The algorithm is derived for the method of lines but can be used in other eigenmode algorithms as well. Photonic floquet media with a complex timeperiodic permittivity. The wave solutions for periodic structures are distinguished from those for the uni form waveguide in that. Combining the two situations, a periodic excitation on a. In this paper, the floquetbloch theorem is used to set up an. We present a new theoretical approach regarding the design of 2d periodic structure at optical frequencies. Details supporting our discussion here can be found.
New formula for evaluating the number of unit cell of a. Floquet engineering with quasienergy bands of periodically. Example from collin of a coaxial line with periodic rings. Wave scattering on lattice structures involving an array. Floquetbloch theorem is widely applied for computing the dispersion properties of periodic structures, and for estimating their wave modes and group velocities. Hence periodic structures have become a field of intense research activities. The most common example of blochs theorem is describing electrons in a crystal. Hence, the periodic structure acts as a slow wave structure in its passband region. Pdf modelling of periodic structures using the finite. The floquet theorem is combined with the finitedifference timedomain fdtd method to obtain sufficiently accurate results for the scattering parameters of a metal strip grating. Floquets unit cell design for periodic structures at optical. Floquetsunitcelldesignforperiodicstructuresat opticalfrequencies.
For the periodic structures, according to floquet theorem, periodic field distributions will be generated on the structure apertures, which can further be seen as the superposition of infinite. Modelling of periodic structures using the finite difference time domain method combined with the floquet theorem article pdf available in electronics letters 295. The model is based on floquet s theory and on the variational equivalent circuit. In section 3 and 4 some general properties of periodic structures floquet bloch theorem and the bragg condition are deduced.
Floquetbloch theory and its application to the dispersion. Department of mechanical engineering, indian institute of technology kanpur, kanpur, india. In section 5 coupled wave theory for periodically layered media based on a perturbation analysis is described, while in section 6 the realization of a. A floquetbloch decomposition of the elastodynamical equations for. The determination of floquet modes for asymmetric periodic structures is presented. Calculation of the mutual coupling parameters and their. However, a blochwave description applies more generally to any wavelike phenomenon in a periodic medium. Floquet waves are waves that naturally propagate in periodic structures and are analogous to the waves that propagate in homogenous structures. However, it seems quite confusing to me since the former involves a second order differential equation schroedinger equation with a periodic potential while the latter is defined for a first order one. If the periodic structure is an infinite array, simple methods based on the floquets theorem 1 or the periodic greens function 10, can be applied, where the characterization of the whole periodic structure can be reduced to the analysis of only a single cell. For example, a periodic dielectric structure in electromagnetism leads to photonic crystals, and a periodic acoustic medium leads to phononic crystals. On the dynamics of periodically restrained flexural. They are best understood by recalling floquets theorem for periodic structures along, say, direction, which states that. Abstract in this article, we present an enhanced broadband common.
We will study here the distribution of electromagnetic. The floquet bloch theory is presented as an alternative to them. Floquet most general representation of a periodic field. The main theorem of floquet theory, floquet s theorem, due to gaston floquet, gives a canonical form for each fundamental matrix solution of this common linear system. Bloch floquet theorem and modal analysis are used to examine the dispersion relation and band structure of the infinitely long periodic beam and the frequency spectrum of the finite beam, respectively. Pdf periodic structures and floquets theorem antonio. Absence of chaos in these systems guarantees longterm stability of structures around us. Blochs theorem, which is widely familiar in solid state physics, is another special case of floquet s theorem, applied to the eigenstates of an electron in a crystal with a potential that has a periodic structure in space. The floquet theorem gives the socalled floquetstate solution of eq. Physics department, suzhou university, suzhou 215006, peoples republic of china. Jan 12, 2017 this proposition is proved and a physical interpretation to the floquet solution is given in terms of counterpropagating waves when the formalism describes one dimensional wave propagation.
The periodic structures made of two, four and six sublayers of two different materials were selected as demonstration cases permanent address. Floquets theorem for periodic structures, which states that the amplitude of free response, vx, obeys the identity vx oct 25, 20 floquet theorem states that a hamiltonian periodic in time has quasistatic eigenstates that are evenly spaced by the drive photon energy. Periodic structures o repeated geometry, defined by a unit cell and a uniform periodic spacing. Floquet theorem can be applied to synthesize periodic structures where all antenna elements are coupled with each other 16 17 31. This approach analyzes 2d dielectric periodic structures. Starting with openor shortcircuit parameter descriptions of the periods a difference equation system is developed that allows the computation of the floquet modes. Floquet theorem article about floquet theorem by the free. Floquetbloch theorem is widely applied for computing the. These socalled floquet states can be regarded as a time analog of bloch states, which are the eigenstates of a hamiltonian periodic in space. Selective scattering between floquetbloch and volkov.
The same example is considered with various damping levels, in. This fact increases the complexity of the problem of the investigated problem and. Floquetbloch decomposition for the computation of dispersion. Setting of periodic structures our aim is to study an application of blochfloquet theory in the multiscale analysis of pde on periodic structures. Pdf floquetbloch theory and its application to the. Bloch proved that waves in such a medium can propagate without scattering, their behavior governed by a periodic envelope function multiplied by a planewave. The source of vibration for example from an elevator, a railway track as in. Solutions to a periodic nonlinear differential equation. Floquetstheory the periodic structure of figure 1a is analyzed by con. Floquet waves in a body with slender periodic structure. Then most of the approaches for periodic structures are based on the floquet theorem. Selective scattering between floquetbloch and volkov states. The distributed circuit model is developed through the use of the microwave network theory and the optical theory of the step discontinuities. If the periodic structure is an infinite array, simple methods based on the floquet s theorem 1 or the periodic greens function 10, can be applied, where the characterization of the whole periodic structure can be reduced to the analysis of only a single cell.
Dispersion curves play a relevant role in nondestructive testing. Floquets seminal paper dealt with the solution of 1d partial differential equations with periodic coefficients. They provide estimations of the elastic and geometrical parameters from experiments and offer a better perspective to explain the wave field behavior inside bodies. The theorem allows reducing computational costs through modeling of a representative cell, while providing a rigorous and wellposed spectral prob. Floquet modal analysis to modelize and study 2d planar.
Floquetbloch hereafter fb theory provides a strategy to analyze the behavior of systems with a periodic structure. The theorem allows reducing computational costs through modeling of a representative cell, while providing a rigorous and wellposed spectral problem representing wave dispersion in. This is a consequence of the floquet theorem 12, which states that a hamiltonian periodic in time with period t has eigenstates that are evenly spaced by the drive energy 2. Power loss due to periodic structures in highspeed packages. Synthesis and optimization of almost periodic antennas using. By using the periodic structure theory pst intro duced by. Analogous to how a bloch state can be decomposed into a. Spatial harmonics and floquet theorem let us expand the above simple description to more general electromagnetic system with a periodic configuration. The study of wave propagation in threedimensionally periodic media was pioneered by felix bloch in 1928, unknowingly extending an 1883 theorem in one dimension by g. Floquet theory is widely used in the analysis of stability of dynamical systems, including the mathieu equation and hills differential equation for approximating the motion of the moon. In this article, we present an enhanced broadband common. We apply floquets theorem to relate the fields in similar periodic structure media and find the dispersion relationship between frequency, propagation constant. This accuracy can be achieved by using a suitable absorbing boundary condition. Two coupled oscillators with periodic parametric excitation.
Because of the pseudoperiodicity of the transformed fields, the conventional grating theory based on the floquet theorem becomes possible to be applied for the. Brillouin diagram for 3d gaas periodic structures with. Let us expand the above simple description to more general. The formulation of the momcem method is applied to this reference cell with periodic walls in modal space, and the distribution of magnetic field differs only by two phases, compared to other cells. Determination of floquet modes in asymmetric periodic structures. Floquets theorem states that the wave in periodic structures consists of an infinite number of space harmonics. Nevertheless stable classical floquet systems are ubiquitous in nature and cover the whole range of scales from living species to planet and galactic systems. It is often stated that blochs theorem and floquet s theorem are equivalent, even the blochs theorem is often referred as floquet bloch theorem. Periodic structures ee625 periodic structures and floquet s theorem.
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