520 CHINESE OPTICS LETTERS / Vol. 8, No. 5 / May 10, 2010
Characterization of tightly focused partially coherent
radially polarized vortex beams
Lina Guo (HHH|||AAA)
1,2
, Zhilie Tang (///)
1∗
, Chongqing Liang (ùùù)
1
,
and Zhiliang Tan ( £££ûûû)
1
1
School of Physics and Telecommunication Engineering, Laboratory of Quantum Information Technology,
South China Normal University, Guangzhou 510006, China
2
School of Electronics and Information, Guangdong Polytechnic Normal University, Guangzhou 510665, China
∗
E-mail: tangzhl@scnu.edu.cn
Received September 14, 2009
Tight focusing properties of partially coherent radially polarized vortex beams are studied based on vecto-
rial Debye theory. We focus on the fo cal properties including the intensity and the partially coherent and
p olarized properties of such partially coherent vortex beams through a high numerical aperture objective.
It is found that the source coherence length and the maximal numerical aperture angle have direct influ-
ence on the focal intensity, as well as coherence and p olarization properties. This research is important in
optical micromanipulation and beam shaping.
OCIS co des: 260.1960, 050.4865, 260.5430.
doi: 10.3788/COL20100805.0520.
Radially polarized beams are well known to create a
strong longitudinally polarized field component near the
focal region by a high numerical aperture (NA) objec-
tive, hence exhibiting special properties
[1−4]
. This fo-
cused light has a tighter spot and can be applied in many
fields such as high resolution microscopy, lithography, op-
tical data storage, material processing, and optical trap-
ping and acceleration
[5]
. In 2003, direct detection and
characterization of this sharp longitudinal field by ex-
perimental demonstration was reported
[5]
. Because of
the unique characteristics of vortex beams
[6,7]
, the tight
focusing of optical vortex beams, such as linearly, circu-
larly, and elliptically polarized vortex beams, as well as
cylindrically p olarized vortex beams, have been exten-
sively studied
[8−11]
.
However, because of the universality and characteris-
tics of partially coherent light
[12,13]
, the partially coher-
ent radially polarized beam seems more important and
promising. Seshadri introduced the cross-spectral den-
sity of partially coherent azimuthally or radially polar-
ized symmetric beams and investigated the effect of the
coherence length on its average Poynting vectors
[13]
. Re-
cently, the propagation properties of partially coherent
radially polarized beams in free space and a turbulent at-
mosphere have been studied
[14,15]
. However, to the best
of our knowledge, there is no paper dealing with the tight
focusing of partially coherent radially polarized vortex
beams. Moreover, it is of great importance to explore
the coherence and polarization properties of the tightly
focused vortex wave fields due to their increasing appli-
cations. In this letter, the focal characteristics including
intensity, and coherence and polarization properties of
partially coherent radically polarized vortex beams are
studied based on vectorial Debye theory. The influence
of the coherence length and the maximal NA angle on
these focusing properties are analyzed in detail.
The electric field of a completely coherent radially po-
larized vortex beam focused by a high NA objective can
be expressed as
[3−11]
E(r, ψ, z) =
"
E
x
(r, ψ, z)
E
y
(r, ψ, z)
E
z
(r, ψ, z)
#
= − i
n+1
E
0
i
¡
I
n+1
e
iψ
− I
n−1
e
−iψ
¢
I
n+1
e
iψ
+ I
n−1
e
−iψ
2I
n
e
inψ
, (1)
where n is the topological charge, E
0
is a constant re-
lated to the intensity of the beam, r, ψ, and z are the
cylindrical coordinates of an observation point in the fo-
cal region. The definition of the variables I
n
and I
n±1
is
given by
I
n
(r, z) =
Z
α
0
P (θ )
√
cos θ sin
2
θ
J
n
(kr sin θ) exp (ikz cos θ) dθ, (2)
I
n±1
(r, z) =
Z
α
0
P (θ )
√
cos θ sin θ cos θ
J
n±1
(kr sin θ) exp (ikz cos θ) dθ, (3)
where P (θ) is the pupil apodization function and J
n
is
the nth Bessel function of the first kind.
Assuming that the field wave is quasi-monochromatic,
the second-order correlation properties of the beam can
be characterized by the 3×3 electric cross-spectral den-
sity matrix W (r
1
, r
2
). The elements of the 3×3 matrix
are given by
[16]
W
ij
(r
1
, r
2
) =
E
∗
i
(γ
1
, ψ
1
, z
1
)E
j
(γ
2
, ψ
2
, z
2
)
®
(i, j = x, y, z), (4)
where E
i
(γ, ψ, z) and E
j
(γ, ψ, z) denote Cartesian com-
ponents of the electric field, the asterisk stands for the
complex conjugate, and the angle brackets represent an
ensemble average.
The explicit expressions of the diagonal elements of W
ij
can be derived from Eq. (1):
1671-7694/2010/050520-04
c
° 2010 Chinese Optics Letters