Optics and Photonics Journal, 2013, 3, 143-147
doi:10.4236/opj.2013.32B035 Published Online June 2013 (http://www.scirp.org/journal/opj)
Study on Turbulence Effects for Beam Propagation in
Turbulent Atmosphere
Pingping Pan
College of physics and communication electronics, Jiangxi normal university, Nanchang, 330022
Received 2013
ABSTRACT
Based on the theory for the turbulence and the Rytov method, the propagation formulas of the scintillation index and the
mean square angle-of-arrival fluctuation for the beam propagation in atmospheric turbulence have been derived respec-
tively. The propagation properties of the two turbulence effects have been investigated, and the effects of the character-
istic parameters of turbulence and the beam parameters have been discussed. The results show that the variation of the
two turbulence effects depends on the structure constant of the refractive index fluctuations Cn2, the inner scale of the
turbulence l0, the waist width of source beam w0 and the wave length λ. Moreover, there are two parameters including
Cn2 and l0 which show more significant effects in atmosphere. Consequently, a new method for determining the char-
acteristics parameters of the turbulence by using the measurement of the scintillation index and the angle-of-arrival
fluctuation has been proposed.
Keywords: Atmospheric Turbulence; Rytov Method; Scintillation Index, Mean Square Angle-of-Arrival Fluctuations;
Constant of the Refractive Index Fluctuations
1. Introduction
The study on the propagation of laser beams through
turbulence is of great importance for many practical ap-
plications, such as the remote sensing, atmospheric opti-
cal communications, and track system [1-5]. There are a
few affecting factors on the beam propagation in atmos-
phere, such as atmosphere attenuation, atmospheric ab-
sorption, scattering, and atmospheric turbulence effect
etc. The atmospheric turbulence effects include the beam
spreading, the angle-of-arrival fluctuations, the scintilla-
tion index and the bit error rate, and they have more ob-
vious influence on the performance of a communication
system. [6-10].
The scintillations and the angle-of-arrival fluctuations
are more significant effects of atmospheric turbulence on
the beam propagation. The effect of the angle of arrival
may increase the field-of-view requirements in a di-
rect-detection optical receiver and degrade the perform-
ance a heterodyne system [11-13]. It is well known that
the atmospheric turbulence affects the received intensity,
resulting in the intensity fluctuations at the receiver plane.
Consequently, the optical scintillation take place in at-
mospheric optic links [14-15]. In this paper, the propaga-
tion properties of the scintillations and the angle-of-ar-
rival fluctuations have been investigated, the effects of
the turbulent parameters and the beam parameters have
been discussed, and a method for estimating the charac-
teristics parameters of the turbulence has been proposed.
2. Turbulence Effects for Beam Propagation
in Atmosphere
2.1. Mean Square Angle-of-Arrival Fluctuations
When light travels through the atmosphere, it experiences
phase fluctuations due to turbulence. The mean square
angle-of-arrival fluctuation is related to the phase struc-
ture function through the expression below [14],
12
2
2
(0, ,
()
s
a
DR
kR

 pp )L
(1)
Here the mean square angle-of-arrival fluctuations are
evaluated at a radial distance R, L is the path length, i.e.,
the length of the atmospheric link, and k = 2π/λ is the
wave number, λ is wavelength. The second phase point
lies at
22 11
221
(,), , (,)(0,0)
xy xy
ppR ppppp (2)
The phase structure function described as
2
121 2
( ,,)[( ,)(,)]
s
DLSLSLPPPP (3)
where <> denotes the ensemble average over the medium
statistics,
1
(, )[ (, )*(, )]
2
SLL L
j

ppp (4)
the asterisk represents the complex conjugate. In Rytov
method, ψ denotes the fluctuations of the complex am-
Copyright © 2013 SciRes. OPJ
P. P. PAN ET AL.
144
plitude in turbulence [4],
1
2
3FS
FS
'
(,)'(,') (,')
2(,)
exp(' )
'
V
k
LdrnzE
EL
ik
pp'
p
rr
rr
zp'
z
(5)
EFS (P, z) is the field in free space and
'(',')(',','), ''''
xyx y
zppzddpdpd rPr (6)
11
(,')(',',')
xy
nznppzp' (7)
expresses the random part of the refractive index, which
can be written in the form of a Fourier–Stieltjes repre-
sentation [5] for a statistically homogeneous and iso-
tropic medium.
The field distribution of Gaussian beam can be written
as,
'2 '2
'' 11
11 2
0
( ,,0)exp()
x
y
Ex yw
 (8)
where w0 is the waist width and (x1
, y1
) is the transversal
coordinates at the z=0 plane. Then, by using the paraxial
form of the extended Huygens-Fresnel principle, the field
distribution of Gaussian beam through free space can be
obtained,
2
22
0
111 1
2
0
22 22
01 1
22
0
(,, )exp[()
2
2
()
]
42
FS kw ik
Exyzikz xy
z
kw iz
kw xy
zikzw

(9)
Assume that the turbulent atmosphere is statistically
homogenous and isotropic, the source and the medium
statistics are independent. The mean square angle-of-
arrival fluctuation of Gaussian beam in turbulence can be
derived,
221/3
0
0
11/62
22 2
0
1
1/62
2
0.1628( 14.45.5663()
1
Re{[]
1(1)
[]})
(1 )
m
an
m
m
CL
dt Lt
k
kj L t


1/3




(10)
Re means the real part, Cn
2 is the structure constant of
the refractive index fluctuations of the turbulence. κm
=5.92/l0, l0 is the inner scale of turbulence; κ0=1/L0, L0 is
the outer scale of turbulence. And
2
0
1222 2
00 0
1,
(/ )
kw jLt
kwL wkwjL



(11)
2.2. Scintillation Index
The scintillation index in atmospheric turbulence at the
L
receiver plane is given by [8-10]
24(,,B
12 )
Iz
pp (12)
121 2
(, ,)(,)(BzL L
,)L

pppp (13)
is the correlation function of the log-amplitude
L
fluctua-
tions at the receiver plane z =L,
(,) 0.5[(,LL) *(,)]

ppp (14)
and Bχ can be written as
y
*
0
[Re()Re()]( )
nx
Bd dd
z

 


(15)
Φn (κ) denotes the spatial power spectra of the refractive

index fluctuations of the turbulent atmosphere, Φn(κ) =0
in free space, and κ= (κx
2+κy
2)1/2.
2
0
2
0
22
0
2
0
exp( )
(0, )2
()(2 )
exp[]
2(2 )
FS
kw
ikikz
Ez izkw
zLt Ltikw
kizkw

(16)
Letting x=0 and y=0 in EFS(x, y, z), the on-axis field
distribution E
FS (0, z) can be obtained. Tatarskii power
spectra is represented as
2
2 11/3
2
( )0.033exp()
nn
m
C

  (17)
After some tedious integral calculation, the scintilla-
tion index of Gaussian beam in turbulence can be de-
rived,
22
1
2222 0
22 2
00
5/6
0
2
0
2(1)
1
0.8816Re {[]
(1 4)
(1)(12)1
[]}
(1 2)
In
m
m
wL t
CkLkLwk
L tiLtwdt
ik iLwk




(18)
3. Numerical Examples and Analysis
f-arrival fluc-
wave propagation through
at
RMS-AOA is zero in the initial plane, and it increases
3.1. Angle-of-Arrival Fluctuations
In this section, the variation of the angle-o
tuations have been studied, and the effects of the turbu-
lent parameters and the beam parameters have been dis-
cussed. Eq. (10) is numerically evaluated and used in
plotting the root mean square of the angle-of-arrival
(RMS-AOA) fluctuations. Figure 1 gives the variation of
the angle-of-arrival fluctuations, and the calculation pa-
rameters are Cn
2 = 10-14 m-2/3, w0 = 30 mm, λ = 1.55 μm, L
= 3 km, l0 = 10 mm, L0 = 25 m.
The wave front tilt of optical
mospheric turbulence gives rise to an angle-of-arrival
fluctuation. Figure 1(a) shows the effects of constant of
the refractive index fluctuations, C = Cn
2/10-15 m-2/3. The
Copyright © 2013 SciRes. OPJ
P. P. PAN ET AL. 145
versus the propagation distance L. The RMS-AOA in-
creases when beams propagate through stronger turbu-
lence, which can be revealed from Eq. (10) straightway.
In addition, the effect of the turbulence is more obvious
for larger L.
The effects of the inner scale of the turbulence l0 and
the outer scale of the turbulence Lare displayed in Fig-
ur 0
e 1(b). The change range of the inner scale l0 in turbu-
lence has been adopted between 1mm and 20mm usually,
so the variation of RMS-AOA under the change range of
l0 is represented. Examining Figure 1(b), the RMS-AOA
decreases for the larger l0 and smaller L0.
Figure 1. Variation of the angle-of-arrival fluctuations in
turbulence.
R
the intensity fluctuations at the re-
Figure 1(c) provides the effects of the beam parame-
ters including the waist width w0 and wavelength λ. The
MS-AOA decreases first and increases then, and there
exits a minimum during the variation of w0. In our graphs,
two infrared wavelengths are employed, representing the
most commonly utilized wavelengths in current space
links. From Figure 1(c), the RMS-AOA of the beam
with λ = 0.85 μm is larger than λ = 1.55 μm. Moreover,
the waist width where the value of minimum for the
RMS-AOA happens is smaller. If the waist width is very
big, the RMS-AOA of the beam will increase. On the
other hand, if w0 is very small, it is not easy to obtain
larger power in the initial plane, so the propagation dis-
tance is limited in practice. Therefore, the appropriate
waist width and wavelength should be selected.
3.2. Scintillations
Scintillations reflect
ceiver plane, and the variation of the scintillation index
in turbulence is provided in Figure 2. The calculation
parameters Cn
2 = 10-14 m-2/3, w0 = 30 mm, λ = 1.55 μmL
= 3 km, l0 = 10 mm are taken. In Figure 2(a), the scintilla-
tion index rises against propagation distance and through
stronger turbulence. Furthermore the effect of the turbu-
lence on the scintillation index is more obvious when the
propagation distance increases. The variation of scintilla-
tion index given in this paper is consistent with the re-
sults of
Figure 2. Variation of the scintillations in turbulence.
Copyright © 2013 SciRes. OPJ
P. P. PAN ET AL.
146
related literature [13]. As observed from Figure 2(b),
there exits a minimum for the scintillation index during
the variation of w0, and both the scintillation index and
the corresponding minimum are larger for the smaller l0.
In a word, the effect of the turbulence on beam propa-
gating in turbulence is very obvious, so the beam quality
may be affected, and especially the effect of the constant
of the refractive index fluctuations is not neglected.
Consequently, it is significative to estimate the structure
constant of the turbulence.
3.3. Estimation of the Structure Constant of the
Turbulence
so
stant of the turbulence can be
estimated roughly. Secondly, if the scintillation index is
scintnd turbulent parameters (see Figure
The relations of the turbulence effects and turbulent pa-
rameters are illustrated in Figure 3 and Figure 4 respec-
tively, which implies a method for the estimation of the
structure constant of the turbulence. We noted that in our
lution, the change range of the inner scale l0 is adopted
between 1 mm and 20 mm, whereas the range of the
structure constant spans several magnitudes. Firstly, if
the RMS-AOA is measured by experiments, by using the
relation of the RMS-AOA and turbulent parameters (see
Figure 3), the structure con
measured by experiments, by using the relation of the
illation index a
2
4), C
n can be estimated once again. Applying the two
change ranges, we can estimate the range of the structure
constant of the turbulence more accurately.
4. Conclusions
In this paper, the propagation formulas of the angle-of-
arrival fluctuation and the scintillation index for the
beam propagation in atmospheric turbulence have been
obtained respectively, indicating that there are similarities
in the derivation of the RMS-AOA and the scintillation
Figure 3. Relation of the RMS-AOA and turbulent parame-
ters.
Figure 4. Relation of the scintillation index and turbulent
parameters.
index. The effects of the turbulent parameters and the
beam parameters have been analyzed quantitatively. The
angle-of-arrival fluctuation and the scintillation index
rise with the increasing propagation distance and the
structure constant of the turbulence, and the decreasing
inner scale, moreover there exits a minimum for them
during the variation of waist width. Finally, the range of
the structure constant of the turbulence can be estimated
from the RMS-AOA and the scintillation index measured
by experiments.
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