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. Starting from maxwells equations the propagation of electromagnetic waves can be described by maxwells equations. Plane waves and free spherical waves both form an onshell complete orthnormal set on with or without the origin. Application of the plane wave expansion method to a two. For the most part, however, it is intended to be applied to structures with some degree of periodicity in the refractive index distribution. The planewave pseudopotential method theory department. Note that emtl was initially developed mostly for fdtd simulations. A plane wave expansion of spherical wave functions for modal. An integral expansion which expresses a plane monochromatic wave as a superposition of gaussian beams is found. A wave is a disturbance which propagates energy and momentum from one place to the other without the transport of matter. In physics, the plane wave expansion expresses a plane wave as a linear combination of spherical waves. 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. Osa planewave expansion method for calculating band.
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. To study defect modes, the same plane wave expansion method can be used. Plane waves can be expanded in terms of free spherical waves by. 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 basic idea is to replace the unit cell by a more complicated unit cell and preserve the periodicity. An expansion method for finding traveling wave solutions to nonlinear pdes. Second, it is shown experimentally that a set of socalled fliege points 14. Consider, first of all, the expansion of a scalar plane wave as a set of scalar spherical waves. Second choice is a core of plane wave expansion pwe method that is very suitable for calculation of photonic crystals band structures. The agreement between theory and experiment is excellent for the lower 95 modes covering a very wide frequency range from dc to 20 khz.
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 expansion can be used to solve many wave propagation problems in complicated structures, including laterally inhomogeneous media with curved interfaces. Some measured normalmode wave amplitudes were compared with the theoretical. Pdf planewaveexpansion method for lamb wave propagation. Planewave expansion of cylindrical functions in lossy media. Plane wave expansion method pwe refers to a computational technique in electromagnetics to solve the maxwells equations by formulating an eigenvalue. Plane wave expansion method for photonic band gap calculation. The plane wave expansion method is one of the numerical methods that has been used. Improved planewave expansion method for band structure. Also, an alternative method to obtain the expansion coefficients of the plane. Expansion of a plane wave into gaussian beams springerlink.
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 thesis the plane wave expansion method and the guided mode. Application of the planewaveexpansionmethod to a photonic. Partialwaveexpansion method how is partialwaveexpansion method abbreviated. Fullpotential linearized augmented planewave method. The planewave pseudopotential method 1 bornoppenheimer approximation 2 apply densityfunctional theory dft to calculate the electronic structure.
A plane wave expansion of spherical wave functions for. From those we will derive a set of equations, whose solutions can be. Progress in electromagnetics research, pier 87, 245262, 2008 on the comparison of the spherical wave expansiontoplane wave expansion and the. Pdf planewave expansion method for calculating band. Lecture 18 cem plane wave expansion method youtube.
In 15, the expression of the spectral representation is derived for a general cylindrical function, i. We derive the expansion of a cylindrical wave into plane waves and we give some examples of applications. Planewaveexpansion method for lamb wave propagation in plate with twodimensional phononic crystal layer coated onuniform substrate march 2008. Plane wave expansion method pwem outline formulation of the basic eigen. Pdf band gap calculation using the plane wave expansion. The larger the rc i, the heavier the terms in direct space.
The plane wave expansion method allows the calculation of dispersion curves i. 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. This equation is sometimes called the master equation 84, and. 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. I am not going to list all source files here, they are pretty short and can be easily modified. Planewave expansion methods for photonic crystal fibres. The theoretical calculations were obtained using the plane wave expansion method applied to the in plane thin plate vibration theory. Photonic crystals are composed of periodic, dielectric structures that affect the propaga tion of electromagnetic waves. The differing signs of the energy in the time behavior will give rise to rapid oscillations. Essentially, there are two difficulties to overcome. That means that one must be able to expand one in terms of the other.
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. We have implemented this method for finding traveling wave solutions of nonlinear burgers equation, kdv equation, rlwburgers equation and hirotasatsuma equation system. A general complex eigenvalue problem is then obtained. The theoretical calculations were obtained using the plane wave expansion method applied to the inplane thin plate vibration theory. In, the expansion into plane waves of a cylindrical wave propagating in a lossless medium has been derived.
A plane wave expansion of spherical wave functions for modal analysis of guided wave structures and scatterers robert h. The plane wave pseudopotential method 1 bornoppenheimer approximation 2 apply densityfunctional theory dft to calculate the electronic structure. The method allows the computation of eigenfrequencies for a photonic crystal to any prescribed accuracy, commensurate with computing time. This algorithm combines the planewave expansion method with perfectly matched layers for the termination of the computational region. In fact, an incoming plane wave along the direction can be expanded in bessel functions. 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. The timeindependent schrodinger equation for the energy eigenstates in the coordinate representation is given by. It was used in some of the earliest studies of photonic crystals 14 and is simple enough to be easily implemented.
Bloch theorem, a plane wave in a periodic structure will be modulated by the periodicity. So that particles can return to their mean position, after having been disturbed. 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. 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. We present a new algorithm for calculation of the band structure of photonic crystal slabs. 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. In addition, the effectivemedium tensor is applied to improve convergence. 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. From those we will derive a set of equations, whose solutions can be approximated by the plane wave expansion method. Pdf planewave expansion method for calculating band structure. 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.
Numerical investigations of twodimensional photonic. So, the complexvalued magnetic field h can be expressed as. Plane wave approximation of homogeneous helmholtz solutions 3. There is no evidence that regularisation improves the planewave expansion method, but with the correct choice of parameters both variants can be used efficiently. 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. We implemented plane wave expansion pwe method as a part of electromagnetic template library emtl. It describes how to construct electromagnetic band diagrams and isofrequency contours. The maxwell equation in a transparent, timeinvariant, source free, and nonmagnetic medium can be written in the following form. Oct 04, 20 this lecture steps the student through the formulation and implementation of the plane wave expansion method.
Pdf we present a new algorithm for calculation of the band structure of photonic crystal slabs. Computational electromagnetics formulation of the plane wave expansion method outline formulation of the basic 3d eigen. Pw expansion of nic has to be converged with density cuto. The mode expansion methods particular cases of which are the plane wave expan. 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. There are some tricks in the derivation process, so we show the process in detail. Plane wave expansion method 1d photonic crystal considering electromagnetic wave in materials with. Expansion of plane waves in spherical harmonics consider a free particle of mass in three dimension. This lecture steps the student through the formulation and implementation of the plane wave expansion method.
1093 231 1214 816 172 1387 585 92 538 832 1292 105 1604 1491 710 1367 1129 1206 1251 720 1000 137 98 1106 199 721 766 845