The basic underlying idea is that a function fx can be expressed as a linear combination of elementary functions speci cally, sinusoidal waves. The method described using the expansion creates spurious modes with zero frequency, which were removed. Furthermore, we have a plane wave, by which we mean that a surface of constant phase is a plane. For the most part, however, it is intended to be applied to structures with some degree of periodicity in the refractive index distribution. Planewave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers. Planewave expansion method for calculating band structure. Plane wave expansion method used to engineer photonic crystal sensors with high efficiency. Timedomain representations of a plane wave with spatial. Plane wave expansion method in computational electromagnetism. Plane wave expansion dielectric slab waveguide waveguideembedded photonic crystals 2.
Can we decompose a spherical wave as a sum of plane waves. Such method has been used for treating the twodimensional scattering by a set of perfectly conducting parallel objects. For that we refer the reader to previous articles 21, for example, or 55, which give the details of the method. The plane wave expansion method uses the set of planes waves expik r as the complete basis.
The plane wave expansion method pwem provides a highly efficient numerical solution to maxwells equations for devices with low to moderate dielectric contrast. 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. The calculated eigenfrequencies and field patterns for a defect in a square array of dielectric rods are consistent with those obtained by the plane wave expansion method. The plane wave expansion method is one of the numerical methods that has been used. Lecture 18 cem plane wave expansion method youtube. Fourier series and fourier transforms the fourier transform is one of the most important tools for analyzing functions. The plane wave expansion method was carried out to construct dispersion curves for along the brillouin zone shown in figure 4.
The theoretical investigation is performed using the plane wave expansion method in both the optical and acoustical regimes. Plane wave approximation since many articles are devoted to the mathematical aspects of the fmm, we will skip the mathematical derivation of the algorithm. This thesis presents the results of numerical studies of scalar and electromagnetic wave propagation in periodic and random dielectric media. 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. Plane wave expansion method for photonic band gap calculation using matlab manual version 1. Growth details and zigzag chain alignment on a ag111 curved vicinal crystal. The larger the rc i, the heavier the terms in direct space.
The plane wave pseudopotential method 1 bornoppenheimer approximation 2 apply densityfunctional theory dft to calculate the electronic structure. 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 the plane wave appproximation, a particle incident on a nucleus has a wavefunction approximated by a plane wave of the form eik. In, the expansion into plane waves of a cylindrical wave propagating in a lossless medium has been derived. Osa planewave expansion method for calculating band. This manual is based on my implementation of pwm plane wave method using matlab. Planewave expansion methods for photonic crystal fibres. Note that emtl was initially developed mostly for fdtd simulations. Introduction to two dimensional scatering 3 let the total wave be the sum of the incident and scattered waves. The condition for solving fors and t in terms ofx and y requires that the jacobian matrix be nonsingular. Band structure of honeycomb photonic crystal slabs taii weng and g. Pdf band gap calculation using the plane wave expansion.
I show how i deal with the 1d, 2d and 3d cases, so you can deal with other problems in. A general complex eigenvalue problem is then obtained. In this document we will apply the plane wave expansion method on a speci c photonic crystal. Introduction to planewave basis sets and pseudopotential. 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. Arpes band structure and photoemission intensity simulations using the electron plane wave expansion epwe method. We present a new algorithm for calculation of the band structure of photonic crystal slabs. That means that one must be able to expand one in terms of the other. We apply a generalization of the plane wave expansion method and calculate the pbs and eigenmodes of mpcs in different magnetization geometries and analyze their polarization state. Yes, a spherical wave can be expanded as a sum of plane waves, and vice versa. 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. Gu, institute of physics, academia sinica china conventional plane wave expansion pwe methods, good for calculating such properties as photonic band gaps for. This method is popular among the photonic crystal community as a method of. 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.
Plane wave 5 represented in the first illustration toward the right is a linearly polarized, electromagnetic wave. To study defect modes, the same plane wave expansion method can be used. 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. The plane wave expansion method allows the calculation of dispersion curves i. The feed lies outside the closed surface, so that the volume v. The plane wave expansion method for the mpcs is briefly considered in sec. In fact, an incoming plane wave along the direction can be expanded in bessel functions. It describes how to construct electromagnetic band diagrams and isofrequency contours. Improved planewave expansion method for band structure. Oct 04, 20 this lecture steps the student through the formulation and implementation of the plane wave expansion method. There are several methods to calculate the dispersion relation and therewith the range of the band gaps. Computational methods for the analysis and design of photonic. Course paperwork syllabus homework course topics resources.
An integral expansion which expresses a plane monochromatic wave as a superposition of gaussian beams is found. Fullpotential linearized augmented planewave method. Plane wave approximation of homogeneous helmholtz solutions a. Pdf we present a new algorithm for calculation of the band structure of photonic crystal slabs. The projector augmented wave paw method was used to describe the ionic cores. In this paper, the timedomain representation for a plane wave is derived based on a inverse fourier transform b laplace domain representation and c plane wave expansion, all of which are mathematically equivalent in the continuous time domain. Simultaneous photonic and phononic band gap, defect. 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.
Plane wave expansion method 1d photonic crystal considering electromagnetic wave in materials with. Second choice is a core of plane wave expansion pwe method that is very suitable for calculation of photonic crystals band structures. Plane wave expansion method for photonic band gap calculation. The basic idea is to replace the unit cell by a more complicated unit cell and preserve the periodicity. The first is energy cutoffs, which is the cutoff for the wave the input files page 4 of 20. Outofplane dispersion and homogenization in photonic. For the planewave expansion, a 450 ev kinetic energy cutoff was used. 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 term is also used, even more specifically, to mean a monochromatic or sinusoidal plane wave. The plane wave spectrum is used to determine the far. 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. Photonic crystals are composed of periodic, dielectric structures that affect the propaga tion of electromagnetic waves. Br effect on the electronic structure of zigzag chains studied by.
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. We implemented plane wave expansion pwe method as a part of electromagnetic template library emtl. The planewave pseudopotential method 1 bornoppenheimer approximation 2 apply densityfunctional theory dft to calculate the electronic structure. The plane wave expansion of a spherical wave is surprisingly hard to find on the web. Pw expansion of nic has to be converged with density cuto. In particular, this allows for the possibility that the projected characteristics may cross each other. Starting from maxwells equations the propagation of electromagnetic waves can be described by maxwells equations. This solution is a wave \traveling in the direction of k in the sense that a point of constant phase, meaning k. The plane wave expansion, or the rayleigh equation, is commonly used in nuclear physics. This method is popular among the photonic crystal community as a method of solving for the band structure dispersion relation of specific photonic crystal. Plane wave expansion pwe method the maxwell equation in a transparent, timeinvariant, source free, and nonmagnetic medium can be written in the following form. The extended plane wave expansion method for 3d anisotropic photonic crystals1 y.
Summary the plane wave expansion method allows the calculation of dispersion curves i. Pdf planewave expansion method for calculating band. Guidedmode expansion method photonic mode dispersion intrinsic diffraction losses convergence tests, comparison with other methods examples of applications books yariv a. One reason for choosing the plane wave basis is that it is the. Photonic band structure and eigenmodes of magnetophotonic. It expands the field into a set of plane waves and converts maxwells equations to matrix formbyassigning each plane wave a complex amplitude. From those we will derive a set of equations, whose solutions can be approximated by the plane wave expansion method. Numerical investigations of twodimensional photonic. By means of it one can find as many defect modes as possible, including those that are inactive to the incident plane wave in the fdtd transmission spectra method. The extended plane wave expansion method for 3d anisotropic. Reformulation of the plane wave method to model photonic crystals. Plane waves can be expanded in terms of free spherical waves by.
Fourier factorization in the plane wave expansion method in modeling photonic crystals 18 willbesetbyintech the numerical ef. Plane wave expansion method used to engineer photonic. Plane wave approximation of homogeneous helmholtz solutions. Planewave expansion method for calculating band structure of.
Research article an enhanced plane wave expansion method to solve piezoelectric phononic crystal with resonant shunting circuits ziyanglian, 1,2 shanjiang, 1,2 hongpinghu, 1,2 longxiangdai, 1,2 xuedongchen, 3 andweijiang 3 department of mechanics, huazhong university of science and technology, wuhan, china. There are some tricks in the derivation process, so we show the process in detail. This file discussess the following topics related to the main manuscript. Research article an enhanced plane wave expansion method. Consider, first of all, the expansion of a scalar plane wave as a set of scalar spherical waves.
The dephasing of the plane wave components results in the diffractive broadening of the beam, as showed in fig. The timeindependent schrodinger equation for the energy eigenstates in the coordinate representation is given by. The matrix method has been implemented, and its potentialities are presented. In particular, a plane to plane method for the determination of the aperture. Digital library of mathematical functions, equation 10. The expansion can be used to solve many wave propagation problems in complicated structures, including laterally inhomogeneous media with curved interfaces.
Because this is a plane wave, each blue vector, indicating the perpendicular displacement from a point on the axis out to the sine. Plane waves and free spherical waves both form an onshell complete orthnormal set on with or without the origin. Fourier factorization in the plane wave expansion method. The photonic energy band structure of periodic composite materials, consisting of regular spherical arrays or scaffoldings of one kind of dielectric material in a different background material, are studied using the plane wave expansion method in order.
This lecture steps the student through the formulation and implementation of the plane wave expansion method. The convergence criterion for the electronic structure iteration was set. There is a further truncation of plane wave expansion in plane wave calculations. 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. Let us examine this set of examples in further detail. The guidedmode expansion method for photonic crystal slabs. Numerical computation of band gaps in photonic crystal fibres. Pdf planewave expansion method for calculating band structure. Plane wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers shouyuan shi, caihua chen, and dennis w. In this paper, we study the approximation of solutions of the homogeneous helmholtz equation. Matlab implementation of the rigorous coupled wave analysis and plane wave expansion method chrisfaddenrcwa.
Unfortunately, the function that describes the material in the. The band structures are calculated by the plane wave expansion method and the. Rami mehrem 2009, the plane wave expansion, infinite integrals and identities involving spherical bessel functions, arxiv. Planewave expansion of cylindrical functions in lossy media.
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. We show through a numerical convergence study and rigorous analysis on a simplified problem that the convergence of the planewave expansion method is. Plane wave expansion of cylindrical functions in lossy media. Also, an alternative method to obtain the expansion coefficients of the plane. The plane wave expansion pwe method have been formulated in a general form for arbitrary 2d photonic crystal and applied for detailed analysis of the. By doing so we will derive the dispersion relation and the density. 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. Acknowledgements this work would not have been successful without the guidance and the help of several individuals. 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. 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 fourier factorization in the plane wave expansion. 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.
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