Ivan D. Maleev and Grover A. Swartzlander, Jr- Composite optical vortices

8/3/2019 Ivan D. Maleev and Grover A. Swartzlander, Jr- Composite optical vortices

1/8

Composite optical vortices

Ivan D. Maleev and Grover A. Swartzlander, Jr.

Optical Sciences Center, University of Arizona, Tucson, Arizona 85721

Received October 7, 2002; revised manuscript received February 19, 2003

Composite optical vortices may form when two or more beams interfere. Using analytical and numerical tech-niques, we describe the motion of these optical phase singularities as the relative phase or amplitude of twointerfering collinear nonconcentric beams is varied. The creation and the annihilation of vortices are found,as well as vortices having translational velocities exceeding the speed of light. 2003 Optical Society ofAmerica

OCIS codes: 350.5030, 260.3160, 260.0260, 030.1670, 030.6140.

1. INTRODUCTION

Optical vortices1 (OVs) or phase defects often occur incoherent radiation, e.g., in LaguerreGaussian laserbeams,2 light scattered from rough surfaces,3,4 opticalcaustics,5,6 and OV solitons.7,8 The center of the vortex ischaracterized by a dark core, within which the intensity

vanishes at a point, assuming the beam is coherent.9,10

The phase front of an OV is helical, and thus the wavevectors have azimuthal components that circulate aroundthe core.11 Owing to this circulation, the optical wavecarries orbital angular momentum.12 In general, whenone beam is superimposed with another, the phase of thecomposite field differs from that of the component beams.Being a topological phase structure, an optical vortex isreadily affected when it is combined with other fields.The new or composite vortex may be repositioned inspace or annihilated, and other vortices may spontane-

ously form in the net field.

1315

The velocity at which a vortex actually moves depends on how quickly param-eters such as the relative phase or amplitude can be var-ied. For example, the rate of change of the amplitudecan be so large for an ultrashort pulse of light that the

vortex speed exceeds that of light. This finding does notviolate the principles of relativity because the vortex ve-locity is a phase velocity.

An interest in OVs traces back to 1974, when Nye andBerry showed that coherent waves reflected from roughsurfaces contain phase defects; namely edge, screw, andphase dislocation.3,5 Practical interest was sparked byLukin et al., who considered phase fluctuations of lightbeams in the atmosphere.16 In 1981, Baranova et al.4

showed that laser speckle contains a large number of ran-domly distributed OVs. In the 1990s, Coullet et al.17

stimulated an interest in nonlinear OVs in laser cavities,while Swartzlander and Law experimentally and numeri-cally discovered the OV soliton in self-defocusing media.7

The latter group was also the first to describe the creationof optical vortices by destabilizing dark-soliton stripes.7,18

Soskin et al.1315 laid groundwork for new experimentalmethods in linear and nonlinear OV phenomenon.Staliunas,19 Indebetouw,20 and Freund21 established anunderstanding of vortex propagation. Owing to theirparticlelike nature, OVs were said to repulse or attract

each other,22 or to split, annihilate, or be born23 as a com-plex system of vortices.6 The propagation dynamics ofoptical vortices was clarified by Rozas et al.24 The fluid-like rotation of propagating optical vortices around a com-

mon center was discussed by Roux25

and observed for thespecial case of small-core OVs by Rozas et al.26 Most re-cently, the phase of globally linked vortex clusters hasbeen described.27

Thus far, the study of beam combinations has been lim-ited to the coherent coaxial superposition of severalbeams.1 For example, Soskin and Vasnetsov showed thata coherent background field changes the position of an OVand can lead to the destruction and creation of vortices.Recent investigations of the presence of incoherent lightwithin the vortex core has been reported.9,10

In this paper, we describe OVs in the superimposedfield of two parallel, noncollinear beams. We map the po-sition and topological charge of the OVs as the relative

phase or distance between the two beams is varied. Forconsistency with experimental approaches, we numeri-cally generate interferograms that allow the determina-tion of the position and topological charge of vortices inthe composite field.

2. SINGLE OPTICAL VORTEX

An optical vortex (Fig. 1) is essentially a phase object, andthus it is necessary to assume that the beam containing itexhibits transverse coherence (at least in the proximity ofthe vortex core). A single optical vortex placed at thecenter of a cylindrically symmetrical coherent beam maybe represented by the complex electric field in the trans-

verse (x , y) plane:

Er, A expi fr expim, (1)

where (r, ) are circular transverse coordinates in the(x r cos , y r sin ) plane (we assume the vortex corecoincides with the origin, r 0), A is a measure of thefield amplitude (assumed to be real), is an arbitraryphase constant, f(r) is a real function representing theprofile of the beam envelope, and m is the topologicalcharge of the vortex (a signed integer). The amplitudeand the phase (A and ) may vary with time. Note that

I. Maleev and G. Swartzlander Vol. 20, No. 6 /June 2003 /J. Opt. Soc. Am. B 1169

0740-3224/2003/061169-08$15.00 2003 Optical Society of America


2/8

the phase factor, exp(im), is undefined at the vortex-coreposition, r 0 [Fig. 1(b)]. For this reason, a vortex issometimes called a phase singularity. In three dimen-sions, the surface of constant phase is a construction ofm helices having a common axis and a pitch equal to thewavelength, . For the special case m 0, the wavefront is planar. In principle, any vortex of nonzero topo-logical charge, m, may be treated as the product of m vor-tices, each having a fundamental charge of 1. Thus forsimplicity, we consider only beams with fundamental val-ues, m 1, 0, or 1. Below, we make use of the rela-tion between the amplitude, A, and power, P, of the beamin Eq. (1):

A P 0

rf 2r dr 1/2

. (2)

Owing to total destructive interference, a characteristicof an optical vortex is a point of zero intensity in the dark

vortex core. Analytically, the position of this zero may befound by considering that both the real and the imaginaryparts of the field must vanish:

ReE Afrcosm Ar 1frx 0, (3a)

ImE Afr sinm Ar

1fry 0, (3b)

where (x, y) (r cos(m ), r sin(m )) are ro-tated coordinates. Both Eqs. (3a) and (3b) vanish identi-cally at the origin for any physically meaningful function,

gr w0 /r fr, (4)

where w0 is the characteristic beam size. In this paper,we restrict ourselves to Gaussian beam profiles: g(r) exp(r2/w0

2).To detect optical vortices in the laboratory, it is not suf-

ficient to simply identify the points of zero intensity. In-stead, one must obtain information about the phase to

verify that it is spatially harmonic (when satisfied, this

condition also satisfies the requirement that the intensityvanishes at the core). Interferometry allows us to do thiswhile also determining the location and the topologicalcharge. The interferogram of a vortex displays a charac-teristic forking pattern [Fig. 1(c)]. For example, whenthe vortex field described by Eq. (1) is interfered with aplanar reference wave given by E A exp(ikxx), wherekx is the transverse wave number of the tilted planewave, the interferogram has a profile given by

E E2 A2f2 A2 2AA fcosm kxx .(5)

Far from the center of the vortex, the interferogram dis-plays lines of constant phase that are nearly parallel tothe y axis. At the center of the core (the origin), the in-tensity of the interferogram has a value of A2. Above(x 0, y 0) and below (x 0, y 0) the phase singu-larity, the value of the factor cos(m kxx) switchessign, thereby creating a forking pattern. In practice, theapproximate locations of vortices are found by locatingthe vertex of these forks, and the precise position is found

by removing the reference wave and determining the po-sition of minimum intensity.

3. COMPOSITE VORTICES

When two or more beams described by Eq. (1) and possi-bly having different centroids are superimposed, the re-sultant zero-intensity points in the composite beam do notgenerally coincide with those of the individual beams.We expect the new composite vortices, being phase ob-

jects, to depend on the relative phase and amplitude ofthe individual beams. Let us first examine the superpo-sition of two mutually coherent beams containing vorticesunder steady-state conditions. The composite field isgiven by

Er, j1

2

Ajgrjrj/w0mj expimjjexp ij,

(6)

where (rj , j) are the transverse coordinates measuredwith respect to the center of the jth beam, and j is thephase of the jth beam. In the laboratory, the location ofthe composite vortices may be found with interferometry,as discussed above, where the intensity profile of the in-terferogram is given by

E A expikxx 2

E 2 A2

2EA cos kxx , (7)

where arctan(ImE/ReE). As we saw in the previ-ous section, the intensity of the interferogram at the loca-tion of a composite vortex has a value given by A2 (sinceE 0), and forking patterns are expected around thesingular points, where is undefined.

For convenience, we consider two beams displacedalong the x axis by a distance s from the origin, asshown in Fig. 2, where beam 1 and beam 2 appear on theright and the left, respectively. The relative displace-ment compared with the beam size is defined by

Fig. 1. Single optical vortex of charge m 1 in a Gaussian

field. (a) Intensity profile. (b) Phase profile showing a singularpoint at the origin. (c) Interferogram showing a forking patternin the vicinity of the vortex.

Fig. 2. Two beams of radial size w 0 separated by a distance 2s.Bipolar coordinates are labeled.

1170 J. Opt. Soc. Am. B /Vol. 20, No. 6 /June 2003 I. Maleev and G. Swartzlander


3/8

(s/w0). One may readily verify that the bipolar co-ordinates (r1 , 1 , r2 , 2) are related to the circular coor-dinates (r, ) by the relations

r1 expi1 r expi s, (8a)

r2 expi2 r expi s. (8b)

Without loss of generality, we define the relative phase, 1 2 , and set 2 0. In many cases the sym-metry of the phase-dependent vortex motion will allow usto simply consider values of between 0 and . Let usnow consider three representative cases of combinedbeams, where the topological charges are identical ( m1 m2), opposite (m1 m2), and different (m1 1,m2 0) .

Case 1

If we superimpose two singly charged vortex beams hav-ing identical charges, say, m1 m 2 1, then the com-posite field in Eq. (6) may be expressed

Er, A1g1r1expi A2g2r2 r/w0expi

A1g1r1expi A2g2r2 s/w0. (9)

The intensity profiles, E 2, for various values of phaseand separation are shown in Fig. 3. Setting the real andthe imaginary parts of Eq. (9) to zero, we find composite

vortices located at the points (x, y) that satisfy the tran-scendental equation

x iy s tanh22x/s 1/2lnA1 /A2 i/2.(10)

Several special cases of Eq. (10) can be readily identifiedand solved.

Let us first consider the in-phase and equal-amplitudecase: 0 and A1 A2 , which allows solution onlyalong the x axis (see the top row in Fig. 3). A solution ofEq. (10) exists at the origin for any value of ; however,

we find its sign changes from positive to negative when increases beyond the critical value:

cr 21/2. (11)

What is more, for cr , two additional m 1 vorticesemerge. One may easily verify this critical value by ex-panding the hyperbolical tangent function in Eq. (10) tothird order, assuming x/s 1, and thereby obtaining

x/s 322 1/861/2. (12)

For well separated beams, cr , a first-order expan-sion of Eq. (10) indicates that the composite vortices aredisplaced toward each other (x/s 1), and, as expected,they nearly coincide with the location of the original vor-

tices:

x/s 1 2 exp42x/s . (13)

Let us next consider the out-of-phase, equal-amplitudecase: , A1 A2 (see bottom row in Fig. 3). In thiscase, we find two vortices for all values of , with an ad-ditional vortex at y . For the special case 0,total destructive interference occurs, and the compositefield vanishes. Again, we find the solutions are con-strained to the x axis, but now the positions are given by

x/s coth22x/s. (14)

The vortices are displaced away from each other and their

original position ( x/s

1), and for

cr ,x/s 1 2 exp42x/s . (15)

These analytical results and the numerically calculatedintensity profiles in Fig. 3 demonstrate that the vortexcore may be readily repositioned by changing the relativeposition or phase of the component beams. From an ex-perimental and application point of view, the later varia-tion is often easier to achieve in a controlled linear fash-ion. For example, the net field may be constructed fromtwo collinear beams from a MachZehnder interferom-eter, whereby the phase is varied by introducing an opti-cal delay in one arm of the interferometer.

A demonstration of the phase-sensitive vortex trajec-

tory is shown in Fig. 4 for the case 0.47 (i.e., cr). We find that a critical value of phase existssuch that, for cr , there is a single composite vortexof charge m 1, and it resides on the y axis. Above thecritical phase, two m 1 vortices split off from the y axis,and, simultaneously, one m 1 vortex remains on the yaxis. The value of the topological charge may be foundby examining numerically generated phase profiles or in-terferograms of Eq. (9). The positively charged vortices(see the black dots in Fig. 4) circumscribe the position ofthe original vortices (indicated by isolated gray dots inFig. 4).

Fig. 3. Phase () and separation ( s/w 0) dependent inten-sity patterns of a composite beam created by superimposing twoequal amplitude m 1 vortex beams. Coaxial componentbeams ( 0) uniformly and destructively interfere. For 0, one or three vortices form when the phase is below orabove a critical value. The third vortex leaves the beam region.



4/8

The net topological charge is mnet 1 for all values of, and, being unequal to the sum of charges of the originalbeams (msum 2), is therefore not conserved in the sensethat mnet msum . From a practical point of view, how-ever, we note that the m 1 vortex escapes toward y when . Furthermore, the field amplitudesurrounding the escaping vortex is negligible and istherefore unobservable. Empirically, this leads one tothe observation that the net topological charge is some-times conserved in a weak sense.

Phase-dependent vortex trajectories for different val-ues of the separation parameter, , are shown in Fig. 4(b).For cr , a single m 1 vortex resides on the y axiswhen cr ; above the critical phase, an m 1 vor-tex continues moving on the y axis toward infinity, whiletwo m 1 vortices form symmetric paths that circum-scribe the positions of the original vortices. For cr , three vortices always exist, with one havingcharge m 1 constrained to the y axis and the othertwo having charge m 1 circling the original vortices.

As discussed above for Fig. 4(a), the net topological chargeis never strictly conserved for the cases in Fig. 4(b).

The value ofcr may be estimated from Eq. (10) by ex-panding the hyperbolical tangent function in the vicinityof x/w0 0. To first order, we find

221 tan2cr/2 1. (16)

For 0.50, we obtain cr 90, in good agreementwith the numerically obtained value.

A qualitative understanding of the composite vortexgeneration can be formed by examining Fig. 3. When 0, we have the simple interference between beamshaving different phases. When 0.63, we find anelongated composite beam having a single vortex when 0, but the emergence of two m 1 and one m 1 vortices when cr .

Case 2For two oppositely charged vortices ( m1 m 2 1) ,Eq. (6) becomes

Fig. 4. Phase-dependent composite vortex trajectories resultingfrom the superposition of two equal-amplitude m 1 componentbeams. (a) 0.47 showing a single m 1 vortex for cr , and two m 1 and one m 1 for cr . Arrowsindicate the direction of motion for increasing values of. Dot-ted curves demark the footprints of the component beam waists.(b) Family of trajectories for different values of . (c) A third vortex appears in all cases when exceeds a separation-

dependent critical value. For

cr

2

1/2

, three vortices al-ways exist, and the trajectory of the two m 1 vortices becomeseparate closed paths.



5/8

Er, A1g1r1expiexpi

A2g2r2expi r/w0

A1g1r1expi A2g2r2 s/w0.

(17)

The intensity profiles for different values of phase andseparation, shown in Fig. 5, are qualitatively different

than those for case 1 (see Fig. 3). To determine the posi-tion of vortices, we set Eq. (17) to zero. It happens thatthe zero-valued field points for both the in-phase and out-of-phase cases, 0 and , must satisfy the equa-tions

x s tanh22x/s 1/2lnA1 /A2 i/2, (18a)

y sinh22x/s 1/2lnA1 /A2 i/2 0. (18b)

For the in-phase equal-amplitude case ( 0, A1 A2), there are zeros that do not correspond to a vortexbut rather to an edge dislocation (i.e., a division betweentwo regions having a phase difference of). The first rowin Fig. 5 shows, for different values of, the dislocation as

a black line bisecting two halves of the composite beam. Vortices do not appear in this in-phase case unless cr [see Eq. (11)], whence they occur on the x axis andare described by the same limiting relations as we foundin case 1 [i.e., Eqs. (12) and (13)]. The out-of-phase equalamplitude ( , A1 A2) solutions are also identicalto those found in case 1 [see Eq. (14)].

The composite vortex positions in case 2 are generallydifferent than case 1 for arbitrary values of and . Forexample, in case 1 we found either one or three vortices,while in case 2 there are always two vortices. Whereasthe vortex positions in case 1 appeared symmetricallyacross the y axis for cr, they appear at radiallysymmetric positions in case 2 for all values of. Finally,we note that, when 0 in case 2, the two original vor-tices interfere for all values of to produce an edge dislo-

cation, as shown in the left-hand column of Fig. 5,whereas in case 1 the beams uniformly undergo progres-sive destructive interference.

The phase-dependent vortex trajectory for 0.47 isshown in Fig. 6(a). Indeed, it differs from the trajectoryfor case 1 shown in Fig. 4(a). At (where 1), wefind in Fig. 6(a) an m 1 vortex at a point on the positive

y axis and an m 1 vortex at the symmetrical point onthe negative y axis. As the phase advances, the vortexpositions rotate clockwise. Curiously, we find that, as thephase is varied slightly from to , the vor-tices suddenly switch signs. This switch may also be in-terpreted as an exchange of the vortex positions. In ei-ther case, this exchange occurs via an edge dislocation atthe phase 0. Edge dislocations are often associatedwith sources or sinks of vortices.7,18,28,29 Unlike case 1,we find that the net topological charge is conserved for all

values of, i.e., mnet msum .Generic shapes of the composite vortex phase-

dependent trajectories are shown in Fig. 6(b) for differentseparation distances. For small values of separation ( 1), the path of each vortex is nearly semicircular, with

vortices on opposing sides of the origin. When cr ,each vortex trajectory forms a separate closed path. Re-gardless of the value of s, the vortex on the right has acharge, m 1, opposite of that of the left. As found incase 1, the critical separation distance delineates open-path and closed-path trajectories. The topological chargeis conserved for all the values of and in Fig. 6, andthus, by induction, we conclude that the net topologicalcharge is always conserved for case 2.

Case 3Last, we consider the interference between beams havinga vortex (beam 1) and planar (beam 2) phase: m1 1, m2 0. Examples of the intensity profiles areshown in Fig. 7. Substitution of the values ofm into Eq.(6) gives

Er, A1g1r1exp ir/w0expi A2g2r2

A1g1r1expis/w0. (19)

Zero field points must satisfy

x iy /s 1 1A2 /A1expiexp42x/s.

(20)

Let us consider the case when the vortex and Gaussianbeams have the same power. From Eq. (2), we obtain therelation between the amplitudes of the vortex, A1 , andGaussian, A2 , beams:

A1 21/2A2 . (21)

In the in-phase case ( 0), Eq. (20) simplifies to

Fig. 5. Same as Fig. 3 except m1 1 and m2 1. Coaxialcomponent beams ( 0) form an edge dislocation, dividing thecomposite beam. This dislocation persists for all values of when 0. Composite vortices appear as an oppositelycharged pair (dipole).



6/8

x iy /s 1 12 1/2 exp42x/s, (22)

which has solutions only along the x axis, and two criticalpoints 1 0.1408 and 2 0.6271, which are the solu-tions of the transcendental equation

23/2 exp42 1. (23)

The in-phase case has no solutions if 1 2 (al-though a dark spot may form in the composite beam), one

solution if 1 (x/s 11.6) or 2 (x/s 0.364), and two solutions otherwise.

Two other cases of special interest exist. An exact so-lution of Eq. (20) is found for any value of when /2:

x s, (24a)

y w0/21/2exp4s2/w0

2. (24b)

For the out-of-phase case, , Eq. (20) simplifies to

x iy /s 1 12 1/2 exp42x/s, (25)

which allows a single composite vortex solution on the xaxis, whose position is displaced away from the Gaussian(m 0) beam.

Numerically determined vortex positions are shown inFig. 8(a) for 0.47. Since this value falls between thetwo critical values, there are no composite vortices for thein-phase case. In fact, we find no vortices over the range cr . The critical phase value depends on the valueof , and in this case cr 40. In general, the criticalphase may be computed from the relation(2 3/2 cos cr)exp(4

2 1) 1. The gray curve with dia-

monds in Fig. 8(a) indicates the position of a dark (butnonzero) intensity minimum that appears when cr . Owing to their darkness, these points could bemistaken for vortices in the laboratory if interferometricmeasurements are not recorded to determine the phase.Beyond the critical phase value, we find an oppositely

Fig. 7. Phase () and separation ( ) dependent intensity pat-terns of a composite beam created by superimposing vortex ( m1 1) and nonvortex ( m2 0) component beams. Both beamshave equal power. The composite beam contains either no vor-tices or a vortex dipole with the m 1 vortex often residing farfrom the beam region.

Fig. 6. Phase-dependent composite vortex trajectories form1 1 and m2 1. (a) 0.47, showing a rotating compos-ite vortex dipole whose orientation flips when 0 and 0 . Arrows indicate the direction of motion for increasingvalues of. Dotted curves demark the footprints of the compo-nent beam waists. (b) Family of trajectories for different values

of . For cr 2

1/2, the trajectories become separateclosed paths.



7/8

charged pair of vortices, with the positively charged vor-tex remaining in close proximity to the m 1 vortex ofbeam 1 and the negatively charged vortex diverging fromthe region as the phase increases. Regardless of the

value of, we find the net topological charge, mnet 0, isnot equal to the sum, msum 1.

Figures 8(b) and 8(c) depict vortex trajectories as thephase is varied for a number of different values of. Thetrajectories for 1 and 2 resemble strophoids, separatingregions having open and closed paths. For example, if

1 2 , a vortex dipole is created or annihilated ata point at some critical phase value; thus the trajectory ofeach vortex is an open path. On the other hand, for 1 or 2 , the dipole exists for all values of ,and the paths are closed. The rightmost vortex in allthese cases has a charge m 1.

Inspections of phase profiles indicate that the net topo-logical charge is mnet 0 for all values of and shownin Fig. 8, except for the degenerate case,1 0 (see left-hand column in Fig. 7), where mnet 1. The case

0.13, shown in Fig. 8(b), suggests that, when the rela-tive separation is small ( 1), one vortex having the

same charge as beam 1 circles the origin, while a secondvortex of the opposite charge appears in the region beyondthe effective perimeter of the beam. As a practical mat-ter, one may ignore the existence of the second vortex if itis surrounded by darkness, and in this case, one maystate that the charge is conserved in a weak sense. How-ever, the second vortex is not always shrouded in dark-ness. Thus the superposition of a vortex and a Gaussianbeam does not generally conserve the net value of the to-pological charge.

4. VORTEX VELOCITY

The superposition of a vortex and Gaussian beam pro-vides a particularly convenient means of displacing a vor-tex, owing to the common availability of Gaussian beamsfrom lasers. By rapidly changing the amplitude of theGaussian beam, the vortex may be displaced at rates thatexceed the speed of light without moving the beams or

varying the phase. The speed of the vortex may be de-termined by applying the chain rule,

Fig. 8. Phase-dependent composite vortex trajectories for m1 1 and m2 0. (a) 0.47, showing the creation or annihi-lation of a vortex dipole at 40. The gray curve markedwith diamonds depicts the path of a nonvortex dark region of thebeam that exists when 40. Arrows indicate the directionof motion for increasing values of. Dotted curves demark thefootprints of the component beam waists. (b), (c) Family of tra-

jectories for different values of. The trajectories change shapeon either side of two critical separations, 1 0.1408 and2 0.6271.



8/8

vx ivy xA2

iy

A2 dA2

dt, (26)

where (x iy ) is given by the transcendental function,Eq. (20). The solution for the velocity simplifies when /2, in which case x s, vx 0, and vy (s/A1)exp(4

2)dA2 /dt . The vortex traverses thebeam at a speed that exceeds the speed of light whendA2 /dt (A2 /0)exp(4

2), where 0 w0 /c is the timeit takes light to travel the distance w0 . A relatively slowlight pulse may be used to achieve this if w0 is large and 0 (note that any value of may be used when 0) . Assuming the amplitude of the Gaussian beam

increases linearly with a characteristic amplitude, A2 ,

and time 2 , A2(t) (A2 /2)t, then light speed may be

achieved when A2 /A1 2 /0 . This may be satisfiedwith a beam having a 100-ps rise time and a 30-mm ra-

dial beam size, assuming A2 /A1 1.

5. CONCLUSION

Our investigation of the superposition of two coherent

beams, with at least one containing an optical vortex, re-veals that the transverse position of the resulting compos-ite vortex can be controlled by varying a control param-eter such as the relative phase, amplitude, or distancebetween the composing beams. The three most funda-mental combinations of beams were explored, namely,beams having identical charges, beams having oppositecharges, and the combination of a vortex and Gaussianbeam. Composite vortices were found to rotate aroundeach other, merge, annihilate, or move to infinity. Wefound that the number of composite vortices and their netcharge did not always correspond to the respective valuesfor the composing beams. As may be expected from theprinciple of conservation of topological charge, the net

composite charge remained constant as we varied the con-trol parameters. Critical conditions for the creation orannihilation of composite vortices were determined.Composite vortex trajectories were found for specialcases. The superposition of oppositely charged vortexbeams produced composite vortices that remained in thebeam, whereas other cases had vortices that diverged to-ward infinity. The speed of motion along a trajectory wasdemonstrated to depend on the rate of change of the con-trol parameter. For example, we described how thespeed of a vortex may exceed the speed of light by rapidly

varying the amplitude of one of the beams.

ACKNOWLEDGMENTThis work was supported by funds from the State of Ari-zona.

REFERENCES1. M. Vasnetsov and K. Staliunas, eds., Optical Vortices, Vol.

228 in Horizons in World Physics (Nova Science, Hunting-ton, N.Y., 1999).

2. A. E. Siegman, Lasers (University Science, Mill Valley, Ca-lif., 1986).

3. J. F. Nye and M. V. Berry, Dislocations in wave trains,Proc. R. Soc. London Ser. A 336, 165190 (1974).

4. N. B. Baranova, B. Ya. Zeldovich, A. V. Mamaev, N. F. Pili-petski, and V. V. Shkunov, Speckle-inhomogeneous fieldwavefront dislocation, Pisma Zh. Eks. Teor. Fiz. 33, 206210 (1981) [JETP Lett. 33, 195199 (1981)].

5. J. F. Nye, Optical caustics in the near field from liquiddrops, Proc. R. Soc. London Ser. A 361, 2141 (1978).

6. A. M. Deykoon, M. S. Soskin, and G. A. Swartzlander, Jr.,Nonlinear optical catastrophe from a smooth initial beam,Opt. Lett. 24, 12241226 (1999).

7. G. A. Swartlander, Jr. and C. T. Law, Optical vortex soli-tons observed in Kerr nonlinear media, Phys. Rev. Lett. 69,25032506 (1992).

8. A. W. Snyder, L. Poladian, and D. J. Michell, Parallel spa-tial solitons, Opt. Lett. 17, 789791 (1992).

9. G. A. Swartlander, Jr., Peering into darkness with a vortexspatial filter, Opt. Lett. 26, 497499 (2001).

10. D. Palacios, D. Rozas, and G. A. Swartzlander, Observedscattering into a dark optical vortex core, Phys. Rev. Lett.88, 103902 (2002).

11. D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., Experi-mental observation of fluid-like motion of optical vortices,Phys. Rev. Lett. 79, 33993402 (1997).

12. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P.Woerdman, Orbital angular momentum of light and thetransformation of LaguerreGaussian laser modes, Phys.Rev. A 45, 81858189 (1992).

13. I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vas-netsov, Optics of light beams with screw dislocations, Opt.Commun. 103, 422428 (1993).

14. V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, Laserbeams with screw dislocations in their wavefronts, PismaZh. Eks. Teor. Fiz. 52, 10371039 (1990) [JETP Lett. 52,429431 (1990)].

15. V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Screwdislocations in light wavefronts, J. Mod. Opt. 39, 985990(1992).

16. V. P. Lukin and V. V. Pokasov, Optical wave phase fluctua-tions, Appl. Opt. 20, 121135 (1981).

17. P. Coullet, L. Gill, and F. Rocca, Optical vortices, Opt.Commun. 73, 403408 (1989).

18. C. T. Law and G. A. Swartzlander, Jr., Optical vortex soli-tons and the stability of dark soliton stripes, Opt. Lett. 18,586588 (1993).

19. K. Staliunas, Dynamics of optical vortices in a laser beam,Opt. Commun. 90, 123127 (1992).

20. G. Indebetouw, Optical vortices and their propagation, J.Mod. Opt. 40, 7387 (1993).

21. I. Freund, Optical vortex trajectories, Opt. Commun. 181,1933 (2000).

22. I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, Interac-tion of optical vortex solitons superimposed on differentbackground beams, Opt. Commun. 130, 385392 (1996).

23. L. V. Kreminskaya, M. S. Soskin, and A. I. Krizhnyak, TheGaussian lenses give birth to optical vortices in laserbeams, Opt. Commun. 145, 377384 (1998).

24. D. Rozas, C. T. Law, and G. A. Swartzlander, Propagationdynamics of optical vortices, J. Opt. Soc. Am. B 14, 30543065 (1997).

25. F. S. Roux, Dynamical behavior of optical vortices, J. Opt.

Soc. Am. B12

, 1215

1221 (1995).26. D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Experimen-tal observation of fluid-like motion of optical vortices,Phys. Rev. Lett. 79, 33993402 (1997).

27. L. C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V.M. Perez-Garcia, and D. Mihalache, Globally linked vortexclusters in trapped wave fields, Phys. Rev. E 66, 036612(2002).

28. C. T. Law and G. A. Swartzlander, Jr., Polarized opticalvortex solitons: instabilities and dynamics in Kerr nonlin-ear media, Chaos, Solitons Fractals 4, 17591766 (1994).

29. H. J. Lugt, Vortex Flow in Nature and Technology (Wiley-Interscience, New York, 1983).


Documents

Ivan D. Maleev and Grover A. Swartzlander, Jr- Composite optical vortices