Osa planewave expansion method for calculating band. The planewave pseudopotential method theory department. Homogenizing metamaterials by the plane wave expansion method. Expansion of a plane wave into gaussian beams springerlink. However, the pwe may have difficulties in simulating pcs with highly contrasting elastic properties 20.
Planewave expansion of cylindrical functions in lossy media. In this thesis the plane wave expansion method and the guided mode expansion. Properties of the transformation from the spherical wave. This algorithm combines the plane wave expansion method with perfectly matched layers for the. The plane wave 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. That means that one must be able to expand one in terms of the other. Plane waves and free spherical waves both form an onshell complete orthnormal set on with or without the origin. The approximation of the exact radial solution by the taylor expansion involving also the second energy derivative of the radial function, except for the first derivative, has two advantages. The red lines are solutions that belong to the transversal electric modes te, whereat the blue lines belong to the transversal magnetic modes tm. Consider, first of all, the expansion of a scalar plane wave as a set of scalar spherical waves. Pdf we present a new algorithm for calculation of the band structure of photonic crystal slabs. This algorithm combines the planewave expansion method with perfectly matched layers for. We are considering the solution to schrodingers equation for scattering of an incoming plane wave in the zdirection by a potential localized in a region near the origin, so that the total wave function beyond the range of the potential has the form.
Pdf study on the convergence of plane wave expansion method. Application of the plane wave expansion method to a two. 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. Plane wave expansion method in computational electromagnetism. Rami mehrem 2009, the plane wave expansion, infinite integrals and identities involving spherical bessel functions, arxiv.
Numerical computation of band gaps in photonic crystal fibres. The plane wave expansion method was carried out to construct dispersion curves for along the brillouin zone shown in figure 4. 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. There are some tricks in the derivation process, so we show the process in detail.
We combine cartesian coordinates and polar coordinates wave number eigenvalue equations based on the plane wave expansion pwe method to calculate and optimize the band structures of the twodimensional 2d metal photonic crystals phcs. Plane wave expansion method pwem outline formulation of the basic eigen. The plane wave expansion, or the rayleigh equation, is commonly used in nuclear physics. Improved planewave expansion method for band structure. Fourier factorization in the plane wave expansion method in modeling photonic crystals. The maxwell equation in a transparent, timeinvariant, source free, and nonmagnetic medium can be written in the following form. The calculations were done by the plane wave expansion method in matlab see code in appendix a. From those we will derive a set of equations, whose solutions can be approximated by the plane wave expansion method. Any periodic function can be expanded using fourier series, so is for e and h. In free space, such a plane wave is not consistent with energy conservation law in the sense of classical electrodynamics. This method is popular among the photonic crystal community as a method of solving for the band structure dispersion relation of specific photonic crystal. In addition, the effectivemedium tensor is applied to improve. Plane wave approximation of homogeneous helmholtz solutions. 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.
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. Digital library of mathematical functions, equation 10. Starting from maxwells equations the propagation of electromagnetic waves can be described by maxwells equations. The plane wave expansion method is one of the numerical methods that has been used. In, the expansion into plane waves of a cylindrical wave propagating in a lossless medium has been derived. The plane wave expansion method allows the calculation of dispersion curves i. Fourier factorization in the plane wave expansion method in. Scattering of a plane wave were now ready to take the ingoing plane wave, break it into its partial wave components corresponding to different angular momenta, have the partial waves individually phase shifted by ldependent phases, and add it all back together to get the original plane wave plus the scattered wave. The plane wave decomposition is mathematically universal. Application of the planewaveexpansionmethod to a photonic. We present a new algorithm for calculation of the band structure of photonic crystal slabs. Numerical investigations of twodimensional photonic. Planewave expansion method for calculating band structure of.
Plane wave expansion method 1d photonic crystal considering electromagnetic wave in materials with. Unfortunately, the function that describes the material in the fibre nx. Plane wave expansion method for photonic band gap calculation. 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. Plane wave expansion method pwe refers to a computational technique in electromagnetics to solve the maxwells equations by formulating an eigenvalue. Oct 04, 20 this lecture steps the student through the formulation and implementation of the plane wave expansion method.
On the dispersive nature of the power dissipated into a lossy. Matrix method for antenna plane wave spectrum calculation. The expansion can be used to solve many wave propagation problems in complicated structures, including laterally inhomogeneous media with curved interfaces. Pdf we study the convergence of plane wave expansion method pwem while calculating the band. Unfortunately, the function that describes the material in the. Matlab implementation of the rigorous coupled wave analysis and plane wave expansion method chrisfaddenrcwa. The plane wave spectrum is used to determine the far. The plane wave expansion method pwe 16,17 and finite difference time domain method fdtd 18,19 have been used to analyze the physical behavior of phononic band gaps. 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.
In addition, the effectivemedium tensor is applied to improve convergence. Consider a free particle of mass in three dimension. In this paper, we study the approximation of solutions of the homogeneous helmholtz equation. Pdf planewave expansion method for calculating band. Plane wave expansion pwe method introduction the maxwell equation in a transparent, timeinvariant, source free, and nonmagnetic medium can be written in the following form. First, the transformation of individual spherical waves is studied in order to understand how these contribute to the different regions of the plane wave spectrum. The developed version of the augmented plane wave method yields eigenvalues and eigenfunctions with higher accuracy than the standard lapw, preserving its computational efficiency. Compared with the traditional pwe methods for metal phcs, the band structures can. For this reason, the plane wave expansion method may not be every. 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. 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. Planar photonic crystals combine 2d periodic structures with a slab waveguide in.
We derive the expansion of a cylindrical wave into plane waves and we give some examples of applications. A general complex eigenvalue problem is then obtained. It describes how to construct electromagnetic band diagrams and isofrequency contours. Pw expansion of nic has to be converged with density cuto.
Plane wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers shouyuan shi, caihua chen, and dennis w. Plane wave approximation of homogeneous helmholtz solutions a. The matrix method has been implemented, and its potentialities are presented. Lecture 18 cem plane wave expansion method youtube. Value problem 3d 2d implementation calculation of band diagrams calculation of isofrequency contours example computing the band diagram for an ebg material with square symmetry slide 2 1 2. The timeindependent schrodinger equation for the energy eigenstates in the coordinate representation is given by.
In the plane wave appproximation, a particle incident on a nucleus has a wavefunction approximated by a plane wave of the form eik. Planewave expansion method for calculating band structure. Plane wave expansion method for photonic band gap calculation using matlab manual version 1. The larger the rc i, the heavier the terms in direct space. The scattering amplitude can be calculated in born approximation for many interesting cases, but as we saw in a few examples already, we need to work out the scattering amplitudes more exactly in certain cases. Pdf planewave expansion method for calculating band structure. Also, an alternative method to obtain the expansion coefficients of the plane. Chapter 2 the wave equation after substituting the. 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.
Combining these equations with the decompositions 2. These unitcell based dispersion techniques lose information regarding the boundary con ditions and mode shapes of the. For the most part, however, it is intended to be applied to structures with some degree of periodicity in the refractive index distribution. An integral expansion which expresses a plane monochromatic wave as a superposition of gaussian beams is found.