Effect of photon-energy-dependent loss and gain mechanisms on polarization switching in vertical-cavity surface-emitting lasers

2106 J. Opt. Soc. Am. B/Vol. 16, No. 11 /November 1999 Ryvkin et al.

Effect of photon-energy-dependent loss and gainmechanisms on polarization

switching in vertical-cavity surface-emitting lasers

B. Ryvkin,* K. Panajotov,† A. Georgievski,‡ J. Danckaert, M. Peeters, G. Verschaffelt,H. Thienpont, and I. Veretennicoff

Vrije Universiteit Brussel, Department of Applied Physics and Photonics, Pleinlaan 2, B-1050 Brussels, Belgium

Received March 30, 1999; revised manuscript received June 7, 1999

We have analyzed the effect of the photon energy and temperature dependence of both the gain and the totallosses inside the cavity to understand the polarization behavior of vertical-cavity surface-emitting lasers. Theassumption that the losses are dominated by free-carrier absorption in the p-doped mirror is made. Devel-oping a new theoretical approach, we are able to predict different polarization switching regimes in whichswitching occurs from the high- to the low-frequency mode, from the low- to the high-frequency mode, or bothconsecutively. All these predictions have been experimentally verified by our measurements on GaAs/AlGaAsproton-implanted vertical-cavity surface-emitting lasers. © 1999 Optical Society of America[S0740-3224(99)00111-3]

OCIS codes: 250.7260, 260.5430, 140.5960.

1. INTRODUCTIONThe polarization behavior of vertical-cavity surface-emitting lasers (VCSEL’s) has attracted a great deal of at-tention recently.1–5 The cylindrical symmetry of an idealVCSEL and its direction of lasing (perpendicular to theactive layer) imply that the polarization state of the emit-ted light should not be fixed a priori. However, real VC-SEL’s do emit linearly polarized light oriented in specificcrystallographic directions. This has been attributed tobirefringence introduced by unintentional stress duringmanufacturing2 and by the electro-optic effect in the cav-ity and the mirrors.6 Furthermore, polarization switch-ing (PS) between the two orthogonal linearly polarizedstates in the fundamental mode is often observed.1,3–5,7,8

Different mechanisms have been proposed to explain thisintriguing polarization behavior.

A first explanation relies on the dependence of thegain on photon energy and temperature.3,4 In 1994,Choquette et al. attributed the switching to the relativeshift of the gain curve with temperature as comparedwith the shift of the two nearly degenerate cavity modes.Indeed, when the temperature is increased, both the gaincurve and the cavity modes redshift (to smaller frequen-cies), but the gain curve is shifting faster. So if onestarts working on the smaller-frequency side of the gainmaximum (where the higher-frequency mode is lasing)and increases the (dc) current (and hence the tempera-ture in the cavity), the two modes can cross the gain maxi-mum, and it will then be the smaller-frequency mode thatwill experience a higher gain. Therefore this thermalmechanism corresponds to a switching from the higher- tothe lower-photon-energy mode (from shorter to longerwavelengths) when the current is increased, a property towhich we will hereafter refer as type I switching.

A second, distinctively different, model9,10 attributesthe switching to saturable dispersion (associated with the

0740-3224/99/112106-08$15.00 ©

linewidth enhancement factor) in combination with spinrelaxation processes in the active semiconductor medium.In the physically interesting parameter domain, thisspin–flip model predicts PS from lower to higher frequen-cies (from longer to shorter wavelengths) with increasingcurrent,11 a property to which we will refer as type IIswitching. Moreover, in the spin–flip model, PS shouldbe accompanied by a transition through a region of eitherbistability or instability, where dynamic effects occur.

Finally, spatial hole burning could also explain PS.12

Indeed, while initially the better-confined (fundamental)mode starts lasing, the gain of this mode could saturateowing to spatial hole burning, therefore favoring the ap-pearance of the lesser-confined (still fundamental) mode.Of course, this effect can occur in combination with one ofthe other mechanisms described above.13

Various groups have indeed reported experimentsshowing type I PS.5,7,8 However, these results also indi-cated that the Choquette mechanism3,4 does not suffice toexplain all the features of the PS process. For instance,type I PS was observed while the laser was continuouslyworking on one side of the gain maximum.5 Further-more, the high-frequency mode was dominating with thelaser operating at both sides of the gain maximum, beforeswitching from the higher- to the lower-frequency modeoccurred.7

So far, when the current is increased, only experimen-tal results of PS from higher to lower photon energy (typeI) have been reported. In this paper we report on experi-mental evidence of both types of PS (type I and II) occur-ring in the same device. Indeed, we have observed type IPS in our GaAs/AlGaAs proton-implanted VCSEL’s whenthey were operating at ;850 nm for relatively low sub-strate temperatures. For higher substrate tempera-tures, we observe switching from higher to lower photonenergy (type I) followed by switching from lower to higher

1999 Optical Society of America

Ryvkin et al. Vol. 16, No. 11 /November 1999 /J. Opt. Soc. Am. B 2107

photon energy (type II) when the current is further in-creased. For still higher substrate temperatures, we ob-serve only type II PS. The frequency splitting betweenthe two polarization modes appears to be approximatelyconstant in all these cases.

These experimental observations, described in Section3, call for an explanation. Although the occurrence ofboth switching types in one and the same device could beexplained by a combination of several of the mechanismsdescribed above,14 we wish to bring up a theoreticalmodel, of a thermal nature, that is able to predict bothswitching types, depending on the values of the substratetemperature and the current heating. Our model, de-scribed in Section 2, takes into account not only the de-pendence of the gain on photon energy and temperaturebut also that of the overall losses inside the device. It isindeed well known that the free-carrier absorption occur-ring in the p-doped distributed Bragg reflector (DBR) mir-ror (or in the p-doped contact layer for intracavity con-tacted devices) is the dominant loss mechanism. Inaccordance with recent work,15 we have evaluated the ef-fect of this absorption and its spectral and temperaturedependence on the PS behavior of VCSEL’s. Finally, wesummarize our results in Section 4 and give an overviewof all possible cases in which switching does or does notoccur.

2. MODELA. PreliminariesIn this paper we focus on the photon energy \v and tem-perature dependence of the net gain G and the total lossesatot in VCSEL’s and their effect on the polarization behav-ior of the device. Our work is an extension of the ideas ofChoquette et al.,3 who took only the photon energy depen-dence of the gain into consideration.

We shall call the two linear polarizations of the funda-mental mode inside the cavity L and H, with photon en-ergies \v l (low photon energy) and \vh (high photon en-ergy), respectively. It is then easy to see that thecriterion of the highest gain for the selection of the lasingmode inside the cavity leads to the selection of the Hmode on the low-energy side of the gain maximum, andvice versa. Or, in other words, the H mode is selectedwhen the condition $dG/@d(\v)#%u\v0

. 0 is satisfied, andvice versa. If, however, we take the losses into consider-ation and use the lasing condition, i.e., G(\v0)5 atot(\v0), then obviously the H mode will be selected,provided that

dG

d~\v!U

\v0

.datot

d~\v!U

\v0

. (1)

By contrast, if the sign of inequality in relation (1) isreversed, the L mode will be generated.

Condition (1) is illustrated in Fig. 1. In this figure weschematically show the dependence of the material gainand the total losses on photon energy for three differentsituations in a small region around the intersectionpoints. If the working point is at the low-energy side ofthe gain maximum, condition (1) is always fulfilled. Ifthe working point is at the high-energy side of the gain

maximum, two different situations are possible. In thecase in which total absorption is given by atot8 , condition(1) is fulfilled, and the H mode is generated. In the atot9case, condition (1) is not fulfilled, and the L mode is gen-erated.

In the following sections we will discuss the functionsG(\v, T) and atot(\v, T) and will see the conditions underwhich the sign of the inequality can be changed and thedirection in which PS occurs. Before we proceed, weshould point out the different assumptions that we havemade:

1. We will always assume that we work in a station-ary regime and that the temperature is uniformly distrib-uted in the device (TQW 5 TDBR 5 T, where QW standsfor the term quantum well).

2. We will disregard all the differences in the trans-verse distribution of the optical field of the two differentpolarization modes that can have important effects on thepolarization state.5,12 In this case the modal gain can bewritten as G 5 (d/L)g, where g is the material gain andd/L is the longitudinal confinement factor (active regionthickness d divided by effective cavity length L).

3. We will assume that internal optical losses are duemainly to the free-carrier absorption in the p-dopedDBR’s or in the p-doped contact layer of the intracavitycontacted VCSEL’s. The high doping level in these lay-ers is necessary to decrease the resistivity16,17 but in-creases absorption losses.

4. We do not take into account any nonlinear effectson the gain (owing to, e.g., spectral or spatial hole burn-ing) and on the dispersion (associated with, e.g., the line-width enhancement factor).

B. Dependence of Optical Losses on Photon Energy andTemperatureWe consider that the optical losses in VCSEL’s can bewritten as the sum of losses dependent on photon energyand on temperature (ap) and independent losses (a0):

atot 5 ap 1 a0 . (2)

As stated above, the internal optical losses ap in VCSEL’sare due mainly to the p-doped DBR, i.e., to free-carrier ab-sorption at transitions between the heavy-hole band andthe split-off band. These are given by18,19

Fig. 1. Schematic representation of the material gain and thecavity losses as a function of photon energy for three differentsituations.

2108 J. Opt. Soc. Am. B/Vol. 16, No. 11 /November 1999 Ryvkin et al.

ap 5A

\vS \v 2 D

kTDBRD 3/2

expS 2B\v 2 D

kTDBRD , (3)

where D 5 0.34 eV is the energy difference between theheavy-hole band and the split-off band and A is propor-tional to the hole concentration. For high photon ener-gies, B has to be determined experimentally. Experi-ments for p-GaAs at hole concentration p > 1.51018 cm23, photon energy \v ' 1.1–1.3 eV, and tempera-ture interval 300–370 K have shown that Eq. (3) gives anexcellent description of free-carrier absorption, with a fit-ted value of B > 0.1.15 A similar value of B is expectedfor AlGaAs layers in the p-doped DBR at a photon energyof ;1.4 eV.

From Eqs. (2) and (3) we can derive

datot

d~\v!5 2ap~\v, TDBR!S B

kTDBR2

3

2

1

\v 2 D1

1

\vD .

(4)

For the parameters mentioned above, Eq. (4) has a nega-tive temperature slope. Rewriting the condition for lasergeneration,

g~\v, TQW , pQW! 5L

datot~\v, TDBR!, (5)

we obtain from Eq. (4) the first ingredient for the imple-mentation of inequality (1):

L

d

datot

d~\v!U

\v0

5 2ap~\v0 , TDBR!

atot~\v0 , TDBR!g~\v0 , TQW , pQW!

3 S B

kTDBR2

3

2

1

\v0 2 D1

1

\v0D . (6)

C. Dependence of Material Gain on Photon Energy andTemperatureFor our further considerations it is very convenient to ex-press the material gain as20

g~\v, TQW , pQW! 5 aQW~\v!@ fn~En , TQW , pQW!

1 fp~Ep , TQW , pQW! 2 1#, (7)

where aQW(\v) is the absorption coefficient of the QW inthe one electron approximation and fn and fp are theFermi–Dirac distributions for electrons and holes, respec-tively. At high injection carrier densities (which is thecase for proton-implanted VCSEL’s21) and close to thegain maximum, fn > 1 and d fn /d(\v) > 0. The mate-rial gain and its derivative with respect to photon energythen become

g~\v, TQW , pQW! ' aQW~\v!fp@C~\v 2 Eg!,TQW ,pQW#,

(8)

dg

d~\v!U

\v0

' f0

daQW

d~\v!U

\v0

2 CaQW~\v0!f0

1 2 f0

kTQW, (9)

where f0 5 fp@C(\v0 2 Eg),TQW , pQW# and C ' 0.1 isdefined by the ratio of the effective masses of electronsand holes. We estimate the gain g and its behavior byusing the dependence of aQW(\v) in a typical

GaAs/Al0.2Ga0.8As 80-Å QW22 in the active layer of a VC-SEL at a carrier concentration of ;1019 cm23 ( f0 * 0.5).

Next, we consider the behavior of d g/@d(\v)# as a func-tion of the active region temperature TQW and TDBR(TDBR 5 TQW 5 T), in analogy with the case in whichonly the carrier temperature changes.15 This behavior iseasy to understand because near the gain maximumaQW(\v) starts to saturate and remains constant forlarger photon energies.22 This means that, at a certaintemperature TQW , daQW /@d(\v)# rapidly drops to zero, ascan be seen in Fig. 2, where the behavior of the two partsof the gain derivative with respect to photon energy [seerelation (9)] are shown. Indeed, the increase of tempera-ture results in a decrease of the bandgap energy Eg , andconsequently the absorption curve and the appropriategain curve shift to smaller energies. Moreover, inas-much as the redshift of the cavity photon energy \v0 withtemperature is much smaller, the lasing frequency v0 willlie in the region where daQW /@d(\v)# 5 0.

Because of this, we have taken into account the factthat the increase of TDBR and TQW results in both an in-crease of losses atot and a decrease of the gain g. To keepthe system lasing, one should increase the carrier concen-tration in the QW. When this enhancement is due toatot , an increase of f0 should occur. Note that the in-crease of the carrier concentration gives rise to an addi-tional redshift of Eg .22 At large temperatures,daQW /@d(\v)# 5 0, and the gain derivative is determinedonly by the second term in relation (9); i.e., the whole de-rivative will be almost independent of the substrate tem-perature (see also Fig. 3, curves 1–8).

The behavior of the gain and loss derivativesd g/@d(\v)# and (L/d)$datot /@d(\v)#%, respectively, areshown in Fig. 3 as a function of the temperature rise DTof the VCSEL. In Fig. 3, dg/@d(\v)# is plotted for differ-ent initial positions on the gain curve, corresponding todifferent substrate temperatures (thin solid curves 1–8 inFig. 3). If (L/d)$datot /@d(\v)#% also depends on the tem-perature, its initial value also changes with substratetemperature, as illustrated by the two solid lines in Fig. 3.

D. Polarization SwitchingWe are now ready to discuss the possible types of PS thatcan be deduced from Fig. 3. We recall that type I PS is

Fig. 2. Temperature dependence of the two terms in the gainderivative [Eq. (9)].

Ryvkin et al. Vol. 16, No. 11 /November 1999 /J. Opt. Soc. Am. B 2109

Fig. 3. Dependence of the gain derivative with respect to the photon energy on DT, the difference between the active region tempera-ture at a certain current and the temperature at threshold (solid curves 1–8). For curve 1, \v0 is close to the gain maximum, andTsub 5 300 K. Each of curves 2–8 corresponds to a relative increase of the substrate temperature of 4.5 K. The dependence of the lossderivative on the active region temperature rise above threshold is shown by the nearly straight lines. The initial points of these thickand thin solid lines are taken at (ap /atot)g ' 2000 cm21 [see Eq. (6)], and Tsub 5 300 K and Tsub 5 350 K, correspondingly. The insetshows an enlargement.

from H to L and that type II PS is from L to H. Fivecases can be distinguished:

Case A. The dashed curve in Fig. 3 illustrates the situ-ation of a large derivative u(L/d)$datot /@d(\v)#%u, whichcould be realized in some cases. Here there is no PS atall, and the higher-photon-energy polarization mode H isalways generated (experimentally, this could be resolvedin the scanning Fabry–Perot spectrum if the lasing modeis almost suppressed with a polarizer, making it possibleto observe the nonlasing mode).

Case B. Only type I PS exists at low substrate tem-peratures in the considered temperature-rise interval:at the intersection (PS I) of gain derivative curves 1–3with the thick solid line showing the loss derivative.

Case C. Increasing the substrate temperature allowsus to observe both types of PS, i.e., type I PS followed bytype II, as is clear from the intersection of gain derivativecurves 1–4 with the thin solid line showing the loss de-rivative (see also the inset in Fig. 3).

Case D. With a further increase of the substrate tem-perature only type II PS is possible, as can be seen fromthe intersection of gain derivative curves 5–8 with thethin solid line showing the loss derivative. From Fig. 3we can see that, if the temperature dependence ofdatot /@d(\v)# is small, i.e., the thick and the thin solidlines almost coincide, and DT (temperature at the PS cur-rent minus the temperature at the threshold current) isalmost constant. In this case it becomes difficult to real-ize type I PS only (case B).

Case E. Note that there can also exist another situa-tion for the temperature interval at hand, that in whichthe gain derivative curve does not intersect the loss de-

rivative curve, i.e., where no PS occurs. In contrast tocase A, the gain derivative is always smaller than the lossderivative for the temperature interval at hand, and thelower-photon-energy mode L is always lasing (gain de-rivative curves 4–8 and the thick solid line for the loss de-rivative).

In our discussion we have assumed that the carriertemperature is the same as the lattice temperature.However, in some cases the heating of the carriers rela-tive to the lattice can also be important for PS.15

3. EXPERIMENTAL RESULTSTo check the validity of our theoretical approach experi-mentally, we used proton-implanted GaAs/AlGaAs QWVCSEL’s, obtained from Vixel Corporation, operatingaround 850 nm.23 In Fig. 4 we show three typical opticaloutput power versus current characteristics for direct cur-rent (dc) at fixed substrate temperatures of 10, 15, and55 °C. From Fig. 4(a) it can be seen that at a low sub-strate temperature (10 °C) only type I PS takes place.Two types of PS take place at Tsub 5 15 °C [Fig. 4(b)]:The first one is from H to L, and the second one is in theopposite direction. At higher substrate temperaturesTsub 5 25–55 °C only type II PS (\v l → \vh) takes place[see Fig. 4(c), Tsub 5 55 °C]. Figure 5 shows the scan-ning Fabry–Perot spectrum taken at the type II PS cur-rent for a substrate temperature of 15 °C. The frequencysplitting between the two modes of orthogonal linear po-larization is 12 GHz and is approximately constant in therange of substrate temperatures for which Fig. 4 was

2110 J. Opt. Soc. Am. B/Vol. 16, No. 11 /November 1999 Ryvkin et al.

measured. Mode hopping between the two orthogonalpolarization modes accompanies the switching process ina small region around the PS current.5

In Fig. 6 the dependencies of the injection current [Fig.6(a)] and the temperature of the active region [Fig. 6(b)]at threshold and at the type II PS point are presented.Since the threshold current is increasing monotonically

Fig. 4. Polarization-resolved optical output power versus dc in-jection current characteristic at three different substrate tem-peratures: (a) Tsub 5 10 °C, (b) Tsub 5 15 °C, (c) Tsub 5 55 °C.The lower (higher) photon energy mode is shown as a full(dashed) curve.

Fig. 5. Typical scanning Fabry–Perot spectrum at the currentof type II switching (J 5 9 mA, Tsub 5 15 °C) showing the direc-tion of the switching and the frequency splitting of 12 GHz.

with the substrate temperature [Fig. 6(a)], our laser is op-erating at the higher-frequency side of the gain maximumover the measured temperature range (15–55 °C). Wemeasured the temperature of the active region by observ-ing the spectral redshift with dc current and by initiallycalibrating this shift for a short-pulse low-duty-cycle cur-rent (50 ns, 10 kHz) as a function of substrate tempera-ture. From Fig. 6(b) it can be seen that DTQW 5 TQW

PS

2 TQWth for type II PS is approximately constant (64%)

over the measured temperature range.Comparing these experimental results with the model

developed in Section 2, we can see that the experimentalPS depicted in Figs. 4(a), 4(b), and 4(c) correspond tocases B, C, and D, respectively, of Subsection 2.D. More-over, the experimentally measured active region tempera-ture rise DTQW is indeed almost constant, as can be de-duced (case D) from the theoretical curves in Fig. 3.

4. CONCLUSIONSIn this paper we have analyzed the effect of the photonenergy and temperature dependence of both the gain andthe total losses inside the cavity to understand the polar-ization behavior of VCSEL’s.

Using the theoretical approach developed in Ref. 15, wewere able to predict different PS regimes in which switch-ing occurs from the high- to the low-frequency mode, fromthe low- to the high-frequency mode, or both types of

Fig. 6. (a) Injection current at the type II PS point JPS (dia-monds) and at the threshold current Jth (squares) as a function ofthe laser substrate temperature. (b) Active region temperatureat the type II PS current (TPS , diamonds) and at the thresholdcurrent (Tth , squares).

Ryvkin et al. Vol. 16, No. 11 /November 1999 /J. Opt. Soc. Am. B 2111

switching consecutively. All these predictions have beenexperimentally verified by our measurements on GaAs/AlGaAs proton-implanted VCSEL’s.

Our results are summarized in Fig. 7. The photon en-ergy dependencies of gain and total losses are schemati-cally depicted for lower and higher cavity temperatures(left-hand-side and right-hand-side plots, respectively).Different cases can be distinguished:

Case A. The laser initially operates at the low-frequency side of the gain maximum, with the high-frequency mode lasing (Fig. 7, left). Although the oper-ating frequency passes through the gain maximum, no PSoccurs, as the photon energy dependence of the losses

overcompensates that of the gain (Fig. 7, right). So themode corresponding to the higher-frequency mode re-mains lasing.

Case B. Again the laser initially operates at the low-frequency side of the gain maximum, with the high-frequency mode lasing (Fig. 7, left). As the lasing fre-quency passes through the gain maximum, switchingoccurs, as the photon energy dependence of the losses issmaller than that of the gain (Fig. 7, right). The laserswitches from the higher- to the lower-frequency mode(type I PS). This was confirmed experimentally in Fig.4(a).

Case D. Now the laser initially operates at the higher-

Fig. 7. Schematic representation of the gain and the loss as a function of photon energy for (a) lower and (b) higher cavity temperatures.The lasing mode in each case is indicated by the vertical arrow. Notice that for higher temperatures the gain has to increase, as it hasto compensate for the losses in the p-doped DBR mirror, which also increase with temperature. The different cases (A, B, D, and E)depicted are discussed in the text.

2112 J. Opt. Soc. Am. B/Vol. 16, No. 11 /November 1999 Ryvkin et al.

frequency side of the gain maximum, with the lower-frequency mode lasing (Fig. 7, left). As the temperatureis increased, switching occurs, essentially because thephoton energy dependence of the losses is overcompensat-ing that of the gain (Fig. 7, right). So type II PS occurscorresponding to our experimental observation, which isshown in Fig. 4(c).

Notice that case D in Fig. 7(a) and case B in Fig. 7(b)are essentially the same, which therefore allows for type Iswitching, described in case B, to be followed by type IIswitching (case D) when the current is increased further.This is in fact what we called case C in Subsection 2.D, asobserved experimentally in Fig. 4(b).

Case E. This is the same as the previous case, but nowthe energy dependence of the losses is not large enough toinduce type II switching. The laser remains lasing in thelower-frequency mode.

The main point that we wish to make here is that ther-mal considerations can explain switching in both direc-tions. The type of switching that will occur depends es-sentially on the initial position on the gain curvedetermined by the substrate temperature and by the ex-tra heat added because of the current flowing through thedevice. Notice that the predictions of our model are es-sentially independent of the magnitude of the frequencysplitting between the two modes. We would also like topoint out that the photon energy and temperature depen-dence of free-carrier absorption need not necessarily belarge. Only the relative value of the photon energy de-pendencies of absorption and gain as stated in Eq. (1) areimportant. The temperature dependence of the free-carrier absorption also need not be large to realize theproposed mechanism, as can be deduced from Fig. 3.

Let us resume the basic approximations made in ourstudy. First, our model is essentially based on steady-state considerations. A number of dynamic features ofthe PS process still require elucidation, both theoreticallyand experimentally. Second, we did not take into consid-eration any nonlinear effects, e.g., on the gain. Indeed,the gain is a nonlinear function of the optical power owingto effects as spatial and spectral hole burning, carrierheating, etc. Such saturation effects (also called gaincompression) could indeed be important, especially whenPS occurs at higher values of optical power. Gain com-pression would influence the position of the switchingpoint, and, more importantly, it would also give rise tohysteresis. Although the hysteresis is difficult to distin-guish clearly because of mode hopping occurring aroundthe switching point, it is sometimes present in our experi-ments. The influence of other nonlinear effects, such assaturable dispersion in combination with spin–flipmechanisms, should be investigated.

In conclusion, we may state that the photon energy de-pendence of gain and loss may indeed be an importantmechanism in understanding the intriguing polarizationbehavior of VCSEL’s, complementing other approachesand opening up new perspectives in this interesting field.

ACKNOWLEDGMENTSThe authors acknowledge the Belgian Office for Scientific,Technical and Cultural affairs for their support, which

was provided within the framework of the InteruniversityAttraction Pole program Photonic Information Systems.J. Danckaert and G. Verschaffelt acknowledge the Fundfor Scientific Research—Flanders for their fellowships.Besides funding from the Fund for Scientific Research—Flanders (for the high-resolution spectrum analyzer), thisresearch benefited from support provided by the Con-certed Research Action Photonics in Computing and bythe Research Council of the Vrije Universiteit Brussel.

J. Danckaert can be reached by telephone at 32-2-629-3659, by fax at 32-2-629-3450, or by e-mail [email protected].

*Permanent address, A. F. Ioffe Physicotechnical Insti-tute, 194021 St. Petersburg, Russia.

†Permanent address, Institute of Solid State Physics,72 Tzarigradsko Chaussee Boulevard, 1784 Sofia, Bul-garia.

‡Permanent address, A. F. Ioffe Physicotechnical Insti-tute, 194021 St. Petersburg, Russia.

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