Flat Band Modes Research Articles

Optical moiré bound states in the continuum

Trapping electromagnetic waves within the radiation continuum holds significant implications in the field of optical science and technology. Photonic bound states in the continuum (BICs) present a distinctive approach for achieving this functionality, offering potential applications in laser systems, sensing technologies, and other domains. However, the simultaneous achievement of high Q-factors, flat-band dispersions, and wide-angle responses in photonic BICs has not yet been reported, thereby impeding their practical performance due to laser direction deviation or sample disorder. Here, we theoretically demonstrate the construction of moiré BICs in one-dimensional photonic crystal (PhC) slabs, where high-Q resonances in the entire moiré flat band are achieved. Specifically, we numerically validate that the radiation loss of moiré BICs can be eliminated by aligning multiple topological polarization charges with all diffraction channels, enabling the strong suppression of far-field radiation from the entire moiré band. This leads to a slow decay of Q-factors away from moiré BICs in the momentum space. Moreover, it is found that Q-factors of the moiré flat band can still maintain at a high level with structural disorder. In experiments, we fabricate the designed 1D moiré PhC slab and observe both high-Q resonances and a slow decrease of Q-factors for moiré flat-band Bloch modes. Our findings hold promising implications for designing highly efficient optical devices with wide-angle responses and introduce a novel avenue for exploring BICs in moiré superlattices.

Low-threshold single-mode nanowire array flat-band photonic-crystal surface-emitting lasers with high-reflectivity bottom mirrors.

A Si-based nanowire array photonic-crystal surface-emitting laser based on a flat band is designed and simulated. By introducing an air gap between the nanowire and substrate, the bottom reflectivity is significantly enhanced, resulting in much lower threshold and smaller cutoff diameter. Through adjusting the lattice constant (the distance between neighboring nanowires) and nanowire diameter, a photonic crystal structure with a flat band is achieved, in which strong interaction between light and matter occurs in the flat band mode. For the device with a small size, single-mode lasing is obtained with a side-mode suppression ratio of 21 dB, high quality factor of 3940, low threshold gain of 624 cm-1, and small beam divergency angle of ∼7.5°. This work may pave the way for the development of high-performance Si-based surface-emitting nanolasers and high-density photonic integrated circuits.

Wide edge state supercontinuum in a Floquet–Lieb topological photonic insulator

Conventional topological photonic insulators typically have narrow nontrivial band gaps truncated by broad dispersive bulk bands, resulting in limited edge mode transmission bandwidths that can be exploited for potential applications. Here, we demonstrate a Floquet–Lieb topological photonic insulator with all flat bands that can support continuous edge mode transmission across multiple Floquet–Brillouin zones. This supercontinuum of edge states results from the coexistence and orthogonality of the localized flat-band modes and the edge states, allowing for continuous excitation of the latter without scattering into the bulk modes. Moreover, we show that these flat bands are perfectly immune to random variations in the on-site potential, regardless of how large the perturbations are, thus ensuring complete robustness of the edge modes to this type of disorder. We realized Floquet–Lieb insulators using 2D microring resonator lattices with perfect nearest-neighbor couplings. Transmission measurements and direct imaging of the scattered light distributions showed an edge mode supercontinuum spanning more than three microring free spectral ranges. The proposed Floquet–Lieb insulator can potentially be used to realize topological photonic devices with wide bandwidths and super robustness for applications in integrated quantum photonics and programmable photonic circuits.

Transport and localization on dendrite-inspired flat band linear photonic lattices

The capacity of a physical system to transport and localize energy or information is usually linked to its spatial configuration. This is relevant for integration and transmission of signals as performed, for example, by the dendrites of neuronal cells. Inspired by recent works on the organization of spines on the surface of dendrites and how they promote localization or propagation of electrical impulses in neurons, here we propose a linear photonic lattice configuration to study how the geometric features of a dendrite-inspired lattice allows for the localization or propagation of light on a completely linear structure. We show that by increasing the compression of the photonic analogue of spines and thus, by increasing the coupling strength of the spines with the main chain (the “photonic dendrite”), flat band modes become prevalent in the system, allowing spatial localization in the linear – low energy – regime. Furthermore, we study the inclusion of disorder in the distribution of spines and show that the main features of ordered systems persist due to the robustness of the flat band states. Finally, we discuss if the photonic analog, having evanescent interactions, may provide insight into linear morphological mechanisms at work occurring in some biological systems, where interactions are of electric and biochemical origin.

Structure-independent flat bands induced by discontinuity plasma-air interface in photonic crystals

The band structures of transverse electric (TE) polarized dispersive photonic crystals are generally more complicated than those of transverse magnetic (TM) polarized photonic crystals. Here, by simplifying the governing equations of a TE polarized dispersion system and conducting the band structure analysis, we deduce the Hermitian eigenstates of the vector electric field. The results exhibit the double degeneracy eigenstates in the triangular lattice for lower eigenfrequencies and numerous plasmon-induced flat band modes in the interface of the plasma and air for higher eigenfrequencies. Moreover, we illustrate that the surface plasmon modes tend to present the quantized eigenfrequencies, which is due to the boundary condition imposed by the cylindrical geometry. However, when the eigenfrequency approaches the surface plasma frequency, the localization of the field on the boundary becomes stronger, which weakens the coupling between sites and reduces the dependence of the flat bands on the lattice structure. Unlike the generation mechanism of the lattice-induced flat bands, such plasma frequency-dependent localization at the interface enables a robust flat band characteristic immune to the lattice disorder and provides a novel degree of freedom to control the energy band. Our findings are expected to be useful for electromagnetic wave manipulation and field enhancement.

Moiré phonons in graphene/hexagonal boron nitride moiré superlattice

We theoretically study in-plane acoustic phonons of graphene/hexagonal boron nitride moir\'e superlattice by using a continuum model. We demonstrate that the original phonon bands of individual layers are strongly hybridized and reconstructed into moir\'e phonon bands consisting of dispersive bands and flat bands. The phonon band structure can be effectively described by a spring-mass network model to simulate the motion of moir\'e domain walls, where the flat-band modes are interpreted as vibrations of independent, decoupled strings. We also show that the moir\'e phonon has angular momentum due to the inversion symmetry breaking by hBN, with high amplitudes concentrated near narrow gap region. Finally, we apply the same approach to twisted bilayer graphene, and we find a notable difference between the origins of the flat-band modes in G/hBN and TBG, reflecting distinct geometric structures of domain pattern.

Magic configurations in moiré superlattice of bilayer photonic crystals: Almost-perfect flatbands and unconventional localization

We investigate the physics of photonic band structures of the moir\'e patterns that emerged when overlapping two uni-dimensional (1D) photonic crystal slabs with mismatched periods. The band structure of our system is a result of the interplay between intra-layer and inter-layer coupling mechanisms, which can be fine-tuned via the distance separating the two layers. We derive an effective Hamiltonian that captures the essential physics of the system and reproduces all numerical simulations of electromagnetic solutions with high accuracy. Most interestingly, \textit{magic distances} corresponding to the emergence of photonic flatbands within the whole Brillouin zone of the moir\'e superlattice are observed. We demonstrate that these flatband modes are tightly localized within a moir\'e period. Moreover, we suggest a single-band tight-binding model that describes the moir\'e minibands, of which the tunnelling rate can be continuously tuned via the inter-layer strength. Our results show that the band structure of bilayer photonic moir\'e can be engineered in the same fashion as the electronic/excitonic counterparts. It would pave the way to study many-body physics at photonic moir\'e flatbands and novel optoelectronic devices.

Moiré Quasibound States in the Continuum.

The novel physics of twisted bilayer graphene has motivated extensive studies of magic-angle flat bands hosted by moiré structures in electronic, photonic, and acoustic systems. On the other hand, bound states in the continuum (BICs) have also attracted great attention in recent years because of their potential applications in the field of designing superior optical devices. Here, we combine these two independent concepts to construct a new optical state in a twisted bilayer photonic crystal slab, which is called as moiré quasi-BIC, and numerically demonstrate that such an exotic optical state possesses dual characteristics of moiré flat bands and quasi-BICs. To illustrate the mechanism for the formation of moiré flat bands, we develop an effective model at the center of the Brillouin zone and show that moiré flat bands could be fulfilled by balancing the interlayer coupling strength and the twist angle around the band edge above the light line. Moreover, by decreasing the twist angle of moiré photonic crystal slabs with flat bands, it is shown that the moiré flat-band mode at the Brillouin center gradually approaches a perfect BIC, where the total radiation loss from all diffraction channels is significantly suppressed. To clarify the advantage of moiré quasi-BICs, enhanced second-harmonic generation (SHG) is numerically proven with a wide-angle optical source. The efficiency of SHG assisted by designed moiré quasi-BICs can be greatly improved compared with that based on dispersive quasi-BICs with similar quality factors.

Flatband mode in photonic moiré superlattice for boosting second-harmonic generation with monolayer van der Waals crystals.

We theoretically investigate boosting second-harmonic generation (SHG) of monolayer van der Waals crystals by employing flatband modes hosted by photonic moiré superlattices. Such a system with high quality factor and a monolayer crystal accommodated on the top of it, provides a unique opportunity to enhance and manipulate SHG emission. We show that employing a doubly resonant diagram on such a moiré superlattice system not only boosts the SHG, but also tunes the directional emission of the second-harmonic wave. Moreover, we demonstrate that a structured beam illumination could further boost SHG, with the phase structure retrieved through a two-beam second-harmonic interference configuration. These results suggest the flatband modes in moiré superlattice as a promising platform for boosting SHG with monolayer van der Waals crystals, offering new possibilities for developing compact nonlinear photonic devices.

Nonlinear compact localized modes in flux-dressed octagonal-diamond lattice

Tuning the values of artificial flux in the two-dimensional octagonal-diamond lattice drives topological phase transitions, including between singular and non-singular flatbands. We study the dynamical properties of nonlinear compact localized modes that can be continued from linear flatband modes. We show how the stability of the compact localized modes can be tuned by the nonlinearity strength or the applied artificial flux. Our model can be realized using ring resonator lattices or nonlinear waveguide arrays.

Controlled imprisonment of wave packet and flat bands in a fractal geometry

The explicit construction of non-dispersive flat band modes and the tunability of has been reported for a hierarchical 3-simplex fractal geometry. A single band tight-binding Hamiltonian defined for the deterministic self-similar non-translationally invariant network can give rise to a countably infinity of such self localized eigenstates for which the wave packet gets trapped inside a characteristic cluster of atomic sites. An analytical prescription to detect those dispersionless states has been demonstrated elaborately. The states are localized over clusters of increasing sizes, displaying the existence of a multitude of localization areas. The onset of localization can, in principle, be ‘delayed’ in space by an appropriate choice of the energy of the electron. The tunability of those states leads to the controlled decay of wave function envelope. The impact of perturbation on the bound states has also been discussed. The analogous wave guide model has also been discussed.

Exciton-polaritons in flatland: Controlling flatband properties in a Lieb lattice

In recent years, novel two-dimensional materials such as graphene, bismuthene and transition-metal dichalcogenides have attracted considerable interest due to their unique physical properties. A range of physical effects can be transferred to the realms of photonics by creating artificial photonic lattices emulating these two-dimensional materials. Here, exciton-polaritons in semiconductor microcavities offer an exciting opportunity to study a part-light, part-matter quantum fluid of light in a complex lattice potential. In this paper, we study exciton-polaritons in a two-dimensional Lieb lattice of buried optical traps. The $S$ and $P_{xy}$ photonic orbitals of such a Lieb lattice give rise to the formation of two flatbands which are of greatest interest for the distortion-free storage of compact localized states. By using a well controlled etch-and-overgrowth technique, we manage to control the trapping as well as the site couplings with great precision. This allows us to spectroscopically monitor the flatness of the flatbands across the full Brillouin zone. Furthermore, we demonstrate experimentally that these flatbands can be directly populated by condensation under non-resonant laser excitation. Finally, using this advanced device approach we demonstrate resonant and deterministic excitation of flatband modes in transmission geometry. Our findings establish the exciton-polariton systems as a highly controllable, optical many-body system to study flatband effects and for distortion-free storage of compact localized states.

Manifestation of a topological gapless phase in a two-dimensional chiral symmetric system through Loschmidt echo

Unlike the edge state of a topological insulator where its energy level lives in the bulk energy gap, the edge state of a topological semimetal hides in the bulk spectrum and is difficult to be identified by the energy. We investigate the sensitivity of bulk and edge states of the gapless phase for a topological semimetal to the disordered perturbation via a concrete two-dimensional chiral symmetric lattice model. The topological gapless phase is characterized by two opposite vortices in the momentum space and nonzero winding numbers, which correspond to the edge flatband when the open boundary condition is applied. For this system, numerical results reveal that a distinguishing feature is that the robustness of the edge states against weak disorder and the flatband edge modes remain locked at zero energy in the presence of weak chiral-symmetry-preserving disorder. We employ the Loschmidt echo (LE) for both bulk and edge states to study the dynamic effect of disordered perturbation. We show that, for an initial bulk state, the LE decays exponentially, whereas it converges to a constant for an initial edge state in the presence of weak disorder. Furthermore, the convergent LE can be utilized to identify the positions of vortices as well as the phase diagram. We discuss the realization of such dynamic investigations in a topological photonic system.

Perfect localization on flat-band binary one-dimensional photonic lattices

The existence of flat bands is generally thought to be physically possible only for dimensions larger than one. However, by exciting a system with different orthogonal states this condition can be reformulated. In this work, we demonstrate that a one-dimensional binary lattice supports always a trivial flat band, which is formed by isolated single-site vertical dipolar states. These flat band modes correspond to the highest localized modes for any discrete system, without the need of any aditional mechanism like, e.g., disorder or nonlinearity. By fulfilling a specific relation between lattice parameters, an extra flat band can be excited as well, with modes composed by fundamental and dipolar states that occupy only three lattice sites. Additionally, by inspecting the lattice edges, we are able to construct analytical Shockley surface modes, which can be compact or present staggered or unstaggered tails. We believe that our proposed model could be a good candidate for observing transport and localization phenomena on a simple one-dimensional linear photonic lattice.

Flux modulated flat band engineering in square-kagomé ladder network

The origin of non-dispersive flat band modes for a quasi-one dimensional square-kagomé ladder network is explored analytically by virtue of the real space renormalization group (RSRG) technique. A section of the eigenstates is non-diffusive i.e., localized within a cluster of sub-lattice sites partly by the destructive type of quantum interference and partly by the physical boundary created by the site with zero wave function amplitude. By making the amplitude vanish at the selective sites it becomes possible to confine the incoming excitation within the trapping cell leading to the formation of compact localized states. The effective mass of the particle becomes infinitely large corresponding to those self-localized modes and hence the mobility of the wave train becomes vanishingly small. This quenched kinetic energy leads to a momentum independent contribution to a dispersion curve. The present analysis is corroborated by numerical calculation of spectral landscape and the corresponding dispersion profile. The application of uniform magnetic flux may lead to a comprehensive engineering of the position as well as the curvature of the band. Also, one-to-one mapping between electronic case and photonic case within the tight-binding framework helps us to study the photonic localization in an analogous single mode wave guide system. The concept of slow light eventually introduces the possibility of spatial compression of light energy.

Compact modes in quasi one dimensional coupled magnetic oscillators

In this work we study analytically and numerically the spectrum and localization properties of three quasi-one-dimensional (ribbons) split-ring resonator arrays which possess magnetic flatbands, namely, the stub, Lieb and kagome lattices, and how their spectra are affected by the presence of perturbations that break the delicate geometrical interference needed for a magnetic flatband to exist. We find that the stub and Lieb ribbons are stable against the three types of perturbations considered here, while the kagome ribbon is, in general, unstable. When losses are incorporated, all flatbands remain dispersionless but become complex, with the kagome ribbon exhibiting the highest loss rate. The stability of flatband modes of certain split-ring resonator arrays suggests that they could be used as components of future stable magnetic storage devices.

New edge-centered photonic square lattices with flat bands

We report a new class of edge-centered photonic square lattices with multiple flat bands, and consider in detail two examples: the Lieb-5 and Lieb-7 lattices. In these lattices, there are 5 and 7 sites in the unit cell and in general, the number is restricted to odd integers. The number of flat bands m in the new Lieb lattices is related to the number of sites N in the unit cell by a simple formula m=(N−1)∕2. The flat bands reported here are independent of the pseudomagnetic field. The properties of lattices with even and odd number of flat bands are different. We consider the localization of light in such Lieb lattices. If the input beam excites the flat-band mode, it will not diffract during propagation, owing to the strong mode localization. In the Lieb-7 lattice, the beam will also oscillate during propagation and still not diffract. The period of oscillation is determined by the energy difference between the two flat bands. This study provides a new platform for investigating light trapping, photonic topological insulators, and pseudospin-mediated vortex generation.

On the stability of flat-band modes in a rhombic nonlinear optical waveguide array

A quasi-one-dimensional rhombic array of waveguides is considered. In the nonlinear case the system of equations describing coupled waves in the waveguides has the particular solutions that represent the superposition of flat-band modes. The stability of these solutions is considered. It was found that in one case the flat-band solution is unstable until the intensity threshold is attained. In another case the stability of the flat-band solution breaks down when the intensity per waveguide exceeds the threshold value.

Nonlinear localized flat-band modes with spin-orbit coupling

We report the coexistence and properties of stable compact localized states (CLSs) and discrete solitons (DSs) for nonlinear spinor waves on a flatband network with spin-orbit coupling (SOC). The system can be implemented by means of a binary Bose-Einstein condensate loaded in the corresponding optical lattice. In the linear limit, the SOC opens a minigap between flat and dispersive bands in the system's bandgap structure, and preserves the existence of CLSs at the flatband frequency, simultaneously lowering their symmetry. Adding onsite cubic nonlinearity, the CLSs persist and remain available in an exact analytical form, with frequencies which are smoothly tuned into the minigap. Inside of the minigap, the CLS and DS families are stable in narrow areas adjacent to the FB. Deep inside the semi-infinite gap, both the CLSs and DSs are stable too.

Electromagnetic field distribution in a quasi-ID rhombic waveguide array

The exact solution of the coupled mode equations is presented in the case of the continuous electromagnetic wave propagation in the quasi-one-dimensional rhombic array of the waveguides. In general case of the boundary conditions the discrete diffraction is described by these solutions. The flat band mode propagation is found as the special case.