University of Central Florida University of Central Florida
STARS STARS
Faculty Bibliography 2000s Faculty Bibliography
12-31-1999
Scintillation model for a satellite communication link at large Scintillation model for a satellite communication link at large
zenith angles zenith angles
Larry C. Andrews
University of Central Florida
Ronald L. Phillips
University of Central Florida
Cynthia Young Hopen
University of Central Florida
Find similar works at: https://stars.library.ucf.edu/facultybib2000
University of Central Florida Libraries http://library.ucf.edu
This Article is brought to you for free and open access by the Faculty Bibliography at STARS. It has been accepted for
inclusion in Faculty Bibliography 2000s by an authorized administrator of STARS. For more information, please
contact STARS@ucf.edu.
Recommended Citation Recommended Citation
Andrews, Larry C.; Phillips, Ronald L.; and Hopen, Cynthia Young, "Scintillation model for a satellite
communication link at large zenith angles" (1999).
Faculty Bibliography 2000s
. 2424.
https://stars.library.ucf.edu/facultybib2000/2424
Scintillation model for a satellite communication
link at large zenith angles
Larry C. Andrews
University of Central Florida
Department of Mathematics and School of
Optics/CREOL
Orlando, Florida 32816
E-mail: landrews@pegasus.cc.ucf.edu
Ronald L. Phillips
University of Central Florida
Department of Electrical and Computer
Engineering
and
Florida Space Institute
Orlando, Florida 32816
Cynthia Young Hopen
University of Central Florida
Department of Mathematics
and
Florida Space Institute
Orlando, Florida 32816
Abstract. A scintillation model is developed for uplink-downlink optical
communication channels applicable in moderate to strong fluctuation
conditions that may arise under large zenith angles between transmitter
and receiver. The model developed here is an extension of a recently
published theory that treats irradiance fluctuations along a horizontal
path as a modulation of small-scale scintillation by large-scale scintilla-
tion. For a downlink path the scintillation index is modeled like that of an
infinite plane wave, and for an uplink path we consider a spherical wave
model. In both cases the scintillation index agrees with conventional
weak-fluctuation-theory results out to zenith angles of 45 to 60 deg. The
covariance function of irradiance fluctuations is also developed under the
same conditions as assumed for the scintillation index. On a downlink
path under small zenith angles the implied correlation length is propor-
tional to the Fresnel-zone scale. For zenith angles exceeding 85 deg, the
downlink correlation length varies directly with the spatial coherence ra-
dius weighted by a factor that depends on changes in
C
n
2
the refractive
index structural parameter with altitude.
©
2000 Society of Photo-Optical Instru-
mentation Engineers.
[S0091-3286(00)02412-0]
Subject terms: atmospheric optics; scintillation; optical wave propagation; laser
satellite communication.
Paper 990315 received Aug. 10, 1999; revised manuscript received July 7, 2000;
accepted for publication July 28, 2000.
1 Introduction
For more than three decades the scientific community has
expressed interest in the possibility of using high-data-rate
optical transmitters for satellite communications.
1–12
This
interest stems from the advantages offered by optical wave
systems over conventional rf systems, such as smaller an-
tennas; less mass, power, and volume; and the intrinsic nar-
row beam and high gain of lasers. Developmental programs
in the USA involving optical communications started in the
1960s with the development of a CO
2
system by NASA
and a Nd:YAG direct detection system by the Air Force. In
the 1970s and 1980s there were a number of follow-on
developmental programs like the Space Flight Test System
共SFTS兲, Airborne Flight Test System 共AFTS兲, Laser
Crosslink Subsystem 共LCS兲, and Follow-on Early Warning
System 共FEWS兲. Unfortunately, most of these government
developmental programs were terminated due to funding
cutbacks.
1
During the 1990s there has been renewed interest in the
use of high-data-rate optical transmitters for satellite com-
munication channels connecting ground/airborne-to-space
or space-to-ground/airborne data links. Although optical
communication systems offer several advantages as stated
above, laser-satellite communication systems are subject to
severe signal fading below a prescribed threshold value,
owing primarily to optical scintillations associated with the
received signal. Several papers have dealt with actual mea-
surements of atmospherically induced scintillation and/or
the development of theoretical models for predicting scin-
tillation levels.
3–12
However, the previously developed the-
oretical models are all based on conditions of weak fluctua-
tions, which generally limits the assumed zenith angle to 45
to 60 deg or less. For greater zenith angles, moderate to
strong-fluctuation theory must generally be used to predict
optical scintillation, but thus far no tractable models have
been developed for this case.
A heuristic scintillation model was introduced recently
that is applicable under moderate to strong irradiance fluc-
tuations, but limited to atmospheric channels along hori-
zontal paths in which the refractive-index structure param-
eter C
n
2
is assumed to be constant.
13
This new model is
based on a modulation process in which small-scale scin-
tillation is modulated by large-scale scintillation. In particu-
lar, it takes into account the continued loss of transverse
spatial coherence of the optical wave as it propagates from
weak to strong fluctuation regimes. The loss of spatial co-
herence is allowed for by the formal introduction of a spa-
tial amplitude filter that eliminates ineffective cell sizes be-
tween the correlation length and scattering disk in multiple-
scattering regimes. In this paper we build upon that model
by introducing a modification of the filter function to ac-
count for changes in C
n
2
as a function of propagation dis-
tance. Our analysis includes models for scintillation and the
covariance function of irradiance fluctuations on both up-
link and downlink laser-satellite communication channels.
2 The Modulation Process
In this section we first briefly review the theory of scintil-
lation developed in Ref. 13 for horizontal paths with con-
3272 Opt. Eng. 39(12) 3272–3280 (December 2000) 0091-3286/2000/$15.00 © 2000 Society of Photo-Optical Instrumentation Engineers
Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 10 Sep 2019
Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
stant refractive index structure parameter. Building on that
theory, we then extend the results to vertical or slant paths.
To begin, we assume the received irradiance of the op-
tical wave can be modeled as a modulation process in
which small-scale 共diffractive兲 fluctuations are multiplica-
tively modulated by large-scale 共refractive兲 fluctuations.
Thus, similarly to previous treatments,
14–16
the received ir-
radiance can be expressed as a product I⫽ xy, where x
arises from large-scale turbulent cells, or eddies, and y from
small-scale cells. If we further assume that x and y are
statistically independent and that
具
I
典
⫽ 1, the scintillation
index
I
2
⫽
具
I
2
典
/
具
I
典
2
⫺ 1 takes the form
I
2
⫽
具
x
2
典具
y
2
典
⫺ 1⫽
共
1⫹
x
2
兲
共
1⫹
y
2
兲
⫺ 1⫽
x
2
⫹
y
2
⫹
x
2
y
2
,
共1兲
where 具典 denotes an ensemble average and where
x
2
and
y
2
are the variances of x and y, respectively. Moreover, we
can express
x
2
and
y
2
in terms of log-irradiance variances
of x and y according to
13
x
2
⫽ exp(
ln x
2
)⫺1 and
y
2
⫽ exp(
ln y
2
)⫺1, which permits us to rewrite Eq. 共1兲 equiva-
lently as
I
2
⫽ exp
共
ln x
2
⫹
ln y
2
兲
⫺ 1. 共2兲
Small-scale scintillation is caused by diffractive turbu-
lence cells on the order of the correlation length of the
irradiance fluctuations, and large-scale scintillation is
caused by refractive cells on the order of the scattering
disk. The scattering disk is defined by the refractive cell
size l at which the focusing angle
F
⬃l/L is equal to the
average diffraction angle
D
, where
D
⬃1/
冑
kL in weak
fluctuations and
D
⬃1/k
0
in strong fluctuations. The cor-
relation length and scattering disk of an optical wave in
weak irradiance fluctuations are both on the order of the
first Fresnel zone (L/k)
1/2
, whereas in strong fluctuations
the correlation length is on the order of the transverse spa-
tial coherence radius
0
and the scattering disk is charac-
terized by L/k
0
. The parameter L represents total path
length between transmitter and receiver, and k is the optical
wave number.
2.1
Spatial Filter Function: Constant-C
n
2
Model
The assumed model for the power spectrum of refractive
index fluctuations is the conventional Kolmogorov spec-
trum ⌽
n
(
,z)⫽ 0.033C
n
2
(z)
⫺ 11/3
, where z represents the
propagation distance from the transmitter. Because of the
assumed modulation process, we formally replace ⌽
n
(
,z)
with the effective spectrum model
⌽
n,e
共
,z
兲
⫽ 0.033C
n
2
共
z
兲
⫺ 11/3
G
共
,z
兲
, 共3兲
where G(
,z) is a spatial filter induced by the propagation
process and z is a propagation distance that varies between
z⫽ 0 and z⫽ L. For propagation environments where C
n
2
is
essentially constant, the induced filter function can be ap-
proximated by
13
G
共
,z
兲
⬇G
x
共
兲
⫹ G
y
共
兲
⫽ exp
冉
⫺
2
x
2
冊
⫹
11/3
共
2
⫹
y
2
兲
11/6
,
共4兲
in which there is no explicit dependence on the propagation
variable z. 共Inner-scale effects were taken into account in
Ref. 13 but are neglected in the present analysis.兲 The
quantity G
x
(
) is the large-scale filter function, and G
y
(
)
is the small-scale filter function, both of which are selected
on the basis of mathematical tractability. Other functional
forms for these filter functions may be just as appropriate
and can lead to similar results to those presented below.
The parameter
x
is a large-scale 共or refractive兲 spatial-
frequency cutoff much like an inner-scale parameter, and
y
is a small-scale 共or diffractive兲 spatial-frequency cutoff
similar to an outer-scale parameter. In this fashion, G(
,z)
acts like an amplitude 共or irradiance兲 spatial filter function
that only permits low-pass spatial frequencies
⬍
x
and
high-pass spatial frequencies
⬎
y
at a given propagation
distance.
The low-pass and high-pass spatial-frequency cutoffs
appearing in the filter function 共4兲 are directly related to the
correlation length and scattering disk of the fluctuating ir-
radiance. Hence, at any distance L into the random me-
dium, we assume the existence of an effective scattering
disk L/kl
x
and an effective correlation length l
y
related,
respectively, to the cutoff wave numbers according to
L
kl
x
⫽
1
x
⬃
再
冑
L/k,
L/k
0
2
Ⰶ 1,
L/k
0
, L/k
0
2
Ⰷ 1,
共5兲
l
y
⫽
1
y
⬃
再
冑
L/k,
L/k
0
2
Ⰶ 1,
0
, L/k
0
2
Ⰷ 1.
共6兲
Here, L/k
0
2
Ⰶ 1 represents weak fluctuation conditions and
L/k
0
2
Ⰷ 1 represents strong fluctuations when C
n
2
is con-
stant.
2.2
Spatial Filter Function: Variable-C
n
2
Model
Propagation along a vertical or slant path requires a C
n
2
(h)
profile model to describe properly the varying strength of
optical turbulence as a function of altitude h. One of the
most widely used models is the Hufnagel-Valley 共HV兲
model described by
17
C
n
2
共
h
兲
⫽ 0.00594
共v
/27
兲
2
共
10
⫺ 5
h
兲
10
exp
共
⫺ h/1000
兲
⫹ 2.7⫻ 10
⫺ 16
exp
共
⫺ h/1500
兲
⫹ A exp
共
⫺ h/100
兲
,
共7兲
where h is in meters 共m兲,
v
is the rms windspeed 共pseudo-
wind兲 in meters per second 共m/s兲, and A is a nominal value
of C
n
2
(0) at the ground in m
⫺2/3
. In the following analysis,
v
⫽ 21 m/s, A⫽ 1.7⫻ 10
⫺ 14
or 3⫻ 10
⫺ 13
m
⫺ 2/3
, h
0
is the
height above ground of an optical transmitter/receiver, H is
the altitude of the satellite 共receiver/transmitter兲,
is the
zenith angle, and the total propagation distance to the sat-
ellite is L⫽ (H⫺ h
0
)sec
.
Andrews, Phillips, and Hopen: Scintillation model...
3273Optical Engineering, Vol. 39 No. 12, December 2000
Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 10 Sep 2019
Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
For an uplink/downlink communication channel, the
spatial-frequency filter function adopted here exhibits the
modified form
G
共
,z;H,h
0
兲
⫽ G
x
共
,z;H,h
0
兲
⫹ G
y
共
;H,h
0
兲
⫽ A
共
H,h
0
兲
exp
再
⫺
冕
0
1
D
冋
0
x
w
共
,z
兲
册
d
冎
⫹
B
共
H,h
0
兲
11/3
共
2
⫹
y
2
兲
11/6
, 共8兲
where
x
and
y
are defined similarly to Eqs. 共5兲 and 共6兲,
and A(H,h
0
) and B(H,h
0
) are weighting constants that
allow for altitude variations of the structure parameter
C
n
2
(h) on the large-scale and small-scale scintillations.
Once again, these particular choices of large-scale and
small-scale filter functions are based on mathematical con-
venience rather than on a rigorous physical basis. The func-
tion D(
) that appears in Eq. 共8兲 is the plane-wave phase-
structure function defined by
D
共
兲
⫽ 2.914
共
k
2
5/3
sec
兲
冕
h
0
H
C
n
2
共
h
兲
dh
⫽ 2.914
0
k
2
5/3
sec
⫽ 2
共
/
0
兲
5/3
, 共9兲
the function w(
,z) depends on propagation distance ac-
cording to
w
共
,z
兲
⫽
再
共
1⫺
⑀
z/L
兲
,
⬍ z/L,
共
z/L
兲
共
1⫺
⑀
兲
,
⬎ z/L
共10兲
(
⑀
⫽ 0 for a plane wave and
⑀
⫽ 1 for a spherical wave兲, and
0
⫽
冕
h
0
H
C
n
2
共
h
兲
dh. 共11兲
Although it is not obvious, the low-pass 共large-scale兲
filter function G
x
(
,z;H,h
0
) in Eq. 共8兲 takes into account
that large-scale effects are strongest near the transmitter
and that a phase variation at propagation distance z⬍ L
induces an amplitude effect at the receiver located at dis-
tance L from the transmitter. Moreover, it is chosen in such
a way that it leads to results consistent with those devel-
oped in Ref. 13 and also agrees with the low-pass filter
function in the asymptotic theory for the saturation
regime.
18–20
In this regard, the weighting constant A(H,h
0
)
reduces to unity in the limiting case of constant C
n
2
, and we
find the scintillation index in the saturation regime takes the
form
20
I
2
共
L
兲
⫽ 1⫹ 32
2
k
2
冕
0
L
冕
0
⬁
⌽
n
共
兲
G
x
共
,z;L,0
兲
⫻ sin
2
冋
2
2k
w
共
z,z
兲
册
d
dz, 共12兲
where
G
x
共
,z;L,0
兲
⫽ exp
再
⫺
冕
0
1
D
冋
L
k
w
共
,z
兲
册
d
冎
. 共13兲
共We should point out that in a recent simulation experi-
ment, Flatte
´
and Gerber
21
showed that their simulation data
for the saturation regime give a somewhat different power-
law behavior than that predicted by the asymptotic
theory.
18–20
兲 Under strong fluctuations, the high-pass filter
function G
y
(
;H,h
0
) depends mostly on small turbulent
cells near the receiver and thus exhibits essentially no ex-
plicit dependence on the distance variable z along the path.
We tacitly assume that this is true of the high-pass filter
function even in weak fluctuations. Like A(H,h
0
), the
weighting constant B(H,h
0
) in the high-pass filter function
also reduces to unity when C
n
2
is constant.
3 Downlink Channel
For a downlink path from a satellite, it is well known that
the ground-level scintillation near the center of the received
wave can be accurately modeled by a plane wave.
22
Based
on weak-scintillation theory and the Kolmogorov spectrum
关Eq. 共3兲 with G(
,z)⬅1], the scintillation index for a
plane wave is simply the Rytov variance defined by
17,22,23
1
2
⫽ 2.606k
2
sec
冕
h
0
H
C
n
2
共
h
兲
冕
0
⬁
⫺ 8/3
⫻
再
1⫺ cos
冋
共
h⫺ h
0
兲
2
sec
k
册
冎
d
dh
⫽ 2.25
1
k
7/6
共
H⫺ h
0
兲
5/6
sec
11/6
, 共14兲
where
1
⫽
冕
h
0
H
C
n
2
共
h
兲
冉
h⫺ h
0
H⫺ h
0
冊
5/6
dh. 共15兲
We see from the form of Eq. 共15兲 that the Rytov variance
共14兲 depends mostly on high-altitude turbulence. Hence, we
generally expect values of the Rytov variance consistent
with weak-fluctuation theory except for the case of large
zenith angles. In the following analysis it is sometimes use-
ful to express the Rytov variance 共14兲 in terms of the plane-
wave transverse spatial coherence radius
17,23
0
⫽ (1.45
0
k
2
)
⫺ 3/5
cos
3/5
which leads to
1
2
⫽ 1.55
冉
L
k
0
2
冊
5/6
1
0
. 共16兲
For constant C
n
2
, Eqs. 共14兲 and 共16兲 reduce to
1
2
⫽ 1.23C
n
2
k
7/6
L
11/6
⫽ 0.847(L/k
0
2
)
5/6
.
3.1
Scintillation Index
By following the approach in Ref. 13, the small-scale log-
irradiance scintillation takes the form
Andrews, Phillips, and Hopen: Scintillation model...
3274 Optical Engineering, Vol. 39 No. 12, December 2000
Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 10 Sep 2019
Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
ln y
2
⫽ 2.606B
共
H,h
0
兲
k
2
sec
冕
h
0
H
C
n
2
共
h
兲
冕
0
⬁
共
2
⫹
y
2
兲
11/6
⫻
再
1⫺ cos
冋
共
h⫺ h
0
兲
2
sec
k
册
冎
d
dh
⬇1.563
关
B
共
H,h
0
兲
0
k
7/6
共
H⫺ h
0
兲
5/6
sec
11/6
兴
y
⫺ 5/6
,
共17兲
or, by the use of Eq. 共14兲,
ln y
2
⫽ 0.695B
共
H,h
0
兲
0
1
1
2
y
⫺ 5/6
, 共18兲
where
y
⫽
L
y
2
k
⫽ 3
共
1⫹ 0.69
1
12/5
兲
. 共19兲
The proper choice of weighting constant in Eq. 共18兲 is not
clear, but by selecting B(H,h
0
)⫽ 1.83(
1
/
0
), we note
that Eq. 共18兲 reduces to the expression
ln y
2
⫽1.272
1
2
y
⫺ 5/6
given in Ref. 13. This choice of weighting
constant B(H,h
0
) also minimizes low-altitude effects in
ln y
2
and reduces appropriately to unity in the limiting case
of constant C
n
2
. Consequently, Eq. 共18兲 becomes
ln y
2
⫽
0.51
1
2
共
1⫹ 0.69
1
12/5
兲
5/6
. 共20兲
To find a comparable expression for the large-scale log-
irradiance variance, we first use Eqs. 共9兲 and 共10兲 with
⑀
⫽ 0 to simplify the low-pass spatial filter, i.e.,
G
x
共
,z;H,h
0
兲
⫽ A
共
H,h
0
兲
exp
再
⫺
冕
0
1
D
冋
0
x
w
共
,z
兲
册
d
冎
⫽ A
共
H,h
0
兲
exp
冋
⫺ 2
冉
x
冊
5/3
5/3
冉
1⫺
5
8
冊
册
,
共21兲
where
⫽ z/L⫽ (h⫺ h
0
)/(H⫺ h
0
) on a downlink channel.
Thus, using the geometrical-optics approximation 共i.e., 1
⫺ cos x⬇x
2
/2), the large-scale log-irradiance variance be-
comes
ln x
2
⫽ 2.606k
2
sec
冕
h
0
H
C
n
2
共
h
兲
冕
0
⬁
⫺ 8/3
G
x
共
,z;H,h
0
兲
⫻
再
1⫺ cos
冋
共
h⫺ h
0
兲
2
sec
k
册
冎
d
dh
⬇1.303A
共
H,h
0
兲
共
H⫺ h
0
兲
2
sec
3
⫻
冕
h
0
H
C
n
2
共
h
兲
2
冕
0
⬁
4/3
exp
冋
⫺ 2
冉
x
冊
5/3
5/3
⫻
冉
1⫺
5
8
冊
册
d
dh
⬇0.263
关
A
共
H,h
0
兲
2
k
7/6
共
H⫺ h
0
兲
5/6
sec
11/6
兴
x
7/6
, 共22兲
where
13
2
⫽
冕
h
0
H
C
n
共
h
兲
⫺ 1/3
共
1⫺
5
8
兲
7/5
dh, 共23兲
x
⫽
L
x
2
k
⫽
1
c
1
⫹ c
2
1
12/5
. 共24兲
The scaling constants c
1
and c
2
in Eq. 共24兲 will be deter-
mined below on the basis of asymptotic behavior under
weak and strong fluctuations. In terms of the Rytov vari-
ance 共14兲, we can rewrite Eq. 共22兲 as
ln x
2
⫽ 0.116A
共
H,h
0
兲
冉
2
1
冊
1
2
x
7/6
. 共25兲
Through arguments analogous to those for the small-
scale log-variance, here we set A(H,h
0
)⫽ 4.68
1
/
2
,in
which A(H,h
0
)⫽ 1 with constant C
n
2
. Also, we set c
1
⫽ 1.09, so that Eq. 共25兲 reduces to
ln x
2
⬇0.49
1
2
under
weak fluctuations (
1
2
Ⰶ 1), giving us
ln x
2
⫹
ln y
2
⫽
1
2
.In
the saturation regime, we impose the asymptotic result
13
A(H,h
0
)
x
7/6
⬇(k
0
2
/L)
7/6
⫽
关
1.7(
1
/
0
)
6/5
1
⫺ 12/5
兴
7/6
,
where we are recalling L/k
0
2
⫽ 0.59(
0
/
1
)
6/5
1
12/5
from
Eq. 共16兲. These results enable us to deduce that
x
⫽
0.92
1⫹ 2
共
1
/
2
兲
6/7
共
0
/
1
兲
6/5
1
12/5
, 共26兲
and the large-scale log-irradiance variance 共25兲 becomes
ln x
2
⫽
0.49
1
2
关
1⫹ 2
共
1
/
2
兲
6/7
共
0
/
1
兲
6/5
1
12/5
兴
7/6
. 共27兲
It is an interesting observation that the coefficient of
1
12/5
in the denominators of Eqs. 共26兲 and 共27兲 is nearly
constant, regardless of the ground-level turbulence strength,
upper atmospheric wind speed, and so forth. That is, using
the Hufnagle-Valley model 共7兲, we find that 共on average兲
2
共
1
/
2
兲
6/7
共
0
/
1
兲
6/5
⬇1 to 1.5. 共28兲
Andrews, Phillips, and Hopen: Scintillation model...
3275Optical Engineering, Vol. 39 No. 12, December 2000
Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 10 Sep 2019
Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
