Floquet-Bloch Modes Research Articles

Analytical formulation of spatiotemporal modulated graphene-based waveguides using Floquet-Bloch theory

In this work, an analytical model to study graphene-based spatiotemporal modulated structures is developed and verified through comparison with full wave numerical simulations. Graphene is an ideal material for realizing spatiotemporal modulated structures at high frequencies of THz and optics. In this analysis, the electromagnetic response of studied structures is expressed in terms of weighted Floquet-Bloch modes supported by the structure, while graphene is modeled by a spatiotemporal modulated surface current that imposes certain boundary conditions on the modes. The developed analytical technique is a comprehensive tool and can be used for accurate modeling of different kinds of spatiotemporal devices including lossy, guided, and leaky wave structures. To demonstrate the accuracy of the model, two plasmonic waveguides with space and time modulated graphene conductivity are analyzed and their interband and intraband transition between modes are thoroughly investigated. Using the developed analytical model, spatiotemporal modulation phenomena such as mode conversion, wave amplification and nonreciprocal response are explored and discussed for the studied structures.

Fast imaging of local perturbations in a unknown bi-periodic layered medium

We discuss a novel approach for imaging local faults inside an infinite bi-periodic layered medium in R3 using acoustic measurements of scattered fields at the bottom or the top of the layer. The faulted area is represented by compactly supported perturbations with erroneous material properties. Our method reconstructs the support of perturbations without knowing or reconstructing the constitutive material parameters of healthy or faulty bi-period layer; only the size of the period is needed. This approach falls under the class of non-iterative imaging methods, known as the generalized linear sampling method with differential measurements, first introduced in [2] and adapted to periodic layers in [25]. The advantage of applying differential measurements to our inverse problem is that instead of comparing the measured data against measurements due to healthy structures, one makes use of periodicity of the layer where the data operator restricted to single Floquet-Bloch modes plays the role of the one corresponding to healthy material. This leads to a computationally efficient and mathematically rigorous reconstruction algorithm. We present numerical experiments that confirm the viability of the approach for various configurations of defects.

Controllable optical Zitterbewegung in PT-symmetric photonic lattices

We present that parity-time-symmetric photonic lattices can support optical Zitterbewegung (ZB) motion with controllable oscillation amplitude. The nonorthogonality of the Floquet–Bloch modes imposed by the non-Hermitian nature of the dynamics substantially affects the wave oscillation dynamics. It is shown that the amplitude of this optical ZB depends on the gain parameter, thus allowing to control the ZB process. The non-reciprocal light propagation induced by the nonorthogonality of the modes can balance the spectral angular broadening of the incident beam and even the strong oscillations with slightly growing amplitude along the propagation can be observed. Robustness of the optical ZB phenomenon to the perturbations of the gain parameter is also demonstrated.

Spatial coherence effects of stochastic optical beams in periodic potentials

The interaction between a partially coherent field with a one-dimensional periodic photonic environment is investigated within the framework of Floquet-Bloch modes. To this end, we describe the interplay between lattice properties and field fluctuations by considering the optical beam as a linear combination of Floquet-Bloch modes, whose coefficients are described by a stationary random process. It is demonstrated that the propagation of partially coherent beams depends not only on the average of the excitation of each band but also on the correlations existent among the various bands supported by the lattice.

Analysis of sampling methods for imaging a periodic layer and its defects

We revisit the differential sampling method introduced in (Haddar and Nguyen 2017 Comput. Math. Appl. 74 2831–55) for the identification of a periodic domain and some local perturbation. We provide a theoretical justification of the method that avoids assuming that the local perturbation is also periodic. Our theoretical framework uses functional spaces with continuous dependence with respect to the Floquet–Bloch variable. The corner stone of the analysis is the justification of the generalized linear sampling method in this setting for a single Floquet–Bloch mode.

Multimodal antireflective coatings for perfecting anomalous reflection from arbitrary periodic structures

Metasurfaces possess vast wave-manipulation capabilities, including reflection and refraction of a plane wave into non-standard directions. This requires meticulously-designed sub-wavelength meta-atoms in each period of the metasurface which guarantee unitary coupling to the desired Floquet-Bloch mode or, equivalently, suppression of the coupling to other modes. Herein, we propose an entirely different scheme to achieve such suppression, alleviating the need to devise and realize such dense scrupulously-engineered polarizable particles. Extending the concept of anti-reflective coatings to enable simultaneous manipulation of multiple modes, we show theoretically and experimentally that a simple superstrate consisting of only several uniform dielectric layers can be modularly applied to \textit{aribtrary} periodic structures to yield perfect anomalous reflection. This multimodal anti-reflective coating (MARC), designed based on an analytical model, presents a conceptually and practically simpler paradigm for wave-control across a wide range of physical branches, from electromagnetics and acoustics to seismics and beyond.

Conical diffraction modulation in fractional dimensions with a [formula omitted]-symmetric potential

This study analyzes conical diffraction in space-fractional parity-time (PT) symmetry, which arises in a fractional Schrödinger equation with a periodic PT-symmetric potential. The peculiar characteristics of the corresponding Floquet Bloch modes are also examined in detail. We find that the band structure in such a model is different from that in regular PT symmetry, especially the critical point, which becomes symmetric linear for the one-dimensional case, and cone-like for the two-dimensional case. During propagation, the linear band structure has peculiar properties, including splitting or diffraction-free propagation, preferential propagation, and unidirectional propagation. Whether the cone-like band structure leads to cone diffraction or nondiffracting propagation depends on which Floquet Bloch modes are excited by the input. Our results may have potential application in light modulation in PT symmetry and provide a new platform to study other exciting topics related to conical diffraction.

Non-volatile MWIR/LWIR beam reconfigurability with all-dielectric metagratings comprising phase-change materials with a high-refractive-index shift

We propose an all-dielectric grating paradigm comprising an optical-phase-change-material (O-PCM), functional in the 5 μm to 10 μm spectral range. This system leverages the capabilities of a newly-discovered O-PCM [Nat. Comm. 10, 4279 (2019)10.1038/s41467-019-12196-4], Ge2Sb2Se4Te1, which can be reliably switched between amorphous and crystalline phases at larger thicknesses close to 1 μm, while exhibiting a high-refractive-index shift of about 1.5 and no optical loss in this spectral range. The amorphous-O-PCM grating predominantly responds as an effectively homogeneous slab, letting light through without perturbing its path. The crystalline-O-PCM grating supports leaky Floquet-Bloch modes, which, at certain wavelengths, can simultaneously interfere destructively into the primary light path and constructively into the back-bent diffraction channel, in transmission. This “accidental” interference effect steers the incident beam in the negative direction. At a slightly detuned wavelength, the output power can be evenly split between the primary light path and the back-bent diffraction channel. Hence, our all-dielectric O-PCM-based metagrating can function as a platform for non-volatile reconfigurable beam steering and splitting. We have designed the metagrating paradigm and predicted its reconfigurable behavior with a semi-analytical calculation method and then verified it with a numerical first-principles experiment. We believe these results are relevant to MWIR/LWIR applications, but can also inspire new means for programmable and reconfigurable photonics across the spectrum as new O-PCMs are being developed.

Extremely low effective impedance in stratified graphene-dielectric metamaterials

The periodic reflections in frequency were observed in a stack of graphene layers and generally reported as a series of mini photonic bandgaps owing to the multiple interference by the graphene layers. In this research, the Floquet-Bloch theory was employed to obtain the effective refractive index and Bloch impedance for understanding the wave propagation characteristic therein. Interestingly, the periodic reflections were found to occur in the frequency band having drastic variation in complex Bloch impedance and effective refractive index as well, wherein a Floquet-Bloch mode having pure real effective refractive index and extremely low Bloch impedance exists.

Triple Ladder Lumped Circuit With Sixth Order Modal Exceptional Degeneracy

We introduce a circuit topology based on a simple triple-ladder circuit realized with lumped reactive components that provides a sixth order degenerate band-edge (6DBE). The 6DBE is a special kind of sixth-order exceptional point of degeneracy in a lossless and gainless periodic ladder. This degeneracy provides a very flat band edge in the phase-frequency dispersion diagram. The proposed topology exhibits unique structured resonance features associated with a high loaded Q-factor. We investigate the Floquet-Bloch modes in an infinite-length periodic triple-ladder and their dispersion relation using the S parameter formalism. We also provide the approximate analytic expressions of the eigenmodes and dispersion relation around the degenerate point based on the Puiseux series expansion. We investigate the filtering characteristics of a finite-length structure terminated with loads to highlight the special properties of the 6DBE compared to ladders with regular band edge (RBE) and fourth order degenerate band edge (DBE). The circuit framework introduced here with a 6DBE can be exploited in designing novel high Q-factor oscillators, filters, sensors, and pulse shaping networks.

Nonlinear conical diffraction in fractional dimensions with a [formula omitted]-symmetric optical lattice

Space-fractional parity-time symmetry, featuring the fractional Laplacian operator rather than the standard operator, continues to be a challenge. This report analytically and numerically assesses the dynamics of wave packets in a space-fractional parity-time symmetric lattice by invoking Kerr nonlinearity. By adjusting the Lévy index, the basic properties of Floquet-Bloch modes in parity-time symmetric optical lattices are examined. It is demonstrated that the width of the first three Floquet-Bloch modes increases as the Lévy index decreases and that the corresponding band structure becomes symmetrically linear. These features result in peculiar properties during propagation, including splitting or diffraction-free propagation, preferential propagation, unidirectional propagation, and phase dislocations. In the two-dimensional fractional case, when the band structure is cone-like, it causes conical diffraction, and non-diffracting propagation occurs when the Floquet-Bloch mode of the upper band is excited by the input beam. Kerr nonlinearity modulates the energy in a certain nonlinear region toward the middle and suppresses the formation of conical diffraction.

Trapping light in a Floquet topological photonic insulator by Floquet defect mode resonance

Floquet topological photonic insulators characterized by periodically varying Hamiltonians are known to exhibit much richer topological behaviors than static systems. In a Floquet insulator, the phase evolution of the Floquet–Bloch modes plays a crucial role in determining its topological behaviors. Here, we show that by perturbing the driving sequence, it is possible to manipulate the cyclic phase change in the system over each evolution period to induce self-interference of a bulk mode, leading to a resonance effect, which can be regarded as a Floquet counterpart of defect-mode resonance in static lattices. This Floquet Defect Mode Resonance (FDMR) is cavity-less since it does not require physical boundaries; its spatial localization pattern is, instead, determined by the driving sequence and is found to be different in topologically trivial and nontrivial lattices. We demonstrated excitation of FDMRs by edge modes in a Floquet octagon lattice on silicon-on-insulator, achieving extrinsic quality factors greater than 104. Imaging of the scattered light pattern directly revealed the hopping sequence of the Floquet system and confirmed the spatial localization of FDMR in a bulk-mode loop. The new Floquet topological resonator could find various applications in lasers, optical filters and switches, nonlinear cavity optics, and quantum optics.

Evidence of asymmetric beaming in a piecewise-linear propagation channel.

Asymmetric beaming in a piecewise-linear propagation channel is demonstrated for a single photonic-crystal prism at Gaussian-beam illumination. The used hybrid refraction-diffraction mechanism exploits oblique incidence, the first-negative-order deflection at the longer interface, and asymmetry in coupling at the exit interfaces and does not need blocking of transmission by dispersion in the backward illumination case. The Floquet-Bloch mode with left-handed behavior and nearly circular equifrequency dispersion contours is utilized. The outgoing waves may have significantly different spatial distributions for the forward and backward illumination cases, yielding asymmetry in the beaming regime.

Nonradiative subdiffraction near-field patterns using metagratings

We present a synthesis scheme to mold periodic nonradiative field patterns in transmission using the recent concept of metagratings (MGs). To this end, we utilize our previously developed analytical model to analyze the interaction of an incoming plane wave with these sparse periodic arrangements of polarizable particles (meta-atoms). As the model reliably predicts coupling to all scattered Floquet–Bloch modes, both propagating and evanescent, desired reactive near-field profiles with deep subwavelength features can be generated. This approach forms an appealing alternative to previously proposed near-field plates based on metasurfaces, where abstract homogenization introduces uncertainties regarding utilization of highly evanescent spectrum, and meta-atom realization incurs full-wave optimization. In contrast, the outlined MG-based methodology, verified via full-wave simulations, directly yields fabrication-ready printed-circuit-board configurations, enabling versatile control of reactive near fields with no interfering radiative components, with potential uses in sensing, selective microwave heating, and wireless power transfer.

Dual-Polarized All-Metallic Metagratings For Perfect Anomalous Reflection

We theoretically formulate and experimentally demonstrate the design of metagratings (MGs) composed of periodic rectangular grooves in a metallic medium, intended for perfect anomalous reflection. Using mode matching, a semianalytical scheme for analysis and synthesis of such MGs, containing multiple, arbitrarily arranged, grooves per period, is derived. Following the typical MG design approach, we use this formalism to identify the relevant Floquet-Bloch (FB) modes and conveniently formulate constraints for suppression of spurious scattering, directly tying the structure's geometrical degrees of freedom (DOFs) to the desired functionality. Solving this set of constraints, in turn, yields a detailed fabrication-ready MG design, without any full-wave optimization. Besides providing means to realize highly-efficient beam deflection with all-metallic formations, we show that the rectangular (two-dimensional) groove configuration enables \emph{simultaneous} manipulation of both transverse electric (TE) and transverse magnetic (TM) polarized fields, unavailable to date with common, printed-circuit-board-based, microwave MGs. In addition, we highlight a physical limitation on the TE-polarization performance, preventing the ability to achieve perfect anomalous reflection in any desired angle. These capabilities are verified using three MG prototypes, produced with standard computer numerical control (CNC) machines, demonstrating both single- and dual-polarized control of multiple diffraction modes. These results enable the use of MGs for a broader range of applications, where dual-polarized control is required, or all-metallic devices are preferable (e.g., spaceborne systems or at high operating frequencies).

Differential imaging of local perturbations in anisotropic periodic media

We develop a differential sampling method to image local perturbations in anisotropic periodic layers, extending earlier works on the isotropic case. We study in particular a new interior transmission problem that is associated with the inverse problem when only a single Floquet–Bloch mode is used. This analysis is then exploited to design an indicator function for the local perturbation. The resulting numerical algorithm is validated for two dimensional numerical experiments with synthetic data.

Experimental demonstration and in-depth investigation of analytically designed anomalous reflection metagratings

We present the design, fabrication and experimental investigation of a printed circuit board (PCB) metagrating (MG) for perfect anomalous reflection. The design follows our previously developed analytical formalism, resulting in a single-element MG capable of unitary coupling of the incident wave to the specified (first order) Floquet-Bloch (FB) mode while suppressing the specular reflection. We characterize the MG performance experimentally using a bistatic scattering pattern measurement, relying on an original beam-power integration approach for accurate evaluation of the coupling to the various modes across a wide frequency range. The results show that highly-efficient wide-angle anomalous reflection is achieved, as predicted by the theory, with a relatively broadband response. In addition, the MG is found to perform well when illuminated from different angles, acting as a multichannel reflector, and to scatter efficiently also to higher-order FB modes at other frequencies, exhibiting multifunctional capabilities. Importantly, the merits of the introduced beam-integration approach, namely, its improved resilience to measurement inaccuracies or noise effects, and the implicit accommodation of different effective aperture sizes, are emphasized, highly relevant in view of the numerous recent experimental reports on anomalous reflection metasurfaces. Finally, we discuss the source for losses associated with the MG; interestingly, we show that these correlate well with edge diffraction effects, rather than (commonly assumed) power dissipation. These experimental results verify our theoretical synthesis scheme, showing that highly-efficient anomalous reflection is achievable with a realistic fabricated MG, demonstrating the practical applicability and potential mutifunctionality of analytically designed MGs for future wavefront manipulating devices.

New interior transmission problem applied to a single Floquet–Bloch mode imaging of local perturbations in periodic media

This paper considers the imaging of local perturbations of an infinite penetrable periodic layer. A cell of this periodic layer consists of several bounded inhomogeneities situated in a known homogeneous media. We use a differential linear sampling method to reconstruct the support of perturbations without using the Green’s function of the periodic layer nor reconstruct the periodic background inhomogeneities. The justification of this imaging method relies on the well-posedeness of a nonstandard interior transmission problem, which until now was an open problem except for the special case when the local perturbation did not intersect the background inhomogeneities. The analysis of this new interior transmission problem is the main focus of this paper. We then complete the justification of our inversion method and present some numerical examples that confirm the theoretical behavior of the differential indicator function determining the reconstructable regions in the periodic layer.

All-angle Negative Reflection with An Ultrathin Acoustic Gradient Metasurface: Floquet-Bloch Modes Perspective and Experimental Verification

Metasurface with gradient phase response offers new alternative for steering the propagation of waves. Conventional Snell’s law has been revised by taking the contribution of local phase gradient into account. However, the requirement of momentum matching along the metasurface sets its nontrivial beam manipulation functionality within a limited-angle incidence. In this work, we theoretically and experimentally demonstrate that the acoustic gradient metasurface supports the negative reflection for all-angle incidence. The mode expansion theory is developed to help understand how the gradient metasurface tailors the incident beams, and the all-angle negative reflection occurs when the first negative order Floquet-Bloch mode dominates inside the metasurface slab. The coiling-up space structures are utilized to build desired acoustic gradient metasurface, and the all-angle negative reflections have been perfectly verified by experimental measurements. Our work offers the Floquet-Bloch modes perspective for qualitatively understanding the reflection behaviors of the acoustic gradient metasurface, and the all-angle negative reflection characteristic possessed by acoustic gradient metasurface could enable a new degree of the acoustic wave manipulating and be applied in the functional diffractive acoustic elements, such as the all-angle acoustic back reflector.

Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers

This paper is dedicated to the design and analysis of sampling methods to reconstruct the shape of a local perturbation in a periodic layer from measurements of scattered waves at a fixed frequency. We first introduce the model problem that corresponds with the semi-discretized version of the continuous model with respect to the Floquet–Bloch variable. We then present the inverse problem setting where (propagative and evanescent) plane waves are used to illuminate the structure and measurements of the scattered wave at a parallel plane to the periodicity directions are performed. We introduce the near field operator and analyze two possible factorizations of this operator. We then establish sampling methods to identify the defect and the periodic background geometry from this operator measurement. We also show how one can recover the geometry of the background independently from the defect. We then introduce and analyze the single Floquet–Bloch mode measurement operators and show how one can exploit them to built an indicator function of the defect independently from the background geometry. Numerical validating results are provided for simple and complex backgrounds.