Page 1
Second-order statistics of a twisted Gaussian
Schell-model beam in turbulent atmosphere
Fei Wang and Yangjian Cai*
School of Physical Science and Technology, Soochou University, Suzhou 215006, China
* yangjian_cai@yahoo.com.cn
Abstract: We present a detailed investigation of the second-order statistics
of a twisted Gaussian Schell-model (TGSM) beam propagating in turbulent
atmosphere. Based on the extended Huygens-Fresnel integral, analytical
expressions for the second-order moments of the Wigner distribution
function of a TGSM beam in turbulent atmosphere are derived. Evolution
properties of the second-order statistics, such as the propagation factor, the
effective radius of curvature (ERC) and the Rayleigh range, of a TGSM
beam in turbulent atmosphere are explored in detail. Our results show that a
TGSM beam is less affected by the turbulence than a GSM beam without
twist phase. In turbulent atmosphere the Rayleigh range doesn’t equal to the
distance where the ERC takes a minimum value, which is much different
from the result in free space. The second-order statistics are closely
determined by the parameters of the turbulent atmosphere and the initial
beam parameters. Our results will be useful in long-distance free-space
optical communications.
©2010 Optical Society of America
OCIS codes: (010.1300) Atmospheric propagation; (030.1670) Coherent optical effects;
(350.5500) Propagation
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1. Introduction
In the past decades, partially coherent beams have been widely investigated and applied in
free space optical communication, optical imaging, nonlinear optics, optical trapping, inertial
confinement fusion, optical projection and laser scanning [1–6]. Gaussian Schell-model
(GSM) beam is a typical and commonly encountered partially coherent beam, whose spectral
density and spectral degree of coherence have Gaussian shapes [1,7]. By scattering a coherent
laser beam from a rotating grounded glass, then transforming the spectral density distribution
of the scattered light into Gaussian profile with a Gaussian amplitude filter, a GSM beam can
be generated [8]. GSM beams can also be generated with specially synthesized rough
surfaces, spatial light modulators and synthetic acousto-optic holograms (c.f [9].). Propagation
properties of a GSM beam have been studied widely [1,10–13]. It has been found that a GSM
beam is less affected by the turbulent atmosphere compared to a coherent Gaussian beam, thus
have important applications in free space optical communication, remote sensing and radar
system [11–13].
A more general partially coherent beam can possess a twist phase, which differs in many
respects from the customary quadratic phase factor. In 1993, Simon and Mukunda first
introduced the twisted Gaussian Schell-model (TGSM) beam [14]. Unlike the usual phase
curvature, the twist phase is bounded in strength due to the fact that the cross-spectral density
function must be nonnegative and it is absent in a coherent Gaussian beam. The twist phase
has an intrinsic chiral property and is responsible for the rotation of the beam spot on
propagation. Friberg et al. first carried out experimental demonstration of TGSM beams [15].
Superposition, coherent-mode decomposition and the analysis of the transfer of radiance of
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the TGSM beam have been investigated in [16,17]. Dependence of the orbital angular
momentum of a partially coherent beam on its twist phase was revealed in Ref [18]. The
conventional method for treating the propagation of TGSM beams is the Wigner-distribution
function [14]. Lin and Cai have introduced a convenient alternative tensor method for treating
the propagation of TGSM beams [19]. With the help of the tensor method, the propagation
properties of a TGSM beam through paraxial ABCD optical system, dispersive media and
nonlinear media were studied in [20–23]. More recently, Ghost imaging with a TGSM beam
was explored in [24]. Zhao et al. studied the radiation force of a TGSM beam on a Rayleigh
particle [25]. Twist phase-induced polarization changes in electromagnetic GSM beam were
studied in [26].
Investigations of the propagation properties of laser beams in a turbulent atmosphere
become more and more important because of their wide applications in e.g. free-space optical
communications and remote sensing [2,3,11–13,27–29]. Average intensity and spreading
properties of a TGSM beam have been studied in [29]. Recently, more and more attention is
being paid to the second-order statistics, such as the propagation factor, the effective radius of
curvature (ERC) and the Rayleigh range, of laser beams in turbulent atmosphere [30–34]. To
our knowledge no results have been reported up until now on the second-order statistics of a
TGSM beam in turbulent atmosphere. The purpose of this paper is to investigate the
propagation factor, the ERC and the Rayleigh range of a TGSM beam in turbulent
atmosphere, and to explore the advantage of a TGSM beam over a GSM beam for overcoming
or reducing the turbulence-induced degradation. Analytical expressions are derived for the
second-order moments of the Wigner distribution function of a TGSM beam in turbulent
atmosphere, and some useful and interesting results are found.
2. Second-order moments of the Wigner distribution function of a TGSM beam in
turbulent atmosphere
A partially coherent beam is generally characterized by the cross-spectral density (CSD)
function, and the CSD function of a TGSM beam in the source plane (z = 0) is expressed as
[14]
1
''
1
01
2
0
42
I
2
''
'2'2
2
r
''''
0
11
2
g
0
, ;0
2
r
exp,
2
T
2
22
ik
W
r
r
rr J r
r
rr
(1)
where
plane, respectively;
and
I
is a scalar real-valued twist factor with the dimension of an inverse distance, limited by the
double inequality
00
0
g
k
coherent limit,
0
g
, the twist factor
0
disappears. In Eq. (1), the symbol J denotes an
anti-symmetric matrix given by [14]
'''
111
( ,
x y
)
r
and
k
''
2
'
22
(,)
x y
is the wave number with being the wavelength of light field.
denote the transverse beam width and spectral coherence width, respectively.
r
2 /
represent two arbitrary position vectors in the source
00
g
0
1
224
due to the non-negativity requirement of Eq. (1). In the
0
1
.
1 0
J
(2)
Under the condition of
a conventional GSM beam without twist phase [7–9]. Due to the existence of the term
12121221
x yx y
rr J rr
in the right side of Eq. (1), the two-dimensional CSD
function cannot be split in a product of two one-dimensional CSD functions.
0
0
, the CSD function in Eq. (1) reduces to the CSD function of
''''''''
T
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Within the validity of the paraxial approximation, based on the extended Huygens-Fresnel
integral, the CSD function of a partially coherent beam propagating in turbulent atmosphere at
z is expressed as [11,30–34]
''''' 2 '
r r , (3)
'
0
22
1
( , ; )
d
r r
( , ;0)exp
d
r r
( ) (
) ( , ; )
d
r r
dddd
ik
z
WzWH z d
d
z
r rrr
where
''''''''
00120
'( , ;0)
r r
( , ;0)
r r
( /2,/2;0)
ddd
WWW
rrrr
. (4)
In the derivation of Eq. (3), we have used following sum and difference notations
''''''
12121212.
dd
r
r
rrrrrrrrrr
,
(5)
The term
and can be written as [11]:
'
exp( ,
r r
; )
z
dd
H
in Eq. (3) is the contributions from atmospheric turbulence,
'
1
22'
0
00
exp ( , ; )
d
r r
4 [1( (1) )]( )
,
d
n
dd
Hz
k z dJd
rr
(6)
where
index-of-refraction fluctuations in turbulent atmosphere, and is the magnitude of the
spatial wave-number.
The Wigner distribution function (WDF) of a partially coherent beam on propagation in
turbulent atmosphere can be expressed in terms of the CSD function by the formula [30,31]
0 J is the Bessel function of zero order,
n
represents the spectral density for the
2
2
1
( , ; )
r θ
( , ; )exp
d
r r
,
dd
hzWz ikd
θ rr (7)
where
k and
Substituting Eq. (3) into Eq. (7), we obtain (after some operation) following expression for
the WDF of a partially coherent in turbulent atmosphere
( ,
k are the wave vector components along the x-axis and y-axis, respectively.
)
xy
θ
denotes an angle which the vector of interest makes with the z-direction;
x
y
2
""
0
2
2 2 "
r
2
1
1
( , ; )
r θ
( ,
r r
;0)exp
(2 )
exp( ,
r r
; )
z
,
ddddd
ddddd
z
k
hzWii ik
z
k
Hddd
r
κκrκ θ r
κκr
(8)
where
Eq. (8), we have used following formula
,
ddxdy
κ
is the position vector in spatial-frequency domain. In the derivation of
''"'"'2 2 "
r
00
2
1
( , ;0)
r r
( , ;0)exp[
d
r r
( )].
(2 )
ddd
WWidd
κrrκ
(9)
We can express
"
0( ,
W
;0)
dd
z
k
r r
κ
of a TGSM beam as follows
2
"2
""
00
2
I
2
I
2
g
(10)
000
11
( ,
r r
;0) expexp().
282
T
dddddd
z
k
z
k
z
k
Wik
r
κrκrκ Jr
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Substituting Eq. (10) into Eq. (8), we obtain (after integration over
expression for the WDF of a TGSM beam in turbulent atmosphere
"r ) the following
2
2
I
2
2
I
0
000
2
2
2
I
2
0
00
2
I
2
g
00
1
1
( , ; )
r θ
2 exp()
(2 )2
11
exp() exp
282
exp
dxd dy
dyd dxdd
d
hzkyz
z
k
kxz
i
r κ
r
κ
22
exp( ,
r r
; )
z
.
dddddd
z
k
ikHdd
θ rκκr
(11)
According to Ref [30], the moments of order
is given by
1212
nnmm
of the WDF of a laser beam
1212121222
1
P
(),
nnm
x
m
y
nnm
x
m
y
x y x yh z d d
r θ
rθ (12)
where
22
().
Ph z d d
r θ
rθ (13)
The second-order statistics of a laser beam, such as the propagation factor, the ERC and
the Rayleigh range, are closely related with the second-order moments of the WDF.
Substituting Eq. (11) into Eq. (12), we obtain (after tedious integration and operation)
following expressions for the second-order moments of the WDF of a TGSM beam
propagating in turbulent atmosphere
2222
I
223
0
( )
z
( )
x z
( )
y z
224 /3,
Az Tz
r
(14)
22
( )
z
( )
z
( )
x z
( )
z
( )
y z
( )
z
22,
xy
Az Tz
r
θ
(15)
2222
( )
z
( )
z
( )
z
24,
xy
A Tz
θ
(16)
where
22
I
22
g
2
0
2
I
000
1/(4 ) 1/(
),
Akk
(17)
3
0
( )
,
n
Td
(18)
2
I
0
2.
P
(19)
In the above derivations, we have used following integral formula [35]:
1
( )
s
exp() ,
2
isx dx
(20)
1
( )
s
( ) exp() , (0, 1, 2),
2
nn
ix isx dx
n
(21)
( )
n
( )
f x
( )
x dx
( 1)
(0), (1, 2).
nn
fn
(22)
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In Eqs. (14) and (16), the symbols
and the squared far-field divergence of the TGSM beam in turbulent atmosphere, respectively.
The ERC of the TGSM beam is closely determined by
2
( ) z
r
and
2
( ) z
θ
represent the squared beam width
( )
z
( )
z
θ
r
in Eq. (15).
3. Propagation factor of a TGSM beam in turbulent atmosphere
The propagation factor (best known as
important property of an optical laser beam [1] being regarded as a beam quality factor in
many practical applications. Based on the second-order moments of the Wigner distribution
function, the
M -factor of a partially coherent beam is defined as [30–32]
2
M -factor) proposed by Siegman is a particularly
2
1/2
2
222
( )
z
( )
z
( )
z
θ
( )
z
( )
z
θ
.
Mk
r
r
(23)
Substituting Eqs. (14)-(16) into Eq. (23), we obtain following expression for the
a TGSM beam in turbulent atmosphere
2
M -factor of
1/2
2
222
I
22322
0
( )
z
(0) (88 /3 4
/3),
MM Az Tzk Tz
(24)
where
source plane given by
2(0)
M
in Eq. (24) represents the M2-factor of the TGSM beam in free space or in the
22
0
24
I
2
I
2
g
000
(0)1 4
4/.
Mk
(25)
Under the condition of
the M2-factor of a TGSM beam in free space. Under the condition of
to the expression for the M2-factor of a GSM beam without twist phase in turbulent
atmosphere. From Eq. (25), it is clear that the M2-factor of a TGSM beam in free space is
independent of the propagation distance, and increases with the increase of the absolute value
of the twist factor. This phenomenon is caused by the fact that the twist factor cause more
rapid spreading of a TGSM beam on propagation.
Now we study the evolution properties of the M2-factor of a TGSM beam in turbulent
atmosphere. In the following numerical examples, we adopt the Tatarskii spectrum for the
spectral density of the index-of-refraction fluctuations, which is expressed as [11]
0
T (without turbulence), Eq. (24) reduces to the expression for
0
0
, Eq. (24) reduces
2
n
11/3
22
m
( )
0.033exp/,
n
C
(26)
where
2
n
C is the structure constant of the refractive index fluctuations of the turbulence and
5.92/
l
with
1060nm
. Substituting Eq. (26) into Eq. (18), we obtain
0
m
0l being the inner scale of the turbulence. In the following text, we set
32
n
1/3
0
0
( )
0.1661.
n
Td C l
(27)
Substituting Eq. (27) into Eq. (24), we can calculate the
turbulent atmosphere numerically.
2
M -factor of a TGSM beam in
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Fig. 1. Normalized M2-factor of a TGSM beam on propagation in turbulent atmosphere for
different values of the structure constant
n
C and the inner scale 0l
2
For the convenience of comparison, we now study the normalized M2-factor of a TGSM
beam defined as
( )/ (0)
MzM
on propagation in turbulent atmosphere. Figure 1 shows the
normalized M2-factor of a TGSM beam on propagation in turbulent atmosphere for different
values of the structure constant
n
C . As illustrated by Fig. 1, the normalized M2-factor of a
TGSM beam in turbulent atmosphere increases on propagation, which is much different from
its propagation-invariant properties in free space (
constant
n
C increases (i.e., turbulence becomes strong) or the value of the inner scale
decreases, the normalized M2-factor increases more rapid on propagation. Figure 2 shows the
normalized M2-factor of a TGSM beam on propagation in turbulent atmosphere for different
values of
0
g
and
0
with 0
0.01m
l
. One finds from Fig. 2(a) that the normalized M2-factor
of a TGSM beam increases slower on propagation as its initial coherence width
decreases, which means that a TGSM beam with lower coherence is less affected by turbulent
atmosphere as expected [30–32]. One finds from Fig. 2(b) that the normalized M2-factor of a
TGSM beam increases slower than that of a GSM beam without twist phase (
propagation in turbulent atmosphere, which means that a TGSM beam is less affected by
atmospheric turbulence than a GSM beam. Furthermore, as shown by Fig. 2(b), the TGSM
beam with larger absolute value of
0
is less affected by the turbulence than that with smaller
absolute value of
0
.
22
2
2
n
0
C
). As the value of the structure
2
0l
0
g
0
0
) on
Fig. 2. Normalized M2-factor of a TGSM beam on propagation in turbulent atmosphere for
different values of
0
g
and
0
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Fig. 3. Deviation percentage of the normalized M2-factor versus the propagation distance z for
different values of
0
In order to show the advantage of a TGSM beam over a GSM beam in turbulent
atmosphere quantitatively, we introduce a parameter
percentage of the normalized M2-factor to show the difference between the normalized M2-
factor of a TGSM beam and that of a GSM beam. The deviation percentage of the normalized
M2-factor is defined as
2( )
Mz
named the deviation
00
0
2222
0
2
22
0
( )/
z
(0)( )/
z
(0)
( )
z
.
( )/
z
(0)
MMMM
M
MM
(28)
The advantage of a TGSM beam over a GSM bam increases with the increase of the
deviation percentage
Mz
.
We calculate in Fig. 3 the deviation percentage of the normalized M2-factor versus the
propagation distance z for different values of
with
. As shown in Fig. 3, the parameter
and it approaches to a constant value in the far field. The constant value increases as the
absolute value of
0
increases. For
parameter
Mz
approaches to 5%, which is quite significant. In practical experiment, we
can convert a GSM beam into a TGSM beam with a six-element astigmatic lens system as
shown in [15], and control the twist phase by controlling the astigmatic lens. A GSM beam
can be generated with the help of a rotating grounded glass and a Gaussian amplitude filter
conveniently [8]. Thus it is economic and realizable to generate a TGSM beam for application
in free-space optical communications.
2( )
0
0
10
I
mm
,
0
10
g
mm
, 0
0.01m
l
and
2
n
142/3
10m
C
2( )
Mz
increases on propagation,
the case of
1
0
1.5km
, the
2( )
4. Effective radius of curvature of a TGSM beam in turbulent atmosphere
According to [33,34], the ERC of a laser beam at z is defined in terms of the ratio of
2
( ) z
r
to
( )
z
( )
z
θ
r
as follows
2
( )( )
z
/ ( )
z
( ) .
z R z
rr
θ
(29)
Substituting Eqs. (14) and (15) into Eq. (29), we obtain following expression for the ERC
of a TGSM beam in turbulent atmosphere
2
I
Az
23
0
22
/3
( ).
Tz
Tz
R zz
(30)
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From Eq. (30), one finds that the ERC of a TGSM beam on propagation are determined by the
beam parameters (i.e., beam width
0
I
, the coherence width
twisted factor
0
) and the parameters of the turbulent atmosphere (i.e., structure constant
and the inner scale 0l ) together. Under the condition of
the ERC of a GSM beam in free space. Equation (30) provides a convenient way for studying
the evolution properties of a GSM beam with or without twist phase in turbulent atmosphere.
We calculate in Fig. 4 the ERC of a TGSM beam in turbulent atmosphere versus the
propagation distance for different values of the structure constant
10
I
mm
and
0
10
g
mm
. One finds from Fig. 4 that the ERC of a TGSM beam
on propagation in free space (
0
n
C
) or in turbulent atmosphere will initially display a
downward trend in the near field, but after reaching a dip, will star to increase. The value of
the ERC on propagation decreases as the structure constant
decreases especially in the far field.
0
g
, the wavelength and the
2
n
C
0
T and
0
0
, Eq. (30) reduces to
2
n
C and the inner scale
0l with
0
2
2
n
C increases or the inner scale 0l
Fig. 4. ERC of a TGSM beam in turbulent atmosphere versus the propagation distance z for
different values of the structure constant
n
C and the inner scale 0l .
2
Fig. 5. ERC of a TGSM beam in turbulent atmosphere versus the propagation distance for
different values of
0
and
0
g
.
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Fig. 6. Deviation percentage of the ERC of a TGSM or GSM beam versus the propagation
distance z.
Figure 5 shows the ERC of a TGSM beam in turbulent atmosphere versus the propagation
distance for different values of
0
and
0
g
with
the difference between the ERC of a TGSM beam in free space and that in turbulence is
smaller than the difference between the ERC of a GSM beam in free space and that in
turbulent atmosphere, which means a TGSM beam is less affected by the turbulent
atmosphere than a GSM beam from the aspect of ERC. From Fig. 5 (b), it is also clear that the
TGSM beam with lower coherence is less affected by the turbulence than that with higher
coherence. To show the advantage of a TGSM beam over a GSM beam quantitatively, we
introduce a parameter
( )
R z
named the deviation percentage of the ERC to show the
difference between the ERC of a TGSM or GSM beam in turbulent atmosphere and that of a
TGSM or GSM beam in free space. The deviation percentage of the ERC is defined as
0
10
I
mm
. One finds from Fig. 5 (a) that
( )( )
( ).
( )
tur free
free
R zR z
R z
R z
(31)
The advantage of a TGSM beam over a GSM beam increases with the increase of the
deviation between the
( )
R z
of a TGSM beam and that of a GSM beam. We calculate in Fig.
6 the deviation percentage of the ERC of a TGSM or GSM beam versus the propagation
distance z with
0
10
I
mm
,
0
10
g
mm
,
l
deviation between the
( )
R z
of a TGSM beam with
increases on propagation, and it approaches to a constant value (about 5%) in far field. This
result agrees well with the result shown in Fig. 3. One also finds from Fig. 5 that evolution of
the ERC of a TGSM beam is little different from that in free space. In free space, the value of
the ERC of a TGSM beam with larger absolute value of
0
0.01m
. One finds from Fig. 6 that the
and that of a GSM beam
1
0
1.5km
0
(or smaller
(or larger
0
) in the near field or
g
) on propagation is
always smaller than that with smaller absolute value of
in the intermediate propagation distance. With the increase of propagation distance, the
difference between the ERC of TGSM beams with different
in the far field, the ERC tends to ( )
R zz
. In turbulent atmosphere, there exists a critical
propagation length
of ERC. For the case of
c
zz
, the value of the ERC of the TGSM beam with larger absolute
value of
0
(or smaller
0
g
) is smaller that that with smaller absolute value of
). For the case of
c
zz
, the reverse situation occurs. From Eq. (30), we obtain following
expression for the critical propagation length
0
0
g
0
or
0
g
becomes smaller, and
cz where the TGSM beams with different
0
or
0
g
have the same value
0
(or larger
0
g
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1/3
2
I
T
0
2
3
.
cz
(32)
One finds from Eq. (32) that
constant
n
C and the inner scale 0l of turbulence. At the critical propagation length, the ERC
of the TGSM beam turns out to be
c
Rz
. By choosing suitable value of the beam width, the
ERC of the TGSM beam can remain invariant at fixed receiving plane if
cz is only determined by the beam width
0
I
, the structure
2
2
n
C or 0l varies.
5. Rayleigh range of a TGSM beam in turbulent atmosphere
The Rayleigh range is an important beam parameter for characterizing the distance within
which the laser beam can be considered effectively non-spreading. The Rayleigh range is
defined as the distance
the place where the area of the cross section is doubled (i.e., the diameter of the spot size
increases by a factor 2 compared to the spot size at the beam waist) [36]. The range of the
minimum effective radius of curvature is defined as the distance
direction of a beam from the beam waist to the place where the ERC of the beam takes the
minimum vale. In free space, the Rayleigh range
effective radius of curvature
the properties of the Rayleigh range
curvature
obtained by solving following equations
Rz along the propagation direction of a beam from the beam waist to
m z along the propagation
Rz equals to the range of the minimum
m z . What will happen in turbulent atmosphere? Now let’s study
Rz and the range of the minimum effective radius of
m z in turbulent atmosphere. Based on the definition of
Rz and
m z [36], they can be
22
()2 (0)0,
Rz
rr
(33)
( )/0.
m
z z
dR zdz
(34)
Substituting Eqs. (14) and (30) into Eqs. (33) and (34) respectively, we obtain
23
R
22
I
0
4 /3 2
2 0,
Tz Az
(35)
224
m
23
m
22
m
22
I
2
I
00
24363 0.
m
T z ATzA z TzA
(36)
Under the condition of T = 0 (free space), Eqs. (35) and (36) reduce to the same quadratic
equation. After some calculation, we obtain following analytical expression for
a TGSM beam in free space
00
1/(4) 1/(
RmII
zzk
Rz and
m z of
1/2
2222
g
2
0
2
I
00
).
k
(37)
Under the condition of
space as shown in [36]. By solving Eqs. (35) and (36), we obtain (after tedious operation and
calculation) following expressions for
0
0
, Eq. (37) reduces to the expression for
Rz of a GSM in free
Rz and
m z of a TGSM beam in turbulent atmosphere,
11
/(2),
AMTM
22
1
2
RzAM
(38)
3422
I
2
3
2
033
6/ /(2),
m
zATNNNAT
(39)
where
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1/3
3422
I
844
I
4322
I
1000
6 2 93
MATT A T
,
2
1
6 2
)
21
44 (54
NNNA
4 1/3
2
1/3
23
3/ /12
NANN
,
3422
I
2
10
54(6)
NAT
. (40)
One finds from Eqs. (38) and (39) that
don’t equal to each other generally.
Rz and
m z of a TGSM beam in turbulent atmosphere
Fig. 7.
and coherence width
Rz and
m z of a TGSM beam in turbulent atmosphere different values of twist factor
0
0
g
We calculate in Fig. 7
values of twist factor
the convenience of comparison, the corresponding results in free space are also shown. As
shown in Fig. 7,
Rz and
m z of a TGSM beam in turbulent atmosphere different
and coherence width
0
g
with
0
0
10
I
mm
and
2
n
14-2/3
10m
C
. For
Rz and
m z don’t coincide with each other due to the influence of turbulence.
m z in turbulent atmosphere is always larger than that in free space, and
atmosphere is always smaller than that in free space. As the absolute value of twist factor
increases or the coherence width
0
g
decreases, the difference between
becomes smaller, which means that a TGSM beam with larger absolute value twist factor or
lower coherence is less affected by the turbulence.
Rz in turbulent
0
Rz and
m z
6. Conclusion
In conclusion, we have derived the analytical expressions for the second-order moments of the
WDF of a TGSM beam in turbulent atmosphere based on the extended Huygens-Fresnel
integral. The second-order statistics, such as the propagation factor, the ERC and the Rayleigh
range, of a TGSM beam propagating in turbulent atmosphere have been studied and compared
with the results in free space. Our numerical results show that a TGSM beam is less affected
by the turbulence than a GSM beam, and a TGSM beam with larger absolute value of twist
factor or lower coherence is less affected by the turbulence than that with smaller twist factor
or higher coherence. Our results will be useful in long-distance free-space optical
communications.
Acknowledgments
Yangjian Cai acknowledges the support by the National Natural Science Foundation of China
under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral
Dissertation of PR China under Grant No. 200928, the Natural Science of Jiangsu Province
under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under
Grant No. 121009 and the Key Project of Chinese Ministry of Education under Grant No.
210081.
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22 November 2010 / Vol. 18, No. 24 / OPTICS EXPRESS 24672
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