Institute of Laser Physics, S. I. Vavilov State Optical Institute Scientific Production Corp., St. Petersburg,Russia�Submitted November 13, 2008�Opticheski� Zhurnal 76, 25–40 �April 2009�
This review discusses the main types of localized lightstructures in systems with substantial energy inflow andoutflow—dissipative optical solitons—mainly on the basis ofwork carried out at the Institute of Laser Physics ScientificProduction Corp. of the S. I. Vavilov State Optical Institute.Because of their increased stability, these solitons are of con-siderable applied interest, especially with regard to data pro-cessing. There are now a large number of systems in whichdissipative optical solitons have been observed or can beformed. Such systems are presented in Section III, but weshall first compare these solitons with conservative ones andshall enumerate their main properties.
II. THE PROPERTIES OF OPTICAL SOLITONS
Optical solitons—stable light structures localized as aconsequence of the balance of linear broadening and nonlin-ear compression in a nonlinear medium1—are divided intotwo classes. Conservative solitons are implemented in trans-parent media in which radiation losses are negligible, there isno energy influx, and the localization is a consequence of abalance of linear broadening �diffraction for a beam and/ordispersion for a pulse� and nonlinear focusing �Fig. 1a�. Op-tical dissipative solitons �autosolitons� are also stable lightstructures but are localized as a consequence of the balanceof energy influx and efflux in a nonlinear medium or system
FIG. 1. Conservative �a� and dissipative �b� solitons as a balance of linearbroadening and nonlinear focusing �a, b� and energy influx and efflux �b�. Iis the intensity, and r and t are the coordinate and time for spatial andtemporal solitons, respectively.
187 J. Opt. Technol. 76 �4�, April 2009 1070-9762/2009/040
�Fig. 1b�.2 The effect by which linear broadening is balancedwith nonlinear focusing can either be present or absent fordissipative solitons.
Conservative and dissipative solitons as essentially non-linear wave objects have both common properties and fun-damental differences. Optical nonlinearity must be present inboth cases—the initial blob of light spreads in the linearregime. Conservative solitons constitute a family with a con-tinually changing parameter—for example, the soliton’swidth or its maximum intensity. The requirement of energybalance for dissipative solitons imposes an additional condi-tion on their parameters, because of which the set of basicparameters becomes not continuous but discrete. This cir-cumstance increases the stability of dissipative optical soli-tons, and this makes them promising for various applica-tions, including optical data processing.
Solitons can be spatial �localization in space of a beamof continuous radiation�, temporal �localization of a pulse intime�, or spatiotemporal �localized in both space and intime�. Both conservative and dissipative solitons are charac-terized by geometrical dimension D. This quantity is thenumber of measurements with respect to which limitation isachieved because of nonlinear factors, whereas such limita-tion can be achieved with respect to the other �3–D� mea-surements by inhomogeneity of the linear characteristics ofthe medium or system—for example, the refractive index.Figure 2 shows examples of temporal �a�, spatial �b and c�,and spatiotemporal �d� solitons of dimension D=1,2 ,3.
FIG. 2. Examples of temporal �a�, spatial �b, c�, and spatiotemporal �d�solitons. �a� Nonlinear single-mode lightguide, �b� planar nonlinear wave-guide, �c, d� bulk nonlinear medium. The dimension of the soliton is D=1�a, b�, 2 �c�, and 3 �d�.
Optical solitons can be coherent or incoherent. The en-ergy influx can be implemented by putting into an opticalsystem a beam of continuous coherent radiation �Fig. 3a� orby an incoherent external signal or pumping, resulting inamplification of the radiation �Fig. 3b�. For coherent solitons,in the former case, the external signal fastens the frequencyand phase to the radiation. In the latter case, the commonphase of the radiation is arbitrary �invariance to a phase shiftby a constant amount�.
During the evolution of solitons, it is possible to estab-lish their constant shape or periodic, quasi-periodic, or evenchaotic pulsations �while maintaining localization�. Coherentoptical solitons are characterized by high-frequency �optical-frequency� oscillations of the electric-field components, withconstancy of the envelope �envelope solitons, Fig. 4a�. How-ever, ultrashort �with a width of the order of the inverse ofthe mean radiation frequency� or ultranarrow �with width ofthe order of the mean wavelength of the radiation or less�solitons are possible even for optical coherent solitons �Fig.4b�.
Initially solitons meant localized structures in a spatiallyhomogeneous �with respect to D measurements� unboundednonlinear medium. In this case, there is translational invari-ance to shift along these D coordinates by an arbitraryamount. However, the literature now shows a tendency tobroaden the definition of solitons, so that inhomogeneity ofthe medium of one sort or another �usually periodic, see Fig.5� is allowed. Of course, the requirement remains that local-ization of the light is caused by optical nonlinearity of themedium.
Yet another terminological remark relates to a certainconventionality of the meaning of localization. Dissipativesolitons in many cases are subjected in an ideal setup to aninfinite plane wave of radiation. The soliton then represents alocal increase �a bright soliton� or decrease �a dark soliton�of some field characteristic �for instance, intensity� on a com-mon homogeneous background �actually, the background canbe weakly modulated�. This definition does not cover transi-tional structures, for instance, the one-dimensional switchingwaves considered below, in which the temperature field ap-proaches a constant value on different sides of an inhomoge-neity, but the values differ from each other. Therefore, here itis impossible to speak of a temperature soliton. However, itis legitimate to use the term soliton in this case, for example,for mechanical stresses, since they are proportional to thetemperature gradient and consequently approach zero as onegoes away from the inhomogeneity on both sides. A soliton
FIG. 3. Geometry of a system with an external coherent signal �a� andwithout it �incoherent external signal or pumping, �b�; z longitudinal and xtransverse coordinates.
188 J. Opt. Technol. 76 �4�, April 2009
for one physical field thereby may not formally be a solitonfor another physical field in the same system under the sameconditions.
Solitons should not be confused with the filaments thataccompany modulation instability—for example, in a me-dium with Kerr nonlinearity �intense threads of radiation�.3 Itis essential that states without and with a soliton should besimultaneously stable. In this sense, it is possible to speak ofbistability or multistability of soliton systems. In otherwords, it must be possible to both create �write� and annihi-late �erase� a spatial soliton in a nonlinear medium or system.Such a property is associated with the possibility of usingsolitons in memory systems.
Solitons, as nonlinearly localized wave objects, haveraised anew the question of the relationship between waves
FIG. 4. Envelope soliton �a� and ultrashort or ultranarrow soliton �b�. Ex isthe x component of the electric field.
FIG. 5. Lightguide systems for forming one-dimensional solitons. �a�Single-mode lightguide with nonlinear core �denoted by shading�, temporalsolitons are formed in the system; �b� single-mode lightguide with a nonlin-ear core and a Bragg grating �longitudinal refractive-index modulation isshown by the darkness of the shading�, fixed and moving solitons can beformed; �c� set of weakly coupled single-mode nonlinear lightguides, inwhich discrete solitons can be formed.
188N. N. Rozanov
and particles. It is reasonable to reformulate this question asfollows: If one correlates solitons with particles, what are theproperties of these particles? Next, for a number of types ofdissipative optical solitons, we shall show that these proper-ties are extremely unusual. In particular, such particles arenot elementary but possess an internal structure, with variouscharacters of the energy fluxes.
One more remark has to do with the presence of noise�fluctuations� in real systems and in the incident radiation.The most fundamental consequence of these, allowing forthe bistability or multistability mentioned above, is the pos-sibility of spontaneous transitions between various solitonand nonsoliton radiation structures. Therefore, in principle, asoliton possesses a finite lifetime �it is metastable�. However,since a soliton is stable relative to small perturbations, tran-sitions can be caused only by large and therefore low-probability fluctuations. Accordingly, a soliton’s lifetime isextremely long except when the parameters are close to itsstability boundaries. Fluctuations effectively reduce the sta-bility range of a soliton.
Quantum fluctuations of radiation are unavoidable inprinciple. Quantum fluctuations of optical solitons are of sig-nificant interest not only from an applied viewpoint but alsofrom a general physical viewpoint, since their analysis re-veals that these macroscopic objects have essentially non-classical properties. It turns out that a soliton can be regardednot only as classical but also as a quantum particle, whichexecutes Brownian motion under the action of vacuum fluc-tuations.
III. SETUPS FOR FORMING DISSIPATIVE OPTICALSOLITONS
At present, a number of setups are known in which it ispossible to form dissipative optical solitons. Naturally, in allsuch setups, at least one of the dimensions must be large bycomparison with the size of the soliton. In the subsequentpresentation, we shall follow not the chronological order, butthe type of setup.
A. Incoherent excitation of a semiconductor layer
In such setups �Fig. 6a�, the medium possesses an ab-sorption coefficient � that increases with increasing radiationintensity I. Continuous coherent radiation with intensity Iin
acts on a thin sample. The thermal mechanism of opticalnonlinearity is more obvious, in which the absorption coef-ficient increases as the sample temperature T increases. Thefollowing situation then arises in some range of the param-eters. If the sample is initially cold, because of the smallabsorption coefficient, it does not warm up and remains inthe cold state. But an initially warm sample at the sameintensity Iin will absorb almost all the radiation and accord-ingly will remain hot. This is the cause of the bistabilityimplemented in a system involving an increase in absorption,known for the concentration mechanism ��n�I�� and the ther-mal mechanism ��T� of absorption-coefficient nonlinearityin semiconductors �n is the charge-carrier concentration, andT is the sample temperature�.4 The spatial-distribution effectsneeded for the formation of the structures appear in the
189 J. Opt. Technol. 76 �4�, April 2009
sample in the form of a long semiconductor rod or plateheated by intense radiation. The system can be homogeneous�in a direction along the axis of the rod or transverse to theplate� or inhomogeneous, with spatial modulation of thecharacteristics in the indicated directions.
This system is clearly dissipative �the energy of the ex-ternal radiation is inserted into the sample, while thethermal-energy losses correspond to the dissipation of heatfrom its surface�. This is a noncavity system and does notcontain a mirror. The optical nonlinearity is nonlocal—theresponse of the medium at a given point depends on theradiation-intensity distribution at other points because ofthermal conductivity. It can be said that this simplest setup isa one-component system, if by the number of components isunderstood the number of independent quantities that com-pletely determine the state of the corresponding point �spa-tially undistributed� system.5
The transverse-distribution effects �with respect to theradiation-propagation direction� in this setup were first con-sidered in Refs. 6 and 7. The results were explained in fairdetail in Refs. 5 and 8. In a transversely homogeneous setupwith an incoherent external signal and classical bistability�two stable, spatially distributed states�, it can be analyticallyshown by means of an obvious “mechanical analogy” thattemperature-switching waves exist and are stable. As ex-plained above, this implies that incoherent dissipative soli-tons of mechanical stress are present. For setups with trans-verse inhomogeneity, spatial bistability and its kinetics—spatial hysteresis—are described. From the general-physicsviewpoint, it is important here that the first example of spa-tial hysteresis—hysteresis in a system with substantial spatialdistributedness—is represented, and many of the results canbe generalized to the case of systems with a phase transitionof first type. At the same time, an incoherent temperaturesoliton is obtained for a system with quasi-periodic trans-verse modulation, and it is shown to be possible to organizemultichannel memory �Fig. 7�. This reveals the importantrole of inhomogeneities in the system and the possibility ofdynamic reconstruction of the setup. Along with the spatial-hysteresis effect, this can serve as a basis for a new approachto nonlinear data processing �see below�.
B. Coherent excitation of a semiconductor layer
This also is a dissipative noncavity and nonmirror setupwith nonlocal response of a thin layer of the medium �Fig.6b�. The difference from the previous setup is that the exter-nal radiation is coherent, and the medium corresponds to amacroscopic quantum state—a Bose–Einstein condensate�BEC� of excitons, implemented at temperatures below thecritical temperature and characterized by their coherence.Accordingly, the system has at least two components �theradiation and the wave function of the condensate are deter-mined by two real quantities—the amplitude and the phase�,and this increases the number of types of spatial structures bycomparison with the previous case.
In such a transversely homogeneous setup, modulationinstability is demonstrated in Ref. 9, while coherent dissipa-tive solitons are found in Ref. 10. A one-dimensional soliton
189N. N. Rozanov
is illustrated in Fig. 8 by the concentration profile n of theexcitons and by the phase � of their collective wave func-tion. A more detailed analysis shows that the soliton pos-sesses internal structure—there are exciton fluxes in it, withvarious directions of motion, and this is typical of dissipativesolitons. It is noteworthy that just one layer of nonlinearmedium with no additional mirrors is suffiicent for the pres-ence of such solitons. The comparatively small size of thesolitons �although not outstanding among those consideredhere� �about 200 nm in the case of Fig. 8� is worthy of at-tention.
C. Molecular chains resonance-excited by coherentradiation
Oriented J-aggregates of cyanine dyes, consisting of alarge number of molecules with identically oriented dipole
FIG. 6. Wide-aperture systems for forming spatial dissipative solitons. �a� Inthin layer of semiconductor, �c� coherent excitation of a molecular chain, �dexcitation of a layer of medium with a feedback chain, �f� coherent excitatmedium, �h� coherent excitation of a nonlinear Fabry–Perot interferometer, �icavity �k� ring laser. The narrow and wide arrows indicate coherent and iadditional linear and nonlinear elements.
190 J. Opt. Technol. 76 �4�, April 2009
moments �Fig. 6c�, can serve as an example. The externallaser radiation oscillates the molecular dipoles so that thedipolar radiation of one molecule acts on the adjacent mol-ecules �the nonlocality mechanism of optical response�. Bi-stability has been theoretically demonstrated in suchchains.11
Even though an extensive literature is devoted to conser-vative molecular solitons, their dissipative analogs have beenanalyzed, as far as we know, only in Ref. 12 �Fig. 9�. Theyare classed as discrete ultranarrow solitons and can have out-standingly small width �about 1 nm—i.e., dissipative “nano-solitons” are real�, and this is of significant scientific andapplied interest.
There are no mirrors or other forms of external feedbackin the setups described above, and radiation propagationplays an auxiliary role in the overall dynamics of the struc-
rent excitation of a thin layer of semiconductor, �b� coherent excitation of aerent excitation of a layer of medium with a feedback mirror, �e� coherentf two layers of medium, �g� coherent excitation of a semi-infinite layer oferent excitation of a nonlinear ring interferometer, �j� laser with a two-mirrorrent radiation, respectively, and M are mirrors. The systems may include
cohe� cohion o� cohncohe
190N. N. Rozanov
tures. For example, in the mean-field approximation, thecharacteristics of the radiation inside the system are averagedin the longitudinal direction and drop out of the dynamicmodel. Not so with the systems presented below, which haveeither external-feedback elements or substantial longitudinaldistributedness of the system.
D. Related layered systems
Spatial structures can also appear in systems with some-what more complex geometry. A broad-aperture system con-sisting of a thin layer of nonlinear medium and a feedbackmirror separated from the layer by some distance �Fig. 6d�can serve as an example. Coherent spatial structures are ex-cited by external laser radiation. Such a setup was proposed
FIG. 7. Spatial bistability and a dissipative temperature soliton in a systemwith spatial intensity modulation of the incident radiation. Quadrant I showsan S-shaped transfer function �the dependence of the stable temperature � inthe point scheme on the incident radiation intensity Iin. Quadrant IV showsthe transverse profile of the intensity of the radiation incident on the sample�a long rod�. Quadrant II shows the temperature profile constructed from thedata of Quadrants I and IV. I0 is the Maxwellian value of the intensity atwhich the switching wave front is fixed, and Tenv is the ambient temperature.
FIG. 8. Transverse exciton-concentration profiles corresponding to a dissi-pative soliton of an excitonic BEC, n�x�= ���x��2 �a� and the phase of thewave front function �b� in dimensionless variables. The horizontal dashedlines in �a� show the concentration levels for three transversely one-dimensional distributions—the lower �stable regime�, intermediate �unstableregime�, and upper �stable regime� branches of the hysteresis transferfunction.10
191 J. Opt. Technol. 76 �4�, April 2009
in 1990 by Firth, who showed that modulation instabilitywas possible for it �see the review in Ref. 13�. Dissipativeoptical solitons were later found and studied in a somewhatmore complicated setup consisting of a sample cell with so-dium vapor in a magnetic field and a feedback mirror, ex-cited by coherent radiation.13
A more general system, consisting of a layer of mediumwith feedback, which need not be purely optical and couldinclude additional operations of field rotation, spatial filter-ing, etc., was studied somewhat earlier.�Fig. 6e�.14,15 As faras we know, the first experimentally discovered dissipativeoptical solitons were in a similar liquid-crystal space-modulator system with feedback and spatial filtering.16
Another example is a system with two parallel nonlinearlayers separated from each other and excited from two sidesby different external signals �Fig. 6f�. The optical nonlinear-ity of the medium here can be regarded as local, because ofthe pronounced longitudinal extension of the setup. For asetup with two layers of nonlinear medium, thecalculations17 also show that dissipative optical solitons arepresent.
It is possible in principle to form localized spatial struc-tures when intense radiation is reflected from a semi-infinitelayer of nonlinear medium, but in this case they are actuallymore likely to be observed when two oblique beams are usedfor excitation, or with spatial modulation of the setup param-eters or of the external radiation �Fig. 6g�.5,8
E. Nonlinear interferometers excited by coherent radiation
In these systems, a nonlinear medium is placed inside atwo-mirror Fabry–Perot interferometer �Fig. 6h� or a ringinterferometer �Fig. 6i�, while coherent laser radiation is fedto the interferometer from outside. Because of resonance am-plification of the field intensity inside a high-Q interferom-eter, coherent spatial structures are already implemented at acomparatively low intensity of the external signal. The opti-cal nonlinearity of the medium can also be considered localfor this cavity setup. The nonlinearity itself can be almostanything—Kerr nonlinearity �cubic or with saturation� orwith quadratic nonlinearity of the refractive index, absorp-tion, etc. A point setup �neglecting transverse distributed-ness� was proposed in 1969 and experimentally studied in1975 �see the history of the question in Ref. 4�.
Such a wide aperture setup was proposed and studied inRef. 18, where modulation instability and the possibility ofsuppressing it by means of spatial filtering was demonstratedfor a ring interferometer. Reference 18 also reported switch-ing waves, the influence of intensity inhomogeneity of theincident radiation �spatial hysteresis�, and features of thecase of oblique incidence �with respect to the interferometeraxis� of the radiation—the geometrical deflection of the rayscorresponding to the phase inhomogeneity of the incidentradiation causes the structures to move in the transverse di-rection. It was shown in Ref. 19 that stable localizedstructures—dissipative optical solitons—can be formed in awide-aperture nonlinear interferometer with periodic inten-sity modulation of the incident radiation. Dissipative spatialsolitons in a transversely homogeneous nonlinear interferom-
191N. N. Rozanov
eter were found and studied initially in Refs. 20–22, as wellas in a large number of theoretical and experimental papers,a review of which can be found in severalmonographs.5,8,23–25
This setup is apparently currently being studied in mostdetail both theoretically and experimentally. The theory ofradiation structures is substantially simplified in the mean-field model �averaging of the envelope in the longitudinaldirection�.26 Dissipative solitons can be found numericallyfor an almost arbitrary form of optical nonlinearity, and thisemphasizes their universal character. Their main propertiescan be determined analytically for nonlinearity of the modeltype �threshold nonlinearity27,28,5,8�. A discrete set of singledissipative solitons is thus detected, as well as a discrete setof their coupled structures. Still another important result firstobtained in the framework of this model but having a generalcharacter is the conclusion that the asymmetry of the struc-ture is one of the sources of its motion �see Fig. 10�. A newapproach to nonlinear-optical data processing, based onswitching waves and diffraction solitons and the dynamiccontrol of the system’s architecture by imposing spatialmodulation on it, was also proposed for this setup.29,30 Forwide-aperture nonlinear interferometers, it is possible to de-velop a theory of quantum fluctuations of dissipative solitonsthat predicts that the temporal growth of the indeterminacyof their position is slower than for conservative solitons.Semiconductor microcavities based on quantum wells gavethe experimentally most impressive results for dissipativesolitons.24,25
F. Laser systems
In this section, we consider systems with no externalcoherent signal �although a laser system can be excited fromoutside by coherent radiation, this version does not substan-tially differ from the passive-interferometer case indicatedabove�. Laser systems can be divided into cavity type �Figs.
FIG. 9. The formation of an ultranarrow dissipative discrete molecular solitoby coherent radiation. n is the molecule number, Rn and Zn are their vibrtransition.12
192 J. Opt. Technol. 76 �4�, April 2009
6j and 6k� and noncavity type �amplifier systems of the typeshown in Fig. 2 in which the medium exhibits nonlinear gainand absorption�. Historically, the former �regardless of spa-tial structures� was used to study the cavity system of a laserwith a saturable absorber �inside a two-mirror or ring lasercavity, besides the amplifying medium, a cell was placedwith a medium in which the absorption decreased as theradiation intensity increased�, so that bistability was experi-mentally observed.31 The bistability mechanism is as fol-lows: The system parameters are chosen so that the loss ex-ceeded the gain for weak radiation, as a consequence of
a result of the collision of two switching waves in a molecular chain excitedal amplitude and the population difference of the levels of the resonance
FIG. 10. Transverse intensity profiles for two-soliton �a, c� and three-soliton�b� complexes. The asymmetric complex �c� moves in a transverse directionat constant velocity.5,22
n asation
192N. N. Rozanov
which the nonlasing regime is stable. The absorption satu-rates more rapidly than does the gain as the radiation inten-sity increases, and therefore intense excitation of the lasingbecomes possible. The case of a laser amplifier with satu-rable absorption is similar. There are no mirrors in this sys-tem, and the radiation propagates along a single-mode light-guide �Fig. 2a�, a planar waveguide �Fig. 2b�, or in acontinuous medium �Figs. 2c and 2d�, with the medium con-taining two sorts of particles—active, with pumping andsaturable gain, and passive, with saturable absorption. These
FIG. 11. Transverse intensity distribution I of stable laser solitons with topoof the soliton as a whole, and x and y are transverse coordinates. In the laybetween the cavity mirrors M, and radiation is generated only in the narrooutgoing laser radiation.
FIG. 12. Intensity distribution �a, c� and energy fluxes �b, d� for weakly cotopological charges. With identical charges, a pair possesses central symmcharges, there is a symmetry axis �d, right-hand arrow�, and the structure m
193 J. Opt. Technol. 76 �4�, April 2009
particles can be uniformly dispersed along the radiation-propagation track or can alternate by layers, with the differ-ence between these two cases not being fundamental for suf-ficiently thin layers.
Laser-type dissipative spatial solitons were first found inthe cavity system of a laser with a saturable absorber.32
Analysis showed that, in the simplest version of inertialessnonlinearity, the same equations describe temporal dissipa-tive solitons in gain systems. The classical bistability oftransversely homogeneous distributions in these systems is a
al charge m=0 �1, 4� and m=1 �2, 3, 5–7�. The arrows indicate the rotationof the laser, a medium with nonlinear gain G and absorption A is placed
lindrical region that represents a laser soliton. The white arrows show the
pairs of vortical autosolitons with identical �a, b� and opposite-sign �c, d�and rotates with constant angular velocity �a, large arrow�. With oppositeas a whole in this direction.
logicout 8w cy
upledetryoves
193N. N. Rozanov
necessary condition for the existence of dissipative solitons�solitons can exist in nonlinear interferometers in the absenceof bistability22�.
The absence of external coherent radiation and the cor-responding invariance to phase shift results in further broad-ening of the set of dissipative solitons in laser systems. Vor-tical �topological� solitons with various values of topologicalcharge can thus become stable �Fig. 11, topological charge isdefined as the phase increment of the field when a point withzero intensity travels around a closed contour, divided by2��. The internal structure of single laser solitons contains alarge number of cells with various forms of energy fluxes.These structures make it possible to clearly distinguish casesof weak �Fig. 12� and strong �Fig. 13� interactions of lasersolitons.33 As can be seen from these figures, all the closedlines inherent to individual solitons are maintained when la-
FIG. 13. Intensity distribution �a, c�, energy fluxes �b, d�, and trajectory of th�a and b; the pair rotates around a fixed center of inertia because of central sthe motion of the center of inertia is curvilinear�.
FIG. 14. Intensity distribution �a� and phase distribution �b� for a rotating “vortical soliton weakly coupled with the nucleus, rotating around the nuccomplex.34
194 J. Opt. Technol. 76 �4�, April 2009
ser solitons weakly interact in the energy-flux distribution,but, when they strongly interact, the external individualclosed lines are lost and are replaced by collective ones.
Yet another important point is how laser solitons andtheir complexes move in the transverse direction �in homo-geneous surroundings�. The literature describes fixed, recti-linearly moving, rotating, and curvilinearly moving trans-verse two-dimensional soliton structures.34,35 Symmetryanalysis helps to systematize the soliton mechanics. Here thechoice of the object of symmetry is itself a nontrivial point.For laser solitons, it is insufficient to speak only of the sym-metry of the intensity distribution. On the other hand, therequirement for the field to have phase symmetry is too bur-densome and not quite physical. Instead of the phase, it isexpedient to consider the transverse radiation-energy flux,defined in conditions of small angular divergence of the laser
ter of inertia �e� for pairs of strongly coupled vortical solitons with identicaletry� and opposite-sign topological charges �c–e; there is no symmetry, and
us” made up of two strongly coupled vortical solitons and a “satellite”—aat different instants t; �b� is the trajectory of the center of inertia of the
e cenymm
nucleleus,
194N. N. Rozanov
radiation as the product of the intensity by the �transverse�gradient of the phase of the field. Accordingly, it is necessaryto speak of the simultaneous symmetry of the intensity dis-tribution and the transverse radiation-energy flux. The fol-lowing assertions then follow from the initial controllingequation �the generalized complex Ginzburg–Landau equa-tion for the envelope of the field�:33,36
• If the transverse distributions of the intensity and the en-ergy flux have a common �mirror� axis of symmetry �sym-metry of the first type�, the component of the velocity ofthe center of inertia of the structure, orthogonal to thisaxis, equals zero. The structure can therefore move onlyalong the symmetry axis and cannot rotate. If there are twosuch axes, the structure is fixed.
• If these two distributions have symmetry with respect torotation by an angle of 2� /N, where N is a whole number�symmetry of the second type�, the center of inertia of thestructure �which then coincides with the center of symme-try� is fixed. It remains possible for the system to rotatearound the center of symmetry in this case.
For “rigid” soliton complexes �in which the distancesand phase differences between the solitons are unchanged�,the symmetry completely determines the character of the
FIG. 15. Trajectory of the center of a pair of weakly coupled vortical soli-tons with opposite-sign topological charges when the gradient of the lengthof the cavity is in direction x.38
FIG. 16. �a� In-phase �above� and out-of-phase �below� pairs of “laser bulleaxis; � is the time in a coordinate system moving with the group velocity; �intensity.39
195 J. Opt. Technol. 76 �4�, April 2009
motion. The versions in Figs. 12a and 12b and in Figs. 13aand 13b �central symmetry� serve as examples of rotationalstructures with a fixed center. In the case of Figs. 12c and12d, rectilinear motion corresponds to the presence of onlyone symmetry axis. Finally, in the absence of symmetry ele-ments, the motion is curvilinear �circular trajectory, Fig.13e�. Note that the mechanics of transverse two-dimensionalrigid soliton complexes are described by a phenomenologicalmodel of the type of the Euler equations for the motion of asolid body.37
Of course, soliton complexes can also be nonrigid. Thus,Fig. 14 illustrates the “planetary model,” consisting of anucleus, strongly coupled rotating pairs of vortical solitons,and a satellite, weakly coupled with the nucleus and moreslowly rotating around it.34,35 The trajectory of this stablesystem has the shape of segments, repeated with some rota-tion �Fig. 14c�. A popular explanation of the mechanics oflaser solitons is contained in Ref. 36.
The introduction of weak inhomogeneities into the sys-tem influences the motion of laser solitons and their com-plexes most of all. In this case, symmetry considerations alsooperate, but should now be differentiated into internal andexternal symmetry. The former relates to the intensity distri-bution and radiation-energy fluxes even for a homogeneoussystem, while the latter is determined by the symmetry of theparameters of the system. Thus, tilting the mirror of the lasercavity introduces external anisotropy into a system with asingle symmetry axis. The interaction of internal and exter-nal symmetries produces new types of motion of the solitoncomplexes �see, for example, Fig. 15�.38
Laser solitons can even be three-dimensional �“laser bul-lets”�, which are possible in a continuous medium with non-linear gain and absorption.5,8 There are various complexes oflaser bullets, some of which are shown in Fig. 16–18. Thecollision of moving soliton complexes with the formation ofa rotating structure is shown in Fig. 19.
In actuality, the inhomogeneities of a system not onlyinfluence the motion and properties of the solitons but alsosharply increase the variety of localized structures. Dissipa-tive Bragg lightguide solitons formed in a single-mode light-guide with nonlinear gain and absorption and induced in a
b� a soliton tetrahedron; �c� a soliton pyramid moving along the symmetrya scale showing the phase of the radiation at the 1% level of the maximum
ts;” �d� is
195N. N. Rozanov
lightguide core by a Bragg refractive-index grating serve as aclear example.40,41 Fixed temporal dissipative solitons andsolitons moving with a discrete set of velocities are possiblein the system. The solitons correspond to a local cavity in-duced by nonlinearity on a section of the lightguide, with theintensity of the counterpropagating waves exponentially de-creasing outside the cavity. For a fixed soliton, the local cav-ity is fixed, since the intensity distribution of the counter-
FIG. 17. A “soliton icosahedron”—a complex of twelve weakly coupled“laser bullets” �a� and a “body-centered soliton icosahedron,” consisting ofthirteen bullets �b�.39
FIG. 18. An axisymmetric seven-soliton complex �a� and its trajectory �b�.39
196 J. Opt. Technol. 76 �4�, April 2009
propagating waves is symmetrical relative to the center ofthe cavity. In moving solitons, the intensity distribution ofthe counterpropagating waves is asymmetric �Fig. 20�. An-other example is offered by dissipative discrete solitons in atwo-dimensional system of coupled single-mode lightguidesalso with nonlinear gain and absorption, which can be fixed,moving, or rotating.42
In laser setups with gain and absorption, the stability ofthe localized structures substantially depends on the relation-ship of the relaxation times of the atoms �ions� responsiblefor the gain and saturable absorption �see Ref. 8 and themore detailed analysis in Ref. 43�. As a consequence, newtypes of dissipative solitons become possible that are distin-guished by high peak intensity and high transverse velocity.It can be difficult to choose actual systems in which the
FIG. 19. Dynamics of the collision of two soliton pyramids with the forma-tion of a rotating eight-soliton structure.39
FIG. 20. �a� Intensity profile If ,b�z� of a forward wave �1� and a counter-propagating wave �2�, �b� the phase difference ��z� between these waves fora dissipative Bragg soliton moving along the axis of a lightguide with rela-tive velocity v=0.024.41
196N. N. Rozanov
stability conditions of the solitons would be met, but it be-comes easier for semiconductor lasers with quantum dots.
Dissipative factors can be used to obtain ultrashort laserpulses �femtosecond or even subfemtosecond level�, and thismakes it possible to speak of dissipative attosolitons. Thesetup proposed and studied in Refs. 44–46 corresponds tothe regime of pulses of self-induced transparency with theintroduction of two sorts of particles into a passive medium:active �with pumping� and passive �with nonlinear absorp-tion�. It is also promising to use nanostructures �quantumdots� as particles in this case. The formation of a dissipativeattosoliton from an initial standard femtosecond pulse is il-lustrated in Fig. 21. A “soliton-collider” system, in which thecollision of weak radiation with counterpropagating solitonsresults in a radical transformation of the frequency and in-tensity of laser radiation,47 may serve as one more exampleof the unexpected use of solitons.
Experimental studies of laser solitons are now being car-ried out in a number of laboratories, and the most promisingcase here appears to involve semiconductor lasers with avertical cavity based on quantum dots. One advantage oflaser solitons by comparison, for example, with solitons innonlinear interferometers is the absence of external coherentradiation and zero background of the soliton �the radiationintensity tends to zero at the periphery of a laser soliton�.This provides high contrast when information is recordedwith solitons.
Studies of dissipative optical solitons thus make it pos-sible to detect a new physics of nonlinear-wave-localizationphenomena and to broaden the scale of these phenomena�including dissipative spatial nanosolitons with dimensionsof the order of 1 nm and dissipative temporal attosolitonswith a duration in the subfemtosecond range�. Such an in-crease of the potential of nonlinear-optical phenomena, tak-ing into account the increased stability of dissipative soli-
FIG. 21. Electric-field distribution at various times t when an ultrashortsoliton is formed from a femtosecond pulse: �a� entry into the active me-dium, �b� propagation in a nonlinear resonance medium and formation of thesoliton. The longitudinal coordinate z is normalized to wavelength �, corre-sponding the frequency 21 of the main transition.44
197 J. Opt. Technol. 76 �4�, April 2009
tons, must soon result in a demonstration of new clearadvantages of these solitons, including advantages in dataprocessing. The requirements on informational laser systems,for example, substantially differ from the requirements onordinary lasers for maximizing the power and maximumbrightness of the laser radiation. Therefore, the combinedefforts of theoreticians, experimentalists, and technologists inphotonics, laser physics and engineering, semiconductor het-erostructures, and nanotechnologies are required to imple-ment efficient soliton systems.
The author is grateful to E. B. Aleksandrov and A. A.Mak for support of this specialization. A substantial contri-bution was made to its development by V. E. Semenov, un-timely taken from us, and his deep physical intuition andunsurpassed calculational skill. The author also recognizesG. V. Khodova, A. V. Fedorov, S. V. Fedorov, A. N. Shatsev,N. V. Vysotina, N. A. Kaliteevski�, A. G. Vladimirov, N. A.Veretenov, Al. S. Kiselev, and An. S. Kiselev for fruitfulcollaboration. This research is being supported by grants ofthe Russian Foundation for Basic Research 07-02-00294a,07-02-12164-ofi, 08-02-9012-Mol-a and the Ministry ofEducation and Science, RNP 2.1.1.4694.
1Yu. S. Kivshar’ and G. P. Agraval, Optical Solitons �Fizmatlit, Moscow,2005�.
2N. N. Rozanov, “Autosoliton,” in Great Russian Encyclopedia, vol. 1.�Moscow, 2004�, p. 171.
3V. I. Bespalov and V. I. Talanov, “The filamentary structure of light beamsin nonlinear liquids,” Pis’ma Zh. Eksp. Teor. Fiz. 3, 471 �1966�.
4H. M. Gibbs, Optical Bistability: Controlling Light with Light �AcademicPress, Orlando, 1985; Mir, Moscow, 1988�.
5N. N. Rozanov, Optical Bistability and Hysteresis in Distributed Nonlin-ear Systems �Nauka, Moscow, 1997�.
6N. N. Rozanov, “Hysteresis of the temperature profile during optical ther-mal breakdown of semiconductors,” Pis’ma Zh. Tekh. Fiz. 6, 778 �1980��Sov. Tech. Phys. Lett. 6, 335 �1980��.
8N. N. Rosanov, Spatial Hysteresis and Optical Patterns �Springer, Berlin,2002�.
9Yu. I. Balkare� and A. S. Kogan, “Delamination of the coherent state ofexcitons subjected to light pumping,” JETP Lett. 57, 286 �1993�.
10N. N. Rozanov, S. V. Fedorov, P. I. Khadzhi, and I. V. Belousov, “Dissi-pative solitons of Bose–Einstein condensate of excitons,” JETP Lett. 85,426 �2007�.
11V. Malyshev and P. Moreno, “Mirrorless optical bistability of linear mo-lecular aggregates,” Phys. Rev. A 53, 416 �1996�.
12Al. S. Kiselev, An. S. Kiselev, and N. N. Rozanov, “Nanosize discretedissipative solitons in resonantly excited molecular J-aggregates,” JETPLett. 87, 663 �2008�.
13T. Ackemann and W. J. Firth, “Dissipative solitons in pattern-formingnonlinear optical systems: Cavity solitons and feedback solitons,” in Dis-sipative Solitons, ed. N. Akhmediev and A. Ankiewicz, vol. 661 of Lec-ture Notes in Physics �Springer, Berlin, 2005�, pp. 55–100.
14S. A. Akhmanov, M. A. Vorontsov, and V. Yu. Ivanov, “Large-scale trans-verse nonlinear interactions in laser beams: New types of nonlinear wavesand the appearance of ‘optical turbulence’,” JETP Lett. 47, 707 �1988��.
15S. A. Akhmanov, M. A. Vorontsov, and V. Yu. Ivanov, “The generation ofstructures in optical systems with two-dimensional feedback: on the pathto the creation of nonlinear-optical analogs of neural systems,” in NewPhysical Principles of Optical Information Processing, ed. S. A. Akh-manov and M. A. Vorontsova �Nauka, Moscow, 1990�, pp. 263–325.
197N. N. Rozanov
16A. N. Rakhmanov, “Transverse diffraction structures in systems with op-tical feedback,” Opt. Spektrosk. 74, 1184 �1993� �Opt. Spectrosc. 74, 701�1993��.
17P. V. Pavlov, I. V. Babushkin, N. A. Lo�ko, N. N. Rozanov, and S. V.Fedorov, “Excitation of localized spatial structures of various symmetriesin a system of two thin films,” Izv. Vyssh. Uchebn. Zaved. Radioelektron.10–11, 1000 �2004�.
18N. N. Rozanov and V. E. Semenov, “Hysteresis variations of the beamprofile in nonlinear Fabry–Perot interferometer,” Opt. Spektrosk. 48, 108�1980� �Opt. Spectrosc. 48, 59 �1980��.
19N. N. Rozanov, V. E. Semenov, and G. V. Khodova, “Transverse fieldstructure in nonlinear bistable interferometers. III. Dependence of thebeam profile on the Fresnel number,” Kvantovaya Elektron. �Moscow�10, 2355 �1983� �Sov. J. Quantum Electron. 13, 1534 �1983��.
20N. N. Rozanov and G. V. Khodova, “Autosolitons in bistable interferom-eters,” Opt. Spektrosk. 65, 1373 �1988� �Opt. Spectrosc. 65 810 �1988��.
21N. N. Rosanov, A. V. Fedorov, and G. V. Khodova, “Effects of spatialdistributivity in semiconductor optical bistable systems,” Phys. StatusSolidi B 150, 545 �1988�.
22N. N. Rosanov and G. V. Khodova, “Diffractive autosolitons in nonlinearinterferometers,” J. Opt. Soc. Am. B 7, 1057 �1990�.
23N. N. Akhmediev and A. Ankevich, Solitons: Nonlinear Pulses andBeams �Fizmatlit, Moscow, 2003�.
24N. Akhmediev and A. Ankiewicz, eds., Dissipative Solitons, vol. 661 Lec-ture Notes in Physics �Springer, Berlin, 2005�.
25N. Akhmediev and A. Ankiewicz, eds., Dissipative Solitons: From Opticsto Biology and Medicine, vol. 751 of Lecture Notes in Physics �Springer,Berlin, 2008�.
26L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passiveoptical systems,” Phys. Rev. Lett. 58, 2209 �1987�.
27N. N. Rozanov, “Diffraction switching waves and autosolitons in an inter-ferometer with threshold nonlinearity,” Opt. Spektrosk. 70, 1342 �1991��Opt. Spectrosc. 70, �1991��.
28N. N. Rozanov, “Pairs of associated diffraction autosolitons in an interfer-ometer with threshold nonlinearity,” Opt. Spektrosk. 71, 816 �1991� �Opt.Spectrosc. 71, 475 �1991��.
29N. N. Rozanov and A. V. Fedorov, “Discrete-analog regime of wide-aperture optical bistable systems,” Opt. Spektrosk. 68, 969 �1990� �Opt.Spectrosc. 68, 565 �1990��.
30N. N. Rozanov, “New types of diffraction switching waves and autosoli-tons in nonlinear interferometers,” Opt. Spektrosk. 72, 447 �1992� �Opt.Spectrosc. 72, 243 �1992��.
31V. N. Lisitsyn and V. P. Chebotaev, “Hysteresis and ‘stiff’ action in a gaslaser,” Pis’ma Zh. Eksp. Teor. Fiz. 7, 3 �1968�.
32N. N. Rozanov and S. V. Fedorov, “Diffraction switching waves and au-tosolitons in a laser with saturable absorption,” Opt. Spektrosk. 72, 1394
198 J. Opt. Technol. 76 �4�, April 2009
�1992� �Opt. Spectrosc. 72, 782 �1992��.33N. N. Rozanov, S. V. Fedorov, and A. N. Shatsev, “The structure of energy
fluxes and its bifurcations for two-dimensional laser solitons,” Zh. Eksp.Teor. Fiz. 125, 486 �2004� �JETP 98, 427 �2004��.
34N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion ofmultivortex laser-soliton complexes with strong and weak coupling,”Phys. Rev. Lett. 95, 053903 �2005�.
35N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Two-dimensional lasersoliton complexes with weak, strong, and mixed coupling,” Appl. Phys. B:Photophys. Laser Chem. 81, 937 �2005�.
36N. N. Rozanov, “The world of laser solitons,” Priroda No. 6, 51 �2007�.37N. N. Rozanov, “Phenomenological equations of motion of dissipative
38N. N. Rozanov, S. V. Fedorov, and A. N. Shatsev, “Motion of dissipativesolitons in a laser with smooth transverse inhomogeneity,” Zh. Eksp. Teor.Fiz. 133, 532 �2008� �JETP 106, 459 �2008��.
39N. A. Veretenov, N. N. Rozanov, and S. V. Fedorov, “Clusters of three-dimensional laser solitons and their collisions,” Opt. Spektrosk. 104, 642�2008� �Opt. Spectrosc. 104, 563 �2008��.
40N. N. Rozanov and X. Tr. Chan, “Dissipative solitons in active Bragggratings,” Opt. Spektrosk. 101, 286 �2006� �Opt. Spectrosc. 101, 271�2006��.
41X. Tr. Chan and N. N. Rozanov, “Conservative and dissipative fiber Braggsolitons �a review�,” Opt. Spektrosk. 105, 432 �2008� �Opt. Spectrosc.105, 393 �2008��.
42N. V. Vysotina, N. N. Rozanov, V. E. Semenov, S. V. Fedorov, and A. N.Shatsev, “Rotating discrete dissipative optical solitons,” Opt. Spektrosk.105, 478 �2008� �Opt. Spectrosc. 105, 436 �2008��.
43N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Dissipative solitons inlaser systems with non-local and non-instantaneous nonlinearity,” in Dis-sipative Solitons, ed. N. Akhmediev and A. Ankiewicz, vol. 751 of Lec-ture Notes in Physics �Springer, Berlin, 2008�, pp. 93–111.
44N. V. Vysotina, N. N. Rozanov, and V. E. Semenov, “Extremely shortpulses of amplified self-induced transparency,” JETP Lett. 83, 279 �2006�.
45N. N. Rosanov, V. E. Semenov, and N. V. Vysotina, “Collisions of few-cycle dissipative solitons in active nonlinear fibers,” Laser Phys. 17, 1311�2007�.
46N. N. Rozanov, V. E. Semenov, and N. V. Vysotina, “Ultrashort dissipativesolitons in active nonlinear lightguides,” Kvantovaya Elektron. �Moscow�38, 137 �2008� �Quantum Electron. 38, 137 �2008��.
47N. N. Rozanov, “Transformation of electromagnetic radiation at movinginhomogeneities of a medium,” JETP Lett. 88, 501 �2008�.
48N. N. Rozanov, “Transformation of electromagnetic radiation by rapidlymoving inhomogeneities of a transparent medium,” Zh. Eksp. Teor. Fiz.135, 154 �2009� �JETP 107, 140 �2009��.
