Vol.2, No.3, 201-204 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.23031
Gaussian beam with non-spiral optical vortex
Xiu-Min Gao1, 2, Song Hu1, Jin-Song Li3, Han-Ming Guo2, Jian Wang1, Song-Lin Zhuang2
1Electronics and Information College, Hangzhou Dianzi University, Hangzhou, China;
2Optics and Electronics College, University of shanghai for Science and Technology, Shanghai, China
3Optics and Electronics College, China Jiliang University, Hangzhou, China
Received 24 November 2009; revised 28 December 2009; accepted 25 January 2010.
ABSTRACT
Optical vortex has attracted much interest re-
cently due to its novel properties and applica-
tions. In this paper, the focusing properties of
Gaussian beam containing one non-spiral opti-
cal vortex are investigated by means of scalar
diffraction theory. Simulation results show that
topological charge of non-spiral optical vortex
affects optical intensity distribution in focal re-
gion considerably, and non-spiral focal pattern
may also occur. Multiple intensity peaks may
appear companying with center main focal spot
under condition of higher topological charge. In
addition, the number of weak intensity peak
outside of the center main intensity peak is re-
lated to the value of topological charge.
Keywords: Focusing Properties; Gaussian Beam;
Optical Vortex; Scalar Diffraction Theory
1. INTRODUCTION
Singular optics studying optical vortex has grown rap-
idly recently because optical vortex has some interest-
ing properties [1-5] and promising applications [6-8].
And common optical vortex refers the phase singularity
in light wavefront. In practice, optical vortex contains
optical orbital angular momentum, which can be used
to construct highly versatile optical tweezers [9]. In the
focal reign of focusing beam that contains optical vor-
tex, the optical momentum can be transferred to mi-
cro-particle, namely, light energy can be transformed
directly to dynamic energy of particle, which has been
used to construct experimentally microscopic optical
rotator. In recent years, there also has been much interest
in optical vortex propagation [10-16]. From a funda-
mental perspective they enable the study of phase singu-
larities in a dynamical context [11,12], where twists,
loops and knots in the path of the vortices have been
shown to appear [13-15].
Spiral optical vortex has attracted much and has been
investigated intensively and extensively, however, non-
spiral optical vortex induces little attention. In fact,
non-spiral optical vortex may own more interesting
properties due to more flexible vortex pattern. The
present paper is aimed at studying focusing properties
of Gaussian beam containing one non-spiral optical
vortex are investigated by scalar diffraction theory. The
principle of the focusing this non-spiral vortex Gaus-
sian beam is given in Section 2. Section 3 shows the
simulation results and discussions. The conclusions are
summarized in Section 4.
2. PRINCIPLE OF THE FOCUSING
GAUSSIAN BEAM WITH ONE
ASYMMET4IC OPTICAL VORTEX
In the focusing system we investigated, the Gaussian
beam contains one non-spiral optical vortex, and con-
vergences through a lens. According to scalar diffrac-
tive theory, the relative amplitude distribution in focal
region of the focusing optical system is [17],
  
2
00
,,,exp() expsin
j
i
Ur zEiikr


 


 


coscos sinikzd d



  
 (1)
where E(θ,φ) indicates the electric the amplitude of
electric field of Gaussian beam that contains one
non-spiral optical vortex, and can be written as,
  
2
022
sin
,exp expsinEA im
NA w




 




 (2)
where A0 is a constant, where w= w
0/rp, w0 is waist
width (defined as radius) of the incident Gaussian
beam.
Parameter rp is the outer radius of the beam, and m is
the topological charge of the non-spiral optical vortex.
The phase is the sine function of azimuthal angle,
which is not spiral distribution and can be changed by
topological charge. The optical intensity distribution in
focal region can be calculated quantitatively by means
of substituting Eq.2 into Eq.1. It should be note that

X. M. Gao et al. / Natural Science 2 (2010) 201-204
202
the intensity distribution in focal region is proportional
to the modulus square of Eq.1, and in this paper the
modulus square of Eq.1 is calculated numerically.
3. RESULTS AND DISCUSSIONS
Without losing generality and validity, the intensity is
normalized by optical intensity maximum, and it was
proposed that 0.6NA , 1w. Firstly, the topological
charge of the optical vortex is chosen as 1. Figure 1(a)
illustrates the phase wavefront distribution under condi-
tion of m = 1 in polar coordinates. The radial coordinate
indicates the phase value. It can be seen that the optical
vortex is non-spiral and turns on off-axis distribution.
The Figure 1(b) shows that optical intensity in focal
region of this kind of vortex Gaussian beam. It should
be noted that Vr and Ψ indicate radial and azimuthal
coordinates, respectively. Ψ ranges from 0 to 2π. And
the distance unit in radial direction is k-1, here k is the
wave number of the focusing beam. We can see that the
focal spot is asymmetric remarkably, and does not locate
at optical axis. The cause of this focal pattern is the
asymmetry characteristics of the optical vortex that was
embedded in the incident Gaussian beam. There the
symmetry property of the optical vortex affects focal
pattern.
(a)
(b)
Figure 1. The (a) phase distribution for m = 1 and
the (b) corresponding focal intensity distribution.
Topological charge is important parameter optical
vortex, here, the effect of different topological charge on
focal pattern is investigated. Figure 2(a) illustrates the
phase distribution for m = 2. It can be seen that there are
four phase maximums and locate around optical axis.
Topological charge m affects phase wavefront distribu-
tion considerably. Figure 2(b) show us the intensity dis-
tribution in focal region under condition of m = 2.
From this figure, we can see that there is one center
main intensity spot on optical axis. Four optical intensity
peaks come into being outside of center main intensity
peak. And the center main intensity peaks also extends
outside to form four intensity branches. Therefore, there
are four local intensity minimums appear between center
main peak and outside four subordinate intensity peaks.
This phenomenon is very interesting and can be used to
restrict those micro particles whose refractive index is
lower than that of the surrounding medium, and this kind
of condition is common in life science. In addition, the
whole focal pattern may also be employed to construct
novel optical trap. By comparing Figure 2 with Figure 1,
the optical intensity distribution in focal region of the
Gaussian beam containing one non-spiral optical vortex
can be altered remarkably by the topological charge.
In order to get insight into the effect of topological
change more deeply, focusing properties for other dif-
(a)
(b)
Figure 2. The (a) phase distribution for m = 2 and
the (b) corresponding focal intensity distribution.

X. M. Gao et al. / Natural Science 2 (2010) 201-204
203
ferent m are also investigated. Figure 3 shows the phase
distribution for m = 3 and the corresponding focal inten-
sity distribution. There are three phase maximums and
turns symmetric distribution around optical axis. From
Figure 3(b), it can be seen that for this case, the focal
pattern contains one center main intensity peak with
three relative weak intensity peaks outside. And there are
no local intensity minimums between center peak and
outside three peaks. Number of subordinate intensity
peaks is related to the number of topological charge m.
Simulation shows that number of subordinate intensity
peaks equals m when m is odd integral number.
We also studied effect of some other value of topo-
logical charge on focal pattern. Figure 4 illustrates phase
distribution for m = 4 and the corresponding focal inten-
sity distribution. It can be seen that there are eight phase
maximums, and also eight weak intensity peaks around
center main intensity peak in focal region, as shown in
Figure 4(b). The number of subordinate intensity peaks
is two times of the parameter m when m is even integral
number.
From all above focal pattern evolution on increasing
topological charge, we can seen that topological charge
of non-spiral optical vortex affects optical intensity dis-
tribution in focal region very obviously, and non-spiral
focal pattern may also occur. Multiple intensity peaks
(a)
(b)
Figure 3. The (a) phase distribution for m=3 and
the (b) corresponding focal intensity distribution.
may appear companying with center main focal spot
under condition of higher topological charge. In addition,
the number of weak intensity peak outside of the center
main intensity peak is related to the value of topological
charge. When topological charge is odd number, the
number of weak intensity peak equals the value of topo-
logical charge, while, the number of weak intensity peak
is two times the value of topological charge if topologi-
cal charge is even number. In optical trapping system, it
is usually deemed that the forces exerted on the particles
in light field include two kinds of forces, one is the gra-
dient force, which is proportional to the intensity gradi-
ent; the other is the scattering force, which is propor-
tional to the optical intensity [18]. Therefore, the tunable
(a)
(b)
Figure 4. The (a) phase distribution for m=4 and
the (b) corresponding focal intensity distribution.
(a) (b)
Figure 5. Holograms for generation this non-spiral optical
vortex for (a) m = 1 and (b) m = 2, respectively.

X. M. Gao et al. / Natural Science 2 (2010) 201-204
204
focal intensity distribution predicts that the focusing
properties of this kind of beam can be employed to con-
struct controllable trap.
We are now going on investigating the focusing prop-
erties of this kind of vortex beam. Next step, the atten-
tion will be focused on the experimental research and
extends study into vector optical domain. Figure 5 illus-
trates holograms for generating this non-spiral optical
vortex, which may be employed in our future experiment.
These holograms can be obtained conveniently by cal-
culating numerically optical interference diagram.
4. CONCLUSIONS
The focusing properties of Gaussian beam containing
one non-spiral optical vortex are investigated by scalar
diffraction theory in this paper. Calculation results show
that topological charge of non-spiral optical vortex af-
fects optical intensity distribution in focal region re-
markably, and asymmetric focal pattern may also occur
for lower topological charge. Multiple intensity peaks
may come into being companying with center main focal
spot under condition of higher topological charge. In
addition, the number of weak intensity peak equals the
value of topological charge under condition of odd
number topological charge, while, the number of weak
intensity peak is twice times the number of topological
charge for even number topological charge.
5. ACKNOWLEDGMENT
This work was supported by National Basic Research Program of
China (2005CB724304), National Natural Science Foundation of
China (60708002, 60878024, 60778022, 60807007), China Postdoc-
toral Science Foundation (20080430086), Shanghai Postdoctoral Sci-
ence Foundation of China (08R214141), and Shanghai Leading Aca-
demic Discipline Project (S30502).
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