A general complex eigenvalue problem is then obtained. In particular, a planetoplane method for the determination of the aperture. Outofplane dispersion and homogenization in photonic. In physics, the plane wave expansion expresses a plane wave as a linear combination of spherical waves. 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. The plane wave pseudopotential method 1 bornoppenheimer approximation 2 apply densityfunctional theory dft to calculate the electronic structure.
Plane wave expansion method pwem outline formulation of the basic eigen. Rami mehrem 2009, the plane wave expansion, infinite integrals and identities involving spherical bessel functions, arxiv. Expansion of plane waves in spherical harmonics consider a free particle of mass in three dimension. Shajesh department of physics and astronomy, university of oklahoma, 440 w. Progress in electromagnetics research, pier 87, 245262, 2008 on the comparison of the spherical wave expansiontoplane wave expansion and the. There are some tricks in the derivation process, so we show the process in detail. 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. Application of the plane wave expansion method to a two. In the method presented, the continuity of the algorithm has been improved by eliminating the jump discontinuity points as well as decreasing the number of wave vectors used.
Planewave expansion method for calculating band structure. For the most part, however, it is intended to be applied to structures with some degree of periodicity in the refractive index distribution. There is a further truncation of plane wave expansion in plane wave calculations. Introduction to planewave basis sets and pseudopotential theory. 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 3. To study defect modes, the same plane wave expansion method can be used. Consider, first of all, the expansion of a scalar plane wave as a set of scalar spherical waves. Thus, convergence results for plane wave based approaches require best approximation estimates in sobolev norms for homogeneous helmholtz solutions by plane waves which are explicit in terms of the mesh size hhversion, and in the number pof plane waves within each element in the approximating spaces pversion. The plane wave expansion method allows the calculation of dispersion curves i. 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 approximation of homogeneous helmholtz solutions.
This equation is sometimes called the master equation 84, and. An improved fast plane wave expansion method for topology. The maxwell equation in a transparent, timeinvariant, source free, and nonmagnetic medium can be written in the following form. Osa planewave expansion method for calculating band. Incoming plane wave expikz outgoing spherical waves expikrr fig. Planewave expansion method for calculating band structure of. Oct 04, 20 this lecture steps the student through the formulation and implementation of the plane wave expansion method. Plane wave expansion method in computational electromagnetism. 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. 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.
For that we refer the reader to previous articles 21, for example, or 55, which give the details of the method. Pdf planewave expansion method for calculating band. 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. Starting from maxwells equations the propagation of electromagnetic waves can be described by maxwells equations. The convergence feature of two types of plane wave expansion methods commonly used for photonic crystals is analyzed. One reason for choosing the plane wave basis is that it is the eigenbasis in free space, so it is required for pseudo q factor considerations 9. The agaifpwem model for topology optimization of pncs is detailed in section 3. 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. From those we will derive a set of equations, whose solutions can be. Plane waves can be seen as a grid basis in momentum space, motivated by blochs theorem. Pwe is traceable to the analytical formulations, and is useful in calculating modal solutions of maxwells equations over an inhomogeneous or periodic geometry. The plane wave expansion method uses the set of planes waves fexpik.
Lecture 18 cem plane wave expansion method youtube. We present a new algorithm for calculation of the band structure of photonic crystal slabs. In principles, an infinite planewave basis sets are required to expand the electronic wave functions. 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. Pw basis set can be truncated to include only plane waves that have kinetic energies less than some. The basic idea is to replace the unit cell by a more complicated unit cell and preserve the periodicity. Eikonal approximation university of oklahoma physics. In the absence of free charge and current densities the maxwell equations are. 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. Furthermore, we have a plane wave, by which we mean that a surface of constant phase is a plane.
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. In section 4, three numerical examples are employed to investigate the efficiency and effectiveness of the proposed 2d agaifpwem model. Plane wave expansion method for photonic band gap calculation. In addition, the effectivemedium tensor is applied to improve convergence. The timeindependent schrodinger equation for the energy eigenstates in the coordinate representation is given by. The planewave pseudopotential method theory department.
An enhanced plane wave expansion pwe method is proposed to solve piezoelectric phononic crystal ppc connected with resonant shunting circuits ppcc, which is named as pweppcc. 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. This solution is a wave \traveling in the direction of k in the sense that a point of constant phase, meaning k. The foe is a candidate linearscaling algorithm if used with realspacelocalized basis functions. Digital library of mathematical functions, equation 10. Osa analysis for the convergence problem of the planewave. Plane wave expansion pwe method the maxwell equation in a transparent, timeinvariant, source free, and nonmagnetic medium can be written in the following form. Also, an alternative method to obtain the expansion coefficients of the plane. Our algorithm utilizes the fermi operator expansion foe method developed by goedecker and teter 15. This is a study report on eikonal approximations which was undertaken under the guidance.
From those we will derive a set of equations, whose solutions can be approximated by the plane wave expansion method. 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 using matlab manual version 1. The coefficients for the pw with small kinetic energy are typically important than those with large kinetic energy. It describes how to construct electromagnetic band diagrams and isofrequency contours. Implementing these changes results in increased computational efficiency when compared to the traditional fast plane wave expansion method fpwem.