15 October 2001
Optics Communications 198 (2001) 49±56
www.elsevier.com/locate/optcom
Transversal optical vortex
V.A. Pas'ko, M.S. Soskin, M.V. Vasnetsov *
Institute of Physics, National Academy of Sciences of Ukraine, Prospect Nauki 46, Kiev 28, Ukraine
Received 14 May 2001; accepted 1 August 2001
Abstract
A structure of wavefront edge dislocations and associated ``transversal'' optical vortices in an interference ®eld of
two two-dimensional Gaussian beams is analyzed. It was shown that the optical vortex rotation is directed toward the
area of higher phase velocity in the interference ®eld (the origin of the phase velocity variation is due to the Gouy eect).
The conditions for the reversal of the sign of rotation were found as well as for annihilation of two edge dislocations.
Topological reaction of ``unfolding'' of an edge dislocation, which happens when vortex collides with a phase saddle, is
studied in details. Ó 2001 Elsevier Science B.V. All rights reserved.
PACS: 42.25
Keywords: Optical vortex; Phase singularity; Wavefront dislocation
1. Introduction
It was shown in last decades that light ¯ux can
produce vortices, similar to vortices in a liquid [1].
The vortex appears around a wavefront dislocation, which is a continuous line in space, where the
®eld amplitude vanishes and the phase is undetermined (singular). According to the classi®cation
introduced by Nye and Berry [2], a monochromatic light wave can possess two main types of
phase singularities: screw wavefront dislocation
and edge dislocation, while mixed edge-screw dislocation is most common situation. Nowadays the
term ``optical vortex'' (OV) introduced in Ref. [3]
became widely used, re¯ecting the general feature
*
Corresponding author. Tel.: +380-44-265-14-22; fax: +38044-265-15-89.
E-mail address: mvas@iop.kiev.ua (M.V. Vasnetsov).
of phase singularities: phase circulation around the
dislocation line. Therefore pure screw dislocation
is a core of a ``longitudinal'' OV, and edge dislocation produces a ``transversal'' OV, with respect to
the wave propagation direction.
In our recent paper [4] we have analyzed how an
edge dislocation of a wavefront can be created in
an interference ®eld of two paraxial Gaussian
beams. Destructive interference of co-axial beams
results in appearance of a zero-amplitude circle,
whose radius and position are determined by relative phases and amplitudes of the beams. Around
the circle, which is a circular edge dislocation, a
sub-wavelength area of circular light ¯ow was
detected within a loop of a separatrix, which divides the light current within and outside the dislocation. The separatrix possesses a self-crossing
in a point of the light current stagnation (phase
saddle) which, depending on the ratio of the
interfering beams amplitudes, can be located as
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 4 8 7 - 0
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V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
within, as outside the zero-amplitude circle. Therefore, two possible directions of the torous ¯ow
within the separatrix loop can be realized. There is
also a situation when the saddle collides with the
vortex, what results in the reversing the sign of
light circulation and a birth of two new circular
edge dislocations.
The purpose of the present paper is to describe
in more details the behavior of a transversal OV
and describe a topological reaction, which occurs
in the saddle±vortex collision. The central question
is to ®nd the condition, which determines the direction of a transversal OV circulation around the
dislocation line.
In contrast to the con®guration used in Ref. [4],
we now simplify the problem to two-dimensional
(2D) (cylindrical) waves. Actually, this approach
gives the same results as for three-dimensional
(3D) Gaussian beams, because the reactions are
governed by general topological laws. However,
the choice of 2D waves for the ®eld description
makes possible creation of linear edge dislocations
and does not require any account for polarization
components but one, what is important for the
Poynting vector calculation. Also, there is a possibility to compare the results with a general constrains derived in Ref. [5].
2. Interference of 2D Gaussian beams
The geometry of waves is explained in Fig. 1. A
structure of a single 2D Gaussian beam is shown in
Fig. 1(a), with the beam waist at z 0 position.
The beam is polarized along Y-axis and propagates along Z-axis, with diraction spread out in
X-axis directions. For this 2D beam of Gaussian
shape the solution satisfying paraxial wave equation looks as follows:
r
w0
x2
kx2
E x; z EG
exp
i
exp
w
w2
2R z
1
z
arctan kz U ;
1
2
zR
where EG is the amplitude parameter, w0 is the
waist parameter, k is the wave number, zR is the
Rayleigh range, zR kw20 =2, R z is the radius of
Fig. 1. (a) A schematic view of a 2D Gaussian beam with the
waist located at z 0 position. (b) Destructive interference of
two 2D Gaussian beams with dierent waist parameters. Solid
lines are Gaussian amplitude distributions in the waist plane.
The resulting amplitude distribution in the waist is shown by
dashed line. Negative amplitude denotes the p phase shift. (c) A
wavefront sequence near the waist (U1 1): there is a lack of
one wavefront between the dislocation lines. The wavefront
ledge before the waist is directed along the direction of the wave
propagation and opposite after the waist.
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V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
the wavefront curvature in the XZ plane,
1=2
w w0 1 z2 =z2R , U is a constant phase term.
The solution (1) is very similar to the description of 3D Gaussian beam, except another degree
of amplitude diminishing with the propagation
path z, w0 =w1=2 instead of (w0 =w) for 3D
Gaussian beam, and coecient 1=2 before the
Gouy phase shift arctan(z=zR ).
To produce a pair of edge dislocations parallel
to the Y-axis in the plane z 0, let us make a superposition of two 2D Gaussian beams with different amplitudes E1 > E2 , waist parameters w2 >
w1 and a phase shift p between their phases U1 and
U2 very similar to how it has been done for 3D
Gaussian beams [4]:
x2
E x; z 0 E1 exp
exp iU1
w2
1 2
x
E2 exp
exp iU2 :
2
w22
At the waist plane, both waves have plane wavefront, and their relative phases are locked as U1
U2 p, thus producing destructive interference, as
shown in Fig. 1(b). The edge dislocation position
(zero amplitude of the ®eld) is determined by
s
w21 w22
x0
ln E1 =E2 :
3
w22 w21
For the analysis below, we de®ne the ratio E2 =E1
as the governing parameter g. To ®nd a wavefront
structure nearby the waist, and, generally, how the
phase behaves at the vicinity of the dark line, we
have to calculate the interference ®eld as a sum
of two complex amplitudes E x; z E1 x; z
E2 x; z and then calculate the resulting phase
U x; z:
tan U x; z
Im E x; z
:
Re E x; z
4
Thus the equation for a wavefront family U x; z
0; 2p; 4p; . . . can be found as ImE x; z 0.
A family of wavefronts (crests) is shown in Fig.
1(c) for g 0:83, w1 10, w2 100, U1 1 (here
and below all ranges are normalized for k 1).
First, it is seen from Fig. 1(c) that there is a lack
of one wavefront inside the interval bordered by
edge dislocations. Before the waist, the wavefront
has a ledge along the direction of propagation,
after the waist the ledge is directed opposite. The
ledge is smoothed far from the waist and the resulting lack of the wavefront results in a small
wavelength dierence within the interval between
the dislocations and outside, or phase velocity
variation. The phase velocity is larger within the
interval between the dislocations, the physical
reason is the Gouy eect [6]: a spatially compressed light beam increases the phase velocity on
the axis. To calculate the phase velocity of a 2D
Gaussian beam in the waist, Eq. (1) can be applied
x2
vc 1 2
2zR
1
2kzR
1
;
5
where c is the speed ofplight.
There is a distance xv
where v c: xv w0 = 2. When x < xv , the phase
velocity exceeds c, and vice versa. Due to this
phase velocity variation, wavefront being plane at
the waist gradually transforms to cylindrical at the
far ®eld. We can also calculate the phase velocity
for the superposition of the beams vR :
vR
E1 x E2 xv1 v2
;
v2 E1 x v1 E2 x
6
where vR , v1 and v2 are functions of the transverse
coordinate x. For the case of constructive interference, minus sign in the numerator and denominator will be replaced by plus. The higher
transversal compression of a beam results in the
higher phase velocity, therefore a narrow beam
with the waist parameter w1 is a little bit faster at
the waist than the wide beam with the waist parameter w2 . We note, however, that the possible
presence of an edge dislocation can drastically
perturb the smooth velocity dependence along the
transversal coordinate nearby the dislocation.
Fig. 2 shows a family of equiphase lines in
the cross-section XZ in the vicinity of the edge
dislocation, for g 0:83. There is an additional
wavefront outside the inner stripe between the
dislocation lines, and a point where it enters the
periodic sequence: here, at the phase saddle, lines
of U x; z 0 and U x; z 2p join.
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V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
the zero-amplitude line is localized in a very narrow region. In comparison with the wavelength,
the separation between the vortex and the saddle is
extremely small and amounts of about 10 2 of the
wavelength.
3. Topological reactions in an interference ®eld of
2D Gaussian beams
Fig. 2. Lines of equal phase of the interference ®eld produced
by two 2D Gaussian beams in the cross-section XZ nearby the
edge dislocation, for g 0:83, U1 0. Saddle point is indicated
as a circle, vortex as a dot. There is a p-hop of the phase at the
zero-amplitude point (core of the vortex).
The equation for determination of the saddle
position xs can be obtained as the coordinate of the
point where a wavefront U x; z 0 splits
kx2
1
kx2
1
g k 2
k 2
2zR2
2zR1 2zR1
2z
R2
2
x
x2
exp
:
7
w21 w22
Analysis shows that the position of the saddle
can be as inside the interval x0 (xs < x0 ), as outside
it (xs > x0 ). First, we shall analyze the case xs < x0
which is similar to the described in Ref. [4] circular
edge dislocation structure. The phase map looks
identical, manifesting the vortices around zeroamplitude lines for g 0:83 (Fig. 2), and we can
seek for some particular details now. The distance
between vortex and saddle Dx depends on the
beam parameters and can be estimated from Eq.
(7) for w2 w1 as:
Dx
2x20 w21
:
2k 2 w21 x0
8
The physical sense of the saddle point is the
condition of the energy current stagnation: vortex
¯ow is exactly compensated by the longitudinal ¯ow, and a line of standing wave is created.
Therefore the vorticity area is ended in the saddle
point, and the whole torous light current around
With increasing g, x0 diminishes according
to Eq. (3), and two dislocation lines come closer
as well as saddles. The saddles, which are located inside the interval between the vortices, will
meet at x 0 point when g zR2 2kzR1 1=
zR1 2kzR2 1. This is the ®rst stage of the collapse
(Fig. 3(a)). Then saddles appear again separately,
but are located now symmetrical on the Z-axis
(Fig. 3(b)). The maximum distance between the
saddles will be reached for g 0:995 and amounts
approximately to 1. In this moment the distance
between vortices becomes equal to the distance
between saddles. With further increase of g the
saddles as well as vortices come closer. Finally,
vortices and saddles collide and annihilate altogether at the point x 0 (Fig. 3(c)).
With decreasing of g, saddle point moves toward the vortex and meets it when g g0
exp w21 w22 =2 w21 w22 (this condition corresponds to the coincidence of the dislocation line
to the line of equality of phase velocities of the
interfering waves, as will be discussed below). In
the moment of this collision a saddle±vortex point
appears, which is a ``pathological'' topological
object. After the crossing event, the saddle appears
outside the dislocation line, and thus the light ¯ow
circulation around the dislocation reverses the direction. This reaction is accompanied with two
similar objects creation outside the plane z 0, as
Fig. 4 shows. These two new dislocations have the
same ``sign'', or direction of the current circulation
around them, as the initial one, so there is again a
lack of one wavefront in the central part of the
beam. If we start from smaller value of g, g < g0
which corresponds to the existence of three edge
dislocations, and then gradually increase g until
it reaches g0 value, the topological reaction will
look as follows: three parallel zero-amplitude lines
V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
53
Fig. 4. Lines of equal phase in the cross-section XZ. Both wave
crests (solid lines) and troughs (dashed lines) are shown, as
calculated from the condition ImE x; z 0, U1 0:1. Position of vortices is shown by dots, saddles are not shown here.
the total number of critical points in the XZ crosssection is six. The signs of vortices, which are out
of the waist plane (``positive''), are opposite to the
sign of vortex localized in the waist plane (``negative''). After the collision, the remaining objects
are one vortex with positive sign and one saddle.
Of course, we discuss the events happening only
in the x > 0 half plane, in the x < 0 half plane
the same process happens symmetrically.
Let us examine now how new edge dislocations
can sprout from the saddle±vortex line. First, we
have to ®nd the conditions for edge dislocation to
exist out of the waist plane. There are two necessary conditions, one for destructive interference,
another for the equality of the interfering waves
amplitudes. The phase condition looks as follows:
kx2
2R1 z
Fig. 3. Collapse of edge dislocations. (a) Equiphase lines around
the dislocations which are close to the beam center, saddle
points have joint. (b) Vortices approach closer, saddles are split
along the Z-axis. (c) All topological objects collide and annihilate at g 1.
accompanied with three lines of stagnation (phase
saddle lines) come closer until the junction. Altogether, there are three vortices and three saddles,
1
z
arctan
2
zR1
kx2
1
z
arctan
0:
zR2
2R2 z 2
9
There is a trivial solution z 0 (the waist plane),
where two initial edge dislocations are located, and
a surface of destructive interference where new
edge dislocations can appear
1 z2 z2R1 z2 z2R2
2
x
kz
z2R2 z2R1
z
z
arctan
arctan
;
10
zR1
zR2
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V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
natural, however, that the axial velocity is usually
larger: diraction is responsible for it. Ordinary
focusing cannot change this feature, and some
tricks are necessary to apply for the construction
of the wanted beam. As it is seen from formula (6),
subtraction of a ``slow'' beam from a ``fast'' one
leads to the increase of its axial velocity, and coherent addition will decrease it. Therefore we have
chosen 2D analog of a Laguerre±Gaussian mode
LG01 as the fast beam
Fig. 5. Solid line: cross-section of the surface of destructive
interference (p phase dierence between interfering beams), for
x > 0. Dashed lines: cross-sections of the surfaces of the amplitude equality. For g > g0 there is only one possible crossing
point and therefore one zero-amplitude line (edge dislocation).
For g < g0 there are three points of the crossing and therefore
three edge dislocations (saddle points are not shown here).
which can be approximated nearby the waist as
x2
zR1 zR2
2z2 2z2R2 zR1 zR2
z2 R1
:
k zR1 zR2
3kzR1 zR2 zR1 zR2
11
The cross-section of the surface of the destructive interference is shown in Fig. 5 by solid line. To
®nd the location of edge dislocation out of the
waist plane, the surface of the amplitude equality
can be found also, as shown in Fig. 5 by dashed
line. Its position depends on the governing parameter g, whereas the line of the destructive interference does not depend on the amplitude ratio
of the beams. This line crosses the X-axis exactly
in the point which corresponds to the equality of
phase velocities of the interfering waves. This coincidence is hardly occasional, and can give a hint
for understanding of the vortex behavior. The
consequence can be made: the vortex rotation is
directed toward the area of higher phase velocity.
4. Inversion of an optical vortex circulation
To prove this assumption, we have to construct
somehow a beam, which possesses a pair of edge
dislocations at the waist and the phase velocity
between them smaller than peripheral one. It is
ELG E1
r 2
w0 4x
1
exp
w w2
5
z
i arctan ikz ;
2
zR
x2
kx2
i
2
w
2R z
12
and 2D Gaussian beam with the amplitude parameter E2 and the same waist parameter w0 as the
slow beam. Again, the amplitude ratio g E2 =E1
will serve as the governing parameter. The 2D LG01
mode possesses increased Gouy phase shift, what
insures higher phase velocity, and inherent phase
defect in a form of a p step along the surfaces
x2 w20 1 z2 =z2R , where amplitude ELG gets zero
value. A coherent in-phase (or out of phase) addition of a 2D Gaussian beam will destroy the
zero-amplitude surfaces, leaving the only two zeroamplitude lines at the waist plane, which are linear
edge dislocations.
Fig. 6(a) shows the result of the destructive interference of the beams. As the amplitude of the
2D LG01 mode in the waist is negative within the
interval 2w0 , the subtraction of the 2D Gaussian
beam will diminish the axial phase velocity here,
whereas outside the interval the interference will
increase the phase velocity, what is necessary for
our task. The resulting wavefront family is shown
in Fig. 6(b). In contrast to Fig. 1(c), an additional
wavefront sheet appears within the interval bordered by edge dislocations.
The detailed structure of the phase around the
dislocation is shown in Fig. 7. The saddle position
now is outside the interval between dislocations,
and the light vortex rotates around dislocations
from interior to outside the interval. Moreover, we
can add the slow 2D Gaussian beam instead of the
subtraction, and obtain the reversed situation.
V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
55
Fig. 7. Lines of equal phase of the destructive interference ®eld
produced by 2D LG01 mode and 2D Gaussian beam in the crosssection XZ nearby the edge dislocation, for g 0:5 (compare
with Fig. 2).
5. Conclusion
Fig. 6. The same as Fig. 1(b) and (c), but for the destructive
interference ®eld produced by 2D LG01 mode and 2D Gaussian
beam. (a) Amplitude distribution in the waist, dashed line is the
resulting amplitude (negative amplitude denotes the p phase
shift). (b) A wavefront sequence near the waist: there is an extra
wavefront between the edge dislocations. The ledge is directed
backward to the direction of the wave propagation before the
waist and along the direction of propagation after the waist.
This veri®cation absolutely agrees with the expectations.
A single edge dislocation can be constructed in
a similar way, if we take a 2D mode
r
w0 x
x2
kx2
exp
i
E x; z E1
w2
2R z
ww
3
z
i arctan ikz
13
2
zR
as the fast beam and add/subtract a 2D Gaussian
beam.
Summarizing, we have found how the transversal OV, created in an interference ®eld of two
2D Gaussian beams, transforms in topological
reactions of edge dislocation collapse and ``unfolding'' [7]. The results obtained are valid as well
for 3D beams.
The rule for determination of the direction of
a transversal OV circulation around an edge dislocation was established: the light vortex rotates
toward the region of the higher phase velocity.
This rule can be used for the explanation of the
natural structure of phase circulation around dark
rings (circular edge dislocations) observed by
Karman et al. [8]. The physical reason of this
feature is not clear enough and this fundamental
property of light propagation seems worthwhile
for further investigations.
Acknowledgements
This work was supported by EOARD, Air
Force Oce of Scienti®c Research and Development, Air Force Research Lab, USA (Partner
Project P051).
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V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
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