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Abstract
We investigate the flat bands in a quasi-one-dimensional rhombic array composed of evanescently coupled microring resonators (MRRs) with non-Hermitian coupling. By changing the relative position of non-Hermitian coupling in each cell, we construct topologically trivial and nontrivial flat bands, where both the real and imaginary parts of energy bands become flat and coalesce into a single band. We show the nontrivial systems are able to support topological boundary modes isolated from the flat bulk bands although there is no band gap. The elusive topology of flat bands can be geometrically visualized by plotting the trajectories of their eigenvectors on Bloch sphere based on Majorana’s stellar representation (MSR). Furthermore, we perform a full wave simulation and show the characteristics of flat bands, associated compact localized modes, and boundary modes are reflected from absorption spectra and field intensity profiles. The study may find potential applications in lasers, narrowband filters, and efficient light harvesting.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The ability to manipulate and steer the flow of light is of fundamental significance and may find great applications in integrated photonics [1,2]. Inspired by condensed matter physics, the concept of band structure is developed in photonics, and the effective control of propagation of light can be achieved by band structure engineering [3–5]. In particular, flat bands, the completely dispersionless energy bands that extend in the entire Brillouin zone, are receiving increasing theoretical and experimental exploration, which are desired in diffraction-less wave propagation, imaging, efficient light trapping, and slow light in aid of zero-group velocity [6–11]. The flat-band localization is usually induced by geometric frustration and the typical cases are reported in rhombic, Kagome, Dice, and Lieb lattices [8,12], different from dynamical [13] and Anderson localization [14,15]. The combination of flat bands and topological phases give rise to novel real-space topology, such as unconventional line states and noncontractible loop states [7,16,17]. Furthermore, the interplay of synthetic gauge fields and certain types of lattice leads to Aharonov-Bohm (AB) caging phenomenon where all bands dramatically develop into flat bands and all states become completely localized as a result of destructive interference [18–22]. The photonic analogies of AB cages are experimentally observed in evanescently coupled waveguides with exhibiting non-quantized bulk topology, whose elusive topological property inherits from a Su-Schrieffer-Heeger model by taking the square-root operation [20].
More recently, the study of flat bands has been extended to non-Hermitian systems in the presence of onsite gain and loss or non-Hermitian couplings [23–28]. The energy bands in non-Hermitian systems are generally complex, but can be real under parity-time (PT) or pseudo-Hermiticity symmetry [29]. Many abnormal optical characteristics are revealed near exceptional points (EPs) where two or more energy bands as well as their eigenstates collapse simultaneously [26]. In non-Hermitian systems, a flat band can be referred to as dispersionless real or imaginary part of band structure or their both [6,30]. The non-Hermiticity provides new way to realize and an additional degree to manipulate flat bands. The simplest non-Hermitian flat bands are directly generated by introducing non-Hermiticity to original Hermitian flat bands. PT-symmetric flat bands are proposed in a rhombic waveguide array, where the associated compact localized modes can be stimulated at arbitrary positions, in contrast to Hermitian cases limited to certain excited channels [31]. Specifically, fine-tuned non-Hermitian couplings give rise to a synthetical gauge flux and the completely flat bands are reported at EPs in a rhombic lattice, which is a non-Hermitian generation of AB cage with gauge fields replaced by balanced gain and loss [32,33]. The intriguing non-Hermitian flat bands can be implemented in coupled waveguide array with topological aspect exactly originating from its squared counterpart [34]. In addition, a local excitation at different input waveguides in this singular flat band displays polynomial power increase while maintaining localized characteristics [35].
In this work, we present another scenario by utilizing evanescently coupled microring resonators (MRRs) to construct non-Hermitian flat bands. The MRRs are able to provide strong confinement of optical fields with the advantage of small size, low energy consumption, and high scalability with other integrated photonic circuits [36–43]. They are widely utilized to demonstrate nontrivial topological phenomenon in aid of flexibly controlled coupling and synthetic gauge field [36,44–51]. The coupling in MRRs can be tuned by varying spatial space and the integrated non-Hermiticity is compatible by current nano fabrication and lasing technology [41]. Here, we utilize transfer matrix approach to analytically derive the non-Hermitian coupling in indirectly coupled MRRs, which agrees with the adiabatic elimination [52–56]. Subsequently, we design topologically trivial and nontrivial flat bands utilizing MRRs arranged in rhombic lattices. They exactly have same energy bands under periodic boundary condition but different open spectra. The real and imaginary parts of band structure of two systems become flat and coalesce into third-order EPs, leading to ill definition of topological invariants via spectral winding and Zak phase. On the contrary, the topology of flat band can be reflected by a geometric representation by plotting their eigenstates on the Bloch sphere based on Majorana’s stellar representation (MSR) [57–61], which are accommodated with the explanation of square-root transformation. Moreover, we perform full wave simulations and show the characteristics of the topological edge modes, flat bands, and associated compact localized states can be observed in absorption spectra and intensity profiles under extra excitation.
2. Theoretical model for a non-Hermitian flat band
We firstly study topologically nontrivial flat bands, whose tight-binding geometry is schematically illustrated in Fig. 1(a), arranged in a rhombic lattice. Each unit cell is composed of three sites labelled as A, B, and C. Site A is coupled to its neighbor site B (C) within the same unit cell with intracell coupling denoted as t. The intercell coupling between B (C) and A is denoted as ±it. Such a scheme acts a non-Hermitian analogy of AB cages as light travelling from different paths will experience different phases. Between adjacent central (A) sites, waves coupling via the upper path accumulate a phase θ = arg(−it*t) = −π/2, while coupling via the lower path accumulates an equal and negative phase −θ. Considering a loop, the scheme emulates a net magnetic flux π per plaquette, forming a non-Hermitian AB cage. The corresponding complex energy bands consequently all flatten out, demonstrating the existence of non-Hermitian flat bands.
Fig. 1. The lattice, eigenvalues, and topological explanation of non-trivial flat band. (a) Schematic of the tight-binding lattice. (b) The energy spectra for a lattice with 13 unit cell with open boundary. Re(E) and Im(E) are plotted in red and blue, respectively. Four edge modes appear at E= ±√2t and ±√2it (highlighted in red dots and blued dots). (c) The trajectory of MSR. (d) The lattice of corresponding H2 model. (e) The real (red) and imaginary (blue) energy spectrum for the corresponding H2 model. Four edge modes appear at E = ±2t2 (highlighted in red dots). (f) The trajectory of the eigenvectors of H2 model on Bloch sphere.
The characteristic of flat bands can be easily obtained by solving the eigenvalue of the system Hamiltonian. The Hamiltonian under periodic boundary condition is written as
It is very important to seek topological invariants to predict the existence of boundary modes. However, the conventional topological invariants of winding number and Zak phase defined by eigenvectors are ill-defined in the non-Hermitian flat bands due to the existence of EPs. Here we show this problem is conquered by utilizing the approach of MSR. MSR is firstly applied to identify the topology of Hermitian systems with multiple bands by transforming an n-band eigenstate into n−1 MSs locating on a Bloch Sphere, each standing for a different spin-1/2 state [57–59]. The method is further extended to non-Hermitian systems [60]. The main point is to figure out the polar and azimuthal angles of MS on a Bloch sphere with the spherical coordinates (1, θ, ϕ), which are determined by the equation [60]
The term Cl(φ) (l = 1, 2, 3) stands for the component of eigenvector |Ψ 〉. In our flat band systems, there is only a single eigenvector due to EPs and thus we only have to calculate one decomposition. By solving Eq. (2), we acquire two roots x at a given φ, which relate to Bloch sphere as xm = tan (θm/2)exp(iϕm) with m = 1, 2, each serving as a MS on a Bloch sphere. A full MSR can be calculated by tracing each MS of xm(φ) on the Bloch sphere via changing φ throughout the Brillouin zone. Subtracting eigenvectors into Eq. (2), we have ${x_{1,2}} = \sqrt 2 \tan ({\varphi /2\; \; \pi /4} )\textrm{exp} ({ - i\pi /2} )$. The corresponding trajectory of MSs is plotted in Fig. 1(c). As φ varies from 0 to 2π, we find MSs form a closed loop passing through north and south poles on the sphere and enclosing the x-axis. The winding number is utilized as a parameter to describe topological invariant. Here we define a winding number for MSs as the total winding of their polar angels, given by
Our explanation agrees well with the previous study to reveal the topological aspects of the model by taking the square of the Hamiltonian matrix (1), which yields [20,21]
The squared Hamiltonian is blocking diagonal and the equivalent lattice model is displayed in Fig. 1(d), representing a single band isolated from a dimer lattice with long-range couplings. In Fig. 1(e), we plot the open-boundary spectra and one can see four topological edge modes are gapped from bulk bands with energy E = ± 2t2. The Hamiltonian H share the same eigenvectors with H2 but the square root of its eigenvalues and thus referred as square-root topological insulator. Alternatively, we can also understand the geometrical meaning of the topological invariant from trajectories of right eigenvectors |ψ2,3〉 of lower block by directly projecting the eigenvectors onto a 2D unit spherical surface [62]. Applying a rotation exp[i(φ/2 − π/4)], the eigenvector can be parametrized as |Ψ2,3〉 = (cos(θ/2), eiφsin(θ/2))T with θ and ϕ corresponding to the azimuthal and polar angles of sphere coordinate, which are analytically calculated by Eq. (4), that is, θ = 3π/2 − φ and ϕ = π/2. In Fig. 1(f), we plot the evolution of the vector |ψ2,3〉 on the Bloch sphere across the Brillouin zone. The trajectory forms a closed curve in accordance with that shown in Fig. 1(c). The winding number is the same as that in Fig. 1(c).
We now propose the tight-binding model for topologically trivial flat bands and show the trajectory of eigenvectors collapse into a single point on Bloch sphere. As shown in Fig. 2(a), we exchange the nearest couplings of four lattices per plaquette and maintain π magnetic flux per plaquette. The Bloch Hamiltonian is transformed into
The periodic eigenvalue spectra are the same as that of the nontrivial system. Both the real and imaginary parts of band structure are dispersionless across the whole Brillouin zone with E1,2,3 = 0. Under open boundary condition, the bulk spectrum remains flat. However, there are no edge modes, as shown in Fig. 2(b). The corresponding eigenvectors are given by |Ψ1,2,3〉 = (0, 1, i)T/2. Substituting this into Eq. (2), we arrive at ${x_{1,2}} ={\pm} \sqrt 2 i$ for trivial system, which are independent of Bloch momentum. As a result, the trajectory of MSs on the sphere just forms two fixed points, as shown in Fig. 2(c). We also explore its square counterpart. Squaring Hamiltonian Htrivial yields
Fig. 2. (a) Schematic of the trivial lattice. (b) The real (blue) and imaginary (red) energy spectra for a lattice with 13 unit cell with open boundary. (c) The Majorana stars of the eigenvectors on Bloch sphere. (d) The corresponding H2 model. (e) The real (red) and imaginary (blue) energy spectrum for the corresponding H2 model. (f) The trajectory of the eigenvectors of H2 model on Bloch sphere.
Here we only discuss two special cases with flat bands in rhombic lattice. The flat bands appear in a more general condition as if the four couplings within a unit cell fulfil certain condition [32]. A general rule to construct non-Hermitian flat band is discussed in [35]. There are three approaches, including spontaneous restoration of non-Hermitian particle-hole symmetry, deviation from a Hermitian system, and a compact Wannier function approach.
We now study the robustness of topological edge modes following the method proposed in [20]. Three kinds of disorders are considered, including the disorder of onsite detuning d on site A, the disorder of coupling α connecting sites B and C, and the presence of both kinds of disorder, as shown in the insert of Fig. 3(a). For all cases, we assume the distribution of disorder has a Gaussian form with standard deviation σ and a vanish mean (〈α〉 = 〈d〉 = 0). The topological protection against disorder can be quantified by the energy offset of edge modes and their corresponding localization length over many disorder realizations. In Fig. 3(a), we plot the average energy offset $\langle{\left|{|{{E_{edge}}/t} |- \sqrt 2 } \right|^2}\rangle$ of four edge modes as a function of disorder strength σ/t. Each point is the average of 1000 calculations with zero average disorder. For the real onsite disorder d (black line) or the real coupling disorder α (red dashed line), we see the energy offset slightly changes as long as the disorder is not too large. At the same time, their corresponding localization length, characterized by the average second momentum [15,10], does not experience prominent change, as shown in Fig. 3(b). These results indicate the edge modes are indeed robust against certain disorder. In contrast, when both types of disorders are introduced at the same time with complex coupling α [Re(〈α〉) = Im(〈α〉) = 〈d〉 = t], the energy offset and localization length increase rapidly with the increase of disorder strength, implying the edge modes are not protected against this general disorder. As pointed in [60], the topological phase transition can emerge without gap closing. The edge states here are not protected by a band gap but only have a topological origin characterized by the winding number based on MSR. The symmetry that protects the edge modes remains an important topic for future exploration.
Fig. 3. The influence of different kinds of disorders on topological edge modes. (a) The average energy offset of edge modes as a function disorder strength σd/t with 〈d〉 = 0. The insert shows the schematic of disorder. (b) The average localization length of edge modes as a function of disorder strength corresponding to (a). The black, red, and blue lines represent the onsite disorder d, coupling disorder α, and both of them, respectively. Each disorder strength is calculated by 1000 times. The total site number of lattices is N = 92.
3. Non-Hermitian couplings in microring resonators
We now utilize indirectly coupled MRRs to implement the proposed non-Hermitian flat bands. The geometry is illustrated in Fig. 4(a) where the site rings are evanescently coupled to their nearest neighbors using a set of auxiliary link rings. We engineer the link rings to acquire different kinds of couplings. As shown in Fig. 4(b), the real coupling t is directly induced by connecting two site rings via a lossless link ring, whose magnitude is determined by the gap distance. In order to realize non-Hermitian couplings, we introduce gain and loss modulation into ring resonators. The positive imaginary coupling it is achieved by setting two site rings with gain (red) and a link ring with loss (black). If we reverse the distribution of gain and loss, the negative imaginary coupling − it can be realized. Note that all the site rings are resonant, the link rings, however, are designed to be anti-resonant (resonant) as the coupling is real-valued (imaginary-valued).
Fig. 4. Schematic of the set-up used to implement non-Hermitian flat band. (a) Simplified sketch illustration of a 1D rhombic chain of coupled MRRs. Each unit cell consists three site rings, evanescently coupled by four link rings. The present gain and loss distribution could reproduce the lattice shown in Fig. 1(a). (b) Coupling model of MRRs to realize real and non-Hermitian couplings. (c) Schematic of a finite 1D array of ring resonators to demonstrate the transfer matrix approach to simulate the transmission characteristic of the system.
We now utilize the transfer matrix method (TMM) to analyze the coupling in 1D MRRs, as shown in Fig. 4(c). Assume each unit is composed of a site ring (black) and a link ring (red), whose length are LSR and LLR, respectively. The propagation modes in link rings are supposed to be clockwise in our calculation and hence the amplitude is amplified by eh1 times as light traveling across half perimeter. Meanwhile, light that propagates in the site ring is decreased by eh2 times. The symbols ɛa, ɛb, …, ɛl sign the wave amplitude at respect positions. The adjacent region between the two rings behaves as a directional coupler. Then, the wave amplitudes between different resonators at the same position are related by TMM
The determinant should be vanished for non-zero solution, that is,
We discuss in two cases. The first case is for real coupling where link rings have longer length than site rings with βLLR − βLSR = 2mπ + π, where β represents the propagation constant of waveguide. The extra length makes link rings anti-resonant such that they simply behave as connecting waveguides. The dispersion relation is acquired by substituting this relation into Eq. (9)
The real coupling fulfills h1 = h2 = 0 with h1 (h2) denoting on-site gain (loss). In the vicinity of resonance of site ring, we have sin(βLSR) ≈ (ω − ω0)LSR/νg with ω0 and νg representing resonant frequency and group velocity, respectively. By denoting coupling J = k2νg/(2LSR), then Eq. (10) is simplified to
The other case is for imaginary coupling as the site and link rings are with equal lengths satisfy βLLR − βLSR = 2mπ. Substituting this into Eq. (9) yields
Equation (12) is simplified to
For large h1 or h2, Eq. (13) is simplified to
The imaginary parameter before cosφ in Eq. (14) implies that the coupling becomes imaginary and the last two terms are the imaginary parts of eigenfrequencies, which can be eliminate by suitably choose h1 and h2. For positive imaginary coupling, we have to introduce gain into site ring and loss into link rings. For negative imaginary coupling, the distribution of gain and loss is reversed. For the ring array, the site rings corresponding to lattice A should be lossless as they connect positive and negative imaginary couplings at the same time. The onsite detuning cancels out. Therefore, the arrangement of gain and loss in Fig. 4(a) exactly reproduces the lattice shown in Fig. 1(a). For the trivial lattice, the gain and loss distribution should be altered.
We now determine the real and imaginary coupling via numerical simulation, which are carried out by COMSOL Multiphysics based on FEM. The practical optical design of real and imaginary couplings is based on the system of two site rings connected by a link ring, as shown in Fig. 4(b). The parameters used in the simulation are given as below. The core width of waveguides is w = 0.27 µm and its refractive index is n0 = 3 with cladding being nair = 1 for simplicity. The gap between site and link rings is d = 0.34 µm. The lengths of horizontal and vertical arms of site ring are L1 = L2 = 8 µm with bending radius of corner R = 3 µm. The resonant frequency of site ring is determined to be ω0 = 196.054 + 5.64 × 10−4i THz, with degenerated clockwise and counterclockwise propagating modes. The real and imaginary couplings can be figured out by calculating eigenfrequencies of a system composed of two site rings connected by a link ring. For real coupling, the extra length η = π/(2neffk0) = 0.175 µm is introduced to the horizontal arm of link ring. The arm length of link ring is the same as that of site ring in the case of imaginary coupling. Then, the real coupling strength is determined to be J = (ω+ − ω−)/2 = 11.1 GHz by the difference of two modes. In order to realize positive imaginary coupling it = 11.1 GHz, the gain-loss coefficients, that is, the imaginary part of refractive index of site and link rings are chosen as γ1 = −2.25 × 10−4 and γ2 = 0.0152, which are determined by the difference of imaginary part of eigenfrequencies. The dissipation added in the link ring greatly exceeds that in the site rings, which is consistent with the results calculated by the adiabatic elimination [52].
To further verify whether the real and imaginary couplings are exactly induced and whether their absolute values are equal, we compare the simulated band structures with tight-binding results. Figure 5(a) illustrates the equivalent lattice model of 1D MRR array with real coupling. The real and imaginary parts of band structures are plotted in Figs. 5(b) and 5(c) as h1 = h2 = 0. The curve Re(ω) oscillates sinusoidally according to ω0 + 2Jcos(ϕ), while Im(ω0) remains unchanged. The theoretical indications (lines) and simulations (dots) agree well with each other, which implies the real coupling J = 11.1 GHz. For imaginary coupling it in Fig. 5(d), we introduce gain coefficient h1 and loss coefficient h2 into site and link rings, meanwhile all the rings are resonant. The band structure becomes complex. To figure out the corresponding coefficient h1 and h2, we should first obtain the relationship between parameters γ and h. As h1 = h2 = 0, the values of Re(ω1) and Re(ω2) are corresponding to φ = 0 and φ = π, according to Eq. (13). Then the parameters can be calculated by tc/k = {Re(ω2) − Re(ω1)}/4J, which is figured out to be k = 0.204, tc = 0.979. The extracted value of Im(ω) reaches maximum at φ = π/2 and the coefficient h (h1 = h, h2 = − h) is calculated by h = 0.5asinh{k2Im(ω3)/(2Jtc2)}, according to Eq. (10). Thus, we get h1 = − 0.0259 and h2 = 0.7216. Substituting all the parameters into Eq. (14), we plot Re(ω) and Im(ω) in the first Brillion zone, as shown in Fig. 5(e) and Fig. 5(f), which are consistent with the simulated results (dots).
Fig. 5. Band structures in 1D Hermitian and non-Hermitian MRR arrays. (a), (b), and (c) are the effective lattice, Re(ω), and Im(ω) for Hermitian case. (d), (e), and (f) are for non-Hermitian array.
4. Simulation of trivial and nontrivial flat bands
In the above section, we show how to achieve non-Hermitian coupling in MRRs. Next, we explore non-Hermitian flat bands in rhombic MRR arrays by analyzing the simulation results of topologically nontrivial and trivial flat bands.
Figure 6(a) illustrates the schematic diagram of the nontrivial lattice, where S1, S2, and S3 denote different input couplers. The distance between couplers and MRR array is denoted as d1. Figures 6(b) and 6(c) plot the simulated real and imaginary parts of eigenfrequencies under open boundary. We observe eight edge modes isolated from the continuous flat bulk spectra. The number of boundary modes is doubled compared with theoretical indication shown in Fig. 1(b) because of the degenerate clockwise and counterclockwise modes. Figures 6(d)–6(f) plot the absorption spectra versus incident frequency ω for three different gap distances as light is injected from S1, S2 and S3, respectively. The gap distances will determine whether the external coupler and the arrays are over, under, or critical coupled, which leads to different absorption spectra. In Fig. 6(d), the boundary modes with real energy $E ={\pm} \sqrt 2 t$ are stimulated. As a result, we observe two absorption peaks at ω = 196.048 and 196.051 THz. Their difference is about $2\sqrt 2 t$ for different d1. The related field distributions are plotted in Fig. 6(g), which are well confined at three site rings at the left termination. In contrast, we only see one prominent absorption peaks at each d1 for center injection from S2, as shown in Fig. 6(e). This indicates the flat-band characteristic as all bulk modes are almost degenerate at single frequencies. The fields are accumulated at the excited MRR and its nearest neighboring seven site resonators [Fig. 6(h)]. There is no energy at all in further resonators, clearly displaying the flat-band localization as a result of destructive interference. For injection from right termination, the boundary modes with imaginary energy $E\; ={\pm} \sqrt 2 it$ are aroused, which correspond to amplified and attenuated modes. The amplified mode will dominate after a long time of evolution when both modes are initially excited. As a consequence, the absorption spectra are negative and have resonant dips at ω = ω0, as displayed in Fig. 6(f). The fields are also well confined at three site rings at right boundary [Fig. 6(i)]. The flatness of energy band is affected by a lot of undesired factors. The method to realize non-Hermitian coupling using auxiliary ring is an approximate approach, which naturally leads to some dispersive feature on the energy bands. Theoretically, the onsite gain and loss incorporating on auxiliary rings should greatly exceed coupling strength to drive them off resonance. However, the onsite gain and loss coefficient is only about ten times larger than the coupling strength. The difference between real and imaginary coupling strength also affects the flatness. Furthermore, the tight-binding condition may be also not strictly satisfied. In addition, the coupling between clockwise and counter-clockwise modes affected the flatness of the energy band.
Fig. 6. Simulation of edge and bulk modes from topologically non-trivial flat band. (a) Schematic of 1D non-trivial lattice. The input waveguide couplers are at three different locations denoted S1, S2 and S3. (b) and (c) plot the real and imaginary parts of eigenfrequencies, respectively. (d), (e) and (f) show the simulated power absorption spectra with three different gap distances as light is injected from S1, S2 and S3, respectively. In all cases, the gap distance between the input coupler and the array is d1 = 0.32, 0.34, and 0.4 µm. (g), (h) and (i) show the simulated spatial intensity distributions (|E|) at resonant peaks indicated by the arrows in (d)−(f).
We now study the trivial non-Hermitian flat band whose schematic of tight-binding lattice is shown in Fig. 7(a). Both the real and imaginary parts of eigenfrequencies are almost flat around ω0 and there are no topological edge modes in the system, as illustrated in Figs. 7(b) and 7(c). In Fig. 7(a), S1, S2, and S3 indicate the injection port. We sweep the input frequency and the corresponding absorption spectra from three different ports are plotted in Figs. 7(d)–7(f). In all cases, the absorption spectra have only single prominent resonant peak at ω = 196.05 THz, implying the flat-band characteristics. We further plot the field profiles at the absorption peaks as gap d1 = 0.34 µm, as shown in Figs. 7(g)–7(i). The field distributions excited from left and right terminations are similar. One can see most light waves are confined at outermost three terminations nearest to the disc waveguide. We emphasize that this edge confinement is a result of flat-band feature via destructive interference rather than topological localization. We have examined the field distributions of eigenmodes for the trivial case and there are no modes localized at two boundaries. For center injection from S2, light is trapped at the initially excited resonator and its nearest resonators, similar to the AB caging phenomena in Hermitian case.
Fig. 7. Simulation of bulk modes in trivial flat-band system. (a) Schematic of lattice model with trivial topology. The locations of input waveguide couplers are denoted as S1, S2 and S3. (b) and (c) are the simulated real and imaginary parts of eigenvalues. (d), (e), and (f) show the simulated power absorption spectra with excitation from S1, S2, and S3, respectively. The gap distance d1 = 0.32, 0.34, and 0.4 µm. (g)-(i) show the simulated spatial intensity distributions (|E|) at the absorption peak indicated by the arrows in (d)−(f) as d1 = 0.34 µm, respectively.
The 2D simulation is used in the manuscript. For 3D simulation, one should consider the realistic structures in the vertical direction, such as the height of resonators, the condition for single mode, and the influence of substrate. Although we have used 2D simulation, it can well predict the full wave dynamics, which should be consistent with 3D simulation after setting same effective refractive index in 2D. According to [63], the proposed model may be implemented in ring resonator arrays made of InGaAsP multiple–quantum well, which is fabricated using electron-beam lithography and plasma etching techniques. The required profile of gain and loss is achieved by laser pumping technology.
5. Summary
In conclusion, we have studied the non-Hermitian flat bands in a rhombic array composed of MRRs where both the real and imaginary parts of energy bands become dispersionless and coalesce into third-order EPs across whole Brillouin zone. The flat bands are implemented via suitably tuning the non-Hermitian coupling to generate synthetical π flux, which is a non-Hermitian analog of AB cage. We utilize the transfer matrix method to analytically derive how to realize non-Hermitian coupling in indirectly coupled MRRs, and the result is consistent with the adiabatic eliminating. By adjusting the position of non-Hermitian coupling in each unit cell, we achieve topologically trivial and nontrivial flat bands, which can be distinguished by whether they are able to support topological boundary states. Furthermore, the topology of flat bands can be geometrically visualized by tracing their eigenvectors on Bloch sphere based on MSR. The trajectory of nontrivial flat band forms a closed loop encircling the x axis while that of trivial band is just a fixed point. The results are accommodated with the previous explanation by taking a square operation of original Hamiltonian. We further show the characteristics of flat bands, associated compact localized modes, and boundary modes are reflected from absorption spectra and field intensity profiles performed by full wave simulation. Our results are helpful to understand the topological origin of non-Hermitian flat bands and may promise potential applications in lasers and efficient light harvesting.
Funding
Guangdong Major Project of Basic and Applied Basic Research (2019B030302003); National Natural Science Foundation of China (12021004, 11804259, 11974124).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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