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Mathematical model for the irradiance probability density function Mathematical model for the irradiance probability density function
of a laser beam propagating through turbulent media of a laser beam propagating through turbulent media
M. A. Al-Habash
L. C. Andrews
University of Central Florida
R. L. Phillips
University of Central Florida
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Recommended Citation Recommended Citation
Al-Habash, M. A.; Andrews, L. C.; and Phillips, R. L., "Mathematical model for the irradiance probability
density function of a laser beam propagating through turbulent media" (2001).
Faculty Bibliography
2000s
. 2907.
https://stars.library.ucf.edu/facultybib2000/2907
Mathematical model for the irradiance probability
density function of a laser beam propagating
through turbulent media
M. A. Al-Habash
*
L. C. Andrews, MEMBER SPIE
University of Central Florida
Department of Mathematics
and
School of Optics/CREOL
Orlando, Florida 32816
R. L. Phillips
Florida Space Institute
and
University of Central Florida
Department of Electrical and Computer
Engineering
Orlando, Florida 32816
Abstract. We develop a model for the probability density function (pdf)
of the irradiance fluctuations of an optical wave propagating through a
turbulent medium. The model is a two-parameter distribution that is
based on a doubly stochastic theory of scintillation that assumes that
small-scale irradiance fluctuations are modulated by large-scale irradi-
ance fluctuations of the propagating wave, both governed by indepen-
dent gamma distributions. The resulting irradiance pdf takes the form of
a generalized
K
distribution that we term the gamma-gamma distribution.
The two parameters of the gamma-gamma pdf are determined using a
recently published theory of scintillation, using only values of the
refractive-index structure parameter
C
n
2
(or Rytov variance) and inner
scale
l
0
provided with the simulation data. This enables us to directly
calculate various log-irradiance moments that are necessary in the
scaled plots. We make a number of comparisons with published plane
wave and spherical wave simulation data over a wide range of turbu-
lence conditions (weak to strong) that includes inner scale effects. The
gamma-gamma pdf is found to generally provide a good fit to the simu-
lation data in nearly all cases tested.
©
2001 Society of Photo-Optical Instrumen-
tation Engineers.
[DOI: 10.1117/1.1386641]
Subject terms: lasers; propagation; irradiance.
Paper 200410 received Oct. 17, 2000; revised manuscript received Feb. 12,
2001; accepted for publication Feb. 27, 2001.
1 Introduction
For more than three decades the scientific community has
expressed interest in the possibility of using high-data-rate
optical transmitters for radar and optical communications.
This interest stems from the advantages offered by optical
wave systems over conventional rf systems such as smaller
antenna: less mass, power, and volume; and the intrinsic
narrow-beam and high-gain nature of lasers. Applications
that could benefit from laser communication, or lasercom,
systems are those that have platforms with limited weight
and space, require very high data links, and must operate in
an environment where fiber optic links are not practical,
such as between buildings across cities or supporting mili-
tary tactical operations. Also, there has been a lot of interest
over the years in the possibility of using optical transmitters
for satellite communications. Although many of the early
developmental programs were terminated due to funding
cutbacks, there was renewed interest during the decade of
the 1990s in the use of optical transmitters for communica-
tion channels connecting ground/airborne-to-space or
space-to-ground/airborne data links.
1,2
The performance of a laser radar or lasercom system can
be significantly diminished by turbulence-induced scintilla-
tion resulting from beam propagation through the atmo-
sphere. Specifically, scintillation can lead to power losses at
the receiver and eventually to fading of the received signal
below a prescribed threshold. The reliability of an optical
system operating in such an environment can be deduced
from a mathematical model for the probability density
function 共pdf兲 of the randomly fading irradiance signal. For
that reason, one of the goals in studying optical wave
propagation through optical turbulence is the identification
of a tractable pdf of the irradiance under all irradiance fluc-
tuation conditions. In addition, it is beneficial if the free
parameters of that pdf can be tied directly to atmospheric
parameters.
The purpose of this paper is to develop a tractable pdf
model for the irradiance fluctuations in homogeneous, iso-
tropic turbulence that can be useful in studying the perfor-
mance characteristics of various laser radar and lasercom
systems that operate in an environment in which the
refractive-index structure parameter C
n
2
and inner scale l
0
can be reasonably estimated. In an effort to validate the use
of this pdf model in practical applications, we make com-
parisons of it with published plane wave pdf simulation
data
3
and spherical wave pdf simulation data
4,5
under a
wide variety of simulated turbulence conditions, including
inner-scale effects.
2 Background
Over the years, many irradiance pdf models have been pro-
posed with varying degrees of success. Under weak irradi-
ance fluctuations, the irradiance pdf is generally accepted to
*
Current affiliation: TeraBeam Networks, 14833 NE 87th St., Bldg. C,
Redmond, WA 98052.
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be log normal. Indeed, measurements of the lower order
normalized irradiance moments under weak conditions gen-
erally fit the values predicted by the lognormal model.
6,7
In
spite of data that support the log-normal model, it has also
been observed that the log-normal pdf can underestimate
the peak of the probability density function and underesti-
mate the behavior in the tails as compared with measured
data.
5,8
Underestimating the tails of a pdf has important
consequences on radar and communication systems where
detection or fade probabilities are calculated primarily over
the tails of the given pdf.
As the strength of turbulence increases and multiple
self-interference effects must be taken into account, greater
deviations from lognormal statistics are present in mea-
sured data.
6–8
Eventually, however, the radiation field of
the wave can be approximated by a zero-mean Gaussian
distribution, and, thus, the irradiance statistics are then gov-
erned by the negative exponential distribution. The nega-
tive exponential distribution is considered a limit distribu-
tion for the irradiance and is therefore approached only far
into the saturation regime.
Some of the most useful of the current pdf models have
evolved from an assumed modulation process as the wave
propagates through optical turbulence.
5,8–15
One of the first
such models to gain wide acceptance for a variety of appli-
cations under strong fluctuation conditions was the K dis-
tribution. This distribution was originally proposed as a
model for non-Rayleigh sea echo,
9
but it was later discov-
ered that it also provides excellent agreement with numer-
ous experimental data involving radiation scattered by tur-
bulent media under strong fluctuation conditions.
10,11
Although usually formulated from a discrete point of view,
the K distribution can be derived from a modulation pro-
cess wherein the pdf of irradiance is assumed governed by
the conditional negative exponential distribution
p
1
共
I
兩
b
兲
⫽
1
b
exp
共
⫺ I/b
兲
, I⬎ 0, 共1兲
where the mean irradiance b⫽
具
I
典
is itself a random quan-
tity. The unconditional pdf for the irradiance is obtained by
calculating the expected value
p
共
I
兲
⫽
冕
0
⬁
p
1
共
I
兩
b
兲
p
2
共
b
兲
d b, I⬎0, 共2兲
where p
2
(b) is the distribution function of the fluctuating
mean irradiance, assumed to be the gamma distribution
p
2
共
b
兲
⫽
␣
共
␣
b
兲
␣
⫺ 1
⌫
共
␣
兲
exp
共
⫺
␣
b
兲
, b⬎ 0,
␣
⬎ 0. 共3兲
In Eq. 共3兲, ⌫共
␣
兲 is the gamma function and
␣
is a positive
parameter related to the effective number of discrete scat-
terers. The resulting distribution arising from Eq. 共2兲 is
given by
p
共
I
兲
⫽
2
␣
⌫
共
␣
兲
共
␣
I
兲
共
␣
⫺ 1
兲
/2
K
␣
⫺ 1
共
2
冑
␣
I
兲
, I⬎ 0,
␣
⬎ 0, 共4兲
where K
p
(x) is a modified Bessel function of the second
kind. Because of the presence of this particular Bessel func-
tion, the pdf 关Eq. 共4兲兴 is known as the K distribution.
The scintillation index predicted by the K distribution
assumes the form
I
2
⫽ 1⫹ 2/
␣
, which always exceeds unity
but approaches it in the limit
␣
→ ⬁. Thus, the K distribu-
tion is not valid under weak irradiance fluctuations for
which the scintillation index is less than unity. One attempt
at extending the K distribution to the case of weak fluctua-
tions led to the I-K distribution.
12,13
However, shortcomings
in both the K and I-K models have been noted when com-
paring these pdf’s with measured irradiance data in ex-
tended turbulence.
8
Later models arising out of an assumed modulation pro-
cess were the log-normally modulated exponential
distribution
14
共valid only under strong fluctuation condi-
tions兲 and the log-normal Rician distribution, also
called
4,5,8,15
Beckmann’s pdf. Both of these models show
excellent agreement with experimental and simulation data
concerning
4,5,8
the pdf. The more general of the two models
is the Beckmann pdf defined by the integral
p
共
I
兲
⫽
共
1⫹ r
兲
exp
共
⫺ r
兲
冑
2
z
⫻
冕
0
⬁
I
0
再
2
冋
共
1⫹ r
兲
rI
z
册
1/2
冎
⫻ exp
再
⫺
共
1⫹ r
兲
I
z
⫺
关
ln z⫹
共
1/2
兲
z
2
兴
2
2
z
2
冎
dz
z
2
,
I⬎ 0, 共5兲
where z is mean irradiance, r is a power ratio,
z
2
is the
variance of the lognormal modulation factor ln z, and I
0
(x)
is a modified Bessel function. Although it provides an ex-
cellent fit to various data, the Beckmann pdf has certain
impediments from a practical point of view. For instance, a
closed-form solution for this integral is unknown. More-
over, the poor convergence properties of its integral form
makes the Beckmann pdf cumbersome for numerical calcu-
lations used in calculating detection or fade probabilities
that are essential in understanding the performance charac-
teristics of an optical communications or radar system. Per-
haps the most serious impediment to its use is that it is
unknown at this time how to deduce the parameters r and
z
2
of the Beckmann pdf directly from atmospheric param-
eters.
3 Gamma-Gamma Distribution
The pdf model we propose in this paper is based on a
modulation process similar to that of the K distribution and
the Beckmann pdf. To calculate the parameters of this pdf
using only measured values of C
n
2
and l
0
, we use a recently
published heuristic theory of scintillation that is applicable
for optical wave propagation through all conditions of irra-
diance fluctuations.
16–18
Because it is central to the devel-
opment of our pdf model, we begin this section with a brief
review of this heuristic theory.
An optical wave propagating through atmospheric turbu-
lence will experience irradiance fluctuations 共scintillation兲
due to small index of refraction fluctuations commonly
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called optical turbulence. Theoretical and experimental
studies of irradiance fluctuations generally involve the scin-
tillation index
I
2
⫽
具
I
2
典
具
I
典
2
⫺ 1, 共6兲
where the quantity I denotes irradiance 共intensity兲 of the
optical wave and the angle brackets 具典 denote an ensemble
average or, equivalently with the assumption of ergodicity,
a long-time average. Weak fluctuation regimes lead to ex-
pressions for the scintillation index that are proportional to
the Rytov variance
1
2
⫽ 1.23C
n
2
k
7/6
L
11/6
, 共7兲
where k⫽ 2
/ is the optical wave number, is wave-
length, and L is the propagation path length between trans-
mitter and receiver. For values less than unity, the Rytov
variance is the scintillation index of a plane wave in the
absence of inner scale effects, and for values greater than
unity, it is considered a measure of the strength of optical
fluctuations.
Under weak fluctuation conditions, the scintillation in-
dex 关Eq. 共6兲兴 increases with increasing values of the Rytov
variance 关Eq. 共7兲兴. The scintillation index continues to in-
crease beyond the weak fluctuation regime and reaches a
maximum value greater than unity 共sometimes as large as 5
or 6兲 in the regime characterized by random focusing.
19
With increasing path length or inhomogeneity strength, the
focusing effect is weakened by multiple self-interference
and the fluctuations slowly begin to decrease, saturating at
a level for which the scintillation index approaches unity
from above.
In a recent series of papers on scintillation theory,
16–18
the irradiance of the received optical wave is modeled as a
product I⫽ xy, where x arises from large-scale turbulent
eddies and y from small scale eddies. It is assumed that x
and y are statistically independent random processes for
which the second moment of irradiance is
具
I
2
典
⫽
具
x
2
典具
y
2
典
⫽
共
1⫹
x
2
兲
共
1⫹
y
2
兲
, 共8兲
where
x
2
and
y
2
are normalized variances of x and y, re-
spectively. For mathematical convenience, we have taken
具
I
典
⫽ 1. Based on Eqs. 共6兲 and 共8兲, the implied scintillation
index is
I
2
⫽
共
1⫹
x
2
兲
共
1⫹
y
2
兲
⫺ 1⫽
x
2
⫹
y
2
⫹
x
2
y
2
. 共9兲
Small-scale contributions to scintillation are associated
with turbulent cells smaller than the Fresnel zone R
F
⫽ (L/k)
1/2
or the coherence radius
0
, whichever is
smaller. Large-scale fluctuations in the irradiance are gen-
erated by turbulent cells larger than that of the first Fresnel
zone or the scattering disk L/k
0
, whichever is larger. Un-
der strong fluctuation conditions, spatial cells having size
between those of the coherence radius and the scattering
disk contribute little to scintillation. That is, because of the
loss of spatial coherence, only the very largest cells nearer
to the transmitter have any focusing effect on the illumina-
tion of small diffractive cells nearer to the receiver, and
eventually even these large cells cannot focus or defocus.
When this loss of coherence happens, the illumination of
the small cells is 共statistically兲 evenly distributed and the
fluctuations of the propagating wave are just due to random
interference of a large number of diffraction scatterings of
the small cells.
To develop a pdf model of the irradiance consistent with
this theory, we make the assumption that both large-scale
and small-scale irradiance fluctuations are governed by
gamma distributions, namely
p
x
共
x
兲
⫽
␣
共
␣
x
兲
␣
⫺ 1
⌫
共
␣
兲
exp
共
⫺
␣
x
兲
, x⬎ 0,
␣
⬎ 0, 共10兲
p
y
共
y
兲
⫽

共

y
兲

⫺ 1
⌫
共

兲
exp
共
⫺

y
兲
, y⬎ 0,

⬎ 0. 共11兲
By first fixing x and writing y⫽ I/x, we obtain the condi-
tional pdf
p
y
共
I
兩
x
兲
⫽

共

I/x
兲

⫺ 1
x⌫
共

兲
exp
共
⫺

I/x
兲
, I⬎ 0, 共12兲
in which x is the 共conditional兲 mean value of I. To obtain
the unconditional irradiance distribution, we form the aver-
age of Eq. 共12兲 over the gamma distribution of Eq. 共10兲,
which leads to
p
共
I
兲
⫽
冕
0
⬁
p
y
共
I
兩
x
兲
p
x
共
x
兲
dx
⫽
2
共
␣
兲
共
␣
⫹

兲
/2
⌫
共
␣
兲
⌫
共

兲
I
共
␣
⫹

兲
/2⫺ 1
K
␣
⫺

关
2
共
␣
I
兲
1/2
兴
, I⬎ 0.
共13兲
We call Eq. 共13兲 the gamma-gamma distribution. The
positive parameter
␣
represents the effective number of
large-scale cells of the scattering process and

similarly
represents the effective number of small-scale cells. When
optical turbulence is weak, the effective number of scale
sizes smaller and larger than the first Fresnel zone is large,
resulting in
␣
Ⰷ 1 and

Ⰷ 1. As the irradiance fluctuations
increase and the focusing regime is approached, both pa-
rameters of Eq. 共13兲 decrease substantially. Beyond the fo-
cusing regime and approaching the saturation regime, we
find that

→ 1, indicating that the effective number of
small-scale cells ultimately reduces to one, determined by
the transverse spatial coherence radius of the optical wave.
On the other hand, the effective number of discrete refrac-
tive scatterers
␣
increases again with increasing turbulence
strength and eventually becomes unbounded in the satura-
tion regime. Under these conditions, the gamma-gamma
distribution approaches the negative exponential distribu-
tion in the deep saturation regime.
From the gamma-gamma pdf 关Eq. 共13兲兴 we find
具
I
2
典
⫽ (1⫹ 1/
␣
)(1⫹ 1/

), and thus we identify the parameters
of this distribution with the large-scale and small-scale
scintillation according to
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␣
⫽
1
x
2
,

⫽
1
y
2
. 共14兲
It follows that the total scintillation index 关Eq. 共9兲兴 is re-
lated to these parameters by
I
2
⫽
1
␣
⫹
1

⫹
1
␣
. 共15兲
4 Comparison with Plane Wave Data
In this section, we compare the gamma-gamma distribution
model with published numerical simulation data for the pdf
by Flatte
´
et al.
3
The simulation data for a plane wave inci-
dent on a random medium characterized by homogeneous,
isotropic Kolmogorov turbulence led to numerous plots of
the log-irradiance pdf as a function of (ln I⫺
具
ln I
典
)/
, cov-
ering a range of conditions that extends from weak irradi-
ance fluctuations well into the saturation regime character-
ized by
1
2
⫽ 25. Here,
具
ln I
典
is the mean value of the log-
irradiance and
⫽
冑
ln I
2
, the latter being the root mean
square 共rms兲 value of lnI. The simulation pdf’s were dis-
played in this fashion in the hope that it would reveal their
salient features. In the saturation regime the simulation data
of Flatte
´
et al.
3
showed that the plane wave pdf lies some-
where between the log-normal and exponential distribu-
tions, and their moments lie between those of a log-normal-
exponential distribution and those of a K distribution.
The model for the refractive-index spectrum used to
generate the simulation data was of the form
⌽
n
共
兲
⫽ 0.033C
n
2
⫺ 11/3
f
共
l
0
兲
, 共16兲
where f (
l
0
) is a nondimensional function that describes
the high wave number spectral bump and dissipation
range.
20
Although the analytic form of this nondimensional
function is unknown, Andrews
21
has shown that it can be
closely approximated by
f
共
l
0
兲
⫽ exp
共
⫺
2
/
l
2
兲
关
1⫹ 1.802
共
/
l
兲
⫺ 0.254
共
/
l
兲
7/6
兴
,
l
⫽ 3.3/l
0
. 共17兲
In past analyses it was common to use the traditional spec-
trum in which f(
l
0
)⫽ exp
关
⫺(
l
0
/5.92)
2
兴
, but Flatte
´
et al.
3
point out that this latter spectrum leads to inaccuracies by
as much as 50% for predicting the scintillation index.
For negligible inner scale, the scintillation theory in
Refs. 16 and 18 for a plane wave leads to the large-scale
and small-scale variances given, respectively, by
x
2
⬵exp
冋
0.49
1
2
共
1⫹ 1.11
1
12/5
兲
7/6
册
⫺ 1, 共18兲
y
2
⬵exp
冋
0.51
1
2
共
1⫹ 0.69
1
12/5
兲
7/6
册
⫺ 1. 共19兲
In the presence of a finite inner scale, the comparable ex-
pression for the large-scale scintillation is
x
2
⫽ exp
再
0.16
1
2
冉
2.61
l
2.61⫹
l
⫹ 0.45
1
2
l
7/6
冊
7/6
⫻
冋
1⫹ 1.753
冉
2.61
2.61⫹
l
⫹ 0.45
1
2
l
7/6
冊
1/2
⫺ 0.252
冉
2.61
2.61⫹
l
⫹ 0.45
1
2
l
7/6
冊
7/12
册
冎
⫺ 1, 共20兲
where
l
⫽ L
l
2
/kl
0
2
⫽ 10.89(R
F
/l
0
)
2
, and the small-scale
variance is still approximated by Eq. 共19兲. From these ex-
pressions the parameters
␣
and

of the gamma-gamma pdf
can be determined through the use of Eq. 共14兲. The selec-
tion of parameters based on Eq. 共14兲 is equivalent to using
only measured values of the refractive index structure pa-
rameter C
n
2
and inner scale l
0
to predict all other parameters
arising in scaling the plots of the simulation data.
We plot in Figs. 1–5 the predicted log-irradiance pdf
associated with the gamma-gamma distribution 共solid line兲
for comparison with the simulation data illustrated in Figs.
4, 5, and 7 of Ref. 3. As representative of typical atmo-
spheric propagation conditions, we use the inner scale
value l
0
⫽ 0.5R
F
(
l
⫽ 44) in Figs. 1 and 2. In Fig. 1, we
use the simulation values
1
2
⫽ 0.1, l
0
⫽ 0.5R
F
, and in Fig. 2
we use the simulation values
1
2
⫽ 2, l
0
⫽ 0.5R
F
, taken from
Figs. 4 and 5, respectively, of Ref. 3. We also plot the
log-normal pdf 共dashed line兲 in Fig. 1 for the sake of com-
parison. Here we find that the gamma-gamma distribution
provides a better fit to the simulation data than does the
log-normal distribution. In fact, the predicted gamma-
gamma pdf generally provides a good fit to the data over
most of the abscissa values in Figs. 1 and 2, except it falls
somewhat under the data for values on the abscissa less
than ⫺3. Rather than use the simulation results, however,
values of the scaling parameters
具
ln I
典
and
required in the
plots were calculated directly from the gamma-gamma pdf.
Fig. 1 The pdf of the scaled log-irradiance for a plane wave in the
case of weak irradiance fluctuations:
1
2
⫽ 0.1 and
l
0
/
R
F
⫽ 0.5. The
open circles represent simulation data, the dashed line is from the
lognormal pdf, and the solid line is from the gamma-gamma pdf [Eq.
(13)] with
␣
and

predicted by Eqs. (14).
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![Fig. 3 The pdf of the scaled log-irradiance for a plane wave in the case of strong irradiance fluctuations: s1 2525 and l0 /RF50. The open circles represent simulation data, the dashed line is the K distribution [Eq. (4)] with parameter a determined from the simulation data, and the solid line is from the gamma-gamma pdf [Eq. (13)] with a and b predicted by Eqs. (14).](/figures/fig-3-the-pdf-of-the-scaled-log-irradiance-for-a-plane-wave-3rot8anr.webp)
