Unidirectional Propagation Research Articles

Pseudospin topological phase transition induced by rotation operation in two-dimensional dielectric photonic crystal with C6 symmetry

We construct a kind of two-dimensional photonic crystal with triangular lattice in which the primitive cell contains six identical rods composed of two half cylinders with different dielectric materials. All the rods have the same rotations around themselves, ensuring that the system preserves the C6 symmetry. The pseudospin topological phase transition (TPT) of the system is realized by rotating the rod elements, and the helical edge states (HESs) are constructed along the interface between two photonic crystals with opposite rotations. Numerical results and simulations reveal that the HESs are topologically protected with the pseudospin-locked unidirectional propagation of electromagnetic waves with robustness against corners and defects along the interface. Adjusting the size of rod elements, the TPT can be engineered at needed frequency.

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.

Orbital stability of smooth solitary waves for the Degasperis-Procesi equation

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L 3 L^3 , but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L 2 L^2 -norm, resulting in L 3 L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L 2 L^2 norm for DP equation, which cannot be used to control the L 3 L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L ∞ L^\infty orbital norm of the perturbation is bounded above by a function of its L 2 L^2 orbital norm, yielding the higher order control and the orbital stability in the L 2 ∩ L ∞ L^2\cap L^\infty space.

Reconfigurable unidirectional propagation of electromagnetic waves in photonic crystal waveguides

We demonstrate reconfigurable unidirectional propagation of electromagnetic waves in waveguide channels sandwiched by two dielectric photonic crystal (PC) slabs or magnetic PC (MPC) slabs, where the extrinsic and intrinsic responses emerge in two kinds of systems. Concretely, the unidirectionality of the MPC system originates from the time-reversal symmetry breaking nature of magnetic material, while in the dielectric PC system, the unidirectionality is achieved by active control of two separated line sources. The results indicate that in the MPC based system, not only the amplitude but also the directionality of the guiding electromagnetic waves is flexibly regulated by controlling the separation and phase difference of two active line sources. However, the directionality in the MPC based system is determined by the magnetization due to the intrinsic unidirectionality of the magnetic system, but for the dielectric PC based system, directionality is controlled by active sources. The introduction of a second line source in the MPC based system can result in two remarkably different consequences. On one hand, the directionality can be switched from forward to backward propagation by reversing magnetization. On the other hand, the same operation results in the transition from unidirectional propagation to localization of the electromagnetic waves by dynamically controlling two active sources. The functionality and flexibility in the present systems might find potential applications in microwave photonics and integrated photonics.

Transverse spin and the hidden vorticity of propagating light fields.

Spatially inhomogeneous fields of electromagnetic guided modes exhibit a complex of extraordinary dynamical properties such as polarization-dependent transverse momentum, helicity-independent transverse spin, spin-associated non-reciprocity and unidirectional propagation, etc. Recently, the remarkable relationship has been established between the spin and propagation features of such fields, expressed through the spin-momentum equations [Proc. Natl. Acad. Sci. USA118, e2018816118 (2021) PNASA60027-842410.1073/pnas.2018816118] connecting the wave spin with the curl of momentum. Here, the meaning, limitations, and specific forms of this correspondence are further investigated, involving physically transparent and consistent examples of paraxial light fields, plane-wave superpositions, and evanescent waves. The conclusion is inferred that the spin-momentum equation is an attribute of guided waves with a well-defined direction of propagation, and it unites the helicity-independent "extraordinary" transverse spin with the spatially inhomogeneous longitudinal field momentum (energy flow) density. Physical analogies with the layered hydrodynamic flows and possible generalizations for other wave fields are discussed. The results can be useful in optical trapping, manipulation, and data processing techniques.

On the Cauchy problem for a class of cubic quasilinear shallow-water equations

In this paper, we mainly investigate the Cauchy problem of the cubic Camassa-Holm-type equations which arise as an asymptotic model with a nonlocal cubic nonlinearity for the unidirectional propagation of shallow water waves, which admit the single peaked solitons and multi-peakon solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model. Then, by overcoming the difficulties cased by the complicated mixed nonlinear structure, we present a precise blow-up scenario. Moreover, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions for Cauchy problem in the Gevrey-Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map. Finally, we investigate the persistence properties of the solution to the Cauchy problem, we prove that the solution maintains the corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

Advances in Metasurfaces: Topology, Chirality, Patterning, and Time Modulation

Advances in metasurface design has given rise to the unique ability of manipulating the phase and propagation of electromagnetic (EM) waves. The application of topological protection, chirality, and other concepts borrowed from condensed matter physics have yielded a rich new source of structures with attractive features. Topological metasurfaces show that symmetry can be used to create unique EM modes that are, for instance, immune to many types of scattering. Similar modes can also be created using defects in homogeneous materials, essentially creating chiral waveguides with polarization-locked propagation. Such defects may have topological protection in real space and/or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -space and lead to modes without global changes to the lattice. This review article explores recent results in chiral and topological metasurfaces, primarily consisting of printed metal patterns, but also planar dielectric structures, at microwave, optical, and acoustic frequencies. The exciting potential applications include nonreciprocal structures, self-collimating waves based on diffusive transport, defect-based amorphous structures, time-modulated metasurfaces, and new kinds of chiral waveguides that exhibit unidirectional propagation without requiring a surrounding periodic structure.

Diverse analytical wave solutions of plasma physics and water wave equations

In plasma physics and water waves, the Gilson-Pickering equation is an important unidirectional wave propagation model. A large number of analytic wave solutions have been established in the form of hyperbolic, trigonometric, exponential, and rational functions using the advanced auxiliary equation approach in this study. The solutions obtained have been compared with the solutions available in the literature, and it is observed that we have established further wave solutions than the solutions determined by the other methods, namely the (G′/G2)-expansion method, sinh-Gordon approach, etc. This method uses a homogeneous balance rule to estimate a polynomial-type solution and delivers an order. A traveling wave transformation has been used to convert the governing equation into a nonlinear differential equation. Each solution includes a variety of parameters related to the model and method. Using different parameter values, the nature of wave solutions is defined by three-dimensional (3D) and two-dimensional (2D) wave profiles. The solitons change their nature and location for the particular values of the dispersion coefficient α and free parameter κ, and shown in 2D figures to illustrate the different forms of solitons.

Y-branch wavelength demultiplexer based on topological valley photonic crystals

Inspired by topological insulators in the condensed matter physics, topological photonics is a new paradigm, which is free of light backscattering at sharp waveguide bends due to the topological protection, and shows unidirectional light propagation based on the valley-chiral locking. Herein, we propose a two-channel Y-branch wavelength demultiplexer operating at the telecom band using the topological photonic integrated circuits for the first time. The proposed Y-branch wavelength demultiplexer is designed as adjoining and separating of two different topological interfaces at its input and output ports, respectively. Topological edge states with spectral interleaving can be excited in the two different topological interfaces, and be split into two output ports of the demultiplexer in short and long wavelengths. The Y-branch topological waveguide can be used as a dichroic mirror or a wavelength filter for separating light at two different wavelengths. The proposed topological integrated circuit is immune to structural defects. It is promising for the practical applications of optical interconnects and signal processing on a silicon photonic chip.

Curvilinear spin-wave dynamics beyond the thin-shell approximation: Magnetic nanotubes as a case study

Surface curvature of magnetic systems can lead to many static and dynamic effects which are not present in flat systems of the same material. These emergent magnetochiral effects can lead to frequency nonreciprocity of spin waves, which has been shown to be a bulk effect of dipolar origin and is related to a curvature-induced symmetry breaking in the magnetic volume charges. So far, such effects have been investigated theoretically mostly for thin shells, where the spatial profiles of the spin waves can be assumed to be homogeneous along the thickness. Here, using a finite-element dynamic-matrix approach, we investigate the transition of the spin-wave spectrum from thin to thick curvilinear shells, at the example of magnetic nanotubes in the vortex state. With increasing thickness, we observe the appearance of higher-order radial modes which are strongly hybridized and resemble the perpendicular-standing spin waves (PSSWs) in flat films. Along with an increasing dispersion asymmetry, we uncover the curvature-induced nonreciprocity of the mode profiles. This is explained in a very simple, general picture for thick curvilinear shells, considering the inhomogeneity of the emergent geometric volume charges along the thickness of the shell. Such curvature-induced mode-profile asymmetry also leads to nonreciprocal hybridization, which can facilitate unidirectional spin-wave propagation. With that, we also show how curvature allows for nonlinear three-wave splitting of a higher-order radial mode into secondary modes which can also propagate unidirectionally. We believe that our study provides a significant contribution to the understanding of the spin-wave dynamics in curvilinear magnetic systems, and also advertises these for novel magnonic applications.

Breaking reciprocity by designed loss

In this paper, we show how designed loss in open quantum systems can break the reciprocity of field propagation, and how non-reciprocal and even unidirectional propagation can be achieved for different kinds of designed loss, both linear and nonlinear. In particular, we show how unidirectional propagation can be achieved for input states of certain symmetry in linear schemes, and demonstrate the possibility of building a single-mode optical insulator by combining two kinds of nonlinear designed losses, and the way to build a non-reciprocal asymmetric field distributor with a planar structure of dissipatively coupled waveguides. We discuss the feasibility of the considered schemes and suggest possible realizations.

Leveraging nonlinear wave mixing in rough contacts-based phononic diodes for tunable nonreciprocal waves

In recent years, phononic material-based diodes or “phononic diodes” have been explored to enable nonreciprocal wave propagation for novel applications such as unidirectional filters and waveguides. Despite extensive studies, the functionality and tunability of these diodes have been restricted to specific frequencies. Here, we exploit mixing of double-frequency (pump and probe) waves in nonlinear phononic materials to design a diode that can control the frequency, mode, and direction of energy propagation of the nonreciprocal waves. Our one-dimensional phononic diode is based on local and periodic nonlinearity, and global asymmetry. The weak nonlinearity is incorporated in the form of precompressed rough contacts, where contacts exhibit a quadratic force–displacement relationship. Using finite element models, we demonstrate tuning of the dispersion branch associated with the nonreciprocal wave by controlling the frequency of the pump waves. This allows propagation of nonreciprocal waves at frequencies lower or higher than the probe wave frequencies. Further, we show the ability to switch between reciprocal and nonreciprocal wave propagation, where the nonreciprocal response can cause either unidirectional or bidirectional energy propagation. This switching is possible through suitable selections of a pair of frequencies, and external precompression that controls the band gaps. The proposed phononic diode exhibits nonreciprocal wave propagation over a broad frequency range yet is simple in design and purely mechanical in construction. Moreover, due to its ability to filter and tune the frequencies of nonreciprocal waves, this diode will have potential applications as mechanical sensors and devices.

Bound valley edge states in the continuum

Topological valley photonics provides a unique way to manipulate the flow of light. In general, valley edge states that exhibit unidirectional propagation and are immune to defects and disorders could be realized at the interface between two valley photonic crystals with opposite valley Chern numbers. Herein, by merging the physics of valley edge states and bound states in the continuum, we propose and numerically demonstrate a novel, to the best of our knowledge, concept of edge states termed bound valley edge states in the continuum, which enjoys the topological features of valley edge states, such as, unidirectional propagation and immunity to disorders, but are formed at the interface between air and a single valley photonic crystal. Our results not only provide an effective way to reduce the size of valley photonic structures but also facilitate new applications where the proposed concept of bound valley edge states in the continuum could be exploited for optical sensing and unidirectional waveguiding.

Analytical solutions of simplified modified Camassa-Holm equation with conformable and M-truncated derivatives: A comparative study

This paper extracts some analytical solutions of simplified modified Camassa-Holm (SMCH) equations with various derivative operators, namely conformable and M-truncated derivatives that have been recently introduced. The SMCH equation is used to model the unidirectional propagation of shallow-water waves. The extended rational sine−cosine and sinh−cosh techniques have been successfully implemented to the considered equations and some kinds of the solitons such as kink and singular have been derived. We have checked that all obtained solutions satisfy the main equations by using a computer algebraic system. Furthermore, some 2D and 3D graphical illustrations of the obtained solutions have been presented. The effect of the parameters in the solutions on the wave propagation has been examined and all figures have been interpreted. The derived solutions may contribute to comprehending wave propagation in shallow water. So, the solutions might help further studies in the development of autonomous ships/underwater vehicles and coastal zone management, which are critical topics in the ocean and coastal engineering.

Bidirectional Whitham type equations for internal waves with variable topography

In the context of oceanic internal gravity waves, one of the most widely-used theories on examining wave dynamics and interpreting observational data is the Korteweg–de Vries (KdV) equation. Nonetheless, the characters of unidirectional propagation and unbounded phase and group velocities restrict its application to some general cases (Benjamin et al., 1972). Thus, using the Dirichlet–Neumann operator with the rigid-lid approximation, we derive both bidirectional and unidirectional Whitham type equations in the Hamiltonian framework, which retain the full linear dispersion relation of the Euler equations. The effect of topography is also incorporated in modeling due to its practical relevance, although the invoked scaling plausibly excludes the accommodation of a significant bottom variation. There are no analytic solutions of internal solitary waves explicitly given in the newly proposed equations, even though these equations possess a concise form. Therefore, a modified Petviashvili iteration method is implemented to obtain the numerical solutions to circumvent this difficulty. Facilitated by these techniques, several numerical experiments are investigated and compared among different models: the KdV equation, the Whitham type equations, and the primitive equations. The discrepancies and similarities between the various models jointly indicate the advantage of full dispersion and bidirectional propagation and, thus, the effectiveness of the Whitham type equations.

Higher-Order Topological Phase of Interacting Photon Pairs.

Topological phases open a door to such intriguing phenomena as unidirectional propagation and disorder-resilient localization at a stable frequency. Recently discovered higher-order topological phases further extend the concept of topological protection enabling versatile control over localization in multiple dimensions. Motivated by the recent advances in quantum technologies such as large coherently operating qubit ensembles, we predict and investigate the higher-order topological phase of photon pairs emerging due to effective photon-photon interaction and described by the extended version of Bose-Hubbard model. Being feasible for state-of-the-art experimental capabilities, the designed model provides an interesting example of interaction-induced topological transitions in the few-particle two-dimensional system.

Orbital stability of the sum of smooth solitons in the Degasperis-Procesi equation

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model as an asymptotic approximation for the unidirectional propagation of shallow water waves. This work is to establish the L2∩L∞ orbital stability of a wave train containing N smooth solitons which are well separated. The main difficulties stem from the subtle nonlocal structure of the DP equation. One consequence is that the energy space of the DE equation based on the conserved quantity induced by the translation symmetry is only equivalent to the L2-norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. Our remedy is to introduce a priori estimates based on certain smooth initial conditions. Moreover, another consequence is that the nonlocal structure of the DP equation significantly complicates the verification of the monotonicity of local momentum and the positive definiteness of a refined quadratic form of the orthogonalized perturbation.

Interfacial delamination-induced unidirectional propagation of guided waves in multilayered media

Unidirectional nonreciprocal wave propagation is an unprecedented phenomenon, which has attracted much research interest. Connecting a phononic crystal with an asymmetric structure to break the spatial inversion symmetry is a popular manner to realize this phenomenon using the wave mode transformation. In this paper, a new model is proposed based on a single periodic structure. The arrays of asymmetric and symmetric interfacial delaminations are intentionally introduced into the top and the bottom part of a stack of periodic elastic layers, respectively. So, the structural spatial inversion symmetry can be broken and the guided waves can pass through the whole structure only from the top side with the changed mode generated by the array of asymmetric interfacial delaminations. Thus, it is indispensable for the part of phononic crystal that the partial band-gaps of symmetric and antisymmetric guided waves have to be separated, which is the reason why we introduce the array of symmetric central or side interfacial delaminations into the stack of periodic elastic layers. The transmission spectra of the guided waves and the dispersion curves for the unit cell imposed by the Bloch–Floquet boundary condition are both calculated by the spectral element method. Then, the interfacial delamination-induced unidirectional propagation of guided waves in the finite stack of periodic elastic layers is numerically confirmed. This paper provides a new concept to control the waves propagating in phononic crystals via the insertion of some interfacial delaminations or cracks.

Diversity of wave structures to the conformable fractional dynamical model

This manuscript examines the recently developed conformable three-dimensional Wazwaz–Benjamin–Bona–Mahony (3D-WBBM) equation’s dynamical behavior in terms of its spatial and temporal variables. The governing equation is stretch for the Korteweg-de-Vries equation that represents the unidirectional propagation of small amplitude long waves on the surface of hydro magnetic and acoustic waves in a channel, especially for shallow water. Solitary wave solutions of various types, such as kink and shock, as well as singleton, combined solitons, and complex solitons, are all retrieved. Additionally, solutions to hyperbolic, exponential, and trigonometric functions are obtained through the use of recently developed methods, namely the Kudryashov method (KM), the modified Kudryashov method (MKM), and the new extended direct algebraic method (NEDAM). The study conducts a comparison of our findings to well-known findings, and concludes that the solutions reached here are novel. Additionally, the earned results are sketched in different shapes to demonstrate their dynamics as a function of parameter selection. We can assert from the obtained results that the applied techniques are simple, vibrant, and quite well, and will be helpful tool for addressing more highly nonlinear issues in various of fields, especially in ocean and coastal engineering. Furthermore, our findings are first step toward understanding the structure and physical behavior of complicated structures. We anticipate that our results will be highly valuable in better understanding the waves that occur in the ocean. We feel that this work is timely and will be of interest to a wide spectrum of experts working on ocean engineering models.

Transversely oriented cylindrically polarized optical fields.

Cylindrical vector (CV) beams have nonuniform polarization vector distribution with a singularity line directed along the optical axis. In this paper, we propose a method to synthesize transversely oriented cylindrically polarized optical fields in the focal region with a singularity line perpendicular to the optical axis. The scheme is based on the time-reversal method, the vectorial diffraction theory, and the 4Pi optical configuration. Both transversely oriented radially polarized and azimuthally polarized optical fields are demonstrated. The superposition of transverse cylindrically polarized optical fields leads to a peculiar distribution carrying controllable transverse spin angular momentum (SAM) and transverse orbital angular momentum (OAM) that may find applications in optical tweezing, light-matter interaction, and unidirectional beam propagation excitation.