with a fine grid. Requirement for computation resources is muchless than conventional FDTD method for same degree of accuracy.The proposed method is very suitable for analyzing circuit prob-lems and scattering problems as long as there exist fine structuralfeatures in the structure.

REFERENCES

1. K.S. Yee, Numerical solution of initial boundary value problems in-volving Maxwell’s equations in isotropic media, IEEE Trans AntennasPropag 14 (1966), 302-307.

2. D.T. Prescott and N.V. Shuley, A method for incorporating differentsized cells into the finite-difference time-domain analysis technique,IEEE Microwave Guided Wave Lett 2 (1992), 434-436.

3. M.W. Chevalier, R.J. Luebbers, and V.P. Cable, FDTD local grid withmaterial traverse, IEEE Trans Antennas Propag 45 (1997), 411-421.

4. M. Okoniewski, E. Okoniewska, and M.A. Stuchly, Three-dimensionalsubgridding algorithm for FDTD, IEEE Trans Antennas Propag 45(1997), 422-429.

5. G. Sun and C.W. Trueman, Unconditionally stable Crank-Nicolsonscheme for solving two-dimensional Maxwell’s equations, Electron Lett39 (2003), 595-597.

6. G. Mur, Absorbing boundary conditions for the finite-difference ap-proximation of the time-domain electromagnetic-field equations, IEEETrans Electromagn Compat EMC 23 (1981), 377-382.

© 2007 Wiley Periodicals, Inc.

CONVOLUTIONAL PERFECTLYMATCHED LAYER FOR WEAKLYCONDITIONALLY STABLE HYBRIDIMPLICIT AND EXPLICIT-FDTDMETHOD

Iftikhar Ahmed and Er-Ping LiElectromagnetics Division, Institute of High Performance Computing,National University of Singapore, Singapore 117528; Correspondingauthor: iahmed@ihpc.a-star.edu.sg

Received 4 May 2007

ABSTRACT: In this article, we developed and implemented the convo-lutional perfectly matched layer (CPML) for the hybrid implicit explicitfinite-difference time-domain (HIE-FDTD) method. The method is vali-dated against the conventional FDTD-PML method and is found promis-ing. The development of CPML is able to extend the HIE-FDTD to opensurface problems, which works at higher CFL numbers in comparing tothe conventional FDTD method. © 2007 Wiley Periodicals, Inc.Microwave Opt Technol Lett 49: 3106–3109, 2007; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22962

Key words: ADI-FDTD; CFL; CPML; FDTD; HIE-FDTD; implicit;explicit

1. INTRODUCTION

The finite-difference time-domain (FDTD) method has been ap-plied to approximately all fields of the electrical engineering,especially elctromagnetics, photonics, RF/microwaves, biodetec-tion, and plasmonics [1]. However, because of its explicit nature,the Courant–Friedrich–Levy (CFL) constraint makes it computa-tionally expensive for high Q and electrically small structures. Toovercome this problem, unconditionally stable implicit methodssuch as alternating direction implicit FDTD (ADI-FDTD) [2],Crank–Nicloson FDTD (CN-FDTD) [3], and locally one-dimen-sional FDTD (LOD-FDTD) [4] methods are presented. Theseimplicit methods have their own constraints. The ADI-FDTDmethod has higher dispersion error [5] at larger time steps. TheCN-FDTD method has less dispersion error at larger time steps,but it is expensive computation wise. The LOD-FDTD methodalso like the ADI-FDTD method has higher dispersion error atlarger time steps [6]. The hybrid implicit explicit FDTD (HIE-FDTD) [7] method is in between implicit and explicit methods andis weakly conditionally stable. It allows larger time steps than thatof the conventional FDTD method.

To model and simulate open structure problems, different re-flectionless absorbing boundary conditions have been introduced[2]. Among absorbing boundaries, the perfectly matched layer(PML) is more common, robust, and widely used. However, thePML is less efficient for evanescent waves. To solve this problem,convolutional PML (CPML) is developed for FDTD and ADI-FDTD methods [8-10]. The main advantage of the CPML is itsgenerality for inhomogeneous, nonlinear, lossy, dispersive, andanisotropic media [9]. The following sections extend the HIE-FDTD method to open structure problems and for this purposeCPML is developed and implemented. It is applied to two-dimen-sional TE case for simplicity, although, on the similar pattern it canbe applied to three-dimensional case. Section 2 in this articlepresents the formulations of the HIE-FDTD and the numericalresults are given in Section 3.

2. FORMULATIONS OF HIE-FDTD CPML

Maxwell’s equations for homogeneous media in general forms are

� � H� ��D�

�t, (1)

� � E� � ��B�

�t. (2)

The matrix form for Eqs. (1) and (2) for the three-dimensionalHIE-FDTD method [8] is given as

�1 0 0 0

�t

�

�

�z�

�t

�

�

2�y0 1 0 0 0 0

0 0 1�t

�

�

2�y�

�t

�

�

�x0

0 ��t

�

�

�z

�t

�

�

2�y1 0 0

0 0 0 0 1 0

��t

�

�

2�y

�t

�

�

�x0 0 0 1

�T2

TABLE 2 Results of the Scattering Problem in Figure 4

Method CPU Time (s) Memory (Kb)

FDTD (Pure coarse grid) 30.6 1056FDTD (Pure fine grid, m � 3) 400.3 2468Hybrid CNDG subgridding (m � 3) 50.2 1212FDTD (Pure fine grid, m � 5) 1523.2 4952Hybrid CNDG subgridding (m � 5) 92.1 1440

3106 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 12, December 2007 DOI 10.1002/mop

� �1 0 0 0 0

�t

�

�

2�y

0 1 0�t

�

�

�z0 �

�t

�

�

�x

0 0 1 ��t

�

�

2�y0 0

0 0 ��t

�

�

2�y1 0 0

��t

�

�

�z0

�t

�

�

�x0 1 0

�t

�

�

2�y0 0 0 0 1

�T1 (3)

where,

T2 � �Exn�1 E

y

n�12 Ez

n�1 Hxn�1 H

y

n�12 Hz

n�1� �,

T1 � �Exn E

y

n�12 Ez

n Hxn H

y

n�12 Hz

n� �.

For simplicity, herein, two-dimensional TE case is consideredand corresponding HIE-FDTD equations are

�Dx

�t�

�

� y�Hzn�1 � Hz

n

2 � , (4)

�Dy

�t� �

�Hzn

� x, (5)

�Bz

�t� ��E

y

n�12

� x�

�

� y�Ex

n�1 � Exn

2�� . (6)

For the two-dimensional TE case, the HIE-FDTD method hastwo extra terms compared to the conventional FDTD method. Oneis the magnetic field term in Eq. (4) and the other is the electricfield term in Eq. (6). The discretized CPML forms of (4)–(6) areexpressed as

Ex�i�12, j

n�1� Ex�i�

12, j

n� �ex� i �

1

2, j��Hz� i�

12, j�

12

n�1� Hz� i�

12, j�

12

n�1 �� �ex� i �

1

2, j��Hz� i�

12, j�

12

n� Hz� i�

12, j�

12

n �� ex� i �

1

2, j�exy

n �i�

12, j

n, (7)

Ey�i, j�

12

n�12

� Ey�i, j�

12

n�12

� �ey� i, j �1

2��Hz� i�12, j�

12

n� Hz� i�

12, j�

12

n �� ey� i, j �

1

2�exy

n �i, j�

12

n, (8)

Hz�i�12, j�

12

n�1� Hz�i�

12, j�

12

n� �hx� i �

1

2, j �

1

2��Ey�i�1, j�

12

n�12

� Ey�i, j�

12

n�12 �

� hx� i �1

2, j �

1

2�hzx

n �i�

12, j�

12

n� �hy� i �

1

2, j �

1

2�

� �Ex� i�12, j�1

n�1� Ex� i�

12, j

n�1� Ex� i�

12, j�1

n� Ex� i�

12, j

n �� hy� i �

1

2, j �

1

2�hzy

n �i�

12, j�

12

n, (9)

where

�ex� i �1

2, j� �

�t

2��yky� j�, ex� i �

1

2, j� �

�t

��yky� j�,

�ey� i, j �1

2� � ey� i, j �1

2� ��t

��xkx�i�,

�hx� i �1

2, j �

1

2� � hx� i �1

2, j �

1

2� ��t

��xkx�i�,

�hy� i �1

2, j �

1

2� ��t

2ky� j���y,

hy� i �1

2, j �

1

2� ��t

��yky� j�,

where subscripts e and h denote the coefficients for electric andmagnetic field, respectively. As compared to the FDTD-CPML inthis method, two field terms in Eqs. (4) and (6) are extra terms.These extra terms can take more computing resources but withadvantage of higher CFL numbers comparatively. However, theHIE-FDTD CPML can take less computer resources as comparedto the ADI-FDTD CPML method, which requires two steps andmore auxiliary parameters (eight as compared to four in HIE-CPML). Two auxiliary equations of CPML for E and H compo-nents are given below and similarly other two can be found.

exy� i�12, j

n� bpexy�

i�12, j

n�12

� ap�Hz� i�12, j�

12

n� Hz� i�

12, j�

12

n � , (10)

hzx� i�12, j�

12

n� bphzy�

i�12, j�

12

n�12

� ap�Ey� i�1, j�12

n� Ey� i, j�

12

n � , (11)

ap ��p

��pkp � kp2�p�

�e���p

kp��p��t

�0 � 1�,

bp � e���p

kp��p��t

�0,

where p � x, y.Equation (7) can be further simplified into the tridiagonal

matrix form. During the simulation process, Eq. (8) is updated firstand then (7) and (9), respectively.

3. NUMERICAL RESULTS

This section will examine the formulation stated in Section 2, andtwo applications are considered. In the first application, a pointsource is placed in free space, while in the second application twoparallel dipole antennas are placed in free space. In first applica-tion, the point source in free space is radiating in all directions andits surrounding is truncated by the eight layers of CPML. Thisstructure is shown in Figure 1 with 50 50 number of cells in x

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 12, December 2007 3107

and y directions, respectively, with the cell size � � �x � �y �0.1 mm.

The following differentiated Gaussian pulse is used as a source:

Hz�ic, jc,t� � � 2� t � to

tw� exp� � �t � to

tw�2�.

The reflection error for the CFLN � 1 is shown in Figure 2 forHIE-FDTD with CPML and FDTD with PML, where the CFLN isdefined as

CFLN ��tHIE

�tCFL�FDTD.

Here �tHIE is the time step taken and �tCFL–FDTD is the time stepwith CFL limit for the conventional FDTD method. The stabilitycondition requirement of the two-dimensional HIE-FDTD method[11] is

�t ��x

c. (12)

This method is useful, when fine mesh is required in onedimension and coarse mesh in the other and vice versa.

Now for example, if �x � 4�y, then �tCFL-FDTD �4�y

c17,

�tHIE �4�y

c, and CFLN �

�tHIE

�tCFL-FDTD� 4.12.

Figure 2 shows that the FDTD-PML and the HIE-FDTD CPMLare in close agreement. After few hundred iterations, the HIE-CPML is better. To calculate the reflection error, the followingformula is used:

Reflection Error � 20 log�Emeasured � Eref�

�Eref max�,

Figure 1 Free space truncated by CPML

Figure 2 Reflection error for HIE-CPML and FDTD-PML for CFLN �1. [Color figure can be viewed in the online issue, which is available atwww.interscience.wiley.com]

Figure 3 Reflection error for HIE-CPML with different CFLN. [Colorfigure can be viewed in the online issue, which is available at www.interscience.wiley.com]

Figure 4 Reflection error for HIE-CPML with different CFLN at twodifferent points. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com]

3108 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 12, December 2007 DOI 10.1002/mop

where Eref is measured by extending the dimensions of the struc-ture to avoid any possible reflection effect from boundaries and150 150 number of cells are taken for the extended dimension.Eref max is the maximum value taken at the same measurementpoint. Figure 3 shows the reflection error for different CFLN. It isclear from this figure that with the increase in the CFLN, reflectionerror also increases and vice versa.

For example, at CFLN � 3.16 the HIE-CPML method has18.46 dB less reflection than that at CFLN � 5.1 and 9.36 dBhigher reflection than that at CFLN � 2.23. Similar to the ADI-CPML method, the increment in reflection error at larger CFLN isdue to dispersion. Figure 4 shows the reflection error at twodifferent positions P1 and P2 (shown in Fig. 1). At CFLN � 2.23,the difference between points A and B is 28.79 dB, while thereflection error at point B is higher than at A.

In the second application, two parallel dipole antennas areplaced in free space and they have 20 cells apart. All otherparameters are same as in the case of first application. The mag-netic field distribution after 62 iterations with CFLN � 1.414 incontour form is shown in Figure 5. This field disappears after fewhundred iterations due to absorbing boundaries.

4. CONCLUSIONS

In this article, CPML is developed and implemented in the hybridimplicit explicit FDTD method, the method has been examinedthrough the numerical simulation examples. This development willenhance the applications of the HIE-FDTD method, which is ableto work at larger time steps compared to the conventional FDTD-PML method, due to its weakly unconditional stability. The CPMLdeveloped in this article can be extended to three-dimensional caseon the similar pattern.

REFERENCES

1. A. Taflove, Computational electrodynamics: The finite-differencetime-domain method, Artech House, Norwood, MA, 2005.

2. F. Zheng, Z. Chen, and J. Zhang, A finite-difference time-domainmethod without the Courant stability conditions, IEEE MicrowaveGuided Wave Lett 9 (1999), 441–443.

3. S.G. Garcia, T. Lee, and S.C. Hagness, On the accuracy of the

ADI-FDTD method, IEEE Antennas Wireless Propag Lett 1 (2002),31–34.

4. J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, Efficientimplicit FDTD algorithm based on locally one-dimensional scheme,Electron Lett 41 (2005), 1046–1047.

5. F. Zheng and Z. Chen, Numerical dispersion analysis of the uncondi-tionally stable 3D ADI-FDTD method, IEEE Trans Microwave TheoryTech 49 (2001), 1006–1009.

6. E. Li, I. Ahmed, and R. Vahldieck, Numerical dispersion analysis withan improved LOD-FDTD method, Microwave Wireless Compon Lett5 (2007), 1–3.

7. J. Chen and J. Wang, A 3-D hybrid implicit–explicit FDTD schemewith weakly conditional stability, Microwave Opt Technol Lett 48(2006), 2291–2294.

8. J.A. Roden and S.D. Gedney, Convolutional PML (CPML): An effi-cient FDTD implementation of the CFS-PML for arbitrary media,Microwave Opt Technol Lett 27 (2000), 334–339.

9. S.D. Gedney, G. Liu, A. Roden, and A. Zhu, Perfectly matched layermedia with CFS for an unconditionally stable ADI-FDTD method,IEEE Trans Antennas Propag 49 (2001), 1554–1559.

10. I. Ahmed and E. Li, The error reduced ADI-CPML method for EMCsimulation, IEEE EMC Symp, in press.

11. B. Huang, G. Wang, Y.S. Jiang, and W.B. Wang, A hybrid implicit–explicit FDTD scheme with weakly conditional stability, MicrowaveOpt Technol Lett 39 (2003), 97–101.

© 2007 Wiley Periodicals, Inc.

AN ACCURATE ANALYSIS OFNUMERICAL DISPERSION FOR 3-DADI-FDTD WITH ARTIFICIALANISOTROPY

Kumar Vaibhav Srivastava, Vishwa V. Mishra, andAnimesh BiswasDepartment of Electrical Engineering, Indian Institute of Technology,Kanpur, Uttar Pradesh, India; Corresponding author:vaibhavs@iitk.ac.in

Received 5 May 2007

ABSTRACT: In this article, an accurate numerical dispersion relation-ship is developed for 3-D alternating direction implicit finite differencetime domain (ADI FDTD) with artificial anisotropy. The numerical dis-persion relation with accurate mathematical model helps to calculatethe anisotropic parameters, which are used to control the error of thenumerical phase velocity. Numerical example shows that the proposedmethod helps to reduce the numerical dispersion significantly. The ef-fects of anisotropic parameters on the numerical dispersion with differ-ent mesh resolution, courant number, and cell dimension ratio, areshown for three-dimensional anisotropic ADI-FDTD. A comparison hasbeen made for anisotropic ADI FTDD and conventional ADI FDTD interms of accuracy and CPU time for the calculation of resonant fre-quencies of rectangular cavity. © 2007 Wiley Periodicals, Inc.Microwave Opt Technol Lett 49: 3109–3112, 2007; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22961

Key words: alternating direction implicit finite difference time domain(ADI-FDTD); artificial anisotropy; numerical dispersion

1. INTRODUCTION

The alternating direction implicit finite difference time domain(ADI-FDTD) method [1], [2] has become more attractive becausethe stability condition of conventional FDTD method [3] can betotally removed in ADI FDTD. Although the ADI-FDTD method

Figure 5 Magnetic field representation of two parallel dipole antennas.[Color figure can be viewed in the online issue, which is available atwww.interscience.wiley.com]

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 12, December 2007 3109