Theory and Modeling of Lasing Modes in Vertical Cavity ... · 88 B. KLEINet al. by the laser....

VLS1 DESIGN1998, Vol. 8, Nos. (1-4), pp. 87-91Reprints 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.

Theory and Modeling of Lasing Modesin Vertical Cavity Surface Emitting Lasers

BENJAMIN KLEIN a’ *, LEONARD F. REGISTER a, KARL HESS and DENNIS DEPPE b

Beckma Institute and Coordinated Science Laborator),, University of lllinois at Urbana-Champaign, Urbana, Illinois 61801;Microelectronics Center, Department of Eledtrical and Computer Engineering, The University of Texas atAustin, Austin, TX 78712-1084

The problem of obtaining the lasing modes and corresponding threshold conditions forvertical cavity surface emitting lasers (VCSELs) is formulated as a frequency-dependenteigenvalue problem in required gain amplitudes and corresponding fields. Both indexand gain guiding are treated on an equal footing. The complex gain eigenvalues definenecessary but not sufficient conditions for lasing. The actual lasing frequencies andmodes that the VCSEL can support are then determined by matching the gain necessaryfor the optical system in both magnitude and phase to the gain available from the laser’selectronic system. Examples are provided.

Keywords: VCSELs, modes, lasing, modeling, gain eigenvalues

1 INTRODUCTION

In the simulation of edge emitting lasers, theoptical field can usually be modeled simply by apredetermined set of Fox-Li quasimodes of thepassive cavity [1]. The threshold condition can thenbe expressed as mode gain times mode lifetimeequals unity. No laser previously has presented thechallenge to optical simulation now presented byvertical cavity surface emitting lasers (VCSELs).The challenge to VCSEL simulation is not just a

much more complicated optical cavity geometryfor analysis-although this problem alone is

significant- but that quasi-mode analysis, itself,can no longer be relied upon. The mirrors aredistributed and lateral confinement may be pro-duced by a combination of index and gain guiding.(See, for example, Ref. [2].) Thus the VCSELoptical cavity boundaries, along with the passivecavity modes and conventional parameters such asthe photon lifetime, are often poorly defined.

2 THEORY

At the threshold for lasing, the optical field will beself-supporting in the presence of the gain supplied

* Corresponding author.

87

88 B. KLEIN et al.

by the laser. Consider an open VCSEL opticalcavity with tensor electric susceptibility (’, co)Xcav(’, co) + Xg(r, co) where Xg(r, co) is the necessa-rily complex susceptibility representing the lasergain inside the active region provided under bias,and cav(Y, co) is the potentially complex suscept-ibility representing the rest of the VCSEL. Withinthe semiclassical approximation, lasing requires [3]

g(r, fJ (1)

where (Y,)is the electric component of thelasing field, Gcav(Y’,) is the tensor Green’sfunction for radiation from a current source withinthe open VCSEL cavity defined by Xcav, andjeoX(r’ ) (’ ) acts as an equivalent cur-rent source.

If the spatial distribution of the gain is known togood approximation, then Eq. (1) reduces to aneigenvalue problem in the necessary gain ampli-tudes for lasing and the corresponding lasing

(0)fields. That is, if Xg(r,) g()Xg (r) where(0)Xg () is known, then Eq. (1) gives

(2)

where () is an integral operator defined by

(3)

Note that this eigenvalue problem need.gnly beevaluated over regions for which Xg (’) issignificant, that is, over the active region of thelaser. The solutions to this eigenvalue problem willbe complex gain eigenvalues and the correspond-ing fields as a continuous function of frequency, incontrast to the more familiar case of finding thediscrete real frequency eigenvalues and modes of aclosed passive cavity.

It is also possible to repartition the VCSELsusceptibility to take part of Xcav out of the

Green’s function and put it in Eq. (2) as a secondsource term. This may assist in the solution ofproblems for which the Green’s function is noteasily obtained. Extending the formulation in thisway changes the ordinary eigenvalue problem ofEq. (2) into a generalized eigenvalue problem.The solutions to Eq. (2) establish necessary but

not sufficient conditions for the VCSEL to lase.The frequencies at which the VCSEL can actuallylase are those for which the complex gainsusceptibility necessary for the optical system canbe matched to the complex gain susceptibilityavailable from the laser electronic system in bothmagnitude and phase.

3 EXAMPLE

To illustrate the use of this formulation, lasing inthe VCSEL cavity diagrammed in Figure wasconsidered [2]. This cavity allows for relativelyeasy analysis while allowing much of the essentialphysics of VCSELs to be modeled. Xcav(’,co)wastaken as that of the planarly layered structureincluding the nominally lossy quantum well layer,but with no lateral confinment. Note, for simpli-city, all susceptibilities were treated as scalars. Thespatial distribution of the gain susceptibility was

3o AWell

AlAs Buffer Layer

20 Period AIAs/GaAs DBR

0.25,0.GaAs

0"25’0,AIAs0,AIGaAs

9 x 10gm Active Region

GaAs

AlAs

AI0.67Ga 0.33As

AI0.67Ga 0.33AsAlAs

GaAs

20 Period AIAs/GaAs DBR

AlAs Buffer Layer

FIGURE Model VCSEL cavity. Ao=980nm.

THEORY AND MODELING OF LASING MODES 89

approximated as uniform inside the active regionand zero elsewhere. Thus for later convenience weset X(g) (Y) -j within the 9 pm 10 gm3 nmquantum well active region, and X(g)(f") -0 else-where.The Green’s functions for such a planarly

layered structure are well known [4], and onlythose with both source and field coordinates withinthe thin quantum well active layer were required.The Green’s functions were obtained in Fourier-space where the only portion of the calculation thatrequires a computer is the calculation of thedistributed-Bragg-reflector (DBR) reflectivities vs.frequency, incident angle and polarization, whichmay be performed using several equivalent meth-ods [5, 6]. Of course, calculation of the Green’sfunction for more complex cavity structures couldrequire computationally intensive numerical meth-ods, while taking advantage of the cylindricalsymmetry in many VCSELs could significantlyreduce the computational load. However, while ofobvious practical importance, how the Green’sfunction is obtained or what coordinate system isused is of no conceptual importance in thisformulation. Note that the potentially difficultproblem of finding the Green’s function wouldonly need to be solved once for all bias conditions,as the Green’s functions are independent of bias aslong as effects such as thermal expansion of thecavity are ignored.Once the Green’s function was calculated, the

gain eigenvalue problem of Eq. (4) was discretizedand solved. Using a moment method, the field,approximated as constant over the width of thewell, was expanded in rectangular testing functionswith unknown complex coefficients in the (x-y)plane of the well, i.e.

’(ff)rect x-nxdx)/\ff(x, y)rect (y nydy

(4)

where rect[(r/-rio)/d] is a rectangular pulse ofamplitude one and full width d centered at r/o, and

the discrete variable ff (nx, ny) labels the grid sites.(For notational convenience, the various depen-dencies on frequency are no longer indicatedexplicitly.) Then the inner products of both sidesof Eq. (4) were taken with the same rectangulartesting functions (Galerkin’s method [7]). The resultwas a finite set of linear equations for the unknownfield coefficients ’(ff) of the same form as Eq. (2),

ZjOjEO(gO ()cav(/’/,). /(/t) !/(/).if,

The corresponding discretized Green’s functionwas obtained from the continuous Green’s func-tion in Fourier-space by

cav(/,/t) f f dkxdkyejkxdx(nx-nx) ejkydy(ny-n)4sin(kxdx/2)sinV(kydy/2) cav(kx ky)222 (6)

for these rectangular testing functions. For thiswork, a commercially available (IMSL) routine forgeneralized complex eigenvalue problems was usedto solve Eq. (5).

Figure (2) shows the first three gain eigenvaluesn(co) (two are essentially degenerate) plotted at

1.51.0006 Xo________

o_

0.5

-0.5 v-

0.2 0.4 0.6 0.8

Real(K)

FIGURE 2 First three complex gain eigenvalues at discreteintervals in frequency; two are essentially degenerate. Lasing ispossible only where the eigenvalue curves crosses the real axis,with the values of the gain eigenvalues at these crossingsdefining the threshold condition for lasing.

90 B. KLEIN et al.

constant intervals in frequency. Approximatingthe gain susceptibility available from the electronicsystem of the laser under bias as purely imaginary,lasing is only possible where one of the gaineigenvalue curves crosses the real axis, such thatXg(F, o) ()X(g) (F) is also purely imaginary.The gain eigenvalue at the crossing frequencydefines the threshold gain susceptibility. Approx-imating the gain susceptibility as purely imaginaryis done here for illustrative purposes only. Inreality the actual lasing modes of the system wouldbe determined by the intersection of the gaineigenvalues with a curve describing the Kramers-Kronig relation between the real and imaginaryparts of the gain susceptibility. The large areaactive region considered in this example producesa very tight spacing of gain eigenvalues infrequency, but the gain amplitudes required forlasing are significantly different because of thedifferent overlaps of the self-consistently calcu-lated lasing modes with the active region. Figure(3) shows the field patterns of the first and thirdlasing modes inside the gain region.

x (microns)

FIGURE 3 Lasing fields at (a) A___

1.0003 Ao and (b) A_

1.0001 Ao as a function of position within the gain region.

4 CONCLUSION

The problem of obtaining the lasing modes andcorresponding threshold conditions for VCSELshas been formulated as a frequency-dependenteigenvalue problem in required gain amplitudesand corresponding fields. Index and gain guidingare treated on an equal footing. The complex gaineigenvalues define necessary but not sufficientconditions for lasing. The actual lasing frequenciesand modes that the VCSEL can support are thendetermined by matching the gain necessary for theoptical system in both magnitude and phase tothat available from the laser electronic system.With this formulation the lasing modes andcorresponding threshold conditions are well de-fined even when the optical cavity boundaries andconventional cavity parameters such as the photonlifetime are not.

This formulation also has the practical advan-tage that the problem of obtaining the self-supporting lasing fields of a VCSEL cavity isseparated into two distinct and already thoroughlystudied problems: that of finding the opticalGreen’s functions for a fixed radiation source inan open cavity and that of solving a complexeigenvalue problem. Thus, this formulation pro-vides a framework for the application of a largepreexisting knowledge base to the relatively newchallenge of modeling VCSELs. For example, inthe specific implementation used for this work, thecavity Green’s function was obtained by textbookmethods [4-6] and the generalized complexeigenvalue problem was solved using a commer-cially available numerical routine.

THEORY AND MODELING OF LASING MODES 91

Acknowledgements

This material is based upon work supported by theOffice of Naval Research.

References

[1] Grupen, and Hess, invited paper IWCE, Tempe Az,October 1995, to be published in VLSI Design.

[2] Lin, C. C. and Deppe, D. G. (1995). J. Lightwave Tech.,13, 575.

[3] Chow, W., Koch, S. and Sargent, M. 111, SemiconductorLaser Physics (Springer-Verlag, Berlin, 1994) p. 55.

[4] Chew, W. C. Waves and Fields in lnhomogeneous Media(Van Nostrand Reinhold, New York, 1990) pp. 57-79,375-418.

[5] Chew, W. C. pp. 45-56.[6] Born, M. and Wolf, E. Principles of Optics, Sixth Edition

(corrected) (pergamon Press, Oxford, 1993) pp. 51-66.[7] Balanis, C. A. Advanced Engineering Electromagnetics

(John Wiley and Sons, New York, 1989) p. 692.

Authors’ Biographies

Benjamin Klein received his B.S. and M.S. degreesin Electrical Engineering from the University ofWisconsin at Madison in 1994 and 1996, respec-tively. He is currently pursuing his Ph.D. inElectrical Engineering at the University of Illinoisat Urbana-Champaign. His thesis research is thenumerical simulation of vertical cavity surfaceemitting lasers.

Leonard F. Register is a Research Scientist in theBeckman Institute and Coordinated Science La-boratory at the University of Illinois at Urbana-Champaign. His current research interests includedissipative quantum transport, laser simulation,and leakage currents in thin oxides.

Karl Hess has dedicated the major portion of hisresearch career to the understanding of electroniccurrent flow in semiconductors and semiconductor

devices with particular emphasis on effects perti-nent to device miniaturization. His theories anduse of large computer resources are aimed atcomplex problems with clear applications andrelevance to miniaturization of electronics. He iscurrently the Swanlund Professor of Electrical andComputer Engineering, Professor of Physics,Adjunct Professor for Supercomputing Applica-tions and a Research Professor in the BeckmanInstitute working on topics related to Molecularand Electronic Nanostructures. He has receivednumerous awards, for example the IEEE DavidSarnoff Field Award for electronics in 1995.

Dennis G. Deppe is an Associate Professor in theElectrical and Computer Engineering Departmentat The University of Texas at Austin. He receivedthe Ph.D. degree in 1988 from the University ofIllinois at Champaign-Urbana, where he studiedatomic diffusion in III-V semiconductor hetero-structures, and semiconductor laser fabricationand characterization. After receiving the Ph.D. in1988 he worked as a Member of Technical Staff atAT&T Bell Laboratories in Murray Hill, NewJersey researching semiconductor lasers, and in1990 accepted a position as Assistant Professor atThe University of Texas at Austin. He waspromoted to Associate Professor in 1994, andholds the Robert and Jane Mitchell EndowedFaculty Fellowship in Engineering. His researchspecialties include III-V semiconductor devicefabrication, optoelectronics, and laser physics. Hehas published over 120 technical journal articlesand holds 6 U.S. patents, and has received theNational Science Foundation Presidential YoungInvestigator and Office of Naval Research YoungInvestigator Awards.

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