694 OPTICS LETTERS / Vol. 31, No. 6 / March 15, 2006

Centrifugal transformation of the transversestructure of freely propagating paraxial light beams

A. Ya. BekshaevI. I. Mechnikov National University, Dvorianskaya 2, Odessa 65026, Ukraine

M. S. Soskin and M. V. VasnetsovInstitute of Physics, National Academy of Sciences of Ukraine, Prospect Nauki 46, Kiev, 03028 Ukraine

Received October 4, 2005; revised November 25, 2005; accepted December 5, 2005; posted December 8, 2005 (Doc. ID 65015)

Superpositions of coaxial Laguerre–Gaussian modes with common waists and the same Gaussian envelopesare known to form beams whose transverse structures experience the self-similar transformation duringfree beam propagation: the beam shape remains the same except for the divergence and rotation around thepropagation axis. We show that under certain conditions this transformation can be represented as if everypoint of the beam cross section performs a centrifugal straight-line fly off. © 2006 Optical Society of America

OCIS codes: 070.4690, 140.3300, 260.2160, 350.5500.

Active studies of phase singularities in light fields,1,2

in particular, investigations of optical vortices (screwwavefront dislocations),3–11 have brought up impor-tant questions on the nature and manifestations ofthe transverse energy circulation in paraxial lightbeams.12–15 In this view, properties of light beamsshowing a sort of rotation around the axis of propa-gation attract particular attention.16–19 It has beenfound that these rotating light beams demonstratesome features of mechanical vortex16 orcentrifugal20–22 motion.

Within the variety of such rotating beams, an im-portant class is formed by superpositions of coaxialLaguerre–Gaussian (LG) modes2,4,5 with coincidingwaist positions and waist parameters; let us refer toany such superpositions as “combined beams” (CBs).If the propagation axis is identified with axis z andthe transverse plane is provided with polar coordi-nates r, �, a CB can be characterized by the complexamplitude distribution (see, e.g., Refs. 2, 19, and 23),

u�r,�,z� = �p=0

�

�l=−�

�

Cplupl�r,�,z�, �1�

where

upl�r,�,z� =1

b� p!

��p + �l��!� r

b��l�

Lp�l�� r2

b2�exp�−r2

2b2

+ ikr2

2R�exp�il� − �2p + �l� + 1��� �2�

is the normalized transverse profile of an individualLGpl mode; p and l are its radial and azimuthal indi-ces, and k is the radiation wavenumber. The param-eters of distribution (2) depend on the propagation

distance in accord with the equations

0146-9592/06/060694-3/$15.00 ©

R�z� =zR

2 + z2

z, b2�z� =

zR2 + z2

kzR, ��z� = arctan� z

zR��3�

(the beam waist is supposed to be situated at theplane z=0), where zR=kb0

2 is the beam Rayleighrange and b0 is the initial (at the waist plane) radiusof the Gaussian envelope.

Now let us focus our attention on those CBs (1) and(2), which propagate in a free space self-similarly—that is, preserving the shape of the transverse ampli-tude distribution, except the smooth overall wave-front distortion (which does not affect the visibleintensity pattern), the beam expansion due to diver-gence, and rotation around the propagation axis.Such situations are realized if, for any pair of super-position (2) members with sets of indices �p , l� and�p� , l��, the following relation takes place19:

B =2�p − p�� + �l� − �l��

l − l�= const �4�

[which, in particular, occurs for arbitrary CB (1) con-sisting of only two terms of form (2)]. Then the azi-muthal orientation of the beam pattern changes withz according to

��z� = �0 + B��z�. �5�

In some cases, this self-similar transformation of apropagating beam appears to obey an interestingregularity that seems to have been unnoticed before.

To show this, let us take in the waist cross section�z=0,�=0� a certain point with coordinates r0, �0. Itcan be some characteristic point (extremum, saddle,zero of amplitude, etc.) or arbitrary point individual-ized by its dislocation with respect to details of thebeam transverse profile. In the course of the self-similar beam propagation, this point, preserving itsindividuality, moves to another location within thetransverse plane; in particular, its new azimuthal co-

ordinate is determined by Eq. (5). Simultaneously,

2006 Optical Society of America

March 15, 2006 / Vol. 31, No. 6 / OPTICS LETTERS 695

the beam divergence causes the point to move awayfrom the axis proportionally to the beam expansion.Therefore its current radial position r�z� obeys the re-lation

r�z� = r0

b�z�

b0. �6�

The explicit representation of the point trajectoryis available after excluding the propagation distancez from Eqs. (5) and (6) with the help of Eqs. (3):

��r� = �0 + B arccos� r0

r � . �7�

This equation describes spirallike curves whose be-havior depends on the absolute value of B. In furtherconsiderations, we restrict ourselves by the mostspectacular and, as it will be shown, physicallymeaningful case:

�B� = 1. �8�

In this case, Eq. (7) defines a straight line

��r� = �0 ± arccos�r0/r�. �9�

Radius vectors of the initial r0 and current r posi-tions of the chosen point of the transverse beam pro-file coincide with the leg and the hypotenuse of aright triangle (see Fig. 1). Consequently, displace-ment r−r0 of the point evolves perpendicularly to theinitial radius vector r0, i.e., along the tangent of thecircumference r=r0. In other words, the visible rota-

Fig. 1. Evolution of the transverse profile for a two-termCB of the form of Eq. (11) with a positive sign in the lastexponent and nonzero coefficients C00=1, C03=1.22. Thecentral inset presents the waist plane, main area plane z=zR. In the initial (waist) cross-section point r0 is chosenthat belongs to the contour (gray) of equal intensity at thelevel 20% of maximum; its current image (point r) lies onthe corresponding contour (black) of the current transverse

profile.

tion of the beam is, in a certain sense, only apparent:Every point of the beam cross section moves in astraight line orthogonal to its radius vector in thewaist plane. It appears as if the whole transversestructure of the beam flies off by inertia due to theaction of a centrifugal force.

The velocity of this fly off can be determined viathe rate with which the value �r−r0� changes in thecourse of the beam propagation. Just as should be ex-pected from the mechanical analogies, it appears tobe constant and proportional to the initial distance ofthe considered point from the rotation center, on thepropagation axis z:

d�r − r0�

dz=

d

dzr0 tan � =

r0

zR. �10�

This result can be provided with the temporal mean-ing. Since the light energy passes the distance dzduring the time interval dt=dz /c, where c is the lightvelocity, it follows that points of the CB transverseprofile move with velocity v=d�r−r0� /dt=cr0 /zR.

An important special situation of centrifugal self-similar transformation takes place when all compo-nents of the considered CB (1) possess a zero radialindex �p�0� while their azimuthal indices have thesame sign. Then the CB contains only LG0l modesthat are widely known as standard carriers of isotro-pic optical vortices2 with topological charges m=−l.Following from Eqs. (1) and (2), its complex ampli-tude distribution can be reduced to the form

u�r,�,z� =1

bexp�−

r2

2b2 + ikr2

2R− i��

��n=0

� C0n

��n! rei�±�−��

b �n

, �11�

where n= �l� and the sign before � in the last expo-nent coincides with the common sign of azimuthal in-dices of the vortex components involved. This class ofCB includes, in particular, aggregates of equallycharged optical vortices embedded within a back-ground Gaussian beam, studied in the context of op-tical vortex dynamics22,24 [compare Eq. (11) to Eq.(14) of Ref. 24]. In this view, the known effect ofstraight-line transverse motion of individual opticalvortices in such superpositions22 becomes a specialcase of the self-similar transformation law (9) in ap-plication to singular points of the CB transverse pro-file. An interesting subclass of family (11) is formedby two-term combinations of LG0l and LG00 (Gauss-ian) modes. Such CBs have a characteristic intensitypattern with n maxima and minima (off-axial vorti-ces) situated symmetrically around the beamaxis4,25,26 (an example is presented as a backgroundin Fig. 1) and are commonly used for visualizing theadditional phase transformations emerging due tothe Gouy phase shift4 or in connection with the rota-tional Doppler effect.25,26

For beams (11), the sign in Eq. (9) is specified bythe sign before � in Eq. (11). This entails that the di-

rection of the CB rotation is always dictated by the

696 OPTICS LETTERS / Vol. 31, No. 6 / March 15, 2006

(common) handedness of its vortex components, evenif their magnitudes are arbitrarily small compared tothe magnitude of the nonvortex �n=0� summand of(11). Representation (11) supplies an additional ex-planation of the self-similar propagation property.Since the beam transverse profile essentially de-pends only on the combination of variables�r /b�expi�±�−��, a chosen point preserves its char-acteristic disposition with respect to the profile de-tails if during the beam propagation this combinationretains the initial value r expi�±�−�� /b=r0 exp�±i�0� /b0, which obviously again leads to Eq.(9). Besides, beams (11) show a somewhat strength-ened version of the self-similar propagation property.One can easily see that, discarding the inessentialbackground Gaussian (exponential prefactor), in thiscase the centrifugal law is applicable to the wholecomplex amplitude profile (not only the amplitudebut also the phase distribution).

In the 3D space the presented centrifugal picturenaturally leads to a beam model in the form of a setof skew straight-line rays (r−r0 in Fig. 1 is the pro-jection of such a ray onto a transverse plane). Modelsof this kind have been used for representation of pureLG modes18,27; now they can be generalized for arbi-trary CB of the form of Eq. (11).

One can recognize a certain analogy between thedescribed mechanism of the transverse beam struc-ture transformation and known facts of centrifugalescape of the vortex solitons (light filaments) in non-linear media20,21 (such solitons are formed in mediawith self-focusing saturable or quadratic nonlinearitywhen a beam with an initial screw dislocation and aringlike intensity distribution breaks off into a sort ofnecklace surrounding the z axis). This analogy is re-inforced by the apparent similarity of the two-termCB profiles (Fig. 1) and the soliton necklace rings,which are formed after breakup of initial ringlikebeam structures caused by azimuthal instability.20

However, the linear situation considered here has asubstantial distinction: in agreement with Eq. (10),the fly-off velocity increases with the initial radiusvector of the point, whereas it was reported in Ref. 20that transverse velocities of soliton filaments are in-versely proportional to the initial radius of the soli-ton ring. The latter property is consistent with themechanical picture16 of the vortex motion of solitons.Consequently, the straight-line motion of points ofthe CB transverse profile has no direct mechanicalmeaning and does not express real energy transpor-tation (for example, it does not reflect the transverseenergy circulation, inevitably existing in the CB withoff-axial vortices shown in Fig. 1). To trace how thepattern of energy flow leads to the visible straight-line transformation of the CB shape seems to be aninteresting problem that deserves special analysis.

Keeping in mind conditions (4) and (8), one can findmany other situations of centrifugal self-similarbeam propagation that admit simple and spectacular

geometric interpretation. The possibility of extending

this idea to more general circumstances, connectedwith combinations of LG modes having differentwaist parameters, also seems to be a productive prob-lem. However, this case is much more complicatedand even for two-23 or four-componentsuperpositions,28 evolution of the beam transversepattern looks rather intricate.

A. Ya Bekshaev’s e-mail address is alec@paco.net.

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