HIGHER-ORDER SPATIAL FDTD SCHEMES FOR EM PROPAGATION IN DISPERSIVE MEDIA

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1 ISEF - XII International Symoium on Electromagnetic Field in echatronic, Electrical and Electronic Engineering Baiona, Sain, Setember -7, HIGHER-ORDER SPATIAL FDTD SCHEES FOR E PROPAGATION IN DISPERSIVE EDIA K.P. Prokoidi and T.D. Tibouki Deartment of Electrical and Comuter Engineering, Aritotle Univerity of Thealoniki, GR-44, Thealoniki, Greece ( krokoi@faraday.ee.auth.gr; tibuki@auth.gr) Abtract A higher-order (,N) (econd-order temoral and Nth-order atial) FDTD cheme with erfectly matched layer (PL) aborbing boundary condition (ABC) i rooed for long-ditance and/or timerolongated electromagnetic (E) wave roagation in Debye and Lorentz media of arbitrary ber of ole. The erical dierion relation i derived and the dierion error are comared with relevant technique. The accuracy and efficiency of the uggeted method i demontrated through erical imulation and comarion to the original FDTD cheme and to the analytic olution. Introduction The conventional finite-difference time-domain (FDTD) method [] ha been extenively ued to imulate electromagnetic (E) wave roagation in dierive media. Although, the FDTD algorithm ha already found wideread ucceful alication, to enure the accuracy of the comuted atial derivative of the field, a fine atial dicretiation (uually more than cell er minimum wavelength) i required, increaing the memory demand. In other word, the inherent erical dierion and aniotroy error render the original econd-order FDTD cheme imroer for long ditance wave roagation or large body roblem. Higher-order (HO) FDTD algorithm have been rooed to imrove the accuracy of the calculation for the above imulation. Several reearcher [-4] have tudied general (,N) (econd-order temoral and Nth-order atial) cheme only in non-dierive media. Neverthele, the tudy of E roagation in dierive media uually demand fine meh reolution and long time duration, rendering HO method a natural choice. Some aer, [,6] combine HO technique with FDTD method for dierive media in order to increae accuracy and reduce dierion error and comutational cot. Here, we rooe a novel methodology, otimized in term of additional variable, that utilize fourthand ixth-order accurate atial aroximation and a modified verion of the Auxiliary Differential Equation (ADE) decribed in [7]. The dierion error aociated with the rooed technique are alo conidered. In addition, we introduce fifth- and ixth-order one-ided aroximation to comlete the (,6) cheme at the interior grid oint near to the boundary. Finally, we rooe a new erfectly matched layer (PL) that can be laced adjacent to a dierive medium and terminate the FDTD comutational domain. Higher-order Satial Scheme for Dierive media The central difference oerator δ N, β of N -order ( N : even ber) ha the general form δ N ( + ) () f = c f f N, β n N, m n m/ n m/ m= ( modd) Thi work wa uorted by IRAKLITOS-Fellowhi for Reearch of the Aritotle Univerity of Thealoniki (under Grant 76), artly funded by the E.U. ISEF'-EE-.7 ISBN (6-Page)

2 where β i one of the atial variable ( x, yz, ) and m i it correonding index. Therefore, the central finite-difference aroximation to a derivative i f / β δn, β f / β. The coefficient c Nm, are calculated through Taylor erie exanion and are given in a cloed form [8] by m m ( ) [( N )!! ] ( ) cnm, =, m=,3,,, N () ( N m)!!( N + m)!! where n!! i the double factorial. Alying the reviou formula, the coefficient c, = for Yee cheme, c 4, = 9/8and c 4,3 = /4 for Fang fourth-order cheme, and c 6, = 7/ 64, c 6,3 = / 384, c 6, = 3/ 64 for a ixth-order cheme are yielded. Relative ermittivity (Real art) 6 Real art Imaginary Part. Relative ermittivity (Imaginary art) Relative ermittivity 4 3 Real art Imaginary art Fig. Comlex relative ermittivity of a Debye medium ( = 4.48, =.48, = 4.7, τ =.386 nec, τ =.9 nec ) and a Lorentz medium ( = 3, =., 9 ω = 4π rad / ec, 9 δ =.ω, G =.4, ω = π rad / ec, δ =.ω G =.6 ) The ermittivity function of a Debye medium with ole and a Lorentz medium with ole j t air are exreed in the frequency domain (auming e ω time deendence) through the relation Gω D( ω) = D, r( ω) = +, L( ω) = L, r( ω) = + ( ) (3) = + jωτ = ω + jωδ ω where, are the infinite and tatic frequency ermittivitie, τ i the relaxation time of the th Debye ole, and ω, δ, Gare the th reonant frequency, the th daming factor and th ole amlitude of the Lorentz medium, reectively. The real and the imaginary art of the relative comlex ermittivitie for a Debye ( Dr, ) and a Lorentz ( Lr, ) medium are hown in Fig.. Following the methodology of [7], an additional variable i introduced for each FDTD cell. The udate equation of the Q variable for the -ole Debye medium i τ t ( ) n n t + n+ n Q = Q + ( E + E ) (4) τ + t τ + t and the Amere law i dicretied a n+ n n+ / n E = cee + cm H + cm Q () = τ + t where c e and c m are contant involving medium arameter and t, given by t t ( ) t σ γ ce =, cm =, and γ =. (6) + σ t+ γ + σ t+ γ = τ + t The atial derivative of the curl are relaced by the HO accurate aroximation. Similar exreion for the Lorentz medium can eaily be derived and are the following ISEF'-EE-.7 ISBN (6-Page)

3 δ t ( t) ω ( t) G ω Q Q Q E (7) n+ n n n = + + δ t+ δ t+ δ t+ σ t t E E H ( Q Q ) (8) n+ n n+ / n+ n = + + σ t + σ t + σ t = Phae Error FDTD(,) Takayama et al FDTD(,) Joeh et al FDTD(,4) Takayama et al FDTD(,4) Joeh et al FDTD(,6) Takayama et al FDTD(,6) Joeh et al Phae Error 3 3 FDTD(,) Takayama et al FDTD(,) Joeh et al FDTD(,4) Takayama et al FDTD(,4) Joeh et al FDTD(,6) Takayama et al FDTD(,6) Joeh et al Phae Error 3 3 FDTD(,4) Takayama et al FDTD(,4) Joeh et al FDTD(,6) Takayama et al FDTD(,6) Joeh et al (c) Fig. The hae error a a function of frequency for the Lorentz medium with arameter of Fig. and x = 37.µm and Courant ber Q =.4 and Q =. and (c) comarion of the HO cheme for -4 very mall time te ( Q = ) The HO tencil require erical boundary condition (excet for the ABC) at the node next to the electric field boundary node. For the (,4) cheme we ued the fourth- and third-order accurate one-ided aroximation of [9]. However, the ixth-order tencil require erical condition for four node next to the boundary. For the electric field node next to the boundary node we obtain, through Taylor exanion, the following fifth-order one-ided aroximation = ( 689 f/ + f3/ + 43 f/ f7/ + 43 f9/ f/ ) +Ο(( β ) ) (9) 3 β and the reective one-ided exreion for all other electric and magnetic node 6 = ( 34 f + 66 f + 47 f 47 f f4 866 f + f6) + O(( β ) ) () 369 β / 6 = ( 6 f 6f+ 93 f + 9 f3 8 f f 9 f6) + O(( β ) ) () 9 β 3/ = ( 7 f/ f3/ + 7 f/ + f7/ 4 f9/ + 9 f/) + O(( β ) ). () 9 β ISEF'-EE-.7 ISBN (6-Page)

4 Numerical Dierion Relation Suoe each field variable F ( EorH) i rereented a a lane wave of the form Fex[ j( ωt k r )], where where k i the erical wave ber given by N k mk, β β = cnm, in β β = xyz,, β m= ( modd) a (3) a β i the unit vector in β -direction. In the dicretied domain, we relace / t with jω, where ω ( ω t ) = in / / t, and with jk in the medium ytem of differential equation in order to extract the erical ermittivity of the medium []. For the Lorentz medium with ole we find Gω = + ( ) (4) ω + jω δ ω where δ δ ( ω ) =, = co t /. It i noted that the erical ermittivity function of (4) i identical to Young method []. It i alo oberved that in Joeh method [], [6] ω, = ω co( ω ) wherea in the reent method ω, = ω. For the Debye medium it i yielded that = + ( t ) in ω / + τ ω ( t + τ ) + jτω /co( ω t/ ) t ω () = Joeh erical ermittivity (given in [6], []) i given by ( ) co( ω t / ) = + (6) = co( ω t/ ) + jτ ω axwell equation in the dicretied domain have the form [ σ co( ω t/ ) + jω ] E = jk H, jω µ H = jk E (7) The erical dierion relation i deduced from (7) and it i found to be σ co ( ω t/ ) + j ω j ω µ = k k (8) In the following, we ue the definition t = Q x/ c, where Q i the o called Courant ber and c i the velocity of light in vacuum. In order to invetigate the dierion error of each technique, the hae error ehae = ln Re{ k k}/ Re{ k} with k the exact waveber, i alo defined. In Fig the fourth- and ixth-order Joeh FDTD cheme give the ame hae error due to the high (for the HO cheme) value of the time te (the ame i valid for the HO Takayama cheme). It i alo noted that econd-order Joeh and Takayama technique coincide and that the HO cheme do not rovide eriou imrovement. In other word, i worthle to ue HO cheme if the time te i not very mall. In Fig., the Courant ber i. (although it i very mall for the econd-order cheme) to indicate the efficiency of the HO cheme with mall time te. In Fig. (c) a comarion of the fourth- and the ixth-order cheme i reented, deignating the accuracy of the lat if very mall time te i alied. Since in dierive media erical accuracy deend on how well the choen time te reolve the hortet timecale (i.e., τ in a Debye medium), one ha to reduce the Courant ber and a a reult the hae error introduced by the FDTD(,) cheme increae. However, in HO cheme uch a reduction of the Courant ber doe not lead to hae error degradation a can be oberved in Fig. In concluion, HO cheme with very low value of the Courant ber are highly recie E olver in dierive media. Perfectly atched Layer for the Termination of Dierive edia in HO Grid Conider an iotroic, inhomogeneou, linear dierive medium. The modified axwell equation in the frequency domain can be written a ISEF'-EE-.7 ISBN (6-Page)

5 σ E = jωµ T H, H = jω + ( ω) T E (9) jω where T = diag{ ζ /( ζ ζ ), ζ /( ζ ζ ), ζ /( ζ ζ )} i the diagonal material tenor, including the PL x y z y z x z x y conductivitie and ζ = /[ κ + σ /( jω)], for = xyz,,. The arameter κ, σ are aumed to be atially variant inide the PL following olynomial rofile of order n. We alo introduce two additional variable R = T E and the variable S uch that ( ζ / ζ ), ( ζ / ζ ), ( ζ / ζ ) S = E S = E S = E () x x y x y y z y z z x z An additional variable B i alo introduced for the magnetic field. The finite difference equation are derived following a methodology imilar to [7]. The udate equation for the variable S x i κ σ t S = S + R R ( ) n+ z z n n+ n x x x x κz + σz t κz + σz t The relation () and (8) can be ued a the udate equation of the variable R. The other udate equation are imilar to () and are omitted here. The efficiency of rooed PL i verified in two dimenional roblem. We conider the D TE wave roagation in a three-ole Debye medium with arameter: =., = 3, =, 3 = and τ = 9.4nec, τ =.nec, τ 3 = µec. We aume an FDTD grid with x= y =.m and Q =. ( t = Q/ c /( x) + /( y) ). The global error for the FDTD(,) method i deicted in Fig. 3 and for the FDTD(,4) method in Fig 4 for different value of PL thickne and for fourth- and econd-order olynomial grading of the n+ / n+ / PL arameter. The global error i a function of time and i defined a ( H H ), where the ubcrit denote the olution under examination in the comutational domain D and the exact the olution obtained in a larger domain with no reflection in the boundarie. It i noted that the rooed PL aborb more effectively the E wave with a fourth-order conductivity rofile and with increaing PL thickne. Numerical Reult Firtly, the rooed cheme were teted out for the wave roagation in a homogeneou mucle, modeled a three-ole Debye medium with arameter: = 4.3, = 974.3, = 3., τ = /(. π ) µec, τ = /(68 π )µec, τ 3 = /(46 π )nec and tatic conductivity σ =.6 S/m. In our imulation, although the FDTD(,) grid wa two time dener than that of the HO cheme, the olution are almot identical a deicted in Fig. 4. The effectivene of the rooed cheme i alo demontrated by conidering the one-dimenional lab roblem. The lab i m thick and i modeled a a two-ole Lorentz medium with arameter: =., = 3, G =.4, G =.6, 7 7 ω = 4π rad /ec, ω = 8π rad / ec, δ =.ω and δ =.ω. The tranmitted electric field, deicted in Fig. 4, eem to be correctly calculated though two time coarer grid were ued for the HO cheme. Concluion A HO FDTD technique with an unlit field PL for the imulation of E wave in dierive media i introduced in thi aer. The erical dierion relation are extracted and the introduced dierion error are etimated for the econd-, fourth- and ixth-order cheme. Additionally, extenive erical verification in roblem with multiole Lorentz and Debye media reveal the advantage and the efficiency of the rooed method. D z, z,exact () ISEF'-EE-.7 ISBN (6-Page)

6 Global error cell, n=4 8 cell, n=4 4 cell, n= 8 cell, n= Global error cell, n=4 8 cell, n=4 4 cell, n= 8 cell, n= 8 8 Electric field (V/m) Time te Time te Fig.3 Global error for the FDTD(,) and FDTD(,4) for different PL thickne and index n. 6 x 4 4 Direction of roagation FDTD(,) FDTD(,4) FDTD(,6) Poition (m) (c) Electric field (V/m) Time (nec) Exact FDTD(,) FDTD(,4) FDTD(,6) Time (nec) Fig.4 Time hitory of the electric field through a three-ole Debye medium at everal time te: (i).834µec (ii).668µec (iii) 3.336µec The tranmitted, exact and the comuted, time-domain olution at ditance 4m away from a two-ole Lorentz lab. Reference [] A. Taflove and S. Hagne, Comutational Electrodynamic: The Finite-Difference Time-Domain ethod, nd ed. Norwood, A: Artech Houe,. [] C. W. anry, S. L. Brochat, and J. B. Schneider, Higher-order FDTD method for large roblem, J. Alied Comutational Electromagnetic Society, Vol., 7-9, 99. [3] K. L. Shlager and J. B. Schneider, Comarion of the dierion roertie of higher order FDTD cheme and equivalent-ized RTD cheme, IEEE Tran. Antenna Proagat., Vol., 9-4, 4. [4] T. T. Zygiridi and T. D. Tibouki, Phae error reduction in general FDTD method via otimum configuration of material arameter, J. ater. Proce. Technol., Vol. 6, 86-9,. [] J. L. Young, A higher order FDTD method for E roagation in a colliionle cold lama, IEEE Tran. Antenna Proagat., Vol. 44, 83-89, 996. [6] K. P. Prokoidi, E. P. Komidou, and T. D. Tibouki, An FDTD algorithm for wave roagation in dierive media uing higher-order cheme, J. Electromagn. Wave Alicat., Vol. 8, 7-94, 4. [7] Y. Takayama and W. Klau, Reinteretation of the Auxiliary Differential Equation method for FDTD, IEEE icrowave Wirele Comonent Lett., Vol., -4,. [8] B. Fornberg and. Ghrit, Satial finite difference aroximation for wave-tye equation, SIA J. Numer. Anal., Vol. 37, -3, 999. [9] A. Yefet and P. G. Petrooulo, A taggered fourth-order accurate exlicit finite difference cheme for the time-domain axwell equation, J. Comut. Phy., Vol. 68, 86-3,. [] J. L. Young, A. Kitticharthayak, Y.. Kwork and D. Sullivan, On the dierion error related to (FD) TD tye cheme, IEEE Tran. icrowave Theory Tech., Vol. 43, 9-9, 99. [] R.. Joeh, S. C. Hagne, and A. Taflove, Direct time integration of axwell equation in linear dierive media with abortion for cattering and roagation of femtoecond electromagnetic ule, Ot. Lett., Vol. 6, no. 8, 4-44, 99. ISEF'-EE-.7 ISBN (6-Page)