Engineering, 2010, 2, 124-128
doi:10.4236/eng.2010.22018 Published Online February 2010 (http://www.scirp.org/journal/eng).
Focusing of Azimuthally Polarized
Hyperbolic-Cosine-Gaussian Beam
Xiumin Gao1,2, Mingyu Gao1, Song Hu1, Hanming Guo2, Jian Wang1, Songlin Zhuang2
1Electronics & Information College, Hangzhou Dianzi University, Hangzhou, China
2University of Shanghai for Science and Technology, Shanghai, China
Received October 3, 2009; revised November 2, 2009; accepted November 8, 2009
Abstract
The focusing properties of azimuthally polarized hyperbolic-cosine-Gaussian (ChG) beam are investigated
theoretically by vector diffraction theory. Results show that the intensity distribution in focal region of azi-
muthally polarized ChG beam can be altered considerably by decentered parameters, and some novel focal
patterns may occur for certain case. On increasing decentered parameters, ring shape of focal pattern can
evolve into four-peak focal pattern, and azimuthal field component affects focal pattern more significantly
than radial field component. Optical gradient force is also calculated to show that the focusing properties
may be used in optical tweezers technique.
Keywords: Focusing Properties, Hyperbolic-Cosine-Gaussian Beam, Vector Diffraction Theory
1. Introduction
Laser beams with cylindrical symmetry in polarization
have attracted many researchers recently for their inter-
esting properties and applications. And these beams are
called cylindrical vector beams and can be generated by
active or passive methods [1–4]. Due to the polarization
symmetry, the electric field at the focus has unique po-
larization properties. K. S. Youngworth and T. G. Brown
calculated cylindrical-vector fields [2], which shows that,
in the particular case of a tightly focused radially polar-
ized beam, the polarization shows large inhomogeneities
in the focal region, while the azimuthally polarized beam
has purely transverse field in focal region. Focus shaping
technique is also reported by using generalized cylindrical
vector beams [3], in which a generalized cylindrical vec-
tor beam can be decomposed into radially polarized and
azimuthally polarized components. And a generalized
cylindrical beam can be generated from a radially polar-
ized or an azimuthally polarized light using a two-half-
wave-plate polarization rotator.
On the other hand, although Gaussian beam is famil-
iar to researchers, it is only one kind of solution of the
Helmholtz equation. The more general solution is Her-
mite-Sinusoidal-Gaussian beams, which was introduced
by Casperson and coworkers [5,6]. Hyperbolic-cosine-
Gaussian beams are regarded as the special case of
Hermite-sinusoidal-Gaussian beams, and are of practi-
cal interest because their profiles can be altered by
choosing suitable beam parameters in cosh parts [7].
Propagation and focusing properties of hyperbolic-cosine-
Gaussian beams have become the object of some works
[8–12]. To investigate focusing properties of this kind
of light beam with azimuthally polarized distribution is
very interesting, which may deepen understanding of its
properties and expand application. In this paper, the
focusing properties of the azimuthally polarized hyper-
bolic-cosine-Gaussian (ChG) beam are investigated
theoretically by vector diffraction theory. In Section 2,
the principle of the focusing system is given. And re-
sults and discussions are shown in Section 3. Conclu-
sions are summarized in Section 4.
2. Principle of the Focusing Azimuthally
Polarized ChG Beam
In the focusing system we investigated, focusing beam is
azimuthally polarized ChG beam whose value of trans-
verse optical field is same as that of the scalar ChG beam,
and its polarization distribution turns on azimuthally
symmetric. Therefore, in the cylindrical coordinate sys-
tem
  0,,  r the field distribution
  0,,  rE
 of the azi-
muthally polarized ChG beam is written as [1–3],
 nrErE 
 0,,0,, 00

(1)
where n
 is the azimuthal unit vector of polarized di-
rection. Term
  0,,
0
 rE is optical field value distribu-

X. M. GAO ET AL.125
tion and can be written in the form [7–9],
 


yxByxE yx  coshcosh0,, )exp( 2
0
22

yx 

(2)
where cosh is hyperbolic-cosine function, 0

is the
waist width of the beam. And B is a constant.
x

and
indicate decentered parameters of ChG beam. Ac-
cording to vector diffraction theory, the electric field in
focal region of azimuthally polarized ChG beam is [13],
y



 eEeEzE 
,, (3)
where

e

and

e

are the unit vectors in the radial and
azimuthal directions, respectively. To indicate the posi-
tion in image space, cylindrical coordinates

 z,,


with origin 0 z

located at the paraxial focus are
employed. Eρ and EΦ are the amplitudes of two orthogo-
nal components and can be expressed as:
  




 
0
2
00sinsincos,, E
iA
zE
    







ddzik cossincosexp (4)
  

 




0
2
00cossincos,, E
iA
zE
  







ddzik cossincosexp (5)
where θ and φ denote the tangential angle with respect to
the z axis and the azimuthal angle with respect to the x
axis, respectively. k is wave number. α=arcsi n(NA) is
convergence angle corresponding to the radius of inci-
dent optical aperture. In order to make focusing proper-
ties clear and simplify calculation process, after simple
derivation, Equation 2 can be rewritten as [11]:
   ,E





 cossincosh 1 
x
NAB
 


 sinsincosh 1 
y
NA








22
2
sin
exp wNA

(6)
where 00 rw
  is called relative waist width. Pa-
rameters βx=rpΩx and βy=rpΩy in this paper are called
decentered parameters of azimuthally polarized ChG
beam. Substitute the Equation 6 into Equations 4 and 5,
we can obtain:
 





 
0
2
0sinsincos,, iAB
zE
 


  iNA xexpcossincosh 1 
 










 
22
2
1sin
expsinsincosh wNA NA y














ddzik 


cossincosexp (7)
 








0
2
0cossincos,, iAB
zE




  iNA xexpcossincosh 1 
 










 
22
2
1sin
expsinsincosh wNA NA y














ddzik 


cossincosexp (8)
The optical intensity in focal region is proportional to the
modulus square of Equation 3. Based on the above equa-
tions, focusing properties of azimuthally polarized ChG
beam can be investigated in detail.
3. Results and Discussions
Without loss of validity and generality, it was supposed
that NA=0.95 and w=1. Firstly, the intensity distributions
in focal region of azimuthally polarized ChG beam under
condition of different decentered parameters are calcu-
lated and illustrated in Figure 1. It should be noted that
β=βx=βy, and the distance unit in all figures in this paper
is k-1, where k is the wave number of incident beam. The
azimuthal angle ranges from -π to π in azimuthal coordi-
nate. It can be seen from Figure 1 that the intensity dis-
tribution turns on one ring shape in focal plane for small
decentered parameter β, which is similar to that for Gaus-
sian beam. On increasing β, the focal pattern changes very
remarkably, from one ring focal pattern to four-peak fo-
cal pattern, and these four intensity peaks overlap very
considerably, as shown in Figure 1(b). Increase decen-
tered parameter β continuously, there appear several
weak overlapping intensity peaks outside of the center
four-peak overlapping focal pattern, in addition, multiple
dark focal spots occur between inner four-peak focal
pattern and outer multiple overlapping intensity pattern.
And dark focal spots get more obvious on increasing β,
such as in Figure 1(f). From above focal pattern evolu-
tion, we can see that intensity distribution in focal region
of azimuthally polarized ChG beam can be altered con-
siderably by decentered parameters, and some novel fo-
cal patterns may occur.
In order to understand the focusing properties deeply,
the different optical field components in focal region are
also investigated. And one calculation example is given
here. Figure 2 illustrates the radial field component for
β=2 and β=17, and the corresponding azimuthal field
component is given in Figure 3.
The radial field component distribution turns multiple
optical intensity peaks outside of optical axis, and changes
slightly under condition of different β. While, azimuthal
field component distribution changes very considerably on
increasing β, evolving from one focal ring pattern to mul-
tiple overlapping optical intensity peaks in focal plane.

X. M. GAO ET AL. 126
(a) (b)
(c) (d)
(e) (f)
Figure 1. Intensity distributions in focal region for (a) β=2, (b) β=5, (c) β=8, (d) β=11, (e) β=14, and (f) β=17, respectively.
By comparing Figures 2 and 3 with Figure 1, it can be
seen that azimuthal field component distribution is very
similar to total focal pattern. Therefore, azimuthal field
component affects focal pattern more significantly than
radial field component, namely, azimuthal field compo-
nent decides optical intensity distribution in focal region
of the azimuthally polarized ChG beam.
Now, the possible application of focusing of azimuth-
ally polarized ChG beam is discussed. Optical tweezers
have been increasingly valuable tool for research and
accelerated many major advances in numerous areas of
science [14–17]. In optical tweezers system, it is usually
deemed that the forces exerted on the particles in light
field include two kinds of forces, one is the gradient
force, which is proportional to the intensity gradient; the
other is the scattering force, which is proportional to the
optical intensity [16]. So, the tunable focal pattern pre-
dicts that the optical trap may be controllable. The gra-
dient force trap is necessary condition for constructing
the optical trap and can be expressed as [16]

2
2
2
32
,,
2
1
2zE
m
m
rn
Fb
grad













 (9)
where r is the radius of trapped particles, nb is the refrac-
tion index of the surrounding medium, and , the rela-
tive index of refraction, equals to the ratio of the refrac-
tion index of the particle np to the refraction index of the
m
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X. M. GAO ET AL. 127
(a) (b)
Figure 2. Radial field component for (a) β=2 and (b) β=17, respectively.
(a) (b)
Figure 3. Azimuthal field component for (a) β=2 and (b) β=17, respectively.
(a) (b)
Figure 4. Optical gradient force for (a) β=2 and (b) β=5, respectively. The arrows indicate the direction of optical gradient
force in focal region.
surrounding medium nb. Gradient force Fgrad points in the
direction of the gradient of the light intensity for np>nb.
The optical gradient force can be computed numerically
by substituting Equation 3 into Equation 9. Figure 4 illus-
trates the optical gradient force for β=2 and β=5. The ar-
rows indicate the direction of optical gradient force in fo-
cal region. There is one ring optical trap for β=2, and mul-
tiple optical traps under condition of β=5. Decentered pa-
rameters can change focal pattern very remarkable. There-
fore, the focusing properties of azimuthally polarized ChG
beam can be constructed to tunable optical traps.
4. Conclusions
The focusing properties of azimuthally polarized ChG
beam are investigated theoretically by vector diffraction
theory. Simulations results show that the intensity distri-
bution in focal region of azimuthally polarized ChG
beam can be altered considerably by decentered parame-
ters and some novel focal patterns may occur. For in-
stance, ring focal pattern, four-peak focal pattern, and
dark focal spot. On increasing decentered parameters,
ring shape of focal pattern can evolve into four-peak fo-
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X. M. GAO ET AL. 128
cal pattern, which may be used to construct tunable opti-
cal tweezers. In addition, azimuthal field component af-
fects focal pattern more significantly than radial field
component.
5. Acknowledgments
This work was supported by National Basic Research
Program of China (2005CB724304), National Natural
Science Foundation of China (60708002, 60807007,
60871088, 60778022), China Postdoctoral Science Foun-
dation (20080430086), Shanghai Postdoctoral Science
Foundation of China (08R214141), and Shanghai Lead-
ing Academic Discipline Project (S30502).
6
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