PHYSICAL REVIEW A NOVEMBER 1998VOLUME 58, NUMBER 5

Polarization fluctuations in vertical-cavity semiconductor lasers

M. P. van Exter, M. B. Willemsen, and J. P. WoerdmanHuygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands

~Received 1 May 1998; revised manuscript received 24 July 1988!

We report, theoretically and experimentally, how polarization fluctuations in vertical-cavity semiconductorlasers are affected by optical anisotropies. We develop a spin-eliminated~class A! description of laser polar-ization and show how the various model parameters can be extracted from the experimental data. In practice,the linear anisotropies are often much stronger than the nonlinear anisotropies, so that the polarization modesdefined by the linear anisotropies form a useful basis. For this case we derive a one-dimensional model forpolarization noise, with simple expressions for the relative strength of the polarization fluctuations and the rateof polarization switches. For the other, more extreme, case where the nonlinear anisotropies are as strong~oreven stronger! than the linear anisotropies, the spin-eliminated description remains valid. However, in this casethe concept of polarization modes is shown to lose its meaning, as a strong four-wave-mixing peak appears inthe optical spectrum and polarization fluctuations become highly nonuniform.@S1050-2947~98!06311-2#

PACS number~s!: 42.55.Px

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I. INTRODUCTION

Polarization fluctuations are present in all lasers, butexceptionally strong in semiconductor vertical-cavsurface-emitting lasers~VCSELs!. The reason for this is twofold. On the one hand, the spontaneous emission nowhich drives the polarization fluctuations, is relatively strodue to the limited size of the device. This is true for asemiconductor laser and leads, among others, to a relatlarge quantum-limited laser linewidth@1#. On the other handthe deterministic forces, being the optical anisotropies indevice, are relatively small due to the nominal cylindricsymmetry of a VCSEL. The combination of strong stochasnoise and weak restoring forces creates relatively largelarization fluctuations. A proper understanding of these fltuations is clearly important from a practical point of viewin any application of VCSELs polarization noise will be coverted into intensity noise by the~unavoidable! polarizationdependence of a practical detection system. Alternativand this is the emphasis of the present paper, a study of tfluctuations constitutes a very useful tool to unravel the vous anisotropies and other laser parameters of pracVCSELs. A preliminary report of our study has appearrecently@2#.

Until recently, it was difficult to compare theory and eperiments on VCSEL polarization. The ‘‘standard’’ theoreical model for the polarization of a quantum-well VCSELthe ‘‘split-inversion model,’’ developed by San MigueFeng, and Moloney@3#. In this model the conduction anheavy-hole valence band are treated as four discrete lewith M56 12 andM56

32 , respectively, and the inversion

split into two transitions~M5 12↔ 32 and M52 12↔2 32 !,each interacting with circularly polarized light of a specihandedness. An important parameter in this model isG,which describes the spin-flip relaxation between the two sinversions~normalized to the inversion decay rate!. Unfortu-nately, the split-inversion model is rather complicated, asdynamics of two population inversions has to be accounfor, so that analytic approaches are very difficult. Numeristudies have concentrated on the issue of polarization st

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ity, depicted in so-called stability diagrams, and more scifically on polarization switching and bistability@4–6#. Al-though polarization switching has been observed in sevexperiments, a quantitative comparison with theory provto be difficult, due to the numerical methodology and duethe fact that only limited information could be extracted frothe experiments reported so far@7,8#. Also, alternative expla-nations for polarization switches seemed equally likely@7#.

Other experimental studies involved the optical spectralight emitted by VCSELs@9#. In general, these spectra cosist of two ~Lorentzian-shaped! components, a strong ‘‘lasing mode’’ and a weak ‘‘nonlasing mode’’ with orthogonapolarization, the two components being related to the tVCSEL polarizations. The differences in center frequenand spectral width between these two components couldalmost completely attributed to linear anisotropies; onsmall deviations between experiment and a linear ‘‘couplmode’’ model hinted at more complicated population dnamics @9#. In practice, nonlinear anisotropies were thfound to be relatively small, corresponding to a large vaof G.

A reconciliation between experiment and theory cawith a simplified theoretical description, which was concurently developed by several authors@6,10,11#, in which thespin inversion was adiabatically eliminated from the lasrate equations. This leads to a first-order separation ofpolarization and intensity/inversion dynamics, so that thelarization dynamics of a VCSEL is that of a class A lasalthough the intensity dynamics is still that of a class B la~with relaxation oscillations!. In this model, the effect of theeliminated spin inversion is still contained in the rate equtions for the optical field, namely, as a nonlinear anisotroor polarization-dependent optical saturation, the saturapower for linearly polarized light being~slightly! larger thanfor circularly polarized light.

As a next step the rate equations are generally lineararound steady state. The simplicity of the linearized speliminated model allows for many analytic expressions;model yields, among others, expressions for a nonlinearshift and for excess damping of the nonlasing mode as c

4191 ©1998 The American Physical Society

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4192 PRA 58M. P. van EXTER, M. B. WILLEMSEN, AND J. P. WOERDMAN

pared to the lasing mode@6,10,11#. More recent predictions@11# concern the appearance of a third~four-wave-mixing!peak in the optical spectrum and asymmetries inpolarization-resolved intensity noise~see below!.

Theoretically, this paper constitutes an extension ofwork on the spin-eliminated model reported in@6# and @11#.We will put special emphasis on the role of noise. The kissue is not so muchwhetherthe lasing polarization is stablebut ratherhowstable it is, what the stability eigenvalues aand how much the polarization still fluctuates aroundequilibrium value. We hereby derive many useful exprsions for VCSEL polarization noise that allow for easy coparison with experiment. As a further extension, we willbeyond the linearized theory, concentrating on the practcase that linear birefringence is the dominant anisotropy,study the dynamics of polarization switching in VCSELExperimentally, we report a multitude of data on VCSEpolarization noise, extending the work reported in@2#. Byanalyzing the measured polarization fluctuations, whichbe exceptionally strong in VCSELs, we extract a seriesVCSEL parameters, with emphasis on the various optanisotropies.

We focus on three experimental tools to study the poization fluctuations. The first tool is a measurement ofpolarization-resolved optical spectrum, where polarizatfluctuations show up in the form of additional spectral peawith a polarization different from that of the lasing peak. Tsecond tool is a measurement of the polarization-resointensity noise, a polarization-type of homodyne detectisuggested by Hofmann and Hess@11# and first demonstratedin @2#, in which the intensity noise, after polarization projetion of the VCSEL output, is frequency analyzed. We wshow how this technique provides information on the polization fluctuations. As a third tool we employ a timdomain study of the polarization-resolved intensity.

In Sec. II we will briefly review the adiabatic model fothe polarization dynamics of VCSELs. In doing so we wgeneralize the earlier theory to the case of nonaligned bfringence and dichroism. After discussing the various paraeters in the problem, we will show how their magnitude cbe determined from experimental data. To facilitate the coparison between theory and experiment, Secs. III andpresent several useful expressions for the polarizatresolved optical spectrum, and the polarization-resolvedtensity noise, respectively.

In Sec. V we isolate the case where the linear birefrgence dominates over all other anisotropies; this is the cencountered for almost any practical VCSEL. We will shohow in this case the adiabatic description in terms of tpolarization variables can be reduced even further, tsimple one-dimensional description, with appealing exprsions for the relative strength of the polarization fluctuatioand the hopping rate in case of polarization switching.

In Secs. VI–X we present and analyze our experimendata, organized via the three basic techniques that we usSec. VI we discuss the experimental setup, in Sec. VIIpolarization-resolved optical spectra, in Sec. VIII tpolarization-resolved intensity noise, and in Sec. IX thelarization switches that occur in some VCSELs. In Sec. Xwill discuss results for VCSELs with a different desig

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II. ADIABATIC DESCRIPTIONOF POLARIZATION FLUCTUATIONS

Since we do not want to copy the derivation of the staing equations we refer to@3–5# for the standard split-inversion model and to@6,11# for the spin-eliminated versionof that model. The validity condition for the adiabatic elimnation has been thoroughly discussed in@6#: the polarizationof the optical field should vary slowly as compared to tmedium response to polarization changes. This means th~i!the optical anisotropies should not be too large, as thesethe time scale of polarization changes, and~ii ! the normal-ized spin-decay rateG should be large enough, as this sethe time scale of the medium response.

We will spend some effort in defining the parameters avariables of the problem~see Table I!, as the literature is farfrom uniform in this respect@3–6,11#. As variables we usethe average inversionN ~normalized to the threshold inversion! and the optical field vector Re@EWe2ivlt#. The latter canbe separated into two complex field componentsEx andEy~or E1 andE2!. This is, however, somewhat inconvenientnonlinear anisotropies create correlations between the fltuations in these components. We will therefore anticipthe expected separation between the polarizationintensity/inversion dynamics, and describe the optical fivector with four real-valued variables, instead. The first vaable is a common phase factor~the phase of the laser fiel

TABLE I. Important parameters and variables, together wtheir symbol and units.

Parameter or variable Symbol Units

Linear birefringence v lin ns21

Linear dichroism g lin ns21

Projected linear dichroism g i5g lincos 2b ns21

Angle between lin. birefringence& lin. dichroism

b rad.

Nonlinear birefringence vnon5agnon ns21

Nonlinear dichroism gnon ns21

Henry’s phase-amplitude coupling factoraEffective birefringence v052pn0 ns

21

Effective dichroism g05g i1gnon ns21

Cavity loss rate~of intracavity optical field!

k ns21

Loss rate of average inversion g ns21

Loss rate of difference inversion gs ns21

Normalized spin decay rate G5gs /gNoise strength D5nspk/S ns

21

Spontaneous emission factor nspNumber of photons in

fundamental cavity modeS

Intracavity intensity~normalized to saturation!

I

Polarization orientation angle f rad.Polarization ellipticity angle x rad.Rotationally averaged polarization anglew rad.

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w l!, which exhibits a diffusive evolution that has no consquences for the other dynamics. The other variables areoptical intensityI 5uEW u2, and two Poincare´ anglesf and xthat characterize the optical polarization@12#, wheref (0

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4194 PRA 58M. P. van EXTER, M. B. WILLEMSEN, AND J. P. WOERDMAN

Equation ~2! shows how polarization fluctuations resufrom a balance between the stochastic driving force of poization noise and the damping and spectral deformacaused by the various anisotropies. The polarization noisa manifestation of the quantum noise that results fromdiscrete character of photons and carriers. For practVCSELsk/g@1, so that photon noise dominates, as theerage number of inverted carrier states is much larger tthe average photon number. As photon noise originates frandom spontaneous emission of photons with arbitrphase and arbitrary polarization (N1'N2), the complexnoise vector fW(t) comprises four independent real-valunumbers, that can be divided into phase noise, intennoise, and two forms of polarization noise. Phase and amtude noise are best known as they also occur in the sinmode~scalar! problem. The two polarization components asimilar uncorrelated real-valued Langevin noise sourcesidentical strength, which satisfy

^ f x~ t1! f x~ t2!&5^ f f~ t1! f f~ t2!&5Dd~ t12t2!, ~5a!

^u f x~v!u2&5^u f f~v!u2&5D5nspk/S, ~5b!

where the noise strength, or diffusion rateD, is inverselyproportional to the photon numberS and proportional to theproduct of cavity loss ratek and spontaneous emission factnsp ~nsp>1 results from incomplete inversion as determinby the finite temperature, which smoothens the sharpnesthe Fermi-Dirac distribution! @15#.

One way to solve the polarization rate equations~2! is viaGreen functions that are based on the eigenvectors of Eq~4!;this was done in@11#. An easier way is to apply a Fourietransformation and solve the equations in the frequencymain, to obtain

f~v!5~ iv2g i22gnon! f f~v!1~v lin12agnon! f x~v!

~v2v02 ig0!~v1v02 ig0!,

~6a!

x~v!52v lin f f~v!1~ iv2g i! f x~v!

~v2v02 ig0!~v1v02 ig0!. ~6b!

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By combining these equations with the expressions forpolarization noise@Eqs.~5!# it is relatively straightforward tocalculate the experimentally accessible polarization-resoloptical spectra and intensity noise. This will be done in tnext section.

For completeness we note that the simplicity of the aboresults is due to the fact that, after spin elimination, thelarization dynamics~f,x! is separated almost completefrom the other dynamics, namely, that of the intensityI ,average inversionN, and optical phasew l . The only cou-pling is via the intensity dependence ofgnon and this cou-pling disappears when the intensity is reasonably consti.e., when fluctuations are limited or at frequencies very dferent from those of the polarization dynamics, so that ocan substitute the average intensity. As a result the polartion dynamics of a VCSEL can be that of a class A laswhereas the~relatively weak! intensity fluctuations that arestill present can be those of a class B laser that exhirelaxation oscillations@15,16#.

III. POLARIZATION-RESOLVED OPTICAL SPECTRA

In this section we will calculate the optical spectruuE(v)u2 of the VCSEL light, as measured after polarizatioprojection. In the linearized description, i.e., forf,x!1, theprojection onto the dominant polarization depends onlythe dynamics of the optical phase and intensity~see below!.On the other hand, if we block this light and project onto torthogonal polarization, we obtain different informationamely, on the polarization dynamics. The optical spectrthus observed is the Fourier transformation ofEy(t)'

2@f(t)1 ix(t)#E(t)exp@2iwl(t)#, where E(t)[uEW (t)u'uEx(t)u. For convenience, we will first assume the opticfield and optical phase to be constant atE(t)exp@2iwl(t)#5E0; later we will remove this restriction. In this practicacase, they-polarized spectrum is dominated by the polariztion dynamics, so that

^uEy~v!u2&'E02^uf~v!1 ix~v!u2&5DE0

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21~v2v lin22agnon!21g i

21~g i12gnon!2

~v22v022g0

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This optical spectrum generally consists of two peaksstrong peak atv'2v0 , which corresponds to the ‘‘nonlasing mode’’ in the coupled-mode description@9#, and a~muchweaker! peak atv'v0 , which is produced in a polarizatiotype of four-wave mixing~FWM! between they-polarizedpeak atv'2v0 and the dominantx-polarized peak atv50 @2#. The y-polarized spectrum can be approximatedthe sum of two Lorentzian curves with the same width whv0@g0 . The position and width~HWHM, half width at halfmaximum! of the two peaks yield the effective birefringencv0 and the effective dichroismg0 , respectively. The intensity of the FWM peak, relative to that of the nonlasing pe

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2 via

^uEy~v0!u2&^uEy~2v0!u2&

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2

4v02 1

g02

4v02 , ~8!

where the second term results from the Lorentzian wingthe nonlasing peak at the position of the FWM peak. Nthat a decomposition of the eigenvectors@Eq. ~4!# in their cwand ccw components gives the same approximate result

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PRA 58 4195POLARIZATION FLUCTUATIONS IN VERTICAL- . . .

It is relatively easy to go beyond the approximation‘‘constant E(t) and w l(t)’’ by noting that the polarization-resolved optical field is the product of the fieldE(t)exp@2iwl(t)# times a function of~f,x!. As a result, in thegeneral case the polarization-resolved spectrum equalsconvolutionof the ideal spectrum@Eq. ~7!# with the spectrumuE(v)2u'uEx(v)u2, as measured for projection onto thdominant polarization. The shape of the latter is similarthat of ‘‘edge-emitting’’ lasers: it has a finite~Schawlow-Townes! laser linewidthg lase, due to diffusion of the opticaphase, and~generally very weak! sidebands due to relaxatiooscillations@16#. After convolution one thus finds that phasdiffusion broadens all spectral peaks by an equal amog lase, being the~HWHM! spectral width ofuEx(v)u2, butthat it does not affect the relative strength of the FWM peas compared to the nonlasing peak, since these have the~intrinsic! width ~for v0@g0!.

IV. POLARIZATION-RESOLVED INTENSITY NOISE

Next we will discuss the polarization-resolved intensnoise. A measurement of this projected noise is extremsimple: the laser light is passed through a rotatablel/4waveplate and subsequently through a rotatable polarizeprojectEW (t) onto a selectable polarization state, after whthe projected intensity noise is measured. Projection ontodominantx or orthogonaly polarization yields informationabout the ‘‘polarization-mode partition noise’’@17#. The in-tensity noise in the orthogonaly projection is generallyrather small, being second order inf andx @see Eq.~1!#. Amuch stronger signal, i.e., first order inf and/orx, is foundfor projection onto a ‘‘mixed’’ polarization likex1y or x1 iy . Such a projection constitutes a polarization homodydetection, because it allows one to observe beats betweex-polarized lasing peak and they-polarized nonlasing andFWM peaks@11#. Through these intensity beats, which gunnoticed without projection, one gets a quantitative msure for the polarization fluctuations in the laser.

An appealing picture of the principle behind polarizatiprojection arises when we introduce the Poincare´ sphere. Onthis sphere each polarization state is depicted as a sipoint, i.e., the normalized Stokes vector (P1 ,P2 ,P3)[(cos 2x cos 2f,cos 2x sin 2f,sin 2x), where the equatocorresponds to all states of linear polarization, the polesthe two states of circular polarization, and the rest to ellipcally polarized light. On the Poincare´ sphere, the polarizationevolution is represented by a time trace and polarization fltuations by a ‘‘noise cloud.’’ Figure 1 sketches how, fdominantlyx-polarized light, this noise cloud is located

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the neighborhood of the equator atf,x!1. When the light ispassed through al/4 plate, with its axes at 45° with respeto the dominant laser polarization, this noise cloud is rotaby 90° on the sphere, to end up around the north pole~right-handed circular polarization!. The projected intensity behinda consecutive polarizer can now be found graphically by pjection of the polarization state onto an axis passing throequator and center of the Poincare´ sphere, with an orientation that depends on the polarizer angle. When the polaraxis is aligned with that of the lasing mode one projects oaxis P1 in Fig. 1 and measuresI project(t)5(I /2)(11sin 2x)'@ I (t)/2#@112x(t)#. When the polarizer axis is aligneunder 45° one projects onto axisP2 and measuresI project(t)'@ I (t)/2#@112f(t)#. Other orientations give linear combnations of these results.

After this discussion, a calculation of the polarizatioresolved intensity noise is straightforward. When the oveintensity is stable enough, the projected noise will be demined by the polarization dynamics only, so that the relatintensity noise, for projection onto thef or x direction re-spectively, is given by

FIG. 1. Principle of noise projection on the Poincare´ sphere. Thepolarization fluctuations around the, almost linearly polarizsteady state are represented as a noise cloud around a positionto the equator. Propagation through al/4 plate and polarizer resultin a 90° rotation towards the north pole and a projection downwaonto an axis, the orientation of which depends on polarizer anBy projecting onto axisP1 or P2 we can measure the noise in thPoincare´ anglesx or f, respectively.

^uDI project~v!u2&^I project&

2 54^uf~v!u2&54DS v21~v lin12agnon!21~g i12gnon!2~v22v022g02!214g02v2 D , ~9a!

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4196 PRA 58M. P. van EXTER, M. B. WILLEMSEN, AND J. P. WOERDMAN

For the case of relatively large birefringence (v0@g0 ,gnon,agnon), these projected noise spectra are qusimilar, both peaking aroundv0 and having a spectral widthof g0 ~HWHM!. From fits to these spectra one can direcobtain the effective birefringencev0 and effective dichroismg0 , without the experimental complication of a finite laslinewidth g lase that occurs when analyzing the optical spetra.

Interestingly enough, the above spectra have the sfunctional form as the relative intensity noise~RIN! spec-trum. When the intensity fluctuations are relatively small,that the intensity rate equation can be linearized, this Rspectrum is given by

^uI ~v!u2&I 0

2 54Dv214g ro

2

~v22v ro2 !214g ro

2 v2, ~10!

where v ro and g ro are the relaxation oscillation frequencand damping rate, respectively@15#. As the diffusion rateDis the same in Eqs.~9a,b! and ~10!, the relative strengths othe polarization fluctuations as compared to the intenfluctuations are approximately equal to the ratio of the relation decay rateg ro , over the polarization decay rateg0 ,where low damping corresponds to a sharp resonancelarge fluctuations.

Once more it is relatively easy to generalize the exprsions for the projected polarization noise to beyond theproximation of stable intensity. In the common case of retively small intensity and polarization fluctuations (D!g0 ,g ro), the time-dependent part of the projected intensis approximately12 DI (t)1I 0x(t) or

12 DI (t)1I 0f(t), where

DI (t) is the deviation from the average intensityI 0 . As theintensity and polarization fluctuations are practically uncrelated, apart from minor interactions via the dichroismg linand gnon, the general projected noise spectrum is equathe sum of the ideal polarization noise spectrum@Eqs. ~9!#and the~scaled! intensity noise spectrum, as measured wiout polarization projection@Eq. ~10!#.

The difference between thef andx projections, i.e., be-tween Eqs.~9a! and ~9b!, is a measure for the ellipticity othe noise cloud on the Poincare´ sphere:

^uf~v!u2&^ux~v!u2&

5v21~v lin12agnon!

21~g i12gnon!2

v21v lin2 1g i

2 ,

~11!

and can be used to estimate the nonlinear anisotropiesgnonand agnon. For relatively large linear birefringence (v lin@g i ,gnon,agnon) the ratio displayed in Eq.~11! approachesunity and the exact result can be approximated as

^uf~v!u2&1/2

^ux~v!u2&1/2'11

2agnonv0

S v02v021v2D , ~12!where we have introduced square roots to facilitate a cparison with the experimental signal on the RF analyzer@2#.Equation~12! shows that the nonuniformity of the polarization fluctuations depends on frequency, being relatively lafor v

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PRA 58 4197POLARIZATION FLUCTUATIONS IN VERTICAL- . . .

birefringence, the polarization-resolved optical and intensnoise spectra become relatively simple, as the strength oFWM peak and the nonuniformity of the polarization flutuations are strongly reduced, being inversely proportionav0

2 andv0 , respectively@see Eqs.~8! and~12!#. One expla-nation for this behavior is that the relatively fast rotationthe Poincare´ sphere, associated with the large linear birefrgence, makes all trajectories look like ‘‘tightly wound corscrews’’ and thereby smooths out the difference betweefandx dynamics. An equivalent explanation is that the larfrequency difference, in the optical spectrum, betweennonlasing and lasing peak reduces the coupling betweentwo, making the orthogonal polarization mode look more amore like a standard nonlasing mode.

As a starting point for our full~nonlinearized! descriptionof the polarization dynamics we could use Eqs.~A1! in Ap-pendix A. Instead, it is more convenient to rewrite the speliminated model in terms of the normalized Stokes vecas @11#

dP1dt

5g lincos 2b~12P12!2g linsin 2bP1P2

12gnonP1P3212agnonP2P3 , ~15a!

dP2dt

52v linP31g linsin 2b~12P22!2g lincos 2bP1P2

12gnonP2P3222agnonP1P3 , ~15b!

dP3dt

5v linP22g lincos 2bP1P3

2g linsin 2bP2P322gnonP3~12P32!. ~15c!

For the case of dominant linear birefringence the prevailevolution over the Poincare´ sphere is a fast rotation arounthe P1 axis, whereP2 and P3 perform a rapid out-of-phasoscillation with approximate frequencyv lin , driven by thefirst terms in Eqs.~15b,c!. On top of this rapid oscillation ofthe P2 andP3 coordinates, there is a much slower evolutiof the P1 coordinate, that can be separated out via a nadiabatic elimination. On the Poincare´ sphere, the slow vari-able measures the position of an almost circular orbit atmost constantP15cos(2w), wherew5f only at x50. Byaveraging Eq.~15a! over the fast rotation just mentionedwe can set ^P1P2&'0, ^P2P3&'0, and ^P1P3

2&'(1/2)P1(12P1

2), to obtain

dP1dt

'~g i1gnonP1!~12P12!. ~16!

As the combination (12P1)/2 is equal to the relative intensity of they-polarized light, the above equation describesdeterministic evolution that underlies the polarization-mopartition noise.

To obtain the full polarization dynamics we will now adnoise to the above equation~16!. For the anglew it is imme-diately clear how much noise should be added: as polartion noise is isotropic on the Poincare´ sphere, the amount onoisef w , perpendicular to the fast orbital evolution, is equ

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2512P22

2P32. Keeping this into account we obtain the followin

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dP1dt

5~g i1gnonP1!~12P12!24DP11~2A12P12! f w ,

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tan~2w!1 f w .

~17b!

These equations show how the dominant linear birefrgence, or fast rotation on the Poincare´ sphere, effectivelyredirects the nonlinear anisotropy, so that the original~non-linear! competition between the two circularly polarizestates is converted into a competition between the linepolarized states aligned along the axes of birefringenEquation~17b! thus has the same form as Eq.~9! in @19#,which was recently derived for the dynamics of the ellipticanglex of an isotropic class A laser with strong competitiobetween its circularly polarized fields.

By transforming the above equations~17a,b! into the cor-responding Fokker-Planck equations we regain the stanproblem of ‘‘diffusion in a potential well,’’ on which thedynamics of a class A laser is usually mapped@20,21#. Thesteady-state probability distributions and potentials of osystem are

P~P1!}expF2 VP1~P1!D G}expF g i2D P12 gnon4D ~12P12!G ,~18a!

P~w!}expF2 Vw~w!D G}sin~2w!expF g i2D cos~2w!1 gnon8D cos~4w!G .

~18b!

The above result can be used to calculate the power rof the nonlasing and lasing modePnonlasing/Plasing, or,equivalently, the mean-square deviation from the steastate polarization, or, equivalently, the size of the noise cloon the Poincare´ sphere @see Eq. ~1!#. For dominantx-polarized emission one finds

PnonlasingPlasing

51

2~12^P1&!5^w

2&5D

g i1gnon5

D

g0. ~19!

For dominanty polarization the expression is the same, apfrom a minus sign in front ofg i . Note that integration of theprojected polarization noise spectrum, Eqs.~9!, over ~posi-tive and negative! frequency, gives the same result, for thcase of dominant linear birefringence considered here. Eqtion ~19! shows, in a very convenient way, how polarizatiofluctuations result from a balance between a stochastic fo

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4198 PRA 58M. P. van EXTER, M. B. WILLEMSEN, AND J. P. WOERDMAN

on the one hand and the restoring forces of the~absorptive!anisotropies on the other hand. More specifically, it shohow the relative power in the nonlasing polarization, or tsize of the noise cloud on the Poincare´ sphere, can be used testimate the noise strengthD, when the dichroismg0 isknown.

Polarization noise can make the laser hop from the potial well of dominant x polarization to the other well odominanty polarization, and back. The present model giva simple expression for the average hopping timegnon/(4D)@1. For the symmetric case (g i50) the averagedwell time in each state is given by@20,21#

^T&'ApDgnon

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In the limit gnon/(4D)@1, the above expression for the hoping time is extremely sensitive to the polarization diffusirate D, so that it can be used to get an accurate meathereof, oncegnon is known.

For completeness we note that, in the spin-eliminamodel, there are actually two different mechanisms thatproduce a polarization switch. One type of switch occwhen we let the linear dichroismg i depend on injectioncurrent@7,9#, in such a way thatg i changes sign at a certaicurrent; at this pointg i'0 and lasing in the two polarizatiodirections is equally favorable. This first type of polarizatiswitch should obey the equations in this section, at leastthe ~common! case of dominant linear birefringence. Dpending on the amount of noiseD and the strength of thenonlinear dichroismgnon, the laser polarization will exhibitfast, or slow hopping@see Eq.~20!#, where extremely slowhopping will experimentally be interpreted as bistabilityhysteresis. The second type of switch is not based on currdependent linear effects, but has an intrinsic nonlinearture. In the spin-eliminated model, Eq.~3b! shows how thisnonlinear switch can occur only in VCSELs with small negtive linear birefringencev lin , where the nonlinear redshifcan pull the~high-frequency! nonlasing mode into the lasinmode, to create polarization instability and switching@6#. Forv0

2,2g02 one of the eigenvalues2g06 iv0 will correspond

to an undamped evolution, the polarization fluctuations wbecome excessively large in one direction, and one has tbeyond the linearized equations to solve the problem. Tphenomenon has been discussed in many theoretical pae.g., in terms of a Hopf bifurcation towards elliptically polarized modes@4,5#, although the exact nature of this switcis often hidden in complicated carrier dynamics. In the eperiments, linear birefringence generally dominates overother anisotropies so that this second type of polarizaswitch, with its different behavior and different statistics,quite rare@22#.

VI. EXPERIMENTAL SETUP

For the experiments we have used a batch of someproton-implanted VCSELs, organized as 1D arrays. Thesers operate around 850 nm and comprise three 8-nm-tGaAs quantum wells in a 1l cavity, sandwiched between aupper and lower Bragg mirror of 19 and 29.5 layer parespectively @24#. The threshold currents of all thes

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VCSELs is around 5 mA, with higher-order modes appeararound 10 mA at an output power of about 2 mW. At locurrent the laser polarization was practically always closevertical, i.e., perpendicular to the array axis. The steady-sellipticity xss was typically 1° or less, with a few exceptionof xss'5210° for lasers with small negative birefringencv0 . The size of the batch allowed us to pick the most intesting VCSELs for further study, namely, those with retively small effective birefringence and those that exhibipolarization switch. In the presentation of the figures we wconcentrate on two specific VCSELs, which we have labeVCSEL 1 and VCSEL 2. Unfortunately, the~current-dependent! VCSEL performance showed small variationfrom day to day, so that the exact numbers for birefringenand dichroism, as obtained for the same VCSEL fromvarious figures, do not always match.

The experimental setup is sketched in Fig. 2. To limit texternal noise to the minimum, the VCSEL is enclosed intemperature-stabilized box~stability'0.1 mK! and driven bya stable current source~stability '0.75mA from dc to 1MHz!. The collimated laser light is first passed through~rotatable! l/4 plate, and subsequently through a combintion of a ~rotatable! l/2 plate and optical isolator, whichtogether effectively act as a rotatable polarizer. By settingangles of thel/4 andl/2 plates we select the polarizatiostate on which the laser light is projected. After projectithe light can be analyzed in three different ways. A planFabry-Pe´rot interferometer, with adjustable free spectrrange, allows for detailed measurements of the optical sptrum. A 6-GHz low-noise photoreceiver~NewFocus 1534!,in combination with a 25-GHz RF analyzer~Hewlett-Packard HP0563E!, allows for measurements of th~polarization-resolved! intensity noise. As a third method wcan also observe this noise in the time domain, using aphotodiode~DC-200 MHz! in combination with a 350-MHzoscilloscope~LeCroy 9450!. In the next sections we willdiscuss the results of these three methods in consecutivder.

VII. POLARIZATION-RESOLVED OPTICAL SPECTRA

Figure 3 shows optical spectra, for VCSEL 1 operatingI 59.0 mA. In Fig. 3~a! the wave plates were set for projetion onto the dominant~horizontal! polarization, whereas thispolarization was largely blocked in Fig. 3~b! ~we intention-ally kept a very small fraction of the lasing peak to servea marker!. These figures show that the optical spectrum csists of three~equidistant! peaks, which~from left to right!

FIG. 2. Experimental setup. After polarization projection wmeasure~i! optical spectra with a Fabry-Pe´rot interferometer,~ii !projected noise spectra with a 6-GHz photodiode and RF analyand ~iii ! time traces with a fast photodiode and oscilloscope.

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PRA 58 4199POLARIZATION FLUCTUATIONS IN VERTICAL- . . .

are denoted the four-wave-mixing~FWM! peak (y2), thelasing peak (x), and the nonlasing peak (y1). Roughlyspeaking, the lasing peak is associated with the steady-polarization of the laser, the nonlasing peak is a resultamplified spontaneous emission in the orthogonal polartion, and the four-wave-mixing peak results from nonlinemixing between these two. Comparison of the vertical scof Figs. 3~a! and 3~b! shows that the lasing peak dominatover the nonlasing peak by roughly 3 orders of magnitudetakes quite some suppression to resolve the latter. The Fpeak is much weaker still and often difficult to observe.fact its presence was first reported only recently@2#.

The optical spectra of Figs. 3~a! and 3~b! contain infor-mation about many laser parameters. First of all thequency difference between the lasing and nonlasing pgives the effective birefringencev0 , whereas the differencein their HWHM spectral width gives the effective dichroisg0 . For VCSEL 1 studied in Fig. 3, the effective birefringence is relatively small atn0[v0 /(2p)521.82(2) GHz~minus sign because the low-frequency mode lases!; this iswhy it has been selected. Its effective dichroism has a mtypical value, namelyg0 /(2p)50.22(2) GHz. For mostother VCSELsn0 ranged between23 and115 GHz ~withtwo exceptions at125 and 140 GHz!; the dichroismg0 /(2p) was always below 1 GHz. In Fig. 3 the measurspectral width of the lasing mode is instrument limited0.06 GHz ~HWHM! by the resolution of the Fabry-Pe´rotinterferometer.

Equation~8! shows how the relative strength of the fouwave-mixing ~FWM! peak, as compared to the nonlasipeak, can be used to quantify the nonlinear anisotropiethe laser. From Fig. 3~b! we find this relative strength to b2.5~2!%. With n0521.82(2) GHz this gives a combine

FIG. 3. Polarization-resolved optical spectra of VCSEL 1 aI59.0 mA, taking with a Fabry-Pe´rot interferometer. Thex-polarized lasing peak, which dominates~a!, is almost completelysuppressed in they-polarized spectrum of~b! ~same arbitrary units!.The latter shows the nonlasing peak at higher frequency and a wFWM peak, as mirror image, at lower frequency.

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nonlinear anisotropy ofAa211gnon53.6(2) ns21. Unfortu-nately, the optical spectrum does not allow a further sepation into nonlinear birefringence and nonlinear dichroismmainly provides information on the nonlinear birefringencas generallya@1 @25#, so thatAa211'a.

Theoretically we expect the relative strength of the FWpeak ~as compared to the nonlasing peak! to be inverselyproportional to the square of the effective birefringencev0@see Eq.~8!#. This is indeed observed: for two other VCSELwe measured a relative strength of 0.63~7!% at n053.45 GHz, and 0.15~3!% at n056.7 GHz. For our ‘‘aver-age’’ VCSEL, with n0'10 GHz, the strength of the FWMpeak was below 0.1% of that of the nonlasing peak athereby below the noise level. This strong dependencebirefringence explains why the FWM peak was not noticuntil recently.

As a last piece of information we calculate the amountpolarization fluctuations, by dividing the sum of the spetrally integrated strengths ofy-polarized nonlasing andFWM peak by the~integrated! x-polarized lasing peak. FromFig. 3 we determine this ratio to be 0.65~5!%. On the Poin-caré sphere, this corresponds to a noise cloud with a s^(2w)2&1/2'9° @see Eq.~19!#, which, on the world globe, isequivalent to an area bigger than Alaska, but smaller tAustralia. At the end of Sec. VIII we will discuss how thabove value can be used to determine the magnitude ofpolarization noise, and thereby the cavity loss ratek.

For VCSEL 2, studied in Fig. 4, the birefringence is etremely small~and negative! at n0520.87 GHz in VCSEL.As a consequence, the strength of the FWM peak namounts to about 20% of that of the nonlasing peak. Forextreme situation the nonlinear and linear anisotropiescomparable in strength and the nonlinear effect can no lonbe treated as a weak perturbation. However, even forextreme situation, the linearized theory developed in Secremains valid; the relative strength of the nonlasing aFWM peak, as compared to the lasing peak, is still o'1%, so thatf,x!1. This is demonstrated by the dashedotted curve in Fig. 4, which is a fit of Eq.~7! to the opticalspectrum, where the fitted width includes the finite widththe lasing peak. The dotted curve shows the Lorentzian fithe nonlasing peak only.

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FIG. 4. The optical spectrum of VCSEL 2 atI 510.0 mA showshow, for VCSELs with very small birefringence~n0520.85 GHzin the present case! the FWM peak can be as much as 20% of tnonlasing peak. The dashed-dotted and dotted lines are fits to~7! and to a single Lorentzian, respectively.

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4200 PRA 58M. P. van EXTER, M. B. WILLEMSEN, AND J. P. WOERDMAN

VIII. POLARIZATION-RESOLVEDINTENSITY NOISE SPECTRA

In this section we will describe measurements ofpolarization-resolved intensity noise, for which the principwas already discussed in Sec. IV~see Fig. 1!. The practicalimplementation is based on a spectral analysis of the insity noise of laser light that has passed through a rotatl/4 plate and a combination of a rotatablel/2 and isolator,which together act as a rotatable polarizer~see Fig. 2!.Figure 5 shows spectra of the projected intensity no^uI project(v)u2&1/2 for VCSEL 2 operating atI 59.0 mA, witha relatively small birefringence ofn0520.85 GHz. Fromtop to bottom, the curves in Fig. 5 show noise spectraprojection onto thex direction, onto thef direction, onto thelasing polarization~labelP!, onto the nonlasing polarizationand the noise in the absence of light~system limit!. As thenoise in the first two projections is much larger than thatprojection onto the lasing polarization, our first conclusionthat polarization noise dominates over pure intensity noOur analysis will concentrate on the noise spectra obsefor the x andf projections.

The dashed curves in Fig. 5 are fits of Eq.~9! to the uppertwo experimental curves over the range 0.3–2.5 GHz. Tfitting range has been limited to avoid both the lofrequency noise tail, as well as the high-frequency nofloor. The high quality of the fits allows us to extract theffective birefringencev0 , the effective dichroismg0 , aconstantC @used to simplify the numerator of Eq.~9! tov21C, see also the discussion just above Eq.~13!#, and aproportionality constant, which contains the detected intsity I , the diffusion rateD, and the system response. Ofitting results are un0u5uv0 /(2p)u50.85(2) GHz,g0 /(2p)50.38(2) GHz, Cf /(4p

2)50.49 GHz2, andCx /(4p

2)53.6 GHz2. The first two parameters,n0 andg0 ,can also be obtained from optical spectra. A big advantagthe present measurement is its extreme resolution: a speanalysis of intensity noise is only limited by the resolutionthe RF analyzer, which can easily be below 1 kHz, wheroptical measurements are limited by the Fabry-Pe´rot resolu-tion of typically 10–100 MHz.

Figure 5 shows that the projected intensity noise in thx

FIG. 5. Projected intensity noise of VCSEL 2 atI 59.0 mA.From top to bottom the curves show noise spectra for projeconto thex direction, onto thef direction, onto the lasing polarization ~label P!, onto the nonlasing polarization, and the noise inabsence of light~system limit!.

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direction is much bigger than that in thef direction (Cx.Cf), or, in other words, that the polarization fluctuatioare highly nonuniform and that the noise cloud on the Pocarésphere is elliptical instead of circular. This differenceintimately related to the presence of the FWM peak inoptical spectrum, and can likewise be used to estimatestrength of the nonlinear anisotropies. To do so we determthe ratio ^uf(v)u2&1/2/^ux(v)u2&1/2 and compare the resuwith Eqs.~11!, ~12!, and~13!. At the resonance frequency o0.85 GHz we find̂ uf(v)u2&1/2/^ux(v)u2&1/250.59. Substi-tution of this ratio in Eq.~12! yieldsagnon'2.2 ns

21. As thevery small birefringence makes the use of this approximexpression disputable, it is better to substitute the fittedCfandCx in Eq. ~13!, using the procedure discussed in Sec. IThis yields estimates ofagnon'2.0 ns

21 on the first try andagnon'2.5 ns

21 upon iteration.The noise spectra observed for the projections onto

lasing and nonlasing polarization contain information onintensity and polarization partition noise. A detailed analyof these spectra will be published elsewhere@26#. The rela-tive strength of the various noise spectra shows how thxand f projection are first order in the polarization fluctutions and how the projections onto the lasing and nonlaspolarization are only second order.

Figure 6 shows spectra of the projected intensity noiseVCSEL 1. This VCSEL exhibits a polarization switch;operates on the high-frequency~vertically polarized! mode atI 58.5 mA @Fig. 6~a!# and on the low-frequency~horizon-tally polarized! mode atI 59.0 mA @Fig. 6~b!#. In both fig-ures the solid and dashed curves denote the intensity nfor projection onto thex and f-direction, respectively,whereas the dash-dotted curve shows the system noise flThe fits to these noise spectra~not shown! were again excel-lent and gave un0u52.96(2) GHz, g0 /(2p)50.23(2)GHz, and agnon52.8(3) ns

21 at I 58.5 mA, and un0u51.75(2) GHz, g0 /(2p)50.23(2) GHz, and agnon53.2(3) ns21 at I 59.0 mA. In Fig. 6 the differences be

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FIG. 6. Projected intensity noise for VCSEL 1 before and afa polarization switch, at~a! I 58.5 mA, and~b! I 59.0 mA.

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PRA 58 4201POLARIZATION FLUCTUATIONS IN VERTICAL- . . .

tweenf and x noise are less prominent than in Fig. 5 asresult of the larger birefringence. The main message offigure is that the nonuniformity of the polarization fluctution is as expected fora@1; when the high-frequency modlases we finduf(v)u.ux(v)u @Fig. 6~a!#; when the low-frequency mode lases we finduf(v)u,ux(v)u @Figs. 5 and6~b!#.

Figure 7 shows again the projected intensity noiseVCSEL 2 ~as in Fig. 5!, but now at an operating current oI 57.0 mA, i.e., closer to threshold (I th55.0 mA), and for awider frequency range. The spectrum for projection ontolasing polarization~solid curve, labelP! is dominated bypure intensity noise; the broad structure around 6 GHzsults from intensity fluctuations associated with the relation oscillations. The dashed-dotted line shows a fit of~10! to this noise spectrum, yielding a relaxation oscillatifrequency of 5.8 GHz and a damping~HWHM! of 1.1 GHz.The f and x curves show the noise spectra for projectionto the corresponding polarization states. From fits inrange 0.4–2.8 GHz we findun0u51.39 GHz, g0 /(2p)50.55 GHz, andagnon52.2 ns

21. This figure clearly showshow intensity noise and polarization noise simply add upthe projection spectrum; the relaxation oscillation iscourse less prominent in thef and x curves because thaverage intensity for polarization projection is about halfintensity for projection onto the lasing polarization.

Next we have measured the correlation between thelarization noise inf andx, which, according to Sec. IV and@11#, should be noticeable as a rotation of the elliptical nocloud on the Poincare´ sphere. For best results we tooVCSEL 2, with its relatively small birefringence and largnonuniformity, and operated it at 9.0 mA. Figure 8 showmeasurement series of the projected intensity noise as a ftion of the angle of the projecting polarizer, where 0° a45° correspond to projection onto thex and f direction,respectively~see dashed vertical lines!. The solid curve is afit, using the square root of Eq.~14a!. Figure 8 shows that thecases of maximum and minimum projection noise docorrespond to purex andf projection, but occur at a slightlysmaller angle. Specifically, the noise ellipse is rotated oan angle ofC rot518(6)° with respect to thex,f coordinate

FIG. 7. Projected intensity noise for VCSEL 2 atI 57.0 mA.Note the presence of the relaxation oscillations around 6 GHz inprojection onto the lasing polarization~labelP! and the corresponding structure in the polarization-resolved intensity noise~f andx!.The dashed curve is a fit based on Eq.~10!.

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system. This agrees very roughly with the rotation angleexpected from Eq.~14b!, which is about 9° for the case odominant birefringence (a53), but as much as 36° for thcase at hand@v lin /(2p)520.85 GHz, gnon'1.0 ns

21, a'3, g i'1.4 ns21#, where the latter estimate is clearly hindered by the uncertainties in the various parameters.

IX. POLARIZATION SWITCHES

For some VCSELs the polarization direction changes sdenly by about 90° when the laser current is varied. A stuof the laser dynamics around such a polarization switchideally suited to determine the various laser parameters. Tis demonstrated in Figs. 9~a! and 9~b!, which show the ef-fective birefringence un0u and dichroism ug0u/(2p) ofVCSEL 1, as obtained from the polarization-resolved intesity noise spectra, as a function of current. This VCSELhibits a polarization switch between 8.9 and 9.1 mA. Tomore specific: at low current the~vertically polarized! high-frequency mode lases, at high current the~horizontally po-larized! low-frequency mode lases, whereas either situatcan occur within the switching region, depending on histo~hysteresis!. Figure 9~a! shows how the frequency splittinbetween the lasing and nonlasing mode changes fromun0u53.16 GHz to 1.93 GHz, when the VCSEL switches polaization. This change is a result of nonlinear birefringence acan be used as a measure thereof@2#. By expanding Eq.~3a!into a linearized expression for the ‘‘spectral redshift of tnonlasing mode’’ we deduce from the switch thagnon'p~3.1621.93! ns

2153.9 ns21. Using the full Eq.~3a!we get a somewhat better estimate,agnon'3.7 ns

21. We notethat VCSEL 1 was also used to obtain the optical spectrof Fig. 3 ~at I 59.0 mA andn0,0, i.e., after the switch!, andthe polarization-resolved intensity noise of Fig. 6~before andafter the switch!.

Figure 9~b! shows how the effective dichroism changwith current and how the vertically polarized mode becomless and less dominant. This is a general trend in allVCSELs: before the switch the dominant polarization isways close to vertical, i.e., perpendicular to the array aafter the switch the dominant polarization becomes horiz

eFIG. 8. Measurement of the projected intensity noise for V

SEL 2 at I 59.0 mA as a function of the orientation angle of thprojecting polarization. The angles 0° and 45° correspond to pjection onto thex and f direction, respectively. Note the angulashift of about29°.

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4202 PRA 58M. P. van EXTER, M. B. WILLEMSEN, AND J. P. WOERDMAN

tal. Furthermore, VCSELs that have a small dichroism at lcurrent exhibit a polarization switch at increasing currewhereas those with larger dichroism do not switch withinrealm of fundamental mode operation. We therefore attribthe occurrence of these switches to a current dependenthe measured effective dichroismg0(I ), and more specifi-cally to the linear part thereof, i.e.,g i(I ), as the nonlinearpart gnon.0 will always favor the lasing polarization ovethe nonlasing one and increase monotonically with curreA measurement ofg0(I ) in fact allows us to predict whetheor not a polarization switch is going to occur at a certacurrent. In the switching region the two polarizations whave almost equal loss (g i'0) so that we conclude for thnonlinear dichroismgnon'g0'2p30.21 ns

2151.3 ns21 @seeFig. 9~b!#. Division of the nonlinear birefringence@in Fig.9~a!# by the nonlinear dichroism@in Fig. 9~b!# yields a'2.9, in agreement with literature values. Similar valuwere found for other VCSELs. As an example, one of thother VCSELs switched its polarization aroundI 58.5 mA,had a frequency splitting of 11.52 GHz and 10.50 GHzfore and after the switch and an effective dichroismg0 /(2p)50.22 GHz within the switching region, so thata'3.1.

We have thus demonstrated how a comparison of spebefore and after a polarization switch allows one to serately determine the nonlinear birefringence and nonlindichroism, irrespective of the VCSEL’s absolute birefrigence. In this respect the analysis presented here is mpowerful then that in Secs. VII and VIII, addressing opticspectra and projected intensity noise; the latter approworked only for smalln0 and gave only a value for th

FIG. 9. The effective birefringenceun0u and dichroismug0u ofVCSEL 1 as a function of current. Note the observed hysteresisthe jump in un0u that occurs upon a polarization switch~aroundI59.0 mA!. From ~b! we conclude that the polarization switch rsults from a current dependence dichroism,g0(I ).

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combined nonlinear birefringence and dichroism. A disavantage, however, of the present technique is that theSEL should actually switch polarization and that one cdetermine the nonlinearities for only one specific currebeing the switching current.

In practice, the VCSELs that switch their polarization chave both positive and negative effective birefringencen0 .In both cases, the observed changes inn0 were consistentwith the expected nonlinear redshift@see Eq.~3b!#: when thehigh-frequency mode dominates (n0.0) at low current, as isgenerally the case in our VCSELs,un0u increased graduallywith current and jumped to a smaller value upon a polarition switch; when the low-frequency mode is dominant (n0,0), un0u decreased with current, to jump to larger valuupon a switch. Furthermore, switches have been observeVCSELs with both small and largen0 . These observationsshow that the nonlinear anisotropies by themselves arethe prime reason for the occurrence of polarization switchas the ‘‘nonlinear’’ explanation predicts only switches frolow to higher frequency operation, and only at relativesmall ~negative! n0 @4,8#.

The physical mechanism behind the polarization switchi.e., the mechanism responsible for the experimentallyserved current dependence ofg i(I ), is not yet known. It istempting to attribute this dependence to a~temperature-induced! shift in frequency detuning between the polarizcavity modes and the gain spectrum@7#. However, this ex-planation seems to be ruled out by our experiments. Apfrom subtleties in the scalar or tensor nature ofg lin , thisexplanation predicts that the mode closest to gain celases and that the current dependence ofg0 is proportional tothe effective birefringencen0 . In practice, we find bothswitches from low-to-high and high-to-low frequencies, awe find hardly any correlation between the slopedg0 /dI @infigures like Fig. 9~b!# andn0 . An alternative explanation hanot yet been found. The observation that the dominant poization is always vertical before and horizontal after tswitch indicates that the physical mechanism behind thelarization switch is linked to either the design layout of tarray or to the orientation of the crystalline wafer.

The diffusion coefficientD can be estimated from threal-time switching dynamics, which was found to depecritically on switching current. VCSELs that switch their polarization above 8–9 mA exhibit the hysteresis shown in F9; for switching at lower current, however, the dominapolarization was not stable all the time, but hopped betwtwo quasistationary polarization states. The time it takesVCSEL to actually switch was found to be very small acould hardly be resolved with our photodiode and oscilscope; we estimate it to be just below 2 ns. On the othand, the average dwell time in the two quasistationary stwas very much larger. This average dwell time was founddepend strongly on switching current; in VCSELs thswitch just below 8.5 mA it was about 1 s, for switchinaround 7 mA it had dropped to~sub!microsecond. The reason for this rapid change is of course the exponential depdence of^T& on gnon/D in Eq. ~20!. As the observed hopping is driven by polarization noise, it can be used to getestimate thereof@see Eq.~20! for the case of dominant lineabirefringence#. At I'8.5 mA an average dwell time of abou1 s combines with a nonlinear dichroismgnon'1.1 ns

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PRA 58 4203POLARIZATION FLUCTUATIONS IN VERTICAL- . . .

give a diffusion coefficient ofD'12 ms21.Alternatively, the diffusion coefficientD can be estimated

from the absolute strength of the polarization fluctuationsgiven by the ratio of power in the dominant polarization athe orthogonal polarization, in combination with the effetive dichroismg0 @see Eq.~19! for case of dominant lineabirefringence#. This power ratio can be obtained most reably from optical spectra like Fig. 3, by integration over tlasing and nonlasing peak, but one can also usefrequency-integrated projection noise, as e.g. in Fig. 6,even the polarization-resolved light-current characteristicthe laser~as long as the higher-order modes remain wea!.We found these estimates to be mutually consistent withfactor 1.5; at a typical current of 8.5 mA they all yieldePnonlasing/Plasing'0.721.0%. Combined withg0'1.1 ns

21

this then corresponds toD'8211 ms21 being in reasonableagreement with the earlier estimate.

As a final step we deduce the cavity loss ratek from thevalue ofD, using Eq.~5b!. We therefore express the intraavity photon numberS in terms of the VCSEL output poweas Pout52hnhkS, where h is the outcoupling efficiencythrough the top mirror. At I 58.5 mA we had D58211 ms21 at an output power of 1.8 mW. For an ideal foulevel laser, wherensp5h51, this would make the estimatecavity loss ratek'200 ns21. A more realistic estimatebased onnsp51.5 andh50.3, givesk'300 ns

21.

X. RESULTS FOR OTHER VCSELS

In order to study the generic validity of our results whave repeated the experiments discussed above on anset of VCSELs, grown at the ‘‘Centre Suisse Electroniqand Microtechnique’’~former Paul Scherrer Institute! in Zü-rich, Switzerland. These were etched-post devices witpost diameter of 17mm ~i.e., no proton implantation! thatcomprise three 8-nm-thick GaAs quantum wells in a 1lcavity. The lower and upper Bragg mirror contain 20 a40.5 pairs of graded AlAs-Al.18Ga.82As layers, respectivelyThe device that was singled out for further study hadthreshold current ofI th54.1 mA, operated in the fundamental transverse mode up to 2I th , and exhibited a polarizationswitch around 5.5 mA, at an output power of 0.30 mW.

Figure 10 shows the effective birefringenceun0u and di-chroism ug0u/(2p) measured as a function of laser curreThe behavior of this etched-post VCSEL is quite similarthat of the proton-implanted VCSEL in Fig. 9. Once mowe observed hysteresis; when the current is increasedVCSEL polarization switches fromy to x at I 55.65 mA;when the current is decreased the VCSEL polarizationgers on inx and switches back atI 55.46 mA. Again, theeffective birefringence exhibits a jump due to the nonlinebirefringence@Fig. 10~a!# and again the switch coincidewith a minimum in the measured dichroism as a functioncurrentg0(I ) @see Fig. 10~b!#. By relating the jump in Fig.10~a! to the nonlinear redshift we find agnon'p~5.1324.68! ns2151.4~1! ns21. By relating the effectivedichroism inside the hysteresis loop to nonlinear effectsfind gnon'2p30.132 ns

2150.83~6! ns21. Combining thesetwo results yieldsa51.7(2),which is relatively low, but notunrealistic for thin quantum wells@25#. As a detail, we notethat the effective dichroism inside the hysteresis loop

s

-

erf

a

there

a

a

.

,he

-

r

f

e

s

asymmetric,g0 being larger after the polarization switcthan before. The reason for this asymmetry is not yet kno

As a next step we tried to observe the effect of the nlinear anisotropies in the polarization-resolved optical aintensity noise spectra. To increase our changes of succand to facilitate the comparison with earlier results, wethe laser current atI 55.55 mA, i.e., inside the hysteresloop, after the polarization switch. At this point, bothn0 andg0 are relatively small, so that both the magnitude of tnonlinear effects and the polarization fluctuations are omized. In this situation the optical spectra showed the ingrated power in the nonlasing peak to be 2.6% of that oflasing peak. What is more important, these spectra ashowed the presence of a four-wave-mixing peak at antensity of 8.0(6)31024 of that of the nonlasing peak. Whewe combine this ratio withun0u54.68 GHz in Eq.~8! wefind Aa211gnon51.7(1) ns21, in good agreement with theearlier estimate based on the observed nonlinear redshif

We also measured the polarization-resolved intennoise. The fits to these spectra were quite good, althothey were somewhat hindered by the presence of a lfrequency relaxation-oscillation peak around 2.3 GHz. Afthe polarization switch the fluctuations in the polarizatianglef were measured to be smaller than in the ellipticangle x, as expected for a VCSEL in which the lowfrequency mode dominates (n0524.68 GHz). At the reso-nance frequency we measurêuf(v)u2&1/2/^ux(v)u2&1/250.92(2). Substitution of this ratio in Eq.~12! yieldsagnon'2.4~6! ns

21. This estimate is somewhat larger thathe previous ones, but still falls within the error bars, whiare relatively large due to the presence of relaxation osctions.

FIG. 10. The effective birefringenceun0u and dichroismug0u ofthe etched-post VCSEL as a function of current. Note the obserhysteresis and the jump inun0u that occurs upon a polarizatioswitch ~aroundI 55.5 mA!. From ~b! we conclude that the polarization switch results from a current dependence dichroism,g0(I ).

on

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29

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4204 PRA 58M. P. van EXTER, M. B. WILLEMSEN, AND J. P. WOERDMAN

Finally we estimate the magnitude of the polarizatinoise from the observed power ratioPnonlasing/Plasing52.6%. Substitution of this ratio, and the fitted valueg050.83 ns

21, into Eq. ~19! yields a diffusion coefficientD522(3) ms21. Just as before, we now insertD, togetherwith the output power of 0.3 mW, into Eq.~5b!, to obtain anestimated cavity loss ratek'120 ns21 for the ideal four-level laser andk'220 ns21 for the casensp51.5 andh50.2.

Comparing the etched-post VCSELs with the protoimplanted VCSELs we note that for both types of devicnonlinear effects were observable in three different ways~i! a nonlinear redshift and extra dichroism,~ii ! a FWM peakin the optical spectrum, and~iii ! a different magnitude of theprojected polarization noise. Although the etched-post devswitched at an output power that was only about 20% of tthe proton-implanted devices, the observed nonlinanisotropies were still sizeable@agnon'1.5~2! ns

21,gnon'0.8 ns

21# at about 50% of the values of the latter dvices. The reason for this is that the cavity loss ratek of ouretched-post device is relatively low~Bragg mirrors withmore periods! so that a given output power corresponds torelatively high internal field. Experimentally, this was alnoticeable in the diffusion coefficientD, which at D'22(3)ms21 for the etched-post device is only a factor2–3 larger than that of the proton-implanted device, desthe factor of 5 lower output power. To account for the dference ink it might be better to relate the degree of satution to the quantity (I /I th)21, which, around the polarizatioswitch, was '0.35 for our etched-post device~I55.55 mA; I th54.1 mA! and 0.420.8 for our proton-implanted devices~I 5729 mA; I th55.0 mA!.

XI. SUMMARY AND CONCLUSIONS

We have presented a general description of polarizafluctuations in VCSELs, allowing direct comparison with eperiment. An overview of the model parameters is givenTable I. In total the model involves four anisotropies. Tphysical mechanism behind these anisotropies is not yet funderstood; we know how the linear birefringence arifrom mechanical strain@27# and internal electrical fields@28#, and how the nonlinear anisotropies result from t~eliminated! spin dynamics@6,11#, but the origin, and in par-ticular the experimental observation of a current dependeof the linear dichroismg lin(I ) is still somewhat of a mystery

In the experimental sections we have shown how the vous parameters can be extracted from the experimentalMore specifically, the effective birefringence and effectidichroism appear as frequency splittings and widths in boptical and projected-intensity spectra. We gave three expmental demonstrations of the presence of nonlinear anisopies. We have shown how they give rise to a four-wamixing peak in the optical spectrum and to a nonuniformin the projected polarization noise. Experimentally, both pnomena can be used to quantify the combined nonlinanisotropies, but both are inversely proportional to~thesquare of! the linear birefringence so that the effects are msurable only for small to moderate birefringence. As a thdemonstration of nonlinear effects we have shown how tgive rise to a spectral redshift and excess width of the n

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lasing peak as compared to the lasing peak. Experimentthese measurements are ideal to separately determinenonlinear birefringence and the nonlinear dichroism, but thonly work for VCSELs that exhibit a polarization switchSpecifically, we have shown how in extreme cases, whthe linear and nonlinear anisotropies are comparable,concept of two polarization modes loses its meaning.

For a batch of proton-implanted VCSEL we have applithe three techniques mentioned above to obtain resultsagreed within about 20%. We have determined the nonlinbirefringence to beagnon'324 ns

21 around I 59 mA andPout51.9 mW and to be'2.5 ns

21 around I 57 mA andPout51.3 mW. The nonlinear dichroism was found to befactor a'3 lower. We have also demonstrated to what etent the fluctuations in polarization directionf and ellipticityx are correlated. In general, we have shown how polarizafluctuations result from a balance between diffusion, duepolarization noise, and a restoring drift, due to dichroisThe diffusion coefficientD, and the related cavity loss ratekcould thus be estimated from the relatively power in tnonlasing polarization and from the average hopping timecase of polarization switching. Repeating the measuremon a batch of etched-post VCSELs from a different suppgave similar results. At lower current (I 55.5 mA) and loweroutput power (P50.3 mW) we now foundagnon'1.5 ns

21,gnon'0.8 ns

21, and a'1.7. Once more the three differenmeasurements were in reasonable agreement. This shthat both the phenomena and the quoted numbers are rgeneral and not limited to a special type of VCSEL.

In conclusion, this work presents an experimental conmation of the validity of the spin-eliminated model for thpolarization behavior of a VCSEL. We have stressed talmost any practical VCSEL satisfies the condition for spelimination. Also, most practical VCSELs satisfy additioally the condition of relatively strong linear birefringencwhich, in turn, greatly simplifies the analytic descriptioThis being said, it remains very interesting, from a theorecal point of view, to study VCSELs which donot satisfy thecondition for spin elimination. In particular, one would likto have a VCSEL with very large birefringence (v lin.gs /a) which exhibits a polarization switch; the switchinbehavior of this VCSEL should violate the framework eposed in the present paper. So far, we have not been abfind such a VCSEL among the many that we have studie

ACKNOWLEDGMENTS

We acknowledge support from the ‘‘Stichting voor Fudamenteel Onderzoek der Materie~FOM!,’’ from the Euro-pean Union under the ESPRIT Project No. 200~ACQUIRE!, and from TMR Network No. ERB4061PL951021 ~Microlasers and Cavity QED!. We thank M.Moser and K. H. Gulden of the ‘‘Center Suisse Electroniqand Microtechnique’’~formerly the Paul Scherrer Institute!in Zürich, Switzerland for kindly providing the etched-poVCSELs.

APPENDIX

In this appendix we will derive the steady-state polariztion and linearized polarization rate equations for a la

aas

ainin

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ofe

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ear

PRA 58 4205POLARIZATION FLUCTUATIONS IN VERTICAL- . . .

where the linear birefringence and linear dichroism makearbitrary angleb. For this case, the full rate equations,found in @6,13,14#, are

2 cos 2xdf

dt52v linsin 2x cos 2f2g linsin 2~f2b!

22agnonsin 2x cos 2x, ~A1a!

2dx

dt5v linsin 2f2g linsin 2x cos 2~f2b!

22gnonsin 2x cos 2x. ~A1b!

These equations are exact in the adiabatic limit; i.e., nosumptions have been made apart from the adiabatic elimtion of the difference inversion. To remove the various sand cosine functions we expand to first order inf,x!1,assuming the intensityI , which codeterminesgnon5kI /G, tobe more or less constant~valid for operation reasonably faabove threshold!. The steady-state angles thus found are

xss'g linsin 2b

2~v lin12agnon!!1, ~A2a!

fss'S g lincos 2b12gnonv lin Dxss!1. ~A2b!

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Equation~A2a! is an extension to the nonlinear regime,Eq. ~18! in @9# that was derived from a linear coupled-modtheory. Note that this equation is asymmetric in~the sign of!v lin ; large ellipticity are most likely for negativev lin , i.e.,for the case where the low-frequency mode lases. Forcase of dominant linear birefringence (v lin@g lin ,gnon) wealso findfss!xss @see Eq.~A2b!#.

For xss,fss!1 the linearized polarization rate equationincluding noise, are

d

dt S f2fssx2xssD5S 2g i 2v lin22agnonv lin 2g i22gnon D S f2fssx2xssD1S f ff x D ,~A3!

whereg i[g lincos 2b, and where we have added the Langvin noise sourcesf f and f x . We want to stress that, as theequations result from a linearization in the adiabatic appromation, they apply to all cases wheref,x,xss!1, includingthe very interesting cases where linear and nonlinanisotropies are comparable in strength. Note that Eq.~A3!becomes identical to Eq.~6! in @11# and Eq.~1! in @2# for thecase of aligned linear birefringence and dichroism.

P.

s.

A

tt.

H.

d,

an,

an,

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