propagation dynamics of optical vortices in laguerre–gaussian beams

Optics Communications 250 (2005) 218–230

www.elsevier.com/locate/optcom

Propagation dynamics of optical vortices inLaguerre–Gaussian beams

F. Flossmann, U.T. Schwarz *, Max Maier

Naturwissenschaftliche Fakultat II, Universitat Regensburg, Universitaetsstr. 31, D-93040 Regensburg, Germany

Received 22 July 2004; received in revised form 8 February 2005; accepted 11 February 2005

Abstract

We have calculated the propagation dynamics of an initial off-axis vortex with topological charge 1 in Laguerre–

Gaussian background beams ðLG01 and LG0

7Þ, which are examples of background beams with non-generic dislocation

surfaces, on which the real and imaginary parts of the light field are zero. When initially a vortex with broad core (e.g.,

r-vortex) is embedded in the background beam, the dislocation surfaces are destroyed during propagation and two vor-

tices with opposite charge are created per dislocation surface in planes perpendicular to the propagation direction. For a

vortex with narrow core (e.g., point vortex) diffraction is important and leads to the birth of more than two vortices per

dislocation surface. These results are also valid for other background beams with dislocation surfaces, e.g., Hermite–

Gaussian and Ince–Gaussian beams. We investigated experimentally the spatial evolution of the intensity distribution

of an initial off-axis vortex with narrow core and topological charge 1 in LG01 and LG0

7 background beams. The exper-

imental results are in good agreement with the calculated intensity distributions.

� 2005 Elsevier B.V. All rights reserved.

Keywords: Phase dislocations; Optical vortices; Laguerre–Gaussian beams

1. Introduction

The propagation and interaction of phase dislo-

cations, in particular of optical vortices, is of inter-

est from a theoretical viewpoint [1–5] and for

0030-4018/$ - see front matter � 2005 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2005.02.032

* Corresponding author. Tel.: +49 941 943 2113; fax: +49 941

943 2754.

E-mail address: [email protected]

(U.T. Schwarz).

practical applications, e.g., atom guiding and

micromanipulation of particles [7,6]. A phase

dislocation is a stationary curved line in three dimen-

sions or a stationary point in two dimensions,

where the phase of the light wave is undetermined

and its amplitude is zero. For a generic dislocation

the integral over the phase change on a circuit that

encircles the line gives 2p [1–4]. In general, thedislocation line forms an arbitrary angle with

the propagation direction of the wave (mixed

ed.

mailto:[email protected]

F. Flossmann et al. / Optics Communications 250 (2005) 218–230 219

edge-screw dislocation). Special cases are non-gen-

eric edge dislocations, which are not lines but sur-

faces in space. In Ref. [4] the nodal surfaces of

higher transverse order modes of a monochro-

matic laser beam are described as examples fordegenerate and non-generic edge dislocation sur-

faces. Special cases of laser modes are Hermite–

Gaussian beams [8], Laguerre–Gaussian beams

[8] and Ince–Gaussian beams [9,10], which have

planes and hyperboloids as zero-amplitude

surfaces, respectively. Further examples of non-

generic dislocation surfaces are the circular and

elliptic cylindrical zero-amplitude surfaces of Bes-sel beams [11] and Mathieu [12,13] beams.

In order to investigate the geometry of the

three-dimensional vortex lines experimentally,

intensity distributions in successive planes perpen-

dicular to the propagation direction z have been

measured. The change of the position of the vortex

in the different planes is often called motion or

propagation dynamics of the vortex in the litera-ture (see next paragraph), even though there is

no time dependence but a dependence of the posi-

tion in the x–y plane on the spatial coordinate z. In

a similar sense the terms interaction and nucle-

ation have been used.

The propagation dynamics of optical vortices in

a Gaussian background beam has been investi-

gated in linear and nonlinear media [14,15].According to a model, the phase gradient of a

rotationally symmetric background beam gives

rise to a radial motion of the vortex in the direc-

tion of the transverse energy flow. The intensity

gradient of the background beam causes motion

of the vortex along contour lines of equal inten-

sity of the background beam [14–17]. So the gradi-

ents of the phase and intensity of the backgroundbeam act like driving forces for the motion of the

vortex. It has been shown that for a Gaussian

background beam an off-axis vortex moves in the

x–y plane (vertical to the propagation direction z)

on a straight line to the periphery [14]. In a repre-

sentation where the x and y coordinates of the

plane are normalised to the beam size w(z) of the

Gaussian beam, the vortex moves on a circle.The interaction of different kinds of dislocations

and the propagation of mixed edge-screw disloca-

tions has been investigated only in a few cases.

Nucleation of additional vortices has been shown

to occur by the interaction of a vortex with an edge

dislocation in a Gaussian background beam [18].

The transformation of a mixed edge-screw disloca-

tion embedded in a Gaussian beam into severalpairs of optical vortices in the near field and single

charge optical vortices in the far field was investi-

gated in Refs. [19,20]. Berry [21] has studied theo-

retically the evolution of helicoidal integer and

fractional phase steps and the resulting optical

vortices. Experimental observation of optical

vortex evolution in a Gaussian beam with an

embedded fractional phase step has been reportedin Ref. [22].

The propagation of an array of off-axis vortices

in a pseudo-nondiffracting background beam was

investigated in Ref. [23]. In particular, the propa-

gation dynamics of an initial J1 Bessel beam with

vortex in the centre, which was off-axis with

respect to the Gaussian envelope, was studied in

great detail [23]. The interaction of an off-axispoint vortex (topological charge 1) with the non-

generic edge dislocation surfaces of a J0 Bessel

beam, which have cylindrical shape, has been stud-

ied in Ref. [11]. It has been shown that the vortex

and the edge dislocation surfaces are combined to

form in a good approximation a mixed edge-screw

dislocation surface (strength 1), the ends of which

consist of a series of oblique dislocation lines.Intensity pictures in the x–y plane were measured

at different propagation distances and found to

be in good agreement with model calculations.

In this paper, we investigate theoretically and

experimentally the interaction of an optical

off-axis vortex with the non-generic dislocation

surfaces of Laguerre–Gaussian beams, which are

rotational hyperboloids. In addition the nucleationof the resulting mixed edge-screw dislocation lines

is studied. This paper is organised as follows. In

Section 2 the basis for the calculations of the

propagation dynamics of the dislocations is given.

Two different theoretical methods are presented.

The electric field distribution is calculated either

by a method using Fourier transformation [24]

or by a method using the expansion of the initialelectric field in terms of orthogonal Laguerre–

Gaussian functions. In Section 3, calculations of

the propagation dynamics of vortices with different

220 F. Flossmann et al. / Optics Communications 250 (2005) 218–230

combinations of cores and background beams

with special emphasis on an r-vortex in Laguerre–

Gaussian beams are presented. In Section 4, the

experimental setup is described. The experimental

results for a narrow-core vortex with charge 1 inLaguerre–Gaussian LG0

1 and LG07 background

beams are presented in Section 5, followed by the

conclusions in Section 6.

2. Theory

The results of the calculations of the following

sections are compared later with the experimental

results, which are obtained in planes perpendicular

to the propagation direction z of the light beam.

Therefore, we use in the following, the terminologyfor vortex points in transverse planes.

2.1. General

In this section, we treat the propagation of dif-

ferent kinds of vortices in Laguerre–Gaussian

background beams. The complex function describ-

ing a vortex can be written as

wðr;/Þ ¼ cðrÞeim/a : ð1ÞHere, c(r) describes the core function. The phase of

a vortex at position r00 = (a,0) is given by

/a ¼ arctany

x� a

� �: ð2Þ

In the literature describing experimental results the

following examples of core functions of vortices

with topological charge m = 1 are used (see e.g.,

[14,25]).

cðrÞ ¼ 1 for a point vortex; ð3Þ

cðrÞ ¼ jr� r00j for an r-vortex; ð4Þ

cðrÞ ¼ tanhjr� r00j

rc

� �for a tanh-vortex: ð5Þ

The r-vortex can also be written as

wðx; yÞ ¼ x� aþ iy ¼ rei/ � a: ð6Þ

The tanh-vortex is interesting because it contains

the point vortex and the r-vortex as limiting cases

[25]. For a Laguerre–Gaussian background beam

with beam radius w the tanh-vortex behaves like a

point vortex for a core with rc � w. In the limit of

large values of rc the first term in the expansion of

the tanh is jr�r00jrc

, i.e., it has the same radial depen-dence as the r-vortex. In practice, the core radius

rc should be substantially larger than the beam ra-

dius w of the background beam. For a

LG01 and LG0

7 Laguerre–Gaussian beam which are

used in the experiments the condition is rc > 2w

and 4w, respectively.

It should be noted that the point vortex (Eq.

(3)) is unphysical and cannot be realised experi-mentally. The term r-vortex (Eq. (4)) can be con-

fusing, since all generic free-space optical vortices

are locally of this type [4]. But this term is used

in the literature (see e.g., [14]). The r-vortex cannot

be realised experimentally, since the field increases

to infinity, but it is useful for describing the local

behaviour of the vortex. We have used the tanh-

vortex (Eq. (5)), because the extension of thevortex core in our experiments is small compared

to the extension of the background beam and it

is most suitable to describe the experimental re-

sults. The most general case for the local structure

of a vortex is described in Refs. [26,27].

We use Laguerre–Gaussian LGlp beams with the

azimuthal index l = 0 and the radial index p 6¼ 0

which have a maximum in the centre of the beam[8,28]. The radial index p determines the number

of bright or dark rings in the radial intensity distri-

bution. The vortex is put into the background

beam by multiplying the vortex function with the

background field.

2.2. Fourier transformation

In the calculation of the propagation of the vor-

tex we start with the electric field distribution

E(x,y, 0) of the light in the plane z = 0, where z

is the propagation direction, and calculate the elec-

tric field distribution E(x,y,z1) in the plane z = z1(near field distribution) [24]. This is done in the fol-

lowing way. Firstly, we make a two-dimensional

Fourier transformation of the light field; secondly,we multiply the Fourier transform with a propaga-

tion factor; thirdly, we perform an inverse Fourier

transformation back into real space, which yields


the field distribution in the plane z = z1. Normally

this is done with numerical methods. However, the

advantage of r-vortices in Laguerre–Gaussian

beams is that the Fourier transformations can be

done analytically in the paraxial approximation,saving computer time and yielding exact results,

e.g., for the number and position of vortices. The

far field distribution (angular spectrum) is calcu-

lated by just performing one two-dimensional

Fourier transformation.

2.3. Expansion in Laguerre–Gaussian functions

Laguerre–Gaussian functions are a complete,

orthogonal set of functions [8]. Therefore any

function f(r,/,z) can be expanded in a series of

Laguerre–Gaussian functions [29].

f ðr;/; zÞ ¼Xl;p

AlpLG

lpðr;/; zÞ: ð7Þ

The Laguerre–Gaussian functions are given by [8]

LGlpðr;/; zÞ ¼ Cl

pe� r2

w2ðzÞ�i kr2

2Rk ðzÞþiUl

pðzÞþil/

� 2r2

w2ðzÞ

� �l2

Llp

2r2

w2ðzÞ

� �: ð8Þ

Here, Llp are the Laguerre polynomials, Cl

p is a nor-

malizing factor, w(z) is the beam radius, Rk(z) is

the radius of curvature of the phase surfaces andUl

pðzÞ is the Gouy phase. w0 is the beam waist

and zR ¼ kw20=2 the Rayleigh range

Clp ¼

1

wðzÞ2p!

pðp þ lÞ!

� �1=2; ð9Þ

w2ðzÞ ¼ w20 1þ z2

z2R

� �; ð10Þ

RkðzÞ ¼z2 þ z2R

z; ð11Þ

UlpðzÞ ¼ ð2p þ lþ 1Þ arctan z

zR

� �: ð12Þ

The coefficients Alp of the expansion are calcu-

lated in the plane z = 0 as the scalar product of

the function f(r,/, 0) with the Laguerre–Gaussian

functions:

Alp ¼ hf ðr;/;0Þ; LGl

pðr;/;0Þi

¼Z 1

0

Z 2p

0

f �ðr;/;0ÞLGlpðr;/;0Þrdrd/: ð13Þ

The integral in Eq. (13) can be calculated analyti-

cally for an off-axis r-vortex with topological

charge m in a Laguerre–Gaussian LGlp back-

ground beam.

A simple closed-form analytical solution is

found for an on-axis r-vortex with charge m P 1embedded in a LG0

q background beam. We

write

f ðr;/; zÞ ¼ LG0qðr;/; zÞrmeim/ ð14Þ

and get for the expansion coefficients

Alp ¼ ð�1Þpþq wm

0

2m2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðp þ mÞ!

p!

sm!

ðq� pÞ!ðp þ m� qÞ!for l ¼ m and q� m 6 p 6 q; ð15Þ

Alp ¼ 0; otherwise: ð16Þ

With this result and Eq. (6) we can calculate the

expansion coefficients for an off-axis r-vortex with

charge m = 1 in a LG0q beam.

For a discussion of the results we present two

examples, which are relevant for the experiments.

1. An initial off-axis r-vortex with topological

charge m = 1 embedded at r00 = (a,0) in a

Laguerre–Gaussian LG01 beam, i.e., l = 0,

q = 1. In this case the beam can be written as

ðx� aþ iyÞLG01ðr;/; zÞ

¼ � w0ffiffiffi2

p LG10ðr;/; zÞ þ w0LG

11ðr;/; zÞ

� aLG01ðr;/; zÞ: ð17Þ

2. An initial off-axis r-vortex with topological

charge m = 1 in a Laguerre–Gaussian LG07

beam, i.e., l = 0, q = 7. The expansion of thisfunction is given by

ðx� aþ iyÞLG07ðr;/; zÞ

¼ �w0

ffiffiffi7

2

rLG1

6ðr;/; zÞ þ 2w0LG17ðr;/; zÞ

� aLG07ðr;/; zÞ: ð18Þ


It turns out that the propagation dynamics is

determined by the z dependent Gouy phases (Eq.

(12)) of the Laguerre–Gaussian functions of the

expansion. The superposition of the Gouy phases

of the three Laguerre–Gaussian beams causes arotation of the phase structure, which determines

the azimuthal position of the vortices. The impor-

tance of the Gouy phase for the propagation of a

superposition of modes has been already empha-

sised by Siegman [8] and Nye [4]. It has been stud-

ied for self-imaging beams in Ref. [30].

3. Calculations for vortices with different cores in

different background beams

3.1. r-vortex inside the zero-intensity circle of a LG01

beam

In the following discussion the calculations

have been done with an initial off-axis r-vortex inLaguerre–Gaussian background beams, because

the number of newly created vortices is lower than

for other vortex cores and it is possible to identify

them unambiguously. In addition, the propagation

Fig. 1. Calculated propagation dynamics of an initial off-axis r-v

zR = 75 cm). (a) a = 0.4w0, (b) a = 0.9w0, (c) a = 1.2 w0. Base plane: in

x/w � y/w. Top plane: trajectories of the vortices. The dotted curve

between: three-dimensional dislocation lines. Light grey curve: dislo

dislocation lines. The different ordinate scale in (b) should be noted.

dynamics is obtained by accurate and fast analyt-

ical calculations.

We start with the discussion of the interaction

of an off-axis vortex initially inside the zero-inten-

sity circle in the x–y plane with the dislocation sur-face of the LG0

1 beam, i.e., with the zero-amplitude

surface of the beam. The calculations have been

carried out for an r-vortex with a = 0.4w0

(w0 = 0.388 mm, zR = 75 cm) on the x-axis. The

original vortex with topological charge +1

destroys the non-generic dislocation surface of the

LG01 beam. Fig. 1(a) shows in a three-dimensional

view that at a distance z = 214.6 cm a new disloca-tion line in U-shape (dark grey) is created in addi-

tion to the dislocation line (light grey) originating

from the initial vortex. Both lines correspond to

mixed edge-screw dislocations. For large distances

(z > 700 cm) these lines become in a good approx-

imation screw dislocation lines because they are

nearly parallel to the propagation direction z.

The intersection points of the lines with planes per-pendicular to the z-axis correspond to vortices

with topological charge +1 of the original vortex

and +1 and �1 of the newly-created vortices.

The total topological charge is conserved.

ortex (m = 1) in a LG01 background beam (w0 = 0.388 mm,

itial intensity distribution in the normalised coordinate system

represents the zero-intensity circle of the initial LG01 beam. In

cation line of initial vortex; dark grey curves: newly-created


The projections of the dislocation lines into the

x-y plane normalised to w(z) are shown on top of

Fig. 1(a). The zero-intensity circle of the initial

LG01 beam is represented by the dotted curve. In

the following we use again the term motion ofthe vortex for characterising the change of the po-

sition of the vortex in successive planes perpendic-

ular to the propagation direction z, which actually

describes the geometry of the three-dimensional

vortex lines. After a short propagation distance z

of the light beam the outer vortices move approx-

imately on circles in opposite directions in the nor-

malised x–y plane because of their different charge,similar to vortices in a Gaussian beam. The origi-

nal inner vortex however moves in the representa-

tion with normalised transverse coordinates

towards the centre of the beam.

The trajectories of the vortices, in particular the

inward motion of the initial inner vortex (Fig. 1(a),

top) are explained in Ref. [31] using a model in

which the phase and intensity gradients of thebackground beam cause the motion of the vortex

[14–17].

3.2. r-vortex outside the zero-intensity circle of a

LG01 beam

We have carried out calculations for an r-vortex

initially outside the zero-intensity circle of the LG01

mode. In this case additional dislocation lines are

created immediately for z > 0 by breaking the non-

generic phase dislocation surface of the LG01 beam.

Three-dimensional pictures of the dislocation lines

are shown in Fig. 1(b) and (c) in coordinates norma-

lised tow(z) for an initial distance of the vortex from

the centre of 0.9w0 and 1.2w0, respectively. For a po-

sition of the initial vortex of w0=ffiffiffi2

p< a < w0 (Fig.

1(b)) the dislocation line (light grey) starting at the

position of this vortex has an inverted U-shape,

while the new dislocation line (dark grey) continues

to the far field. After some propagation distance a

new U-shaped dislocation line is created (see Fig.

1(b)). Therefore, one vortex or three vortices are ob-

served in planes perpendicular to the propagation

direction z. For a position of the initial vortex ofa > w0 three dislocation lines are observed from

the beginning to the far field (see Fig. 1(c)). On top

of Fig. 1(c) the projection of the dislocation lines

into the normalised x–y plane is shown. The original

vortex and one of the new vortices move after a

short distance approximately on a circle, while the

other new vortexmoves to the y axiswith decreasing

distance to the centre.

3.3. Comparison for a LG01 beam

A comparison of the dynamics of initial vortices

inside the zero-intensity circle ða < w0=ffiffiffi2

pÞ and

outside the beam waist (a > w0) in the representa-

tion, where the x and y coordinates are normalised

to the beam diameter w(z) shows the following twomain differences.

1. An initially inside vortex moves to a point close

to the centre on the y/w(z) axis (Fig. 1(a), top).

When the initial vortex is outside, it propagates

on a quarter circle to the y/w(z)-axis (Fig. 1(c),

top).

2. When the initial vortex is inside the zero-intensity circle, one new three-dimensional dis-

location line in U-shape, corresponding to two

vortices in a plane, is created at a distance zn,

which is the larger the smaller the off-axis dis-

tance a of the initial vortex is (Fig. 1(a)).

When the initial vortex is outside the beam

waist, two new independent three-dimensional

dislocation lines are created immediately inthe beginning of the propagation (for z > 0)

(Fig. 1(c)).

For a discussion of this different behaviour, we

have plotted the zero-curves of the real and imag-

inary parts of the field at propagation distances

z = 0 and 3 cm for two examples: a = 1.2w0 and

a = 0.4w0 in Fig. 2. On the zero-intensity circle atz = 0 cm both the real and imaginary part of the

light field are zero. In order to distinguish between

them the light grey circle for the real part has been

plotted at z = 0 cm (Fig. 2(a) and (c)) slightly smal-

ler than the dark grey circle of the imaginary part.

For the initial outside vortex V (Fig. 2(a)) the

two crossings of the dark grey zero-circle and the

dark grey zero-line along the x-axis at z = 0 cmare avoided for z > 0 (Fig. 2(b)). Due to the

destruction of the dislocation surface two cross-

ings V1 and V2 of the zero-curves of the real and

Fig. 2. Zero-curves of the real and imaginary parts of the electric field of an initially off-axis vortex in a LG01 beam. On the zero-

intensity circle at z = 0 cm ((a) and (c)) both the real and imaginary part of the light field are zero. In order to distinguish between them

the light grey circle for the real part has been plotted slightly smaller than the dark grey circle of the imaginary part. Initial shift a of the

vortex and propagation distance z: (a) a = 1.2w0, z = 0 cm; (b) a = 1.2w0, z=3 cm; (c) a = 0.4w0, z = 0 cm; (d) a = 0.4w0, z=3 cm. V is the

initial vortex. The newly created vortices V1 and V2 are marked by circles.


imaginary parts (light and dark grey curves,

respectively), corresponding to two vortices,

appear immediately. For the initially inside vortex

V (Fig. 2(c)) there is not only an avoided crossing

of the zeros of the imaginary part (dark grey

curves), but also of the zeros of the real part (light

grey circle and vertical line) of the field (see Fig.

2(d)). As a consequence, in the beginning of thepropagation there are no additional crossings of

the zero-curves of the real and imaginary parts

and the new vortices are created at larger

distances.

3.4. r-vortex in a LG0p beam (p > 1)

Calculations of the interaction of an initialr-vortex in higher order LG0

p beams (p > 1) with

p dislocation surfaces have shown the following re-

sults. An r-vortex in a Laguerre–Gaussian LG0p

background beam disturbs the phase dislocation

surfaces of the beam and produces one three-

dimensional U-shaped dislocation line or two

separate dislocation lines per dislocation surface,

corresponding to two vortices with opposite

charge in a plane. Finally, in total 2p + 1 vortices

are observed and the total charge is conserved.

We have calculated the trajectories of the vortices

for the example of a LG07 beam in the normalised

x–y plane and found that almost all of the vortices

move approximately on circles to the y/w(z) axisexcept the innermost, which moves to the beam

center. They reach the y/w(z) axis in the far field.

3.5. Different vortex cores

We have also carried out calculations for point

vortices and vortices with a tanh(r/rc)-core. It

should be emphasised that the tanh-vortex is espe-cially interesting, because the variation of the core

diameter rc provides also the limiting cases of a

point vortex and an r-vortex in a good approxima-

tion (last paragraph of Section 2.1). The general

result of the calculations is that an initial narrow

tanh-vortex or a point vortex generates more

Fig. 3. LG07 background beam with an initial off-axis tanh-

vortex (rc = 0.2 mm) at a propagation distance of z = 30 cm

(corresponding to Fig. 7(D)). Central part of the calculated

zero-curves of the real and imaginary parts of the light field.

The crossings of the light and dark grey curves correspond to

vortices. Some vortices are marked by circles.


new vortices than an r-vortex. Fig. 3 shows as an

example a LG07 background beam. For clarity only

the central part of the zero-curves of the real and

imaginary parts of an initial tanh-vortex with nar-

row core (rc = 0.2 mm) in a LG07 beam is given at a

propagation distance of z = 30 cm (corresponding

to the experimental situation in Fig. 7(D)). One

vortex V is clearly seen. A closer inspection ofthe figure shows additional very flat crossings of

the light and dark grey zero-curves of the real

and imaginary parts of the electric field, represent-

ing non-canonical vortices. To simplify identifica-

tion of these vortices, we have marked eight of

them by circles. Further vortices are created during

propagation to larger distances.

Finally, we would like to discuss the differ-ences between a narrow tanh-vortex (correspond-

ing approximately to a point vortex) and an

r-vortex or a tanh-vortex with broad core in a

LG beam. The disturbance by the strong diffrac-

tion of the narrow tanh-vortex (or the point

vortex) leads to the generation of many non-

canonical vortices, but causes a rather small shift

of the zero-curves of the real and imaginary

parts (see e.g., Fig. 3) of the non-generic phase

dislocations of the background beam. This

means that the non-generic dislocation surfaces

are partially maintained in a good approxima-tion. The effect of diffraction of the r-vortex is

negligible, but the disturbance of the phase dislo-

cations is much stronger. Therefore, the disloca-

tion surfaces are completely destroyed and a

well-defined number of vortices is created (2p

vortices for a LG0p background beam).

3.6. Different background beams

There exist many light beams which have non-

generic dislocation surfaces, on which the real

and imaginary parts of the light field are zero.

Examples are shape-invariant beams like Her-

mite–Gaussian beams [8] and Ince–Gaussian

beams [9,10] and propagation-invariant beams like

Bessel [11], Mathieu [12,13] and parabolic [32]beams.

A theoretical analysis of the above mentioned

shape-invariant beams has shown, that, similar

to Laguerre–Gaussian beams, an initial off-axis

r-vortex creates two vortices of opposite charge

per dislocation surface in a plane perpendicular

to the propagation direction. This has been dem-

onstrated as follows. Each of these beams can berepresented by an expansion into Laguerre–

Gaussian beams [8,10]. Therefore, each beam

can be written as a Gaussian envelope and a

polynomial in x and y of degree d = 2p. The

insertion of an off-axis r-vortex with topological

charge 1 increases the overall polynomial degree

by 1. Performing Fourier transformations on this

function (as described in Section 2.2), does notchange its polynomial degree, so there are up to

2p + 1 vortices for z > 0 and in the far field of

the beam, similar to a Laguerre–Gaussian LG0p

beam.

For propagation-invariant beams the analysis

is complicated. But we expect also that an initial

off-axis vortex with broad core creates two vorti-

ces of opposite charge per dislocation surface.This has been shown for a special case of a qua-

si-propagation-invariant beam [23]. The propaga-

tion dynamics of an initial J1-Bessel beam with


r-vortex in the centre, which was off-axis with re-

spect to the Gaussian envelope, was calculated

analytically. It was shown that two vortices are

created per dislocation surface of the Bessel beam

[23].The evolution of the intensity distribution of

high order Laguerre–Gaussian beams with an

off-axis vortex (see Section 5.3) is remarkable sim-

ilar to that of a Bessel beam with off-axis vortex

[11] despite the fact that the phase and far-field

intensity distribution of both beams are completely

different.

4. Experimental setup

For the experimental observation of vortices in

different planes perpendicular to the propagation

direction we use the setup shown in Fig. 4. The

diameter (2w0 = 1.1 mm) of a HeNe laser beam is

increased by telescope T1 consisting of lenses L1and L2 to a diameter of 18.3 mm. After passing

the aperture B1 with diameter 4 mm it illuminates

an amplitude grating G1 for the production of La-

guerre–Gaussian beams as described in [28]. The

amplitude grating is positioned at the beam waist

and only introduces the p phase jumps at the

zero-intensity curves of the LG modes. It thus

generates a superposition of several modes withdifferent beam waists which all have one of their

zero-intensity curves at this position. If the ratio

of the diameter of the innermost circular p phase

jump in the grating and the diameter of the illumi-

nating beam is adjusted to �2 by use of the aper-

ture B1, the LG01 component in the superposition

Fig. 4. Experimental setup. HeNe laser; T1, T2 telescopes

consisting of lenses L1, L2, L3, L4 with focal lengths 1.2, 20,

100, 30 cm, respectively; B1, B2 apertures; G1 amplitude

grating; G2 phase grating; CCD camera.

reaches over 80% [28]. In the case of the LG07 beam

the diameter ratio for the highest efficiency is �4.

Directly behind the grating G1 we use a 4-level

phase grating G2 for the production of a narrow

(rc = 0.15 mm) vortex of topological chargem = 1. The telescope T2 consisting of lenses L3

and L4 is then used to separate the refraction or-

ders of the gratings by aperture B2 and to adjust

the diameter of the beam waist and the divergence

of the beam. The beam intensity is observed with a

CCD camera behind the back focal plane of lens

L4, in which the Gouy phase shift of the beam is

zero and the vortex is relative to the beam still atits initial position. In the experiments we measure

the intensity distribution of the beam in successive

planes perpendicular to the propagation direction

z. Therefore, we use the terminology for vortices

in planes, e.g., vortex motion, interaction (see Sec-

tion 1).

5. Experimental results

5.1. Vortex outside the zero-intensity circle of a LG01

beam

We start with the simplest case of a vortex in a

Laguerre–Gaussian beam LG01, which has one

non-generic edge dislocation surface. In the begin-ning at z = 0 the vortex is localised at the point

r00 = (a,0), with a = 1.0 w0 (w0 = 0.388 mm, Ray-

leigh range zR = 75 cm), i.e., at the bright ring.

Fig. 5 presents the intensity patterns at different

propagation distances z. The last column shows

the intensity in the far field. The x and y coordi-

nates of the intensity patterns are normalised to

the beam diameter w(z) at the respective distancez. The experimental results, the calculated results

for a tanh-vortex (rc = 0.15 mm) and an r-vortex

are shown in rows 1, 2, and 3, respectively. In

the calculated pictures the zeros of the imaginary

and real parts of the light field are shown as dark

and light grey curves, respectively. The crossing

points of these curves correspond to the positions

of vortices.There is good agreement between the experi-

mental (row 1) and the calculated intensity pic-

tures (row 2). We find only minor differences

Fig. 5. Propagation dynamics of an off-axis vortex (m = 1, a = 1.0w0) in a LG01 background beam (w0 = 0.388 mm, zR = 75 cm). First

row: measured intensity distributions; second and third row: calculated intensity distributions for a tanh-vortex (rc = 0.15 mm) and an

r-vortex, respectively. Propagation distances: (A) z = 0 cm, (B) 30 cm, (C) 60 cm, (D) 90 cm, (E) far field. The light and dark grey

curves correspond to the zeros of the real and imaginary part of the light field. The closely spaced light and dark grey curves in the

margin of (b), (c), and (d) are artifacts of the numerical calculations. The boxes have a width of approximately 8.5w0.

Fig. 6. Intensity distributions at a propagation distance

z = 60 cm (corresponding to Fig. 5(C)). (a) Measured intensity

distribution. (b) Calculated zero-curves of the real and imag-

inary part of the field. (c) Calculated interference pattern,

(d) measured overexposed intensity distribution.


between the calculated intensity patterns for a

tanh-vortex (with rc = 0.15 mm) and an r-vortex(rows 2 and 3, respectively). However, the problem

with intensity patterns is that it is difficult to see

enough details. Therefore we included the calcu-

lated curves of the zeros of the imaginary and real

parts of the field, to show the exact positions of the

vortices.

In various papers [18–20,29] vortices are identi-

fied by performing interference experiments. How-ever, in the case of Laguerre–Gaussian beams the

unequivocal interpretation of interference pictures

is difficult. Problems in the localization of vortex

points investigated with interferometric methods

have been discussed in Ref. [33]. Fig. 6(a) and

(b) show as an example the measured intensity dis-

tribution at z = 60 cm (same as in Fig. 5(C)) and

the calculated zero-curves of the real and imagi-nary parts of the field, respectively. In the center

of the beam five crossings V1–V5 of the dark

and light grey curves, corresponding to vortices,

are seen. The very flat crossings V2 and V3 lie in

a dark zone and it is practically not possible to


identify them in the experiments. In this region the

initial edge dislocation surface is retained in a

good approximation. Vortices V1, V4 and V5

can be principally identified in interference pic-

tures. However, inspection of Fig. 6(c), where thecalculated interference pictures corresponding to

Fig. 6(a) are given, shows that it is difficult to lo-

cate these vortices. Therefore, we did not carry

out interference experiments.

We have tried to identify the vortices V1, V4

and V5 from overexposed intensity pictures. In

Fig. 6(d) the overexposed measured intensity dis-

tribution corresponding to the case of Fig. 6(a) isshown. The dark spots representing the vortices

V1, V4 and V5 are clearly seen. Their positions

in the measured intensity distribution are in good

agreement with the positions of the calculated

crossings of the dark and light grey curves. Similar

results have been obtained in the comparison be-

tween experiments and calculations at various

propagation distances.The crossings of the dark and light grey zero

curves in Fig. 5 show that during propagation

additional vortices are created. For example, there

are four vortices in addition to the initial narrow

tanh-vortex (Fig. 5(c) and (d)) and two vortices

in addition to the initial r-vortex (Fig. 5 (c)–(�)).The reason for this difference is the stronger dif-

fraction [21] of the narrow tanh-vortex, which

Fig. 7. Propagation dynamics of an initial off-axis vortex (m = 1, a = 0

First row: measured intensity distribution. Second row: calculated int

distances: (A) z = 0 cm, (B) 7cm, (C) 15 cm, (D) 30 cm, (E) 70 cm. Th

leads to more crossings of the zero-curves of the

real and imaginary parts of the field. There is a fur-

ther difference between the development of the

beam with an initial narrow tanh-vortex and r-vor-

tex. The tanh-vortex represents a weaker distur-bance of the non-generic edge dislocation than

the r-vortex. Therefore the dark and light grey zero

curves are practically not separated by the tanh-

vortex along the lower part of the zero-intensity

circle (Fig. 5(b) and (c)), i.e., this part of the

non-generic edge dislocation surface survives in a

good approximation.

5.2. Vortex inside the zero-intensity circle of a LG01

beam

We have done experiments with a vortex in a

Laguerre–Gaussian LG01 beam located at

a = 0.6w0 (w0 = 0.388 mm), i.e., just inside the

zero-intensity circle. During propagation from

z = 0–70 cm a small part of the initial zero-inten-sity circle of the LG0

1 background beam gets

bright and a dark spot, corresponding to the

initial vortex, is observed. In the x–y plane the

vortex moves away from the x-axis, first towards

the y-axis and then parallel to the y-axis to the

periphery of the beam. In the x–y plane norma-

lised to the beam diameter w(z) this vortex

moves away from the x/w-axis and approaches

.9w0) in a LG07 background beam (w0 = 0.48 mm, zR = 114 cm).

ensity distribution for a tanh-vortex (rc = 0.2 mm). Propagation

e boxes have a width of approximately 8w0.


a point on the y/w-axis close to the centre of the

beam (see top of Fig. 1(a)). The behaviour of

this vortex in the LG01 beam is in contrast to

that of a vortex in a Gaussian beam, which

moves in the x–y plane on a straight line parallelto the y-axis outwards and in the normalised x–y

plane on a quarter circle to the y-axis. We dis-

cussed this motion in detail in Ref. [31]. The

measured intensity patterns at different distances

z were in good agreement with that calculated

for a tanh-vortex.

5.3. Vortex in a LG07 beam

In Fig. 7 the results for a LG07 background

beam with a vortex at position r00 = (a,0), with

a = 0.9w0 (w0 = 0.48 mm, zR = 114 cm), i.e., on

the second bright ring of the LG07 beam, are pre-

sented for different propagation distances z. The

first and second row show the experimental results

and the calculated intensity distribution for a tanh-vortex (rc = 0.2 mm), respectively. The measured

and calculated intensity pictures agree very well.

In the beginning of the propagation range a dark

spot is observed on the second bright ring (Fig.

7(A) and 1). During propagation a bright or dark

spiral is formed, the ends of which move along the

x-axis in opposite directions, one to the beam cen-

ter and the other outwards (Fig. 7(B) and (C)). Inthis way the number of windings of the spiral

increases. When the inner end of the spiral has

reached the beam centre (Fig. 7(D)), a vortex is left

in the centre and the end of the spiral changes its

orientation by 180�. Then it propagates away from

the centre and the winding number decreases (Fig.

7(E)).

6. Conclusions

In this paper, the propagation dynamics of vor-

tices with different types of core functions (point,

tanh, r) in Laguerre–Gaussian background beams

was investigated. The intensity distributions of a

narrow initial off-axis vortex with topologicalcharge 1 in LG0

1 and LG07 background beams were

measured at different propagation distances and

found to be in good agreement with calculations.

The electric field distribution was calculated with

two methods, one using Fourier transformation

and the other expansion in terms of the orthogonal

set of Laguerre–Gaussian functions.

The interaction of an off-axis r-vortex with aLG0

1 beam was calculated for three cases: the

vortex initially inside the zero-intensity circle of

the beam, in between the zero-intensity circle

and the beam waist and outside the beam waist.

When the vortex was initially inside, after some

propagation distance, which depends on the dis-

tance of the initial vortex from the beam center,

a U-shaped three-dimensional dislocation lineappears corresponding to two new vortices in

planes perpendicular to the propagation direc-

tion. In contrast, for a vortex initially outside

the beam waist two separate new dislocation

lines are formed immediately for z > 0 and

extend to the far field region. When the initial

vortex is in between the zero-intensity circle

and the beam waist, the situation is more com-plex. At z > 0 first a new dislocation line and

an inverted U-shape dislocation line (with one

end at the initial vortex) are observed. Later at

larger distances a U-shaped dislocation line is

created. The newly-created vortices are often dif-

ficult to observe experimentally, because they are

in regions of very low intensity.

We have calculated the propagation dynamicsof a narrow tanh-vortex and compared it with an

r-vortex in a Laguerre–Gaussian background

beam. Due to the strong diffraction of the narrow

vortex the number of newly-created vortices in a

plane is larger than for an r-vortex, which gener-

ates p pairs of oppositely charged vortices in a La-

guerre–Gaussian LG0p beam. The original

dislocation surfaces of the Laguerre–Gaussianbeam are partially maintained in a good approxi-

mation, when a narrow vortex is embedded. But

they are completely destroyed by an r-vortex.

We have also considered other background

beams with non-generic dislocation surfaces, e.g.,

Hermite–Gaussian, Ince–Gaussian, Bessel and

Mathieu beams. Similar to a LG background

beam we expect an initial off-axis vortex to destroythe dislocation surfaces and create at least two vor-

tices per surface in a plane perpendicular to the

propagation direction.


Acknowledgements

The authors thank Dr. S. Sogomonian for his

contributions in the initial phase of this work.

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