670 OPTICS LETTERS / Vol. 35, No. 5 / March 1, 2010

Propagation of vector fractional chargeLaguerre–Gaussian light beams in the thermally

nonlinear moving atmosphere

Maxim A. Molchan,1,2,3,* Evgeny V. Doktorov,1 and Rafael A. Vlasov1

1B.I. Stepanov Institute of Physics, Minsk 220072, Belarus2Department of Mathematics, University of Cape Town, Rondebosch 7701, South Africa

3National Institute for Theoretical Physics (NITheP), Stellenbosch 7600, South Africa*Corresponding author: m.moltschan@gmail.com

Received October 16, 2009; revised January 5, 2010; accepted January 26, 2010;posted January 27, 2010 (Doc. ID 118650); published February 24, 2010

We study numerically the mutual propagation of two fractional topological charge Laguerre–Gaussian (LG)light beams in the moving atmosphere with the thermal nonlinearity. The fractional charge beam is con-structed as a weighted superposition of a finite number of the standard LG modes with integer charges.Numerical simulations demonstrate an enhanced stability of the cross-section intensity distribution of thefractional charge LG beams against the thermal blooming, as compared with the case of integer chargebeams. The dominant mechanism of such a stability results from the multimodal structure of the fractionalcharge beams. © 2010 Optical Society of America

OCIS codes: 010.1300, 010.3310, 190.5940.

Optical spatial vortices, i.e., finite-size beams with aphase singularity at the beam center, represent botha fundamental physical object [1,2] and a tool forvarious applications in optical communications [3],optical tweezers [4], optical-vortex coronography [5],etc. The phase of the vortex beam rotates around thesingularity and the circulation of the phase gradientdefines the topological charge of the vortex.

An active area of beam applications relates withthe optical sensing and communication in the atmo-sphere [6,7]. A minimization of the thermal bloomingof beams is still a significant problem of atmosphericoptics [8,9]. Adding angular momentum to the beamintroduces rotation that slows down the beam expan-sion during the propagation, but never stops it com-pletely. As a result, the ring-shaped beam with a he-lical phase structure having an integer topologicalcharge eventually disintegrates.

New prospects of the (quasi-)stable optical beampropagation can be opened with using fractionalcharge beams (FCBs). Fractional charges are gener-ated by a noninteger phase step [10] or by the holo-graphic technique [11]. Notice that the occurrence ofa vortex chain in a dark line related to the edgephase dislocations still causes problems with thebeam stability. In [12] a different method for prepar-ing FCBs was proposed. The authors used a re-stricted set of the Laguerre–Gaussian (LG) modeswith integer charges to synthesize a beam with afractional charge. Evidently, a finite superposition ofthe integer charge LG modes does not produce the ex-act state with a fractional charge. Nevertheless, acontribution of states outside a very small amount ofthe leading ones is negligible. In [13] we extendedthis approach to the beam propagation in nonlinearmedia. Namely, considering the simplest case of thescalar beam propagation, we showed that the chargefractionality benefits to the remarkable improvement

of the beam stability.

0146-9592/10/050670-3/$15.00 ©

In the present Letter we demonstrate further ad-vancement of the beam quality owing to using mul-tiple beams. Namely, we consider a (nearly) horizon-tal propagation of two co-axial equal wavelengthmutually orthogonal light beams E= �E1 ,E2� in theatmosphere in the presence of the thermal nonlinear-ity and a cross wind. Our model is governed by theparaxial vector equation

2ik�E

�Z+ ��E + k2� i�

k+

T

�0

��

�T�E = 0. �1�

The beams propagate along the Z axis, Y denotes thevertical direction, ��=�2 /�X2+�2 /�Y2, k is the lightwavenumber, � is the temperature-dependent air per-mittivity, �0 is the linear part of permittivity, and thewind blows along the X axis at the velocity v. Thetemperature T changes along the X axis in accor-dance with the equation

v�T

�X=

�I

�0cp, �2�

where I=c��0�E�2 / �8�� is the beam intensity, � is theabsorption coefficient, �0 is the air density, and cp isthe specific heat capacity of air. Neglecting the ther-mal conductance of the atmosphere along the Z axisand considering the limit of weak absorption, wetransform Eqs. (1) and (2) to a sort of the vector non-local nonlinear Schrödinger equation in the dimen-sionless variables,

2i��

�z+ q� �2

�x2 +�2

�y2�� −2

q��

−�

x

���2dx = 0. �3�

Here �= �u ,v�T and ���2= �u�2+ �v�2, where u and v aredimensionless complex field amplitudes of the beamsnormalized to the maximum input amplitudes, and z

and �x ,y� are coordinates normalized to the self-

2010 Optical Society of America

March 1, 2010 / Vol. 35, No. 5 / OPTICS LETTERS 671

action length and the input beam radius, respec-tively. The parameter q is the ratio of the thermalself-action length and diffraction length and variesfrom 0.05 (highly nonlinear regime) to 0.8 (weak non-linearity) for typical parameters of the atmosphere,while the distance z=1 varies in the range of10–103 m. Equation (3) admits the conservation law����2dxdy=�.

Following the procedure described in [12] we con-struct the LG beam with the fractional charge Mwritten in cylindrical coordinates � and � as

uM��,�,z = 0� = A m=mmin

mmax

cmlpm��,�,z = 0�, �4�

where A is the amplitude of the beam, cm�� ,0� areweight coefficients, and lp

m are the standard LGmodes with integer charges. In Fig. 1 we present theLG beam with the charge of M=1.5. The shape of thebeam is defined by the number of modes in Eq. (4),nmodes=mmax−mmin+1, the edge dislocation �, thestarting angle 0 (onward we set nmodes=20, �=0, and0=−�), and the choice of the indices p. We follow [12]and choose p in order to minimize the set of Gouyphases in the constituents of the FCB. For the distri-bution of p presented in Fig. 1(c) the Gouy phases aredefined by two numbers only: 2p+ �m�+1=11 or 12.Besides, we take equal powers of the fractionalcharge components u and v of the vector LG beam �(i.e., ��u�2dxdy=��v�2dxdy=� /2) by appropriately cal-culating the amplitude A in Eq. (4).

Our numerical simulations are based on the split-step fast Fourier transform method, with the accu-racy being controlled via the calculation of the con-served beam power. First, we consider the dynamicsof the vector beam consisting of ordinary LG modesA�e−�2/2+iM� with integer charges of Mu=1(u-component) and Mv=2 (v-component). From theresults in Fig. 2 it follows that the thermal effects

Fig. 1. (Color online) FCB with M=1.5, q=0.3, andnmodes=20: (a) intensity distribution, (b) phase, (c) distribu-tion of the index p, (d) values of the weight coefficients

2

�cm� .

and the wind considerably affect the initial ring-shaped total intensity distribution of the beam lead-ing to the significant beam distortion.

The situation with the beam integrity drasticallychanges when using the FCBs. In Fig. 3 we presentthe cross-section intensity distribution for the propa-gating vector beam with fractional charges of Mu=1.5 [Fig. 3(a)] and Mv=3.5 [Fig. 3(b)]. From com-parison with Fig. 2 it is immediately seen that thevector beam with the fractional topological chargesexhibits pronounced stability and almost preservesits initial circular cross-section configuration [Fig.3(c)].

Next, we considered the pair Mu=−1.5 and Mv=3.5, as well as the pairs Mu= ±1.5 and Mv=7.5when the topological charges are substantially differ-ent. For both cases we revealed the same enhancedstability of the beams during their propagation.

The central question here is the origin of the en-hanced stability of the FCBs during the propagation.We believe that the dominant mechanism that is re-sponsible for such a stability stems from the multi-modal structure of the FCBs. The multimodal struc-

Fig. 2. (Color online) Evolution of the total intensity ���2of the vector beam consisting of ordinary LG modesA�e−�2/2+iM�, with q=0.3 and charges of M=1 �A=0.71� andM=2 �A=0.71�.

Fig. 3. (Color online) Intensity evolution of the vector LGbeam with the fractional charges of (a) Mu=1.5 �A=1.3�and (b) Mv=3.5 �A=1.3�, and (c) total intensity ���2= �u�2

2

+ �v� . Here q=0.3.

672 OPTICS LETTERS / Vol. 35, No. 5 / March 1, 2010

ture favors the suppression of the instability byaveraging significantly the effects of small-scale self-action of single modes. We demonstrate in Fig. 4 theevolution of the FCB constructed from two [Fig. 4(a)],six [Fig. 4(b)], and 20 [Fig. 4(c)] integer charge LGmodes. It is seen that the greater the mode amountinvolved in the FCB construction, the more regular isthe cross-section intensity distribution. Notice thatthe 20-mode construction of the FCB practically coin-cides with the exact (infinite modes) synthesis of thebeam with the same charge (see Fig. 8 in [12]).

The additional mechanism of the beam stability isdictated by the vector character of the beams whentheir interaction via, e.g., cross-phase modulation, re-sults in the mutually supported quasi-stable propa-gation [14]. For typical atmospheric parameters,however, the vector mechanism is much weaker thanthe multimodal one. Nevertheless, a close look at Fig.5 shows that the beam becomes more compact inter-acting with its counterpart.

The numerical experiments presented above weremostly carried out for q=0.3 that corresponds to ge-neric values of the parameters of the atmosphere andthe light beams in the regime of moderate nonlinear-ity [13]. By decreasing (increasing) the value of q weenhance (suppress) the nonlinearity effects and de-crease (increase) the diffraction divergence. As fol-lows from the numerical data, for q0.06 (highlynonlinear regime) the FCBs are distorted during thepropagation and tend to decay into crescent-like frag-ments.

Fig. 4. (Color online) Evolution of the total intensity ���2of the vector beam with the fractional charge componentsMu=1.5 and Mv=3.5 constructed from (a) two modes in Eq.

(4), (b) six modes, and (c) 20 modes. On all plots, q=0.3.

In conclusion, we have shown numerically that thevector FCBs demonstrate considerable stabilityagainst the thermal blooming when propagating inthe moving atmosphere with the thermal nonlinear-ity. This is the multimodal nature of charge fraction-ality that permits the beam profiles to be conservedduring the mutual beam channeling. This property ispromising in the context of a high laser power andoptical information transfer. A generalization to Nbeams is straightforward.

This work has been done within the Program“Photonics-1.19.”

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Fig. 5. Intensity cross-section configuration at z=1.5 for(a) scalar beam with the charge of M=1.5 and (b) fractionalcharge of Mu=1.5 component of the vector beam (anothercomponent Mv=3.5). For both plots, q=0.07.

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