Using coherence theory, the partially coherent flattopped vortex hollow beam is introduced. The analytical equation for propagation of a partially coherent flattopped vortex hollow beam in turbulent atmosphere is derived, using the extended HuygensFresnel diffraction integral formula. The influence of coherence length, beam order
N
, topological charge
M
, and structure constant of the turbulent atmosphere on the average intensity of this beam propagating in turbulent atmosphere are analyzed using numerical examples.
I. INTRODUCTION
Recently, much attention has been paid to the propagation properties of a laser beam in turbulent atmosphere
[1]
. It is found that the intensity and spreading of a laser beam are affected by atmospheric turbulence
[2

8]
, and the laser beam with a vortex has been widely studied, due to its potential applications in freespace laser communication. In past years, Wang
et al.
studied the focusing properties of a Gaussian Schellmodel vortex beam in experiments
[9]
. Zhou
et al.
studied the partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere
[10]
. Wang and Qian studied the spectral properties of a random electromagnetic partially coherent flattopped vortex beam in turbulent atmosphere, based on the extended HuygensFresnel principle
[11]
. Gu studied the transverse position of an optical vortex upon propagation through atmospheric turbulence
[12]
. Zhou and Ru studied the angular momentum density of a linearly polarized LorentzGauss vortex beam
[13]
. Huang
et al.
studied the intensity distributions and spectral degree of polarization of partially coherent electromagnetic hyperbolicsineGaussian vortex beams through nonKolmogorov turbulence using numerical examples
[14]
. Wu
et al.
developed an expression for the wandering of random electromagnetic GaussianSchell model beams propagating in atmospheric turbulence, and studied the properties of the beams
[15]
. Recently, a new dark hollow beam called the partially coherent flattopped vortex hollow beam has been proposed, which has advantages over a flattopped hollow beam, and which has potential applications in freespace wireless laser communication. However, to the best of our knowledge, the propagation properties of a partially coherent flattopped vortex hollow beam in turbulent atmosphere have not been reported.
In this work, we first introduce the partially coherent flattopped vortex hollow beam based on the theory of coherence, and then investigate the its propagation properties in turbulent atmosphere.
II. PROPAGATION OF A PARTIALLY COHERENT FLATTOPPED VORTEX HOLLOW BEAM IN TURBULENT ATMOSPHERE
In the Cartesian coordinate system with the
z
axis set as the axis of propagation, a circular or elliptical flattopped vortex hollow beam in the source plane can be described as
[16]
where
N
is the order of the elliptical flattopped vortex hollow beam,
M
is the topological charge,
w_{x}
and
w_{y}
are the beam width in the
x
and
y
directions respectively, and
denotes the binomial coefficient.
Based on the theory of coherence, a fully coherent flattopped vortex hollow beam can be extended to a partially coherent flattopped vortex hollow beam. The secondorder correlation properties of an electromagnetic beam can be characterized by the crossspectral density function introduced by Wolf
[17]
,
where
g
(
x
_{1}

x
_{2}
,
y
_{1}

y
_{2}
) is the spectral degree of coherence, assumed to have a Gaussian profile, and
where
σ
is the transverse coherence length
Substituting Eq. (1) into Eq. (2), the partially coherent flattopped vortex hollow beam can be written as
where
r
_{10}
= (
x
_{10}
,
y
_{10}
) and
r
_{20}
= (
x
_{20}
,
y
_{20}
) are the position vectors at the source plane
z
= 0.
According to the extended HuygensFresnel principle, the spectral density of a laser beam propagating through turbulent atmosphere can be expressed as follows
[1

9]
:
where
k
= 2
π
/
λ
is the wave number;
ψ
(
x
_{0}
,
y
_{0}
,
x
,
y
) is the solution to the Rytov method that represents the random part of the complex phase (the asterisk denoting complex conjugation), and
r
= (
x
,
y
) and
r
_{0}
= (
x
_{0}
,
y
_{0}
) are respectively the position vectors at the output plane
z
and the input plane
z
=0. The ensemble average in Eq. (5) can be expressed as
[5]
where
ρ
_{0}
is the sphericalwave lateral coherence radius due to the turbulence, and
with
is the constant of refraction index structure, which describes the turbulence strength.
Upon substituting Eq. (4) into Eq. (5), and recalling the integral formulas
[18]
after tedious integral calculations, we can obtain
with
and
Eqs. (11)~(15) make up the main analytical expression for a partially coherent circular or elliptical flattopped vortex beam propagating in turbulent atmosphere. Using the derived equations we can investigate the propagation and transformation of a partially coherent circular or elliptical flattopped vortex hollow beam in turbulent atmosphere.
The degree of coherence of the laser beam is written as
[19]
and the position of coherence vortices at the propagation L is expressed as
[20]
where Re and Im are respectively the real and imaginary parts of
µ
(
r
_{1}
,
r
_{2}
,
z
).
III. NUMERICAL EXAMPLES AND ANALYSIS
In this section we study the propagation properties of a partially coherent flattopped vortex hollow beam in turbulent atmosphere. In this work, the calculation parameters
λ
and
w_{x}
throughout the text are set to be
λ
= 1064
nm
(Nd:YAG laser) and
w_{x}
= 20
mm
.
Figures 1
and
2
show the normalized average intensity and corresponding contour graphs of, respectively, partially coherent circular and elliptical flattopped vortex hollow beams propagating in turbulent atmosphere; the calculation parameters are
,
N
=2,
M
=1,
σ
=10
mm
and
w_{y}
= 40
mm
(
Fig. 2
). As can be seen from
Figs. 1
and
2
, a partially coherent circular or elliptical flattopped vortex hollow beam can keep its original intensity pattern over a short propagation distance (
Figs. 1(a)
and
2(a)
), and with increasing propagation distance either beam loses its initial dark, hollow centre, and the flattopped vortex dark hollow beam evolves into a flattopped beam (
Figs. 1(b)
,
1(c)
,
2(b)
, and
2(c)
). The partially coherent circular and elliptical flattopped vortex hollow beams eventually evolve respectively into circular and elliptical Gaussian beams in the far field, due to the influence of the coherence length.
Normalized average intensity of a partially coherent circular flattopped vortex hollow beam propagating in turbulent atmosphere with , N =2, and M =1, w_{x} = w_{y} = 20 mm, and σ =10 mm. (a) z = 100 m, (b) z = 300 m, (c) z = 600 m, (d) z = 2000 m.
Normalized average intensity of a partially coherent elliptical flattopped vortex hollow beam propagating in turbulent atmosphere with , N =2, and M =1, w_{x} = 20 mm, w_{y} = 40 mm, and σ =10 mm. (a) z = 100 m, (b) z = 300 m, (c) z = 600 m, (d) z = 2000 m.
Figure 3
shows the cross section (
y
=0) of normalized average intensity for a partially coherent circular flattopped vortex hollow beam propagating in turbulent atmosphere for various values of the coherence length
σ
, with
,
N
=2, and
M
=1. It can be seen from
Fig. 3
that a partially coherent circular flattopped vortex hollow beam spreads more rapidly than a fully coherent beam (
σ
=inf), with the initial coherence length
σ
decreasing during propagation;, and that a partially coherent beam with a small coherence length will evolve into a Gaussian beam faster than a beam with a large coherence length in the far field (
Fig. 3(d)
).
Cross section (y=0) of the normalized average intensity for a partially coherent circular flattopped vortex hollow beam propagating in turbulent atmosphere with , N =2, and M =1. (a) z = 100 m, (b) z = 300 m, (c) z = 600 m, (d) z = 2000 m.
Figure 4
presents the cross section (
y
= 0) of normalized average intensity for a partially coherent circular flattopped vortex hollow beam propagating in turbulent atmosphere for various values of
, with
σ
=10
mm
,
N
=2, and
M
=1. It can be seen that a beam propagating in turbulent atmosphere and free space
can almost keep its initial dark hollow profile over a short distance (
Figs. 4(a)
and
(b)
), and with increasing propagation distance a partially coherent flattopped vortex hollow beam loses its initial dark hollow profile faster with increasing structure constant
in the far field (
Fig. 4(d)
).
Cross section (y=0) of the normalized average intensity for a partially coherent circular flattopped vortex hollow beam propagating in turbulent atmosphere with σ =10 mm, N=2, and M=1. (a) z = 100 m, (b) z = 300 m, (c) z = 600 m, (d) z = 2000 m.
Figure 5
depicts the cross section (
y
=0) of the normalized average intensity for a partially coherent circular flattopped vortex hollow beam propagating in turbulent atmosphere for different values of
M
and
N
with
σ
=10 mm and
. As can be seen, abeam with higher order
M
(
Figs. 5(a)
and
(b)
) loses its initial dark hollow center more slowly, while a beam propagating in turbulent atmosphere with different orders
N
has similar evolution properties with increasing propagation distance.
Cross section (y=0) of the normalized average intensity of a partially coherent circular flattopped vortex hollow beam propagating in turbulent atmosphere for different M and N with σ =10 mm. (a) z = 100 m, (b) z = 1000 m, (c) z = 100 m, (d) z = 1000 m.
Figure 6
presents the curves for Re
µ
=0 and Im
µ
=0 for a partially coherent flattopped vortex hollow beam propagating in turbulent atmosphere with
,
σ
=10
mm
,
N
=2, and
M
=1, and
r
_{2}
= (10
mm
, 20
mm
). From
Fig. 6(a)
it can be seen that the beam propagation at a distance of
z
= 10
m
has a coherent vortex, and with increasing propagation distance the beam at a distance of
z
= 500
m
has two coherent vortices. Thus a partially coherent flattopped vortex hollow beam will experience a change in its number of coherent vortices with increasing propagation distance in turbulent atmosphere.
The curves for Re µ=0 and Im µ=0 for a partially coherent circular flattopped vortex hollow beam propagating in turbulent atmosphere with σ =10 mm, N=2 and M=1. (a) z = 100 m, (b) z = 500 m.
IV. CONCLUSION
In this paper the partially coherent flattopped vortex hollow beam is introduced, and then the propagation Eq. for a partially coherent flattopped vortex hollow beam in turbulent atmosphere is derived. The average intensity of a beam propagating in turbulent atmosphere is examined using numerical examples. It is found that a partially coherent flattopped vortex hollow beam will evolve into a Gaussian beam in the far field, and that a beam propagation in turbulent atmosphere with small coherence length or large structure constant
will evolve into a Gaussian beam more rapidly. We also find that a beam with higher order
M
loses its initial dark, hollow center more slowly, while beams of different order
N
have similar evolution properties.
Acknowledgements
This work was supported by the Fundamental Research Funds for the Central Universities (Grants No. 3132015152, 3132015233, 3132014337, 3132014327), and National Natural Science Foundation of China (11404048, 11375034), the program for Liaoning Educational Committee (L2015071).
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