Dmitri V. Petrov- Vortex-edge dislocation interaction in a linear medium

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Vortex±edge dislocation interaction in a linear medium

Dmitri V. Petrov *

Department of Fundamental Chemistry, University Federal of Pernambuco, Recife, CEP 50670-901, Brazil

Received 30 August 2000; received in revised form 6 November 2000; accepted 11 December 2000

Abstract

The interaction of a vortex and an edge phase dislocation nested in a Gaussian beam by its free-space propagation is

studied experimentally and theoretically in the paraxial regime. This interaction induces the nucleation of additional

vortices of both topological charges. Their positions and number depend on the initial shift of the edge dislocation

relative to the vortex. Ó 2001 Published by Elsevier Science B.V.

PACS: 42.25.Bs; 42.25.Hz

Keywords: Optical phase singularities; Optical vortex; Optical edge dislocation; Gauss±Hermite modes; Angular momentum of light

Optical ®elds may contain topological defects of phase, that is, points or lines on the wavefront

surface where both the real part and imaginary

part of the ®eld are equal to zero [1]. There are two

types of phase defects or dislocations. A screw

dislocation or vortex is a spiral phase ramp around

the point of phase singularity where the phase of

the wave is unde®ned thus its amplitude must

vanish. The order of the singularity multiplied by

its sign is referred to as the topological charge of

vortex. The doughnut mode of a laser cavity is an

example of an optical vortex. An edge dislocation

is the p-shift in the wave phase located along a line

in the transverse plane. The Gauss±Hermite mode

H 1;0 is the simplest example of a ®eld with edge

dislocation.

The propagation dynamics of a vortex in both

linear and nonlinear media is aected by any

source of the phase gradient and (or) the intensitygradient, and it has been the subject of numerous

publications (see, e.g., Refs. [2±16]). In particular,

in a linear medium two vortices interact and their

positions relative to each other and relative to the

host beam change as they propagate [2,3,10±12].

Publications on the propagation dynamics of

edge dislocations are few. The nonlinear evolution

of an edge dislocation was studied in an aniso-

tropic photorefractive material [17,18], and in a

defocusing Kerr medium [19]: at high intensities

this dislocation is unstable and break up into pairs

of vortices.

The aim of this paper is to study experimentally

the interaction between a single-charged vortex

and an edge dislocation nested in a smooth Gauss-

ian beam by its propagation in a linear medium.

This work was stimulated by a recent paper of

Kivshar et al. [20] on vortex±stripe soliton inter-

actions in a nonlinear defocusing medium. It was

observed a vortex-induced dark stripe soliton

(containing an edge dislocation) bending and its

15 February 2001

Optics Communications 188 (2001) 307±312

www.elsevier.com/locate/optcom

*Tel.: +55-81-3271-8440; fax: +55-81-3271-8442.

E-mail address: [email protected] (D.V. Petrov).

0030-4018/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V.P II: S0 0 3 0 -4 0 1 8 (0 1 )0 0 9 9 3 -2



break up into vortices. This rises the question how

this interaction manifests itself in a linear medium

because the above mentioned mechanisms of in-

teraction between dislocations present in this case

too.

The experiments were designed with the fol-

lowing guidelines. First, only a vortex will be

nested into a Gaussian beam and the phase dis-

tribution in the beam after free propagation will be

monitored using interferometric technique. Sec-

ond, only an edge dislocation will be nested in the

Gaussian beam and its phase structure will be also

analyzed. Then, the both dislocations will be em-

bedded in the beam and the beam's intensity and

phase distributions will be studied at dierent ini-

tial positions of the edge dislocation relative to thevortex being always at the center of the Gaussian

beam.

A single-charge vortex in a Gaussian beam

from a He±Ne laser k 0:6328 lm was gener-

ated using an on-axis computer-generated spiral

zone plate (60 cm focal length) [21]. A diameter of

the mask was 0.6 cm. To separate the beam with

vortex from the undiracted light and high-order

Fresnel images, a lens with 10 cm focal length was

placed at the fundamental focus of the zone plate.

An image of the beam or an interference patternobtained with a reference beam was observed on a

screen located at about one Rayleigh length from

the lens and was recorded with a charge-coupled

device camera. Fig. 1a and b shows the intensity

distribution of the beam with the vortex and cor-

responding interference pattern. The extra fringe

generated at the phase dislocation is clearly visible.

An edge phase dislocation was created by a p-

phase jump imposed across the beam by a thin

glass plate [17,18]. Interference of the reference

beam with the light passed through the plate was

used to adjust the tilt of the glass plate to ensure a

phase shift of about 2pn p between the two

halves of the beam (Fig. 1c and d). One can readily

observe this one-half fringe shift of the interference

fringes.

To generate the both phase dislocations in the

Gaussian beam the thin glass plate was introduce

in the beam just before the computer-generated

spiral zone plate as shown in Fig. 2. No dierencein experimental results was observed whether the

glass plate was placed before or after the spiral

zone plate.

When the both phase dislocations are on-axis of

the host Gaussian beam, the intensity distribution

and corresponding interference pattern (Fig. 3a

and b) change considerably. Two vortices of the

same topological charge (which is equal to the sign

of the input vortex) present now as a result of the

interaction of the dierent types of phase disloca-

tions. The edge dislocation alone does not ob-served.

If the vortex is on-axis but the edge dislocation

is oset to the right (the shift from the center of the

vortex is about 0.1 cm), a new vortex (3) opposite

in sign to other vortices (1) and (2) is observed as

well as an asymmetry of the intensity distribution

(Fig. 3c and d). By further shift of the edge dis-

Fig. 1. Experimental images of intensity and interferogram for

the output beam (a, b) with vortex or (c, d) with edge disloca-

tion.

Fig. 2. Diagram of the experimental setup. GP, glass plate;

MASK, computer-generated hologram; L, lens.

308 D.V. Petrov / Optics Communications 188 (2001) 307±312



location the vortex (3) moves towards the vortex

(1), then the vortex (3) changes its trajectory and

moves towards the vortex (2) (Fig. 3e and f). At

last a limited edge dislocation may be observed

(Fig. 3g and h). It is notable that the vortex (1)

seen in Fig. 3c and d moves downwards by the

shift of the edge dislocation (compare its position

at Fig. 3g and h).

Hence, we observed experimentally that the

total topological charge is conserved (two positive

vortices and one negative vortex) in the case when

the vortex is on-axis and the initial edge disloca-

tion is o-axis but does not appear to be conserved

when the both phase dislocations are on-axis ± two

positive vortices present in this case. Before to

discuss this result we describe a simple model in

the paraxial approximation.

Let us consider an input ®eld at z 0 that

consists of a single positive charge vortex, located

at x 0 and y 0, and an edge dislocation located

at x x

0, both nested in a Gaussian beam of waist

w0:

E x; y ; z 0 E 0 x i y

w0

xÀ x0w0

Â exp

À x2 y 2

w20

: 1

There are dierent ways to represent the ®eld of an

edge dislocation: as the Gauss±Hermite mode H 1;0as in Eq. (1), as a tanh xÀ x0 function or as a step

function giving a p-shift at the wavefront at x x0.

The two last functions are best ®tted to our ex-

perimental conditions, however they do not allowto analyze analytically the ®eld free-space prop-

agation. For this reason we chose the Gauss±

Hermite mode H 1;0 to model our experiments.

In order to calculate how the ®eld E x; y ; z 0transforms upon propagation in a linear medium,

we use the standard paraxial method [22]. First we

calculate the Fourier transform of E x; y ; z 0,then use the paraxial transfer function for free-

space propagation and, ®nally, an inverse Fourier

transform gives the ®eld E x; y ; z after propaga-

tion of a distance z :

E x; y ; z E 0 1

2

in

1 in2

" x i y xw2

01 in3

À x i y x0

w201 in2

#exp

À x2 y 2w2

01 in;

2where n z =kw2

0=2 and k is the wave number.

The same results were obtained using the substi-

tution method proposed by Freund [23]. This ®eld

may contain several phase defects. Their positionscan be found by equating the real and imaginary

part of the ®eld (2) to zero. We ®nd these to be

given by

x1 x0 À nD; y 1 x0n D;

x2 x0 nD; y 2 x0nÀ D;

x3 0; y 3 w20

2 x0

n;

3

with D w2

0=2 À x20

p :

Fig. 3. Experimental images of intensity and interferogram for

the output beam with vortex and edge dislocation at dierentshift of the edge dislocation x0: (a, b) x0 0, (c, d) x0 1 mm,

(e, f) x0 2 mm, (g, h) x0 4 mm.

D.V. Petrov / Optics Communications 188 (2001) 307±312 309



Below we analyze trajectories of these vortices

by changing x0 assuming that the propagation

distance is constant. Our experiment was per-

formed under this condition.

If x0 0 the output ®eld contains two vortices

x1; y 1 and x2; y 2 and this was observed in ex-

periments (Fig. 3a and b). The third vortex x3; y 3is at in®nity y 3 3I. I f 0 < x0 < w0=

2

p xÃ0,

the output ®eld contains three vortices. Using the

de®nition of vortex topological sign introduced by

Freund [4,5], two vortices x1; y 1 and x2; y 2 are

positive (i.e. they have the same sign as the input

vortex), and the sign of the vortex x3; y 3 is neg-

ative. If x0 xÃÃ0 nw0=21 n21=2

the vortex

x1; y 1

annihilates with the vortex

x3; y 3

, however

by further increase of x0 the ®eld contains againthree vortices. Two of them are positive and the

third is negative. At x0 xÃ0 the vortex x3; y 3 an-

nihilates with the vortex x2; y 2, and only one

positive vortex x1; y 1 survives.

Fig. 4a shows the trajectories of the three vor-

tices calculated by w0 1 and at n 1. Fig. 4b±g

illustrates the intensity distributions and interfer-

ence patterns, calculated from Eq. (3) at dierent

positions of the edge dislocation x0.

At given shift x0 the initially straight edge dis-

location bents when propagation distance in-creases. In two nulls located at the bent curve a

pair of screw dislocation appears.

Numerical simulations of the full governing

equation describing the beam propagation in a

linear medium in the paraxial approximation was

performed with a split-Fourier algorithm [24]. We

used three function to model the edge dislocation:

the Gauss±Hermite H 1;0 function xÀ x0=w0 as in

Eq. (1), the tanh-like beam tanh xÀ x0=b, where

b describes the amplitude pro®le near x x0, and

the step function P xÀ x0

, where P

xÀ x0

1 if

x > x0, and P xÀ x0 À1 if x < x0, that gives a

p-phase shift of the ®eld at x x0.

Some examples of numerical simulations (Fig.

5) show very good agreement with the predictions

of Eq. (3) within the resolution that was achievable

with split-step meshes. The number of vortices and

their positions are almost independent on the type

of functions used to describe the edge dislocation.

It means that the intensity gradient (that is dier-

ent in both cases) does not contribute considerably

in the interaction between the dislocations. Com-

paring these plots with the experimental data on

Fig. 3, an agreement between theory and experi-

ment may give us some con®dence to the model

proposed.

Fig. 4. (a) Calculated trajectories of the vortices with positive(solid lines) and negative sign (dash line). Intensity and inter-

ferogram for dierent shift of the edge dislocation x0: (b, c)

x0 0, (d, e) x0 0:3, (f, g) x0 0:7. Conditions: n 1, w0 1.




One can explain a break up of the edge dislo-

cation and the generation of additional screwdislocations in the following simple way. The ex-

pression (1) can also be written as

E x; y ; z 0 E 0 x2 i yxÀ xx0 À i yx0

Â exp

À x2 y 2

w20

; 4

which is a superposition of Gauss±Hermite modes

H mn x; y : E x; y ; z 0 H 20 i H 11 H 10 i H 01

H 00. As well known the amplitude distribution

of these modes have numerous intersections with

zero. Therefore the conditions of vanishing both

the real and imaginary parts can be satis®ed at

several points. These modes will acquire dierent

phase shifts (proportional to the indices of modes)

upon propagation. The positions of the defects are

thus not expected to remain stationary.

Therefore, the calculations based on the simple

model also showed that the total topological

charge does not conserved in free-space propaga-

tion when the both phase dislocations are on-axis.

For the ®rst time the violation of topological

charge conservation was mentioned by Soskin

et al. [14]. It was shown that during free-space

propagation of a light wave, which is a combina-tion of a Gaussian beam and an optical vortex

within a Gaussian envelope, additional vortices

appear or annihilate. In spite of any variation in

the number of vortices, the total angular momen-

tum is constant during the propagation. Even

when all vortices may be suppressed, the wave-

front of the beam still has folds, but in a smooth

surface without defects. These folds and not just

screw phase dislocations produce components of

angular momentum. Moreover, it was shown that

by some choices of parameters the combination of a Gaussian beam and a single-charge optical vor-

tex gives a wave with two single-charge vortices of

the same sign. In our case of interaction between

two topological phase defects we observed the

similar eect. By parametric wave mixing the vio-

lation of topological charge conservation was ob-

served in Refs. [25,26].

Now let us compare our results obtained in the

linear case with the nonlinear case studied in Ref.

[20]. In the nonlinear case by propagation of each

of the phase dislocation the diraction eect is

suppressed due to the nonlinear change of refrac-

tive index: they propagates as a vortex dark soliton

or as a stripe dark soliton. In the linear case the

diraction is strong enough (compare the initial

and ®nal beam sizes in Fig. 5). Nevertheless all

basic eects observed in the nonlinear case due to

the interaction between the two dislocations (the

stripe bending, the generation of a pair of new

vortices and the shift of initial vortex) present also

in the linear case. As it was mentioned in Ref. [20]

Fig. 5. (a, c, e, g) Numerically obtained beam intensity distri-

butions and (b, d, f, h) corresponding interferograms at a ®xed

propagation distance n 0 (a, b, e, f) and n 1 (c, d, g, h).

Plots were calculated with the edge dislocation modeled as

xÀ x0=w0 (a±d) or tanh xÀ x0=b with b 0:01 (e±h). Re-

sults found with step function P xÀ x0 are not shown because

they are very similar to ones obtained with tanh xÀ x0=b:Parameters: w0 1, x0 0:3:

D.V. Petrov / Optics Communications 188 (2001) 307±312 311



``the main eect produced by a vortex on the dark-

soliton stripe is associated with the vortex phase

gradient'' and apparently this is right for the linear

propagation.

In conclusion, the interaction of the vortex and

the edge phase dislocation nested in a Gaussian

beam induces nucleation of additional vortices of

both topological charges. Their positions depend

on the initial shift of the edge dislocation relative

to the vortex. The total topological charge is

conserved in the case when the edge dislocation is

o-axis and the initial vortex is on-axis, but does

not appear to be conserved in the on-axis case for

the both dislocations.

Acknowledgements

I greatly appreciate I. Freund for enlightening

discussions.

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Dmitri V. Petrov- Vortex-edge dislocation interaction in a linear medium

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