foldingand twisting

waves of lightYURI S. KIVSHAR AND ELENA A. OSTROVSKAYA

OPTICAL VORTICES:

Intensity distribution in an optical beam carrying a vortex (mesh) and its helical wave front (color surface).

In physics, wave propagationis traditionally analyzed bymeans of regular solutions

of wave equations. These solu-tions often possess singularities,the points or lines in space atwhich mathematical quantitiesthat describe physical proper-ties of waves become infinite orchange abruptly.1 For example,at the point of phase singularity,the phase of the wave is undefined andwave intensity vanishes.

Phase singularities are now recognizedas important features common to allwaves. They were first discussed in depthin a seminal paper by Nye and Berry.2

However, the earliest known scientific de-scription of phase singularity was made inthe 1830's by Whewell, as discussed byBerry in Ref. 1. While Whewell studied theocean tides, he came to the extraordinaryconclusion that rotary systems of tidalwaves possess a singular point at which allcotidal lines meet and at which tide heightvanishes. Waves that possess a phase sin-gularity and a rotational flow around thesingular point are called vortices. They canbe found in physical systems of differentnature and scale, ranging from waterwhirlpools and atmospheric tornadoes toquantized vortices in superfluids andquantized lines of magnetic flux in super-conductors.

In a light wave, the phase singularityforms an optical vortex: The wave rotatesaround the vortex core in a given direc-tion; at the center, the velocity of this rota-tion is infinite and the light intensity van-ishes. The study of optical vortices and as-sociated localized objects is extremely im-portant from the viewpoint of both fun-damental and applied physics. Theunique, robust nature of vortex fields isexpected to lead to applications in areasthat include optical data storage, distribu-tion, and processing. Another area inwhich optical vortices could prove usefulis in the establishment of free-space opti-cal interconnects between electronic chipsand boards.3 The ability to use light vor-tices to create robust, reconfigurable pat-terns of complex intensity in an opticalmedium could aid laser cooling by opticaltrapping of particles in a vortex field,4 andcould enable light to be guided by the lightitself, or in other words by the waveguidescreated by optical vortices.5 It is not sur-prising that singular optics, the study of

wave singularities in optics,6 is nowemerging as a new discipline.7

Here, we summarize recent advances inthis exciting field associated with the studyof nonlinear effects that include frequencyconversion, self-trapping of light, and vor-tex solitons.8

Vortices: an overviewIn a broad perspective, the study of opti-

cal vortices brings to light similarities be-tween different and seemingly disparatefields of physics; the comparison of singu-larities of optical and other origins leadsto theories that transcend the confines ofspecific fields. Vortices play an importantrole in many branches of physics, eventhose not directly related to wave propaga-tion. An example is the Kosterlitz–Thou-less phase transition9 in solid-state physicsmodels, characterized by creation of tight-ly bound pairs of pointlike vortices that re-store the quasi-long-range order of a two-dimensional model at low temperatures.Such vortex-induced phase transitions canbe observed in superfluid helium films,

thin superconducting films,and surfaces of solids, as well asin models of interest to particlephysicists and cosmologists. Itis also believed that the study ofvortex generation under a rapidquench could shed new light onthe early stages of the evolutionof the universe.

The Bose–Einstein conden-sate (BEC), a state of matter in

which a macroscopic number of particlesshare the same quantum state, constitutesa well-researched example of a superfluidin which topological defects with a circu-lating persistent current are observed.Nearly 75 years ago, Bose and Einstein in-troduced the idea of condensate of a dilutegas at temperatures close to absolute zero.The BEC was experimentally created in1995 by the JILA group,10 who trappedthousands (later, millions) of alkali 87Rbatoms in a 10-�m cloud and then cooledthem to a millionth of a degree above ab-solute zero. The study of vortices with theBEC11 promises a deeper understanding ofa possible link between the physics of su-perfluidity, condensation, and nonlinearsingular optics.

Wave-front dislocations and optical vorticesTo explain the physics of optical vortices,we recall that a light wave can be repre-sented by a complex scalar function �(e.g., an envelope of an electric field),which varies smoothly in space and/ortime. Phase singularities of the wave func-tion � appear at the points (or lines inspace) at which its modulus vanishes, i.e.,when Re � = Im � = 0. Such points are re-ferred to as wave-front screw dislocationsor optical vortices, because the surface ofconstant phase structurally resembles ascrew dislocation in a crystal lattice, andbecause the phase gradient direction swirlsaround the singular line much like fluid ina whirlpool. Optical vortices are associatedwith zeros in light intensity (black spots)and can be recognized by a specific helicalwave front. If the complex wave functionis presented as, �(r,t) = �(r,t) exp[i� (r,t)],in terms of its real modulus � (r,t) andphase � (r,t), the dislocation strength (orvortex topological charge) is defined bythe circulation of the phase gradientaround the singularity, S = (2�)-1 � ∇ �dr.The result is an integer because the phasechanges by a multiple of 2� (see Fig. 1). It

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Figure 1. Diagram of phase distribution aroundthe core of a single-charge optical vortex. Theblack circles represent a contour plot of vortex in-tensity.

Optical vortices—phase singularities or phase defects in electromagnetic waves—constitute a

unique and fascinating class of phenomena withinthe context of the physics of light, displaying a deepsimilarity to their close relatives, quantized vortices

in superfluids and Bose–Einstein condensates.

also measures an orbital angular momen-tum of the vortex associated with the heli-cal wave-front structure.

If a light wave is characterized by an ex-tra parameter, or, in other words, by wavepolarization, its mathematical representa-tion is no longer a scalar but a vector field.In vector fields, several types of line singu-larity exist; for example, those analogousto disclinations in liquid crystals, whichcould be edge type, screw type, or mixed-edge-screw type, that could move relativeto background wave fronts and could in-teract in several different ways.6,7 In thelinear theory of waves, each wave disloca-tion could be understood as a simple con-sequence of destructive wave interference.

Vortices in nonlinear mediaA laser beam with a phase singularity gen-erally has a doughnutlike shape and dif-fracts when it propagates in a free space.However, when the vortex-bearing beampropagates in a nonlinear medium, a vari-ety of interesting effects can be observed.Nonlinear optical media are characterizedby the electromagnetic response that de-pends on the strength of the propagatinglight. The polarization of such a mediumcan be described as P = �1E + �2E

2 + �3E3,

where E is the amplitude of the lightwave's electric field, and the coefficientscharacterize both the linear and the non-linear response of the medium. The �1 co-efficient describes the linear refractive in-dex of the medium. When �2 vanishes (ashappens in the case of centrosymmetricmedia), the main nonlinear effect is pro-duced by the third term that can be pre-sented as an intensity-induced change of

the refractive index proportional to �3E2.

An important consequence of such inten-sity-dependent nonlinearity is the sponta-neous focusing of a beam that is due to thelensing property of a self-focusing (�3 > 0) medium. This focusing action of anonlinear medium can precisely balancethe diffraction of a laser beam, resulting inthe creation of optical solitons, which areself-trapped light beams that do notchange shape during propagation.8

A stable bright spatial soliton is radiallysymmetric and has no nodes in its intensi-ty profile. If, however, a beam with elabo-rate geometry carries a topological chargeand propagates in a self-focusing nonlin-ear medium, it decays into a number ofbright spatial solitons. The resulting fielddistribution does not preserve the radialsymmetry. Figure 2 shows the experimen-tal data obtained at the Laser Physics Cen-tre of the Australian National University,in which a doughnut-shaped beam with asingle topological charge S = +1 propa-gates in a Rb-vapor nonlinear medium.The vortex beam decays into a pair of out-of-phase solitons that repel and twistaround one another as they propagate.12

The rotation is due to the angular mo-mentum of the vortex transferred to thesoliton pair.

Remarkably, the behavior of a laserbeam in a self-defocusing nonlinear medi-um (�3 < 0) is distinctly different. Such amedium cannot produce a lensing effectand therefore cannot support bright soli-tons. Nevertheless, a negative change ofthe refractive index can compensate for aspreading dip in light intensity, thus creat-ing a dark soliton,8 a self-trapped, local-

ized low-intensity state (a dark hole) in auniformly illuminated background. Fig-ures 3(a) and 3(b) show the formation of adark-soliton stripe created by a �-phasemask in a defocusing medium when self-trapping occurs in the transverse dimen-sion only. The stripe presents an edge dis-location that is however unstable againsttransverse modulations. It subsequentlydecays into localized dark spots or opticalvortex solitons [see Figs. 3(c)-3(f)], as re-ported by Tikhonenko et al.13 and in a re-cent review paper.14 These dark spots ap-pear in pairs inasmuch as each is associat-ed with a helical phase distribution andtwo adjoined vortices carry opposite topo-logical charges.

Similar effects of the stripe break-upand generation of optical vortex solitonscan also be observed as a result of the in-teraction of the stripe with a different op-tical vortex.15 The presence of a vortex in-duces a phase shift in the stripe propor-tional to the vortex topological charge Sthat initiates the stripe's transverse insta-bility and subsequent break-up. In its lin-ear limit, this phenomenon is analogous tothe spirallike distortion of the lines withconstant phase caused by interaction of awave packet with the magnetic vector po-

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28 Optics & Photonics News ■ April 2001

Figure 2.Top: simulation of the break-up of a sin-gle-charge optical vortex in a self-focusing sat-urable nonlinear material.The frames show the ef-fect of increasing nonlinearity on the beam at theoutput of the material. Note that the vortex beambreaks into a pair of off-axis solitons which thenrepel and rotate. Bottom: experimental resultsshowing remarkable agreement with the numericalsimulations.

tential, the famous Aharonov–Bohm effectin mesoscopic physics.

Vortices of single and multiple topo-logical charges can be created in both lin-ear and nonlinear media by use of com-puter-generated holograms.7,8 Propagatingthrough a nonlinear self-defocusing medi-um, such as a vortex-carrying beam, cre-ates a self-trapped state, a vortex soliton.Vortex solitons have been observed exper-imentally in different materials with self-defocusing nonlinearity, such as slightlyabsorbent liquids, vapors of alkali metals,and photorefractive crystals.16

Parametric topology conversionA stable optical vortex that propagates in alinear or nonlinear medium usually re-tains its topological charge or orbital an-gular momentum as well as its helicalwave-front structure. However, certainnonlinear wave interactions in optical me-dia make it possible to transform proper-ties of the vortex field in a controlled man-ner. The most interesting transformationsof this kind occur in optical parametricprocesses, in which, under appropriateconditions imposed on the wave vector ofthe pump and signal beams (phase-matching conditions), multiple new fre-quencies can be generated from a singleinput beam. Depending on the frequencyof the output wave, parametric processescan be either the upconversion (sum-fre-quency generation) type or the downcon-version (fractional frequency generation)type.

One of the simplest parametric upcon-version processes is second-harmonic gen-eration (SHG) in a noncentrosymmetricmedium with quadratic nonlinearity (�2 ≠ 0). In an experiment first conductedby Basistiy and colleagues,17 a singularfundamental wave with a helical wavefront (a vortex) generated a second-har-monic (SH) beam with a double-helixwave front, automatically doubling bothits frequency and its topological charge,S2� = 2S� . The transformation of the vor-tex topology is caused by the conservationof the total orbital angular momentum ofthe coupled harmonic beams in the sum-frequency generation processes.18 In gen-eral, if two input vortices have differentcharges of S1 and S2, the sum-frequencyvortex will have a combined charge ofS1+S2. However, there is still debate overwhat happens to a vortex in parametricdownconversion processes such as, for ex-

ample, degenerate half-frequency conver-sion. There is also debate over whether theconservation law of angular momentum isuniversal.

Parametric instabilities make it difficultto observe vortices in parametric process-es. In the SHG process with a singular fun-damental-frequency beam, for example, alow-intensity SH beam possesses thedoughnutlike shape of the input beam. Athigher intensities, however, the effectivecascade nonlinearity of the medium actsas a self-focusing cubic nonlinearity, andboth input and output vortex beams un-dergo azimuthal instability, subsequentlybreaking up into a set of bright spatial soli-tons.19 Even in the event that the nonlin-earity effectively defocuses in the cascad-ing limit, strong parametric instability ofthe two-frequency beams that carry phasesingularities poses a serious obstacle to ex-perimental observations. A successful ex-perimental demonstration of a stable, lo-calized two-component optical vortex wasrecently reported by Di Trapani et al.20 Theresearchers used the combined effect oftransverse walk-off and finite beam size toeliminate parametric instability and to al-low observation of stable structures withangular momentum in optical wave mix-ing.

Vortex or dipole?In a nonlinear regime, self-trapped opticalvortices exist on a broad backgroundbeam (in a self-defocusing medium) ordecay because of azimuthal instability (ina self-focusing medium). Stabilization of abeam with a finite extension carrying anangular momentum could occur if the op-tical vortex were to copropagate with abright soliton in a self-focusing nonlinearmedium. Through a nonlinear change ofthe medium refractive index, the solitonbeam creates an effective waveguide thatcan trap and guide a copropagating beam.As follows from the linear waveguide theo-ry, which is applicable when a trappedbeam is weak, guided modes of differentshapes can be trapped by such a soliton-induced radially symmetric waveguide.However, as the intensity of the trappedbeam grows, the guided mode can nolonger be considered linear, affecting thewaveguide itself because of nonlinear in-teraction with the soliton beam. As a re-sult, both beams form a composite (orvector) soliton with mutually trappedcomponents of a complex shape. The exis-

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April 2001 ■ Optics & Photonics News 29

Figure 3. Output beam intensity profiles demon-strating the instability of a dark-soliton stripe withthe growing input intensity in the experiments withthe Rb vapor. The vapor concentration increasesfrom small in (a) to the order of 1013 cm-3 in (f).

(a)

(b)

(c)

(d)

(e)

(f)

tence of such composite soliton structuresthat carry an angular momentum (thespin of the beam) would allow observa-tion of an optical version of the spin-or-bital interaction of light waves.21

Two examples of such composite soli-tons are shown in Fig. 4. The first featuresa component with the doughnut-shapedstructure characteristic of the Laguerre–Gaussian (LG0

1) vortexlike mode of a lin-ear radially symmetric optical waveguide,which carries an angular momentum.21

The second type of vector soliton origi-nates from use of a soliton-induced wave-guide to trap a dipolelike Hermite-Gauss-ian (HG01) mode.22 The two lobes of thedipole-mode component have a relativephase difference of �. Contrary to whatmight have been assumed before publica-tion of the most recent research on thistopic,22 the radially asymmetric dipole-mode soliton is more stable than the radi-ally symmetric vortex-mode soliton. Thelatter, in fact, undergoes a nontrivial sym-metry-breaking instability and transformsinto a rotating dipolelike structure that re-sembles two spiraling solitons. This rota-tion is caused by the angular momentumimparted by the decaying vortex mode to

its residuals. In contrast to its unstablecounterpart, a dipole-mode soliton is a ro-bust—albeit complex—object that can belikened to a molecule of light. It can pre-serve its structural integrity in collisionswith other molecules and atoms and withscalar soliton beams, or can display morecomplicated dynamics that involve the ex-citation of the molecular degrees of free-dom.

Conclusion We have presented an overview of excitingresearch in the field of nonlinear singularoptics. It is clear that understanding andcontrolling the properties of optical vor-tices could lead to numerous applicationsin the future, ranging from optical com-munications and data storage to the trap-ping, control, and manipulation of parti-cles and cold atoms. Indeed, optical vor-tices provide an efficient way to controllight by creating reconfigurable waveguidesin bulk media. The study of phase singular-ities in optical parametric processes notonly suggests new directions of fundamen-tal research in optics but also provides linksto other branches of physics. For example,

the recent discovery of a rich variety of ex-otic topological defects in unconventionalsuperfluids (such as 3He-A) and supercon-ductors points to the likelihood that deepanalogies exist between vortices in com-plex superfluids and multifrequency lightwaves. Recent theoretical analysis23 of two-color parametric optical vortices hasdemonstrated that such vortices possessunexpected properties that make themeven more exotic than quantized vorticesin unconventional superfluids.AcknowledgmentsWe thank many of our colleagues for theircollaboration and useful discussions, mostespecially Barry Luther-Davies, Morde-chai Segev, Marat Soskin, Lluis Torner,Paolo Di Trapani, and Vladimir Tikho-nenko for sharing their knowledge with usand for their constant support. This re-search was supported by the Performanceand Planning Fund and the AustralianPhotonics Cooperative Research Centre.

References1. M. Berry, "Making waves in physics," Nature (Lon-

don) 403, 21 (2000).2. J.F. Nye and M.V. Berry, "Dislocations in wave

trains," Proc. R. Soc. London Sect.A 336, 165-90(1974).

3. J. Scheuer and M. Orenstein, "Optical vortices crys-tals: Spontaneous generation in nonlinear semicon-ductor microcavities," Science 285, 230-3 (1999);see also the special feature in Photon.Technol.News, No. 11, 1999 (www.photonics.com).

4. K.T. Gahagan and G.A. Swartzlander, "Simultaneoustrapping of low-index and high-index microparticlesobserved with an optical-vortex trap,'' J.Opt. Soc.Am. B 16, 533-7 (1999).

5. See, for example,A.G.Truscott, M. E. J. Friese, N. R.Heckenberg, and H. Rubinsztein-Dunlop,"Opticallywritten waveguide in an atomic vapor,"Phys. Rev.Lett. 82, 1438-41 (1999); C.T. Law, X. Zhang, andG.A. Swartzlander, "Waveguiding properties of opti-cal vortex solitons," Opt. Lett. 25, 55-7 (2000).

6. J. F. Nye, Natural Focusing and Fine Structure ofLight: Caustics and Wave Dislocations (Author: isthis the name of a book published by theIOP?) (Institute of Optics, Bristol, England, 1999);M.Vasnetsov and K. Staliunas, eds., Optical vortices(Nova, New York, 1999).

7. M. Soskin and M.Vasnetsov, "Singular optics as anew chapter of modern photonics," Photonics Sci.News 4, 22-42 (1999).

8. Y.S. Kivshar and B. Luther-Davies, "Optical dark soli-tons: physics and applications," Phys. Rep. 298, 81-197 (1998).

9. J.M. Kosterlitz and D.J.Thouless, "Ordering, metasta-bility and phase transitions in two-dimensional sys-tems," J. Phys. C 6, 1181-203 (1973).

10. M. H.Anderson, J. R. Ensher, M. R. Matthews, C. E.Wieman, and E.A. Cornell, "Observation of Bose-Einstein condensation in a dilute atomic vapor," Sci-ence 269, 198-201 (1995).

11. J. E.Williams and M. J. Holland, "Preparing topologi-cal states of a Bose-Einstein condensate," Nature(London) 401, 568-72 (1999); M.R. Matthews, B. P.Anderson, P. C. Haljan, D. S. Hall, C. E.Wieman, andE.A. Cornell, "Vortices in Bose–Einstein conden-sate," Phys. Rev. Lett. 83, 2498-501 (1999).

12. V.Tikhonenko, J. Christou, and B. Luther-Davies,"Spiraling bright spatial solitons formed by the

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30 Optics & Photonics News ■ April 2001

Figure 4. Left: intensity components of a composite vortex-mode soliton in a self-focusing medium (thesecond beam carries a vortex). Right: intensity components of a composite dipole-mode soliton.

breakup of an optical vortex in a saturable self-fo-cusing medium," J. Opt. Soc.Am B 12, 2046-53(1995).

13. V.Tikhonenko, J. Christou, B. Luther-Davies, and Y. S.Kivshar, "Observation of vortex solitons created bythe instability of dark soliton stripes," Opt. Lett. 21,1129-31 (1996).

14. Y. S. Kivshar and D. E. Pelinovsky, "Self-focusing andtransverse instabilities of solitary waves," Phys. Rep.331, 117-95 (2000).

15. Y. S. Kivshar,A. Nepomnyashehy,V.Tikhonenko, J.Christou, and B. Luther-Davies, "Vortex-stripe soli-ton interaction," Opt. Lett. 25, 123-5 (2000).

16. See, for example, G.A. Swartzlander and C.T. Law,"Optical vortex solitons observed in Kerr nonlinearmedia," Phys. Rev. Lett. 69, 2503-6 (1992);A.V. Ma-maev, M. Saffman, and A.A. Zozulya, "Vortex evolu-tion and bound pair formation in anisotropic non-linear optical media," Phys. Rev. Lett. 77, 4544-7(1996); Z. Chen, M. Segev, D.W.Wilson, R. E. Muller,and P. D. Maker,"Self-trapping of an optical vortex byuse of the bulk photovoltaic effect," Phys. Rev. Lett.78, 2948-51 (1995).

17. I.V. Basistiy et al., "Optics of light beams with screwdislocations," Opt. Commun. 103, 422-8 (1993).

18. K. Dholakia, N. B. Simpson, M. J. Padgett, and L.Allen,"Second-harmonic generation and the orbital angu-lar momentum of light," Phys. Rev.A 54, R3742-5(1996).

19. L.Torner and D.V. Petrov, "Splitting of light beamswith spiral phase dislocations into solitons in bulkquadratic nonlinear media," J. Opt. Soc.Am B 14,2017-23 (1997); D.V. Petrov, L.Torner, J. Martorell, R.

Vilaseca, J. P.Torres, and C.Cojocaru, "Observationof azimuthal modulational instability and formationof patterns of optical solitons in a quadratic nonlin-ear crystal," Opt. Lett. 23, 1444-6 (1998).

20. P. Di Trapani,W. Chinaglia, S. Minardi,A. Piskarskas,and G.Valiulis,"Observation of quadratic opticalvortex solitons," Phys. Rev. Lett. 84, 3843-6 (2000).

21. Z.H. Musslimani, M. Segev, D. N. Christodoulides,and M. Soljacic, "Composite multihump vector soli-tons carrying topological charge," Phys. Rev. Lett.84, 1164-7 (2000); J.N. Malmberg,A. H. Carlsson, D.anderson, M. Lisak, E.A. Ostrovskaya, and Y. S.Kivshar, "Vector solitons in (2+1) dimensions," Opt.Lett. 25, 643-5 (2000).

22. J. J. Garcia-Ripoll,V. M. Perez-Garcia, E.A. Ostro-vskaya, and Y. S. Kivshar, "Dipole-mode vector soli-tons," Phys. Rev. Lett. 85, 82-5 (2000);W. Krolikows-ki, E.A. Ostrovskaya, C.Weilnau, M. Geisser, G. Mc-Carthy,Y. S. Kivshar, C. Denz, and B. Luther-Davies,"Observation of dipole-mode vector solitons,"Phys. Rev. Lett. 85, 1424-7 (2000).

23. T. J.Alexander,Y. S. Kivshar,A.V. Buryak, and R.A.Sammut, "Optical vortex solitons in parametricwave mixing," Phys. Rev. E 61, 2042-9 (2000).

Yuri S. Kivshar and Elena A. Ostrovskaya are with the Research School of Physical Sciences and Engi-neering at the Australian National University,Canber-ra, Australia. They can be reached by e-mail atysk124@rsphysse.anu.edu.au

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