July 10, 2007 / Vol. 5, No. 7 / CHINESE OPTICS LETTERS 379
Generation of partially coherent vortex bottle beams
Lianzhou Rao (
)
1,2
and Jixiong Pu (
ÍÍÍ
)
1
1
Department of Electronic Science & Technology, Huaqiao University, Quanzhou 362021
2
Department of Physics and Electromechanical Engineering, Sanming University, Sanming 365004
Received January 4, 2007
Intensity distribution of the partially coherent Bessel vortex beams fo cused by an aperture lens is inves-
tigated. It is found that the intensity distribution in the neighborhood of the geometrical focus is not
only dependent on the topological charge and the radial frequency of the incident partially coherent Bessel
vortex beam, but also on its coherence length. Based on this, the desired partially coherent vortex bottle
beams can be obtained by choosing appropriate values of parameters. Because such bottle beams possess
characteristics of low coherence and vortex, it may be used in microscopic particles guiding, trapping, and
inducing rotation.
OCIS codes: 050.1970, 030.1640, 999.9999 (vortex bottle beam).
In recent years, there have been increasing interests in
the generation of an optical bottle beam, in which a dark
focus is surrounded by regions of higher intensity
[1−11]
.
Various applications of a bottle beam in atom guid-
ing, atom trapping, and optical tweezers have been
explored
[1−3]
. In a blue-detuned bottle beam, the atoms
are guided to the dark or the low-field region and ma-
nipulated in the dark center
[1]
, and the storage time
can approach the order of 1 s
[3]
. Several techniques for
generating an optical bottle beam were also described.
Kaplan et al. proposed a new scheme for constructing
a single-beam dark optical trap that minimizes light-
induced perturbations of the trapped atoms
[6]
. Yelin
et al. presented an optical setup for generating three-
dimensional (3D) dark focus
[7]
. Recently, a new method
for generating partially coherent bottle beams has been
presented
[8]
. In addition, it is well recognized that an
optical vortex beam with a helical phase structure of
exp (inφ), where n represents the topological charge and
φ is the azimuthal angle, carries orbital angular momen-
tum (OAM)
[12]
. Such beam was also applied to induce
rotation of particles due to the transfer of OAM from
the light to the particles
[13]
. Here, we notice that all the
techniques for generating partially coherent bottle beams
above do not take into account the beams carrying opti-
cal vortex. Can partially coherent vortex bottle beams
be generated by focused partially coherent high order
Bessel vortex beams? This question is interesting, be-
cause the vortex bottle beams of low coherence may show
some advantages over those of complete coherence
[14]
and
possess the characteristics of the optical vortex. In this
letter, we present a novel method for generation of par-
tially coherent vortex bottle beams. It is shown that the
size of dark focus is adjustable by modulating the spatial
coherence length and the topological charge etc..
Suppose that a partially coherent high order Bessel
vortex beam is incident upon an aperture lens with full
width 2a and focal length f at the z = −f plane, as
shown in Fig. 1. The field distribution of the high order
Bessel vortex beams in the polar coordinate system is
written as
[15,16]
E
(0)
(r,ω)=E
0
J
n
(αr)exp(inφ)exp(iωt)exp(iβ),
n =1, 2, 3, ··· , (1)
where J
n
is the nth-order Bessel function of the first kind,
r is the position vector of a point p in the aperture lens,
α is a radial frequency, and β is an arbitrary phase (as a
spatially distributed random variable).
The cross-spectral density of a partially coherent wave
field can be written as
W
(0)
(r
1
, r
2
,ω)=E
∗
(r
1
,ω)E(r
2
,ω) , (2)
where the angle bracket denotes an ensemble average
monochromatic realization of the field. Substituting
Eq. (1) into Eq. (2) and using Gaussian-Schell coherent
model
[17]
, we obtain the expression for the cross-spectral
density in the z = −f plane
W
(0)
(r
1
, r
2
,z = −f)=E
2
0
J
n
(αr
1
) J
n
(αr
2
)
× exp
− (r
1
− r
2
)
2
2σ
2
exp [−in(φ
1
− φ
2
)] , (3)
where σ is the transverse coherence length, E
0
the con-
stant amplitude factor.
According to the generalized Huygens-Fresnel
diffraction integral, the cross-spectral density in the
focused field can be expressed as
[17]
Fig. 1. Notation relating to the aperture-lens system.
1671-7694/2007/070379-04
c
2007 Chinese Optics Letters