Note that emtl was initially developed mostly for fdtd simulations. Application of the plane wave expansion method to a two. From these plane waves, the current induced by the array in a single. Rami mehrem 2009, the plane wave expansion, infinite integrals and identities involving spherical bessel functions, arxiv. In the plane wave expansion method, the impedance sum is ned by transforming the vector potential of an infinite array. Fullpotential linearized augmented planewave method. In addition, the effectivemedium tensor is applied to improve convergence. We have implemented this method for finding traveling wave solutions of nonlinear burgers equation, kdv equation, rlwburgers equation and hirotasatsuma equation system. It expands the field into a set of plane waves and converts maxwells equations to matrix formbyassigning each plane wave a complex amplitude. This algorithm combines the plane wave expansion method with perfectly matched layers for the termination of the computational region in the direction out of the plane. A plane wave expansion of spherical wave functions for modal analysis of guided wave structures and scatterers robert h. More recently, the plane wave discontinuous galerkin pwdg method has been studied by hiptmair et al. You can download this pdf file and related source file from my personal homepage.
The pwe method can therefore not account for materials with loss, or. Plane wave approximation since many articles are devoted to the mathematical aspects of the fmm, we will skip the mathematical derivation of the algorithm. We start by recalling the theory of classical plane wave spectrum for regularly distributed near. The dephasing of the plane wave components results in the diffractive broadening of the beam, as showed in fig. This solution is a wave \traveling in the direction of k in the sense that a point of constant phase, meaning k. The theoretical investigation is performed using the plane wave expansion method in both the optical and acoustical regimes. There are several methods to calculate the dispersion relation and therewith the range of the band gaps. The condition for solving fors and t in terms ofx and y requires that the jacobian matrix be nonsingular. Summary the plane wave expansion method allows the calculation of dispersion curves i. Plane wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers shouyuan shi, caihua chen, and dennis w. In this document we will apply the plane wave expansion method on a speci c photonic crystal.
Macphie, life fellow, ieee, and keli wu, senior member, ieee abstract a new finite plane wave series expansion for spherical wave functions of the first kind is presented in this paper. Plane wave approximation of homogeneous helmholtz solutions a. A plane wave expansion of spherical wave functions for modal. This method is popular among the photonic crystal community as a method of solving for the band structure of specific photonic crystal geometries. He regarded this as proof that fresnels wave theory was nonsense, and that light must be a particle and not a wave. From those we will derive a set of equations, whose solutions can be approximated by the plane wave expansion method.
This method is popular among the photonic crystal community as a method of solving for the band structure dispersion relation of specific photonic crystal geometries. We implemented plane wave expansion pwe method as a part of electromagnetic template library emtl. Any periodic function can be expanded using fourier series, so is for e and h. I show how i deal with the 1d, 2d and 3d cases, so you can deal with other problems in. For that we refer the reader to previous articles 21, for example, or 55, which give the details of the method. In this thesis the plane wave expansion method and the guided mode expansion method have been. This algorithm combines the plane wave expansion method with perfectly matched layers for the.
Usual methods of solving the plane wave expansion of the kohnsham equations is by diagonalisation of the hamiltonian matrix whose elements are given by the term in curly brackets. Furthermore, we have a plane wave, by which we mean that a. Planewave expansion method for calculating band structure. Improved planewave expansion method for band structure. Another useful representation of the dyadic greens function is its fourier transform where the. The plane wave pseudopotential method 1 bornoppenheimer approximation 2 apply densityfunctional theory dft to calculate the electronic structure. Osa planewave expansion method for calculating band. Planewave expansion of cylindrical functions in lossy media. Lecture 18 cem plane wave expansion method youtube.
In 1818, poisson used fresnels theory to predict this phenomenon. That is, when the average interaction energy between the incident particle and the scattering potential is much smaller than the particles incident kinetic energy, the scattered wave can be considered to be a plane wave. Fjam es,as tudngio fourier transforms optics played a major role in all the physics revolutions of the 20th century, so weoll do some. Pdf a planewave expansion method for analyzing propagation in. Pdf we present a new algorithm for calculation of the band structure of photonic crystal slabs. The first is energy cutoffs, which is the cutoff for the wave the input files. Photonic band structure and eigenmodes of magnetophotonic. The feed lies outside the closed surface, so that the volume v. Unfortunately, the function that describes the material in the. The method described using the expansion creates spurious modes with zero frequency, which were removed. In this case the problem is cast in the form of a discontinuous galerkin method, with possibly mesh dependent penalty parameters. Spherical wave expansion of vector plane wave in discussing the scattering or absorption of electromagnetic radiation by localized systems, it is useful to be able to express a plane electromagnetic wave as a superposition of spherical waves.
The plane wave expansion method is one of the numerical methods that has been used. Plane waves and free spherical waves both form an onshell complete orthnormal set on with or without the origin. Expansion of a plane wave into gaussian beams springerlink. Inoue1 1 t oy h as iu nv er t f cl g,1m pku a 4 85 0 j.
This algorithm combines the planewave expansion method with perfectly matched layers for the termination of the computational region in the direction out of the plane. Publishers pdf, also known as version of record link back to dtu orbit citation apa. Outofplane dispersion and homogenization in photonic. By doing so we will derive the dispersion relation and the density. Consider, first of all, the expansion of a scalar plane wave as a set of scalar spherical waves. Plane wave approximation of homogeneous helmholtz solutions. From the geometrical point of view, the wave diffraction can be described as fourier decomposition of light beam into plane waves which, in propagationacquire, phase shifts depending on their propagation direction. Planewave expansions pwes based on fourier transform and their physical. Also, an alternative method to obtain the expansion coefficients of the plane. The planewaveexpansion pwe approach dedicated to the simulation of periodic devices has. The method allows the computation of eigenfrequencies for a photonic crystal to any prescribed accuracy, commensurate with computing time. This expansion in terms of plane waves is useful since the scattering solution of many problems to plane wave excitation is known. It expands the field into a set of plane waves and converts maxwells equations to matrix formbyassigning each plane wave a. On the relationship between the spherical wave expansion.
The larger the rc i, the heavier the terms in direct space. It describes how to construct electromagnetic band diagrams and isofrequency contours. Waves, the wave equation, and phase weoll start with optics. The spatial frequency spectrum of the cws can obviously be obtained through numerical fourier transform techniques, however, the knowledge of the analytical closedform expression of the spectral representation gives substantial benefits. Numerical investigations of twodimensional photonic. Second choice is a core of plane wave expansion pwe method that is very suitable for calculation of photonic crystals band structures. Besides, we also provide an example for testing and explaining, and we also compare the results with the band structure derived from conventional plane wave expansion method, then we finally find that there is a good consistency between them.
Introduction to planewave basis sets and pseudopotential. What links here related changes upload file special pages permanent link page. The wave function of a two spin 12 particle system is either symmetric or antisymmetric. It follows that the size of the hamiltonian matrix is. Matlab implementation of the rigorous coupled wave analysis and plane wave expansion method chrisfaddenrcwa. In this paper, we study the approximation of solutions of the homogeneous helmholtz equation. Matrix method for antenna plane wave spectrum calculation. In particular, a plane to plane method for the determination of the aperture. Pw expansion of nic has to be converged with density cuto. In this paper, an additional advantage to the plane wave expansion is presented, whereby the expansion itself is used to derive analytical solutions to integrals over spherical bessel functions and identities involving them.
The planewave pseudopotential method 1 bornoppenheimer approximation 2 apply densityfunctional theory dft to calculate the electronic structure. Plane waves can be expanded in terms of free spherical waves by. This manual is based on my implementation of pwm plane wave method using matlab. The matrix method has been implemented, and its potentialities are presented. Oct 04, 20 this lecture steps the student through the formulation and implementation of the plane wave expansion method. In particular, this allows for the possibility that the projected characteristics may cross each other. Compared with the traditional pwe methods for metal phcs, the band structures can be calculated directly in our method and no further procedures are needed to handle the folded band structures.
Pdf the planewave expansion method has been the basis for much of the understanding of photonic and electrical behavior in crystals. Plane wave expansion method in computational electromagnetism. Gaussian, hermitegaussian, and laguerregaussian beams. Pdf planewave expansion method for calculating band structure. Plane wave expansion method refers to a computational technique in electromagnetics to solve the maxwells equations by formulating an eigenvalue problem out of the equation. Application of the plane wave expansion method to a photonic crystal in form of a twodimensional, triangular arrangement of air holes by hermann edlbauer, lukas schickhofer and markus autscthnig 1 introduction photonic crystals are composed of periodic, dielectric structures that a ect the propagation of electromagnetic waves. There is a further truncation of plane wave expansion in plane wave calculations. The plane wave expansion method pwem provides a highly efficient numerical solution to maxwell us s equations for devices with low to moderate dielectric contrast. Pdf on jun 1, 2009, vincent laude and others published plane wave expansion method for phononic crystals. Planewave expansion method for calculating band structure of. This lecture steps the student through the formulation and implementation of the plane wave expansion method.
The maxwell equation in a transparent, timeinvariant, source free, and nonmagnetic medium can be written in the following form. Namely, only the reciprocal lattice vectors whose kinetic energy lower than a predefined maximum cutoff energy, are kept in the expansion, while the rest of the coefficients are set to zero. An expansion method for finding traveling wave solutions to nonlinear pdes. In, the expansion into plane waves of a cylindrical wave propagating in a lossless medium has been derived. For this reason, the plane wave expansion method may not be every. Scott, the spectral domain method in electromagnetics, norwood. Pdf plane wave expansion method for phononic crystals. This method is popular among the photonic crystal community as a method of solving for the band structure dispersion relation of specific photonic crystal.
The plane wave expansion method pwem provides a highly efficient numerical solution to maxwells equations for devices with low to moderate dielectric contrast. The plane wave spectrum is used to determine the far. We need outgoing wave so we pick hankel function of the first kind summary. The plane wave expansion method was carried out to construct dispersion curves for along the brillouin zone shown in figure 4.
It expands the field into a set of plane waves and converts maxwells equations to matrix form by assigning each plane wave a complex amplitude. I am not going to list all source files here, they are pretty short and can be easily modified for your own purpose if you know a little bit knowledge about matlab programming. There are some tricks in the derivation process, so we show the process in detail. Digital library of mathematical functions, equation 10. We present a new algorithm for calculation of the band structure of photonic crystal slabs. That means that one must be able to expand one in terms of the other. The plane wave expansion method allows the calculation of dispersion curves i.
Wave functions waveguides and cavities scattering separation of variables the special functions vector potentials the spherical bessel equation each function has the same properties as the corresponding cylindrical function. An integral expansion which expresses a plane monochromatic wave as a superposition of gaussian beams is found. Photonic crystals are composed of periodic, dielectric structures that affect the propaga tion of electromagnetic waves. Acknowledgements this work would not have been successful without the guidance and the help of several individuals. A general complex eigenvalue problem is then obtained. Plane wave expansion method for photonic band gap calculation. Numerical computation of band gaps in photonic crystal fibres. It expands the field into a set of plane waves and converts maxwells equations to matrix form by assigning each plane wave a. The term is also used, even more specifically, to mean a monochromatic or sinusoidal plane wave. Plane wave expansion method pwe refers to a computational technique in electromagnetics. Implications in the interpretation of planewave expansions in lossy.
In other words this work is an attempt to devise a formula for wave propagation from cylindrical surface in spectral domain and apply it to digital holography. Pwe is traceable to the analytical formulations, and is useful in calculating modal solutions of maxwells equations over an inhomogeneous or periodic geometry. The matrix method for plane wave expansion in this section we present the mathematical basis of the matrix method to calculate the pwe from a rectangular irregular near. Plane wave expansion method 1d photonic crystal considering electromagnetic wave in materials with. Plane wave expansion method pwe refers to a computational technique in electromagnetics to solve the maxwells equations by formulating an eigenvalue problem out of the equation. A full 3d planewaveexpansion model for piezoelectric. Simultaneous photonic and phononic band gap, defect states. For the most part, however, it is intended to be applied to structures with some degree of periodicity in the refractive index distribution. Problems with the plane wave expansion method the pwe has the twin advantages of accuracy and efficiency.
Pdf planewave expansion method for calculating band. And waves and the fourier transform play major roles in all of science, so weoll do that, too. In actual calculations, we have to limit the plane wave expansion at some cutoffs are always given in energy units such as rydberg or ev corresponding to the kinetic. This manual is based on my implementation of pwmplane wave method using matlab. Application of the planewaveexpansionmethod to a photonic. Photonic band structure and eigenmodes of magnetophotonic crystals a. The expansion can be used to solve many wave propagation problems in complicated structures, including laterally inhomogeneous media with curved interfaces. Furthermore, we have a plane wave, by which we mean that a surface of constant phase is a plane. Fourier factorization in the plane wave expansion method in modeling photonic crystals 3 and 7 will propose how to factorize anisotropic and threedimensional 3d photonic crystals, respectively. Hitherto, since the focus has been to understand plane wave. Starting from maxwells equations the propagation of electromagnetic waves can be described by maxwells equations.
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