spatial solitons in semiconductor microresonators

488 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 3, MAY/JUNE 2002

Spatial Solitons in Semiconductor MicroresonatorsV. B. Taranenko and C. O. Weiss

Invited Paper

Abstract—We describe experiments showing the existence ofspatial solitons in semiconductor microresonators. We describe themanipulation of such solitons in view of technical applications:switching solitons on and off by coherent, as well as incoherentlight, reducing the light power necessary to sustain and switch asoliton by optical pumping, and the limit in which independent soli-tons become so strongly bound that they form a coherent hexagonalpattern, marking the maximum information storage density.

I. INTRODUCTION

T HE EXISTENCE of spatial solitons in optical resonatorscontaining a nonlinear medium was predicted theoretically

some time ago [1]. Such spatial solitons can be viewed as smallstable domains of one field state surrounded by another state ofthe field. The two different field states can be a high and a lowfield (bright, dark solitons), positive and negative field (phasesolitons), or high and zero field (vortices). They can exist in avariety of nonlinear resonators, such as lasers (vortices), drivennonlinear resonators (bright, dark solitons), resonators with gaindue to parametric interaction (phase solitons) or laser resonatorscontaining a saturable absorber (bright solitons).

Vortex solitons differ from other soliton types in that they pos-sess “structural stability” in addition to “dynamical stability,”the only stabilizing mechanism of the other solitons. The exis-tence of vortices in lasers initially shown in [2] and later alsothe existence of vortex solitons [3]. Bright solitons in laser res-onators containing a nonlinear absorber were initially shown toexist in [4], which was limited to single stationary solitons. Ex-istence of moving solitons with velocities quantized and simul-taneous existence of large numbers of stationary solitons wasshown in [5], [6]. Phase solitons in degenerate parametric wavemixing resonators were predicted in [7] and demonstrated in[8]. Theoretically, it was shown in [9] that not only two-dimen-sional spatial resonator solitons exist, but that also in three-di-mensional (3-D) such structures can be stable, linking the fieldof optical solitons with elementary particle physics [10].

All types of solitons in resonators are bistable as already evi-dent from the fact that they can be moved around: at a given loca-tion a soliton can exist or not. Thus, spatial resonator solitons aresuited to carry information. In particular, they constitute mobileinformation carriers. This means that information can be written

Manuscript received February 8, 2002. This work was supported by DeutscheForschungsgemeinschaft under Grant We 743/12-1.

The authors are with the Physikalisch-Technische Bundesanstalt, 38116Braunschweig, Germany.

Publisher Item Identifier S 1077-260X(02)05482-5.

Fig. 1. Structure of a semiconductor multiple quantum-well (MQW)microresonator.

in the form of a spatial soliton, somewhere, and then transportedaround at will; finally being read out somewhere else, possiblyin conjunction with other solitons. In this respect, the spatial res-onator solitons have no counterpart in any other kind of infor-mation-carrying elements and lend themselves therefore to op-erations not feasible with conventional electronic means, suchas an all-optical pipeline storage register (“photon buffer”) oreven processing in the form of “cellular automata.”

Experiments on the manipulation of bright solitons as re-quired for such processing tasks were first carried out on a slowsystem: laser with (slow) saturable absorber. In particular, it wasdemonstrated how to write and erase solitons and how to movethem or localize them. For reviews see [11], [12].

In order to be applicable to technical tasks, it is mandatory tooperate in fast miniaturized systems. For compatibility and in-tegrability with other information processing equipment, it isdesirable to use semiconductor systems. We chose the semi-conductor microresonator structure as commonly used for ver-tical-cavity surface-emitting lasers [13] consisting of a few pairsof quantum wells (QWs) sandwiched between Bragg mirrors(Fig. 1).

The resonator length of such a structure is only typically1 m, while the area is typically 5 cm, so that an enormous

Fresnel number results which allows a very large numberof spatial solitons to coexist. The resonator is obviously ofthe plane mirror type, implying frequency degeneracy of alltransverse modes, and thus allowing arbitrary field patterns tobe resonant inside the resonator. This is another prerequirementfor existence and manipulability of spatial solitons.

Since these spatial solitons are bistable, their existence isclosely linked with the plane wave resonator bistability. In fact,using semiconductor structures precisely as these, the early ex-periments on semiconductor resonator bistability were carriedout [14].

1077-260X/02$17.00 © 2002 IEEE

TARANENKO AND WEISS: SPATIAL SOLITONS IN SEMICONDUCTOR MICRORESONATORS 489

Fig. 2. (a) Bistability loop in reflection. (b) Switching on and off of a drivensemiconductor MQW microresonator.

Fig. 3. (a) Bright (dark in reflection) and (b) dark (bright in reflection)hexagonal patterns spontaneously formed in the semiconductor microresonatorfor the dispersive/defocusing case and for different illumination intensities.

Mirror reflectivities of 99.7% (resulting in a Finesse of400)in conjunction with the absorptive or dispersive (or mixed) non-linearity constituted by 10 QW pairs result in resonator bista-bility. Fig. 2(a) shows the measured bistable characteristic ofsuch a GaAs–GaAlAs microresonator. Fig. 2(b) shows that theresonator field can be switched between the two states by shortlight pulses.

Working in the dispersive nonlinearity regime (i.e., wave-length longer than the bandgap wavelength), we observed ini-tially hexagonal patterns (Fig. 3), but not spatial solitons. Thesewe found subsequently working at a wavelength close to thebandedge, where the nonlinearity is mixed defocusing/absorp-tive.

Fig. 4 shows bright solitons, dark solitons, as well as collec-tions of several of the bright and dark solitons, several solitonsexisting at the same time.

II. M ECHANISMS OFSPATIAL SOLITON FORMATION IN

SEMICONDUCTORRESONATORS

Whereas the formation of propagation solitons (also called“self-trapped beams”) is always describable as a balance of alinear effect (spreading of a light beam by diffraction) with anonlinear effect (self-focusing), the conditions in a resonator aremore complex. The soliton formation is, therefore, not alwaysdescribable as such a simple balance.

For linear effects in a resonator there is also the spreading oflight by diffraction. The nonlinear material susceptibility whichcan balance this diffraction can do this in various ways. It has

Fig. 4. Solitons in the semiconductor microresonator. (a) Switched area. (b)Dark soliton in switched area. (c) Bright soliton on unswitched background. (d)Two bright solitons. (e), (f) 2,3 dark solitons.

a real (refractive) and imaginary (dissipative) part and can actlongitudinally and transversely.

1) Longitudinally the most important effect is the change ofthe refractive index with field intensity.

2) The transverse effect of the nonlinear refractive index canbe:

a) self-focusing (favorable for bright and unfavorablefor dark solitons);

b) self-defocusing (favorable for dark and unfavorablefor bright solitons).

3) Absorption (or gain) saturation (“bleaching”) leads to“nonlinear gain guiding” in laser parlance, a transverseeffect.

Longitudinal and transverse effects can work oppositely, orcooperate. It appears that the (longitudinal) effect of the non-linear index plays the strongest role in soliton formation. Thus,it was found in [15] that bright solitons exist under conditionsof defocusing nonlinearity.

The phenomenon of intensity-dependent resonator lengthgoes under the name “nonlinear resonance” [16]. Even if therewas no transverse influence of an intensity-dependent index(self-focusing or self-defocusing) the longitudinal effect aloneallows to stabilize a soliton.

Generally, one uses a detuning of the light from the resonatorfrequency which is reducing with increasing intensity. Thenfor illumination intensities not quite sufficient for reaching theresonance condition for the whole resonator area, the system“chooses” to concentrate the light intensity in the resonator inisolated spots where the intensity is then high enough to reachthe resonance condition thus forming bright solitons.

Instead of saying “the system chooses,” one would moremathematically express this by describing it as an instability.The detuned plane wave field without spatial structure withintensity insufficient to reach the resonance condition is un-stable against structured solutions i.e., solutions where spatialsolitons exist at arbitrary locations.

According to our numerical tests, a very large number of suchstructured solutions are stable and coexist.


It is this large number of coexisting field states which givesthe system the capability of processing and memorizing infor-mation. (Remember that an optical system can recognize [17]as many optical patterns as it has coexisting spatial field states).

In a different picture, solitons can be seen as a small circularswitching front.

A switching front connects two stable states. In an opticalresonator e.g., the high transmission and the low transmissionstate. Such a front can, in two-dimensions (2-D), surround a do-main of one state. When this domain is comparable in diameterto the “thickness” of the front, then each piece of the front in-teracts with the piece on the opposite side of the circular smalldomain, which can lead, particularly if the system is not far froma modulational instability, to a stabilization of the diameter ofthe small domain [1]. In which case, the small domain becomesan “isolated structure” or in less strict mathematically parlancea “dissipative soliton.”

The art of finding the most stable spatial solitons for appli-cations (or less stable ones if dictated by the application) thenconsists in playing with the nonlinear effects 1)–3) by choice ofthe wavelength, the resonator detuning, and finally by the pop-ulation inversion.

We recall that all nonlinearities 1)–3) change their sign attransparency i.e., at the point where in the valence and theconduction band populations are equal. Going from belowtransparency (absorption) to above transparency (populationinversion, producing light amplification), nonlinear absorptionchanges to nonlinear gain, self-focusing changes to self-defo-cusing and vice versa, and decrease of resonator length withintensity changes to increase (and vice versa). It follows thatbright solitons are more stable above transparency than below,since the transverse nonlinearity (defocusing) counteractsthe longitudinal one (the nonlinear resonance effect) belowtransparency, while the two cooperate above transparency. Italso follows that around the transparency point no solitonswill exist since the magnitudes of the nonlinearities diminish,approaching the transparency point from both sides.

The population of the bands can be controlled by “pumping”i.e., transferring electrons from the valence band to the con-duction band. This can be done by optical excitation [19], withradiation of wavelength shorter than the bandedge wavelengthor even more conveniently—if the structure is suited to sup-port electrical currents (i.e., if it is a real VCSEL-structure)—byelectrical excitation.

III. SWITCHING OF SEMICONDUCTORRESONATORSOLITONS

Spatial resonator solitons are always bistable. If they are tobe used as information carriers, it is necessary to “write” and“erase” them anywhere in the resonator cross section.

Writing a soliton at some point in the cross section is straight-forward: Fig. 5 shows the plane wave characteristic of such aresonator together with the existence area of spatial solitons.

Suppose the uniform background illumination intensity issomewhere in the middle of the bistability range. Increasingthe intensity locally, beyond the switch-on intensity, bringsthe resonator field (locally) to the upper branch of the hys-teresis curve. Reducing then the intensity back to the uniform

Fig. 5. Steady-state plane wave characteristics for reflected light as functionof incident light. The soliton shown exists for incident intensities correspondingto the shaded area.

background value brings the system then into the range wheresolitons exist. With a little luck a solitons form. Concretely,experimentally one applies a short pulse of sharply focusedlight in addition to the uniform background field to switch asoliton on. “With a little luck” above translates more preciselyinto: the spatial shape of the focused switching beam must besuch that the field configuration near the end of the switchingpulse is (in phase space) “inside the basis of attraction of thesoliton solution.” What that means precisely is not expressibleanalytically. It was found numerically [20]1 that Gaussian laserbeams when focused to about the width of a soliton, do thejob and this corresponds to the experimental experience. Onecan understand that the precise shape of the switching beamwill not be particularly critical. The system has to settle onto asolution, of which, given the proper choice of the backgroundintensity there are in general only a few, i.e.,

1) the low intensity plane wave solution (to which the systemcould in principle revert after the switching pulse!);

2) the high intensity plane wave solution;3) possibly a patterned solution; and4) the soliton solution.It is plausible that the system will by approximate choice of

the switching beam shape be guided to the soliton solution, be-cause the initial conditions leading to the different solutions areprobably very qualitatively different.

Erasing an existing soliton appears to be a more difficult task.Initially, investigations were carried out which used the idea thatwriting a second soliton very near to the existing one would killboth in a competition process [21]. This scheme can work forparticular conditions, however, only for small parameter ranges,and probably not particularly fast.

We suggested a conceptionally simpler, and also experimen-tally realistic more general method: Apply a focused switching

1Numerical tests show that pulses of Gaussian beams with the width of thesoliton are suitable to switch bright solitons.


Fig. 6. Experimental setup. AOM: Acoustooptic modulator (is sometimesreplaced by mechanical chopper),�=2: halfwave plate. PBS: Polarizing beamsplitters. EOM: Electrooptical amplitude modulators. BE: Beam expander.PZT: Piezoelectric transducer. L: lenses, BS: beam splitters, PD: photodiode, P:polarizer. Polarizer P combines coherently the injection and background fields.

pulsed beam as in the case of writing a soliton, at the location ofthe existing soliton, but of course with opposite optical phase.This should locally decrease the intensity by destructive inter-ference so that the system has no choice than to fall back onto thelow intensity plane wave solution (i.e., the soliton disappears).

The argument initially advanced against this method: thatit requires a supposedly difficult control of the phase of theswitching field, is really only a theoretical misconceptioncaused by customarily writing down an optical field with afixed phase (typically ). In reality, switching on by anadditional pulse requires constructive interference, and thus,equally, a control of the phase.

For this case of “coherent switching” a phase control ofthe switching field is, thus, always needed: for switching onas well as switching off a soliton. Fig. 6 shows the opticalarrangement used for the semiconductor soliton experiments,and in particular for their switching on or off. Light from a laserof suitable wavelength illuminates the semiconductor resonatorsample (reasonably homogeneously in an area of100 mdiameter). Part of the laser light is split off the illuminationbeam, focused sharply, and directed to some particular locationin the illuminated area, where a soliton is to be switched on/off.

For experimental convenience to limit thermal effects, weperform the whole experiments within a few microseconds, byadmitting the light through an acoustooptical modulator. Theswitching light is opened only for a few nanoseconds using anelectrooptic modulator. The observation is done in reflectionby a CCD camera combined with a fast shutter (another elec-trooptic modulator), which permits to take nanosecond snap-shots at a given time of the illuminated area on the resonatorsample. Recording movies on this nanosecond time scale is alsopossible. To follow intensity in time in certain points (e.g., at thelocation of a soliton) a fast photodiode can be imaged onto ar-bitrary locations within the illuminated area.Fig. 7 shows the following soliton switching observations

1) Switching a soliton on. After the background light isapplied the local switching pulse is applied (1.2 s).It is in phase with the background light, as visible fromthe constructive interference. A bright soliton results,showing up in the intensity time trace as a strong reduc-tion of the reflected intensity.

Fig. 7. Recording of coherent (a) switching-on and (c) switching-off of abright soliton. “Failed switches” with the address beam intensity too low toswitch soliton (b) on or (d) off. Heavy arrows mark the application of switchingpulses. Dotted traces: Incident intensity. The insets show intensity snapshots,namely soliton and unswitched state.

2) “Failed” switch on. If the intensity of the localizedswitching pulse is too small no soliton is created.

3) Switching a soliton off. The background light is increasedto a level where a soliton is formed spontaneously. Thelocalized switching pulse is then applied in counterphaseto the background light, as visible from the destructive in-terference. The soliton then disappears, showing up in theintensity time trace as reversion of the reflected intensityto the incident intensity value.

4) Failed switch off. The soliton persists after the switchingpulse if the latter’s intensity is too small.

The Fig. 7 insets show 2-D snapshots before and after theswitching pulses for clarity.

With the help of an electrooptic phase modulator in theswitching path, it is possible to show repeated switching on andoff of a soliton. Fig. 8 shows the corresponding recording. Thephase of the switching beam was modulated sinusoidally in arange of as visible from the constructive and destructiveinterference at the soliton location. No pulse-amplitude mod-ulator was used in the switching beam. The switching on andoff occurs when the local intensity crosses the upper and lowerswitching value.

Besides this coherent switching, a soliton can be switchedwithout control of the relative phase between background- andswitching light: A soliton can be created by increase of the localcarrier density which can be effected by switching light inco-herent with the background field. For this type of switching, weused the fields mutually incoherent by perpendicular polariza-tions.

Fig. 9 shows the incoherent switch-on of a soliton, where theperpendicularly polarized switching pulse is applied at s.As apparent, a soliton forms after this incoherent light pulse.The working wavelength was in this case near the bandedge.

As discussed in [22] and [23], temperature effects lead in thiscase to a slow formation of solitons, associated with the shiftof the bandedge by temperature, while when working inside the


Fig. 8. Repeated switching on and off of a soliton.

Fig. 9. Recording of incoherent switching on of a soliton. Snapshot picturesshow circular switched domain (left) and a soliton (right). Dotted trace: Iincidentintensity.

band as in [23], [24] the shift plays no role. The slow formationof the soliton is apparent in Fig. 9 (using roughly the time from

to s). It should be emphasized that this thermaleffect is not instrumental for switching a soliton on. However, itallows to switch a soliton off incoherently.

This is shown in Fig. 10(a) where the background light isinitially raised to a level at which a soliton forms spontaneously(Note again the slow soliton formation due to the thermal effect).The incoherent switching pulse is then applied which leads todisappearance of the soliton.

The soliton can thus be switched on and also off by an inco-herent pulse. The reason for the latter is thermal. Initially thematerial is “cold”. A switching pulse leads then to the creationof a soliton. Dissipation in the material raises the temperatureand the soliton is slowly formed. At the raised temperature, thebandedge (and with it the bistability characteristic) and the ex-istence range of solitons is shifted so that a new pulse brings thesystem out of the range of existence of solitons. Consequently,the soliton is switched off.

Thus, switching on a soliton is possible incoherently with the“cold” material and switching off with the “heated” material.The proof of this picture is given in Fig. 10(b). A soliton is cre-ated initially spontaneously by raising the background illumi-nation high enough. An incoherent pulse can then switch the

Fig. 10. (a) Recording of incoherent switching off of a soliton. The insets left,center, right show intensity snapshots, namely switched domain [as Fig. 4(a)],soliton [as Fig. 4(c)] and unswitched state, respectively. (b) Soliton switchesback on spontaneously after the material has cooled for�20 �s. Dotted traces:Incident intensity.

soliton off (at s) even though the background intensityis high enough to produce a soliton spontaneously. It followsthat a new soliton must be created spontaneously after the mate-rial has cooled off. As Fig. 10(b) shows the material has cooledsufficiently at s and a soliton is then indeed formedspontaneously.

In conclusion, coherently the soliton can be switched on andoff. Incoherently, it can be switched on, but it can be switchedoff only with the aid of thermal shift of the bandedge, and onlytemporarily. In the experiment Fig. 10(b), the soliton remainsoff for about 20 s, which may be long enough for certain ap-plications.

For the coherent switching case, we tested if the mechanismallowing the incoherent switch off i.e., thermal shift of the band-edge is possibly also instrumental. Fig. 11 shows that this is notthe case. At s, a soliton is spontaneously created. Justas in Fig. 10(a) we then increase the intensity locally (by aninphase pulse). The result as opposed to Fig. 10(a) is that thesoliton is not switched off. Thus, we conclude that here thermaleffects are unimportant and the switching is purely electronic.

IV. SPATIAL SOLITONS IN PUMPED RESONATOR

The thermal effects discussed above result from the “local”heating caused by the high intracavity intensity at the brightsoliton location. They limit the switching speed of solitons andthey will also limit the speed at which solitons could be movedaround, limiting applications. The picture is that a soliton carries


Fig. 11. Demonstration of the nonthermal character of coherent switching. Asopposed to the incoherent switching off (see Fig. 10), the soliton is stable againstlocal increase of intensity. Dotted trace: Incident intensity.

with it a temperature profile, so that the temperature becomes adynamic and spatial variable influencing the soliton stability.

As opposed, a spatially uniform heating will not cause suchproblems, as it shifts parameters but does not constitute a vari-able in the system. The largely unwanted heating effects are di-rectly proportional to the light intensity sustaining a soliton. Forthis reason and quite generally, it is desirable to reduce the lightintensities required for sustaining solitons.

Conceptually, this can be expected if part of the power sus-taining a soliton could be provided incoherently, i.e., indepen-dent of the background field. As a means to provide such “in-coherent” power, we have pumped the semiconductor resonatoroptically (pumping would also be possible electrically, however,our resonator samples contained no provisions for passing cur-rent through them). Pumping was done by a semiconductor laserat 810- m wavelength (above the band gap energy) of 600-mWpower focused to an area of200 m.

Fig. 12(a) shows the switch-on intensity and the switch-offintensity of a soliton as a function of pump intensity. We notethat the intensity sustaining a soliton is reduced by more than afactor of ten when going from zero pump intensity to 2 kW/cm(the maximum available in the experiment). The power neededto sustain soliton is reduced from 1 mW at zero pumping to70 W at 2-kW/cm pump and all thermal effects are reducedcorrespondingly.

Fig. 12(b) shows a numerical simulation agreeing well withthe measurements. We remark that the unphysical crossing ofswitch-on and switch-off intensities in Fig. 12(a) is an artifactfrom the measurement method. For the measurements, the back-ground intensity was raised until a soliton forms. This yieldsthe “on” curve in Fig. 12(a). Later, the background intensity isreduced so that the soliton switches off. This yields the “off”curve.

Fig. 12. (a) Measured and (b) calculated switching on and off intensities ofthe bistable semiconductor microresonator as a function of pump strength. Forthe unphysical crossing of curves (a) see text.

Fig. 13. (a) Bright (view from “bottom”) and (b) dark (normal or “top”-view)resonator solitons in the pumped semiconductor MQW microresonator.

Fig. 14. Calculated plane wave pumped resonator characteristics. Bright anddark soliton solutions shown on top exist close to the bistability range.

Now again, the switch-on intensities refer to on the “cold”material while the switch-off intensities refer to the heated mate-rial, the latter having parameters, shifted with respect to the coldmaterial. The result is the apparent crossing of the switchingcurves, which is evidently unphysical.

Just as in the unpumped material, we find that bright anddark solitons exist (Fig. 13). For comparison, Fig. 14 shows anumerical calculation done for the maximum pump intensityof 2 kW/cm . Bright and dark solitons exist for the same pa-rameters (in particular, for the same resonator detuning) at only


Fig. 15. Switching of individual spots of hexagonal structures with focusedaddress pulses aimed at different places (marked 1 and 2) of the pattern.

slightly different background intensities. Pumping, done hereoptically, thus permits to reduce considerably the power neces-sary to sustain a soliton. As a byproduct, it practically eliminatesthe thermal effects which would limit practical applications.

We mention here that during the writing of this manuscript,bright solitons have also been observed in semiconductor res-onators with electrical pumping [25].

Concluding on thermal effects, it is amusing to mention theeffects on dark solitons. Bright solitons are accompanied by alocal temperature increase. The increasing temperature shiftsmaterial parameters so that soliton stability is increased. Darksolitons, as opposed, are accompanied by a local temperaturedecrease. This in general leads to a parameter shift, which de-creases soliton stability. A curious effect can result analogouslyto what we termed a “restless” structure in [26]: As soon as thedark soliton is formed, it cools locally, producing at its locationmaterial parameters unsuitable for soliton existence. As a conse-quence, the soliton will move sideways to where the material ishot and where it can exist stably. Alas, however, as soon as it hasmoved, it will cool again. Again destroying the conditions forits own existence. For this reason, the soliton moves on, etc. adinfinitum. This is the curious case in which a system can neverfind a stable point of existence, resulting in what we call “rest-less” motion [26], which can be periodic or chaotic.

V. MAXIMUM INFORMATION DENSITY IN SOLITON ARRAYS:COHERENT ANDINCOHERENTHEXAGONAL PATTERNS

When thinking of spatial resonator solitons for informationprocessing, information density becomes an important consid-eration. This translates into the question of how close one canplace spatial solitons while maintaining their independence (i.e.,being able to switch them on and off independently). The firstexperiments on pattern formation of semiconductor resonators[18] yielded hexagonal patterns (not entirely unexpected sincethe system is not very different from a Kerr-resonator for whichhexagonal patterns had been predicted for some time already[27]).

Some typical hexagon structures observed are shown inFig. 15. We found experimentally that the bright spots insuch hexagonal patterns can be addressed by focused optical

Fig. 16. Numerical results for pattern formation in the dispersive/defocusingcase. Solid/dotted line: Plane wave characteristic. Dark/bright spots correspondto bright/dark spots in the experiment where observation is in reflection.

(incoherent) pulses and can be switched independently [18].Fig. 15 shows the experimental results.

a) Shows the hexagonal pattern formed. The focused lightpulse can be aimed at individual bright spots such as theones marked “1” or “2.”

b) Shows that after the switching pulse aimed at “1” spot “1”is off.

c) Shows the same for spot “2.”We remark that in these experiments, we speak of true logicswitching: The spots remain switched off after the switchingpulse (if the energy of the pulse is sufficient, otherwise the brightspot reappears after the switching pulse).

Fig. 15 (d)–(f) demonstrate that not only single spots can beswitched. The address pulse was aimed in this case betweenthree spots with the effect that different “triples” of spots(marked “1” and “2”) are switched off.

These observations indicate that in these hexagonal patternsof bright spots, the individual spots are rather independent, evenat this dense packing where the spot distance is about the spotsize (which in turn is the size of individual solitons). Thus, theinformation density is as high as can be possibly expected.

This experimental finding was puzzling and not understoodfor some time so that we performed model calculations [29] inorder to elucidate the effects. Fig. 16 shows the bistable planewave characteristic of the semiconductor resonator for condi-tions roughly corresponding to the experimental conditions. Atthe intensities marked a) to d) patterned solutions exist.


Fig. 17. Stable hexagonal arrangements of dark spatial solitons: (a) withoutdefects, (b) with single-soliton defect, and (c) with triple-soliton defect.

The driving field is detuned with respect to the resonator res-onance and this is reflected in the low intensity solution a): thepattern period in a) corresponds precisely to the detuning in thefollowing way: If the driving field is detuned, the resonancecondition of the resonator (resonator length being a multiple ofthe half wavelength) cannot be fulfilled by plane waves travel-ling exactly perpendicularly to the mirror plane. However, theresonance condition can be fulfilled if the wave plane is some-what inclined with respect to the mirror plane (tilted wave so-lution [28]). The nonlinear system chooses, therefore to sup-port resonant, tilted waves. Fig. 16(a) is precisely the superposi-tion of six tilted waves which support each other by (nonlinear)four-wave mixing. The pattern period corresponds precisely tothe detuning. Thus, the pattern formation in Fig. 16(a) is mostlya linear process. In this pattern, the bright spots are not indepen-dent. Individual spots cannot be switched.

On the high intensity pattern d), the pattern period is remark-ably different from a) even though the detuning is the same. Thisis indication that the internal detuning is smaller and means thatthe resonator length is nonlinearly changed by the intensity-de-pendent refractive index (nonlinear resonance). From the ratioof the pattern periods of a) and d) one sees that the nonlinearchange of detuning is about half of the external detuning. Thatmeans the nonlinear detuning is by no means a small effect. Thisin turn indicates that by spatial variation of the resonator field in-tensity the detuning can vary substantially in the resonator crosssection. In other words, the resonator has at the higher inten-sity a rather wide freedom to (self-consistently) arrange its fieldstructure. One can expect that this would allow a large number ofpossible stable patterns between which the system can choose.

Fig. 17 shows that the high intensity conditions of Fig. 16(d)allow precisely to reproduce the experimental findings onswitching individual bright spots and switching “triples” ofright spots. Fig. 17(a) is the regular hexagonal pattern asFigs. 16(d), 17(b) shows one bright spot switched off as a stablesolution and Fig. 17(c) shows a “triple” of bright spots switchedoff as a stable solution, just as observed in the experiments.Numerically, we have found that almost any arrangementof bright and switched-off spots is stable and in the limit ofswitching the majority of bright spots off, the remaining brightspots manifest themselves as isolated free spatial solitons.

Thus, while Fig. 16(a) is a completely coherent spacefillingpattern, Fig. 16(d) is really a cluster of (densest packed) indi-vidual solitons. The increase of intensity from a) to d) allows thetransition from the space-filling pattern to the localized struc-tures by the increased nonlinearity, which gives the system an

additional internal degree of freedom. We note that the tran-sition from the “coherent” low intensity pattern to the “inco-herent” higher intensity structure proceeds through stripe-pat-terns as shown in Fig. 16(b) [29]. For the intensity of Fig. 16(c),the individual spots are still not independent.

The remaining regularities in the high intensity caseFig. 16(d) such as pattern period, or well-defined size of spatialsolitons etc. shows that the freedom of the system is not acomplete one. External parameters such as detuning or valueof intensity still influence the system, although in rather subtleways, such as determining the size of the elements, but leavingtheir arrangement completely free. One would extrapolate, thatas nonlinearity becomes stronger (with increasing externalintensity), the system would gain more and more freedomfor the spatial (and probably temporal) arrangement of itsinternal field. One may guess that at high intensity the externalconstraints will be completely overruled by the nonlinearityand the system state would probably be simply space-timeturbulent.

It follows for practical applications (where a certain degreeof regularity and predictability is required because we have notyet learned to put turbulence to use in information processing)that highest intensities and nonlinearities are unsuitable. An in-formation processing system has to have “just the right amount”of internal freedom, not too much and not too little. This meansthe working light intensities will have to be in some interme-diate range.

Returning to the question of information density, we seethat—although it is well known that such solitons interact bytheir mutual field gradients, solitons can be packed as denselyas conceivable and still retain their information-carryingcapacity if the nonlinearity is high enough.

Eventually, as our understanding of nonlinear systems ad-vances, we may be able to put to use more complex structuresthan solitons for information processing, working at higher non-linearity. At our present level of insight into nonlinear physics,such more complex structures appear simply as “unmanage-able” and we are, therefore, at a loss of ideas of how to put themto use. Evidently, such a research direction aims at nothing lessthan something comparable to the human brain.

We learn from these experiments on simple nonlinear res-onators what the key element of such brain structures is: nonlin-earity. Nonlinearity gives a system the freedom to organize itselfinternally rather independently from environmental parameters.This feature, incidentally, being the prime characteristic of bio-logical beings.

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V. B. Taranenko, photograph and biography not available at the time of publi-cation.

C. O. Weiss, photograph and biography not available at the time of publication.

spatial solitons in semiconductor microresonators

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