Compact RBF Meshless Methods for Photonic Crystal Modelling

8/3/2019 Compact RBF Meshless Methods for Photonic Crystal Modelling

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Compact RBF meshless methods for photonic crystal modelling

E.E. Hart , S.J. Cox, K. DjidjeliCED Group, School of Engineering Sciences, University of Southampton, SO17 1BJ, UK

a r t i c l e i n f o

Article history:

Received 15 September 2010Received in revised form 14 January 2011Accepted 6 March 2011Available online 12 March 2011

Keywords:Photonic crystalMeshless methodsCompactly supported radial basis functions

a b s t r a c t

Meshless methods based on compact radial basis functions (RBFs) are proposed for model-ling photonic crystals (PhCs). When modelling two-dimensional PhCs two generalisedeigenvalue problems are formed, one for the transverse-electric (TE) mode and the otherfor the transverse-magnetic (TM) mode. Conventionally, the Band Diagrams for two-dimensional PhCs are calculated by either the plane wave expansion method (PWEM) orthe nite element method (FEM). Here, the eigenvalue equations for the two-dimensionalPhCs are solved using RBFs based meshless methods. For the TM mode a meshless localstrong form method (RBF collocation) is used, while for the tricker TE mode a meshlesslocal weak form method (RBF Galerkin) is used (so that the discontinuity of the dielectricfunction xcan naturally be modelled). The results obtained from the meshless methodsare found to be in good agreement with the standard PWEM. Thus, the meshless methodsare proved to be a promising scheme for predicting photonic band gaps.

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1. Introduction

Photonic crystals (PhCs) are periodically-structured electromagnetic media, which generally possess photonic band gapsi.e. ranges of frequency in which light cannot propagate through the structure. Photonic band gap (PBG) materials have at-tracted considerable attention recently due to a variety of important expected applications, such as, high efciency lasers,optical circuits and optical communications [1] . Accurate computations of PBG materials are thus necessary for the devel-opment of optical bres and optoelectronic devices [24] . By combining the fundamental laws of electromagnetism andmathematics it is possible to create methods of numerical analysis for modelling PhCs. The traditional method used for mod-elling photonic crystals is the plane wave expansion method (PWEM) [57] as it is simple to implement and provides goodresults. However, in spite of successful computations, there are many problems with Fourier-based methods. Firstly, in PhCs,it is much more common that the underlying medium is discontinuous, so in this case Gibbs-type phenomena (from Fourier-type expansions) may lead to slow convergence of the truncated eld [8] . Secondly, many precautions should be taken in

order to ensure that the calculated spectra are correct. For example, it was found that the discontinuous nature of the dielec-tric function severely limits the accuracy of PWEM [8] . Thirdly, there is eld localisation due to the complexity of geometry.In addition, the method creates large dense matrices so it is slow to converge, thus computationally expensive. The niteelement method (FEM) has been used to model novel PhCs [9] , but as it relies on elements that are connected by nodesin a predened way its applications are limited. It requires a complex mesh that is computationally expensive and often re-quires re-meshing in order to achieve the required accuracy. Recently, meshless methods have been suggested as an alter-native due to their capability of solving partial differential equations (PDEs) using a set of nodes (that have no pre-speciedconnectivity between each other) within the domain of interest. A key feature of meshless methods is that they do not

0021-9991/$ - see front matter 2011 Elsevier Inc. All rights reserved.doi: 10.1016/j.jcp.2011.03.010

Corresponding author.E-mail address: [email protected] (E.E. Hart).

Journal of Computational Physics 230 (2011) 49104921

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require a grid or mesh, and it is computationally easy to add or remove points from a pre-existing set of nodes. This is not thecase for mesh-based methods; where the addition or removal of a point/element would lead to remeshing and hence addi-tion computational complexity. The emergence of meshless methods in engineering/electromagnetics is in its early stages,but its suitability for a variety of problems (particularly for uid/structural problems) has been demonstrated [1012] .Meshless methods can be categorised according to their formulation procedures, which fall into three groups: weak-form,strong-form or collocation and a combination of weak-form and strong-form [11] .

A number of meshless methods have been proposed to date. Smooth Particle Hydronamics (SPH) [13,14] , the diffuse ele-ment method (DEM) [15] , the element free Galerkin (EFG) method [16] , the reproducing kernel particle method (RKPM) [17] ,the boundary node method (BNM) [18] , the meshless local PetrovGalerkin (MLPG) method [19] , the point interpolationmethods (PIM) [20] and the meshfree weak-strong (MWS) form method [21] . In recent years, another group of meshlessmethods which are based on so-called RBFs, have become attractive for solving PDEs [12,2224] . Initially, RBFs were devel-oped for multivariable data and function interpolation, especially for higher dimension problems. Using RBFs as a meshlesscollocation method to solve PDEs possesses some advantages: it is a truly meshless method, it is space dimension indepen-dent, and in the context of scattered data interpolation it is known that some RBFs have spectral convergence orders. More-over, the use of RBFs for problems with a discontinuity are found to reduce the complications of Gibbs phenomenoncompared to the use of the Fourier basis [25,26] .

The most commonly used globally supported RBFs (GSRBF) in the literature for solving PDEs are multiquadratics (MQ),inverse multiquadractis (IMQ), thin plate splines (TPS) and Gaussian [27] . MQ, IMQ and Gaussian RBFs include a shapeparameter, whose numerical value can be varied to control the domain of inuence of the basis function (for example, inthe case of the Gaussian RBF, increasing the value of the shape parameter leads to atter basis functions). However, theseglobal RBFs produce dense matrices, which tend to become poorly conditioned, as the number of collocation points in-creases. There are currently several ways to overcome the disadvantages of using GSRBFs for solving PDEs, such as domaindecompression, preconditioning and ne tuning the shape parameters of MQs [2729] . Compactly supported RBFs (CSRBFs)provide a promising approach, and were introduced by Wu [30] , Wendland [31] and Buhmann [32] . The CSRBFs have a do-main of inuence that extends over a nite region of the domain as opposed to the GSRBFs whose inuence extends over theentire domain. The CSRBFs kernels contain a support size parameter by which we can adjust the sparsity of the matrix, thusmaking it well-conditioned [33] .

This paper proposes new meshless methods for solving Maxwells equations for PhC modelling with periodic boundaryconditions. These methods are naturally suited to handle discontinuous media, using local RBF approximations which con-form to the material interfaces; and are found to reduce the Gibbs phenomenon. This paper builds on the work of [24] wherean eigenvalue problem with a periodic domain is solved using a meshless method with CSRBFs. In this work, numerical re-sults are presented for two well known crystal structures in the literature and compared to the results from the PWEM.

This paper is organised as follows: Section 2 presents the equations used in modelling PhCs. In Section 3 a basic idea of RBFs is given. In Section 4 the RBF meshless methods are formulated. Results are presented in Section 5 for comparison withthe PWEM before conclusions are drawn in Section 6.

2. Two-dimensional photonic crystals

By making the standard assumptions Maxwells equations can be rewritten in terms of the magnetic eld as [34] :

r 1

r r H r xc

2

H r : 1

A two-dimensional PhC is periodic in the xy-plane and homogenous in the z -direction. When looking at light propagation inonly the xy-plane, mirror reection symmetry in the z -direction allows separation into two distinct polarisations: trans-verse-electric (TE) and transverse magnetic (TM). Eqs. (2) and (3) show the TE mode and TM mode respectively:

r 1

xr w kw; 2

1

xD w kw; 3

where xis the dielectric constant, w is the scalar eld intensity and k is the spectral parameter:k

xc

2

; 4

where c is the speed of light and x is the frequency. The PhC is modelled as innite, by imposing periodic boundary condi-tions on the unit cell, then the BlochFloquet theory can be applied [35] . Consequently the wave function can be representedas:

w ek x u x; 5

where k k1k2 is the quasimomentum vector and x

x y .

E.E. Hart et al. / Journal of Computational Physics 230 (2011) 49104921 4911
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3. Radial basis functions

The use of RBFs to solve PDEs can be viewed as a generalisation of the multivariate interpolation problem. For scattereddata xi; f i 2R d1 ; 1 6 i 6 N , the approximation F xto a function f xcan be written as:

F x XN

j1n j/ k x x jk; 6

where k x x jkis the Euclidean distance between points x and x j; N is the total number of points, / k:kis a RBF with centre x jand x is a point in R d . The unknown coefcients a j, j 1; 2; . . .; N can be determined by setting F xi f i; i 1 ; 2 ; . . .; N . Thisyields the system of linear equations:

A n F ; 7

where A / k xi x jkis an N N matrix, n n j and F F xiare N 1 matrices. Clearly, there exists a unique solution if and only if Ais non-singular. Micchelli [36] gaveconditions on / which guarantee thenon-singularityof A. These conditionsarequitegeneraland can be checked formanyRBFs. In particular, thereconditions aresatised forthe choicesof / given in Table1 .

The RBFs introduced in the table are GSRBFs. One of the most popular classes of CSRBFs is the one introduced by Wu [30]and Wendland [31] . Theses CSRBF are strictly positive denite in R d for all d less or equal to some xed value d0 . The basicdenition of the CSRBF / l;j r have the form:

/ l;j r 1 r n pr ; for j P 1 8

with the following conditions:

1 r n 1 r n if 0 6 r < 1 ;0 if r P 1 ;( 9

where l d2 j 1 is a dimension number, 2 j is the smoothness of the function and pr is a prescribed polynomial. Table2 list some of the Wu and Wendland CSRBFs when d 3. Note that unlike GSRBFs, the inuence of CSRBFs is local in [0,1]and the inuence vanishes in 1 ;1. Also, we can scale the basis function with compact support on 0; d by replacing r with r dwhere d is referred to as the support parameter of the CSRBF.

4. Meshless RBF methods

4.1. Meshless local strong form method

For the TM mode, substituting Eq. (5) into (3) gives:

r k r ku xku : 10

Representing u Pn j1 n j/ j where / j is a CSRBF and / j xi / jj xi x jjjgives rise to a generalised eigenvalue problem of theform:

Akn kBn; 11

where n are the eigenvectors of each eigensystem corresponding to the nodal eld values of allowable modes of propagationthrough the PhC and k are the respective eigenvalues that correspond to the frequencies of the mode [37] . The A and B matri-ces are:

Aij r k r k/ j xi; 12

Bij x/ j xi: 13 By expanding the equation that makes up the matrix Aij it is possible to construct the following eigensystem matrices:

E ij r2/ j xi; 14

F ij r / j xi; 15 H ij / j xi: 16

Table 1

Global radial basis functions.

/ r er 2c Gaussian

/ r c 2 r 21=2 Multiquadratic (MQ)

/ r c 2 r 21=2 Inverse multiquadratic (IMQ)

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Hence for the TM mode:

Aij E ij 2 k F ij k2 H ij : 17

The domain of the system can be seen in Fig. 1. The domain is made periodic by imposing the conditions:

u x; 0 u x; b 18

and

u0 ; y ua ; y; 19

where a and b are the length of the edges of the domain.For the TE mode, substituting Eq. (5) into (2) leads to:

r k 1

xr ku ku: 20

Expanding the left of Eq. (20) gives:

r 1

xr u r 1 xku k 1 xr u k 1 xku ku : 21

As the dielectric constant xis discontinuous for PhCs (having different dielectric constants at the interface) the mesh-less local strong form method (MLSFM) could not be applied to Eq. (21) . Thus, an alternative method based on the meshlessweak form method (which is naturally suited to handle discontinuous media) is applied to the TE mode. This method is de-scribed in the next section.

4.2. Meshless local weak form method

The solution to the differential Eq. (20) , given the periodic boundary conditions, is found using Galerkins method and themeshless RBF method. The Galerkin method calculates a discretised approximation to the true solution of the boundary va-lue problem, which gives the following integral for TE:

Z 1 xr ku r kv dx kZ uv dx: 22 Representing u P

n j1 n j/ j and

v Pnl1 nl/ l (where / is a CSRBF) gives rise to a generalised eigenvalue problem of the form:

C kn kDn; 23

Table 2

Wu and Wendland CSRBFs.

/ r 1 r 5

8 40 r 48 r 2 25 r 3 5r 4 Wu-C2/ r 1 r

6

6 36 r 82 r 2 72 r 3 30 r 4 5r 5 Wu-C4/ r 1 r

4

1 4r Wendland-C2/ r 1 r

6

3 18 r 35 r 2 Wendland-C4

Fig. 1. Domain of the periodic system.



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where the C and D matrices are for the TE mode:

C jl Z 1 xr k/ j r k/ ldx; 24 D jl Z / j/ ldx: 25

By expanding the integral that makes up the matrix C jl it is possible to construct the following eigensystem matrices:

S jl Z 1 xr / j r / ldx; 26 P jl Z 1 xk/ j r / ldx; 27 Q jl Z 1 xk/ l r / jdx; 28 T jl Z 1 x/ j / ldx: 29

Hence for the TE mode polarisation:

C jl S jl P jl Q jl k2 T jl: 30

This approach can also be applied to the TM mode. The solution to the differential Eq. (10) using Galerkins method and themeshless RBF method gives the following integral for TM:

Z r ku r kv dx kZ xuv dx; 31 which is another generalised eigenvalue problem with C and D matrices dened as:

C jl Z r k/ j r k/ ldx; 32 D jl Z x/ j / ldx: 33

By expanding the integral that makes up the matrix C jl it is possible to construct the following eigensystem matrices:

S 1 jl

Z r / j r / ldx; 34

P 1 jl Z k/ j r / ldx; 35 Q 1 jl Z k/ l r / jdx; 36 T 1 jl Z / j / ldx: 37

Hence for the TM mode polarisation:

C jl S 1

jl P 1

jl Q 1

jl k2 T 1 jl: 38

Although it is possible to solve TM using the Meshless Local Weak Form Method (MLWFM), the MLSFM is preferred as itrequires less computation time.

When a unit cell is discretised by a set of nodes, the nodes can be located exactly on the interface. Unfortunately in thissituation, it is not clear which dielectric constant should be assigned to the interface nodes, as the two materials have dif-ferent dielectric constants ( xdiscontinous). Using the RBF based meshless method will avoid this problem, as it employs aset of integration points (Gaussian points), apart from the set of nodes, to calculate the integrals in Eqs. (25)(29) and (33)(37) . These integration points will be located in either one of the materials, but not on the interface, and thus a given value of dielectric constant can be easily assigned to them. This will make the numerical implementation easy, and not requiringadditional methods, such as, for example averaging or smoothing the dielectric function. In the next section we describethe Gauss quadrature approach for the MLWFM.

4.2.1. Gauss quadrature for the MLWFM The numerical integration in the MLWFM is solved using Gauss quadrature. In two dimensions the integration over a

quadrilateral with a 6 x 6 b and c 6 y 6 d is given by:

Z b

a Z d

c f x; ydxdy Z 1

1 Z 1

1 f xf; ygj J jdfdg; 39

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where j J jis the Jacobian Matrix, and f is given by 1 xr / j r / l for Eq. (26) , 1 xk/ j r / l for Eq. (27) , 1 xk/ l r / j for Eq. (28) and1

x/ j / l for Eq. (29) respectively (for the TE mode). Substituting x ba2 f ba2 , and y dc 2 g dc 2 into Eq. (39) leads

to:

Z b

a Z d

c f x; ydxdy

b a2 d c 2 Z

1

1 Z 1

1 f

b a2

f b a

2 ;d c

2 g d c

2 dfdg: 40 Fig. 2 shows a quadrilateral with centre f 0; g 0 and 4 Gauss points. Evaluating the integral in Eq. (40) using Gauss

quadrature leads to:

b a2 d c 2 Z

1

1 Z 1

1 f

b a2

f b a

2 ;d c

2 g d c

2 dfdg

b a2 d c 2 X

n x

i1 Xn x

j1W ix W jy f

b a2

f i b a

2 ;d c

2 g j d c

2 : 41 When for 4 Gauss points: n x n y 2 ; W ix W jy 1 and f 1 g1

ffiffi3p

3 ; f 2 g2 ffiffi

3p 3 .

A set of uniform nodes is created that cover the unit cell. For two-dimensions there should be between 3 and 9 times moregauss points than there are nodes [11] . Then a set of background cells is generated that covers the unit cell and each back-ground cell has 4 gauss points and its own local support domain with radius r . Therefore for the rst background cell withGauss points 14, if nodes 6 and 7 were both in the support domain, Eq. (34) can be written as:

S 167 / 6 Gp1 / 7 Gp1 / 6 Gp2 / 7 Gp2 / 6 Gp3 / 7 Gp3 / 6 Gp4 / 7 Gp4 ; 42

where / is a CSRBF. However nodes 6 and 7 will also be included in other support domains so the values for each S 167 must besummed together to assemble the global S 1 matrix. A similar approach is used to assemble the other global matrices.

5. Results

The results from the new algorithms can be validated by comparison with other theoretical results presented in the lit-erature. The computational experiments in this section were carried out using MATLAB. A uniform point layout was used for

Fig. 2. Gauss point layout for the 4 point rule.

Fig. 3. A section of the two-dimensional PhC structure for a square array of rods with radius 0:2a , the black dashed line highlights the unit cell used.

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all of the experiments as in was found in a previous study to have a slightly better convergence rate than a random or sobolpoint layout of equal size [24] . The same study also found that the Wus C2 and C4 CSRBFs have a better convergence ratethan Wendlands C2 and C4 CSRBFs, which is why they are used in this work. In this section, the Band Diagrams for two dif-ferent crystals structures are calculated, rst for a square array of cylindrical rods in air and then for a square array of con-nected veins in air. The PWEM data was obtained using the MIT Photonic-Bands package (MPB) [38] .

Fig. 3 shows a section of the crystal structure for a square array of rods with radius 0:2a (a is the lattice constant), theblack dashed line highlights the unit cell used. Fig. 4(a) shows a Band Diagram of the TM mode, produced by the MLSFMusing Wus C2 CSRBF and uniform grid of 60 by 60 nodes, for square array of alumina ( 8 :9) rods with radius 0:2a inair 1 :0. Fig. 4(b) shows the Band Diagram of the TM mode, produced by the MLSFM using Wus C4 CSRBF and uniformgrid of 60 by 60 nodes, for the same square array of dielectric columns. Both gures are in good agreement with the standardPWEM gure, it can be seen from Table 3 that the average relative error is 1%. The relative error E E p E m=E p was cal-culated for all the eigenvalues in the rst four bands, where E p is the value of the eigenvalue from the PWEM (number of plane waves is 2048 and mesh size is set to 128) and E m is the value of the eigenvalue from the meshless method. InFig. 4, star markers are for the MLSFM results and open circles for PWEM. Both gures show the correct band gap betweenmode 1 and mode 2.

Fig. 5(a) shows the Band Diagram of the TM mode, produced by the MLWFM using Wus C4 function, a uniform grid of 30by 30 nodes and 36 by 36 background cells each with 4 Gauss points (5184 gauss points). The gure shows the PBG betweenmodes 1 and 2 and is in good agreement with the PWEM (with an average relative error of 0.1% Table 3 ). Fig. 5(b) shows theBand Diagram for the TE mode. This gure shows that unlike the MLSFM the MLWFM can also produce the correct gure forTE mode which is in good agreement with the PWEM (with an average relative error of 1%). Fig. 5(c) shows the TE and TMmodes and correctly has no complete PBG. Table 4 shows the eigenvalues for the rst four bands, calculated at each of thecorners of the irreducible Brillouin zone ( C, X and M ), using the meshless methods and the PWEM, for the square array of alumina rods. The results for the meshless methods and the PWEM show good agreement.

Convergence rates for the averaged eigenvalues of TM and TE modes are given in Fig. 6 for the relative errorE E r E m=E r , in which E r is the high resolution solution to the meshless method and E m is the result at the current res-olution. The order of accuracy for the MLSFM and MLWFM are Oh

2

and Oh4

for the TM and TE modes, respectively (theorder is found by expressing the error E h %Ch

p in dependence of h or n 1=h with loglog). The results of the TE modesshow better accuracy over the TM modes. This may be due to using weak form meshless methods which are found to have

usually better accuracy than the strong form [11] . In comparison with the FEM [37] , similar results (linear trends) were ob-tained using the MLSFM for the TM mode but for the TE mode the MLWFM has better convergence rates.

Fig. 4. Various band diagram showing the TM modes for a square array of dielectric columns 8:9with r 0:2a in air 1. PWEM results aredenoted by open circles and MLSFM results by star markers.

Table 3The average relative error of the Meshless methods compared to MPB (PWEM).

Average Relative Error

TM-MLSFM TM-MLWFM TE-MLWFM

C2 C4 C4 C4

Rods 0.01 0.01 0.001 0.01Veins 0.03 0.03 0.01 0.01



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Table 4

Eigenvalues for the rst four bands (calculated at each of the corners of the irreducible Brillouin zone ( C , X ,M )) for a square array of dielectric columns 8 : 9with r 0 : 2a in air 1; using meshless methods and the PWEM.

Band No: 1 2 3 4

C TM-MLSFM C2 0 0.583050 0.635560 0.637934TM-MLSFM C4 0 0.582480 0.635130 0.636200TM-MLWFM C4 0 0.582353 0.629054 0.629054

PWEM 0 0.582310 0.627800 0.627800X TM-MLSFM C2 0.275894 0.443988 0.643476 0.777156

TM-MLSFM C4 0.277114 0.445875 0.643433 0.776309TM-MLWFM C4 0.275117 0.443014 0.637289 0.773058PWEM 0.274706 0.442519 0.635953 0.772230

M TM-MLSFM C2 0.323557 0.552335 0.554360 0.689682TM-MLSFM C4 0.325103 0.554545 0.555431 0.694813TM-MLWFM C4 0.322850 0.549939 0.549940 0.693245PWEM 0.322395 0.548831 0.548831 0.693589

C TE-MLWFM C4 0 0.647549 0.832420 0.832420PWEM 0 0.627827 0.823526 0.823526

X TE-MLWFM C4 0.417970 0.466823 0.714904 0.868447PWEM 0.417558 0.461679 0.701203 0.854968

M TE-MLWFM C4 0.563892 0.602208 0.602209 0.681982

PWEM 0.548855 0.601888 0.601888 0.681155

Fig. 5. Various band diagram showing the modes for a square array of dielectric columns 8 :9with r 0:2a in air 1. PWEM results are denoted byopen circles and MLSFM results by star markers. The dashed lines represent the TM mode and the solid line the TE mode.



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Fig. 7 shows a section of the crystal structure for a square array of dielectric veins with thickness 0:165 a , the whitedashed line highlights the unit cell used. Fig. 8(a) shows a Band Diagram of the TM mode, produced by the MLSFM usingWus C2 CSRBF and uniform grid of 60 by 60 nodes, for a square array of alumina 8:9veins in air 1:0. Fig. 8(b)

Fig. 6. Convergence rates of the averaged eigenvalues for the TM (MLSFM) and TE (MLWFM) modes. The dashed line represents the TM mode and the solidline the TE mode.

Fig. 7. A section of the two-dimensional PhC structure for a square array of dielectric veins with thickness 0:165 a , the white dashed line highlights theunit cell used.

Fig. 8. Various band diagram showing the TM modes for a square array of dielectric veins 8 :9with thickness 0 :165 a in air 1 :0. PWEM resultsare denoted by open circles and MLSFM results by star markers.



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Fig. 9. Various band diagram showing the modes for a square array of dielectric veins 8 :9with thickness 0:165 a in air 1 :0. PWEM results aredemoted by open circles and the MLWFM results by star markers. The dashed lines represent the TM mode and the solid line the TE mode.

Table 5

Eigenvalues for the rst four bands (calculated at each of the corners of the irreducible Brillouin zone ( C , X ,M )) for a square array of dielectric veins 8 : 9with thickness 0 : 165 a in air 1; using meshless methods and the PWEM.

Band No: 1 2 3 4

C TM-MLSFM C2 0 0.452676 0.498638 0.529626TM-MLSFM C4 0 0.452586 0.498543 0.529519TM-MLWFM C4 0 0.435746 0.484633 0.508652

PWEM 0 0.439363 0.487806 0.513489X TM-MLSFM C2 0.236703 0.308276 0.486806 0.535565

TM-MLSFM C4 0.238450 0.310206 0.487308 0.535991TM-MLWFM C4 0.230579 0.299361 0.470726 0.515386PWEM 0.231376 0.300945 0.474696 0.51991

M TM-MLSFM C2 0.305745 0.360987 0.361283 0.645264TM-MLSFM C4 0.307888 0.363437 0.363437 0.645350TM-MLWFM C4 0.297665 0.349016 0.350020 0.618693PWEM 0.299445 0.352086 0.352086 0.629336

C TE-MLWFM C4 0 0.616463 0.618431 0.693236PWEM 0 0.618249 0.618249 0.694420

X TE-MLWFM C4 0.255452 0.439252 0.704695 0.718234PWEM 0.249566 0.438048 0.708642 0.716878

M TE-MLWFM C4 0.368238 0.529366 0.533327 0.540246

PWEM 0.361152 0.530968 0.530968 0.539335



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shows the Band Diagram of the TM mode, produced by the MLSFM using Wus C4 CSRBF and uniform grid of 60 by 60 nodes,for the same square array of dielectric veins. Both gures are in good agreement with the standard PWEM gure and cor-rectly show no PBG (average relative error of 3%).

Fig. 9(a) shows the Band Diagram of the TM mode, produced by the MLWFM using Wus C4 function, a uniform grid of 30by 30 nodes and 36 by 36 background cells each with 4 Gauss points (5184 gauss points). The gure shows no PBGs and is ingood agreement with the PWEM (average realtive error of 1%). Fig. 9(b) shows the Band Diagram for the TE mode. The gurecorrectly shows PBG between modes 1 and 2. Fig. 9(c) shows the TE and TM modes and correctly has no complete PBG (aver-age relative error of 1%). Table 5 shows the eigenvalues for the rst four bands, calculated at each of the corners of the irre-ducible Brillouin zone ( C, X and M ), using the meshless methods and the PWEM, for the square array of dielectric veins. Theresults for the meshless methods and the PWEM show good agreement.

6. Conclusion

In this paper a new algorithm based on a meshless compact RBF method was presented for use in PhC modelling. TheMLSFM is able to obtain accurate TM mode bands that are in good agreement with the standard PWEM. The MLWFM is ableto obtain accurate results for both the TM and TE modes that are in agreement with the standard PWEM. The RBF basedmeshless methods are shown to be a promising alternative scheme for predicting extended photonic band gaps. Future workis planned to extend the proposed meshless methods to three-dimensional PhC modelling.

Acknowledgments

This research was funded by the Microsoft Institute for High Performance Computing, Southampton. The authors are verygrateful to the referees for providing valuable comments and suggestions which are incorporated into this paper.

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Compact RBF Meshless Methods for Photonic Crystal Modelling