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    Journals >Photonics Research >Volume 9 >Issue 4 >Page 574 > Article
    • Photonics Research
    • Vol. 9, Issue 4, 574 (2021)
    Sensitivity of topological edge states in a non-Hermitian dimer chain
    Zhiwei Guo1、2、*, Tengzhou Zhang1, Juan Song1, Haitao Jiang1, and Hong Chen1、3、*
    Author Affiliations
  • 1MOE Key Laboratory of Advanced Micro-structured Materials, School of Physics Sciences and Engineering, Tongji University, Shanghai 200092, China
  • 2e-mail: 2014guozhiwei@tongji.edu.cn
  • 3e-mail: hongchen@tongji.edu.cn
    • show less
    DOI: 10.1364/PRJ.413873 Cite this Article Set citation alerts
    Zhiwei Guo, Tengzhou Zhang, Juan Song, Haitao Jiang, Hong Chen. Sensitivity of topological edge states in a non-Hermitian dimer chain[J]. Photonics Research, 2021, 9(4): 574 Copy Citation Text show less

    Abstract

    Photonic topological edge states in one-dimensional dimer chains have long been thought to be robust to structural perturbations by mapping the topological Su–Schrieffer–Heeger model of a solid-state system. However, the edge states at the two ends of a finite topological dimer chain will interact as a result of near-field coupling. This leads to deviation from topological protection by the chiral symmetry from the exact zero energy, weakening the robustness of the topological edge state. With the aid of non-Hermitian physics, the splitting frequencies of edge states can be degenerated again, with topological protection recovered by altering the gain or loss strength of the structure. This point of coalescence is known as the exceptional point (EP). The intriguing physical properties of EPs in topological structures give rise to many fascinating and counterintuitive phenomena. In this work, based on a finite non-Hermitian dimer chain composed of ultra-subwavelength resonators, we propose theoretically and verify experimentally that the sensitivity of topological edge states is greatly affected when the system passes through the EP. Using the EP of a non-Hermitian dimer chain, we realize a new sensor that is sensitive to perturbation of on-site frequency at the end of the structure and yet topologically protected from internal perturbation of site-to-site couplings. Our demonstration of a non-Hermitian topological structure with an EP paves the way for the development of novel sensors that are not sensitive to internal manufacturing errors but are highly sensitive to changes in the external environment.

    1. INTRODUCTION

    Topological insulators, an interesting research topic in physics, have greatly improved the understanding of the classification of states in condensed matter physics. The fully occupied electronic band structure has the topological characteristics identified by the topological invariants [1]. Topological insulators have also opened up a new research stream in the development of new semiconductor devices to be used in quantum computing, high-fidelity quantum communication, and so on [2,3]. Inspired by the topological properties of electronic band structures, scientists designed a photonic counterpart and observed the charming photonic edge states in the artificial photonic structures [4–6]. It is of great scientific significance to use topology to control the motion of photons, and this unique research has been extended to quasiperiodic systems [7–11]. Photonic topological edge states can overcome the scattering losses caused by structural defects and disorders and realize topologically protected photonic devices, such as unidirectional waveguides and single-mode lasers [4–6].

    As one of the simplest topological structures, the one-dimensional (1D) dimer chain has been widely used in the study of photonic topological excitation. In this structure, the topological invariant can be directly identified by comparing the relative magnitude of intra-cell and inter-cell coupling coefficients [12]. Notably, in 2009, Malkova et al. experimentally revealed the linear Shockley-like surface states in an optically induced semi-infinite photonic superlattice, and it is the first, to the best of our knowledge, demonstration of 1D topological states in photonics [13]. Recently, the edge states will appear symmetrically at two ends of the finite topological chain [14], and the topological order—winding number—will be directly observable in the microwave regime [15]. At present, research on the advantages of 1D dimer chains has been extended to nonlinear [16–21] and active [22–25] systems. However, non-Hermitian topological photonics is also a very interesting research topic in topological physics. Advances in the field of non-Hermitian photonics based on parity–time (PT) symmetry have greatly improved the ability to design new photonic topological insulators in previously inaccessible ways [26,27]. In general, the eigenvalues of open optical non-Hermitian systems are generally complex. However, the PT-symmetric structure with real eigenvalues belongs to a very special non-Hermitian system. The most noticeable feature of a non-Hermitian system is that there are degenerate points in the Riemannian surface of the parameter space, where eigenvalues and corresponding eigenvectors simultaneously coalesce [28,29]. These non-Hermitian degeneracies, are also called exceptional points (EPs). Remarkably, in 2009, Guo et al. firstly highlighted the ability of PT-symmetric systems to guide light and then uncover the interesting loss-induced transparency due to an EP in a passive PT-symmetric experiment performed with coupled passive waveguides [30]. In addition, EPs have turned out to be the origin of many counterintuitive phenomena, such as band merging [31], dynamic wireless power transfer [32,33], the chirality-reversal phenomenon [34], and mode transfer [35–38]. EPs provide a new way to design new highly sensitive sensors beyond the linear response, and performance can be optimized as the order of the EPs increases [39–49]. Non-Hermitian topological structures contain new physical phenomena not usually found in normal Hermitian topological structures [50–60]. To date, the unique topological order [52,55], phase transition [50,57,58,60], and edge state [51,52,54,56] of the non-Hermitian 1D dimer chain have been studied. For example, the non-Bloch winding number as the topological invariant has been theoretically proposed [53], and topological protection of the edge states has been demonstrated experimentally [52]. Topological non-Hermitian systems provide an effective avenue for studying the intriguing properties of topological photonics involving EPs and developing new functional devices.

    Near-field mode coupling is a fundamental physical effect that plays an important role in controlling electromagnetic waves [61,62]. Researchers who have studied the near-field coupling of topological edge states have found many interesting phenomena, such as robust topological Fano resonance [63,64] and Rabi splitting [65]. Specifically, in a finite non-Hermitian dimer waveguide array, the coupling effect of edge states leads to deviation from the topological zero mode and thus weakens the robustness of the edge states [66,67]. In general, mode splitting induced by near-field coupling can be eliminated by increasing the length of the chain [61]. To recover topological protection, the coupling of the two edge states must be significantly reduced by increasing the chain length, which will cause the splitting edge state modes to return to zero energy. However, in a non-Hermitian system, the splitting frequencies can be degenerated again at the EPs by directly altering the gain or loss strength while keeping the length of the chain unchanged [66,67]. Although the effects of near-field coupling on the robustness of topological edge states have been confirmed qualitatively from field distributions [66,67], the behavior of novel EPs in non-Hermitian topological systems has not been reported. Topological edge states are generally thought to be robust to structural perturbations, as they result from nonlocal response based on the bulk-boundary correspondence. In contrast, the EP is often used to achieve highly sensitive sensors and is sensitive to slight variation in the environment. Thus, a question naturally arises: can topological edge states be used to design new highly sensitive sensors by combining EPs?

    Recently, topological circuit has been widely used as a versatile platform to study the abundant topological physics [68–72]. In this work, we study experimentally the properties of the EP in a finite non-Hermitian topological circuit-based dimer chain. The coupling between two edge states is presented, which is particularly relevant to the realization of second-order EPs. By adding loss and gain to both ends of the dimer chain, we can obtain the non-Hermitian topological chain that satisfies PT symmetry and then observe the EP by increasing the loss or gain of the system. Moreover, we also study the sensitivity of topological edge states to disturbances in the environment before and after the EP. As a result, a new highly sensitive sensor with topological protection is realized based on the EP of topological edge states. In sharp contrast to traditional sensors, this new sensor based on non-Hermitian and topological characteristics has unique advantages. It is immune from disturbances of site-to-site couplings in the internal part of the structure and is very sensitive to perturbation of on-site frequency at the end of the structure. By combining non-Hermitian systems with topology photonics, we design a sensor that has both the robustness of topology and the sensitivity of EPs. In addition, an even more sensitive topological sensor could be designed in the future considering the high-order EPs realized by the synthetic dimension [73,74]. Our findings not only present a novel photonic sensor with topological protection but also may be very useful for a variety of applications with non-Hermitian properties, including wireless power transfer [75–78], energy harvesting [79,80], and antennas [81].

    2. EPs OF EDGE STATES IN A FINITE NON-HERMITIAN DIMER CHAIN

    Composite resonator with tunable gain and loss designed in the current study. (a) Effective circuit model of the composite resonator, composed of a simple LC resonator, a negative resistance convertor (NIC) component, and a tunable resistor. (b) Details of the composite resonator, where the gold and blue structures indicate the top and bottom copper layers, respectively. Here, d1=46.2 mm, d2=48 mm, and w=1.12 mm; the thickness of the substrate is h=1.6 mm. The lumped circuit elements and vias are marked by the green rectangles and red dots, respectively. (c) Circuit model of the NIC component. The effective gain is tuned by the external direct current (DC) voltage source. (d) Schematic of the realization of the NIC component based on RF chokes and metal-oxide-semiconductor field-effect transistors (MOSFETs).

    Figure 1.Composite resonator with tunable gain and loss designed in the current study. (a) Effective circuit model of the composite resonator, composed of a simple LC resonator, a negative resistance convertor (NIC) component, and a tunable resistor. (b) Details of the composite resonator, where the gold and blue structures indicate the top and bottom copper layers, respectively. Here, d1=46.2  mm, d2=48  mm, and w=1.12  mm; the thickness of the substrate is h=1.6  mm. The lumped circuit elements and vias are marked by the green rectangles and red dots, respectively. (c) Circuit model of the NIC component. The effective gain is tuned by the external direct current (DC) voltage source. (d) Schematic of the realization of the NIC component based on RF chokes and metal-oxide-semiconductor field-effect transistors (MOSFETs).

    Measured reflection spectrum of the composite resonator. (a) The reflection spectrum from changing the resistance without the external bias voltage. The resonant frequency, which is marked by the pink dashed line, is almost unchanged. (b) Similar to (a), but the external voltage changes while the resistance is fixed at R=2.8 kΩ. The slight frequency shift of 0.086 MHz is marked by gray shading.

    Figure 2.Measured reflection spectrum of the composite resonator. (a) The reflection spectrum from changing the resistance without the external bias voltage. The resonant frequency, which is marked by the pink dashed line, is almost unchanged. (b) Similar to (a), but the external voltage changes while the resistance is fixed at R=2.8  kΩ. The slight frequency shift of 0.086 MHz is marked by gray shading.

    1D non-Hermitian topological dimer chain. (a) Schematic of a topological dimer chain with 10 resonators. Effective loss and gain are added into the left and right resonators, respectively. (b) The real eigenfrequencies of the finite chain as a function of parameter gL. As gL increases, the splitting edge states gradually coalesce in the EP, which is marked by the black arrow. (c) The enlarged eigenfrequencies of two edge states as a function of parameter gL and frequency detuning ε. (d), (e) Normalized wave functions of two splitting edge states (ω+ and ω−).

    Figure 3.1D non-Hermitian topological dimer chain. (a) Schematic of a topological dimer chain with 10 resonators. Effective loss and gain are added into the left and right resonators, respectively. (b) The real eigenfrequencies of the finite chain as a function of parameter gL. As gL increases, the splitting edge states gradually coalesce in the EP, which is marked by the black arrow. (c) The enlarged eigenfrequencies of two edge states as a function of parameter gL and frequency detuning ε. (d), (e) Normalized wave functions of two splitting edge states (ω+ and ω−).

    3. EXPERIMENTAL OBSERVATIONS OF THE SENSITIVITY OF TOPOLOGICAL EDGE STATES AT DIFFERENT PHASES AROUND THE EP

    Measured reflection spectrum of the 1D non-Hermitian dimer chain. (a) Photo of the non-Hermitian topological dimer chain. The sample is put on a PMMA substrate with a thickness hs=1 cm. (b) Measured reflection spectrum as the dissipative loss of the lossy resonator, which is controlled by the tunable resistor at the right end of the chain, increases. Dots denote the frequencies of the edge states. Resistance is given on a logarithmic scale.

    Figure 4.Measured reflection spectrum of the 1D non-Hermitian dimer chain. (a) Photo of the non-Hermitian topological dimer chain. The sample is put on a PMMA substrate with a thickness hs=1  cm. (b) Measured reflection spectrum as the dissipative loss of the lossy resonator, which is controlled by the tunable resistor at the right end of the chain, increases. Dots denote the frequencies of the edge states. Resistance is given on a logarithmic scale.

    To further explore the intriguing properties of the EP in the non-Hermitian topological system, we study the sensitivity of the edge states in three different regimes around the EP. The robustness of edge states in a degenerating regime was recently demonstrated experimentally in a waveguide array with passive PT symmetry [66]. However, the EP property of edge states in non-Hermitian systems has not been considered. Here, we quantitatively study the sensitivity of topological edge states. In particular, the EP for this system is expected to realize a new type of sensor. Within the context of coupled mode theory, the effective second-order non-Hermitian system realized by the two edge states in the topological dimer chain can be described by the effective Hamiltonian [78], Hee=(ω0+iγaκeeκeeω0−iγb),where κee denotes the effective coupling coefficient of two edge states. γa and γb are the gain and loss of the left and right states, respectively. The eigenvalues of Eq. (3) are ωe−=12[iγa−iγb−−(γa+γb)2+4κee2+2ω0],ωe+=12[iγa−iγb+−(γa+γb)2+4κee2+2ω0].Once γa+γb=2κee, two eigenstates coalesce, ωe−=ωe+, which corresponds to the EP. When a small perturbation ε affects the edge state on the right end of the chain, the effective Hamiltonian can be written as Hee′=(ω0+iγaκeeκeeω0−iγb+ε).The eigenvalues of Eq. (5) are ωe−′=12[iγa−iγb+ε−−(γa+γb+iε)2+4κee2+2ω0],ωe+′=12[iγa−iγb+ε+−(γa+γb+iε)2+4κee2+2ω0].Next, we study the sensitivity of the edge states in three different regimes around the EP. Considering the case γa=0.5κee,γb=0.5κee in the splitting region, the series expansion of δω=ωe+′−ωe−′ can be found as δω∼ε. Therefore, the response of edge states to the on-site frequency perturbation in the splitting region is linear. Similarly, considering γa=0.5κee,γb=2.5κee in the degenerating region, the series expansion of the frequency splitting of edge states can also be found as δω∼ε. However, for the EP (γa=0.5κee,γb=1.5κee), the series expansion of the frequency splitting of edge states can be found as δω∼ε, confirming the square-root behavior for small perturbations. Comparing the results of the edge states in the three regions, we can find that the edge states are more sensitive to the on-site frequency perturbation in EP than in the splitting and degeneracy regions. In addition, due to the increase of the incoherence effect, the asymmetric edge states in the degenerate region keep good localization along the left boundary of the dimer [76]; thus, the topological edge states in the degenerate region have stronger robustness to the perturbation on the right end of the chain than those in the splitting region.

    Measured frequency splitting of edge states on frequency detuning of the right resonator, which is controlled by the loaded capacitors. The results are given on a logarithmic scale. The green circles, blue triangles, and pink stars indicate results from the EP, degenerating region, and splitting region, respectively. Green, pink, and blue dashed lines with slopes of 1/2, 1, and 1, respectively, are displayed for reference.

    Figure 5.Measured frequency splitting of edge states on frequency detuning of the right resonator, which is controlled by the loaded capacitors. The results are given on a logarithmic scale. The green circles, blue triangles, and pink stars indicate results from the EP, degenerating region, and splitting region, respectively. Green, pink, and blue dashed lines with slopes of 1/2, 1, and 1, respectively, are displayed for reference.

    Frequency splitting as a function of disorder strength in the non-Hermitian dimer chain. The disorder is introduced by randomly moving four coil resonators 1.0, 1.5, 2.0, or 2.5 cm in the center of the chain. Each case is averaged by 20 realizations. Green, blue, and pink dashed lines indicate results from the EP, degenerating region, and splitting region, respectively. The standard deviation is represented by error bars.

    Figure 6.Frequency splitting as a function of disorder strength in the non-Hermitian dimer chain. The disorder is introduced by randomly moving four coil resonators 1.0, 1.5, 2.0, or 2.5 cm in the center of the chain. Each case is averaged by 20 realizations. Green, blue, and pink dashed lines indicate results from the EP, degenerating region, and splitting region, respectively. The standard deviation is represented by error bars.

    4. CONCLUSION

    In summary, using a finite non-Hermitian topological dimer chain, we study the sensitivity of edge states in three different regimes: the splitting regime, the EP, and the degenerating regime. According to conventional wisdom, the edge states in a topological structure are topologically protected, which makes them robust to structural perturbations. In this work, we show experimentally that the edge states in the degenerating regime after the EP can enhance topological protection in a finite system. However, this scenario breaks down at the EP, and the degenerating regime becomes very sensitive to perturbation at the end of the non-Hermitian chain. Our results for the EP of edge states not only improve understanding of the robustness of topological states but also provide a new scheme for designing a new type of sensor with topological protection against internal disturbances of site-to-site couplings and high sensitivity to boundary on-site frequency perturbations.

    References

    [1] D. Xiao, M. C. Chang, Q. Niu. Berry phase effects on electronic properties. Rev. Mod. Phys., 82, 1959-2007(2010).

    [2] M. Z. Hasan, C. L. Kane. Topological insulators. Rev. Mod. Phys., 82, 3045-3067(2010).

    [3] X. L. Qi, S. C. Zhang. Topological insulators and superconductors. Rev. Mod. Phys., 83, 1057-1110(2011).

    [4] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit. Photonic Floquet topological insulators. Nature, 496, 196-200(2013).

    [5] A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, G. Shvets. Photonic topological insulators. Nat. Mater., 12, 233-239(2013).

    [6] L. Lu, J. D. Joannopoulos, M. Soljačić. Topological photonics. Nat. Photonics, 8, 821-829(2014).

    [7] Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, O. Zilberberg. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett., 109, 106402(2012).

    [8] M. A. Bandres, M. C. Rechtsman, M. Segev. Topological photonic quasicrystals: fractal topological spectrum and protected transport. Phys. Rev. X, 6, 011016(2016).

    [9] Z. W. Guo, H. T. Jiang, Y. Sun, Y. H. Li, H. Chen. Asymmetric topological edge states in a quasiperiodic Harper chain composed of split-ring resonators. Opt. Lett., 43, 5142-5145(2018).

    [10] S. Longhi. Topological phase transition in non-Hermitian quasicrystals. Phys. Rev. Lett., 122, 237601(2019).

    [11] D. Varjas, A. Lau, K. Pöyhönen, A. R. Akhmerov, D. I. Pikulin, I. C. Fulga. Topological phases without crystalline counterparts. Phys. Rev. Lett., 123, 196401(2019).

    [12] W. P. Su, J. R. Schrieffer, A. J. Heeger. Solitons in polyacetylene. Phys. Rev. Lett., 42, 1698-1701(1979).

    [13] N. Malkova, I. Hromada, X. Wang, G. Bryant, Z. Chen. Observation of optical Shockley-like surface states in photonic superlattices. Opt. Lett., 34, 1633-1635(2009).

    [14] J. Jiang, Z. W. Guo, Y. Q. Ding, Y. Sun, Y. H. Li, H. T. Jiang, H. Chen. Experimental demonstration of the robust edge states in a split-ring-resonator chain. Opt. Express, 26, 12891-12902(2018).

    [15] J. Jiang, J. Ren, Z. W. Guo, W. W. Zhu, Y. Long, H. T. Jiang, H. Chen. Seeing topological winding number and band inversion in photonic dimer chain of split-ring resonators. Phys. Rev. B, 101, 165427(2020).

    [16] Y. Hadad, J. C. Soric, A. B. Khanikaev, A. Alù. Self-induced topological protection in nonlinear circuit arrays. Nat. Electron., 1, 178-182(2018).

    [17] D. A. Dobrykh, A. V. Yulin, A. P. Slobozhanyuk, A. N. Poddubny, Y. S. Kivshar. Nonlinear control of electromagnetic topological edge states. Phys. Rev. Lett., 121, 163901(2018).

    [18] Y. Wang, L. J. Lang, C. H. Lee, B. L. Zhang, Y. D. Chong. Topologically enhanced harmonic generation in a nonlinear transmission line metamaterial. Nat. Commun., 10, 1102(2019).

    [19] S. Kruk, A. Poddubny, D. Smirnova, L. Wang, A. Slobozhanyuk, A. Shorokhov, I. Kravchenko, B. Luther-Davies, Y. Kivshar. Nonlinear light generation in topological nanostructures. Nat. Nanotechnol., 14, 126-130(2019).

    [20] D. Smirnova, D. Leykam, Y. D. Chong, Y. Kivshar. Nonlinear topological photonics. Appl. Phys. Rev., 7, 021306(2020).

    [21] S. Xia, D. Jukić, N. Wang, D. Smirnova, L. Smirnov, L. Tang, D. Song, A. Szameit, D. Leykam, J. Xu, Z. Chen, H. Buljan. Nontrivial coupling of light into a defect: the interplay of nonlinearity and topology. Light Sci. Appl., 9, 147(2020).

    [22] Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, S. Iwamoto. Active topological photonics. Nanophotonics, 9, 547-567(2020).

    [23] H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, L. Feng. Topological hybrid silicon microlasers. Nat. Commun., 9, 981(2018).

    [24] M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, M. Khajavikhan. Edge-mode lasing in 1D topological active arrays. Phys. Rev. Lett., 120, 113901(2018).

    [25] Y. Ota, R. Katsumi, K. Watanabe, S. Iwamoto, Y. Arakawa. Topological photonic crystal nanocavity laser. Commun. Phys., 1, 86(2018).

    [26] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, D. N. Christodoulides. Non-Hermitian physics and PT symmetry. Nat. Phys., 14, 11-19(2018).

    [27] L. Feng, R. El-Ganainy, L. Ge. Non-Hermitian photonics based on parity–time symmetry. Nat. Photonics, 11, 752-762(2017).

    [28] K. Özdemir, S. Rotter, F. Nori, L. Yang. Parity–time symmetry and exceptional points in photonics. Nat. Mater., 18, 783-798(2019).

    [29] M. A. Miri, A. Alù. Exceptional points in optics and photonics. Science, 363, 7709(2019).

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

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

    [32] H. Xu, D. Mason, L. Jiang, J. G. E. Harris. Topological energy transfer in an optomechanical system with exceptional points. Nature, 537, 80-83(2016).

    [33] S. Assawaworrarit, X. F. Yu, S. H. Fan. Robust wireless power transfer using a nonlinear parity–time-symmetric circuit. Nature, 546, 387-390(2017).

    [34] H.-Z. Chen, T. Liu, H.-Y. Luan, R.-J. Liu, X.-Y. Wang, X.-F. Zhu, Y.-B. Li, Z.-M. Gu, S.-J. Liang, H. Gao, L. Lu, L. Ge, S. Zhang, J. Zhu, R.-M. Ma. Revealing the missing dimension at an exceptional point. Nat. Phys., 16, 571-578(2020).

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

    [36] A. U. Hassan, B. Zhen, M. Soljačić, M. Khajavikhan, D. N. Christodoulides. Dynamically encircling exceptional points: exact evolution and polarization state conversion. Phys. Rev. Lett., 118, 093002(2017).

    [37] X. L. Zhang, S. B. Wang, B. Hou, C. T. Chan. Dynamically encircling exceptional points: in situ control of encircling loops and the role of the starting point. Phys. Rev. X, 8, 021066(2018).

    [38] Q. J. Liu, S. Y. Li, B. Wang, S. L. Ke, C. Z. Qin, K. Wang, W. W. Liu, D. S. Gao, P. Berini, P. X. Lu. Efficient mode transfer on a compact silicon chip by encircling moving exceptional points. Phys. Rev. Lett., 124, 153903(2020).

    [39] J. Wiersig. Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection. Phys. Rev. Lett., 112, 203901(2014).

    [40] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-ganainy, D. N. Christodoulides, M. Khajavikhan. Enhanced sensitivity at higher-order exceptional points. Nature, 548, 187-191(2017).

    [41] W. J. Chen, K. Özdemir, G. M. Zhao, J. Wiersig, L. Yang. Exceptional points enhance sensing in an optical microcavity. Nature, 548, 192-196(2017).

    [42] P. Y. Chen, M. Sakhdari, M. Hajizadegan, Q. S. Cui, M.-C. Cheng, R. El-Ganainy, A. Alù. Generalized parity–time symmetry condition for enhanced sensor telemetry. Nat. Electron., 1, 297-304(2018).

    [43] S. B. Wang, B. Hou, W. X. Lu, Y. T. Chen, Z. Q. Zhang, C. T. Chan. Arbitrary order exceptional point induced by photonic spin-orbit interaction in coupled resonators. Nat. Commun., 10, 832(2018).

    [44] K. Ding, G. C. Ma, Z. Q. Zhang, C. T. Chan. Exceptional demonstration of an anisotropic exceptional point. Phys. Rev. Lett., 121, 085702(2018).

    [45] Z. Dong, Z. P. Li, F. Yang, C.-W. Qiu, J. S. Ho. Sensitive readout of implantable microsensors using a wireless system locked to an exceptional point. Nat. Electron., 2, 335-342(2019).

    [46] Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, K. Özdemir, R. El-Ganainy. Sensing with exceptional surfaces in order to combine sensitivity with robustness. Phys. Rev. Lett., 122, 153902(2019).

    [47] Z. C. Xiao, H. N. Li, T. Kottos, A. Alù. Enhanced sensing and nondegraded thermal noise performance based on PT-symmetric electronic circuits with a sixth-order exceptional point. Phys. Rev. Lett., 123, 213901(2019).

    [48] Y. J. Zhang, H. Kwon, M.-A. Miri, E. Kallos, H. Cano-Garcia, M. S. Tong, A. Alù. Noninvasive glucose sensor based on parity-time symmetry. Phys. Rev. Appl., 11, 044049(2019).

    [49] J.-H. Park, A. Ndao, W. Cai, L. Hsu, A. Kodigala, T. Lepetit, Y.-H. Lo, B. Kanté. Symmetry-breaking-induced plasmonic exceptional points and nanoscale sensing. Nat. Phys., 16, 462-468(2020).

    [50] J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, A. Szameit. Observation of a topological transition in the bulk of a non-Hermitian system. Phys. Rev. Lett., 115, 040402(2015).

    [51] C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, H. Schomerus. Selective enhancement of topologically induced interface states in a dielectric resonator chain. Nat. Commun., 6, 6710(2015).

    [52] S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, A. Szameit. Topologically protected bound states in photonic parity–time-symmetric crystals. Nat. Mater., 16, 433-438(2017).

    [53] S. Y. Yao, Z. Wang. Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett., 121, 086803(2018).

    [54] B. K. Qi, L. X. Zhang, L. Ge. “Defect states emerging from a non-Hermitian flatband of photonic zero modes. Phys. Rev. Lett., 120, 093901(2018).

    [55] K. Takata, M. Notomi. Photonic topological insulating phase induced solely by gain and loss. Phys. Rev. Lett., 121, 213902(2018).

    [56] S. Longhi, L. Feng. Non-Hermitian multimode interference. Opt. Lett., 45, 1962-1965(2020).

    [57] S. Lieu. Topological phases in the non-Hermitian Su–Schrieffer–Heeger model. Phys. Rev. B, 97, 045106(2018).

    [58] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, M. Ueda. Topological phases of non-Hermitian systems. Phys. Rev. X, 8, 031079(2018).

    [59] A. Cerjan, M. Xiao, L. Yuan, S. Fan. Effects of non-Hermitian perturbations on Weyl Hamiltonians with arbitrary topological charges. Phys. Rev. B, 97, 075128(2018).

    [60] S. Liu, S. J. Ma, C. Yang, L. Zhang, W. L. Gao, Y. J. Xiang, T. J. Cui, S. Zhang. Gain- and loss-induced topological insulating phase in a non-Hermitian electrical circuit. Phys. Rev. Appl., 13, 014047(2020).

    [61] Z. W. Guo, H. T. Jiang, Y. H. Li, H. Chen, G. S. Agarwal. Enhancement of electromagnetically induced transparency in metamaterials using long range coupling mediated by a hyperbolic material. Opt. Express, 26, 627-641(2018).

    [62] Z. W. Guo, H. T. Jiang, H. Chen. Hyperbolic metamaterials: from dispersion manipulation to applications. J. Appl. Phys., 127, 071101(2020).

    [63] W. Gao, X. Y. Hu, C. Li, J. H. Yang, Z. Chai, J. Y. Xie, Q. H. Gong. Fano-resonance in one-dimensional topological photonic crystal heterostructure. Opt. Express, 26, 8634-8644(2018).

    [64] F. Zangeneh-Nejad, R. Fleury. Topological Fano resonances. Phys. Rev. Lett., 122, 014301(2019).

    [65] Q. S. Huang, Z. W. Guo, J. T. Feng, C. Y. Yu, H. T. Jiang, Z. Zhang, Z. S. Wang, H. Chen. Observation of a topological edge state in the X-ray band. Laser Photon. Rev., 13, 1800339(2019).

    [66] W. Song, W. Z. Sun, C. Chen, Q. H. Song, S. M. Xiao, S. N. Zhu, T. Li. Breakup and recovery of topological zero modes in finite non-Hermitian optical lattices. Phys. Rev. Lett., 123, 165701(2019).

    [67] W. G. Song, W. Z. Sun, C. Chen, Q. H. Song, S. M. Xiao, S. N. Zhu, T. Li. Robust and broadband optical coupling by topological waveguide arrays. Laser Photon. Rev., 14, 1900193(2020).

    [68] J. Ningyuan, C. Owens, A. Sommer, D. Schuster, J. Simon. Time- and site-resolved dynamics in a topological circuit. Phys. Rev. X, 5, 021031(2015).

    [69] V. V. Albert, L. I. Glazman, L. Jiang. Topological properties of linear circuit lattices. Phys. Rev. Lett., 114, 173902(2015).

    [70] S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling, F. Schindler, C. H. Lee, M. Greiter, T. Neupert, R. Thomale. Topolectrical-circuit realization of topological corner modes. Nat. Phys., 14, 925-929(2018).

    [71] Y. Li, Y. Sun, W. W. Zhu, Z. W. Guo, J. Jiang, T. Kariyado, H. Chen, X. Hu. Topological LC-circuits based on microstrips and observation of electromagnetic modes with orbital angular momentum. Nat. Commun., 9, 4598(2018).

    [72] T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, R. Thomale. Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits. Nat. Phys., 16, 747-750(2020).

    [73] X. F. Zhang, K. Ding, X. J. Zhou, J. Xu, D. F. Jin. Experimental observation of an exceptional surface in synthetic dimension with magnon polaritons. Phys. Rev. Lett., 123, 237202(2019).

    [74] F. X. Zhang, Y. M. Feng, X. F. Chen, L. Ge, W. J. Wan. Synthetic anti-PT symmetry in a single microcavity. Phys. Rev. Lett., 124, 053901(2020).

    [75] J. Feis, C. J. Stevens, E. Shamonina. Wireless power transfer through asymmetric topological edge states in diatomic chains of coupled meta-atoms. Appl. Phys. Lett., 117, 134106(2020).

    [76] L. Zhang, Y. H. Yang, Z. Jiang, Q. L. Chen, Q. H. Yan, Z. Y. Wu, B. L. Zhang, J. T. Huangfu, H. S. Chen. Topological wireless power transfer(2020).

    [77] J. Song, F. Q. Yang, Z. W. Guo, Y. Q. Chen, H. T. Jiang, Y. H. Li, H. Chen. One-dimensional topological quasiperiodic chain for directional wireless power transfer(2020).

    [78] J. Song, F. Q. Yang, Z. W. Guo, X. Wu, K. J. Zhu, J. Jiang, Y. Sun, Y. H. Li, H. T. Jiang, H. Chen. Wireless power transfer via topological modes in dimer chains. Phys. Rev. Appl., 15, 014009(2021).

    [79] T. Hofmann, T. Helbig, F. Schindler, N. Salgo, M. Brzezińska, M. Greiter, T. Kiessling, D. Wolf, A. Vollhardt, A. Kabaši, C. H. Lee, A. Bilušić, R. Thomale, T. Neupert. Reciprocal skin effect and its realization in a topolectrical circuit. Phys. Rev. Res., 2, 023265(2020).

    [80] S. Weidemann, P. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, A. Szameit. Topological funneling of light. Science, 368, 311-314(2020).

    [81] V. Peano, M. Houde, F. Marquardt, A. A. Clerk. Topological quantum fluctuations and traveling wave amplifiers. Phys. Rev. X, 6, 041026(2016).

    [82] J. Ran, Y. W. Zhang, X. D. Chen, H. Chen. Frequency mixer based on Doppler effect. IEEE Microw. Wireless Compon. Lett., 28, 43-45(2018).

    [83] Y. Sun, W. Tan, H. Q. Li, J. Li, H. Chen. Experimental demonstration of a coherent perfect absorber with PT phase transition. Phys. Rev. Lett., 112, 143903(2014).

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    Zhiwei Guo, Tengzhou Zhang, Juan Song, Haitao Jiang, Hong Chen. Sensitivity of topological edge states in a non-Hermitian dimer chain[J]. Photonics Research, 2021, 9(4): 574
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