IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 21, NO. 5, MAY 2011 225

A Laguerre-FDTD Formulation forFrequency-Dependent Dispersive Materials

Myunghyun Ha, Student Member, IEEE, and Madhavan Swaminathan, Fellow, IEEE

Abstract—In this letter, Laguerre finite-difference time do-main (FDTD) formulation for linear dispersive materials isproposed. Laguerre-FDTD is a time-domain method using La-guerre polynomials that ensures unconditional stability. Beingfree from Courant–Friedrich–Lewy (CFL) stability condition,Laguerre-FDTD is advantageous in the simulation of multiscalestructures. The derived formulation has been implemented in 3-Dand verified with a test structure containing frequency-dependentdispersive materials.

Index Terms—Dispersive materials, finite-differencetime-domain (FDTD) method, Laguerre-FDTD, Laguerrepolynomials.

I. INTRODUCTION

T HE finite-difference time-domain (FDTD) method hasbeen widely used to analyze the interactions of electro-

magnetic waves with various structures since its introductionby Yee [1]. However, the time-step in the FDTD method is con-strained by the well-known Courant–Friedrich–Lewy (CFL)criteria to ensure the stability of the FDTD simulation [2].To overcome these limitations, unconditionally stable FDTDmethod has been proposed using Laguerre polynomials byChung et al. [3], which is called the Laguerre-FDTD method inthis letter. Due to its unconditional stability, Laguerre-FDTDis free from CFL condition and is very efficient for analyzingmultiscale structures that require a fine time-step [4], [5].

Luebbers introduced the frequency-dependent FDTD formu-lation for dispersive materials in 1990 [6]. However, due to CFLcondition, it has been difficult to simulate multiscale structuressuch as chip-package structures using FDTD by including thefrequency-dependent dielectric loss.

In this letter, frequency-dependent Laguerre-FDTD formula-tion to model wave propagation is introduced using Laguerretransform of the convolution operator. The proposed formula-tion is verified with a numerical example and compared to the re-sults from the FDTD formulation. The comparison reveals thatthe frequency-dependent FDTD formulation for dispersive ma-terial is properly implemented in Laguerre domain so that fre-quency-dependent dispersive materials can be analyzed usingLaguerre-FDTD simulation method.

Manuscript received October 12, 2010; revised January 07, 2011; acceptedFebruary 11, 2011. Date of publication March 28, 2011; date of current versionMay 11, 2011. This work was supported by the Interconnect and PackagingCenter through Semiconductor Research Corporation.

The authors are with the Interconnect and Packaging Center and School ofElectrical and Computer Engineering, Georgia Institute of Technology, Atlanta,GA 30332 USA (e-mail: m.ha@gatech.edu; madhavan.swaminathan@ece.gatech.edu ).

Color versions of one or more of the figures in this letter are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LMWC.2011.2119296

II. MATHEMATICAL FORMULATION

A. Time-Domain Formulation for Frequency-DependentMaterials

In this letter, materials are assumed to be linear and isotropic.The frequency domain information (such as permittivity andpermeability) is Fourier-transformed to a time-domain suscepti-bility function [7]. For simplicity, only the permittivity (electricsusceptibility) is discussed in this letter. The extension to mag-netic permeability is similar. In the time domain, we have

(1)

where , , and represent electric susceptibility, permit-tivity of free space, and infinite frequency relative permittivity,respectively.

B. Transform of Convolution From Time Domain IntoLaguerre Domain

In order to represent (1) in the Laguerre domain, transform ofthe convolution term in (1) into Laguerre domain is required.

Laguerre domain is based on orthonormal Laguerre basisfunctions using Laguerre polynomials. The th Laguerre basisfunction is defined as the product of the th Laguerrepolynomial and exponential function [3], given by

(2)

where is the th Laguerre polynomial and is the time-scaling constant.

Let and be arbitrary time-domain waveforms de-fined for . They can be represented in Laguerre domainas a sum of Laguerre basis functions scaled by Laguerre basiscoefficients and as follows:

(3)

(4)

Let be the convolution of and . Applying thetemporal testing procedure with , ’s th Laguerre basiscoefficients can be obtained as

(5)

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226 IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 21, NO. 5, MAY 2011

Summation of from to is

(6)

In (6), the following property of Laguerre polynomial hasbeen used:

(7)

Defining and substituting it in (6), we get

(8)

From the summation of in (8), the th-order Laguerrebasis coefficient for the convolution of and is

(9)

C. Transform of Derivatives From Time Domain Into LaguerreDomain

The first derivative of the Laguerre basis function is

(10)

In the derivation of (10), the following property of Lagurrepolynomial has been used:

for (11)

For a casual function , the summation of all of itsLaguerre basis coefficients goes to zero as

(12)

Using (12) and the fact that is a uniformly con-vergent series, the first derivative of with respect to timecan be derived as

(13)

The second derivative can be derived similarly as

(14)

D. Laguerre-Domain Formulation for Frequency-DependentMaterials

Under the assumption that is defined only for , (1)can be rewritten using the infinite integral as

(15)

This assumption is reasonable since the Laguerre transformis for functions of defined for . Laguerre-domainrepresentation of (15) can be written as

(16)

where , , and represent th Laguerre basis coefficientsof , , and , respectively, .

Maxwell’s curl equations in differential form are

(17)

(18)

In (18), we assume that for linear and isotropic media.By applying the curl operation on (18) and substituting (17),

we get

(19)

HA AND SWAMINATHAN: LAGUERRE-FDTD FORMULATION FOR FREQUENCY-DEPENDENT DISPERSIVE MATERIALS 227

Fig. 1. Test structure: microstip transmission line.

Now, (19) is transformed into Laguerre domain with consid-eration of frequency-dependent behavior of dispersive materialusing (13), (14), and (16), which results in

(20)

resulting in

(21)

The double curl on the left side in (21) can be represented indiscretized form using Yee’s space lattice and the central differ-ence scheme.

It is important to note that the right side of (21) consists of La-guerre basis coefficients up to th order for and Laguerrebasis coefficients for source current , while the left side hasonly th-order Laguerre basis coefficients for electric field .Therefore, (21) enables recursive calculation of Laguerre coef-ficients using previous coefficients for electromagnetic waveswithin structures containing dispersive material.

III. NUMERICAL RESULTS

The proposed formulation is applied on a simple microstriptransmission line, as shown in Fig. 1. Two ports are defined atboth ends of the microstrip line made of PEC whose length is 90mm with conductor width and thickness of 3 mm and 10 m, re-spectively. Dielectric thickness is 200 m. The substrate mate-rial is FR-4, and it is assumed to be dispersive and modeled withthe first-order Debye model. Parameters for the Debye modelare , , and s [7]. Thestructure is discretized into 46 400 cells that contain 278 400 un-knowns. The Gaussian derivative current source is used to excitethe structure at port1.

Thin thickness of the conductor compared to its lateral di-mension results in a very long simulation time of 11 h usingFDTD due to the small time-step size of 33 fs. However, La-guerre-FDTD could be used to solve the same structure in 8 minon the same computer using 665 Laguere basis functions. In thisexample, Laguerre-FDTD showed 80 speedup over FDTD.

Fig. 2. Simulated time-domain waveforms.

Fig. 3. Comparison between simulated time-domain waveforms with disper-sive and nondispersive FR-4 model.

The simulated time-domain waveforms of E-field at the portsshow very good agreement between FDTD and Laguerre-FDTD,as shown in Fig. 2. For comparison, the same structure has beensimulated without frequency-dependent dispersive materialproperty using a constant permittivity of 4.301 with no loss.Fig. 3 shows the time-domain waveform variation as the mate-rial property changes from dispersive to nondispersive.

IV. CONCLUSION

A Laguerre-FDTD formulation for frequency-dependentdispersive materials is presented. The proposed formula hasbeen verified using a test example. Simulation results showthat Laguerre-FDTD with the proposed formulation solves theexample structure with dispersive material properties moreefficiently than FDTD while maintaining accuracy.

REFERENCES

[1] K. S. Yee, “Numerical solution of initial boundary value problemsinvolving Maxwell’s equations in isotropic media,” IEEE Trans.Antennas Propag., vol. AP-14, no. 3, pp. 302–307, Mar. 1966.

[2] A. Taflove and S. C. Hagness, Computational Electrodynamics: TheFinite-Difference Time-Domain Method, 3rd ed. Boston, MA: ArtechHouse, 2005.

[3] Y.-S. Chung, T. K. Sarkar, B. H. Jung, and M. Salazar-Palma, “Anunconditionally stable scheme for the finite-difference time-domainmethod,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp.697–704, Mar. 2003.

[4] K. Srinivasan, P. Yadav, E. Engin, M. Swaminathan, and M. Ha, “FastEM/circuit transient simulation using Laguerre equivalent circuit(SLeEC),” IEEE Trans. Electromagn. Compat., vol. 51, no. 3, pt. 2,pp. 756–762, Aug. 2009.

[5] M. Ha, K. Srinivasan, and M. Swaminathan, “Transient chip-packagecosimulation using the Laguerre-FDTD scheme,” IEEE Trans. Adv.Packag., vol. 32, no. 4, pp. 816–830, Nov. 2009.

[6] R. Luebbers, F. Hunsberger, K. Kunz, R. Standler, and M. Schneider,“A frequency-dependent finite-difference time-domain formulation fordispersive materials,” IEEE Trans. Electromagn. Compat., vol. 32, no.3, pp. 222–227, Aug. 1990.

[7] M. Koledintseva, K. Rozanov, A. Orlandi, and J. Drewniak, “Extrac-tion of Lorentzian and Debye parameters of dielectric and magneticdispersive materials for FDTD modeling,” J. Elect. Eng., vol. 53, no.9/s, pp. 97–100, 2002.