Digital library of mathematical functions, equation 10. Planewave expansion of cylindrical functions in lossy media. Rami mehrem 2009, the plane wave expansion, infinite integrals and identities involving spherical bessel functions, arxiv. Introduction to planewave basis sets and pseudopotential.
We have implemented this method for finding traveling wave solutions of nonlinear burgers equation, kdv equation, rlwburgers equation and hirotasatsuma equation system. It describes how to construct electromagnetic band diagrams and isofrequency contours. Let us examine this set of examples in further detail. The planewave pseudopotential method 1 bornoppenheimer approximation 2 apply densityfunctional theory dft to calculate the electronic structure. We present a new algorithm for calculation of the band structure of photonic crystal slabs. Inoue1 1 t oy h as iu nv er t f cl g,1m pku a 4 85 0 j. This manual is based on my implementation of pwmplane wave method using matlab. The expansion can be used to solve many wave propagation problems in complicated structures, including laterally inhomogeneous media with curved interfaces. By doing so we will derive the dispersion relation and the density. Pdf on jun 1, 2009, vincent laude and others published plane wave expansion method for phononic crystals. Pwe is traceable to the analytical formulations, and is useful in calculating modal solutions of maxwells equations over an inhomogeneous or periodic geometry. 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. The term is also used, even more specifically, to mean a monochromatic or sinusoidal plane wave. It expands the field into a set of plane waves and converts maxwells equations to matrix formbyassigning each plane wave a.
Plane waves can be expanded in terms of free spherical waves by. What links here related changes upload file special pages permanent link page. In addition, the effectivemedium tensor is applied to improve. Pdf the planewave expansion method has been the basis for much of the understanding of photonic and electrical behavior in crystals. An integral expansion which expresses a plane monochromatic wave as a superposition of gaussian beams is found. It follows that the size of the hamiltonian matrix is. A general complex eigenvalue problem is then obtained. This algorithm combines the plane wave expansion method with perfectly matched layers for the. 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. Pw expansion of nic has to be converged with density cuto. 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. We need outgoing wave so we pick hankel function of the first kind summary. The method described using the expansion creates spurious modes with zero frequency, which were removed.
Acknowledgements this work would not have been successful without the guidance and the help of several individuals. Oct 04, 20 this lecture steps the student through the formulation and implementation of the plane wave expansion method. On the relationship between the spherical wave expansion and the plane wave expansion for antenna diagnostics. The plane wave expansion method pwem provides a highly efficient numerical solution to maxwells equations for devices with low to moderate dielectric contrast.
Plane wave expansion method in computational electromagnetism. Matlab implementation of the rigorous coupled wave analysis and plane wave expansion method chrisfaddenrcwa. Plane wave expansion method pwe refers to a computational technique in electromagnetics. 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. In this paper, we study the approximation of solutions of the homogeneous helmholtz equation. Pdf planewave expansion method for calculating band structure. 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. In addition, the effectivemedium tensor is applied to improve convergence. The matrix method has been implemented, and its potentialities are presented. An expansion method for finding traveling wave solutions to nonlinear pdes. Plane wave approximation since many articles are devoted to the mathematical aspects of the fmm, we will skip the mathematical derivation of the algorithm. There are some tricks in the derivation process, so we show the process in detail. Fjam es,as tudngio fourier transforms optics played a major role in all the physics revolutions of the 20th century, so weoll do some.
You can download this pdf file and related source file from my personal homepage. 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. Planewave expansion method for calculating band structure of. Pdf a planewave expansion method for analyzing propagation in. In particular, this allows for the possibility that the projected characteristics may cross each other. Plane waves and free spherical waves both form an onshell complete orthnormal set on with or without the origin.
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. 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. The planewaveexpansion pwe approach dedicated to the simulation of periodic devices has. 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. We implemented plane wave expansion pwe method as a part of electromagnetic template library emtl. Consider, first of all, the expansion of a scalar plane wave as a set of scalar spherical waves. Furthermore, we have a plane wave, by which we mean that a. Numerical computation of band gaps in photonic crystal fibres. In, the expansion into plane waves of a cylindrical wave propagating in a lossless medium has been derived. Also, an alternative method to obtain the expansion coefficients of the plane. And waves and the fourier transform play major roles in all of science, so weoll do that, too. Gaussian, hermitegaussian, and laguerregaussian beams.
Note that emtl was initially developed mostly for fdtd simulations. The pwe method can therefore not account for materials with loss, or. Photonic band structure and eigenmodes of magnetophotonic. This method is popular among the photonic crystal community as a method of solving for the band structure of specific photonic crystal geometries. Application of the plane wave expansion method to a two. Publishers pdf, also known as version of record link back to dtu orbit citation apa.
In this case the problem is cast in the form of a discontinuous galerkin method, with possibly mesh dependent penalty parameters. A plane wave expansion of spherical wave functions for modal analysis of guided wave structures and scatterers robert h. There are several methods to calculate the dispersion relation and therewith the range of the band gaps. The dephasing of the plane wave components results in the diffractive broadening of the beam, as showed in fig. Scott, the spectral domain method in electromagnetics, norwood. Numerical investigations of twodimensional photonic.
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. In particular, a plane to plane method for the determination of the aperture. Problems with the plane wave expansion method the pwe has the twin advantages of accuracy and efficiency. Hitherto, since the focus has been to understand plane wave. That means that one must be able to expand one in terms of the other. The feed lies outside the closed surface, so that the volume v.
Plane wave expansion method 1d photonic crystal considering electromagnetic wave in materials with. For this reason, the plane wave expansion method may not be every. Pdf plane wave expansion method for phononic crystals. Starting from maxwells equations the propagation of electromagnetic waves can be described by maxwells equations. Plane wave expansion method for photonic band gap calculation. The condition for solving fors and t in terms ofx and y requires that the jacobian matrix be nonsingular. A full 3d planewaveexpansion model for piezoelectric. Any periodic function can be expanded using fourier series, so is for e and h. The plane wave expansion method was carried out to construct dispersion curves for along the brillouin zone shown in figure 4. Waves, the wave equation, and phase weoll start with optics. Photonic band structure and eigenmodes of magnetophotonic crystals a. The wave function of a two spin 12 particle system is either symmetric or antisymmetric.
The plane wave pseudopotential method 1 bornoppenheimer approximation 2 apply densityfunctional theory dft to calculate the electronic structure. 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. 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. Osa planewave expansion method for calculating band. The timeindependent schrodinger equation for the energy eigenstates in the coordinate representation is given by. In 1818, poisson used fresnels theory to predict this phenomenon. 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. A plane wave expansion of spherical wave functions for modal. Planewave expansions pwes based on fourier transform and their physical. This expansion in terms of plane waves is useful since the scattering solution of many problems to plane wave excitation is known.
The theoretical investigation is performed using the plane wave expansion method in both the optical and acoustical regimes. Fullpotential linearized augmented planewave method. 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. Simultaneous photonic and phononic band gap, defect states. The mode expansion methods particular cases of which are the plane wave expan sion method and the.
Improved planewave expansion method for band structure. For that we refer the reader to previous articles 21, for example, or 55, which give the details of the method. Another useful representation of the dyadic greens function is its fourier transform where the. The method allows the computation of eigenfrequencies for a photonic crystal to any prescribed accuracy, commensurate with computing time. 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 plane wave expansion method is one of the numerical methods that has been used. This solution is a wave \traveling in the direction of k in the sense that a point of constant phase, meaning k. He regarded this as proof that fresnels wave theory was nonsense, and that light must be a particle and not a wave. Unfortunately, the function that describes the material in the. I show how i deal with the 1d, 2d and 3d cases, so you can deal with other problems in. 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. The larger the rc i, the heavier the terms in direct space. From these plane waves, the current induced by the array in a single. Expansion of a plane wave into gaussian beams springerlink. 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. The first is energy cutoffs, which is the cutoff for the wave the input files. In contrast, numerical simulations which accurately account for the. 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. In the plane wave expansion method, the impedance sum is ned by transforming the vector potential of an infinite array. Matrix method for antenna plane wave spectrum calculation. Summary the plane wave expansion method allows the calculation of dispersion curves i. The maxwell equation in a transparent, timeinvariant, source free, and nonmagnetic medium can be written in the following form. In this document we will apply the plane wave expansion method on a speci c photonic crystal. Fourier factorization in the plane wave expansion method. Planewave expansion method for calculating band structure. Outofplane dispersion and homogenization in photonic. We start by recalling the theory of classical plane wave spectrum for regularly distributed near. From those we will derive a set of equations, whose solutions can be approximated by the plane wave expansion method. For the most part, however, it is intended to be applied to structures with some degree of periodicity in the refractive index distribution. Plane wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers shouyuan shi, caihua chen, and dennis w. More recently, the plane wave discontinuous galerkin pwdg method has been studied by hiptmair et al.
Furthermore, we have a plane wave, by which we mean that a surface of constant phase is a plane. Pdf we present a new algorithm for calculation of the band structure of photonic crystal slabs. 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. Lecture 18 cem plane wave expansion method youtube. It was used in some of the earliest studies of photonic crystals 14 and is simple enough to be easily implemented. The plane wave spectrum is used to determine the far. Application of the planewaveexpansionmethod to a photonic.
It expands the field into a set of plane waves and converts maxwells equations to matrix form by assigning each plane wave a. There is a further truncation of plane wave expansion in plane wave calculations. Photonic crystals are composed of periodic, dielectric structures that affect the propaga tion of electromagnetic waves. It expands the field into a set of plane waves and converts maxwells equations to matrix formbyassigning each plane wave a complex amplitude. Plane wave approximation of homogeneous helmholtz solutions a. 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. 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. 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 pwm plane wave method using matlab. 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. This method is popular among the photonic crystal community as a method of solving for the band structure dispersion relation of specific photonic crystal. On the relationship between the spherical wave expansion. The plane wave expansion method allows the calculation of dispersion curves i. Second choice is a core of plane wave expansion pwe method that is very suitable for calculation of photonic crystals band structures.
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