couplers [24–31], Bragg gratings [32–34], microcavities [35–42], and so on.

In fact, the highlight of PT symmetric system is the presence of EP where a series of fascinating phenomena, such as optical isolation [12], power oscillation [25, 26], loss-induced transparency [27], nonreciprocal light propagation [26, 32, 33, 35], coherent perfect absorption [43], nonlinear effects [44, 45], laser-absorber [46–53], unidirectional reflectionlessness [30, 34, 38, 54] occur. Especially, the unidirectional reflectionless light propagation near the PT symmetry breaking point has been researched extensively. For example, in 2012, Feng et al. [34] experimentally demonstrated an unidirectional reflectionless optical metamaterial near the spontaneous PT symmetry phase transition point where reflection from one side was significantly suppressed. Their results provided the feasibility for creating on-chip PT metamaterials and optical devices based on PT properties. Huang et al. [30, 38] constructed a non-PT-symmetric plasmonic waveguide-cavity system to form an EP and realize the unidirectional reflectionless propagation at optical communication wavelength. Their structure would open new possibilities to potentially applications in highly compact unidirectional integrated nanoplasmonic devices. In 2016, Yang et al. [54] reported the unidirectional reflectionless phenomenon in periodic ternary layered material with the dimension of 30µm based on balanced gain and loss. Their work has great potential applications in on-chip integrated optics. However, so far the work on the PT properties in a periodic metasurface with the sub-wavelength nanoscale has rarely been mentioned.

In this work, a non-ideal PT sub-wavelength nanoscale metasurface system without gain consists of two silver ring resonators embedded in photopolymer is firstly designed to research the unidirectional reflectionless propagation properties at EP based on far field coupling. The results based on numerical simulations and analytical calculations consistently indicate that the unidirectional reflectionless propagation can be manipulated by adjusting the angle of incident wave and distance s between two silver rings. Furthermore, the structure is polarization independent due to the two-ring structure. Besides, the presence of non-ideal PT metasurface structure near EP can act as an absorber with absorption of ∼96% and quality factor (Q-factor) of ∼41.

2. Structure

Figure 1 shows the schematic of unit cell of the unidirectional reflectionless non-ideal PT metasurface structure that place on a glass substrate. Two silver ring resonators with the thickness h are embedded in photopolymer. The periods t of unit cell in both x and y directions are 640nm. The inner radius r1 (r2) and outer radius R1 (R2) of up (down) ring are 61nm (63nm) and 122nm (170nm), respectively. The distance s between two rings is variables. The incident wave is in xz plane and has an angle θ with +z axis. The permittivities of photopolymer and glass substrate are 2.4025 and 2.25, respectively. Moreover, the permittivity of silver in near-infrared can be described by Drude model with plasmon frequency ωpl = 1.366 1016rad/s and collision frequency ωc = 3.07 × 1013Hz [57]. The numerical simulations are carried out by using a finite-integration package (CST Microwave Studio).

 figure: Fig. 1

Fig. 1 Schematic of unit cell of the non-ideal PT metasurface structure. The parameters are h = 20nm, r1 = 61nm, R1 = 122nm, r2 = 63nm, R2 = 170nm and t = 640nm, respectively. The distance s is variable. The incident wave is in xz plane and has an angle θ with +z axis.

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3. Results and discussions

To verify the unidirectional reflectionless phenomena, we make use of the scattering matrix to analyze the reflection spectra. The scattering properties corresponding to the presence of non-ideal PT metasurface system in Fig. 1 can be given by the transfer matrix Tall [57, 58] in a certain frequency ω

Tall=Ts1×Tp×Ts2=(T11T12T21T22),
where
Ts1(2)=(1iγ1(2)ωω1(2)+iΓ1(2)2iγ1(2)ωω1(2)+iΓ1(2)2iγ1(2)ωω1(2)+iΓ1(2)21iγ1(2)ωω1(2)+iΓ1(2)2),
and
Tp=(exp(iϕ)00exp(iϕ)),
here Ts1(2) and Tp are the transfer matrices for up (down) silver ring and the phase shift of wave propagation from up silver ring to down silver ring. Γ1(2) and ω1(2) are the dissipative losses and resonant frequency of up (down) silver ring, respectively. γ1(2) is the width of resonance for up (down) ring coupled with the incident wave. ϕ is accumulated phase shift of wave propagation between the two rings and it depends on distance s. ω is the frequency of incident wave. Moreover, the phase shift ϕ1(2) for up (down) ring can be expressed as [57, 58]
ϕ1(2)=arctan[Im(Ts,211(2)/Ts,221(2))Re(Ts,211(2)/Ts,221(2))]=(ωω1(2))(γ1(2)+Γ1(2)2),
based on Eq. (2).

In the present structure, the phase difference ϕall between two silver rings is composed of three parts: the phase shift of up silver ring, the phase shift of down silver ring and the phase shift between two silver rings. In other words, ϕall is equal to ϕ1(2)ϕ2(1) + 2ϕ in excitation along +z (−z)

Additionally, the transmission and reflection coefficients can be defined by Eq. (1), as

t=t+z=tz=1T22,r+z=T21T22,rz=T12T22.
Then the optical properties of our non-ideal metasurface system can be simply obtained by scattering matrix S
S=(trzr+zt).

Figure 2 shows the simulated (analytical) reflections in excitation along +z and −z of the incident wave by optimizing the distance s (phase ϕ). The black solid and red solid lines show the simulated reflection spectra in excitation along +z and −z corresponding to distance s = 485nm, 455nm, 444nm, 430nm and 395nm, respectively, in Figs. 2(a)–2(e). From Figs. 2(a)–2(e), the reflection peak values in excitation along +z increase gradually with the decreasing distance s from 485nm to 395nm, and the reflection peaks have the blue-shift. On the contrary, the reflection peak values in excitation along −z decrease gradually with the decreasing distance s from 485nm to 395nm. The reflectivities in excitation along +z and −z for s = 485nm (395nm) are ~0 (0.8) and ∼0.67 (0) at wavelength 1441nm (1387nm), respectively. These phenomena indicate that the bilateral unidirectional reflectionlessness at wavelength 1441nm and 1387nm appear with the appropriate distance s, as shown in Figs. 2(a) and 2(e). Based on Eq. (5), the reflection spectra for excitation along +z and −z are shown in Fig. 2. The blue dot and dark cyan dot lines depict the analytical reflection spectra corresponding to different phase ϕ. The results from Fig. 2 indicate the good agreement between numerical simulation (based on distance s) and analytical calculation (based on phase ϕ). The bilateral unidirectional reflectionless phenomena can be obtained in our structure. The relevant fitting parameters Γ1(2) and γ1(2) are shown in Fig. 2(f) by using the analytical calculation. We notice that Γ1(2) and γ1(2) nearly remain invariant with the variation of s.

 figure: Fig. 2

Fig. 2 (a)–(e): Dependency of the simulated (analytical) reflection spectra on the distance s (phase ϕ) and the wavelength of incident wave. (f) The relevant fitting parameters Γ1(2) and γ1(2) versus the distance s.

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In order to analyze the unidirectional reflectionlessness in the non-ideal PT metasurface system better, the z-component distributions of electric field of two silver rings for the incident wave at two EPs are shown in Fig. 3. When s is 485nm, both silver rings at EP (1441nm) are excited simultaneously by the incident wave of excitation along +z(−z) and the induced currents are the same (opposite) directions as shown in Figs. 3(a) (3(c)) and 3(b) (3(d)), which means that the phase differences of induced currents of two silver rings are approximately 2π (Figs. 3(a) and 3(b)) and π (Figs. 3(c) and 3(d)), respectively. Therefore, the reflections at wavelength 1441nm are ∼ 0 (black solid line) and a high value (red solid line) corresponding to the phase difference of ∼2π in excitation along +z and ∼π in excitation along −z, respectively, based on the Fabry-Pérot (FP) resonant coupling as shown in Fig. 2(a). When the distance s reduces to 395nm, the unidirectional reflectionless propagation appears again. It can also be seen from Figs. 3(e)–3(h), when s = 395nm at the second EP (1387nm), the phase differences between induced currents of two silver rings in excitation along +z and −z are ∼π and ~2π, respectively, and the corresponding reflections are a high value (black solid line) and ~0 (red solid line), respectively, as shown Fig. 2(e). Therefore, the unidirectional reflectionless phenomena appear at two EPs (1441nm and 1387nm) for s = 485nm and 395nm, respectively. Based on Eq. (4) and some appropriate parameters, ϕall of about 2π can be obtained, which indicates that the directions of induced currents of both silver rings at two EPs are the same as shown in Figs. 3(a)–(b) and Figs. 3(g)–3(h). Therefore, we have confirmed the unidirectional reflectionless phenomena in our non-ideal PT metasurface system further by analytical calculation.

 figure: Fig. 3

Fig. 3 The z-component distributions of electric field of two silver rings for s = 485nm ((a)–(d)) and s = 395nm ((e)–(h)) at the wavelengths 1441nm and 1387nm in +z and −z directions, respectively.

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Then we discuss the influences of distance s and accumulated phase shift ϕ between two silver ring resonators on the bilateral reflectionlessness in our structure by using the numerical simulations (Figs. 4(a) and 4(b)) and analytical calculations (Figs. 4(c) and 4(d)). From Figs. 4(a) and 4(b), the low reflection peaks have blue-shifts with the decreasing s in excitation along +z and −z. Moreover, the low reflection region (Fig. 4(a)) in the ranges of wavelength from 1421nm to 1463nm and distance s from 470nm to 500nm in excitation along +z corresponds to the high reflection region (Fig. 4(b)) in excitation along −z. Similarly, the low reflection region (Fig. 4(b)) in the ranges of wavelength from 1367nm to 1401nm and distance s from 380nm to 410nm in excitation along −z corresponds to the high reflection region (Fig. 4(a)) in excitation along +z. Therefore, the bilateral unidirectional reflectionless propagations can be manipulated by adjusting distance s in our non-ideal PT metasurface structure in a wide wavelength range. According to Eq. (5), the dependency of reflection on phase ϕ and wavelength in excitation along +z and −z are shown in Figs. 4(c) and 4(d), respectively. A comparision of the Figs. 4(a)–4(b) with Figs. 4(c)–4(d) shows the consistency of results based on numerical simulations and analytical calculations, and the dispersions of reflection versus phase ϕ match well with that versus distance s in the same direction.

 figure: Fig. 4

Fig. 4 Reflection as the functions of distance s ((a) and (b)) and phase ϕ ((c) and (d)) in excitation along +z and −z based on the numerical simulation (Sim) and analytical calculation (Ana), respectively.

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Next, we discuss the dependency of reflection on the incident angle of excitation along +z and −z. Figures 5(a) and 5(c), and 5(b) and (5(d) show the reflections as the functions of incident angle and wavelength for s = 485nm (395nm) in excitation along +z and −z, respectively. Increasing the angle of incident wave in excitation along +z (Fig. 5(a)) and −z (Fig. 5(d)), red-shifts occur gradually in the low reflection region. When incident angle is smaller than 25°, high reflections appear in the same region for excitation along −z (Fig. 5(b)) and +z (Fig. 5(c)), respectively. From Figs. 5(a)–5(b) and Figs. 5(c)–5(d), the unidirectional reflectionless phenomena appear corresponding to the incident angle in a wide range of nearly ±25°. Not only that, the unidirectional reflectionless phenomena can be controlled by adjusting incident angle from 0 to ∼40° for s = 485nm and 395nm, respectively. Actually, when the incident angle is 0 for s = 485nm (395nm), the unidirectional reflectionless phenomena appear, while when the incident angle is ∼40°, the reflections in both excitation along ±z are nearly the same, thus the unidirectional reflectionless phenomena disappear.

 figure: Fig. 5

Fig. 5 Dependency of reflection on the angle of incident wave in excitation along +z and −z and wavelength for s = 485nm ((a) and (b)) and s = 395nm((c) and (d)), respectively.

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In order to discuss the relevant physics phenomena at two EPs in detail, we can calculate the eigenvalues of scattering matrix S as

λ1(2)=t±r+zrz,
based on Eq. (6). Here, the non-Hermitian system is called an ideal PT or non-ideal PT system if t is real or complex [43]. In our metasurface system, t is complex, which corresponds to a non-ideal PT system. From Eq. (7), one can determine that the non-ideal metasurface system is symmetry when r+zr−z > 0 and the symmetry is broken when r+zr−z < 0. The EP occurs in r+zr−z = 0. This means that when r+z or r−z is 0, two eigenvalues coalesce, which indicates that the unidirectional reflectionlessness appear at the EP. According to Eq. (7), the real and imaginary part curves of eigenvalue λ1(2) with respect to the wavelength are shown in Fig. 6 for ϕ = 3.31, π and 2.88, respectively. Obviously, the real and imaginary parts of two eigenvalues coalesce at a non-zero point (EP) at 1441nm and 1387nm for ϕ = 3.31 and 2.88, respectively, according to the Figs. 6(a)–6(b) and 6(e)−6(f). In this case, t is the complex and r+zr−z is 0. This means that the phase transition from non-ideal PT symmetry to symmetry breaking appears at EP. Hence, by appropriately adjusting the phase difference between two silver rings, the unidirectional reflectionlessness can be obtained in our non-ideal PT metasurface system. And from Figs. 6(c)–6(d), the imaginary parts of λ1 and λ2 coalesce at the zero point at wavelength 1416nm for ϕ = π. In this case, t is real and r+zr−z is bigger than 0. That is to say, the phase transitions from Hermitian to non-Hermitian occur at wavelength 1416nm.

 figure: Fig. 6

Fig. 6 Real and imaginary parts of eigenvalues of scattering matrix S as a function of wavelength in different ϕ. The black solid and red dash lines correspond to the two eigenvalues λ1 and λ2, respectively.

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In fact, our non-ideal PT metasurface system is not only applicable to the unidirectional reflectionless propagation at EPs, but also can act as an absorber near EPs where coherent perfect absorptions (CPAs) emerge, as shown in Figs. 6(a)–6(b) and Figs. 6(e)–6(f). From Figs. 6(b) and 6(f), the imaginary part curves of λ1 and λ2 have two intersections near EP with the zero axis line (blue), respectively. When r+zr−z < 0, one of eigenvalues for non-ideal PT symmetric broken system is real and the other is complex, in this case, two CPAs near each EP appear. Moreover, due to that the phase differences ϕall of induced currents for two silver ring resonators are ~2π near two EPs based on FP resonant coupling, the corresponding reflection close to 0. Together with the low transmissions near EP, the absorptions near 1441nm are ~83% with Q-factor of ~32 for ϕ = 3.29 according to A = 1 − RT. While for ϕ = 2.878 and 2.9, the absorptions at 1388nm and 1385nm (near EP 1387nm) are ∼96% with Q-factor of 41. Therefore, the present non-ideal PT metasurface system is applicable to light absorption within a narrow spectral range. In addition, our structure can be fabricated by using standard nanofabrication procedures [59].

4. Conclusion

We have designed a non-ideal PT metasurface system consisted of two silver ring resonators to realize the unidirectional reflectionless propagation based on far field coupling. The unidirectional reflectionless propagation can be manipulated by adjusting the angle of incident wave and the distance s between two silver ring resonators. Moreover, the bilateral unidirectional reflectionless propagations is insensitive to a wide range of incident angle (~25°). And the structure is polarization independent due to the two-ring structure. Besides, the ultra-narrow-band absorptions of ∼96% with Q-factor of ∼41 appear near EPs in non-ideal PT symmetric broken metasurface system. We believe that the unidirectional reflectionless non-ideal PT metasurface will have potential applications to the filter, sensor and highly compact intergrated nanophotonic devices.

Funding

National Natural Science Foundation of China (Grant No. 11364044); the Education Department of Jilin Province Science and Technology Research Project (Grant No. 2015-09 and JJKH20170455KJ).

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54. E. Yang, Y. Lu, Y. Wang, Y. Dai, and P. Wang, “Unidirectional reflectionless phenomenon in periodic ternary layered material,” Opt. Express 24, 14311 (2016). [CrossRef]   [PubMed]  

55. H. Alaeian and J. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014). [CrossRef]  

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

57. X. R. Jin, Y. Q. Zhang, S. Zhang, Y. P. Lee, and J. Y. Rhee, “Polarization-independent electromagnetically induced transparency-like effects in stacked metamaterials based on Fabry-Pérot resonance,” J. Opt. 15, 125104 (2013). [CrossRef]  

58. J. J. Chen, C. Wang, R. Zhang, and J. H. Xiao, “Multiple plasmon-induced transparencies in coupled-resonator systems,” Opt. Lett. 37, 5133 (2012). [CrossRef]   [PubMed]  

59. 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,” Nature Mater. 8, 758 (2009). [CrossRef]  

References

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  31. B. He, S. B. Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
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    [Crossref]
  51. M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
    [Crossref] [PubMed]
  52. M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
    [Crossref] [PubMed]
  53. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
    [Crossref] [PubMed]
  54. E. Yang, Y. Lu, Y. Wang, Y. Dai, and P. Wang, “Unidirectional reflectionless phenomenon in periodic ternary layered material,” Opt. Express 24, 14311 (2016).
    [Crossref] [PubMed]
  55. H. Alaeian and J. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
    [Crossref]
  56. M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87, 053824 (2013).
    [Crossref]
  57. X. R. Jin, Y. Q. Zhang, S. Zhang, Y. P. Lee, and J. Y. Rhee, “Polarization-independent electromagnetically induced transparency-like effects in stacked metamaterials based on Fabry-Pérot resonance,” J. Opt. 15, 125104 (2013).
    [Crossref]
  58. J. J. Chen, C. Wang, R. Zhang, and J. H. Xiao, “Multiple plasmon-induced transparencies in coupled-resonator systems,” Opt. Lett. 37, 5133 (2012).
    [Crossref] [PubMed]
  59. 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,” Nature Mater. 8, 758 (2009).
    [Crossref]

2017 (2)

Y. Aurégan and V. Pagneux, “PT-symmetric scattering in flow duct acoustics,” Phys. Rev. Lett. 118, 174301 (2017).
[Crossref]

M. H. Teimourpour, A. Rahman, K. Srinivasan, and R. El-Ganainy, “Non-Hermitian enfineering of synthetic saturable absorbers for applications in photonics,” Phys. Rev. Appl. 7, 014015 (2017).
[Crossref]

2016 (8)

K-H. Kim, M-S. Hwang, H-R. Kim, J-H. Choi, Y-S. No, and H-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref] [PubMed]

M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep. 6, 33253 (2016).
[Crossref] [PubMed]

P. Y. Chen and J. Jung, “PT symmetry and singularity-enhanced sensing based on photoexcited graphene metasurfaces,” Phys. Rev. Appl. 5, 064018 (2016).
[Crossref]

Y. Huang, C. Min, and G. Veronis, “Broadband near total light absorption in non-PT-symmetric waveguide-cavity systems,” Opt. Express 24, 22219–22231 (2016).
[Crossref] [PubMed]

C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. 7, 11110 (2016).
[Crossref] [PubMed]

J. Christensen, M. Willatzen, V. R. Velasco, and M. H. Lu, “Parity-time synthetic phononic media,” Phys. Rev. Lett. 116, 207601 (2016).
[Crossref] [PubMed]

Y. Y. Fu, Y. D. Xu, and H. Y. Chen, “Zero index metamaterials with PT symmetry in a waveguide system,” Opt. Express 24, 1648 (2016).
[Crossref] [PubMed]

E. Yang, Y. Lu, Y. Wang, Y. Dai, and P. Wang, “Unidirectional reflectionless phenomenon in periodic ternary layered material,” Opt. Express 24, 14311 (2016).
[Crossref] [PubMed]

2015 (7)

B. Baum, H. Alaeian, and J. Dionne, “A parity-time symmetric coherent plasmonic absorber-amplifier,” J. Appl. Phys. 117, 063106 (2015).
[Crossref]

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

B. He, L. Yang, and M. Xiao, “Cyclic permutation-time symmetric structure with coupled gain-loss microcavities,” Phys. Rev. A 91, 033830 (2015).
[Crossref]

R. Fleury, D. Sounas, and A. Alu, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6, 5905 (2015).
[Crossref] [PubMed]

B. He, S. B. Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
[Crossref]

Y. Choi, J. K. Hong, J. H. Cho, K. G. Lee, J. W. Yoon, and S. H. Song, “Parity-time-symmetry breaking in double-slab surface-plasmon-polariton waveguides,” Opt. Express 23, 11783–89 (2015).
[Crossref] [PubMed]

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).
[Crossref]

2014 (10)

M. H. Teimourpour and R. El-Ganainy, “Light transport in PT-invariant photonic structures with hidden symmetries,” Phys. Rev. A 90, 053817 (2014).
[Crossref]

Y. L. Xu, L. Feng, M. H. Lu, and Y. F. Chen, “Unidirectional transmission based on a passive PT symmetric grating with a nonlinear silicon distributed Bragg reflector cavity,” IEEE. Photonics. J 6, 0600507 (2014).
[Crossref]

B. Peng, S. K. Özdemir, 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, 394 (2014).
[Crossref]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref] [PubMed]

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 1439032014).
[Crossref] [PubMed]

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89, 023842 (2014).
[Crossref]

H. Alaeian and J. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref] [PubMed]

2013 (3)

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

X. R. Jin, Y. Q. Zhang, S. Zhang, Y. P. Lee, and J. Y. Rhee, “Polarization-independent electromagnetically induced transparency-like effects in stacked metamaterials based on Fabry-Pérot resonance,” J. Opt. 15, 125104 (2013).
[Crossref]

N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110, 053901 (2013).
[Crossref] [PubMed]

2012 (6)

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-symmetric Talbot effects,” Phys. Rev. Lett. 109, 3 (2012).
[Crossref]

J. J. Chen, C. Wang, R. Zhang, and J. H. Xiao, “Multiple plasmon-induced transparencies in coupled-resonator systems,” Opt. Lett. 37, 5133 (2012).
[Crossref] [PubMed]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

H. Ramezani, J. Schindler, F. M. Ellis, U. Gunther, and T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85, 062122 (2012).
[Crossref]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
[Crossref] [PubMed]

2011 (6)

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, 213901 (2011).
[Crossref] [PubMed]

H. Benisty, A. Degiron, A. Lupu, A. D. Lustrac, S. Chénais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Lérondel, “Implementation of PT symmetric devices using plasmonics: principle and applications,” Opt. Express 19, 18004–18019 (2011).
[Crossref] [PubMed]

D. D. Scott and Y. N. Joglekar, “Degrees and signatures of broken PT symmetry in nonuniform lattices,” Phys. Rev. A 83, 050102 (2011).
[Crossref]

E-M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A 84, 013818 (2011).
[Crossref]

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (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, 093902 (2011).
[Crossref] [PubMed]

2010 (4)

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

C. E. Rüer, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

2009 (4)

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,” Nature Mater. 8, 758 (2009).
[Crossref]

S. B. Lee, J. Yang, S. Moon, S. Y. Lee, J. B. Shim, S. W. Kim, J. H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref] [PubMed]

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A: Math. Theor. 42, 153001 (2009).
[Crossref]

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

2008 (3)

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. E. Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. E. Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

2005 (2)

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38, L171–L176 (2005).
[Crossref]

M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13, 3068–3078 (2005).
[Crossref] [PubMed]

2004 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

2002 (1)

B. Bagchi and C. Quesne, “Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework,” Phys. Lett. A 300, 18–26 (2002).
[Crossref]

1998 (2)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamitonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

I. Y. Goldsheid and B. A. Khoruzhenko, “Distribution of eigenvalues in non-Hermitian Anderson models,” Phys. Rev. Lett. 80, 2897 (1998).
[Crossref]

Aimez, V.

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

Alaeian, H.

B. Baum, H. Alaeian, and J. Dionne, “A parity-time symmetric coherent plasmonic absorber-amplifier,” J. Appl. Phys. 117, 063106 (2015).
[Crossref]

H. Alaeian and J. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
[Crossref]

Almeida, V. R.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
[Crossref] [PubMed]

Alu, A.

R. Fleury, D. Sounas, and A. Alu, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6, 5905 (2015).
[Crossref] [PubMed]

An, K.

Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

S. B. Lee, J. Yang, S. Moon, S. Y. Lee, J. B. Shim, S. W. Kim, J. H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref] [PubMed]

Aurégan, Y.

Y. Aurégan and V. Pagneux, “PT-symmetric scattering in flow duct acoustics,” Phys. Rev. Lett. 118, 174301 (2017).
[Crossref]

Azaña, J.

Bagchi, B.

B. Bagchi and C. Quesne, “Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework,” Phys. Lett. A 300, 18–26 (2002).
[Crossref]

Barbillon, G.

Baum, B.

B. Baum, H. Alaeian, and J. Dionne, “A parity-time symmetric coherent plasmonic absorber-amplifier,” J. Appl. Phys. 117, 063106 (2015).
[Crossref]

Bélanger, N.

Bender, C. M.

B. Peng, S. K. Özdemir, 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, 394 (2014).
[Crossref]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamitonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Benisty, H.

Bersch, C.

A. Regensburger, C. Bersch, M-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

Besbes, M.

Blaize, S.

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamitonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Brandstetter, M.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref] [PubMed]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

Bruyant, A.

Cao, H.

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, 213901 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Cerjan, A.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

Chan, C. T.

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).
[Crossref]

Chen, H.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 1439032014).
[Crossref] [PubMed]

Chen, H. Y.

Chen, J. J.

Chen, P. Y.

P. Y. Chen and J. Jung, “PT symmetry and singularity-enhanced sensing based on photoexcited graphene metasurfaces,” Phys. Rev. Appl. 5, 064018 (2016).
[Crossref]

Chen, Y.

C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. 7, 11110 (2016).
[Crossref] [PubMed]

Chen, Y. F.

Y. L. Xu, L. Feng, M. H. Lu, and Y. F. Chen, “Unidirectional transmission based on a passive PT symmetric grating with a nonlinear silicon distributed Bragg reflector cavity,” IEEE. Photonics. J 6, 0600507 (2014).
[Crossref]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
[Crossref] [PubMed]

Chénais, S.

Cheng, L.

C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. 7, 11110 (2016).
[Crossref] [PubMed]

Cho, J. H.

Choi, J-H.

K-H. Kim, M-S. Hwang, H-R. Kim, J-H. Choi, Y-S. No, and H-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref] [PubMed]

Choi, Y.

Y. Choi, J. K. Hong, J. H. Cho, K. G. Lee, J. W. Yoon, and S. H. Song, “Parity-time-symmetry breaking in double-slab surface-plasmon-polariton waveguides,” Opt. Express 23, 11783–89 (2015).
[Crossref] [PubMed]

Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

Chong, Y. D.

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

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Christensen, J.

J. Christensen, M. Willatzen, V. R. Velasco, and M. H. Lu, “Parity-time synthetic phononic media,” Phys. Rev. Lett. 116, 207601 (2016).
[Crossref] [PubMed]

Christodoulides, D. N.

M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep. 6, 33253 (2016).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-symmetric Talbot effects,” Phys. Rev. Lett. 109, 3 (2012).
[Crossref]

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, 213901 (2011).
[Crossref] [PubMed]

C. E. Rüer, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

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

K. G. Makris, R. E. Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. E. Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

Dai, Y.

Degiron, A.

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38, L171–L176 (2005).
[Crossref]

Deutsch, C.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref] [PubMed]

Ding, K.

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).
[Crossref]

Dionne, J.

B. Baum, H. Alaeian, and J. Dionne, “A parity-time symmetric coherent plasmonic absorber-amplifier,” J. Appl. Phys. 117, 063106 (2015).
[Crossref]

H. Alaeian and J. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
[Crossref]

Dubois, M.

C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. 7, 11110 (2016).
[Crossref] [PubMed]

Duchesne, D.

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

Eichelkraut, T.

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, 213901 (2011).
[Crossref] [PubMed]

El-Ganainy, R.

M. H. Teimourpour, A. Rahman, K. Srinivasan, and R. El-Ganainy, “Non-Hermitian enfineering of synthetic saturable absorbers for applications in photonics,” Phys. Rev. Appl. 7, 014015 (2017).
[Crossref]

M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep. 6, 33253 (2016).
[Crossref] [PubMed]

M. H. Teimourpour and R. El-Ganainy, “Light transport in PT-invariant photonic structures with hidden symmetries,” Phys. Rev. A 90, 053817 (2014).
[Crossref]

Ellis, F. M.

H. Ramezani, J. Schindler, F. M. Ellis, U. Gunther, and T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85, 062122 (2012).
[Crossref]

Fan, S.

B. Peng, S. K. Özdemir, 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, 394 (2014).
[Crossref]

Fegadolli, W. S.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
[Crossref] [PubMed]

Feng, L.

Y. L. Xu, L. Feng, M. H. Lu, and Y. F. Chen, “Unidirectional transmission based on a passive PT symmetric grating with a nonlinear silicon distributed Bragg reflector cavity,” IEEE. Photonics. J 6, 0600507 (2014).
[Crossref]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
[Crossref] [PubMed]

Fleischhauer, M.

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,” Nature Mater. 8, 758 (2009).
[Crossref]

Fleury, R.

R. Fleury, D. Sounas, and A. Alu, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6, 5905 (2015).
[Crossref] [PubMed]

Forget, S.

Fu, Y. Y.

Ganainy, R. E.

C. E. Rüer, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. E. Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. E. Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Ge, L.

M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep. 6, 33253 (2016).
[Crossref] [PubMed]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (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, 093902 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Gianfreda, M.

B. Peng, S. K. Özdemir, 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, 394 (2014).
[Crossref]

Giessen, H.

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,” Nature Mater. 8, 758 (2009).
[Crossref]

Goldsheid, I. Y.

I. Y. Goldsheid and B. A. Khoruzhenko, “Distribution of eigenvalues in non-Hermitian Anderson models,” Phys. Rev. Lett. 80, 2897 (1998).
[Crossref]

Graefe, E-M.

E-M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A 84, 013818 (2011).
[Crossref]

Gunther, U.

H. Ramezani, J. Schindler, F. M. Ellis, U. Gunther, and T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85, 062122 (2012).
[Crossref]

Günther, U.

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

Guo, A.

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

Han, J. L.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89, 023842 (2014).
[Crossref]

He, B.

B. He, L. Yang, and M. Xiao, “Cyclic permutation-time symmetric structure with coupled gain-loss microcavities,” Phys. Rev. A 91, 033830 (2015).
[Crossref]

B. He, S. B. Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
[Crossref]

Hong, J. K.

Huang, C. Y.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89, 023842 (2014).
[Crossref]

Huang, Y.

Hwang, M-S.

K-H. Kim, M-S. Hwang, H-R. Kim, J-H. Choi, Y-S. No, and H-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref] [PubMed]

Jin, X. R.

X. R. Jin, Y. Q. Zhang, S. Zhang, Y. P. Lee, and J. Y. Rhee, “Polarization-independent electromagnetically induced transparency-like effects in stacked metamaterials based on Fabry-Pérot resonance,” J. Opt. 15, 125104 (2013).
[Crossref]

Jing, H.

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref] [PubMed]

Joglekar, Y. N.

D. D. Scott and Y. N. Joglekar, “Degrees and signatures of broken PT symmetry in nonuniform lattices,” Phys. Rev. A 83, 050102 (2011).
[Crossref]

Jones, H. F.

E-M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A 84, 013818 (2011).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

Jung, J.

P. Y. Chen and J. Jung, “PT symmetry and singularity-enhanced sensing based on photoexcited graphene metasurfaces,” Phys. Rev. Appl. 5, 064018 (2016).
[Crossref]

Kang, M.

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

Kang, S.

Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

Kästel, J.

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,” Nature Mater. 8, 758 (2009).
[Crossref]

Khoruzhenko, B. A.

I. Y. Goldsheid and B. A. Khoruzhenko, “Distribution of eigenvalues in non-Hermitian Anderson models,” Phys. Rev. Lett. 80, 2897 (1998).
[Crossref]

Kim, H-R.

K-H. Kim, M-S. Hwang, H-R. Kim, J-H. Choi, Y-S. No, and H-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref] [PubMed]

Kim, J. R.

Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

Kim, K-H.

K-H. Kim, M-S. Hwang, H-R. Kim, J-H. Choi, Y-S. No, and H-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref] [PubMed]

Kim, S. W.

S. B. Lee, J. Yang, S. Moon, S. Y. Lee, J. B. Shim, S. W. Kim, J. H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref] [PubMed]

Kim, W.

Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

Kip, D.

C. E. Rüer, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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Klaiman, S.

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Velasco, V. R.

J. Christensen, M. Willatzen, V. R. Velasco, and M. H. Lu, “Parity-time synthetic phononic media,” Phys. Rev. Lett. 116, 207601 (2016).
[Crossref] [PubMed]

Veronis, G.

Vitebskiy, I.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-symmetric Talbot effects,” Phys. Rev. Lett. 109, 3 (2012).
[Crossref]

Wang, C.

Wang, J.

B. He, S. B. Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
[Crossref]

Wang, P.

Wang, Y.

E. Yang, Y. Lu, Y. Wang, Y. Dai, and P. Wang, “Unidirectional reflectionless phenomenon in periodic ternary layered material,” Opt. Express 24, 14311 (2016).
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C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. 7, 11110 (2016).
[Crossref] [PubMed]

Weiss, T.

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,” Nature Mater. 8, 758 (2009).
[Crossref]

Willatzen, M.

J. Christensen, M. Willatzen, V. R. Velasco, and M. H. Lu, “Parity-time synthetic phononic media,” Phys. Rev. Lett. 116, 207601 (2016).
[Crossref] [PubMed]

Xiao, J. H.

Xiao, M.

B. He, L. Yang, and M. Xiao, “Cyclic permutation-time symmetric structure with coupled gain-loss microcavities,” Phys. Rev. A 91, 033830 (2015).
[Crossref]

B. He, S. B. Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
[Crossref]

Xu, J. Q.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89, 023842 (2014).
[Crossref]

Xu, Y. D.

Xu, Y. L.

Y. L. Xu, L. Feng, M. H. Lu, and Y. F. Chen, “Unidirectional transmission based on a passive PT symmetric grating with a nonlinear silicon distributed Bragg reflector cavity,” IEEE. Photonics. J 6, 0600507 (2014).
[Crossref]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
[Crossref] [PubMed]

Yan, S. B.

B. He, S. B. Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
[Crossref]

Yang, E.

Yang, J.

S. B. Lee, J. Yang, S. Moon, S. Y. Lee, J. B. Shim, S. W. Kim, J. H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref] [PubMed]

Yang, L.

B. He, L. Yang, and M. Xiao, “Cyclic permutation-time symmetric structure with coupled gain-loss microcavities,” Phys. Rev. A 91, 033830 (2015).
[Crossref]

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, 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, 394 (2014).
[Crossref]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

Yilmaz, H.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

Yoon, J. W.

Zhang, J.

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref] [PubMed]

Zhang, R.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89, 023842 (2014).
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J. J. Chen, C. Wang, R. Zhang, and J. H. Xiao, “Multiple plasmon-induced transparencies in coupled-resonator systems,” Opt. Lett. 37, 5133 (2012).
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Zhang, S.

X. R. Jin, Y. Q. Zhang, S. Zhang, Y. P. Lee, and J. Y. Rhee, “Polarization-independent electromagnetically induced transparency-like effects in stacked metamaterials based on Fabry-Pérot resonance,” J. Opt. 15, 125104 (2013).
[Crossref]

Zhang, X.

C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. 7, 11110 (2016).
[Crossref] [PubMed]

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

Zhang, Y. Q.

X. R. Jin, Y. Q. Zhang, S. Zhang, Y. P. Lee, and J. Y. Rhee, “Polarization-independent electromagnetically induced transparency-like effects in stacked metamaterials based on Fabry-Pérot resonance,” J. Opt. 15, 125104 (2013).
[Crossref]

Zhang, Z. Q.

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).
[Crossref]

Zheng, J.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89, 023842 (2014).
[Crossref]

Zhu, J.

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

Zhu, X.

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

IEEE. Photonics. J (1)

Y. L. Xu, L. Feng, M. H. Lu, and Y. F. Chen, “Unidirectional transmission based on a passive PT symmetric grating with a nonlinear silicon distributed Bragg reflector cavity,” IEEE. Photonics. J 6, 0600507 (2014).
[Crossref]

J. Appl. Phys. (1)

B. Baum, H. Alaeian, and J. Dionne, “A parity-time symmetric coherent plasmonic absorber-amplifier,” J. Appl. Phys. 117, 063106 (2015).
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J. Opt. (1)

X. R. Jin, Y. Q. Zhang, S. Zhang, Y. P. Lee, and J. Y. Rhee, “Polarization-independent electromagnetically induced transparency-like effects in stacked metamaterials based on Fabry-Pérot resonance,” J. Opt. 15, 125104 (2013).
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J. Phys. A: Math. Gen. (1)

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38, L171–L176 (2005).
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J. Phys. A: Math. Theor. (1)

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A: Math. Theor. 42, 153001 (2009).
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Nat. Commun. (4)

R. Fleury, D. Sounas, and A. Alu, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6, 5905 (2015).
[Crossref] [PubMed]

C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. 7, 11110 (2016).
[Crossref] [PubMed]

K-H. Kim, M-S. Hwang, H-R. Kim, J-H. Choi, Y-S. No, and H-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref] [PubMed]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref] [PubMed]

Nat. Mater. (1)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
[Crossref] [PubMed]

Nat. Phys. (2)

B. Peng, S. K. Özdemir, 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, 394 (2014).
[Crossref]

C. E. Rüer, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Nature (1)

A. Regensburger, C. Bersch, M-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

Nature Mater. (1)

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,” Nature Mater. 8, 758 (2009).
[Crossref]

Opt. Express (7)

Opt. Lett. (1)

Phys. Lett. A (1)

B. Bagchi and C. Quesne, “Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework,” Phys. Lett. A 300, 18–26 (2002).
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Phys. Rev. A (11)

H. Ramezani, J. Schindler, F. M. Ellis, U. Gunther, and T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85, 062122 (2012).
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D. D. Scott and Y. N. Joglekar, “Degrees and signatures of broken PT symmetry in nonuniform lattices,” Phys. Rev. A 83, 050102 (2011).
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E-M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A 84, 013818 (2011).
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M. H. Teimourpour and R. El-Ganainy, “Light transport in PT-invariant photonic structures with hidden symmetries,” Phys. Rev. A 90, 053817 (2014).
[Crossref]

B. He, S. B. Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
[Crossref]

H. Alaeian and J. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
[Crossref]

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

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89, 023842 (2014).
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S. Longhi, “PT-symmetry laser absorber,” Phys. Rev. A 82, 031801 (2010).
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L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

B. He, L. Yang, and M. Xiao, “Cyclic permutation-time symmetric structure with coupled gain-loss microcavities,” Phys. Rev. A 91, 033830 (2015).
[Crossref]

Phys. Rev. Appl. (2)

P. Y. Chen and J. Jung, “PT symmetry and singularity-enhanced sensing based on photoexcited graphene metasurfaces,” Phys. Rev. Appl. 5, 064018 (2016).
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M. H. Teimourpour, A. Rahman, K. Srinivasan, and R. El-Ganainy, “Non-Hermitian enfineering of synthetic saturable absorbers for applications in photonics,” Phys. Rev. Appl. 7, 014015 (2017).
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Phys. Rev. B (1)

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).
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Phys. Rev. D (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
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Phys. Rev. Lett. (18)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamitonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
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Z. H. Musslimani, K. G. Makris, R. E. Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
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J. Christensen, M. Willatzen, V. R. Velasco, and M. H. Lu, “Parity-time synthetic phononic media,” Phys. Rev. Lett. 116, 207601 (2016).
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S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. V. Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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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, 213901 (2011).
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S. B. Lee, J. Yang, S. Moon, S. Y. Lee, J. B. Shim, S. W. Kim, J. H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
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Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
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Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 1439032014).
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H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-symmetric Talbot effects,” Phys. Rev. Lett. 109, 3 (2012).
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N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110, 053901 (2013).
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Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
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M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
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Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref] [PubMed]

Phys. Rev. X (1)

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

Sci. Rep. (1)

M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep. 6, 33253 (2016).
[Crossref] [PubMed]

Science (1)

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1 Schematic of unit cell of the non-ideal PT metasurface structure. The parameters are h = 20nm, r1 = 61nm, R1 = 122nm, r2 = 63nm, R2 = 170nm and t = 640nm, respectively. The distance s is variable. The incident wave is in xz plane and has an angle θ with +z axis.
Fig. 2
Fig. 2 (a)–(e): Dependency of the simulated (analytical) reflection spectra on the distance s (phase ϕ) and the wavelength of incident wave. (f) The relevant fitting parameters Γ1(2) and γ1(2) versus the distance s.
Fig. 3
Fig. 3 The z-component distributions of electric field of two silver rings for s = 485nm ((a)–(d)) and s = 395nm ((e)–(h)) at the wavelengths 1441nm and 1387nm in +z and −z directions, respectively.
Fig. 4
Fig. 4 Reflection as the functions of distance s ((a) and (b)) and phase ϕ ((c) and (d)) in excitation along +z and −z based on the numerical simulation (Sim) and analytical calculation (Ana), respectively.
Fig. 5
Fig. 5 Dependency of reflection on the angle of incident wave in excitation along +z and −z and wavelength for s = 485nm ((a) and (b)) and s = 395nm((c) and (d)), respectively.
Fig. 6
Fig. 6 Real and imaginary parts of eigenvalues of scattering matrix S as a function of wavelength in different ϕ. The black solid and red dash lines correspond to the two eigenvalues λ1 and λ2, respectively.

Equations (7)

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T all = T s 1 × T p × T s 2 = ( T 11 T 12 T 21 T 22 ) ,
T s 1 ( 2 ) = ( 1 i γ 1 ( 2 ) ω ω 1 ( 2 ) + i Γ 1 ( 2 ) 2 i γ 1 ( 2 ) ω ω 1 ( 2 ) + i Γ 1 ( 2 ) 2 i γ 1 ( 2 ) ω ω 1 ( 2 ) + i Γ 1 ( 2 ) 2 1 i γ 1 ( 2 ) ω ω 1 ( 2 ) + i Γ 1 ( 2 ) 2 ) ,
T p = ( exp ( i ϕ ) 0 0 exp ( i ϕ ) ) ,
ϕ 1 ( 2 ) = arctan [ Im ( T s , 21 1 ( 2 ) / T s , 22 1 ( 2 ) ) Re ( T s , 21 1 ( 2 ) / T s , 22 1 ( 2 ) ) ] = ( ω ω 1 ( 2 ) ) ( γ 1 ( 2 ) + Γ 1 ( 2 ) 2 ) ,
t = t + z = t z = 1 T 22 , r + z = T 21 T 22 , r z = T 12 T 22 .
S = ( t r z r + z t ) .
λ 1 ( 2 ) = t ± r + z r z ,

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