Modeling of Radiation Fields Sub-Picosecond Photo...

VLSI DESIGN1998, Vol. 8, Nos. (1-4), pp. 407-412Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under

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Printed in India.

Modeling of Radiation Fields in a Sub-PicosecondPhoto-Conducting System

K. A. REMLEY A. WEISSHAAR V. K. TRIPATHI and S. M. GOODNICKb’,

Departrent of Electrical and Computer Engineering, Oregon State University, Corvallis, Oregon, USA 97331,"Department of Electrical Engineering, Arizona State University, Tempe, Arizona, USA 85287

FDTD and Monte Carlo methods are combined to simulate the terahertz radiationfrom a coplanar photoconducting structure. The simulation tool under considerationallows calculation of potentials, particle distributions, current densities, and the nearfield electromagnetic fields anywhere in the computational domain. To model the farfield radiation, it is not efficient nor, in many cases, physically possible at present to usethe FDTD technique directly because of the excessive computational burden.Techniques are discussed for modeling both the near field and the far field radiation.Computational results showing the far field radiation are in qualitative agreement withpublished experimental results.

Keywords." Terahertz radiation, subpicosecond systems, photo-conducting systems, FDTD, timedomain analysis, computational modeling, electromagnetic radiation, far-field radiation, near-to-far-field transformation, Hertzian dipole

INTRODUCTION

Characterization of the electromagnetic radiationarising from ultrafast electronic structures whosedimensions approach the wavelength of operationis a subject of continued research. The design ofterahertz radiators [1, 2] as well as applicationssuch as electrooptic sampling [3], switching [4], andterahertz spectroscopy [5] all require an accuratemodel of the radiated electromagnetic field. In

some simple cases the radiated field may becalculated directly or approximated from a knownexcitation. Alternatively, numerical techniquessuch as the finite-difference time domain (FDTD)method allow computation of the fields radiatedfrom structures with arbitrary physical geometryor material composition. For an accurate portraitof the radiated fields, the open region simulationspace used in the FDTD should be terminatedwith absorbing boundary conditions (ABCs).

* Corresponding author.

407

408 K.A. REMLEY et al.

ABCs eliminate reflections at the edges of thecomputational domain, allowing simulation ofpropagation in infinite free space.The computational burden of the FDTD

method increases with the dimensions of thephysical simulation space, and thus becomes lessefficient for calculations far from the source. TheFDTD also suffers from numerical grid dispersionfor large computational domains. Additionally,the larger the simulation space, the larger thecomputational burden involved in the ABCcalculations. It is, therefore, desirable to usealternate techniques for determination of the farfield radiation.

FORMULATION

POISSONINIT FDTD,,J

MONTE CARLOSCATTER PARTICLES

FDTD SOLVEB-FIELDS

MONTECARLOSCATTER PARTICLES

PARTICLE CURRENT

FDTD SOLVEE-FIELDS

DETERMINEFAR FIELD

FIGURE Computer algorithm used in the present work.

Simulation Tool

The present work utilizes a simulation tool whichhas been developed at Oregon State University [6].The tool couples a Monte Carlo particle simulatorto either a Poisson solver or an FDTD electro-magnetic solver to determine device characteristicssuch as potential, particle distribution, and currentdensity anywhere in the computational domain. Inthe present case, the FDTD solver is used,enabling determination of the electromagnetic(EM) fields throughout the simulation space. TheMonte Carlo simulator and the FDTD solver arecoupled together in a leap frog manner, allowingcontinual updates of both the particle distributionand the EM fields, see Figure 1.The Monte Carlo particle simulator allows

stochastic solution to the Boltzmann transportequation for particle motion. Particle motion ismodeled as a series of free flights subject tomagnetic and electric forces. Motion is terminatedby instantaneous, random scattering events. Ran-dom scattering times are generated using a randomnumber generator and the calculated quantummechanical scattering cross-section. Importantscattering mechanisms for GaAs are included ina three band model.

The FDTD scheme used in the present workallows modeling of complex, three dimensionalstructures. Centered differences are used in time andspace, utilizing the Yee cell formulation [7]. Thistechnique allows broadband response predictionsand access to field components throughout thecomputational domain. The absorbing boundaryconditions used in the simulations space areBerenger’s Perfectly Matched Layer (PML) [8].Impedance matching to a highly lossy medium isaccomplished by utilizing electric and magnetic lossterms to attenuate only the outgoing portion of thewave. In the PML, absorption is theoreticallyindependent offrequency and angle ofincidence [8].

Photo-Conducting Experiment

The radiation arising from a photo-conductingsystem is modeled using the simulation tooldescribed above. In this system, a sub-picosecondlaser pulse incident on a GaAs substrate createselectron-hole pairs, as shown in Figure 2. Biasedelectrodes on either side of the pulsed area causethe electrons and holes to migrate away from eachother, creating a dipole moment. A time-varyingcurrent density is induced in the GaAs substrate.

MODELING OF RADIATION FIELDS 409

Laser Pulse

GaAs

biased electrodes

0.06

0.04

0.02

0

(a)0 0.1 0.2 0.3 0.4 0.5

Time (ps)

FIGURE 2 Typical structure for an electrooptic samplingexperiment.

The rise time of the current pulse is extremely fast,on the order of a few hundred femtoseconds, asillustrated in Figure 3a. This current density is thesource of the EM radiation in the photo-conduct-ing experiment. Once the excitation is removed,the biased electrodes continue to attract theelectrons and holes, therefore recombinationoccurs over a timescale much longer than thesimulation time.

6

(b)0.1 0.2 0.3 0.4 0.5

Time (ps)

FIGURE 3 (a) Current density in the GaAs substrate for aninjection rate of lel5/cm and bias of 40 volts. (b) Numericallysmoothed time derivative of the current pulse shown in (a).

RESULTS

Figures 4a and 5a show the near field radiationalong a surface above and parallel to the GaAssubstrate of the structure shown in Figure 2. Thesefields were calculated using the simulation tooldescribed above, with a bias of 40 volts on themetal electrodes, and an injection rate of e15/cm3.In this simulation it was assumed that the laserpulse had a Gaussian wavefront with beam radiusstandard deviation of 2 gm, a Gaussian-distrib-uted energy variation between 1.42 eV and 1.62 eV(larger than the bandgap of GaAs), and a pulseduration of approximately 20 fs.

If the source is modeled as a Hertzian dipole,i.e., a dipole with negligible length, the radiationmay be found using the well-known equations foran infinitesimal dipole, repeated, for convenience,below. See e.g. [9], for the derivation.

fvJ(t-7) dv fv 7)dv’ (1)cR2 + c2R

Here R is the distance from the dipole to the farfield observation point, J is the volume currentdensity induced by the dipole, and 0 is measuredoff the dipole axis. Figures 4b and 5b show thefields from an ideal dipole for comparison to thatof the near field radiation found using the FDTD


(a) 5 lO -lO

10

FIGURE 4 Eo: (a) Near field radiation from the photo-conducting structure found using FDTD. (b). Near fieldradiation from an ideal Hertzian dipole.

4003

(a) 5 10 -10

10

400]

-400------____

FIGURE 5 ER: (a) Near field radiation from the photo-conducting structure found using FDTD. (b) Near fieldradiation from an ideal Hertzian dipole.

simulation in Figures 4a and 5a. The effects of thevarious material parameters in the simulationspace may be seen in Figure 4a. Specifically, the

GaAs/air interface causes some variation in thefield distribution near the center of the surface.To model the far field radiation, only terms

which decrease as 1/R are retained from Eq. (1). Inthis case, the electric field is given by:

sin 0 0Eo(R, t)-4r-oc 20-t .f f fvJvdV’ (2)

The far field radiation is proportional to thetime derivative of the excitation. Eq. (2) is oftenused to approximate the far field radiation arisingfrom photoconducting antennas and arrays [1].However, (2) assumes a dipole source of negligiblelength and free space propagation. The use of (2)in the photo-conducting experiment also neglectsthe effects of the GaAs substrate and metalelectrodes. In addition, numerical implementationof the time differentiation emphasizes the highfrequency noise components. Figures 3a and 3bshow a typical current pulse and two numericallysmoothed versions of its derivative with respect totime. Smoothing is accomplished using a movingaverage filter. Figure 3b compares to measuredresults such as [10].

It is expected that a more accurate representa-tion of the far field radiation would be given with anear-to-far field transformation. This technique,which has been utilized by microwave and RFengineers for the analysis of radiation from, forexample, horn antennas [11], utilizes an equivalentsource methodology. The fields found on a closedvirtual surface surrounding the source are used asnew sources of radiation, replacing the originalsource, as illustrated in Figure 6. In this technique,the near field radiation arising from a complexstructure may be calculated using the FDTDmethod, and the far field radiation may be foundusing the equivalent source in a homogeneousexternal problem space. Several formulations foran equivalent source representation exist, inclu-ding vector forms such as the Schelkunoff equiva-lence theorem [12], the Stratton-Chu formulation[13], the Franz formulation [14], and the scalarKirchhoff Surface Integral Representation, de-

MODELING OF RADIATION FIELDS 411

Far fieldobservationpoint

\\ Equivalent

\\ r-r’ surface current

\\ densities

-_-Original i)source-------’T-,:

/--/’

Monte Carlo particle simulator with an FDTDelectromagnetic solver was used to numericallydetermine the near field radiation. Computationalresults showing near field radiation patterns wereshown. Techniques for approximating the far fieldradiation were discussed. Simulation results for a

technique based on calculation of the derivative ofthe excitation were presented.

FIGURE 6 Equivalent current source model. Acknowledgement

scribed, for example, in [15]. All provide arelationship between the electromagnetic fields ona surface surrounding a source and the field at apoint outside the surface. The field produced bythe equivalent source at the observation point willbe the same as that produced by the originalsource. The Kirchhoff formulation would be thepreferred technique in modeling the photo-con-ducting structure because the technique requiresonly the values of the electric fields on theequivalent surface. This is an advantage whenused with the FDTD since the electric andmagnetic fields are offset spatially by one half cell[16, 17]. Additionally, since this is a scalarformulation, each field component may be calcu-lated independently, allowing placement of thevirtual surfaces to coincide with each fieldcomponent’s placement in the Yee cell. Applica-tion of this technique to the photoconductingstructure is currently being implemented by thepresent authors.

CONCLUSION

Techniques for describing and modeling theterahertz radiation from a photo-conductingstructure have been discussed. The physical sourceof radiation for laser excitation on a GaAssubstrate was described, and an approximationof the source in terms of an ideal Hertzian dipolewas presented. A simulation tool which couples a

This work is supported by grant #ECS-9312240from the National Science Foundation. K. A.Remley is also supported under a National ScienceFoundation Graduate Research Fellowship.

References

[10]

[1] Froberg, N., Hu, B., Zhang, X. and Auston, D. H.,"Terahertz radiation from a photoconducting antennaarray", IEEE J. Quantum Electron., 28, 2291 2301, Oct.1992.

[2] Smith, P. R., Auston, D. H. and Nuss, M., "Subpicose-cond photoconducting dipole antennas", IEEE J. Quan-tum Electron., 24, 255-260, Feb. 1988.

[3] Valdmanis, J. and Mourou, G., "Subpicosecond electro-optic sampling: principles and applications", IEEE J.Quantum Electron., QE-22, 69-78, Jan. 1986.

[4] E1-Ghazaly, S. M., Joshi, R. P. and Grondin, R. O.,"Electromagnetic and transport considerations in sub-picosecond photoconductive switch model", IEEE Trans.Microwave Theory Tech., 38, 629-637, May 1990.

[5] Greene, B., Federici, J., Dykaar, D., Levi, A. and Pfeiffer,L., "Picosecond pump and probe spectroscopy utilizingfreely propagating terahertz radiation", Opt. Lett., 16,48-49, Jan. 1991.

[6] Goodnick, S., Pennathur, S., Ranawake, U., Lenders, P.and Tripathi, V. (1995). "Parallel implementation of aMonte Carlo particle simulation coupled to Maxwell’sequations", Intl. J. of Num. Modelling, 8, pp. 205-219.

[7] Yee, K., "Numerical solution of initial boundary valueproblems involving Maxwell’s equations in isotropicmedia", IEEE Trans. Antennas Propagat., AP-14, 302-307, Dec. 1966.

[8] Berenger, J. P. (1994). "A perfectly matched layer for theabsorption of electromagnetic waves",J. ComputationalPhysics, 114, 185 200.

[9] Franceschetti, G. and Papas, C. H., "Pulsed Antennas",IEEE Trans. Antennas Propagat., AP-22, 651-661, Sept.1974.Son, J., Sha, W., Norris, T., Whitaker, J. and Mourou, G.,"Transient velocity overshoot dynamics in GaAs forelectric fields > 200 kV/cm", Appl. Phys. Lett., 63, 923-925, Aug. 1993.


[11]

[12]

[13]

[14]

[15]

[16]

[17]

Balanis, C. (1989). Advanced Engineering Electromag-netics. New York: Wiley.Schelkunoff, S. A. (1936). "Some equivalence theorems ofelectromagnetics and their application to radiation prob-lems", Bell System Tech. J., 15, 92-112.Stratton, J. A., and Chu, L. J. (1939). "Diffraction Theoryof Electromagnetic Waves", Phys. Rev., 56, 99-107.Franz, V. W. (1948). "Zur Formulierung des Huygens-schen Prinzips", Z. Naturforsch. A, 3a, 500-506.Jackson, J. D. (1962). Classical Electrodynamics. NewYork: Wiley.De Moerloose, J. and Dezutter, D. "Surface IntegralRepresentation Radiation Boundary Condition for theFDTD Method", IEEE Trans. Antennas Propagat., 41,890- 896, July 1993.Ramahi, O. "Near- and Far-Field Calculation in FDTDSimulations Using Kirchhoff Surface Integral Representa-tion", IEEE Trans. Antennas Propagat., 45, 753-759,May 1997.

Authors’ Biographies

Kate Remley was born in Ann Arbor, Michigan inDecember, 1959. She worked as a broadcastengineer in Eugene, Oregon between 1983 and1992, and was Chief Engineer of an AM/FMbroadcast station from 1989-1992. She receivedthe B.S. degree magna eum laude from OregonState University in 1993 and the M.S. degree in1995. She is currently working toward the Ph.D.degree at Oregon State University as a NationalScience Foundation Graduate Research Fellow.Her research interests include computational mod-eling and measurement of guided-wave optical,microwave, and millimeter-wave devices, modelingof quantum waveguide structures, and the theoryof semiconductor device fabrication technology.Andreas Weisshaar received the Diplom-Inge-

nieur (Dipl.-Ing.) degree in electrical engineeringfrom the University of Stuttgart, Germany, in1987, and the M.S. and Ph.D. degrees in electricaland computer engineering from Oregon StateUniversity in 1986 and 1991, respectively. Since1992 he has been an Assistant Professor at theDepartment of Electrical and Computer Engineer-ing at Oregon State University. His current

research interests are in the areas of CAD ofmicrowave circuits and components, interconnectsand packaging, fiber optic sensors, and wirelesscommunications.

Vijai K. Tripathi received the B.Sc degree fromAgra University, India, M.Sc Tech. degree inelectronics and radio engineering from AllahabadUniversity, India and the M.S.E.E. and Ph.D.degrees in electrical engineering from the Uni-versity of Michigan, Ann Arbor, in 1958, 1961,1964 and 1968, respectively. He is a Professor andHead of Electrical and Computer Engineering at

Oregon State University in Corvallis, Oregon.Prior to joining Oregon State in 1974, he hadbeen with the Indian Institute of Technology inBombay, the University of Michigan in AnnArbor and the University of Oklahoma in Nor-man, Oklahoma. Over the years he has been aconsultant to many industrial organizations in-cluding AVANTEK, EEsof Inc., Teledyne MMICand Tektronix. His research activities are in thegeneral areas of microwave and distributed para-meter circuits and systems, computations electro-magnetics, electronic packaging and inter-connects.

Stephen M. Goodniek received his B.S. degree inEngineering Science from Trinity University in1977, and his M.S. and Ph.D. in ElectricalEngineering from Colorado State University in1979 and 1983 respectively. He was an Alexandervon Humboldt Fellow at the Technical Universityof Munich, Germany and the University ofModena, Italy in 1985 and 1986. He was theMelchor visiting chair at the University of NotreDame in 1991. He was a faculty member from1986 to 1997 in the Department of Electrical andComputer Engineering at Oregon State Univer-sity, Corvallis, Oregon, where he held the rank ofProfessor. He is presently Chair and Professor ofElectrical Engineering at Arizona State University.

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