It describes how to construct electromagnetic band diagrams and isofrequency contours. 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. The mode expansion methods particular cases of which are the plane wave expan. In, the expansion into plane waves of a cylindrical wave propagating in a lossless medium has been derived. From those we will derive a set of equations, whose solutions can be. The differing signs of the energy in the time behavior will give rise to rapid oscillations. For example, a 4 x 4 supercell with a central defect can give reasonable accuracy because the missing holes are spaced four lattice units apart. The theoretical calculations were obtained using the plane wave expansion method applied to the in plane thin plate vibration theory. Improved planewave expansion method for band structure. Application of the plane wave expansion method to a two.
Plane wave expansion method 1d photonic crystal considering electromagnetic wave in materials with. For construction of the dispersion equation, various methods have been proposed such as finite element method 6, plane wave expansion method 7, and chebyshev interval method 8. It was used in some of the earliest studies of photonic crystals 14 and is simple enough to be easily implemented. Application of the planewaveexpansionmethod to a photonic. In this thesis the plane wave expansion method and the guided mode. Partial wave analysis of scattering we can take a quick look at scattering from a potential in 3d we assume that far from the origin so the incoming and outgoing waves can be written in terms of our solutions for a constant potential. A general complex eigenvalue problem is then obtained. Numerical computation of band gaps in photonic crystal fibres.
In addition, the effectivemedium tensor is applied to improve convergence. 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. There is no evidence that regularisation improves the planewave expansion method, but with the correct choice of parameters both variants can be used efficiently. There are some tricks in the derivation process, so we show the process in detail. The planewave pseudopotential method theory department. Essentially, there are two difficulties to overcome. We present a new algorithm for calculation of the band structure of photonic crystal slabs. Second, it is shown experimentally that a set of socalled fliege points 14. Partial wave analysis of scattering we can take a quick look at scattering from a potential in 3d we assume that far from the origin so the incoming and outgoing waves can be written in terms of our solutions for a constant potential in fact, an incoming plane wave along the direction can be expanded in bessel functions. 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. Expansion of a plane wave into gaussian beams springerlink. Planewave expansion of cylindrical functions in lossy media.
The timeindependent schrodinger equation for the energy eigenstates in the coordinate representation is given by. 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. Photonic crystals are composed of periodic, dielectric structures that affect the propaga tion of electromagnetic waves. Plane waves can be expanded in terms of free spherical waves by. A wave is a disturbance which propagates energy and momentum from one place to the other without the transport of matter. So that particles can return to their mean position, after having been disturbed. This equation is sometimes called the master equation 84, and. Pdf we present a new algorithm for calculation of the band structure of photonic crystal slabs. We have implemented this method for finding traveling wave solutions of nonlinear burgers equation, kdv equation, rlwburgers equation and hirotasatsuma equation system. Pdf planewaveexpansion method for lamb wave propagation. Starting from maxwells equations the propagation of electromagnetic waves can be described by maxwells equations. Oct 04, 20 this lecture steps the student through the formulation and implementation of the plane wave expansion method.
Plane wave expansion method for photonic band gap calculation. Plane wave approximation of homogeneous helmholtz solutions 3. 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. I am not going to list all source files here, they are pretty short and can be easily modified. The agreement between theory and experiment is excellent for the lower 95 modes covering a very wide frequency range from dc to 20 khz. Planewave expansion methods for photonic crystal fibres. This lecture steps the student through the formulation and implementation of the plane wave expansion method. 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.
Note that emtl was initially developed mostly for fdtd simulations. 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. This algorithm combines the planewave expansion method with perfectly matched layers for the termination of the computational region. Fullpotential linearized augmented planewave method. The expansion can be used to solve many wave propagation problems in complicated structures, including laterally inhomogeneous media with curved interfaces.
Blochs theorem and plane wave basis sets as yet there has been no mention of how to handle the infinite number of interacting electrons moving in the static field of an infinite number of ions. 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 plane wave expansion method is one of the numerical methods that has been used. Pdf band gap calculation using the plane wave expansion. To study defect modes, the same plane wave expansion method can be used. The larger the rc i, the heavier the terms in direct space. In fact, an incoming plane wave along the direction can be expanded in bessel functions. Bloch theorem, a plane wave in a periodic structure will be modulated by the periodicity. Also, an alternative method to obtain the expansion coefficients of the plane. Pdf planewave expansion method for calculating band. That means that one must be able to expand one in terms of the other. For the most part, however, it is intended to be applied to structures with some degree of periodicity in the refractive index distribution. An expansion method for finding traveling wave solutions to nonlinear pdes. So, the complexvalued magnetic field h can be expressed as.
A plane wave expansion of spherical wave functions for modal analysis of guided wave structures and scatterers robert h. Pw expansion of nic has to be converged with density cuto. Consider, first of all, the expansion of a scalar plane wave as a set of scalar spherical waves. Plane waves and free spherical waves both form an onshell complete orthnormal set on with or without the origin. In physics, the plane wave expansion expresses a plane wave as a linear combination of spherical waves. 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. Progress in electromagnetics research, pier 87, 245262, 2008 on the comparison of the spherical wave expansiontoplane wave expansion and 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. The method allows the computation of eigenfrequencies for a photonic crystal to any prescribed accuracy, commensurate with computing time. A plane wave expansion of spherical wave functions for modal. Osa planewave expansion method for calculating band. Expansion of plane waves in spherical harmonics consider a free particle of mass in three dimension. The plane wave pseudopotential method 1 bornoppenheimer approximation 2 apply densityfunctional theory dft to calculate the electronic structure. Some measured normalmode wave amplitudes were compared with the theoretical.
The plane wave expansion method allows the calculation of dispersion curves i. Planewaveexpansion method for lamb wave propagation in plate with twodimensional phononic crystal layer coated onuniform substrate march 2008. The theoretical calculations were obtained using the plane wave expansion method applied to the inplane thin plate vibration theory. We implemented plane wave expansion pwe method as a part of electromagnetic template library emtl.
In 15, the expression of the spectral representation is derived for a general cylindrical function, i. Lecture 18 cem plane wave expansion method youtube. 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. Computational electromagnetics formulation of the plane wave expansion method outline formulation of the basic 3d eigen. The maxwell equation in a transparent, timeinvariant, source free, and nonmagnetic medium can be written in the following form. Plane wave expansion method pwe refers to a computational technique in electromagnetics to solve the maxwells equations by formulating an eigenvalue. The basic idea is to replace the unit cell by a more complicated unit cell and preserve the periodicity. A plane wave expansion of spherical wave functions for. An integral expansion which expresses a plane monochromatic wave as a superposition of gaussian beams is found. Partialwaveexpansion method how is partialwaveexpansion method abbreviated. Pdf by using the plane wave expansion method, we have calculated the band gap for a metallic slab with air rods in the transverse magnetic tm mode.
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. Unfortunately, the function that describes the material in the. Partial wave analysis for elastic scattering 19 using 17 we can write the asymptotic limit of the scattered wave function 5 as this wave function 19 is known as a distorted plane wave, which differs from a plane wave by the phase shifts. Pdf planewave expansion method for calculating band structure. We derive the expansion of a cylindrical wave into plane waves and we give some examples of applications. 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. From those we will derive a set of equations, whose solutions can be approximated by the plane wave expansion method.