GENERALIZATION OF BERENGER S ABSORBING BOUNDARY CONDITIONS FOR 3-D MAGNETIC AND DIELECTRIC ANISOTROPIC MEDIA

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1 CW Light Injection, J. Lightwae Technol., Vol. 12, Mar. 1994, pp J. Wang, M. K. Haldar, L. Li, and F. V. C. Mendi, Enhancement of Modulation Bandwidth of Laer Diode b Injection Locing, IEEE Photon. Technol. Lett., Vol. 8, Jan. 1996, pp T. B. Simpon and J. M. Liu, Enhanced Modulation Bandwidth in Injection-Loced Semiconductor Laer, IEEE Photon. Technol. Lett., Vol. 9, Oct. 1997, pp R. S. Tucer, Large Signal Circuit Model for Simulation of Injection Laer Modulation Dnamic, Proc. Int. Elect. Eng., Vol. 128, Part I, 1981, pp R. Hui, Optical PSK Modulation Uing Injection-Loced DFB Semiconductor Laer, IEEE Photon. Technol. Lett., Vol. 2, 199, pp Figure 3 Photon number of the injection-loced laer diode under modulation 1998 John Wile & Son, Inc. CCC Figure 4 Frequenc modulation of the injection-loced laer diode GENERALIZATION OF BERENGER S ABSORBING BOUNDARY CONDITIONS FOR 3-D MAGNETIC AND DIELECTRIC ANISOTROPIC MEDIA I. Villo Pere, 1 S. Gonale Garcıa, 1 R. Gome Martın, 1 and B. Garcıa Olmedo 1 1 Electromagnetic Group of Granada Departamento de Fıica Aplicada Facultad de Ciencia Univerit of Granada Fuentenueva / n Granada, Spain Receied 15 December 1997 ABSTRACT: A general method i deeloped which etend Berenger perfect matching laer ( PML) aborbing boundar condition ( ABC) to achiee a perfect matching of wae propagating in 3-D magnetic and or dielectric aniotropic media. The method i alidated with the finite-difference time-domain method John Wile & Son, Inc. Microwave Opt Technol Lett 18: 12613, Ke word: aborbing boundar condition; aniotropic media; FDTD method; perfect matching laer Figure 5 4. CONCLUSION Phae modulation of the injection-loced laer diode A circuit model of an injection-loced laer diode ha been developed. The model ha been verified b comparing the reult of circuit imulation with reult reported earlier. With thi circuit model, imulation of direct intenit modulation of an injection-loced laer diode can be carried out uing circuit imulation pacage. The circuit model provide a fleible method for large-ignal imulation. REFERENCES 1. S. Mohrdie, H. Burhard, and H. Walter, Chirp Reduction of Directl Modulated Semiconductor Laer at 1 Gb b Strong I. INTRODUCTION The propagation of wave in aniotropic material, which are commonl ued in microwave component, microtrip waveguide, and fiber optic, i an area of current interet. The numerical imulation of thee device require the truncation of the computational domain. In previou contribution 1, 2, the author have tudied the etenion of Berenger perfect matching laer Ž PML. aborbing boundar condition Ž ABC. for matching aniotropic media. In the econd paper, the propoed continuit condition to achieve the PML, and pecific equation were found for the aborption of both the ordinar and etraordinar wave in a bidimenional, dielectric medium. The preent paper offer a tematic procedure to obtain different verion of Berenger PML for matching wave propagating in magnetic andor dielectric aniotropic media. The well-nown 3-D PML 3, 4 for iotropic media, and the PML for aniotropic one recentl propoed b Zhao, Juntunen, and Raianen in 5, are obtained a pecial cae in a imple and traightforward manner. 126

2 2. MAXWELL S CURL EQUATIONS IN ANISOTROPIC MEDIA Mawell curl equation for aniotropic, nondiperive, dielectric, and magnetic media can be written in matri form a H RE, E RH Ž 1. t t where E Ž E. t, E, E, H H, H, H t, the uperinde t indicate the tranpoitionof matrice, R Ž 2. i the rotational matri, and Ž. and Ž. ij ij repre- ent the dielectric and magnetic tenor of the medium, henceforth taen a real and mmetric. Taing the pace and time Fourier tranform of Ž. 1, the following et of algebraic equation i obtained: H K E, E KH 3 where K i the wavevector matri K. Ž 4. Thee equation are fulfilled b plane-monochromatic field jžtr. r,t e Ž 5. of amplitude and frequenc that propagate in the ˆn-direction with wavevector n. ˆ 3. EQUATIONS IN THE EPML MEDIUM Following Berenger idea, we will propoe the etended perfect matching laer Ž EPML. in the following manner: let u define the plit field e and h in thi numerical medium, each one with i component, e Ž e, e, e, e, e, e. Ž 6. t h Ž h, h, h, h, h, h. Ž 7. and the compact field e Ž e. t, e, e and h h, h, h t, with three component and related to the plit field b e C e, h C h, 1 1 C / Each compact component i the um of two plit one, i.e., e e e. In order to generalie the PML medium, we will relate the plit field b mean of two differential equation formall t 8 Ž. imilar to 1 including lo conductivit term: h h R e, t Ž 9. ee Rh t where,,, and are real and mmetric Ž 6 6. matrice whoe coefficient give the propertie of the EPML medium, and R i the plit rotational: Rr C, r. Ž 1. Notice that Eq. Ž. 9 are an etenion of Berenger 3-D PML equation 6. Thee equation admit monochromatic plane wave olution of the form jžwtr. r,t e Ž 11. with the comple wavevector in the EPML medium. The fulfill the tem of equation where M h G e, E e G h Ž M, E Ž 13. j j are the comple 6 6 matrice defining the EPML, and G i the comple wavevector matri Gg C, g. Ž PERFECT MATCHING BETWEEN BOTH MEDIA If a wave propagating in the firt medium occur on an interface with an EPML medium, no reflection will occur at the interface for an angle of incidence, nor for an frequenc, if there i a perfect matching between both media. Thi perfect matching i attained b etting appropriate boundar condition at the interface. Since the length of the 127

3 EPML medium i intended to be finite, it i necear, furthermore, for the tranmitted wave to be attenuated in it. A ufficient condition that achieve the matching i obtained b enforcing the continuit of the component of the field in the medium with thoe of the compact field. For intance, for the interface plane, Ž Ee., Ž Hh.. Ž 15. The continuit condition implie Ž j r jr 1. phae continuit e e. 2. amplitude continuit E e, H h. The firt condition require, 16 a mutuall independent. Thee condition are atified if A B. Ž 23. If, a in Ž 16., the interface lie in the plane, then 1 and 1. Similar relation are obtained for an interface in an other plane. The general condition Ž 23. can be epreed in term of the electric and magnetic parameter of the EPML b multipling, in Ž 21., A b M and B b E : and impoe no retriction on. The econd condition i eplicitl handled a follow: aume a wave with amplitude E and H and arbitrar wavevector occurring on the interface and propagat- with compact field amplitude e and h ; taing into account Ž.Ž 8, 12., and Ž 14., we can write ing into theepml h C M g e, e CE g h Ž 17. Ž. and from Eq. 3, there i continuit in the amplitude if 1 1 K C M g/ E Ž K C E g/ H. Ž 19. A M C, B E C. Ž 24. Eplicitl, the element of M are then related to thoe of b the following tem of 18 equation: M M, M M M M, M M M M, M M M M, M M Thee relation hold for ever value of E and H if the coefficient matrice are identicall null, thati, Defining 1 1 K C M g, K C E g A C M, B C E Ž 21. with A and B Ž 3 6. matrice, Ž 2. i written a M M, M M M M, M M M M, M M M M, M M M55 M 65, 33 M56 M K Ag, K Bg A B. Ž 22. For an incident wave of arbitrar frequenc, traveling in an arbitrar direction,,, and are mutuall indepen- dent variable. Moreover, if the aborption in the EPML i to be realied without puriou reflection from the edge and corner of the interface,,, and hould alo be taen Since matrice A and B are analogou, the electric parameter of the EPML medium are obtained in a imilar wa, b ubtituting in the above relation Mij Eij and ij ij Iotropic Cae. Aume the incidence medium i iotropic uch that I, I with I being the identit matri. It can be een that there are everal wa to chooe the param- 128

4 eter that obtain matching condition. Berenger equation are obtained b etting to ero all of the nondiagonal element of M and E, with the following reult for the interface at : Mij E ij, for i j M22 E22 M33 E33 M E M E M55 E55 M66 E66. Ž 27. So, in thi cae, M11 M44 M55 M66 and E11 E44 E55 E66 are real, o conductivit i ero. The remaining diagonal element can be choen a arbitrar comple number fulfilling Ž 26.. Thu, we will tae M22 M33 Ž 1j. and E E Ž 1j , with, , 22 33,. Given that, in free pace, n Ž c n, c' with n the -component of the unit wavevector ˆn, the propagation contant i written a Ž n c. Ž A jb. with A, B. Ž 28. B and A are, in conequence, frequenc-independent parameter that can be tuned to aborb propagative 7 and nonpropagative mode 8. The field in the EPML i then Ž 11. jžtž1c.ž n n n A.. Ž n c. B r,t e e. Ž Aniotropic Cae. Different pecific condition for the perfect matching of an electric and magnetic aniotropic medium are obtained from Ž 25. in a imilar wa to that ued for the iotropic cae. Auming that and are mmetric matrice, and without pecifing the plane of the interface, one imple choice for M i A imilar procedure i performed for E, with ij ij. For incidence on the plane, Ž. Ž. 1 and E22 E24 E26 E31 E33 E M22 M24 M26 M31 M33 M35. Ž A in the iotropic cae, we can epre Ž n.ž A jb. with A, B ij ij ij ij l l l l ij, l4 22, 114, 24, 124, 26, 134, 31, 214, 33, 224, 35, and where,, the phae velocit in the medium, i a function of direction. 5. IMPLEMENTATION AND VALIDATION Thee condition have been implemented in a three-dimenional FDTD code, baed on a two-node cheme 9 to imulate aniotropic media. The conductivitie of the EPML medium urrounding the computational domain have been choen with a patial profile, a in 7, taing the form ij l m B / n ij, l4 22, 114, 24, 124, 26, 134, 31, 214, 33, 224, 35, where Bm i a maimum value for B, i the ditance from the beginning of the EPML, and i the total depth of the EPML medium. The electric conductivitie are treated in an equivalent wa, from Ž 32.. The propagated wave in the EPML i attenuated, and after croing the EPML, it i reflected b perfectl conducting condition which end the domain. The theoretical reflection coefficient, a a function of the direction Ž,. of incidence, i then Ž2n1.Ž B m Ž,.. co R, e. 34 The ret of the EPML implementation detail Žcorner, etc.. are treated in the ame wa a 7. We have validated the above reult with a dielectric and magnetic aniotropic material, in which the relative contant are 1.64,.137,.162, ,.222, 1.34 and 1.58, ,.179, 1.66, , and The finite-difference time-domain method, with a two- node-baed alternative cheme for aniotropic media 9, i ued to imulate the problem. An EPML interface located at wa illuminated with a Gauian puled propagation mode with a time width to produce a deca of.1% in the frequenc domain at fma 1 GH, ampled with a time increment t Ž 15.Ž 1f. and pace increment ct Ž table in the two-node cheme., occurring normall on the interface with phae peed.639 c. The EPML comprie eight cell, and a quadratic ma 129

5 Figure 1 Reflection coefficient a a function of the frequenc and the reolution interpolation conductivit wa choen in order to obtain a normal theoretical reflection coefficient of.1% 1 db. Reult for the reflection coefficient a a function of frequenc are preented in Figure CONCLUSIONS In thi paper, a general method for the contruction of EPML aborbing boundar condition ha been developed. The general matching condition for three-dimenional electric and magnetic aniotropic media have been found. A imple continuit hpothei of the field at the boundar ha enabled u to encloe the general theor under imple algebraic condition. The method ha been validated in the time domain through the ue of the FDTD method, although the condition obtained here are alo uitable for other frequenc- and time-domain method. ACKNOWLEDGMENT Thi wor wa upported in part b the National Reearch Project TIC C4-1. REFERENCES 1. I. Villo Pere, S. Gonale Garcıa, R. Gome Martın, and B. Garcıa Olmedo, Etenion of Berenger Aborbing Boundar Condition to Match Dielectric Aniotropic Media, Microwae Guided Wae Lett., Vol. 7, Sept S. G. Garcıa, I. V. Pere, R. G. Martın, and B. G. Olmedo, On the Applicabilit of the PML Aborbing Boundar Condition to Dielectric Aniotropic Media, Electron. Lett., Vol. 32, No. 14, 1996, pp J.-P. Berenger, Three-Dimenional Perfectl Matched Laer for the Aborption of Electromagnetic Wave, J. Comput. Ph., Vol. 127, Sept. 1996, pp D. S. Kat, E. T. Thiele, and A. Taflove, Validation and Etenion to Three Dimenion of Berenger PML Aborbing Boundar Condition for FD-TD Mehe, IEEE Microwae Guided Wae Lett., Vol. 4, Aug. 1994, pp A. P. Zhao, J. Juntunen, and A. Raianen, Material Independent PML Aborber for Arbitrar Aniotropic Dielectric Media, Electron. Lett., Vol. 33, No. 18, 1997, pp J.-P. Berenger, Perfectl Matched Laer for the FDTD Solution of Wave-Structure Interaction Problem, IEEE Tran. Antenna Propagat., Vol. 44, Jan. 1996, pp J.-P. Berenger, A Perfectl Matched Laer for the Aborption of Electromagnetic Wave, J. Computat. Ph., No. 114, 1994, pp B. Chen, D. Fang, and B. Zhou, Modified Berenger PML Aborbing Boundar Condition for FDTD Mehe, IEEE Microwae Guided Wae Lett., Vol. 5, Nov. 1995, pp S. G. Garcıa, T. M. Hung-Bao, R. G. Martın, and B. G. Olmedo, On the Applicaton of Finite Method in Time Domain to Aniotropic Dielectric Waveguide, IEEE Tran. Microwae Theor Tech., Vol. 44, Dec. 1996, pp John Wile & Son, Inc. CCC A DUAL-FREQUENCY RECTANGULAR MICROSTRIP ANTENNA WITH A CIRCULAR SLOT Horng-Dean Chen 1 1 Department of Electronic Engineering Nan-Tai Intitute of Technolog Tainan, Taiwan 71, R.O.C. Receied 18 Noember 1997 ABSTRACT: B cutting a circular lot in the microtrip patch and uing a ingle probe feed, a compact rectangular microtrip antenna with dual-frequenc operation can be implemented. The propoed deign with ariou lot ie i eperimentall tudied. A 2% antenna ie reduction ha been demontrated b comparing the preent dual-frequenc deign with the conentional deign. Detail of the eperimental reult are preented and dicued John Wile & Son, Inc. Microwave Opt Technol Lett 18: 13132,