topological charge and angular momentum of light beams …57989/uq57989.pdf · 2019. 10. 9. ·...

PHYSICAL REVIEW A NOVEMBER 1997VOLUME 56, NUMBER 5

Topological charge and angular momentum of light beams carrying optical vortices

M. S. Soskin, V. N. Gorshkov, and M. V. VasnetsovInstitute of Physics, National Academy of Sciences of the Ukraine, Kiev 252650, Ukraine

J. T. Malos and N. R. HeckenbergDepartment of Physics, University of Queensland, Brisbane 4072, Australia

~Received 17 March 1997!

We analyze the properties of light beams carrying phase singularities, or optical vortices. The transforma-tions of topological charge during free-space propagation of a light wave, which is a combination of a Gaussianbeam and a multiple charged optical vortex within a Gaussian envelope, are studied both in theory andexperiment. We revise the existing knowledge about topological charge conservation, and demonstrate possiblescenarios where additional vortices appear or annihilate during free propagation of such a combined beam.Coaxial interference of optical vortices is also analyzed, and the general rule for angular-momentum densitydistribution in a combined beam is established. We show that, in spite of any variation in the number ofvortices in a combined beam, the total angular momentum is constant during the propagation.@S1050-2947~97!09910-1#

PACS number~s!: 42.65.Sf, 42.50.Vk

ronddhvasengcit

ls

esin

fulewobrgo

ers

tia

a-

dau

eoute

y

islo-

n-ar-ign

e-

rtiescal

ce,y a

thele

tam

in agi-

eo-dy,ons

onsn for

INTRODUCTION

Light beams possessing phase singularities, or wave-fdislocations@1,2# have been studied intensively in linear anonlinear optics@3–21#, to reveal their basic properties, anfor the sake of possible applications. At the singularity tphase becomes undetermined and the wave amplitudeishes, resulting in a ‘‘dark beam’’ within a light wave. Phasingularities appear on wave fronts in diffuse light scatteriwhen a light wave has a form of speckle field, and easpeckle has one screw wave-front dislocation in the vicin@3#, also known as an optical vortex@4#. At present, severadifferent techniques are used to generate ‘‘singular beamsynthesized holograms@5,6#, phase masks@7–9#, active lasersystems@10–13#, and low-mode optical fibers@14#. Recentlythe generation of phase singularities was observed as a rof light wave-front deformations caused by self-actionnonlinear media@15,16#.

The general result of these investigations is the origin onew chapter of modern optics and laser physics—singoptics, which operates with terminology and laws quite nto traditional optics. Phase singularities are topologicaljects on wave-front surfaces, and possess topological chawhich can be attributed to the helicoidal spatial structurethe wave front around a phase singularity. This structursimilar to a crystal lattice defect, and therefore was at fiknown as a wave-front screw dislocation@1,2#. The interfer-ence of a wave possessing such wave-front screw dislocawith an ordinary reference wave produces a spiral fringe ptern @12,17,18#, or, in the case of equal wave-front curvtures, radial fringes@19#. The number of fringes radiatingfrom the center of the interference pattern equals the molus of the topological charge, and the direction of the spiring is determined by the sign of the charge and relative cvature of the wave fronts.

Optical vortices embedded in a host light beam behavsome degree as charged particles. They may rotate arthe beam axis, repel and attract each other, and annihila

561050-2947/97/56~5!/4064~12!/$10.00

nt

en-

,hy

:’’

ult

aar

-esfist

ont-

u-l-r-

inndin

collision @20,21#. Other types of wave-front defects also maoccur, such as edge or mixed screw-edge dislocations@22#.The possible transformations between edge and screw dcations~in laser mode terms, 01 and doughnut modes! maybe performed by modal converters@11,23,24#.

The light field of a singular beam carries angular mometum @23# which may be transferred to a captured micropticle causing its rotation in a direction determined by the sof the topological charge@25#. In nonlinear optics, so-calledvortex solitons, which are singular beams in nonlinear mdium, are a subject of growing interest@26,27#.

However, singular beams demonstrate unusual propeeven in linear optics, in free-space propagation. Optiphase singularities as morphological objects~tears of wavefront! are robust with respect to perturbations. For instanaddition of a small coherent background does not destrovortex, but only shifts its position to another place wherefield amplitude has a zero value. For vortices with multipcharge this operation will split an initiallym-charged vortexinto umu single-charge vortices@21#. Intuition suggests thathe total topological charge would be conserved in a bepropagating in free space@1,2#. Our goal is to analyze themain properties of beams containing phase singularitiesgeneral way, and demonstrate the limitations on the topolocal charge conservation principle for real beams, both thretically and in experiment. On the basis of the present stuwe establish the rules of angular-momentum transformatiin light beams with phase singularities.

WAVE EQUATION SOLUTIONS POSSESSING PHASESINGULARITIES

In this section we demonstrate some particular solutipossessing phase singularities of the scalar wave equatioa uniform isotropic medium

¹2E51

c2]2E

]t2, ~1!

4064 © 1997 The American Physical Society

ro

e-

s

l

i-teli-

hi

sc

eith

oet

orre-

awein

has

the

ams

ofmallam.heamer-ical

larin

the-

ing

-

56 4065TOPOLOGICAL CHARGE AND ANGULAR MOMENTUM OF . . .

whereE is the wave amplitude,c is the speed of light, andtis time. The existence of these solutions for a monochmatic light wave was first emphasized by Nye and Berry@1#.The derived complex amplitude of a light wave with frquencyv, wavelengthl, and wave vectork52p/l orientedalongz axis has the form

E~r,w,z,t !} H r umu exp~ imw1 ikz2 ivt !,r2umu exp~ imw1 ikz2 ivt !, ~2a!~2b!wherer, w, andz are cylindrical coordinates. Both solutionhave a phase depending on the azimuth anglew multipliedby an integerm ~positive or negative! called the topologicalcharge of the phase singularity~optical vortex!. The wave-front ~equal phase surface! forms in space part of a helicoidasurface given by the equalitymw1kz5const. After oneround trip of the wave front around thez axis there is acontinuous transition onto the next~or preceding! wave-frontsheet separated byml, which results in a continuous helcoidal wave-front surface. The topological charge attributo this wave-front structure is positive for a right-screw hecoid (m.0), and vice versa. In the caseumu.1, the wave-front structure is composed fromumu identical helicoidsnested on thez axis and separated by the wavelengthl.

The solutions in forms~2a! and ~2b! cannot describe anyreal wave because of the radial amplitude dependence wgrows proportionally tor for Eq. ~2a!, and tends to infinitywhen r→0 for Eq. ~2b!. To avoid unwanted amplitudegrowth, we may combine the solution in Eq.~2a! with aGaussian beam. Using the paraxial approximation of thelar wave equation in the form

1

r

]

]r S r ]E]r D1 1r2 ]2E

]w222ik

]E

]z50, ~3!

we obtain the corresponding solution for a ‘‘singular’’ wavin a Gaussian envelope carrying an optical vortex wchargem @17#:

E~r,w,z!5Esrsws

S rwsDumu

expS 2 r2ws2Dexp@ iFs~r,w,z!#,~4!

where Es is the amplitude parameter, andrs is the beamwaist parameter. The phaseFs is

Fs~r,w,z!52~ umu11!arctan2z

krs2 1

kr2

2Rs~z!1mw1kz,

~5!

the transversal beam dimension is

ws5Ars21~2z/krs!2, ~6!

and the radius of the wave front curvature is

Rs~z!5z1k2rs

4/4z. ~7!

The amplitude distribution in a transverse cross sectionthe beam has a form of an annulus, and the waist paramrs is connected with the radius of maximum amplitude az50 by a relation

-

d

ch

a-

fter

rmax5rsS umu2 D1/2

, ~8!

The maximum amplitude value of a singular wave atz50,r5rmax, therefore amounts to

Esm5EsS umu2e Dumu/2

. ~9!

The phase singularity disappears whenm50, and solution~4! becomes an ordinary Gaussian beam:

E~r,z!5Egrgwg

expS 2 r2wg2Dexp@ iFg~r,z!#, ~10!where the propagating parameters for Gaussian beam cspond to Eqs.~5!–~7! with m50.

As an example we show that a solution in form~2b! hav-ing an amplitude singularity atr→0 may be used to createsolution with only a phase singularity. For this reasontake a similar solution with an amplitude singularity withGaussian envelope,

E~r,w,z!5Esrsr

expS 2 r2ws2Dexp@ iFs~r,w,z!#, ~11!where the phaseFs is

Fs~r,w,z!5kr2

2Rs~z!1w1kz. ~12!

The combination of solutions~11! and ~2b! for m51 re-moves the amplitude singularity and gives a wave whicha phase singularity atr50:

E~r,w,z!5Esrsr

$12exp@2r2/ws21 ikr2/2Rs~z!#%

3exp~ iw1 ikz!. ~13!

The amplitude of a wave created this way is zero atcenter (r50) and decreases on periphery}1/r. Other pos-sible solutions of the scalar wave equation are Bessel be@28# and Bessel-Gauss beams@29# carrying optical vortices.

OPTICAL VORTICES IN COMBINED BEAMS

In any practical realization of singular beams by usesynthesized holograms or special optical elements, a scoherent background is always present in a singular beThe origin of this background may be a scattering in tdirection of the singular beam propagation or readout bediffraction by the fundamental spatial frequency of an impfect hologram. This background causes splitting of an optvortex with chargeumu.1 into umu single-charged vortices@21#. Another case is the interference between a singuwave and copropagating reference wave which is usedanalyzing the value and sign of the topological charge ofphase singularity@18,21#. We shall now generalize the problem of coherent coaxial addition of singular beams carryoptical vortices with different charges~including the vortex-free wave,m50!. Our goal is to establish principles of to

en

ssB

dsu

pooft

in

-si

iorene

ava

m-

at

glem-

ing

her

of

f the

or-ence

eam.s

the

o

ly

are

-

e

4066 56M. S. SOSKINet al.

pological charge addition and subtraction, and revise theisting knowledge about topological charge conservation ilight beam propagating in free space.

Without loss of generality, we shall take a coaxial Gauian wave as a coherent background for a singular wave.varying the amplitudeEg and waist parameterrg of theGaussian beam, we may examine its influence in near anzones, as plane or spherical waves in limiting cases. Theof the singular beam with the coaxial Gaussian beam~oranother singular beam! we shall call a combined beam.

The presence of a coherent background changes thetion of a vortex which was initially localized at the centersingular beam Eq.~4!. To find the positions of the vortices incombined beam, we need to write two equations, one givthe radius of the zero-amplitude point~amplitudes of thesingular and Gaussian waves are equated!, and another giv-ing the angular coordinatew, which corresponds to the destructive interference between the singular and Gauswaves:

Egrgwg

expS 2 r2wg2D 5Es rsws S rwsDumu

expS 2 r2ws2D ,Fg~r,z!5Fs~r,w,z!6p. ~14!

Equations~14! are the basis for analysis of vortex behavin a combined beam. The first equation is easy to analyzshow the number of possible amplitude zeros in a combibeam.

To simplify the calculations, we suppose both beams ha waist atz50, and use a normalized transverse coordinr 5r/rs and distancej5z/LR , where LR is the Rayleighlength of the singular beam,LR5krs

2/2. The first equation ofsystem~14! may be rewritten as

r umu5EgEs

C~j!exp~ar 2!, ~15!

where

C~j!5~11j2! umu/2S 11j211j2/k4D1/2

, ~16!

a51

11j22

1

k21j2/k2, ~17!

andk is the ratio of waist parametersk5rg /rs .If ak, ~18a!

k

nteb-tas

pli-

e ofas

xi-of

aiti-

or

to

l

-


,1.39 ~no vortices in the beam!, two intersections and twooppositely charged vortices for 1.39,j,2, and finally oneintersection and thus a single vortex forj.2.

In a practical situation we may not have informatioabout the initial beam amplitude ratio and waist paramerg andrs , or the optical paths of beams from waist to oservation plane may be different. The only experimenmeasured values at the plane of observation are inten

FIG. 1. Diagrams~a! and ~b! show the number of vortices incombined beam.~a! The solid curve is the dependence of the crcal amplitude ratio (Eg /Esm)cr on the normalized distancej plottedfor k50.5 andumu51. The area above the curve contains no vtices. The regionj,k separated by the vertical line containsumuvortices, and the part below the curve corresponds to 2umu vorticesin a combined beam. The horizontal dashed line correspondsparticular value of the amplitude ratio,h54.05. This line crosses inturn regions withumu, 2umu, 0, 2umu vortices. The total topologicacharge ism, while j

thvahe

o

a

hth

at

aca

si

raaln

esth

-

alsec-

esith

andularedsiteam

allyhe

hens,

satnys

m.

at

amToo

ine

e-inve

ts

d at


the singular waveAsm by the same relation as Eq.~9!. Tak-ing into account the previous analysis, we may formulatefollowing consequences important for experimental obsertion of vortex transformations in a combined beam: If tsingular beam is broader than the Gaussian beam at theservation plane,ws.wg , Eq. ~21b! has oneumu-times de-generate root, the total topological charge is conserved,the originalm-charged vortex breaks intoumu vortices. If theGaussian beam is broader than the singular,wg.ws , thetotal topological charge of the combined beam is zero. Tnumber of vortices in a combined beam is determined byrelation between amplitudes. IfAg /Asm.(Ag /Asm)cr , thecombined beams will contain no vortices. IfAg /Asm,(Ag /Asm)cr , an additionalumu vortices will appear in thecombined beam, as discussed above. The critical r(Ag /Asm)cr is

S AgAsmD cr5S wg2

wg22ws

2D umu/2. ~22!The number of vortices in the combined beam for this prtical situation may be determined from the diagram shownFig. 3. The hatched area corresponds to the casews.wgwhen the topological charge remains the same as in thegular beam. In the right part of the diagram a solid curveplotted according to Eq.~22! for m51 and a dotted curve fom53. The region under the curves corresponds to thepearance ofumu additional vortices and zero total topologiccharge. No vortices exist in the region above the correspoing curves.

To determine the position of vortices on a planer ,w weneed to use the second equation of system~14!. Alterna-tively, we may calculate directly the position of the vorticas points of intersection of lines representing the zeros of

FIG. 3. Diagram showing the number of vortices in a becombined from arbitrary Gaussian and coaxial singular beams.horizontal coordinate is the transverse size ratio, the vertical cdinate is the amplitude ratio. The dashed area corresponds tocase of conservation of the total topological charge in the combbeam ~the singular beam is wider than the Gaussian!. The solidcurve at the right side is the critical amplitude ratio forumu51, andthe dotted curve is forumu53. The area under the curves corrsponds to 2umu vortices in the combined beam. No vortices arethe combined beam with parameters in the area above the cur

e-

b-

nd

ee

io

-s

in-s

p-

d-

e

real and imaginary parts of the complex amplitudeE(r ,w,j)of the combined beam: Re(E)50 and Im(E)50 @3,15,30,31#.Figure 4 shows these lines inx,y coordinates~normalized onrs! at cross sectionj51.405 of a combined beam with parametersm51, k52, andEg /Esm51.05. The positions ofvortices are shown by dots~intersections of zero lines!. Theoriginal vortex is shifted from the center and an additionvortex with opposite charge is located at the second intertion point. Both lines Re(E)50 and Im(E)50 are closed.

Figure 5 exhibits transformations of zero-amplitude linand corresponding vortices map for a combined beam wm53, k52, andh51.75. At near field (j50.5) all vorticesare suppressed by Gaussian wave with larger amplitudewaist parameter. The faster transverse spread of singbeam leads to the equalizing of amplitudes at combinbeam periphery and appearance of three pairs of oppocharged vortices. Three vortices move away from the beand disappear whenj52. Finally, whenj.2, combinedbeam contains three single vortices located symmetricwith the conservation of the initial topological charge of tsingular beam.

The trajectories of vortices within the cross section of tcombined beam are shown in Fig. 6 for two main situatiok.1 and k,1. The combined beam~m51, k52, andh50.5! starts with two vortices@Fig. 6~a!# which move ontheir trajectories as shown in Fig. 6~a! in opposite directions,and the negative vortex leaves the beam atj52. For anotherinitial amplitude ratioh51.05, the behavior of the vorticeis somewhat different. Two vortices annihilate in collisionj'0.655@Fig. 6~b!#, and then the beam does not contain asingularity until j'1.39. Born as a pair, two new vorticewith opposite charges repel each other and finally one~nega-tively charged! disappears at infinity (r→`) at j52, and theremaining one carries the initial charge of the singular beaIn the casek,1 the combined beam~m51, k50.5, andh54.05! starts with only the primary vortex~shifted from thecenter!, and an additional negative vortex enters the beam

her-

thed

s.

FIG. 4. Phase map shows the lines Re(E)50 ~thick line! andIm(E)50 ~thin line! in (x,y) coordinates normalized onrs at across sectionj51.405 of a combined beam with parametersm51, k52, andh51.05. Positions of vortices are shown by do~intersections of zero lines!. The original vortex is shifted from thecenter, and an additional vortex with opposite charge is locatethe second intersection point.

lli-

omilaromepli-

e oftios

ary

add

er.ith

the

o-

hep-.hic

are,

S2

and-va-

BS3nter-ve.

ror

the

gn-co-farinas

-

e

ym


FIG. 5. Phase maps for different distancesj of a combinedbeam propagation with parametersm53, k52, andh51.75. ~a!j50.5, lines Re(E)50 and Im(E)50 do not intersect, and no vortices exist in the combined beam.~b! j51.25, lines Re(E)50 andIm(E)50 intersect in points shown by dots, and six single-chargvortices~three pairs! exist in the beam.~c! j52.5, three intersec-tion points are shown by dots, and three vortices are located smetrically around the beam center.

j50.5 coming from infinity@Fig. 6~c!#. After several roundtrips it meets the prime vortex, and both annihilate in cosion atj'0.81. A new vortices pair originates atj'1.16,and exists within the beam up to infinity.

The analysis of vortex behavior in a beam combined frtwo coaxial singular beams may be performed in a simmanner. However, the combined beam may possess snew features in this case. First, both beams have zero amtude at the center. Ifum1uÞum2u, the combined beam willhave at the center a vortex with the smaller absolute valuthe charge. On the beam periphery, depending on the raof amplitudes and sizes, the number of vortices may vbetween zero,um12m2u, and 2um12m2u. In the particularcase whenum1u5um2u, there are two situations:m15m2 andm152m2 . When charges have the same sign, wavescoherently producing circular interference fringes@32#, andthe only m-charged vortex is located at the beam centWhen vortices have opposite charges, they compete weach other for the central position. For smallr, we mayneglect the Gaussian envelope of the beams, and writeamplitude at the core as

E~r,w!}r umu@As1eimw1As2e

2 imw#, ~23!

which determines the phase near the core as arctan@(As12As2)/(As11As2)tan(mw)#. This means that the vortex withlarger host wave amplitude will win, but becomes anistropic @30#. Finally, if As15As2 , vortices annihilate eachother. The interference pattern displays 2umu fringes radiat-ing from the center.

EXPERIMENTAL OBSERVATION OF VORTICES INCOMBINED BEAM

The setup for our experiment is shown in Fig. 7. Tlinearly polarized output beam of a 10-mW He-Ne laser oerating on the TEM00 mode is split first at beamsplitter BS1The directly transmitted beam is diffracted at the holograpgrating, and the first diffracted order~singular beam! is se-lected with the iris aperture. The holographic gratingsphase holograms restoring charge21 and 3 singular beamsboth blazed for greatest efficiency into first order@25#. Thereflected beam from beamsplitter BS1 is further split at Binto the ‘‘background’’ Gaussian beam~reflected beam! andreference wave~transmitted beam!. LensesL1 –L4 are usedto control the sizes and radii of curvature of the Gaussianreference wave. LensL2 is finely controlled with a translation stage to allow fine adjustments of the wave-front curture. Polarizing beamsplitters~PBS’s! are used to control therelative intensities of the singular and Gaussian beams.recombines the singular and Gaussian beams, and BS4 iferes with the combined beam and the reference wa~Double reflection of the singular beam from BS3 and mirM does not change the sign of its topological charge.! Fi-nally, lensL5 creates a magnified image of the beam onscreen, recorded with a charge-coupled device~CCD! cam-era.

The experimental technique consisted, first, of an aliment of the singular and Gaussian beams to make themaxial. The resulting interference pattern, observed in thefield, was then used to determine the relative differencecurvature of wave fronts between the two beams. This w

d

-

tnt,the

ndges,S’sr ofin-ian

d-asnce

nal

ionthrte

r

t

amis

ssianThe

to


FIG. 6. Vortex trajectories in a combined beam cross sect~a! For m51, k52, andh50.5. The combined beam starts witwo oppositely charged vortices shown by dots. The negative voleaves the beam atj52. ~b! Two vortices in a beam withh51.05 annihilate in collision atj'0.655. A pair of two new vor-tices with opposite charges appear atj'1.39. They repel each otheand finally one~negatively charged, dashed line! disappears at in-finity ( r→`) at j52. ~c! Combined beam~m51, k50.5, h54.05! starts with only the primary vortex~shifted from the cen-ter!, and an additional negative vortex enters the beam atj50.5~dashed line! coming from infinity. After several round trips imeets the prime vortex, and both annihilate in collision atj'0.81. New vortices’ pair originates atj'1.16 ~dotted lines!.

adjusted using the position of lensL2, so that there was amost one round of dark fringe in the pattern. At this poithe wavefronts of two beams were effectively matched atobservation screen.

The profiles of energy distribution in the Gaussian asingular beams were obtained from separate CCD imaand the relative adjustment in beam intensities using PBcould then be carried out to obtain the necessary numbeamplitude graph intersection points. The correspondingtensity distributions are shown in Figs. 8 and 9 for Gaussand singular beams~charges21 and 3!. Once the two beamswere roughly the required relative intensity and size for aditional vortices to be observable, the reference wave wused to interfere the intensity pattern to observe the preseof fringe dislocations, and hence the vortices in the origipattern.

.

x

FIG. 7. Sketch of the experimental setup. The He-Ne laser besplits into three channels. In the first channel a singular beamcreated by the hologram, the second channel carries the Gauwave, and the reference wave is formed in the third channel.interference patterns are observed on a screen.

FIG. 8. Experimental intensity distributions for Gaussian~m50, open circles! and singular~m521, closed circles! beams atthe observation plane. Solid lines are numerical fits accordingformulas~24!.

thntut

enisaad

atit

ul

e

icom

,

is

eex.di-

lar

anaseo-

sian3ale

wo-

-

an

ian

nedtri-

ofnd a

an

the


The reference beam was simply adjusted throughchoice of lensesL3 andL4 to be of much larger size thathe combined beam cross section. The interference picproduced by the superposition of the reference beam andcombined beam at BS4 could be roughly controlled to givreasonable number of fringes for sufficient resolution of afringe dislocations present. To avoid reduction of fringe vibility due to a loss of coherence through multilongitudinmode operation of the laser, all path differences were minteger multiples of the laser cavity length.

The parameters obtained from a numerical fit of the dwere used for a comparison of the experimental results wtheoretical predictions. The following functions were usedgenerate the intensity profiles of Gaussian and singbeams numerically:

Pg~x!5H Ag expF2 ~x2x0!2wg2 G J2

,

P21~x!5H A21w21 @~x2x0!21y02#1/2expF2 ~x2x0!21y0

2

w212 G J 2,

P3~x!5H A3w33 @~x2x0!21y02#3/2expF2 ~x2x0!21y0

2

w32 G J 2,

~24!

whereAg , A21 , andA3 are the amplitudes of the respectivwaves~Gaussian, singular charges21 and 3! andwg , w21 ,andw3 are the beam sizes. The functionPg(x) is the inten-sity profile of a Gaussian beam defined in thex,y plane,where the profile is taken along thex direction with y50.P21,3 are the intensity profiles alongx axis of singularbeams, charges21 and 3, aty5y0 , in order to take intoaccount a small misalignment of the beam centers, whprevents the central minimum of the singular beam frgoing to zero. Parametersx0 and y0 determine the beamcenter. The data obtained from the CCD~in relative units!were fitted to functions~24! with fitting parameters:Ag

FIG. 9. The same distributions as in Fig. 8, for Gaussiancharge 3 singular beams.

e

reheay-le

athoar

h

58.86, wg585.22, andx0565.76 for the Gaussian beamand A21519.23, (A21m58.25), w21545.81, x0562.25,andy0516.73 for the singular~charge21! beam. The cal-culated curves are shown in Fig. 8~solid lines!. The ratio ofthe beam sizes iswg /ws51.86, i.e., the Gaussian beamwider than the singular. The amplitude ratio isAg /A21m51.07, and according to the theoretical predictions~see thediagram in Fig. 3! an additional vortex should appear in thcombined beam with a sign opposite to the original vortThe presence of intersection points of the curves also incates two vortices existence in the combined beam.

For the profiles of the Gaussian and charge 3 singubeam, the fitting parameters wereAg56.29, wg5115.92,andx0566.30 for the Gaussian beam, andA3515.82 (A3m56.49), w3541.23,x0566.05, andy0533.91 for the sin-gular beam~Fig. 9!. The Gaussian beam is again wider ththe singular, and the amplitude ratio corresponds to the cof three additional vortices appearing, with the total toplogical charge of the combined beam being zero.

For the two superpositions investigated, namely, Gausbeam with charge21 singular and Gaussian with chargesingular, the intensity patterns were calculated in gray scand are shown in Figs. 10 and 11. To generate the tdimensional intensity distributionI (x,y) in a beam com-bined from Gaussian and charge21 singular beams numerically, the expression forI (x,y) is represented as follows:

I ~x,y!5Pg~x,y!1P21~x,y!12APg~x,y!P21~x,y!

3cosS x21y2R2 2arctanyx 1d D , ~25!whereR is the radius of relative curvature of the Gaussiwave front with respect to the singular wave,d is an adjust-able phase factor, and arctan(y/x) is the azimuth angle. Theintensity distribution for the combined beam with Gaussand charge 3 singular wave has the same form as Eq.~25!,only the azimuth factor becomes 3 arctan(y/x). Figures 10and 11 also show the experimental patterns of combibeam intensity in the far field and calculated intensity disbutions with lines of zeros for the functions Re@E(x,y)#50~solid line! and Im@E(x,y)#50 ~dashed line!. The points ofintersection of the zero lines correspond to the positionsvortices. For the case of the combination of a Gaussian a

d

FIG. 10. Experimental~a! and numerically generated~b! inten-sity distributions in a combined~Gaussian plusm521 singular!beam cross section. Zero-amplitude lines are shown in~b!, when thesolid line represents the real part of the complex amplitude, anddashed line the imaginary part.

ehe

rti

s

iged

nene1

winn

th

reinic

0nl-th

thanu

caco

s a

omhas

nt ation

of

tedbe

arss

m

fo

xwi

isga-


charge21 singular beam, a single dark spiral fringe is se@Fig. 10~a!#. The curved shape of the fringe is due to sligdifference in wave-front curvature. The zero-amplitude linhave two points of intersection, corresponding to two voces existing within the dark fringe@Fig. 10~b!#. For theGaussian and charge 3 singular beam, three dark fringediating from the center of the pattern are present@Fig. 11~a!#.There are six intersection points of the zero lines in F11~b!, indicating two vortices with opposite charges locatin each dark fringe.

To demonstrate the existence of vortices in the combibeams clearly, the interference patterns of the combibeam with the reference wave are shown in Figs. 12 andThe results of experimental observations are comparedcalculated intensity distributions. For the Gaussian acharge 21 combined beam, both theory and experimeshow a dark spiral fringe which begins at the center ofpattern and ends after nearly two turns@Figs. 12~a! and 12~b!#. Thereafter, the interference fringes form rings corsponding well with the theory: the phase has no rotatcomponent at this area. Thus two oppositely charged vortmake the total topological charge zero.

For the Gaussian and charge 3 combined beam, therethree inner interference ‘‘forks,’’ equally separated by 12and directed clockwise, and three outer ‘‘forks’’ directed aticlockwise ~Fig. 13!. Hence there are six single vortices atogether, or three pairs of oppositely charged vortices. Atarea outside, the fringes again form rings.

The experimental observations clearly demonstrateappearance of additional vortices in combined beams, ching the total topological charge of a beam. We have thshown in both theory and experiment that the topologicharge of a beam containing phase singularities is not astant while propagating in free space.

FIG. 11. The same intensity distributions as in Fig. 10, butGaussian plusm53 singular combined beam.

FIG. 12. The interference pattern of the combined beam~Gauss-ian plusm521 singular! and reference wave. The primary vorteis located near the center, and an additional vortex is seenpositive charge:~a! experimental picture;~b! numerical simulation.

nts-

ra-

.

dd3.thdte

-ges

are°-

e

eg-sln-

ANGULAR MOMENTUM OF LIGHT BEAM CARRYINGOPTICAL VORTICES

The rotation of phase around the vortex axis causenonzero value of angular momentum of a beam@23,33#. Theorigin of the angular momentum may be easy explained frsimple evaluations. As the wave front of a singular beama helicoidal shape, the Poynting vectorP(r,w,z) which isperpendicular to the wave front surface has at each poinonzero tangential component. In the paraxial approximathis component equalsP'(r,w,z)52mP/kr, and theangular-momentum density in thez-axis direction isMz(r,w,z)5(r/c

2)P'(r,w,z). As the Poynting vector isproportional to a light wave intensity,P}uEs(r,w,z)u2, wemay obtain the expression for angular-momentum densitya singular beam:

Mz~r,w,z!}2muEs~r,w,z!u2. ~26!

The time-averaged density of angular momentum direcalong thez axis in a combined beam cross section maycalculated in a general form@23,33# as

Mz5i

2ve0FxS E* ]E]y2E ]E*]y D2yS E* ]E]x2E ]E*]x D G ,

~27!

wheree0 is the permittivity of free space. The total angulmomentumLz of the beam is an integral over the beam crosection

Lz5E2`

` E2`

`

Mzdx dy. ~28!

The simplest combined beam is a sum of singular@Eq. ~4!#and Gaussian@Eq. ~10!# beams, and its angular momentucalculated according to Eq.~27! is

Mz~r,w,z!52ve0m$uEs~r,w,z!u2

1uEs~r,w,z!Eg~r,z!ucos@Fs~r,w,z!

2Fg~r,z!#%, ~29!

which attains a simple form for a pure singular beam (Eg50):

Mz~r,w,z!52ve0muEs~r,w,z!u2, ~30!

r

th

FIG. 13. The interference pattern of the combined beam~Gauss-ian plusm53 singular! and reference wave. The primary vortexsplit into three vortices, and an additional three vortices with netive charge produce, with them, three pairs:~a! experimental pic-ture; ~b! numerical simulation.

ity

entum


FIG. 14. ~Color! Distribution of energy and angular momentum in a combined beam.~a! and ~b! m51, Eg /Esm51.05, k52, andj50.3. Green is for energy distribution, red for positive angular momentum densityMz.0, and blue for negative angular-momentum densMz,0. The positions of two oppositely charged vortices are shown by white crosses.~c! and ~d! The same distributions form53,Eg /Esm52.5, andk52. No vortices exist in the combined beam with this amplitude ratio, but a modulation of the angular-momdensity is clearly seen. The transverse coordinates arex andy, normalized onrs .

t

l

f aiali-

anic

itn

cal-ms

r-

be-gralve

ingeraledial

tioncalbutularem.do

med

coinciding with Eq.~26!, and

Lz522pve0mEs2S rswsD

2E0

`

rdrS rwsD2umu

expS 2 2r2ws2 D52

pmumu!2umu11

ve0rs2Es

2. ~31!

The density of the angular momentum of a singular beamMzis therefore proportional to the beam intensity, and the toangular momentumLz to the beam energy. In a mediumwithout losses, the total momentumLz is conserved, as welas the beam energy.

The relation obtained for the angular momentum ocombined beam~29! may be generalized as a law of coaxaddition ofN beams carrying optical vortices with topologcal chargesmi :

Mz~r,w,z!52ve02 (i , j

N

~mi1mj !uEi~r,w,z!Ej~r,w,z!u

3cos@F i~r,w,z!2F j~r,w,z!#. ~32!

This law is valid not only for singular waves with a Gaussienvelope, but for all kinds of waves with axially symmetramplitude distributions.

An interesting consequence following from Eq.~29! isthat even a small amount of a singular wave combined wa strong Gaussian wave produces a substantial modulatio

al

hof

the angular-momentum density. Figure 14 represents theculation of angular-momentum density of combined beawith parametersm51, k52, andh51.05 ~a! andm53, k52, andh52.5 ~b!.

For a combined beam with a density of angulamomentum distribution given by Eq.~29!, the total angularmomentum is equal to that of the singular beam alone,cause an interference term gives a zero amount in inte~28!. This demonstrates that coherent addition of a wawith zero angular momentum does not affect the resultangular momentum of the combined beam. Using the genform of the angular-momentum distribution in a combinbeam~32!, we are able now to calculate the result of coaxinterference ofN arbitrary singular beams,

Lz5(i

N

Lzi , ~33!

which establishes a rule: In a beam combined fromN coaxialbeams, the angular momentaLzi add arithmetically.

This rule gives interesting consequences for the addiof singular beams with equal and opposite topologicharges. Two identical singular beams with equal energyopposite charges form a vortex-free beam with zero angmomentum, independent of the phase relation between thIdentical beams with equal topological charges of vorticesnot obey the general rule~33!, because the interference teris a constant in this case, and the energy of the summ

pula

th

tumoma

tectoeiab

inavti

ailaiou

triplin

en-etla

heb-

a-

m-, theeen

m-ave

e,or-

e of

amon-ainof

lard a

sityam.

nalt of


beam is strongly dependent on relative phases of comnents. Further, adding a vortex-free wave with zero angmomentum cannot change the total angular momentumLz ofa beam, even though it may suppress all the vortices incombined beam.

To understand the transformations of angular momenassociated with anm-charged vortex in a combined beawhere the number of vortices may change and even beczero, we need to imagine the wave front of a combined bewhich is not a symmetricalm-pitched helicoid as for a puresingular beam. When all the vortices are suppressed bystrong background wave, the wave front of the combinbeam still has folds, but is a smooth surface without defeThe inclination of local ‘‘rays’’ which are perpendicular tthe surface produces both positive and negative componof the angular momentum. With an increase of Gaussbackground wave amplitude, the folds become smaller,the beam amplitude grows proportionally, and resultnegative and positive parts of angular momentum hnearly the same value. The difference between the negaand positive components of the angular momentum remexactly equal to the initial angular momentum of the singubeam, but is now due to small residual amplitude modulatover the combined beam cross section. In the case of mtiple vortices localized within a combined beam, the disbution of the angular momentum becomes more comcated, but the resulting total angular momentum remaconstant.

CONCLUSIONS

Our analysis of coherent coaxial addition of optical wavcarrying ~at least one! optical vortices has revealed the geeral properties of combined beams. We have studied in dthe behavior of vortices in a beam combined from singu

V

.

a

n-

,

nd

o-r

e

m

em

heds.

ntsnutgevensrnl-

-i-s

s

ailr

and ‘‘background’’ Gaussian waves in order to check tprinciple of topological charge conservation. In brief, we otained the following results.

~1! Addition of a coherent coaxial vortex-free wave tosingular wave with am-charged vortex may change the number of vortices and the total topological charge in the cobined beam. Depending on background wave parametersnumber of vortices in the combined beam may vary betwzero, umu, and 2umu. The total topological charge ism orzero.

~2! In free-space propagation fromz50 to infinity, thetotal topological charge and number of vortices in a cobined beam do change for any choice of background wparameters, except the casek51 ~equal transverse sizes!.

~3! We demonstrate for the first time to our knowledgthe possibility, in free-space propagation, of additional vtices appearing from the far periphery of a beam~from in-finity!, and their disappearance at a beam periphery.

~4! The obtained results are also applicable to the cascombinations of singular beams.

~5! We show that the total angular momentum of a beis conserved in all cases in free-space propagation, in ctrast with the total topological charge. We establish the mrules for addition and subtraction of angular momentalight beams.

~6! We analyzed the transverse distribution of the angumomentum in a combined beam cross section, and founstrong spatial modulation of the angular-momentum deneven in the case of absence of vortices in a combined be

ACKNOWLEDGMENTS

This research was supported, in part, by the InternatioScience Foundation, and by the Australian DepartmenIndustry, Science and Tourism.

ori,

.

d.

in-d.

.

@1# J. F. Nye and M. V. Berry, Proc. R. Soc. London, Ser. A336,165 ~1974!.

@2# M. V. Berry, in Les Houches Lecture Series Session XXX,edited by R. Balian, M. Klae´man, and J.-P. Poirier~North-Holland, Amsterdam, 1981!, p. 453.

@3# N. B. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. VShkunov, and B. Ya. Zel’dovich, J. Opt. Soc. Am.73, 525~1983!.

@4# P. Coullet, L. Gil, and F. Rocca, Opt. Commun.73, 403~1989!.

@5# V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, Pis’mZh. Eksp. Teor. Fiz.52, 1037 ~1990! @JETP Lett. 52, 429~1990!#.

@6# N. R. Heckenberg, R. McDuff, C. P. Smith, H. RubinszteiDunlop, and M. J. Wegener, Opt. Quantum Electron.24, S951~1992!.

@7# S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soiferand G. V. Uspleniev, J. Mod. Opt.39, 1147~1992!.

@8# S. Tidwell, G. Kim, and W. Kimura, Appl. Opt.32, 5222~1993!.

@9# M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristiensen, aJ. P. Woerdman, Opt. Commun.112, 321 ~1994!.

@10# J. M. Vaughan and D. V. Willetts, J. Opt. Soc. Am.73, 1018~1983!.

@11# C. Tamm and C. O. Weiss, J. Opt. Soc. Am. B7, 1034~1990!.@12# F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Resid

Phys. Rev. Lett.67, 3749~1991!.@13# S. R. Liu and G. Indebetouw, J. Opt. Soc. Am. B9, 1507

~1992!.@14# M. Ya. Darsht, I. V. Kataevskaya, N. D. Kundikova, and B

Ya. Zel’dovich, Zh. Eksp. Teor. Fiz.107, 1 ~1995! @Sov. Phys.JETP80, 817 ~1995!#.

@15# A. V. Ilyenkov, A. I. Khizniak, L. V. Kreminskaya, M. S.Soskin, and M. V. Vasnetsov, Appl. Phys. B62, 465 ~1996!.

@16# T. Ackemann, E. Kriege, and W. Lange, Opt. Commun.115,339 ~1995!.

@17# V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, J. MoOpt. 39, 985 ~1992!.

@18# A. G. White, C. P. Smith, N. R. Heckenberg, H. RubinszteDunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, J. MoOpt. 38, 2531~1991!.

@19# O. Bryngdahl, J. Opt. Soc. Am.63, 1098~1973!.@20# G. Indebetouw, J. Mod. Opt.40, 73 ~1993!.@21# I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V

-

nd

od

tt

n.

ss,

P.


Vasnetsov, Opt. Commun.103, 422 ~1993!.@22# I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Com

mun.119, 604 ~1995!.@23# M. W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, a

J. P. Woerdman, Opt. Commun.96, 122 ~1993!.@24# E. Abramochkin and V. Volostnikov, Opt. Commun.83, 123

~1991!.@25# H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. M

Opt. 42, 217 ~1995!.@26# G. Swartzlander, Jr. and C. Law, Phys. Rev. Lett.69, 2503

~1992!.@27# B. Luther-Davies, R. Powles, and V. Tikhonenko, Opt. Le

.

.

19, 1816~1994!.@28# J. Durnin, J. Opt. Soc. Am. A4, 651 ~1987!.@29# F. Gori, G. Guatari, and C. Padovani, Opt. Commun.64, 491

~1987!.@30# I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commu

101, 247 ~1993!.@31# N. R. Heckenberg, M. Vaupel, J. T. Malos, and C. O. Wei

Phys. Rev. A54, 2369~1996!.@32# M. Harris, C. A. Hill, and J. M. Vaughan, Opt. Commun.106,

161 ~1994!.@33# L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J.

Woerdman, Phys. Rev. A45, 8185~1992!.

topological charge and angular momentum of light beams …57989/uq57989.pdf · 2019. 10. 9. ·...

Documents