While the relevant topological invariants for the quantum hall effect are the chern numbers of the respective bands of the hamiltonian, the invariants. Here, the authors report a predictive floquet engineering of various topological phases. Floquetbloch modes excited inside the acoustic gradient metasurface. The theorem allows reducing computational costs through modeling of a representative cell, while providing a rigorous and wellposed spectral prob. Research article floquetbloch theory and its application to. Floquet theory for discontinuously supported waveguides. Details regarding the boundary condition inputs you must supply and the internal treatment at boundaries are provided. Considering the appropriate boundary conditions for the wave function and its derivative, we derive the determining equations for the floquet bloch eigenstates, which. Floquetbloch theory and topology in periodically driven lattices.
We revisit the floquet bloch eigenstates of a onedimensional electron gas in the presence of the periodic kronigpenny potential and an oscillating electronic field. The unique characteristic of allangle negative reflection is that both. Bloch wave is inonite in the semiinonite layered medium and exponentially decays in the adjacent homogeneous medium. Making use of floquet theory, we focus on translationally invariant situations in which the singleparticle dynamics can be described in terms of spatially extended floquet bloch waves. Mar 26, 2015 however, i dont know how to setup periodic bloch floquet boundary conditions so that i can loop over a set of kx and ky wavevectors. This procedure allowed the retrieval of the dispersion relationships. Creating stable floquetweyl semimetals by laserdriving. Incident plane wave arbitrary propagation direction floquetbloch boundaries in the frequency domain. The emphasis is on the development of a good physical understanding of the nature of these waves, using the wavevector diagram to summarize their spatial dispersion and spectra. Observation of floquet bloch states on the surface of a topological insulator.
The effect of a shear flow on an acoustic liner consisting of a perforated plate backed by cavities is studied. We present an absorbing boundary condition for electromagnetic frequency domain simulations of photonic crystals and metamaterials. The photonic crystal or metamaterial in question can include lossy, active, anisotropic, and even bi. To impose such dirichlet boundary conditions an extrusion coupling variable maps of displacements from the source face is exported to the destination face corresponding to the. A bloch wave also called bloch state or bloch function or bloch wavefunction, named after swiss physicist felix bloch, is a type of wavefunction for a particle in a periodicallyrepeating environment, most commonly an electron in a crystal. Optics of floquetbloch waves in dielectric gratings.
A t x, \displaystyle \dot xa tx, a t \displaystyle \displaystyle a t a piecewise continuous periodic function with period. Floquetbloch waves in onedimensional photonic crystals. We develop a general formalism for the scattering of these floquet bloch waves. Pdf floquetbloch theory and topology in periodically. Overall, the results add to a better understanding of the behavior of light.
By definition, if a wave is a bloch wave, its wavefunction can be written in the form. The stability regions together with the dispersion relation are found in terms of the floquet theory through the discriminant. Observation of floquetbloch states on the surface of a topological insulator. The floquetbloch theory is presented as an alternative to them. Jun 09, 2010 therefore im getting used to the periodic boundary conditions and the floquet bloch phase shift. It is shown that floquetbloch theory offers some advantages. The floquetbloch boundary conditions are imposed on a reduced order linear model in the only dynamically active degreesoffreedom. Propagation of light in a onedimensional photonic crystal. Floquet theory applicable to linear ordinary di erential equations with periodic coe cients and periodic boundary conditions. The fourier transform will change a di erential equation in to an algebraic equation.
Floquet bloch periodic boundary conditions are enforced. Oct 23, 2017 floquet bloch modes excited inside the acoustic gradient metasurface. Idea if a linear di erential equation has periodic coe cients and periodic boundary conditions, then the solutions will generally be a. Special attention is given to the cases where floquetbloch solutions become periodic functions. Periodic boundary conditions and floquetbloch phase shift.
They provide estimations of the elastic and geometrical parameters from experiments and offer a better perspective to explain the wave field behavior inside bodies. For floquetbloch systems in two spatial and one time dimension, the w 3 invariant is of primary interest. Boundary conditions floquetbloch periodic boundaries have to be applied for periodic objects gratings, photonic crystals h x e y e z physical grid. Computing floquetbloch modes in biperiodic slabs with. Therefore im getting used to the periodic boundary conditions and the floquet bloch phase shift. Due to the periodic boundary conditions, these spaces contain functions which.
The applied boundary conditions are equalities of all 3d displacements on the two pairs of lateral faces. In section 3 and 4 some general properties of periodic structures floquetbloch theorem and the bragg condition are deduced. Gratingassisted coupling of light between semiconductor. In fact, it has been shown that in such structures buckling can be explored considering. Due to the complexity, approximate theories were developed with a limited applicability, but which lead to a solution in a faster and easier way. The function u in is a floquetbloch mode in if the propagation vectors satisfy.
Floquetbloch theory and its application to the dispersion. Theory for the transmission line boundary conditions. In this paper, there is a periodic laser pulse hitting the sample which gives a timeperiodic hamiltonian. Dispersion equations and oeld distributions of surface waves modes localised near the boundary of the semiinonite layered medium are derived and conditions of their appearance are.
Floquetbloch solutions are found and illustrated for the bandgaps, allowed bands, and bandedges of the crystal. Sofar my implementation looks as shown below, where i calculate eigenmodes for the unit cell with outer edges held constant at u0. Bloch proved that waves in such a medium can propagate without scattering, their behavior governed by a periodic envelope function multiplied by a planewave. Efficient computation of the w 3 topological invariant and. Observation of floquetbloch states on the surface of a. Floquet eigenmodes of a periodic crystal, including both propagating and evanescent modes. Since the resonators can be designed to open and shift band gaps, an optimal design, focused on the largest possible gap in the lowfrequency range, is achieved by solving a maximization problem in the bounded. Keywords sawtooth photonic crystal band structure floquetbloch solutions 1 introduction. The information in this chapter is divided into the following sections. Bloch floquet boundary conditions in weak form constraint. East, west, north, south, top, and bottom denote the sides of the ve from 6. We revisit the floquetbloch eigenstates of a onedimensional electron gas in the presence of the periodic kronigpenny potential and an oscillating electronic field. Hence the kvalues found for any are also part of the floquetbloch spectrum of. Additionally, it has n such j and together they satisfy e1t e2t ent exp z t 0 tras ds 3.
For instance, u ftranforms to, on applying fourier transform. Although bloch studied quantum mechanics, leading to the surprising result that electrons in a conductor scatter only. Creating and controlling topological states of matter has become a central goal in condensed matter physics. A floquetbloch decomposition of maxwells equations, applied. The first set of boundary conditions accounts for the periodicity of the fractures, and the second set accounts for the mechanical. Boundary conditions this chapter describes the boundary condition options available influent. Floquetbloch theorem is widely applied for computing the dispersion properties of periodic structures, and for estimating their wave modes and group velocities. It is shown that floquet bloch theory offers some advantages conceptually over the commonly used coupled.
Motivated by recent experimental implementations of artificial gauge fields for gases of cold atoms, we study the scattering properties of particles that are subjected to timeperiodic hamiltonians. M 0e 0, where e 0 is a solution in a background medium m 0,byintroducing sources j. Sep 27, 20 bloch floquet boundary conditions in weak form constraint. Floquetbloch decomposition for the computation of dispersion. Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form. An asymptotic approximation to the lowest cutoff frequency is given. Boundary conditions floquetbloch periodic boundaries applied for calculating the band structure of a pc geometry of a pc unit cell with cylindrical inclusion periodic boundary with a particular andeinc. Considering the appropriate boundary conditions for the wave function and its derivative, we derive the determining equations for the floquetbloch eigenstates, which are represented by a single.
In particular, it gives the number of anomalous edge states that can not be predicted with other invariants such as the simpler w 1 invariant, which is directly related to the winding number of the eigenvalues of a unitary map, or the. A floquetbloch decomposition of maxwells equations. The behavior of light in dielectric gratings is discussed in terms of the optical floquetbloch waves or modes. Research article floquet theory for discontinuously supported.
Jan 17, 2017 creating and controlling topological states of matter has become a central goal in condensed matter physics. Space harmonic distribution at bragg condition in periodical dielectric waveguides naihsiang sun, yuwei liu, mingyu tsai, liti kung, jungsheng chiang. Butler, fellow, ieee, naihsiang sun,member, ieee, gary a. Floquetbloch periodic boundary conditions are enforced. Topological invariants have gained considerable interest in solid state physics and related fields through their application to the quantum hall effect 1, 2, topological insulators 36, and more recently, floquetbloch systems 719. Evans, fellow, ieee, lily pang, member, ieee, and philip congdon, member, ieee abstract floquetbloch theory is used to calculate the elec. The behavior of light in dielectric gratings is discussed in terms of the optical floquet bloch waves or modes.
Floquetbloch solutions in a sawtooth photonic crystal. For a periodic medium according to the floquet bloch theorem, the light propagation can be represented as a. Pdf periodic structures and floquets theorem antonio. Dispersion curves play a relevant role in nondestructive testing. The floquet bloch theory is presented as an alternative to them. Research article floquetbloch theory and its application. Elastic wave propagation along a set of parallel fractures. This boundary condition can simultaneously absorb multiple blochfloquet eigenmodes of a periodic crystal, including both propagating and evanescent modes. The unique characteristic of allangle negative reflection is that both the incident wave and reflected wave appear at the. Creating stable floquetweyl semimetals by laserdriving of.
A bloch wave also called bloch state or bloch function or bloch wavefunction, named after swiss physicist felix bloch, is a kind of wave function which can be written as a plane wave modulated by a periodic function. Making use of floquet theory, we focus on translationally invariant situations in which the singleparticle dynamics can be described in terms of spatially extended floquet. Periodicity compatible mixed boundary conditions pmubc applied on the sides of a volume element for six independent uniform strain load cases. Research article floquet theory for discontinuously. The exact boundary conditions as well as the extension operators are computed by solving local problems on a single periodicity cell. Osa absorbing boundary condition for blochfloquet eigenmodes. First, the duct and the liner are considered as a periodic system while in the second approach the liner is considered as homogeneous and described by an impedance. However, i dont know how to setup periodic blochfloquet boundary conditions so that i can loop over a set of kx and ky wavevectors. It is however often unclear where the applied approximate theory is no longer valid. Periodic blochfloquet boundary conditions for calculating. Perfectly matched layer pml boundary conditions are implemented in the numerical scheme to compute the outgoing radiation 25.
The first approach computing the floquet propagators leads to compute non standard eigen solutions of a problem where the eigenvalues i. Yields knowledge of whether all solutions are stable. The photonic crystal or metamaterial in question can include lossy, active. Sep 27, 2016 the effect of a shear flow on an acoustic liner consisting of a perforated plate backed by cavities is studied. Pdf floquetbloch waves in onedimensional photonic crystals.
1344 1374 596 181 371 243 604 688 1521 1425 1450 777 1531 496 865 1355 421 91 1374 159 124 894 612 698 827 960 1394 1436 252 1179 502 906 1035 284 874 258 832 980