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We demonstrate stable Dirac points (DPs) in low dimensions by taking advantage of non-Hermiticity in an optical lattice composed of two coupled Su–Schrieffer–Heeger chains. The occurrence of DPs stems from the constraints of pseudo-Hermiticity and charge-conjugation parity symmetry, which force the system to support both real bands and orthogonal eigenmodes despite its non-Hermitian nature. The two characteristics hold even at spectral degeneracies of zero energy, giving rise to non-Hermitian DPs. We show that DPs are stable with the variation of dissipation since they are topological charges and can develop into nodal rings in two dimensions. Moreover, we investigate the beam dynamics around DPs and observe beam splitting with stable power evolution. The study paves the way for controlling the flow of light to aid dissipation together with high stability of energy.

The gauge fields dramatically alter the algebraic structure of spatial symmetries and make them projectively represented, giving rise to novel topological phases. Here, we propose a photonic Möbius topological insulator enabled by projective translation symmetry in multiorbital waveguide arrays, where the artificial π gauge flux is aroused by the inter-orbital coupling between the first (s) and third (d) order modes. In the presence of π flux, the two translation symmetries of rectangular lattices anti-commute with each other. By tuning the spatial spacing between two waveguides to break the translation symmetry, a topological insulator is created with two Möbius twisted edge bands appearing in the bandgap and featuring 4π periodicity. Importantly, the Möbius twists are accompanied by discrete diffraction in beam propagation, which exhibit directional transport by tuning the initial phase of the beam envelope according to the eigenvalues of translation operators. This work manifests the significance of gauge fields in topology and provides an efficient approach to steering the direction of beam transmission.

We investigate the topological edge modes in a rhombic waveguide array with imaginary coupling, which is realized by incorporating auxiliary waveguide between the coupled waveguides. By suitably tuning both real and imaginary couplings to generate an effective π flux, a non-Hermitian analog of Aharonov-Bohm (AB) cage is formed as the band structures become flat and coalesce into third-order exceptional points (EPs). We show the array can support gapped topological edge modes, which can be explained by mapping the system Hamiltonian into the square root of an anti-parity-time (PT) -symmetric waveguide array. In contrast to the linear power increase of bulk modes, the propagation of edge modes can be conserved or exponentially amplified depending on which termination is initially excited. Our study provides a promising way to realizing robust light propagation.

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.

We investigate the excitation of surface plasmon polaritons (SPPs) using a metallic nanoaperture array illuminated by circularly polarized Laguerre-Gaussian (LG) vortex beams. The direction of SPP excitation is tunable by changing the circular polarization and topological charge of LG beams. The left- or right-handed circular polarization determines SPP propagation on either side of the nanoaperture array. Furthermore, varying the topological charge of LG beam will result in beam splitting of SPPs. We also utilize a composite nanoaperture array with different periods to achieve unidirectional excitation of SPPs. The study provides an interesting approach to control the excitation direction of SPPs and may find great applications in SPP generators and optical switches.

We investigate the second-order topological phases in a two-dimensional ring resonator array with each plaquette occupied by \ensuremath{\pi} gauge flux and imaginary gauge field. The real and imaginary gauge fields are induced by shifting the displacement and integrating gain or loss into the two half perimeters of the auxiliary rings. The system supports topological corner modes with their emergence being determined by the non-Bloch topological invariant due to skin effects. The bulk modes, exhibiting second-order skin effects in both trivial and nontrivial phases, are accumulated at opposite corners depending on whether clockwise or counterclockwise modes are excited. By introducing an interface with different imaginary gauge fields, we show the bulk modes exist at the interface while the topological corner modes are localized at the physical corners. Furthermore, the skin effects are also presented in the passive ring resonators. The study may find applications in lasers and broadband light trapping.

We investigate the topological skin effect in a ring resonator array which can be mapped into the square root of a Su-Schrieffer-Heeger (SSH) model with non-Hermitian asymmetric coupling. The asymmetric coupling is realized by integrating the same amount of gain and loss into the two half perimeters of linking rings that effectively couple two adjacent site rings. Such a square-root topological insulator inherits the properties from its parent Hamiltonian, which has the same phase transition points and exhibits non-Bloch features as well. We show the band closing points for open chain are different from that of periodic chain as a result of the skin effect. Moreover, the square-root insulator supports multiple topological edge modes as the number of band gaps is doubled compared to the original Hamiltonian. The full-wave simulations agree well with the theoretical analyses based on a tight-binding model. The study provides a promising approach to investigate the skin effect by utilizing ring resonators and may find potential applications in light trapping, lasers, and filters.

We study the plasmonic second-order topological modes in nanowire waveguides which are covered by monolayer graphene. The waveguide arrays are arranged in the kagome lattice. In a triangle-shaped lattice, the topological modes are localized at the corner of the triangle by tuning the spatial spacing between the different waveguides. The corner modes also depend on the corner shape, which only exist at one corner in a rhombic lattice. In addition to the corner modes, both structures also support the topological edge mode. We show that the corner modes experience a smaller modal wavelength, longer propagation distance, and smaller mode volume than the edge modes. The study may be utilized to explore the topological bound modes at the nanoscale.

Exceptional points (EPs) are branch point singularities of self-intersecting Riemann sheets, and they can be observed in a non-Hermitian system with complex eigenvalues. It has been revealed recently that dynamically encircling EPs by adiabatically changing the parameters of a system composed of lossy optical waveguides could lead to asymmetric (input-output) mode transfer. However, the length of the waveguides had to be considerable to ensure adiabatic evolution. Here we demonstrate that the parameters can change adiabatically along a smaller encircling loop by utilizing moving EPs, leading to significant shortening of the structures compared to fixed EPs. Meanwhile, the mode transmittance is remarkably improved and the transfer efficiency persists at ∼90%. Moving EPs are very promising for applications such as highly integrated broadband optical switches and convertors operating at telecommunication wavelengths.

We investigate the topological bound modes in a binary waveguide array with alternating positive and negative couplings. The negative coupling is realized by incorporating additional waveguides into the bare array and suitably tuning the refractive index. We show the band structures close and Dirac point emerges at the Brillouin zone center when the absolute values of real and negative couplings are equal, indicating topological phase transition. The topological bound modes emerge at the edge of topological non-trivial arrays or at the interface of two arrays with different topologies. Moreover, we show the topological bound mode can be isolated from the trivial defect mode by tuning the coupling strength, which is reflected from wave propagation. The study provides a promising approach to realizing robust light transport.

The application of energy band theory in optics provides an effective approach to modulating the flow of light. The recent discovery of non-Hermitian skin effect promotes the development of traditional energy band theory, which further enables an alternative way to realize light localization and unidirectional propagation. However, how to effectively generate and steer the non-Hermitian skin effect is still an important topic, especially in integrated optical systems. Here, we investigate the non-Hermitian skin effect in quasi-one-dimensional rhombic optical lattice with synthetic gauge potential. By calculating the eigenenergy spectra, spectral winding number, and wave dynamics, the gauge potential can be utilized to effectively tune the localization strength of skin modes. In particular, the skin effect is completely suppressed when the gauge potential in each plaquette is equal to π, while the flat-band localization caused by Aharonov-Bohm caging effect is dominant. By utilizing the indirectly coupled micro ring resonator array, the gauge potential and asymmetric coupling can be generated at the same time, which provides a potential experimental scheme to explore the competition between Aharonov-Bohm cage and skin effect. The present study provides an alternative way to steer the skin effect, which offers an approach to achieving the on-chip non-magnetic unidirectional optical devices.

We investigate the topological interface modes of surface plasmon polaritons in a multilayer system composed of graphene waveguide arrays. The topological interface modes emerge when two topologically distinct graphene multilayer arrays are connected. In such multilayer system, the non-trivial topological interface modes and trivial modes coexist. By tuning the configuration of the graphene multilayer arrays, the associated non-trivial interface modes present robust against structural disorder. The total number of topological modes is related to that of graphene layers in a unit cell of the graphene multilayer array. The results provide a new paradigm for topologically protected plasmonics in the graphene multilayer arrays. The study suggests a promising approach to realize light transport and optical switching on a deep-subwavelength scale.

We investigate the light propagation in a double-waveguide coupler with imaginary-valued coupling coefficient. The imaginary coupling is realized by incorporating an additional waveguide between the coupled waveguides and considering both gain and loss. Both symmetric and asymmetric couplers with imaginary coupling are analyzed. The coupling strength can be controlled by varying the dielectric gain and loss. As the coupling is imaginary, the energy of the system is no longer conserved and the light intensities in individual waveguides should oscillate in phase. It is also found that the exceptional points can emerge in the asymmetric couplers, which are reflected from the beat length of the intensity oscillation. The study may find great applications in optical switches and splitters.

We theoretically investigate the time-domain dynamics in a non-Hermitian electric circuit network with anti-parity-time (anti-PT) symmetry, which is consisted of amplified resonant LRC circuits connected by resistors. The effective coupling strength and on-site potential can be flexibly tuned via controlling the resistance value and resonant frequencies of LRC circuits, respectively. Therein, we are able to realize all three symmetry phases of the system, including anti-PT symmetric, intermediate, and broken phases, which are distinguished from their band structures. We show the time evolution of the envelope of voltage is analogous to discrete diffraction as a single injection stimulates more and more channels over time. Specifically, the system exhibits constant intensity diffraction at the boundary between intermediate and broken phases as the voltages at different channels are of the same value during the evolution. The reason lies in the dispersionless band structure near exceptional points (EPs) and the related linear energy amplification. The results enrich the study of discrete diffraction and are helpful to further exploration of topological phenomena in anti-PT-symmetric systems.

We show that the synthetic gauge fields for photons provide a versatile approach to generate and control the non-Hermitian skin effect. By utilizing indirectly coupled optical ring resonator arrays with long-range couplings and on-site gain and loss, we find that the skin effect appears once the gauge field is not an integer multiple of π. In addition to tunable localization direction, the skin modes display anisotropic behaviors with frequency-dependent decay length, which can be explained by the split subregion of the generalized Brillouin zone (GBZ) and an effective model under adiabatic elimination. Through numerical simulation, we can also demonstrate exotic features in propagation effects enabled by the skin effect, including asymmetric transmission and reconfigurable accumulation interface. Our study paves the way to dynamically steer skin modes, which may find applications in laser, optical switch, and signal processing.

We investigate the absorption property of a single layer of periodically rippled graphene adhering to dielectric substrate. By varying the geometric parameters of the ripple and chemical potential of graphene, the absorption can remarkably exceed that of planar graphene and reaches over 50%. It is found that the absorption peak undergoes a redshift as the ripple height increases. The variation of the incident angle will basically not influence the absorption wavelength but may yield additional absorption peaks due to higher order oscillation. The study reveals the optical response of graphene in the intrinsically nonplanar profile and will benefit to the design of graphene-based absorbers and photodetectors.

We investigate the spatial Goos–Hanchen (GH) shifts in one-dimensional defective photonic lattices by modulating the imaginary part of refractive index, namely, the imaginary modulation. The reflection spectra possess two singular points under imaginary modulation when the reflectance is infinite or vanishing. These two points correspond to the coherent-perfect-absorption-laser (CPA-LP) point and defect mode (MD) in the parameter space composed of the incident angle and refractive index. We show the spatial GH shifts are extremely giant in the vicinity of CPA-LP and MD as the phase of reflection coefficients dislocate. Moreover, the GH shifts are sensitive to the incident angle and the refractive index of defect. The sensitivity coefficients can reach as high as 106. The study may find potential applications in highly sensitive sensors.

The cascaded dual-chirped optical parametric amplification (DC-OPA) is presented for efficient generation of few-cycle infrared (IR) laser pulses. The input pulses are strategically chirped to optimize the phase-matching bandwidth in each nonlinear crystal, and four regions of the signal spectrum are amplified in cascaded crystals with different cutting angles, enabling flexible manipulation of the output spectrum. Broadband gain and high conversion efficiency are simultaneously achieved owing to the cascaded-crystal arrangement, the signal pulse duration of 4.2 cycles is obtained with 11.7-mJ pulse energy, corresponding to a conversion efficiency of 39.0%. The proposed scheme offers a robust and simple approach to pushing the phase-matching bandwidth limits introduced by the nonlinear crystal, which manifests great prospect in various researches involving ultrafast optics and strong-field physics.

We investigate the exceptional points (EPs) in a non-Hermitian system composed of a pair of graphene sheets with different losses. There are two surface plasmon polaritons (SPPs) modes in the graphene waveguide. By varying the distance between two graphene sheets and chemical potential of graphene, the EPs appear as the eigenvalues, that is, the wave vectors of the two modes coalesce. The cross conversion of eigenmodes and variation of geometric phase can be observed by encircling the EP in the parametric space formed by the geometric parameters and chemical potential of graphene. At the same time, a certain input SPP mode may lead to completely different output. The study paves a way to the development of nanoscale sensitive optical switches and sensors.

We studied the topological properties of an extended Su&ndash, Schrieffer&ndash, Heeger (SSH) model composed of a binary waveguide array with alternating real and imaginary couplings. The topological invariant of the periodic structures remained quantized with chiral symmetry even though the system was non-Hermitian. The numerical results indicated that phase transition arose when the absolute values of the two couplings were equal. The system supported a topological zero mode at the boundary of nontrivial structures when chiral symmetry was preserved. By adding onsite gain and loss to break chiral symmetry, the topological modes dominated in all supermodes with maximum absolute value of imaginary energy. This study enriches research on the SSH model in non-Hermitian systems and may find applications in optical routers and switches.

We explored exceptional points (EPs) in one dimensional non-Hermitian photonic crystals incorporated with a defect. The defect was asymmetric with respect to the center. Two EPs could be derived by modulating the normalized frequency and the gain-loss coefficient of defect. The reflection coefficient complex phase changed dramatically around EPs, and the change in complex phase was &pi, at EPs. The electric field of EPs was mainly restricted to the defect, which can induce a giant Goos&ndash, H&auml, nchen (GH) shift. Moreover, we found a coherent perfect absorption-laser point (CPA-LP) in the structure. A giant GH shift also existed around the CPA-LP. The study may have found applications in highly sensitive sensors.

We investigate the energy transfer of surface plasmon polaritons (SPPs) based on adiabatic passage in a non-Hermitian waveguide composed of three coupled graphene sheets. The SPPs can completely transfer between two outer waveguides via the adiabatic dark mode as the waveguides are lossless and the coupling length is long enough. However, the loss of graphene can lead to breakdown of adiabatic transfer schemes. By utilizing the coupled mode theory, we propose three approaches to cancel the nonadiabatic coupling by adding certain gain or loss in respect waveguides. Moreover, the coupling length of waveguide is remarkably decreased. The study may find interesting application in optical switches on a deep-subwavelength scale.

We investigate the mode conversion in a dielectric waveguide under spatially complex modulation, which refers to modulating both the real and imaginary parts of the dielectric permittivity. The complex modulations are applied separately in two distinct regions in the waveguide. The mode conversion can be controlled by adjusting the initial phase difference of the modulations in the two cascaded regions and by varying the phase difference of real and imaginary modulations in each region. The former is analogous to an Aharonov–Bohm (AB) phase and the latter suggests a symmetric propagation phase. Comparing to one modulated region, the cascaded modulations could benefit to realizing laser, coherent perfect absorber, and bidirectional invisibility. The study may contribute to the applications in optical switches and mode converters.

We investigate the exceptional points (EPs) in plasmonic waveguide arrays, including metallic waveguide arrays (MWAs) and graphene sheet arrays (GSAs). The EPs emerge at the boundary of strong and weak coupling ranges in both systems. The cross conversion of Bloch modes and variation of geometric phase can be observed by encircling an EP in the parametric space. We also show the Bloch modes exhibit strong absorption in the vicinity of EPs in GSAs, which originates from the enhanced longitude electric field along the propagation direction. The abnormal absorption and field enhancement also arise in ultrathin MWAs and disappear when the thickness of metal film increases. Our results may find applications in optical switches and sensors at the nanoscale.

We investigate the lateral shift of optical beam in complex dielectric multilayers with nearly parity–time (PT) symmetry. The gain and loss of dielectrics are modulated to achieve nearly PT symmetry. As light impinges on the multilayers, a laser point (LP) and exceptional points (EPs) appear in the parameter space composed of the incident angle and refractive index. The phase of reflection dislocates at the LP and EPs, inducing large lateral shift around these points. The shift can be positive and negative, and the polarities of the lateral shift convert at the LP and EPs. The shift, which is sensitive to the incident angle and refractive index of dielectrics, could be tuned flexibly. The study may find potential applications in highly sensitive sensors.

We theoretically investigate the Goos-Hanchen (GH) shifts of optical beam in a defective photonic crystal composed of dielectric multilayers and graphene. The system is non-Hermitian and possesses exceptional points (EPs) as the scattering matrix becomes defective at the zero points of reflection. The reflective wave at EPs experiences an abrupt phase change and there the eigenvalues of scattering matrix coalesce. The GH shifts are extremely large near EPs in parametric space composed of dielectric refractive index and incident angle. The positive and negative maxima of GH shifts could be as high as 103 times of the incident wavelength. The direction of GH shifts switches at EPs and the EPs position can be readily controlled by the chemical potential of graphene. Moreover, the GH shifts should remarkably change as the incident waves impinge on the structure from opposite directions. The study of GH shifts in the graphene incorporated multilayers may find great applications in highly sensitive sensors.

We investigate the topological edge modes of surface plasmon polaritons (SPPs) in a non-Hermitian system composed of graphene pair arrays with alternating gain and loss. The topological edge modes emerge when two topologically distinct graphene arrays are connected. The gain and loss present in the system provide additional ways to control the propagation loss and field distributions of the topological edge modes. Moreover, the existence of the topological edge modes is related to the broken parity-time (PT) symmetry. We show the beam diffraction can be steered by tuning the chemical potential of graphene. Thanks to the strong confinement of SPPs, the topological edge modes can be squeezed into a lateral width of ~λ/70. We also show such modes can be realized in lossy graphene waveguides without gain. The study provides a promising approach to realizing robust light transport and optical switches on a deep-subwavelength scale.

We investigate the switching between two surface plasmon polariton modes in a non-Hermitian system composed of a pair of graphene sheets using topological operations. The topological operation is implemented by suitably designing the waveguide geometry and the chemical potential of graphene, which is equivalent to changing the system parameters along a closed loop in the parameter space. Efficient and robust mode switching takes place as the loop encloses an exceptional point and the wave experiences a sufficiently large propagation length. Moreover, we show this mode switching is chiral, in the sense that the output modes are different for choosing different loop directions. The study provides a promising approach to robust mode switching on a deep-subwavelength scale.

We investigate the optical exceptional points (EPs) in the graphene incorporated multilayer metamaterial manifesting Fano resonance. The system is non-Hermitian and possesses EPs where both the eigenvalues and eigenvectors of the Hamiltonian coalesce. In the aid of Fano resonance, the reflection may reach minimum approaching to zero, resulting in the degeneration of both eigenvalues and eigenvectors and thus the emergence of EPs. The transmission and reflection of light through the metamaterial change sharply by varying slightly the incident wavelength and chemical potential of graphene in the parameter space when encircling the EPs. In addition, the unidirectional invisibility can be achieved at EPs. The study paves a way to precisely controlling the transmission and reflection through metamaterials and may find applications in optoelectronic switches, modulators, absorbers, and optical sensors.

We investigate the topological bound modes of surface plasmon polaritons (SPPs) in a graphene pair waveguide array. The arrays are with uniform inter-layer and intra-layer spacings but the chemical potential of two graphene in each pair are different. The topological bound modes emerge when two arrays with opposite sequences of chemical potential are interfaced, which are analogous to Jackiw-Rebbi modes with opposite mass. We show the topological bound modes can be dynamically controlled by tuning the chemical potential, and the propagation loss of topological bound modes can be remarkably reduced by decreasing the chemical potential. Thanks to the strong confinement of graphene SPPs, the modal wavelength of topological bound modes can be squeezed as small as 1/70 of incident wavelength. The study provides a promising approach to realizing robust light transport beyond diffraction limit.

We investigate the optical bistability of graphene in parity-time-symmetric (PT-symmetric) photonic lattices incorporated with a defect at terahertz frequencies. The field localization of the defect mode can strengthen the nonlinearity of graphene to achieve low-threshold bistability. The nonlinearity is further enhanced, and the bistability threshold decreases, by increasing the gain-loss factor in the PT-symmetric structure. The interval of upper and lower bistability thresholds is broadened as the exceptional points split. Moreover, we show the phase transition between bistability and nonbistability by modulating the incident wavelength and chemical potential of graphene. The study may find great applications in all-optical switches and optical storage.

We investigate the topological bound modes in a binary optical waveguide array with anti-parity-time (PT) symmetry. The anti-PT-symmetric arrays are realized by incorporating additional waveguides to the bare arrays, such that the effective coupling coefficients are imaginary. The systems experience two kinds of phase transition, including global topological order transition and quantum phase transition. As a result, the system supports two kinds of robust bound modes, which are protected by the global topological order and the quantum phase, respectively. The study provides a promising approach to realizing robust light transport by utilizing mediating components.

We investigate the frequency conversion in a waveguide by dynamically modulating the real and imaginary parts of permittivity, namely, the complex modulation. A single frequency of incident light could convert into many discrete frequencies with equivalent intervals, which is analogous to discrete diffraction in spatial waveguide arrays. We develop a coupled mode theory to analyze the frequency conversion process and show the complex modulation results in a complex-valued coupling coefficient between adjacent frequency channels. The coupling strength can be flexibly tuned by controlling the modulation amplitudes. Moreover, the complex coupling becomes asymmetric with increasing the modulation phase difference between the real and imaginary parts of permittivity. The capability to control the complex coupling leads to new features in discrete diffraction in the frequency dimension, including the unidirectional transport and equalization of light intensities at different channels. The study may find interesting applications in frequency comb generators and frequency controllers.

We investigate the Bloch mode conversion of surface plasmon polaritons in a periodic array of graphene pairs with each consisting of two separated parallel graphene sheets. The employment of graphene pair as a unit cell in the array yields two Bloch modes belonging to different bands. By periodically modulating the permittivity of dielectrics between graphene along the propagation direction, the interband transitions occur and the modes will alternatively couple to each other, similar to traditional Rabi oscillations in quantum systems. The indirect Rabi oscillations can also be observed through introducing transverse modulation momentum. The period of Rabi oscillations can be optimized by taking advantage of the flexible tunability of graphene. The study suggests that the structure have applications in optical switches and mode converters operating on deep-subwavelength scale.

We investigate the periodically patterned cross-shaped graphene arrays from far-infrared to THz region as wavelength tunable absorbers. By increasing the cross-arm width, the absorption of the single-layer array can essentially exceeds the continuous graphene sheet due to the surface plasmon resonance, even for small graphene filling ratio. The absorption can be further enhanced with higher chemical potential and relaxation time. The complementary structure brings about more absorption of incident light compared to the original graphene array. Moreover, the wavelength of absorption maximum shows angle independence for both TE and TM polarizations. The absorption efficiency can be significantly improved with double layers of the cross-shaped graphene arrays, which are helpful to design dual-band and broadband absorbers.