JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. X, NO. Y, Z 2006 1

Cavity solitons in a driven VCSEL above thresholdX. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J.R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand,

L.A. Lugiato, I. Protsenko, and M. Brambilla

(Invited Paper)

Abstract— We experimentally demonstrate the existence andthe control of cavity solitons in externally driven vertical-cavity semiconductor lasers above threshold. A model includingmaterial polarization dynamics is used to predict and confirmthe experimental findings.

Index Terms— Cavity solitons, VCSEL, pattern formation

I. I NTRODUCTION

A Considerable interest has by now been consolidated inthe field of Nonlinear and Quantum Optics, concerning

the researches on spatial pattern formation and pattern controlin optical systems. The motivation mainly resides in theperspective of encoding and treating optical information inthe spatial domain, just as it is now customary to manipulateinformation in the temporal (or spectral) shape of opticalsignals, relying on the coupled effects of e.g. dispersion andnonlinearities.

In the spatial domain, though, the possibility of encodingindividual informational elements is generally frustrated by thehigh degree of spatial correlation with which most patternsspontaneously emerging from modulational instabilities (MI)are endowed. To beat this phenomenon, recent advances inthis line exploited the well-known phenomenon of spatiallocalization [1] to obtain self-confined, non-diffracting spatialstructures in the coherent field profile. Spatial solitons [2] andCavity Solitons (CS) [3], [4] are two of the most appealingentities for such purposes. CS in particular, have uniquefeatures which make them appealing for information treatmentin a stationary configuration, while spatial solitons generallyoccur in propagative configurations and are thus intrinsicallytransient.

CS appear as isolated intensity peaks in the field emitted bya nonlinear dissipative system, generally, but not only, a drivennonlinear cavity, and they are embedded in a homogeneous,dim background field which acts as a sort of a skating ringacross which the CS is free to shift, following intrinsic orexternally-imposed gradients of the coherent field. CS canmoreover be switched on and off at arbitrary locations in-dependently and this makes them good candidates to realize

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J.R. Tredicceare with Institut Non-lineaire de Nice, UMR 6618 Centre National de laRecherche Scientifique - Universite de Nice Sophia-Antipolis, 06560 Val-bonne, France

F. Prati, G. Tissoni, R. Kheradmand, L.A. Lugiato are with INFM-CNR,Dipartimento di Fisica e Matematica, Universita dell’Insubria, Via Valleggio11, 22100 Como, Italy

I. Protsenko is with Lebedev Physics Institute, Moscow, and RussiaScientific Center of Applied Research, JINR, Dubna, Russia

M. Brambilla is with INFM-CNR, Dipartimento di Fisica Interateneo,Universita e Politecnico di Bari, Via Amendola 176, 70126 Bari, Italy

dynamically reconfigurable arrays of self-assembled pixels,not affected by the rigidity unavoidable in e.g. etched arraysof microlasers [5].

A breakthrough towards implementation of CS-based appli-cations was provided by the joint experimental and theoreticalevidence of manipulable CS in a VCSEL in the amplifyingregime (i.e. with pump current between transparency andlasing threshold) [6], soon sided by multiplication of theencodable CS [7] and confirmation of viable timescales forCS encoding [8].

In the course of some 10-12 years, the first theoreticalpredictions of CS in optical resonators were mainly basedon prototypical models, centered on the interplay amongdispersive/absorptive nonlinearities, paraxial diffraction anddissipation/feedback, as provided by the resonator [5], [9]. Inmany of such works, the energy to the intracavity medium wasconveyed just by an external plane-wave beam. Of course,in the perspective of applications it would be, and still is,highly desirable to exploit CS properties within classes ofdevices where an external, non-optical, energy intake can beconveyed to the optical structures and -so to say- convertedinto the self-confined, manageable, optical power associated tothe bright CS. The most natural class of devices where to workindeed would have been the lasers, trying to obtain a devicewhich could emit (spontaneously or under some deterministiccontrol) CS at or close to threshold.

Unluckily, early in the game it was predicted that a two-level Free-Running-Laser was unavoidably deemed to emit atime-dependent pattern in the form of travelling waves, directlyat threshold, whenever a pattern-forming instability could beachieved (i.e. with positive atomic detuning) [10], [11]. Suchpredictions were later confirmed in lateral and vertical emit-ting semiconductor lasers [12]–[14]. These conditions wereintuitively quite unfavorable to the realization ofstationaryself-confined structured, the background being unstable andthe dynamics growing more and more irregular as the devicewas pumped beyond threshold.

These studies were soon generalized to the laser with aninjected signal, and it was shown that in general the dynamicsis not substantially different from that of the free running laser[15], [16], except when the frequency of the injected field istuned close to the atomic resonance [17], [18]. The laser withinjected signal was analyzed more recently in [19], and it waspredicted that models based on a complex Ginzburg Landauequation, which describes lasers near threshold, could exhibitstable phase solitons under the application of external forcingfrequencies.

On the basis of a different approach, in order to predict,characterize and study the onset of dissipative solitons in a

JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. X, NO. Y, Z 2006 2

vast number of pioneering works, the group of N.N. Rosanovconsidered the case of a laser with saturable absorber, i.e. aresonator where two media shared the cavity, one providinggain due to external pumping and the other purely absorptive.This system is known to exhibit a generally stable emissionprofile in the limit of fast media response and was thussuited to combine lasing with stationary background emissionwhereupon solitons could be stably realized [20].

Only lately, it was theoretically/numerically demonstratedthat more involved lasing schemes were indeed able to exhibitthe formation and the control of CS (writing, erasing, shifting)[21]–[23], and that self-confinement appears indeed quiterobust even in presence of radiating patterned waves [21].Such indications triggered back to actuality the hunt for CS inemitters, and in semiconductor based emitters as an electiveclass for applicative purposes; of course, the absence of asecond, passive, nonlinear medium could be considered as abonus in simplifying the device’s architecture.

In this paper we report on the joint experimental andtheoretical evidence of stable and addressable CS in a broad-area VCSEL driven well above the lasing threshold. Thedevice type is the same where CS were observed in theamplifying regime [6] and we offer proof that they existon top of a dynamically unstable background with no harmfor their stability or independence. In Sec. II, the setup andthe characterization of the lasing conditions in the spatiallyextended cross-section of the device are reported, along withthe analysis of the solitary laser emission; then we describethe laser emission and dynamics when an external holdingbeam with a homogeneous profile is injected into the devicein order to provide the bistable behavior which is knownto be beneficial for the CS stability and finally we showthe stable CS and prove their characteristic properties, alongwith an analysis of their addressing dynamics. In Sec. III atheoretical model is provided, suited to describe the spatio-temporal dynamics of a broad-area semiconductor device withfast polarization dynamics (as is the case for a VCSEL),a condition where it is well-known that it is impossible toperform a standard adiabatic elimination of the fast variables[24]. While we leave to a future publication the thoroughdiscussion of the model’s derivation, descriptive span and ofthe various regimes that were studied, in this work the focusis posed on the analysis of the main phenomena found in theexperiment, namely the coexistence of different states in thetransverse plane due to the spatial frequency gradient in thedevice, and the existence of bistability between CS and the(unstable) low intensity state of emission. Sec. IV offers someconclusions and perspectives.

II. EXPERIMENT

The experimental set-up is similar to the one described in[6], [25] and it is shown in Fig. 1. The injected laser hasbeen provided by Ulm university and is an oxidized bottom-emitter VCSEL with Bragg mirrors consisting of 20.5 pairs onthe bottom side and 30 pairs on the top side, and 3 quantumwells emitting around 970 nm at threshold [27]. Its diameteris 150µm.

S

I

T °

T °

CCD

PD

G

M

I AOM1

AOM2

PZT

C

P2

S.F 1

S.F 2

P2

B .S

B .S

B .S

B .S

L

P.M

O.I

B .S

O.F

OSA

SA LC

L L

LO.F

Fig. 1. Experimental set-up. M: high power edge emitter laser, I: currentdriver stabilized up to 0.01 mA, To: temperature controller, G: grating, O.I:optical diode, OSA optical spectrum analyzer, AOM: acousto-optic modu-lators, LC: digital oscilloscope, SA: power spectrum analyzer, S.F1: beamexpander-configurator with spatial filtering, S.F2: beam reducer-configuratorwith spatial filtering. S: broad-area vertical cavity surface emitting laser, C:collimator, CCD camera, PD: photo-detector, PZT: piezo-electric ceramic,BS: beam-splitters, P.M: power meter (optional),λ/2: λ/2 waveplates, P:polarizes, OF: optical fiber.

The master laser is an external grating laser which can betuned in the range 960-980 nm by steps of 41 GHz and, foreach step, it can be tuned continuously on a range of 6 GHz.This laser provides the injection beam (Holding Beam, HB)whose linewidth is less than 1 MHz on long time acquisitions(larger than 1 minute). The injection beam is spatially filteredand it is collimated to a waist of about 200µm. An opticalisolator (return loss> 30 dB) is placed at the beam exitof the external-cavity laser in order to avoid any disturbancefrom back reflections and from the slave output. The intensityof the injection beam entering in the VCSEL (before thecollimator) can be adjusted up to 20 mW. The HB poweris controlled by an acousto-optic modulator together with apolarizer. The master laser provides also the Writing Beam(WB). This beam is prepared to obtain a waist of 10-15µmwhile its maximum injectable power in the VCSEL is 100µW.The WB phase relationship with respect the HB is controlledby piezo positioning of a mirror on its own path.

The reflected output of the VCSEL is monitored by a CCDcamera Pulnix 765E, on which we form the image of theemitted near-field. A photodetector Thorlabs PDA8GS (DC-8GHz bandwidth) monitor a small portion of the VCSELtransverse plane in order to detect the dynamics of a CS.A digital oscilloscope Lecroy Wavemaster 8600A (6 GHzanalogue bandwidth, 20 GHz digital bandwidth) is used formonitoring the detector output together with a spectrum an-alyzer (bandwidth: .1 MHz - 20 GHz). For measuring theoptical spectrum emission of the slave laser we use an opticalspectrum analyzer Agilent 86140B (0.5A resolution), thedetection can be performed either on the total emission ofthe slave laser, either on a small region in order to detect thespectral information relative to a single CS.

The solitary VCSEL light intensity output as function of thepumping current (J), (L/I curve), indicates a laser threshold ofJth = 195 mA. In fact, the usual definition of laser threshold,where the intensity output is integrated through the wholetransverse plane of the device, is not meaningful when dealingwith such broad-area laser. More information can be obtained

JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. X, NO. Y, Z 2006 3

a b c

d e f

Fig. 2. Average intensity profiles of the solitary VCSEL under differentpumping current. A set of neutral density filter is used to avoid CCD camerasaturation: the power attenuation factor is reported with respect the situation ina). a)J = 200 mA, b) J =300 mA (attenuation factor: -10dB), c)J = 350mA (attenuation factor: -10dB), d)J = 400 mA (attenuation factor: -20dB),e) J = 550 mA (attenuation factor: -25dB), f)J = 750 mA (attenuationfactor: -25dB).

by plotting the near-field transverse emission profile of thesolitary VCSEL as a function of the pumping current (Fig. 2).ForJ < 200 mA we observe the spontaneous emission profile.At J = 200 mA (Fig. 2a) the lasing emission starts on the leftborder of the device and progresses, asJ is increased, alongthe circular border on the left side of the VCSEL (Fig. 2b).Further increase of the current makes bright spots appear atthe center of the device, (Figs. 2c,d). These spots are randomlydistributed and no regular patterns can be recognized in theemission profile. The transverse distribution of emission doesnot change for current higher than 500 mA, while the emittedpower continues to increase linearly (Figs. 2e,f). It is importantto remark the existence of a region at the right side of thedevice where the emission level is extremely low, even forcurrent values larger thanJ > 750 mA. This region hasappeared only recently and it is a sign of ageing of this device.For current J> 400 mA (Figs. 2d-f) it is possible to recognizetwo vertical lines along which the bright spots are more dense.These lines are the consequence of the presence of defects onthe surface of the wafer. In order to match the lattice constantof the active region (InGaAs) with the pure GaAs, the InGaAsis grown with a certain amount of strain. The strain tends torelax in presence of defect and creates a defect line that canextend over distances of the order of several millimeters. Thedefect, which is the origin of the line can be therefore far awayfrom the active zone of the laser [26].

The plots of Fig. 2 show that three different regions canbe distinguished in the transverse plane of the device. Thefirst is the circular border at the left of the device where,due to the current crowding [27], the emission occurs at thelowest level of pumping current. It is important to remarkthat the current crowding, apart from the pumping currentlocalization, engenders also thermal inhomogeneities in thetransverse plane of the device. This affects the transversedistribution of the VCSEL emission frequency. The secondregion, on the extreme right part of the transverse plane,shows an extremely low emission level in the current rangeexplored. Finally, a central region where the emission is ratherhomogeneous untilJ = 250 mA. We will work in this regionof the transverse plane and we will bias the device in order toobserve lasing emission from this region (J > 329 mA).

Another important characteristic of our device is that thecavity resonance varies along an horizontal axis of the trans-verse plane. This has been described in [25], [28] and it isdue to the non-parallel layers forming the cavity, owing to thestandard epitaxial growth techniques. By consequence, in oursystem, the lasing wavelength varies along one direction of thetransverse plane. In this particular device, a lasing wavelengthgradient of 0.4 GHzµm−1 is measured along the horizontaldimension, which leads to a resonance frequency differenceof 60 GHz between one side and the other of the amplifier.

We inject along its vertical axis the HB generated by themaster laser set to a power of 20 mW while the VCSEL currentis set toJ = 450 mA. As a function of the frequency detuningbetween the laser cavity resonance and the frequency of theinjected radiation (θ), we may distinguish experimentally threekinds of responses: i) the VCSEL emission is not perturbedby the application of the HB, which means that there isno interaction between the injected field and the device dueto an injected field not resonating with the VCSEL cavity(|θ| exceedingly large), ii) the VCSEL emission is modifiedwhen the injected field is applied and its transverse profileappears homogeneous, at least as detected from the time–averaging CCD camera, iii) the VCSEL emission developsspatial structures as a result of the injection. In fact, due tothe cavity resonance frequency gradient and to the currentcrowding the frequency of laser emission is not homogeneousin the transverse plane, and the same occurs to the parameterθ. As a result these three responses coexist in the transverseplane of the device, for the same value of injection frequency.In Fig. 3 we plot the near field of VCSEL emission forincreasing injected field frequencies. This sequence can becompared to the solitary laser emission plotted in Fig. 3a. InFig. 3b interaction is observed and it determines an increase ofthe emission level of the homogeneous background especiallyevident in the central part of the device, while the structuresremains as in Fig. 3a1. For lower injection frequencies nointeraction is observed and the emission is everywhere like inFig. 3a. For higher injection frequencies (Fig. 3c) structuresform at the left side of the device while the emission in the restof the device remains similar to Fig. 3b. As the frequency isincreased, the pattern region expands in the device especiallyaround the border (Fig. 3d) and in Fig. 3e the patterns regionis filling the left side of the device while the central part isquite homogeneous with two bright spots that reminds theCS observed below threshold [6]. In Fig. 3f we observe thatthe patterns covers the whole transverse plane (except forthe ”dark region” at the right border of the device). Furtherincrease of the injected frequencies (Fig. 3g-i) makes thespatial frequency of the structures to increase and finally theinteraction is very weak and the transverse profile is againsimilar to the one of the solitary VCSEL.

In analogy to what observed below threshold, the curveseparating the pattern emission region and the homogeneous(after time-averaging) region can be interpreted as the locusof the spatial positions where the local values ofθ andfield intensity meet the condition for the onset of a (pattern-

1We remind that this profile is time-averaged by the CCD camera.

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a b c

d e f

g h i

Fig. 3. Average intensity profiles of the VCSEL under different injectionfrequency around 970 nm progressing towards the blue side of the spectrum(the injection frequency is quantified in each situation with reference to theinjection frequency used in b) which is the lowest for which interaction isobserved. The wavelength of the injected field in b) is 971.71 nm.) a) solitarylaser, b) 0 GHz, c) +34 GHz, d) +71 GHz, e) +111 GHz, f) +148 GHz, g)+188 GHz, h) +225 GHz, i) +269 GHz.J = 450 mA, PHB = 20 mW

inducing) modulational instability (MI). As the injection fre-quency is increased the pattern region progressively fill thewhole transverse profile of the device. This is due to the shiftof the MI boundary to the new locus of points where thecritical detuning condition is satisfied.

We can estimate that, in order to observe interaction be-tween the light injected and the VCSEL, the range of variationof the injected frequency (and, by consequence ofθ) isapproximately 270 GHz starting from the lowest frequencyvalue for which interaction is observed (Fig. 3b). If we increasethe pumping current, the laser cavity resonance shifts towardsthe red, due to the Joule heating of the semiconductor medium.Then, for maintaining the detuning value in the range whereinteraction is observed, the master wavelength needs to beincreased as well. On the other hand, the qualitative behaviorshown in Fig. 3 as a function of the injected frequencyobtained forJ = 450 mA does not change fixingJ at differentvalues above threshold.

The theoretical prediction (see discussion of Fig. 11g in thenext Section) and the analogy with the case below thresholdindicate that the most favorable region in the transverseplane for finding CS is the one in the neighborhood of theboundary where the modulation instability occurs. As in thesub–threshold case [6], the presence of the cavity resonancegradient limits the existence of the CS to a small regionaround the boundary separating the patterned region from thehomogeneous region.

The CS can be created by injecting the WB. We fix allparameter values in order to obtain a situation similar to theone shown in Fig. 3e, where the MI boundary is on the leftside of the device, and we inject the WB into the homogeneousregion. Starting with no spot, the WB is capable of generatinga high intensity spot with a diameter on the order of 15µmwhen it is in phase with the HB. If we remove the WB, thebright spot remains on indefinitely. We then apply this beamin a different location without changing any parameter value

Fig. 4. Intensity distribution of the output field. The HB is always on, and allparameters are kept constant:PHB = 18 mW, J = 462.4 mA. a) the WB isblocked; b) the WB (PWB = 20 µW) targets a point into the homogeneousregion; it induces the appearance of a single CS and when the WB is blockedagain, the CS remains; c) the WB is displaced in position and switched-onagain and it generates a second CS that persists when the WB is blocked; d)the WB targets again the second CS, but the relative phase of WB with respectto HB has been changed ofπ and the CS is switched off and it remains offwhen the WB is blocked; e) the WB targets again the first CS, but the relativephase of WB with respect to HB has been changed ofπ and even the firstCS is erased. Once the WB is blocked, the intensity distribution is identicalto a).

and a second spot is generated without perturbing the first onewe have created. This one will also persist after removal of theWB. We reach, then, the situation in which two spots exist.Changing the phase of the WB byπ in respect with the HB andre-injecting it successively at each location, where the spotshave been created, we erase each of them in an independentway. The full series is displayed in Fig. 4 and this sequenceassesses these spots being CS.

CS can also appear decreasing the HB power from valueswhere only pattern solutions are stable to values where onlyCS and the background solution are stable. We have recordedexperimentally, as a function of the HB power, the localintensity output of the VCSEL monitoring a spot of 15µmcentered on a position where a CS can be ignited by theWB. Starting from PHB ≈ 0 and increasing power, thereis a critical value ofPHB at which the background solutionlooses its stability and a bright spot appears and, for decreasingPHB, there is a second critical value for which the CS solutionlooses its stability and the system switches to the backgroundsolution. The difference between these two values forPHB

defines the bistability region in term of the HB power. In thisregion the CS can be addressed by a WB and they can beswitched on and off.

In Fig. 5 we plot the local emission of the VCSEL as afunction of the injection power for three sweeping cycles ofthe HB power. The bistability is assessed by a hysteresis cyclewhose lower branch correspond to the background solution,while the upper branch correspond to the CS solution. Closeto the parameter values where the bifurcation occurs, the noisemay induce spontaneous jumps of the system towards the othersolution. This results in a relative uncertainty of parametervalues of about 20% for the borders of the bistability regionand determines, at every cycle, a slightly different positions ofthe transition lines (see Fig. 5). The bistability region, i.e. theregion where the CS can be controlled, depends critically onthe parameterθ. Fixing the pumping current of the VCSEL,and setting the injection frequency in order to observe abistable behavior as the one in Fig. 5, the injection frequencycan be varied by less than 6 GHz before loosing the bistability.This means that the region ofθ where CS can be observed isless than 6 GHz wide. The tight condition on the value of theparameterθ in order to observe controllable CS, makes very

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Fig. 5. left: Local intensity output from the VCSEL as a function of the in-jection power. right: transverse profile emission of the VCSEL correspondingto the upper branch (top) and to the lower branch (bottom). The monitoredregion of the transverse plane is indicated with a grey square marked on theprofiles. The parameters are like in Fig. 4.

sensible the operation of setting the injection frequency whenvarying the VCSEL pumping current.

In order to gain more insights on the properties of theobserved CS, we measure the local average optical spectrumemission of the slave laser at the place where the CS is created(the emission is detected in a spot of 10 micron around thepoint targeted by the WB). In the case below threshold, werecall that the CS optical spectrum is the same (except for anamplitude factor) as the spectrum emitted by the backgroundand both are locked to the master injection frequency. Thedetuning range between the injection frequency and the lasercavity resonance for the onset of the CS is only few GHz widearound the critical value [25]. In Fig. 6 we plot, for differentvalues of the pumping current, the local emission spectrumof the VCSEL without injection, the local emission spectrumof the injected VCSEL when the CS is off (background) andthe local emission spectrum of the injected VCSEL when theCS is on. The master laser injection frequency is marked aswell. It is important to notice that the local emission signalis brought to the spectrum analyzer by mean of a monomodeoptical fiber. The light coupling into the fiber engenders spatialfiltering of the field and therefore, in the optical spectra, thetransverse components present in the emission signal appearsthe more attenuated the more they deviate from the on-axiscomponent.

When the VCSEL is biased slightly above threshold (Fig.6a,b), the local spectra of the background and of the CS havethe same characteristics as forJ < Jth: both are lockedto the injection frequency. Moreover, both spectra are reddetuned with respect the peak in the optical spectrum of thesolitary VCSEL and the detuning range where CS can begenerated is, as already pointed out, only few GHz wide.For higher pumping currents (Fig. 6c,d) the local spectrumof the background spectrum shows the presence of two peaksone corresponding to the injection frequency, the second closeto the solitary VCSEL spectrum peak. The spectrum of theCS shows a narrow and high peak centered on the injectionfrequency and detuned with respect to the solitary VCSELemission.

In Fig. 6c,d the detuning between the injected wavelengthand the lasing wavelength is about35±8 GHz, it is important

a

c

b

d

e

Fig. 6. Local optical spectra as a function of the pumping current. Theregion monitored is a spot of 15 micron around the point targeted by the WB(see the profiles in Fig. 5. The lowest curve is the spectrum of the solitaryVCSEL, the curve in the middle is the emission spectrum of the VCSEL withinjection and the CS off, while the upper curve is the emission spectrum ofthe VCSEL with injection and CS on.PHB = 9 mW, a) J = 316 mA, b)J = 352 mA, c) J = 362 mA, d) J = 420 mA, e) J = 498 mA.

to stress that the possibility of creating CS vanishes ifθ isvaried of more than 6 GHz. So it appears that this detuningis a severe constraint for the presence of CS. If we changethe pumping current, the width of the range inθ where CSexist remains constant, at least within the resolution of theoptical spectrum analyzer (0.5A, 15 GHz). Instead, since thelaser wavelength (and the cavity resonance) is increasing, themaster wavelength needs to be increased as well in order tomatch this critical detuning.

The presence of these double peaks in the backgroundoptical spectra reveals that the pedestal of the CSs, which wasan homogeneous state in the amplifier configuration, has a verydifferent nature when the injected VCSEL is operated wellabove threshold. The double peak structure in the backgroundspectrum indicates that the background is oscillating in timeat a frequency corresponding to the peak separation (35GHz). The bandwidth limitation of our detectors does notallow to resolve such high frequency signal. Measurements ofpossible lower frequency beatings have been performed withthe fastest detector available (8 GHz bandwidth). As expected,no modulated signal was evidenced within that frequencyrange, either in presence or in absence of a CS. We interpretthis as a signature that no oscillatory behavior is present apartfrom that evidenced in the optical spectrum, while the peakseparation lies well beyond the detector’s capability.

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III. T HEORY

The rate equation model adopted to describe the formationof patterns and CS in the driven VCSEL below threshold [6]is no longer adequate for the VCSEL above threshold. In fact,as already evidenced for two-level lasers [24], the standardadiabatic elimination of the polarization variable, which leadsfrom the Maxwell-Bloch equations to the rate equations,introduces unphysical effects in presence of diffraction. Inthe case of a laser with injected signal, the rate equationmodel predicts that below the threshold of injection locking alltransverse wave-numbers are unstable, as a consequence of thefact that the standard adiabatic elimination of the polarizationimplicitly assumes that the gain width is infinite.

In the case of a two-level laser such a wrong result dis-appears as soon as one considers the full set of Maxwell-Bloch equations. For a semiconductor laser the solution is lessstraightforward, because in principle for such lasers the bandstructure of the active material imposes to adopt microscopicMaxwell-Bloch equations, with an enormous increase of thecomplexity of the problem, which makes the analysis ofpattern formation unrealistic.

However, the use of effective Maxwell-Bloch equations forsemiconductor lasers has been suggested, where a collectivemacroscopic polarization variable is introduced, and a dy-namical equation is cast for its evolution, although suitablymodified in order to capture the typical features of the sus-ceptibility of a semiconductor medium [29]–[31]. In this waypattern formation in semiconductor lasers can be studied withthe same degree of complexity as for two-level lasers. Weconsider the following effective Maxwell-Bloch equations

∂E

∂t= σ

[EI + P − (1 + iθ)E + i∇2

⊥E]

, (1)

∂P

∂t= Γ (1 + i∆) [(1− iα)DE − P ] , (2)

∂D

∂t= b

[J −D − 1

2(EP ∗ + E∗P ) + d∇2

⊥D

], (3)

whereE andP are the slowly varying envelopes of the electricfield and of the effective macroscopic polarization, andD is apopulation variable proportional to the excess of carriers withrespect to transparency. Time is scaled to the dephasing rateτd of the microscopic dipoles and the decay ratesσ andb aredefined asσ = τd/τp and b = τd/τc, whereτp and τc are,respectively the photon lifetime and the carrier recombinationtime.

As already evidenced in the previous Section, an importantparameter isθ, which represents the difference between thecavity longitudinal mode frequency and the frequency of theinjected field, multiplied byτp. The amplitude of the injectedfield is denoted byEI , while J is a parameter related to thepump current, normalized in such a way that the threshold forthe solitary laser isJth = 1 when it emits on axis. Finally,dis the diffusion coefficient for carriers.

This model differs from the two-level Maxwell-Bloch equa-tion in the equation for the macroscopic polarizationP where,as in [29], the right hand side of the equation is multiplied bya complex term where two real parameters,Γ and∆, appearand determine the shape of the effective susceptibility. The two

parameters are assumed to depend on the population variableD, and the dependence can be phenomenologically derived bya linear fitting of the gain curves calculated with a microscopicmodel. In this paper we setΓ(D) = 0.276 + 1.016D and∆(D) = −α+2δ(D)/Γ(D), with δ(D) = −0.169+0.216D.Γ(D) is associated with the gain linewidth, whileδ(D) isthe detuning between the reference frequency (in our casethat of the injected field) and the frequency where the gain ismaximum. Our model differs from that of [29] mainly becausewe write explicitly the gain term in the equation forP as(1 − iα)ED, where α is the usual linewidth enhancementfactor.

The full derivation of the model, its comparison with themicroscopic susceptibility and an extended analysis of themodulational instabilities characterizing the system, beyondthe specific experimental match sought in this article, cannotbe presented here for reasons of space and will be the subjectof a further publication. Here, we just point out that a standardadiabatic elimination ofP allows to recover straightforwardlythe rate equation model [6], which makes the physical com-parison with the amplifier configuration more straightforward.

Throughout this paper we kept fixed the parametersσ =4 · 10−2, b = 10−4, α = 3, d = 0.052 . Assumingτd =100 fs, this meansτp = 2.5 ps, andτc = 1 ns. The choiceproceeds from values found in literature or mutuated fromthe experimental measurements throughout the research workon the samples under investigation. The free parameters werethe intensityE2

I of the HB, the pump parameterJ and thedetuningθ.

In Figs. 7 and 8 we show the homogeneous stationarycurve and the instability domains, focussing on two regimescompatible with the experimental conditions to validate theobserved evidences. In Fig. 7 the laser is pumped close tothreshold (J = 1.05) while in Fig. 8 it is about 20% abovethreshold (J = 1.2). Choosing properly the detuningθ, the ho-mogeneous stationary state shows the typical S-shaped curveof bistable systems (Figs. 7a and 8a). The stability analysisreveals the existence of two instability domains, associated,respectively, with a Turing and a Hopf instability.

The Turing instability (Figs. 7b and 8b) affects the negativeslope branch (K = 0) and part of the upper branch, and it isresponsible for the formation of patterns and eventually CS .It does not depend on the parametersΓ and∆, and thereforeit is independent from the way we describe the semiconductorsusceptibility. In fact, it can be found also in the rate equationmodel.

The Hopf instability (Figs. 7c and 8c) is associated with thewell-known instability of a laser with injected signal whichappears whenever the frequency of the injected field differsfrom that of the free running laser. This instability existsfor low injection powers, and it disappears at the injectionlocking point, where the injected field is strong enough tolock the laser to its frequency. In a class-B laser that point isapproximately given by the condition|ES |2 = J − 1, whichmeans that locking is achieved as soon as the intensity emittedby the laser with injected signal equals that of the free runninglaser. As mentioned above, this instability cannot be describedcorrectly within the rate equation model because in presence of

JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. X, NO. Y, Z 2006 7

0.00 0.02 0.040.0

0.2

0.4

0.6

|Es|2

EI

2

a

0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

2.0

stable

K2

|Es|2

unstable

b

0.00 0.01 0.02 0.03 0.04 0.05 0.060.0

0.5

1.0

1.5

stable

σ K2

|Es|2

unstable

c

Fig. 7. Stationary curve and instability domains for the homogeneousstationary solution.J = 1.05, θ = −2.3. a) stationary intensityE2

s vs.the injected intensityE2

I ; b) instability domain for the Turing instability;c) instability domain for the Hopf instability.K is the modulus of thetransverse wavevector against which the stability of the homogeneous solutionis analyzed.

a flat gain profile, arbitrarily large transverse wave-vectors areexcited. Instead in our model only a finite band of transversewave-vectors are excited. Here on the vertical axis we plotthe quantityσK2, which has the meaning of the frequencyassociated with the transverse mode whose wavevector hasmodulusK. Hence, the shape of the unstable domain is aconsequence of the shape of the gain in our model in the blueside of the spectrum.

Since the position of the injection locking point depends onthe injected currentJ , two different scenarios are possible: inFig. 7a a portion of the lower intensity homogeneous branchbetween the injection locking point and the right turningpoint is stable, while in Fig. 8a the whole lower intensityhomogeneous branch of the steady-state curve is unstable. Inthe former case CS are not substantially different from thosefound when the device was kept below threshold [6], becausethey still sit on a stable background.

In Figs. 9a and 9b we show the results of the dynamical

0.00 0.02 0.04 0.06 0.080.00

0.25

0.50

0.75

|Es|2

EI

2

a

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2.5

stable

K2

|Es|2

unstable

b

0.00 0.05 0.10 0.15 0.200

1

2

3

stable

σ K2

|Es|2

unstable

c

Fig. 8. Same as Fig. 7, but withJ = 1.2 andθ = −2.5.

simulations for the parameters of Fig.7 and 8, respectively. Thesquares indicate the intensity averaged in time in a point of thetransverse plane, which is chosen as the center of a CS whena CS exist. The vertical bars represent the oscillations aroundthe average intensity. The two scenarios are quite different.

In Fig. 9a we show that a branch of stable CS exist incorrespondence with the stable part of the lower branch.Apart from the small oscillations observed before the injectionlocking point is met, the behavior is very similar to that foundfor the laser below threshold [6].

The CS characteristics below and close to the VCSELthreshold may induce to think that, in order to exist and bestable, the CS need a stable background. Hence, they couldnot exist when the whole lower branch is unstable. Yet, thisis not the case, as shown in Fig. 9b. Although the lowerbranch is never stable, we found that CS exist for a rangeof injected intensity even larger than in Fig. 9a. In this caseboth the lower branch and the CS display strong oscillations.The amplitude of these oscillations for the CS increases asthe injected intensity decreases, until aroundE2

I = 0.019 theCS precipitates to the lower branch. However, until that point

JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. X, NO. Y, Z 2006 8

0.00 0.01 0.02 0.03 0.04

0.0

0.5

1.0

1.5

Inte

nsity

EI

2

a

CS

0.00 0.01 0.02 0.03 0.04 0.05 0.06

0

1

2

3

Inte

nsity

EI

2

b

CS

Fig. 9. The results of the dynamical simulations are superimposed to thestationary curves of Figs. 7a and 8a. The parameters are the same as for thosefigures. The meaning of the squares and of the vertical lines is explained in thetext. a) CS coexist with the stable part of the lower branch from the injectionlocking point to the right turning point. b) CS coexist with the unstable lowerbranch in the intervalE2

I = 0.019 to E2I = 0.047.

the CS are well visible, stable and robust (e.g. versus additionof a spatio-temporally distributed stochastic processes whichmimics white noise in the device), although they are embeddedin an oscillating background. They can also be written anderased in the usual manner.

We have also calculated the local optical spectra of the fieldfor the two values ofJ . The results are shown in Fig. 10. Asin Fig. 6 the lower traces correspond to the solitary laser, themiddle traces to the laser with injected signal and CS off, andthe upper traces to the laser with injected signal and CS on.The intensity of the injected field isE2

I = 0.03, a value forwhich CS exist for both values ofJ .

In our model the reference frequency is the frequency ofthe injected field, to which it corresponds the peak atν =0 displayed in the two upper traces. The frequency of theplane wave emitted by the solitary laser is, in normalized units,

0 100 200 300

0.0

0.5

1.0

1.5

2.0

Inte

nsity

Frequency (GHz)

a

0 100 200 300

0.0

0.5

1.0

1.5

2.0

Inte

nsit

y

Frequency (GHz)

b

Fig. 10. Local optical spectra for the same parameters as in Figs. 7 for panela) and 8 for the panel b). The meaning of the three traces is the same as inFig. 6

σ(α+θ)/2π. In order to find CS we had to chooseθ negativeand smaller in absolute value thanα, hence this frequency isalways positive. This agrees with the experimental observationthat the HB must be red-detuned with respect to the solitarylaser.

The frequency of the plane wave corresponds to the leftmostpeak in the lower traces of Figs. 10a and 10b. The otherpeaks visible in this figure are associated with the modes withtransverse vectorK, whose frequencies are displaced to theright by an amountσK2. Apart from the fact that a largernumber of transverse modes with large intensity is present forJ = 1.2, the two lower traces do not differ significantly.

Instead, the middle traces are completely different, becausein Fig. 10a only the peak atν = 0 appears, which meansthat locking has been achieved, while in Fig. 10b the wholespectrum of the solitary laser coexist with the peak atν =0, although the peaks are attenuated. We notice also that thepresence of the injected field slights push towards the rightthe spectrum of the solitary laser.

Finally, in the upper traces the peaks atν = 0 become

JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. X, NO. Y, Z 2006 9

a b c

d e f

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

-4 .0 -3 .5 -3 .0 -2 .5 -2 .0

C S

H o p f

g

fe

dc

b

θ

|Es|2

a

Fig. 11. a-f: average intensity profiles calculated taking into account that thecavity frequency varies along the horizontal axis and varying the frequencyof the injected field. g: stationary homogeneous intensity as a function of thedetuningθ. The horizontal bars indicate the ranges covered byθ in a-f. Thelimit of the Hopf instability domain and the existence region of CS are alsoindicated.J = 1.2, EI = 0.15.

higher, and in Fig. 10b only few frequencies survive in theblue side of the spectrum, associated with the oscillations ofthe CS peak.

On the overall, the main features of these spectra qualita-tively agree with those of the experiment, for instance, Figs.6b and 6d. Notice especially that the detuning between theleftmost peak of the solitary laser spectrum and the frequencyof the injected field both in Fig. 10a and in Fig. 10b lyesinside the experimental range35±8 GHZ. In the comparison,one must consider that, as explained in the previous Section,the frequencies of the transverse modes in the experiment areprobably suppressed by the spatial filtering action of the fiberused in the detection apparatus.

All the previous numerical results have been obtainedimposing periodic boundary conditions in the numerical code.This amounts to consider only a small portion of the VCSELtransverse section. To reproduce the coexistence of differentoutput profiles in the transverse plane we had to drop thatassumption and introduce in the equations the spatial depen-dence of the pumpJ and of the parameterθ. We assumed thatJ is different from 0 only in a circle whose diameter is aboutthe 90% of the integration window. The parameterθ varies of1 along the horizontal axis. With our choice ofτp = 2.5 ps,this corresponds to a variation of about 60 GHz of the cavityfrequency.

In Figs. 11a-f we show the six averaged patterns obtainedas the value ofθ at the right extremum of the laser varies from-1.66 to -3.25, as illustrated in Fig. 11g. This amounts to a

variation of about 100 GHz of the frequency of the injectedfield. Hence the six figures cover more or less the same rangeas in Figs. 3d-g. In Fig. 11a we observe the appearance ofthree CS in the left region of the VCSEL. From Fig. 11b toFig. 11d the CS shift to the right. During this translation somepatterns are created to the left of the CS, and new CS appear.In Fig. 11e only patterns are visible, and finally in Fig. 11f thepatterns are confined in the right region, and in the remainingpart of the VCSEL the output is almost homogeneous.

The sequence can be easily understood looking at Fig. 11g,where we show the intensity of the unstable homogeneousstationary state as a function ofθ instead ofE2

I . Here J =1.2, hence the locking threshold is|Es|2 = 0.2. Below thatthreshold the stationary state is Hopf unstable, and this resultsin temporal oscillations which, averaged in time, give thealmost homogeneous output of the right part of Fig. 11a andthe left part of Fig. 11f.

Beyond the locking threshold the stationary state is unstableagain, but this time the instability is of the Turing type and itleads to the patterns observed in Figs. 11b-e. The vertical stripewhere CS exist in these figures corresponds to the interval ofθ shown in Fig. 11g, whose width is about 0.2, which meansabout 12 GHz, a value twice larger than the experimental one.This explains why in Fig. 11 the region where CS can befound is more extended than in Fig. 3.

IV. CONCLUSION

In this paper, we provided the first joint experimental andtheoretical evidence of stable and independently addressableCavity Solitons in a VCSEL driven by a homogeneous co-herent beam, with current injection above the solitary laserthreshold. The experimental analysis globally shows a patternscenario not dissimilar from the previous cases where theVCSEL was kept below the threshold and operated as anamplifier.

On the other hand, the relevant issue is that observa-tions clearly indicate the presence of CS simultaneously withthe existence of a competing field component at differentwavelength, associated to the background homogeneous field.Despite this ”beating”, the CS are quite stable, their bistablecharacter is unspoiled by the dynamical competition and twoof them have been independently and repeatedly turned on andoff.

The theory supports and validates the findings and theirinterpretation. It relies on a model for collective variables,adopted especially for this quest; although the analytic study ofthe model has been limited to the regimes under investigations,the indication is clear that (at least) two cases exist whereinCS can be stable: one is the case where, at given pump, theinjected field intensity locks the the laser to its own frequency,and the field is stationary; the other is the case where, inthe unlocked regime, a dynamically oscillating backgroundappears as the manifestation of a Hopf modulational instabilitywhich activates transverse modes in the emission profile.

Remarkably, the CS are predicted to be stable even onthis wobbly pedestal and the extension of the stability regionis sizeable. The experiment confirmed the expectations that

JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. X, NO. Y, Z 2006 10

stationary, self-localized structures in the field profile of amicrocavity exist despite the onset of a spatially-modulatedbackground, irregularly oscillating on very fast timescales.

This opens up new classes of operating regimes and ofbroad-area optical microdevices where the search for CS mightlead to more appealing, flexible and/or exploitable applicationsto the manipulation of information encoded in the spatial fieldprofile. The previous experiments all showed that CS existedon top of a stationary, purely homogeneous background.

While the relevance of the achieved evidences is apparent,we are still confronted with a number of open questions both ofapplicative and fundamental flavor: on the experimental side,one has to cope with the rather strict resonance conditionswith the laser field (estim. 6GHz), and a better understandingof the origin of such constraint may lead to easen it. Anotherissue is the analysis of the more specific behavior of the laserwith injected signal, to better understand the locking processand characterize the local field dynamics.

On the theoretical side, this model must be investigatedin depth to comprehend the complicated phenomenology ofthe field spatiotemporal dynamics that this work just begununveiling. In particular the behavior of the laser with injectedsignal at the Hopf MI threshold seems much more compli-cated than the one predicted for the 2-level laser, while thedifferences between the experimentally measured spectra andthe theoretical ones (quite likely to be associated to this issue)should be understood in such terms and possibly mitigated.

We thus expect that gaining more insight into the spatiotem-poral field dynamics of the VCSEL with injected signal, willdrive us towards the assessment of a model for semiconductorlaser oscillators which could be exported towards differentclasses of devices, similarly operating as emitters. While weare very well aware that the complexity of the semiconductorsusceptibility even close to threshold will probably forbid theuse of a microscopic model, we are confident that the basics ofthe field spatiotemporal dynamics can be grasped even by sim-pler approaches [6]. On a more fundamental note, the analyseswill hopefully provide a (possibly intuitive) justification asfor the CS stability, and for the peak suppression induced inthe spectra by the CS switch-on as it was observed in thesimulations.

ACKNOWLEDGMENT

These results have been worked out in the framework of theESPRIT Project FUNFACS.

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Xavier Hachair was born in France 28 years ago.He joined the Institut Non Linaire de Nice (INLN,France) as a Ph.D. student. He received the Ph.D.degree from the Universit de NiceSophia Antipoliswith a thesis entitled ”Solitons de cavit - tudeexprimentale”. During his Ph. D, he stayed sixmonth in the ”Center for optoelectronic computingsystems and department of electrical engineering”(Colorado state university, Fort Collins, Colorado,USA). After one year with a contract of ”A.T.E.R”at the Universit de Nice-Sophia Antipolis, he is

currently post-doc at the Laboratoire Physique de Nanostructures in Paris.His activity revolves around the experimental analysis of spatio-temporaldynamics of semiconductor lasers and cavity solitons.

Francesco Pedaci was born in Milan, Italy in1977. He obtained the ”Laurea in Fisica” from theUniversity of Milan in 2002. At the moment he isPhD student at the Institut Non Linaire de Nice(France). His work, funded by three years grantby the French Ministry of the Research, is focusedon the spatio-temporal dynamics of semiconductorlasers and optical amplifiers.

Emilie Caboche was born in France 23 years ago. She contributes to thispaper in the frame of her project of ”Diplme d’Etudes Approfondies”. She iscurrently PhD student at the Institut Non Linaire de Nice (France).

Stphane Barlandgot his PhD in 2001 at the InstitutNon Linaire de Nice, in collaboration with theInstitut Mediterrani d’Estudis Avanats (IMEDEA).He stayed at IMEDEA for a short-term postdoctoralfellowship working on the control of transversestructures in semiconductor lasers. He then joinedas a postdoctral fellow the Istituto di Fisica dellaMateria (Florence, Italy) for a two year stay in theframework of the VISTA european project, workingon semiconductor laser dynamics and characteriza-tion. He is now working as a researcher at the Istitut

Non Linaire de Nice. His present research interests include semiconductorlaser dynamics and optical matter.

Massimo Giudici was born in Italy 35 years ago; hereceived the ”Laurea in Fisica” from the Universityof Milan in 1995. After one year at Pirelli CaviSpA, in Milan, working on the development of lasermodule packages, he joined the Institut Non Linairede Nice (INLN, France) as a Ph.D. student. Hisstudies were funded by a two years TMR grant of theEuropean Union. In 1998, he joined the Universidadde las Islas Baleares (Spain) with a one year teachingcontract. In 1999, he received the Ph.D. degreefrom the Universit de NiceSophia Antipolis with a

thesis entitled ”Non Linear Dynamics in Semiconductor Lasers with Opticalfeedback”. After one year Post Doc at the Instituto Mediterrneo de EstudiosAvanzados (Spain), where he investigated the role of noise in excitableoptical systems, he returned to the INLN as Associate Researcher in orderto work on spatial localized structures in semiconductor microcavities. Heis actually ”Matre de Confrences” at the Universit de Nice-Sophia Antipolisand his activity revolves around the experimental analysis of spatio-temporaldynamics of semiconductor lasers.

J.R. Tredicce is Professor at Universit de Nice-Sophia Antipolis” and he is the present Directorof the ”Institut Non-Lineaire de Nice” Nice whereseveral experiments are actually running on opticalsystems, liquid crystals, and hydrodynamical sys-tems. In particular he is leading the experimentdevoted to the investigation of localized structure inbroad area semiconductor lasers. He is co-author ofthe paper showing experimental evidence of perioddoubling and chaos in lasers. He is one of thepioneers in using dimension and entropy calculations

applied to data obtained from laser systems. He has been also responsible forexperimental works on space time chaos in lasers and the use of bifurcationtheory applied to symmetry groups to understand and classify intensitypatterns. He is author of more than 100 publications in refereed scientificjournals.

Franco Prati received his Ph. D. degree in 1993 at the University of Zurich forhis studies on spatio-temporal dynamics in lasers. In 1996 he got a permanentposition as researcher at the University of Milano and presently he is AssociateProfessor in Physics of Matter at the University of Insubria, Como, Italy. Hisresearch activity is theoretical and concerns nonlinear optics, and, in particular,pattern formation and dynamics in active and passive systems. In the last yearshis interests focussed on semiconductor lasers. Since 1989 he participated inseveral national and European projects and he is co-author of about 50 papersin international refereed journals.

Giovanna Tissoni received the laurea degree inphysics in 1995 and the Ph.D. degree in physicsfrom the University of Milan, Italy, in 1998. Sheis currently a researcher at the INFM Research Unitin Como, Italy. Her research activities began withthe study of transverse and polarization properties ofthe light emitted by Vertical Cavity SemiconductorLasers (VCSELs), and, since 1997, she has beendealing with semiconductor modelling and applica-tions to transverse pattern and cavity soliton for-mation in semiconductor microresonators. She co-

authored about 30 publications in international journals and proceedingsand has participated in several international Conferences and national andEuropean projects. Dr Tissoni has been a member of the National Institute ofPhysics of Matter (INFM, now CNR) since 1995.

Reza Kheradmand received the Ph. D degree inPhysics in 2005 from University of Tabriz-Iran,where he is currently working as a researcher. Hewas a ICTP (Abdus Salam International Center forTheoretical Physics, Trieste-Italy) Research fellowat the University of Insubria in Como-Italy fromNovember 2002 to November 2003 for a first timeand from February 2005 to July 2005 for a secondtime. He was associated with the INFM- NationalInstitute for the Physics of Matter-Italy in thoseperiods. His activity concerns the theory of Cavity

Solitons in Semiconductor microresonatros.

JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. X, NO. Y, Z 2006 12

Luigi A. Lugiato received the laurea degree inphysics from the University of Milan, Milan, Italy, in1968. He is currently a Professor of Quantum Elec-tronics at the University of Insubria, Como, Italy,and the Director of the INFM (National Institute ofPhysics of Matter) Research Unit in Como, Italy. Hisresearch activities are mainly in the fields of nonlin-ear and quantum optics, contributing to the topicsof superfluorescence, optical bistability, squeezing,optical instabilities, optical pattern formation, cavitysolitons, and quantum imaging. He is the author of

more than 350 publications in international refereed journals and conferenceproceedings. Prof. Lugiato is a Fellow of the American Physical Society andthe Optical Society of America. He received the Albert A. Michelson medal(Franklin Institute) in 1987 and the Willis E. Lamb Medal for Laser Scienceand Quantum Optics (Physics of Quantum Electronics Inc.) in 2002.

Igor Protsenko received the Ph.D. degree in physicsin Lebedev Physical Institute of RAS in Moscow in1989. He is currently senior researcher in LebedevPhysical Institute, Joint institute of nuclear research(Dubna) and lecturer of Theoretical quantum elec-tronics course in Moscow Engineering and Physicsinstitute. His research interests are in the theory ofquantum radiation-matter interaction, laser nonlineardynamics and new optical materials (nanoplasmon-ics). He is a co-author of about 50 publications ininternational refereed journals and 10 patents.

Massimo Brambilla received the master in Physicsat the University of Milan and the Ph.D. degree inphysics from the University of Zurich, Switzerland,in 1992. He is currently an Associate Professor at theEngineering Faculty of the Polytechnic of Bari, Italyand has co-authored about 90 publications in interna-tional journals and proceedings. Massimo Brambillacoordinated or participated in several national andEuropean projects with the groups of Milano (1988-1995) and Bari (1995-present), in particular he hasbeen active in ESPRIT or FET projects. His research

activities cover dynamical instabilities in optical bistability and lasers, patternformation in nonlinear optical systems; in the last eight years the researchmainstreams have converged towards the studies of localized structuresformation in semiconductor-based microresonators and the application thereofto all-optical information treatment.