Optical vortex solitons: experiment versus theory

Luther-Davies et al. Vol. 14, No. 11 /November 1997 /J. Opt. Soc. Am. B 3045

Optical vortex solitons: experiment versus theory

Barry Luther-Davies, Jason Christou, and Vladimir Tikhonenko

Laser Physics Centre, Australian Photonics Cooperative Research Centre, Research School of Physical Sciences andEngineering, Australian National University, Canberra, ACT 0200, Australia

Yuri S. Kivshar

Optical Sciences Centre, Australian Photonics Cooperative Research Centre, Research School of Physical Sciencesand Engineering, Australian National University, Canberra, ACT 0200, Australia

Received January 30, 1997; revised manuscript received April 28, 1997

We present a brief overview of recent experimental and theoretical results on optical vortex solitons (in par-ticular, those recently obtained at the Australian National University). Special attention is paid to a directcomparison between the experimental data on the structure and motion of vortex solitons created as localizedstructures in a diffracting Gaussian beam and the theory based on the generalized nonlinear Schrodingerequation. We also analyze, for the first time to our knowledge, the effect of strong nonlinearity saturation onthe transverse (or snake-type) instability of a dark-soliton stripe and show that saturation leads to a drasticsuppression of the instability. © 1997 Optical Society of America [S0740-3224(97)00811-4]

1. INTRODUCTIONSpatiotemporal evolution of light in nonlinear media andstable propagation of optical solitons have been subjectsof considerable theoretical and experimental research innonlinear optics during past years. Spatially localizedsolutions of nonlinear propagation equations in defocus-ing media are dark solitons, which correspond to an inten-sity dip within a uniform background. Temporal darksolitons described by the (1 1 1)-dimensional (one tem-poral and one spatial) nonlinear Schrodinger (NLS) equa-tion have been predicted theoretically1 and subsequentlyhave been observed in optical fibers.2 The spatial analogof these dark solitons were also observed experimentallyas spatial dark-stripe solitary waves in a bulk nonlinearmedium.3 Despite strong similarities between darkstripes and their temporal relatives, the stripe differsfrom the (1 1 1)-dimensional soliton in that it is uncon-strained in (2 1 1) spatial dimensions, whereas the tem-poral soliton is confined by the fiber in higher transversedimensions. Linear stability analysis developed for a de-focusing Kerr medium4 shows that a spatial dark-solitonstripe is unstable to transverse (long-wavelength) modu-lations. The effects produced by this transverse modula-tional (or snake-type) instability have been observed ex-perimentally, and it is well established that in thestrongly nonlinear regime the instability leads to the gen-eration of pairs of optical vortex solitons with alternatingpolarities, either as isolated spots spawned from theedges of the dark-soliton stripes5 or as a sequence of vor-tex solitons as the result of the stripe decay.6–8 The in-stability of the dark-soliton-stripe beams and subsequentvortex generation have been investigated bothnumerically5,9,10 and by means of the asymptotic analyti-cal theory.11

Optical vortex solitons are stable stationary structuresthat exist in a defocusing bulk nonlinear optical medium

0740-3224/97/113045-09$10.00 ©

on a uniform background. Vortex soliton solutions of the(2 1 1)-dimensional cubic NLS equation were introducedand analyzed in the pioneering paper by Pitaevskii12 astopological excitations within superfluids. In the contextof nonlinear optics, they were theoretically suggested bySnyder et al.,13 and experimental observations of opticalvortex solitons have been reported by severalgroups5,14–16 for different optical materials. In experi-ments, however, vortex solitons are of finite total energy,are necessarily nonuniform, and are often created in ma-terials that exhibit saturating or diffusive nonlinearities.Vortex motion is observed to be strongly dependent on theinhomogeneities in the background field. In contrast, thetheory of optical vortex soliton motion, often in a searchfor a precedent in fluid dynamics, has assumed a cubicnonlinearity and a background of constant amplitude(analogous to fluid incompressibility) and has been un-able to capture a number of obvious features of vortex dy-namics, e.g., the drift and rotation of a vortex around abeam—effects that also exist for linear propagation butwhich become dependent on light intensity in the nonlin-ear regime. Understanding this dynamics is importantfor any future application of vortex solitons.

In this paper we present an overview of the experimen-tal results on the generation and dynamics of optical vor-tex solitons (basically those obtained at the Laser PhysicsCentre, Australian National University) and comparethem with the recent analytical results based on the gen-eralized NLS equation that takes into account a back-ground beam of finite extent and non-Kerr nonlinearity.We start our consideration by introducing the model (Sec-tion 2) and then analyzing, for the first time to our knowl-edge, the transverse instability of a dark-soliton stripe ina defocusing (e.g., saturable) non-Kerr medium (Section3). Then, in Section 4, we describe several reported dem-onstrations of optical vortex solitons that are generated

1997 Optical Society of America

3046 J. Opt. Soc. Am. B/Vol. 14, No. 11 /November 1997 Luther-Davies et al.

by special phase masks. We also compare the experi-mentally measured parameters of the vortex soliton withthose obtained from the generalized NLS equation. Fi-nally, we describe experimental results on the vortex driftand rotation and also compare them with the theoreticalpredictions based on the recently introduced analyticalmodel of vortex motion (Section 5).

2. MODEL AND DARK SOLITONSWe consider the propagation of a monochromatic scalarelectric field E in a bulk optical medium with anintensity-dependent refractive index, n 5 n0 1 nNL(I),where n0 is the uniform refractive index of the unper-turbed medium and nNL(I) describes the variation in theindex that is due to the field intensity I 5 uEu2. In theparaxial approximation, Maxwell’s equation can be re-duced to the generalized NLS equation for the slowlyvarying envelope E of the electric field:

22ik0n0]E]Z

1 ¹'2E 1 G~I !E 5 0, (1)

where k0 is the free-space wave number, n0 is the linearrefractive index, I 5 uEu2 is the beam intensity, and ¹' isa gradient operator defined in the transverse plane(X, Y). The function G(I) 5 2k0

2n0nNL(I) describes thecharacter of nonlinearity, and it is determined by theintensity-dependent correction nNL(I) to the refractive in-dex. The latter is usually introduced phenomenologi-cally, and, generally speaking, it should be characterizedby parameters that can then be measured in experiment,such as the Kerr coefficient n2 and the maximum changeof the refractive index for large intensities nmax . In thecase of saturable nonlinearity these values have a simplephysical meaning. Indeed, for I ! Isat , where Isat is thecharacteristic saturation intensity, we can assume a fa-miliar linear dependence known as the Kerr effect,nNL(I) ' 2un2uI, where n2 is the Kerr coefficient of an op-tical material (here we assume a defocusing medium withn2 , 0). For large intensities I @ Isat the refractive in-dex saturates and approaches its maximum value,@nNL(I)#I5` [ nmax . To make explicit calculations and acomparison with the experimental results, below we con-sider a standard model of the nonlinearity saturation:

nNL~I ! 5n2I

1 1 I/Isat, (2)

so that n2 is the Kerr coefficient and nmax 5 n2Isat .Equation (2) describes the refractive-index saturationsimilar to that derived for the standard two-level model.

We assume that for X, Y → ` the modulus of the fieldenvelope E tends to a nonzero value AI0 and look for a so-lution in the form E 5 AI0 exp(ibNLZ)u, where

bNL 5k0un2uI0

1 1 I0 /Isat(3)

is the nonlinearity-induced shift of the background beampropagation constant b 5 b0 1 bNL . The equation forthe function u (uuu → 1 for x, y → 6`) can then be pre-sented in the dimensionless form

i]u

]z1

1

2 S ]2u

]x2 1]2u

]y2 D 5 S 1

1 1 s2

uuu2

1 1 suuu2Du,

(4)

where the parameter s 5 I0 /Isat characterizes the effectof the nonlinearity saturation. The normalized propaga-tion coordinate z and the transverse coordinates (x, y)are measured in the units of k0un2uI0 and (n0un2uk0

2I0)1/2,respectively.

In a general case, NLS equation (1) can be written inthe following normalized form:

i]u

]z1

1

2 S ]2u

]x2 1]2u

]y2 D 1 @F ~q ! 2 F ~ uuu2!#u 5 0,

(5)

where F (q) is proportional to the nonlinearity-inducedshift of the background propagation constant, providedthat q is the background intensity. For the model ofsaturable nonlinearity [Eq. (4)], we have F (uuu2)5 2uuu2/(1 1 suuu2) at q 5 1.

A. Dark-Soliton StripesFor functions F (I) such that F 8(q) . 0, the cw back-ground uuu2 5 q is modulationally stable, and Eq. (5) ad-mits of a (1 1 1)-dimensional dark-soliton solution, adark-soliton stripe. We assume that this plane dark soli-ton is directed along the y axis, so it can be presented inthe one-dimensional form

us~j! 5 U~j!exp@iw~j!#, (6)

where j 5 x 2 vz and two real functions, U(j; v, q) andw(j; v, q), depend on two parameters, the soliton velocityv (or the sinus of the soliton steering angle, in the spatialcase) and the dimensionless background intensity q.These functions satisfy the ordinary differential equa-tions

dw

dj5 vS 1 2

q

U2D , (7)

d2U

dj2 1 v2S U 2q2

U3D 1 2@F ~q ! 2 F ~U2!#U 5 0, (8)

with the boundary conditions U → Aq and w(j) → w06 1/2 Ss as j → 6`, where Ss is the total phase shiftacross the dark-soliton stripe. Equations (7) and (8) canbe always presented in quadratures, and the solution fora dark soliton can be always found in an implicit form.

For v 5 0, solution (6) looks much simpler, us5 U(x), and it describes the so-called black soliton,which in the case of a defocusing nonlinearity has zero in-tensity and p-phase jump at its center, x 5 0.

B. Vortex SolitonsAs is known, in linear optics and acoustics a vortex canappear as a particular mode associated with a wave-frontscrew dislocation that carries a phase singularity of thelinearly diffracting field (see, e.g., Refs. 17–20). In a self-defocusing nonlinear medium this screw dislocation cancreate a stationary beam structure of circular symmetrywith a phase singularity known as a vortex soliton (andsometimes called a black soliton of circular symmetry13).


To find the structure of the stationary solution for thevortex soliton, we look for a solution of normalized NLSequation (5) in the polar coordinates r and u :

u~r, u! 5 U~r !exp~imu!, (9)

where m is the so-called winding number (or the vortexcharge). The modulus function U(r) satisfies the follow-ing boundary-value problem:

d2U

dr2 11

r

dU

dr2

m2

r2 U 1 2@F ~1 ! 2 F ~U2!# 5 0

(10)

for positive r and the boundary conditions

U~0 ! 5 0, U~`! 5 1.

Usually only asymptotics of the function U(r) can befound analytically, so the profiles U(r; m) should be ob-tained by numerical integration of ordinary differentialequation (10), for example, as it was calculated in Ref. 21for the Kerr medium and in Ref. 22 for a saturable me-dium.

3. TRANSVERSE INSTABILITY OF DARK-SOLITON STRIPESA. Asymptotic AnalysisWe are interested in the analysis of stability of a planedark soliton with respect to small transverse perturba-tions, i.e., perturbations localized in the direction of the xaxis but periodic in the direction of the y axis. Thereforewe are looking for solutions to Eq. (5) in the formu 5 us(x 2 vz) 1 c (x 2 vz)exp(l z 1 ipy), where c (x)represents a small localized perturbation with the param-eters l, which describes the propagation evolution of thisperturbation [with the value Re(l) . 0, which defines theinstability growth rate], and p, which is the transversewave number of the periodic perturbation. In the linearapproximation this problem reduces to a construction oflocalized eigenfunctions c (j), corresponding to some ei-genvalue l(p), as a solution of the following linear eigen-value problem:

Lc 5 ~ p2 2 2il!c, (11)

where L is the one-dimensional (complex) linear operatorevaluated at the dark-soliton solution:

L 5]2

]x2 2 2iv]

]x1 2@F ~q ! 2 F ~ uusu2!#

2 2F 8~ uusu2!@ uusu2 1 us2~* !#.

The linear homogeneous equation Lcn 5 0 has four com-plex solutions (n 5 1, 2, 3, 4). Their roles are differentin our analysis. First, there is an exponentially localizedsolution, c1 5 ]us /]x, and the related solution c2 , whichis exponentially diverging for x → 6`. Because of theseeigenfunctions of the homogeneous problem, the inhomo-geneous linear system [Eq. (11)] possesses a constraintthat requires the eigenfunction c (x) to be spatially local-ized:

ilE2`

1`S ]us

]xc* 2

]us*

]xc Ddx

112

p2E2`

1`S ]us

]xc* 1

]us*

]xc Ddx 5 0. (12)

As a matter of fact, this constraint can be treated as thedependence l(p), provided that the profile of the localizedeigenfunction c (x) is given.

Next, there are also two eigenfunctions of the homoge-neous problem that are secular at infinity, i.e., the eigen-function c3 5 ius , which tends to a constant value asx → 6`, and eigenfunction

c4 5 ixus 2]us

]v1

qv

c2

]us

]q,

which grows linearly in x. Here we have introduced thelimiting speed c 5 AqF 8(q), so the dark soliton propa-gates with a velocity less than c. The existence of thesecular eigenfunctions makes analysis of linear problem(11) complicated because solutions of Eq. (11) becomeweakly localized in space. For instance, to apply theasymptotic methods for solving this linear system wehave to introduce two different spatial scales for the func-tion c (x) and have to use the technique of the matchingasymptotic expansions (see, e.g., Refs. 11 and 23).

Let us analyze the problem [Eq. (11)] and construct theapproximate solutions in the limit of small p. Thisasymptotic limit for system (11), but only for the case of aKerr nonlinearity when F (I) ; I, was first considered byKuznetsov and Turitsyn.4 Here we extend this analysisto the case of non-Kerr (or generalized) nonlinearities.

First, to describe the region of small wave numbers weintroduce a formal small parameter e ! 1, making the re-placements l → el and p → ep. Then we seek solutionsto Eq. (11) in the form of the asymptotic series

c~x ! 5 (n50

`

enc~n !~x !.

In the leading order of O(1) we have the homogeneousproblem Lc (0) 5 0, and we choose the localized solutionto this problem as c (0) 5 c1 . In the next order we havethe problem Lc (1) 5 22ilc (0), which admits of an ex-plicit solution in the form

c ~1 ! 5 2l]us

]v1 ac3 ,

where we have taken into account the secular eigenfunc-tion c3 with an undetermined constant a because theterm ]us /]v has nonzero asymptotics as x → 6`.

Using the asymptotic series for c (x) up to the first-order terms in relation (12), we find the dependence l(p)in the leading order (;e 2):

e2S l2]Ms

]v2 p2HsD 1 O~e3! 5 0, (13)

where Ms and Hs are two invariants of the generalizedNLS equation, the renormalized momentum and theHamiltonian, respectively, calculated for the plane darksoliton according to the expressions


Ms 5i

2E

2`

1`S ]us

]xus* 2

]us*

]xusD S 1 2

q

uusu2D dx,

Hs 5 E2`

1`U]us

]x U2

dx . 0.

As is well known (see, e.g. Ref. 23 and references therein),if the derivative ]Ms /]v is positive, a dark soliton isstable with respect to x-dependent (longitudinal) pertur-bations. In this case, it immediately follows from Eq.(13) that dark solitons in a non-Kerr medium are alwaysunstable with respect to transverse perturbations, pro-vided that ]Ms /]v . 0. This means that there exists areal positive value l(p) that defines the exponentiallygrowing solutions and the corresponding instabilitygrowth rate. In the limit of small p we find from Eq. (13)that

l2~ p ! 5 ap2, a 5 HsS ]Ms

]u D 21

. (14)

The instability band is characterized by the maximum(cutoff) modulation wave number pmax , so only long-wavelength modulations, 0 , p , pmax , will lead to in-stability. However, to determine the value pmax for ageneral nonlinearity F (I) would require a much moreelaborate analytical technique, and the correspondinganalytical results will be reported elsewhere.

B. Numerical SimulationsWe carried out simulations of a dark-stripe instability ina saturable medium, using the dimensionless, generalizedNLS equation for propagation [Eq. (4)], which includesthe effect of the nonlinearity saturation. Dark-soliton-stripe solutions to Eq. (4), us(z, x) have been found nu-merically for different values of the dimensionless satura-tion parameter s, as solutions of stationary equations (7)and (8). Here we report the numerical results for themost important case of a black soliton, when v 5 0. Aperturbation in the form of a spectral band of uniform in-tensity and random phase was added to the(1 1 1)-dimensional dark soliton, and the effect on thesubsequent soliton propagation was observed. We thenexamined the evolution of noise (u 2 us) in the trans-verse spatial-frequency domain to determine the growthrates l at different transverse wave numbers p. Figure1 shows the maximum growth rate calculated for trans-verse wave numbers at which the noise was observed togrow at its fastest, as a function of the dimensionlesssaturation s. Measurements were made for a number ofdifferent saturation parameters, and it can be seen thatas the saturability increases from Kerr (s 5 0) behavior,the growth rates tend to decrease. This indicates thatthe breakup of the stripe that is due to noise will beslower and will display larger spatial features as the satu-ration of nonlinearity is increased.

To make a comparison with the theory presented inSubsection 3.B, we carried out direct measurements ofthe long-wavelength instability of the dark-soliton stripeobtained from examining the rate of change of the growthrate with wave number p for small wave numbers. Fig-ure 2 shows both analytical and numerical results that in-

dicate the decrease of this slope with increasing satura-tion, thereby indicating the stabilizing influence ofsaturation on long-wavelength instability.

C. Experimental StudyDark solitons observed in optical fibers2 do not includethe additional dimension, and therefore they are stable,as described by the cubic NLS equation.1 In contrast,since the first experiment for a bulk,3 dark spatial soli-tons have been reported as (1 1 1)-dimensional dark-soliton stripes in a (2 1 1)-dimensional geometry. Asnoted above, such stripes should be unstable to a trans-verse modulational instability that leads to stripebreakup4 and to the eventual creation of optical vortexsolitons.5,10 The instability was avoided in the early ex-periments by the use of finite-sized background beamsand weak nonlinearity.

By increasing the nonlinearity, however, we should beable to observe the transverse instability even with finite-sized beams, as was reported in Ref. 5. Here we discussthe experiments carried out not only to verify the exis-tence of this transverse instability but also to observe thedecay of the dark-soliton stripe and creation of pairs of op-tical vortex solitons. The experiments were performedby Tikhonenko et al.,6 who used a cw Ti:sapphire laser

Fig. 1. Simulations of dark-soliton-stripe instability in a satu-rable medium. The maximum growth rate, corresponding to thetransverse wave number at which energy grew from noise mostquickly, is plotted as a function of the dimensionless saturationparameter s. The solid curve is a simple rational function in-terpolation to show the trend of the data more clearly.

Fig. 2. Slope (small wave numbers) versus saturation. Simu-lation (crosses) and theory (solid curve) for the long-wavelengthchange of growth rate with wave number in a saturable medium.The theoretical curve is determined by Eq. (14).


and a nonlinear medium composed of atomic-rubidiumvapor. Very similar observations, with less evidence ofthe stripe decay into vortex solitons, were made simulta-neously by Mamaev et al.7,8

In the experiments in which rubidium vapor6 was used,the laser output was a linearly polarized slightly ellipticalGaussian beam with a wavelength tuned close to therubidium-atom resonance line at 780 nm. A p-phasejump was imposed across the beam center by a mask, andthe resulting beam was imaged into the nonlinear me-dium. The rubidium-vapor concentration could be in-creased to as much as 1013 cm23 through a change in thecell temperature. Images of the beam at the output ofthe cell were recorded by a CCD camera–frame-capturesystem.

The important step in observing the instability was toresonantly enhance the value of nonlinearity of the me-dium by tuning the laser frequency close (20.4 to21.0 GHz) to the rubidium-atom D2 line and using themaximum vapor pressure consistent with tolerable ab-sorption. The power in the beam at the input face of thecell was 240 mW with a 1/e2 waist of 0.3 mm. A maxi-mum nonlinear refractive-index change of the order of1024 was achieved. For vanishingly small vapor concen-tration, the beam underwent linear propagation throughthe medium shown in Fig. 3(a), right-hand column. Withincreasing temperature (i.e., increasing nonlinearity) theoutput beam developed a vertically uniform dark-solitonstripe, as shown in Fig. 3(b). Further increase in thetemperature led to the growth of a periodic modulation ofthe uniformity of the stripe [Fig. 3(c)]. As the tempera-ture was further increased, the breakup of the stripe be-gan, initially appearing as a growing, snake-type bending[Fig. 3(d)], and then as breaking, with the field coalescinginto dark spots at the inflection points in the bends [Fig.3(e)]. At the highest nonlinearity the dark spot assumedclose to circular symmetry, consistent with the predictedformation of a pair of optical vortex solitons, as can beseen in Fig. 3(f).

The process of breakup was found to be sensitive to thesize of the phase step on the input beam.6 Misalignmentof the phase mask from the optimum (which provided aphase jump closest to p) caused the final stages in evolu-tion of the instability, involving the creation of opticalvortex pairs, to disappear. Under such conditions, onlythe snake-type bending of the soliton stripe was observed.This result supported the theoretical prediction (Subsec-tions 3.A and 3.B) that the growth rate of the instabilityshould be lower for gray-soliton stripes.

Tikhonenko et al.6 also carried out numerical simula-tions based on the generalized NLS equation, includingsaturation and dissipative effects for comparison with theexperimental results. These results are presented inFigs. 3(a)–3(f) (left-hand column). The calculated outputintensity distributions showed the same dynamics ob-served experimentally with increasing nonlinearity. Thebeam defocusing, power depletion, and instability growthrates seen in experiments appeared to be well approxi-mated by the simulations. The observed variation of thedistance between the created vortex solitons can be un-derstood from the results of Subsection 3.B, suggestingthat the distance should vary with the value of the local

background intensity. However, the average period ofthe transverse perturbation corresponding to the maxi-mum growth rate appeared to be smaller than that seenin experiment. This discrepancy is most likely due to thephysically complicated nonlinear response of rubidiumvapor, which was only approximated by the model used inthe simulations, and to the difficulty in accurately char-acterizing the initial field, which was found to sensitivelyaffect the simulations. Note that this sensitivity was notobserved in the experiments, suggesting that the breakupprocess may have been partially stabilized by some physi-cal mechanisms (e.g., nonlocality in the form of diffusion)not included in the model.

Similar experimental results were reported by Mamaevet al.7 (see also Ref. 8, where the instability of a bright-soliton stripe was also presented). In those experiments

Fig. 3. Output beam intensity profiles demonstrating the insta-bility of a dark-soliton stripe as the nonlinearity is increased(based on the results reported in Ref. 6). The vapor concentra-tion increases from small in (a) to the order of 1013 cm23 in (f).Left, results of numerical simulations; right, experimentalresults.6


a biased photorefractive strontium barium niobate crys-tal, irradiated with a 10-mW He–Ne laser beam contain-ing a phase step, was used as the nonlinear medium.Numerical simulations that used the generalized NLSequation with a saturable nonlinearity demonstrated asimilar breakup of the initial stripe into pairs of opticalvortex solitons, as reported by Tikhonenko et al.6 The ef-fectively nonlinearity could be varied by an increase of thebias voltage on the crystal. When zero voltage was ap-plied, the dark stripe spread as the result of diffraction,as did the background beam. As the applied voltage in-creased, the background beam underwent self-defocusing,and a dark-stripe soliton was clearly formed. A furtherincrease in the voltage to 990 V [the maximum voltage re-ported was 1410 V (Ref. 7) and 2000 V (Ref. 8)] led to theappearance of the snakelike bending of the dark-solitonstripe. However, the final state was markedly differentfrom that shown in Fig. 3(f), and it did not present clearevidence of the creation of optical vortex soliton pairs, aswere observed in related numerical simulations. How-ever, the same authors also reported8 the observation ofzeros in the electromagnetic field from interferometricmeasurements of the output beam, with distances be-tween the zeros of ;40 mm. This measurement certainlyindicates that single vortices were being formed.

4. GENERATION OF VORTEX SOLITONSEarly experiments confirmed5 the existence of vortex soli-tons without attempting to describe the observed phe-nomena with an appropriate analytical theory. The im-portant step in bringing about this descriptive gap lay inthe development of effective methods for the creation ofphase singularities that readily generate vortex solitons.In an earlier experiment5 a thermally nonlinear mediumwas irradiated with an argon laser beam that had passedthrough a mask used to impose an approximation on thehelical phase structure of an optical vortex. This maskcontained regions of 0-, p- and 2p- phase thickness sur-rounding a single point in the plane of the mask. At theexit from the medium a dark spot localized at this pointwas observed, and interferometric measurements demon-strated the presence of a phase dislocation at that point,supporting the idea that an optical vortex soliton hadformed. The authors also demonstrated that a(2 1 1)-dimensional induced waveguide existed in the vi-cinity of the vortex by copropagating a He–Ne beam as aguided mode through the induced structure.

A more practical method of creating the helical vortexfield in the input beam was demonstrated by Bazhenovet al.18 By numerically calculating the interference pat-tern between either an on-axis spherical wave or an off-axis plane wave and the optical vortex, they computer-generated diffracting masks that could be used to createsingle optical vortices in the first-order diffracted beamsfrom a Gaussian input. This method was used byLuther-Davies et al.,14 Tikhonenko et al.,6 and Christouet al.24 The advantage of this method is that beams witharbitrarily placed numbers of vortices with selectablechirality (clockwise or anticlockwise) can be generated.The final method of vortex creation recognizes that the far

field of a single vortex is identical to the p 5 1 Gauss–Laguerre mode of an optical resonator.15

A number of other papers have dealt with the creation,dynamics, and waveguiding properties of optical vortexsolitons in saturating nonlinear media. Duree et al.15

provided the first experimental demonstration of the cre-ation of optical vortex solitons in photorefractive media,using the nonlocal response in a biased strontium bariumniobate crystal. They either used an input beam thatwas the coherent doughnut-shaped mode of a laser or con-structed this mode by summing two beams: one with avertical notch and the other with a horizontal notch, withan appropriate p/2 relative phase between them. Bothmethods produced a Gauss–Laguerre beam that pos-sessed the desired azimuthal phase dependence to createan optical vortex soliton at the output of the medium. Inspite of the anisotropy of the nonlinear response in thephotorefractive material, nearly circular solitons were ob-tained, although it was noted that the degree of circular-ity would depend on the input conditions.

Tikhonenko and Akhmediev25 studied the creation ofsingle optical vortices in rubidium vapor (for experimen-tal details see Luther-Davies et al.14), and they comparedexperimental and numerical predictions of the outputfield distributions. They also demonstrated that satura-tion and absorption must be taken into account to yieldagreement. A detailed comparison between the theoret-ical and the experimentally measured vortex profiles in asaturable defocusing medium recently was carried out byTikhonenko et al.22 First, the authors developed an ana-lytical theory to find the stationary, radially symmetriclocalized solitons of generalized NLS equation (10) with asaturating nonlinearity. The vortex profile and diameterdepended strongly on the degree of saturation s, and theFWHM diameter of the vortex increased almost linearlywith s (see Fig. 4). To link the theory, which deals withstationary solutions on an infinite uniform background,and the experiments, in which an input beam with asomewhat arbitrary intensity profile and helical phase is

Fig. 4. Comparison between theory and experiment for the di-ameters of the vortex soliton versus dimensionless saturation s(see Ref. 22). Shown are the experimentally measured beam di-ameter (thick solid curve) and the same value obtained from thetheory based on NLS equation (4) (long-dashed curve). Theshort-dashed curve displays the variation of the dimensionlesspropagation distance.22


used, Tikhonenko et al.22 analyzed the transition fromseveral typical input profiles to a vortex soliton. Forcomparison, the dynamics of the vortex propagation wereinvestigated experimentally, with rubidium vapor as amedium with a variable saturating nonlinearity. Somenovel phenomena were reported, such as rotation of aninitially elliptical vortex core as the soliton formed. Mea-surements of the vortex diameter as a function of the ef-fective saturation showed (Fig. 4) that the almost lineargrowth with s, predicted by the theory, could be observedonly in the region of high saturation where the effectivepropagation distances were long enough to approach thestationary propagation. For lower saturations, the mea-sured vortex diameters were observed to be less than thecorresponding stationary values, as is clearly shown inFig. 4.

5. VORTEX ROTATION, DRIFT, ANDSTEERINGIn earlier experiments by Luther-Davis et al.14 a diffract-ing mask was used to create pairs of like-charged vorticesin a single beam to demonstrate rotation of the vortexsolitons around the beam axis at the output of a nonlinearmedium with increasing nonlinearity. An off-axis vortexrotates around the axis of a Gaussian beam by 90° as itpropagates from the beam waist to infinity.26 This rota-tion exactly matches the so-called Guoy shift that charac-terizes the change of on-axis phase of the Gaussian beamwith propagation relative to that of a plane wave. Thedefocusing action of the nonlinear medium acts to flattenthe phase fronts of the background beam, which reducesthe effective Guoy shift at a given distance. Hence theaction of the nonlinearity is to subtract from the naturalrotation that the vortex experiences during linear propa-gation in a nonuniform Gaussian beam (the effect is ab-sent in a uniform plane wave).

To aid in understanding the effect of nonlinearity onthe vortex motion, Christou et al.24 introduced a simpleanalytical model to describe vortex radial and angularmotion. This analytical model was recently rigorouslyderived by means of the asymptotic analytical approach.27

The analytical model for vortex motion describes the be-havior of the vortex in terms of the transverse gradientsof the background beam at the position of the vortex, pro-vided that these gradients are shallow, with respect to thenonlinear length scale. With R to denote the vortex po-sition vector, Ib to indicate the background beam inten-sity, and ub to indicate the background phase, the theoryfor vortex motion states that24,27

k0n0dRdz

5 2S ¹ub 1m2

CJ¹ ln IbD Ur5R

, (15)

where J is the operator of rotation by p/2, defined by thematrix

J 5 F0 21

1 0 G ,The constant C is a slowly varying function of propaga-tion distance given by

C 5 2lnH cegu¹ ln Ibu

4k0n0@2nnl~Ib!/n0#1/2J .

The transverse drift velocity of a vortex, according tothe model [Eq. (15)], has two components that arise sepa-rately from the phase and intensity gradients of its back-ground field. The first component is directed normal tothe wave front of the background, giving rise to the radialmotion of a vortex in a Gaussian beam. The second com-ponent is directed along the intensity contour (isophote)of the background on which the vortex is positioned, withthe sense of direction given by the vortex charge, m. Fora Gaussian background the isophote in any transverseplane is a circle, so the second component of velocity de-scribes the angular motion of the vortex. The flatteningof the intensity profile under nonlinear action reduces theintensity gradient and thus subtracts from the rotationexperienced by a vortex in linear propagation. For a flat-ter intensity profile (plane waves) the motion of the vortexbecomes more dependent solely on the background wavefront, and results of previous research that examined thisregime19,28 can be recovered.

Fig. 5. Position of the vortex at the output of the cell as a func-tion of the detuning below resonance. The graphs show resultsfor cell temperatures of 88 °C (squares) and 108 °C (crosses).The corresponding behavior predicted by the theory is shown bythe solid curves. Note that, for the higher temperature, the va-por concentration is increased, thereby strengthening nonlinear-ity at all the detunings.27


Experimental results have already confirmed the pre-diction of this model for the radial vortex drift24 even un-der conditions in which the background deviated from theassumed Gaussian form. Here we report the results offurther experiments that have been undertaken to mea-sure both radial and angular motion of a vortex with non-linear propagation and to compare this behavior with anew theory of vortex motion [Eq. (15)]. The nonlinearmedium was, again, a 20-cm-long cell containing atomic-rubidium vapor. A Ti:sapphire laser provided a cwGaussian beam tuned to one of the hyperfine 5s –5p3/2resonances of the rubidium atom, at a wavelength of 780nm, for a strong nonlinear response to propagationthrough the cell. The self-defocusing regime of nonlin-earity was entered by detuning of the laser to the lower-frequency region of the resonance.

We generated the initial condition by imaging the waistof the Gaussian beam onto a phase mask, similar tomasks used in previous investigations of screwdislocations,18,19 thereby introducing a singly chargedphase dislocation into the wavefront. The beam waistand dislocation were then imaged onto the input windowof the nonlinear cell. The beam at the input had a 1/e2

radius of 0.16 mm and a power of 52 mW, with the dislo-cation located 0.11 mm from the beam axis. The imageat the output of the cell was captured with a CCD camera,and the position of the vortex was measured as function ofthe detuning away from the resonance. The results areplotted in Fig. 5 for both an intermediate vapor concen-tration (squares) and a high vapor concentration(crosses).

To test the predictions of this theory against the experi-mental results shown in Fig. 5, we simulated the nonlin-ear evolution of the background and obtained the positionof the vortex at the output of the cell by integrating theequation for vortex velocity over the 20-cm propagationdistance. The model for saturable defocusing nonlinear-ity was chosen as in Ref. 6 (see also Section 2), and modelparameters were obtained for each detuning by the pro-cess of output profile matching. The output positionspredicted by theory are plotted in Fig. 5 for comparisonwith experimental results. As was found previously,24

Fig. 5(a) shows that the radial motion of the vortex in thebeam is well described by Eq. (15). In addition, Fig. 5(b)shows that there is a also good correlation between theoryand experiment with regard to rotation. Discrepanciesin the results arise from transient effects in the beam evo-lution, particularly from the reshaping and radiation as-sociated with the formation of a vortex soliton from aphase dislocation. These effects were not incorporatedinto this essentially asymptotic theory for vortex motion.

The analytical model [Eq. (15)], which predicts vortexmotion on nonplanar background beams, can be useful indeveloping vortex steering methods that employ the influ-ence of the background gradients in the control of vortexposition. Christou et al.24 demonstrated a steeringmethod that used a coherent background wave whose in-tensity was ;1% of that of a Gaussian beam containing avortex. Model (15) was able to quantitatively describethe amplification of vortex displacement from its initialposition, set by the strength of the interfering beam, to its

end position after nonlinear propagation. Also, adjustingthe relative phase of the interfering wave could move theposition of the vortex to any selectable angular position inthe output beam.

6. CONCLUSIONSWe have presented a brief overview of the recent experi-mental results on the modulational instability of a dark-soliton stripe and also on the generation and dynamics ofoptical vortex solitons, dark solitons of circular symmetrythat can exist in a self-defocusing nonlinear non-Kerr me-dium. We have demonstrated the recently developedtheory that accounts for the effects of a nonplanar back-ground and saturation of the medium nonlinearity, per-mitting the capture, qualitatively and even quantita-tively, of a number of interesting features ofexperimentally observed vortex dynamics, e.g., the driftand rotation of a vortex soliton in the background Gauss-ian beam. Understanding this dynamics is important forany future application of vortex solitons in steerable all-optical switching devices based on the concept of lightguiding light.

ACKNOWLEDGMENTSThe authors thank Dmitry Pelinovsky, Len Pismen, andVictoria Steblina for collaboration on this topic and MotiSegev and Allan Snyder for useful and encouraging dis-cussions. This study was conducted within the frame-work of experimental and theoretical research projects onlight guiding light of the Australian Photonics Coopera-tive Research Centre.

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Optical vortex solitons: experiment versus theory