# Scintillation of Partially Coherent Light in Time Varying Complex Media

Josselin Garnier Centre de Mathématiques Appliquées, Ecole Polytechnique,
91128 Palaiseau Cedex, France<EMAIL_ADDRESS>Knut Sølna
Department of Mathematics, University of California, Irvine CA 92697
<EMAIL_ADDRESS>
###### Abstract

We present a theory for wave scintillation in the situation with a time-
dependent partially coherent source and a time-dependent randomly
heterogeneous medium. Our objective is to understand how the scintillation
index of the measured intensity depends on the source and medium parameters.
We deduce from an asymptotic analysis of the random wave equation a general
form of the scintillation index and we evaluate this in various scaling
regimes. The scintillation index is a fundamental quantity that is used to
analyze and optimize imaging and communication schemes. Our results are useful
to quantify the scintillation index under realistic propagation scenarios and
to address such optimization challenges.

###### Contents

  1. 1 Introduction
  2. 2 Probing Time-Dependent Complex Media with Partially Coherent Sources
    1. 2.1 Source and Medium Modeling
    2. 2.2 The Itô-Schrödinger Equation
    3. 2.3 Measurements and the Challenge of Understanding Scintillation
  3. 3 Fourth-order Field Moment and Scintillation
  4. 4  Scintillation in Canonical Scaling Regimes
    1. 4.1 Source with Large Correlation Radius
    2. 4.2 Source with Intermediate Correlation Radius
    3. 4.3 Source with Small Correlation Radius
  5. 5 Example with Experimental Data
  6. 6 Conclusions
  7. A Analysis of Fourth-order Moment Equations
  8. B Derivation of Scintillation Results
    1. B.1 Scintillation Regime with a Large Correlation Radius of the Source
    2. B.2 Scintillation Regime with an Intermediate Correlation Radius of the Source
    3. B.3 Scintillation Regime with a Small Correlation Radius of the Source

##  1 Introduction

We consider the fundamental problem of characterizing the scintillation of
optical measurements with a time-dependent partially coherent source and a
time-dependent random medium. The scintillation index corresponds to a measure
of the signal-to-noise ratio or relative strength of fluctuations in the
intensity. If $I$ is the measured intensity (irradiance) then we define the
scintillation index by

${\cal S}=\frac{\mathbb{E}[I^{2}]-\mathbb{E}[I]^{2}}{\mathbb{E}[I]^{2}},$ (1)

where $\mathbb{E}[\cdot]$ stands for the statistical expectation obtained by
averaging over repeated measurements under independent and identically
distributed conditions. Modeling and analysis of laser speckle and
scintillation is a classic challenge in optics [10, 32, 1]. A rigorous
mathematical analysis and quantification of scintillation has been a long
standing open question despite the long history and importance of this
challenge. General insight about what governs the scintillation is important
in the design of optical systems, for instance for imaging and communication
through the turbulent atmosphere [27] and through oceanic turbulence [36]. The
challenge of choosing appropriate sources for scintillation control has
received a lot of attention [26].

In [19, 20] we presented an analysis of the scintillation problem for
deterministic coherent beams and plane wave sources and time-independent
media. In this paper we consider the scintillation problem when the source is
partially coherent in time and space and the medium has time and space random
fluctuations. Partially coherent sources have indeed been promoted for
reducing scintillation at a receiving end in the context of laser propagation
[29, 2, 35, 26]. Most of these studies rely on physical experiments or
numerics and Monte Carlo simulations to evaluate the scintillation index. From
the theoretical point of view, analysis of wave propagation can be carried out
in a perturbative regime using in particular Rytov theory with small
fluctuations in the wave field to obtain insight about the scintillation [30,
3]. The fluctuations of intensity over different receiver response times have
been studied in [12, 6, 7] in the limits of very slow or very fast detectors.
The effect of temporal coherence on scintillation for weak turbulence was
considered in [13]. An analysis of scintillation and how it depends on the
smoothness of deterministic initial condition is presented in [5] with a focus
on understanding self-averaging situations with a vanishing scintillation
index. Issues related to aperture averaging is also considered in [31] in the
context of deterministic sources by using a path integral approach for
modeling the effect of turbulence.

Here we consider the high-frequency and far-field situation where the effect
of the random medium can be captured by a white-noise term in the Itô-
Schrödinger equation that governs the evolution of the wave field [8, 9]. This
equation can address both weak intensity fluctuations and strong intensity
fluctuations (the saturated regime). The response time of the photodetector,
the coherence times of the source and the random medium can be arbitrary,
provided that they are larger than the travel time of the field from the
source to the detector through the medium. Under such circumstances, the
effective reduced system (45) for the fourth-order moments of the wave field
can be derived from the Itô-Schrödinger equation and used for numerical
evaluation of the scintillation index in the general high-frequency and far-
field situation. Based on this system we obtain explicit expressions of the
scintillation index in three scaling regimes determined by the ratio of the
correlation radius of the source over the correlation radius of the medium. We
then explicitly characterize scintillation with partially coherent sources and
time-dependent random media and quantify how the space-time statistical
parameters of the source and medium affect the scintillation index. An
important aspect of our analysis is that we allow the medium to be time-
dependent, so that it changes on a time scale that is slow relative to the
travel time of the optical field. This is the situation in the context of
laser beam propagation in the atmosphere with turbulence creating slow
temporal changes of the medium. The detector in our modeling has a finite
response time which can be on the time scale of the changes in the medium. The
averaging at the detector can have a strong impact on the scintillation index
depending on the characteristic time scales involved.

The configuration that we consider is illustrated in Figure 1 with a partially
coherent source field impinging from the left and propagating through a random
medium and then the scintillating intensity pattern is recorded at the
receiver end. Our objective is to characterize the scintillation index of the
observed transmitted intensity pattern shown to the right in the figure.

Figure 1: The figure shows the configuration that we consider. A partially
coherent source fluctuating randomly in space and time is impinging on a
complex medium. The complex medium is modeled as random and changes in time.
The time changes in the medium happen on the recording time scale of the
detector, but are slow relative to the travel time of the wave over the
considered range. Due to time averaging the detector measures a smoothed
version of the intensity and we seek to characterize the scintillation index
of this measurement, which corresponds to a signal-to-noise ratio.

We comment on a special, but important, case corresponding to the wave field
having a Gaussian distribution. Indeed it is a well-accepted conjecture that
the statistics of the complex wave field becomes circularly symmetric complex
Gaussian when the wave propagates through the turbulent atmosphere [33, 37]
and the conjecture can be proved in certain situations [14, 4, 28]. In the
Gaussian case the intensity is the sum of the squares of two independent
Gaussian random variables, which up to a scaling has $\chi$-square
distribution with two degrees of freedom, that is, an exponential
distribution. This situation gives a unit scintillation index. Based on an
analysis of the fourth moment of the wave field we identify in this paper
regimes that correspond to a unit value for the scintillation index and which
are consistent with the Gaussian conjecture. The case with a Gaussian field
distribution is the critical situation with the signal-to-noise ratio of the
intensity being one. In general the scintillation index can be below one for
small fluctuations in the intensity and can reach values beyond one when the
intensity distribution has heavier tails than those corresponding to the
exponential distribution. We encounter both situations in this paper and
discuss what type of scaling regime may lead to such situations.

The outline of the paper is as follows. We formulate the problem in Section 2.
This involves defining the statistical models for the source and the medium
and deriving the stochastic partial differential equation, the Itô-Schrödinger
equation, that characterizes the wave field. We then relate the solution of
the stochastic partial differential equation to the measured scintillation
index. The main theoretical foundation for analyzing the scintillation is a
framework for analyzing the fourth-order moment of the wave field and we
discuss this in Section 3. In Section 4 we give the main results which
characterize the scintillation index in various scaling regimes. In Section 5
we present an example involving data presented in [29]. Technical calculations
associated with the fourth-order moment equations are presented in the
appendices.

##  2 Probing Time-Dependent Complex Media with Partially Coherent Sources

In this section we outline the modeling and the problem that we will consider.
In section 2.1 we describe the statistical modeling of the source and of the
random medium. In section 2.2 we give the Itô-Schrödinger equation that
describes the evolution of the wave field in the random medium. In section 2.3
we relate the random transmitted wave field to the quantity of interest which
is the scintillation index of the measurements.

###  2.1 Source and Medium Modeling

The time-harmonic field $U(z,{{\itbf x}},t)$ satisfies the Helmholtz equation

$\Delta{U}+k_{o}^{2}{\rm n}^{2}(z,{{\itbf x}},t){U}=-\delta(z)f({{\itbf
x}},t),\quad\quad(z,{{\itbf x}})\in\mathbb{R}\times\mathbb{R}^{2},$ (2)

where $k_{o}=2\pi/\lambda_{o}$ is the central wavenumber ($\lambda_{o}$ is the
central wavelength). Here $t$ is the slow time corresponding to the time at
which the random medium and the source change. The coherence times of the
medium and source are assumed to be much larger than the travel time from the
source to the detector through the medium so that $t$ is a frozen parameter in
(2).

The source $f$ in the plane $z=0$ is partially coherent, statistically
stationary in space and time. We model it as a complex Gaussian process with
mean zero, variance one, and covariance

$\mathbb{E}\Big{[}f\big{(}{{\itbf x}}+\frac{{{\itbf
y}}}{2},t+\frac{\tau}{2}\big{)}\overline{f}\big{(}{{\itbf x}}-\frac{{{\itbf
y}}}{2},t-\frac{\tau}{2}\big{)}\Big{]}=F\left(\frac{\tau}{\tau_{\rm
s}}\right)\exp\Big{(}-\frac{|{{\itbf y}}|^{2}}{4\ell_{\rm s}^{2}}\Big{)},$ (3)

where $\ell_{\rm s}$, resp. $\tau_{\rm s}$, is the correlation radius, resp.
the coherence time, of the source and the time covariance function $F$ is
normalized so that $F(0)=1$ and $\int_{0}^{\infty}F(s)ds=1$. Here the
correlation radius of the source $\ell_{\rm s}$ is assumed to be small
relative to the range $L$ and the coherence time $\tau_{\rm s}$ is assumed to
be much larger than the propagation time $L/c_{o}$, where $c_{o}$ is the
background speed of propagation and $L$ is the distance from the source to the
detector. For convenience we use here a Gaussian correlation function for the
spatial source correlations, but we remark that we could have used a more
general form. A more detailed model for the source, in particular a discussion
of realization via Spatial Light Modulators (SLMs) can be found in [24, 2].
For other approaches to the generation of the partially coherent source we
refer to [34] for instance.

The medium is random and we denote by $\nu$ the relative fluctuations in the
square index of refraction: ${\rm n}^{2}(z,{{\itbf x}},t)=1+\nu(z,{{\itbf
x}},t)$. The stochastic process $\nu$ is stationary in space and time and
zero-mean, and its covariance function is of the form

$\mathbb{E}[\nu(z^{\prime}+z,{{\itbf x}}^{\prime}+{{\itbf
x}},t+\tau)\nu(z^{\prime},{{\itbf x}}^{\prime},t)]=\sigma_{\rm
m}^{2}G\left(\frac{\tau}{\tau_{\rm m}}\right){\cal C}_{\rm
m}\left(\frac{z}{\ell_{\rm m}},\frac{{{\itbf x}}}{\ell_{\rm m}}\right),$ (4)

where $\ell_{\rm m}$, resp. $\tau_{\rm m}$, is the correlation radius, resp.
the coherence time, of the random medium fluctuations, $\sigma_{\rm m}$ is the
standard deviation of the fluctuations of the square index of refraction, and
the functions $G$ and ${\cal C}_{\rm m}$ are normalized so that $G(0)=1$,
$\int_{0}^{\infty}G(s)ds=1$, ${\cal C}_{\rm m}(0,{\bf 0})=1$,
$\int_{\mathbb{R}}{\cal C}_{\rm m}(\zeta,{\bf 0})d\zeta=1$, and
$\int_{\mathbb{R}^{2}}{\cal C}_{\rm
m}(0,\boldsymbol{\chi})d\boldsymbol{\chi}=1$. The special case where the
correlation function corresponds to Kolmogorov turbulence is discussed in
[25]. Here the correlation radius of the random medium $\ell_{\rm m}$ is
assumed to be small relative to the range $L$ and the coherence time of the
medium $\tau_{\rm m}$ is assumed to be much larger than the propagation time
$L/c_{o}$. Thus, the ‘turnover time’ of the medium is long compared to the
propagation time, however, we assume that it is on the scale of the coherence
time of the source $\tau_{\rm s}$. Our interest is now in determining how the
characteristics of these source and medium statistics determine the
scintillation index of the transmitted wave field. We discuss next the
equation that can be used to describe the evolution of the statistics of the
wave field, that is, the equation that describes how the random scattering
modifies the statistical distribution of the wave field from those of the
source as the wave field propagates through the complex medium.

###  2.2 The Itô-Schrödinger Equation

The complex amplitude field $u$ which modulates the carrier plane wave:

$U(z,{{\itbf x}},t)=\frac{i}{2k_{o}}\exp(ik_{o}z)u(z,{{\itbf x}},t)$

satisfies the Itô-Schrödinger equation [21]:

$d{u}(z,{{\itbf x}},t)=\frac{i}{2k_{o}}\Delta_{{{\itbf x}}}{u}(z,{{\itbf
x}},t)dz+\frac{ik_{o}}{2}{u}(z,{{\itbf x}},t)\circ d{B}(z,{{\itbf x}},t),$ (5)

with the initial condition in the plane $z=0$:

${u}(z=0,{{\itbf x}},t)=f({{\itbf x}},t).$

Note that here $\Delta_{{\itbf x}}$ is the transverse Laplacian and $t$ is the
slow time scale corresponding to the time at which the source and the random
medium change and that it is a frozen parameter in (5). This is a consequence
of our assumption that the coherence times of the source and of the medium are
long relative to the travel time of the field over the range $L$. The
derivation of Eq. (5) from Eq. (2) follows the lines of the proof presented in
[21, 19] which deals with the case of a time-independent medium. It is valid
in the white-noise paraxial regime, when the wavelength is much smaller than
the correlation radii of the source and of the medium, which are themselves
much smaller than the propagation distance. Note that in (5) the symbol
$\circ$ stands for the Stratonovich stochastic integral, moreover, that
$B(z,{{\itbf x}},t)$ is a real-valued Brownian field over
$[0,\infty)\times\mathbb{R}^{2}\times\mathbb{R}$ with a covariance that
derives from the model for the medium fluctuations in (4)

$\mathbb{E}[{B}(z,{{\itbf x}},t){B}(z^{\prime},{{\itbf
x}}^{\prime},t^{\prime})]=\sigma_{\rm m}^{2}\ell_{\rm
m}{\min\\{z,z^{\prime}\\}}G\left(\frac{t-t^{\prime}}{\tau_{\rm
m}}\right){C}\left(\frac{{{\itbf x}}-{{\itbf x}}^{\prime}}{\ell_{\rm
m}}\right),$ (6)

where $C(\boldsymbol{\chi})=\int_{\mathbb{R}}{\cal C}_{\rm
m}(\zeta,\boldsymbol{\chi})d\zeta$, which is such that $C({\bf 0})=1$. The
first- and second-order moments of the wave field solution of (5) has been
studied in [21, 22]. The first-order moment of the wave field is zero. The
second-order moment of the wave field (mutual coherence function) defined by

$\mu_{2}(z,{{\itbf x}},{{\itbf y}};\tau)=\mathbb{E}\Big{[}u\big{(}z,{{\itbf
x}}+\frac{{{\itbf y}}}{2},t+\tau\big{)}\overline{u\big{(}z,{{\itbf
x}}-\frac{{{\itbf y}}}{2},t\big{)}}\Big{]}$ (7)

satisfies [19]

$\frac{\partial\mu_{2}}{\partial z}=\frac{i}{k_{o}}\nabla_{{{\itbf
x}}}\cdot\nabla_{{{\itbf y}}}\mu_{2}+\frac{k_{o}^{2}\sigma_{\rm
m}^{2}\ell_{\rm m}}{4}{U}_{2}\big{(}{{\itbf x}},{{\itbf
y}};\tau\big{)}\mu_{2},$ (8)

with the potential $U_{2}({{\itbf x}},{{\itbf y}};\tau)=G(\tau/\tau_{\rm
m})C({{\itbf y}}/\ell_{\rm m})-1$ and the initial condition
$\mu_{2}(z=0,{{\itbf x}},{{\itbf y}};\tau)=\mathbb{E}[f({{\itbf x}}+{{\itbf
y}}/2,t+\tau)\overline{f({{\itbf x}}-{{\itbf y}}/2,t)}]$ that are both
independent on ${{\itbf x}}$. The second-order moment is given by

$\mu_{2}(z,{{\itbf x}},{{\itbf y}};\tau)=F\left(\frac{\tau}{\tau_{\rm
s}}\right)\exp\Big{[}-\frac{|{{\itbf y}}|^{2}}{4\ell_{\rm
s}^{2}}-\frac{\sigma_{\rm m}^{2}k_{o}^{2}\ell_{\rm
m}z}{4}\Big{(}1-G\left(\frac{\tau}{\tau_{\rm m}}\right)C\left(\frac{{{\itbf
y}}}{\ell_{\rm m}}\right)\Big{)}\Big{]}.$ (9)

By inspection of the behavior of the second-order moment when $\tau=0$:

$\mathbb{E}\Big{[}u\big{(}z,{{\itbf x}}+\frac{{{\itbf
y}}}{2},t\big{)}\overline{u\big{(}z,{{\itbf x}}-\frac{{{\itbf
y}}}{2},t\big{)}}\Big{]}=\exp\Big{[}-\frac{|{{\itbf y}}|^{2}}{4\ell_{\rm
s}^{2}}-\frac{\sigma_{\rm m}^{2}k_{o}^{2}\ell_{\rm
m}z}{4}\Big{(}1-C\left(\frac{{{\itbf y}}}{\ell_{\rm m}}\right)\Big{)}\Big{]},$

we find that the scattering mean free path $\ell_{\rm mfp}$ (that is the
typical propagation distance over which a coherent wave becomes incoherent) is

$\ell_{\rm mfp}^{-1}=\frac{\sigma_{\rm m}^{2}k_{o}^{2}\ell_{\rm m}}{8}.$ (10)

Note that the scattering mean free path therefore is inversely proportional to
the medium correlation length $\sigma_{\rm m}^{2}\ell_{\rm m}$ characterizing
the strength of the scattering. When $C$ is smooth at zero:
$C(\boldsymbol{\chi})=1-c_{2}|\boldsymbol{\chi}|^{2}+o(|\boldsymbol{\chi}|^{2})$,
the correlation radius $\rho_{\rm c}(z)$ of the wave field is

$\rho_{\rm c}^{-2}(z)=\ell_{\rm s}^{-2}+\frac{c_{2}\sigma_{\rm
m}^{2}k_{o}^{2}z}{\ell_{\rm m}}.$ (11)

By inspection of the behavior of the second-order moment when ${{\itbf
y}}={\bf 0}$:

$\mathbb{E}\Big{[}u\big{(}z,{{\itbf
x}},t+\tau\big{)}\overline{u\big{(}z,{{\itbf
x}},t\big{)}}\Big{]}=F\left(\frac{\tau}{\tau_{\rm
s}}\right)\exp\Big{[}-\frac{\sigma_{\rm m}^{2}k_{o}^{2}\ell_{\rm
m}z}{4}\Big{(}1-G\left(\frac{\tau}{\tau_{\rm m}}\right)\Big{)}\Big{]},$

we can see that, when $F(s)=\exp(-s^{2}/4)$ and $G$ is smooth at zero,
$G(s)=1-g_{2}s^{2}+o(s^{2})$, the coherence time $\tau_{\rm c}(z)$ of the wave
field is

$\tau_{\rm c}^{-2}(z)=\tau_{\rm s}^{-2}+\frac{g_{2}\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm m}z}{\tau_{\rm m}^{2}}.$ (12)

Therefore, for deep probing both the correlation radius and coherence time of
the wave field are proportional to the reciprocal of the square root of the
propagation distance. We discuss next the measurements of intensity associated
with the field $u$ and the associated scintillation index.

###  2.3 Measurements and the Challenge of Understanding Scintillation

The intensity at lateral location ${{\itbf x}}$ in the plane $z=L$ of the
photodetector is

$I_{T}({{\itbf x}})=\frac{1}{T}\int_{0}^{T}|u(L,{{\itbf x}},t)|^{2}dt.$ (13)

The intensity profile forms a smoothed speckle pattern. This smoothed speckle
pattern depends in particular on the values of the integration time $T$, the
coherence times of the source $\tau_{\rm s}$ and of the medium $\tau_{\rm m}$
and we aim to understand how.

The empirical scintillation index measured by the photodetector of total
aperture $A$ is

${\cal S}=\frac{\frac{1}{|A|}\int_{A}I_{T}({{\itbf x}})^{2}d{{\itbf
x}}-\Big{(}\frac{1}{|A|}\int_{A}I_{T}({{\itbf x}})d{{\itbf
x}}\Big{)}^{2}}{\Big{(}\frac{1}{|A|}\int_{A}I_{T}({{\itbf x}})d{{\itbf
x}}\Big{)}^{2}}.$ (14)

A more detailed model for the detector, in particular a discussion of the role
of finite sized pixels of a (CCD) camera can be found in for instance [23,
29]. If the diameter of the photodetector aperture $A$ is large (much larger
than the speckle radius, i.e. the correlation radius), then

${\cal S}=\frac{\mathbb{E}[I_{T}({\bf 0})^{2}]-\mathbb{E}[I_{T}({\bf
0})]^{2}}{\mathbb{E}[I_{T}({\bf 0})]^{2}},$ (15)

which is equal to

${\cal S}=\frac{2}{T}\int_{0}^{T}\big{(}1-\frac{\tau}{T}\big{)}\frac{{\rm
Cov}\big{(}|u(L,{\bf 0},0)|^{2},|u(L,{\bf
0},\tau)|^{2}\big{)}}{\mathbb{E}[|u(L,{\bf 0},0)|^{2}]^{2}}d\tau,$ (16)

which is our quantity of interest. Note that the expectation here and below
refers to expectation with respect to both the randomness of the medium and of
the source. Note also that it follows from [19] Section 5 that
$\mathbb{E}[|u(L,{\bf 0},t)|^{2}]=1$. Therefore, in order to analyze ${\cal
S}$, it remains to compute the fourth-order moment $\mathbb{E}[|u(L,{\bf
0},0)|^{2}|u(L,{\bf 0},\tau)|^{2}]$. We discuss the task of computing this
moment next.

##  3 Fourth-order Field Moment and Scintillation

It is convenient to introduce the notation

$f_{\tau}=F\left(\frac{\tau}{\tau_{\rm s}}\right),\quad
g_{\tau}=G\left(\frac{\tau}{\tau_{\rm m}}\right).$ (17)

We also introduce a notation for the fourth moment

$\mu_{4}(z,{{\itbf x}}_{1},{{\itbf x}}_{2},{{\itbf y}}_{1},{{\itbf
y}}_{2};\tau)=\mathbb{E}\big{[}u(z,{{\itbf
x}}_{1},t+\tau)\overline{u(z,{{\itbf y}}_{1},t+\tau)}u(z,{{\itbf
x}}_{2},t)\overline{u(z,{{\itbf y}}_{2},t)}\big{]}.$ (18)

Here we focus on the fourth moment, while in [15] moments of all orders were
considered under some simplifying assumptions of a different type. The fourth
moment in (16) is a special case of the general fourth moment in (18)
corresponding to evaluation at one spatial point only. The motivation for
introducing the general fourth moment is that we can identify a partial
differential equation satisfied by this general moment and we will
subsequently discuss the simplification that follows from evaluating this at
particular values for the arguments. The general fourth moment satisfies the
equation

$\frac{\partial\mu_{4}}{\partial z}=\frac{i}{2k_{o}}\Big{(}\Delta_{{{\itbf
x}}_{1}}+\Delta_{{{\itbf x}}_{2}}-\Delta_{{{\itbf y}}_{1}}-\Delta_{{{\itbf
y}}_{2}}\Big{)}\mu_{4}+\frac{k_{o}^{2}\sigma_{\rm m}^{2}\ell_{\rm
m}}{4}{U}_{4}\big{(}{{\itbf x}}_{1},{{\itbf x}}_{2},{{\itbf y}}_{1},{{\itbf
y}}_{2};\tau\big{)}\mu_{4},$ (19)

with the generalized potential

$\displaystyle{U}_{4}\big{(}{{\itbf x}}_{1},{{\itbf x}}_{2},{{\itbf
y}}_{1},{{\itbf y}}_{2};\tau\big{)}={C}\left(\frac{{{\itbf x}}_{1}-{{\itbf
y}}_{1}}{\ell_{\rm m}}\right)+{C}\left(\frac{{{\itbf x}}_{2}-{{\itbf
y}}_{2}}{\ell_{\rm m}}\right)+g_{\tau}{C}\left(\frac{{{\itbf x}}_{1}-{{\itbf
y}}_{2}}{\ell_{\rm m}}\right)$
$\displaystyle\hbox{}+g_{\tau}{C}\left(\frac{{{\itbf x}}_{2}-{{\itbf
y}}_{1}}{\ell_{\rm m}}\right)-g_{\tau}{C}\left(\frac{{{\itbf x}}_{1}-{{\itbf
x}}_{2}}{\ell_{\rm m}}\right)-g_{\tau}{C}\left(\frac{{{\itbf y}}_{1}-{{\itbf
y}}_{2}}{\ell_{\rm m}}\right)-2,$ (20)

and the initial condition:

$\mu_{4}(z=0,{{\itbf x}}_{1},{{\itbf x}}_{2},{{\itbf y}}_{1},{{\itbf
y}}_{2};\tau)=\mathbb{E}\big{[}f({{\itbf x}}_{1},t+\tau)\overline{f({{\itbf
y}}_{1},t+\tau)}f({{\itbf x}}_{2},t)\overline{f({{\itbf y}}_{2},t)}\big{]}.$

This follows from (5) using Itô calculus for Hilbert space valued processes
[20, 11]. Using the Gaussian property of the source and Isserlis formula, the
initial condition for the fourth-order moment is:

$\displaystyle\mu_{4}(z=0,{{\itbf x}}_{1},{{\itbf x}}_{2},{{\itbf
y}}_{1},{{\itbf y}}_{2};\tau)=$ $\displaystyle\exp\Big{(}-\frac{|{{\itbf
x}}_{1}-{{\itbf y}}_{1}|^{2}}{4\ell_{\rm s}^{2}}-\frac{|{{\itbf
x}}_{2}-{{\itbf y}}_{2}|^{2}}{4\ell_{\rm s}^{2}}\Big{)}$
$\displaystyle+f_{\tau}^{2}\exp\Big{(}-\frac{|{{\itbf x}}_{1}-{{\itbf
y}}_{2}|^{2}}{4\ell_{\rm s}^{2}}-\frac{|{{\itbf x}}_{2}-{{\itbf
y}}_{1}|^{2}}{4\ell_{\rm s}^{2}}\Big{)}.$ (21)

We can now express the quantity of interest, the scintillation index (16), in
terms of the general fourth moment

${\cal
S}=\frac{2}{T}\int_{0}^{T}\big{(}1-\frac{\tau}{T}\big{)}\frac{\mu_{4}(L,{\bf
0},{\bf 0},{\bf 0},{\bf 0};\tau)-\mu_{2}(L,{\bf 0},{\bf
0};0)^{2}}{\mu_{2}(L,{\bf 0},{\bf 0};0)^{2}}d\tau,$ (22)

where $\mu_{2}(L,{\bf 0},{\bf 0};0)=\mathbb{E}[|u(L,{\bf 0},t)|^{2}]=1$. Thus,
it is the special fourth moment $\mu_{4}(L,{\bf 0},{\bf 0},{\bf 0},{\bf
0};\tau)$ that is needed to analyze the scintillation index. The explicit
solution of the problem (19) is not known. In some scaling regimes we can,
however, identify asymptotic solutions using the framework introduced in [20].
In Appendix A we discuss a fundamental transformation of the fourth moment
equation in (19) to a simplified problem from which the special fourth moment
$\mu_{4}(L,{\bf 0},{\bf 0},{\bf 0},{\bf 0};\tau)$ derives.

##  4  Scintillation in Canonical Scaling Regimes

We discuss here the three scaling regimes for the scintillation. In these
regimes we can solve for the fourth moment in (19) explicitly. This allows us
to get quantitative insight about the behavior of the scintillation and how it
depends on the characteristic parameters in the problem and we comment on this
in detail. The first two regimes are particular cases of the far-field (or
Fraunhofer) regime $\frac{\lambda_{o}L}{\min(\ell_{\rm s},\ell_{\rm
m})^{2}}\gg 1$, when scattering is moderate or strong $L\gtrsim\ell_{\rm
mfp}$. The third regime is a special Fresnel regime
$\frac{\lambda_{o}L}{\min(\ell_{\rm s},\ell_{\rm m})^{2}}\sim 1$ when
scattering is strong $L\gg\ell_{\rm mfp}$. The parameter determining the
different regimes of scintillation is the ratio of the correlation radius
$\ell_{\rm s}$ of the source over the correlation radius $\ell_{\rm m}$ of the
medium fluctuations.

###  4.1 Source with Large Correlation Radius

We consider first the regime in which the correlation radius $\ell_{\rm s}$ of
the source is larger than the correlation radius of the medium $\ell_{\rm m}$
so that $\frac{\lambda_{o}L}{\ell_{\rm m}^{2}}\gg 1$ but
$\frac{\lambda_{o}L}{\ell_{\rm m}\ell_{\rm s}}\lesssim 1$. We carry out the
analysis of this regime in Appendix B.1 where we derive the following
expression for the scintillation index (22):

${\cal
S}=\frac{2}{T}\int_{0}^{T}\big{(}1-\frac{\tau}{T}\big{)}\Big{[}f_{\tau}^{2}\exp\Big{(}-\frac{(1-g_{\tau})\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm m}L}{2}\Big{)}+{\cal
Q}_{g_{\tau}}(L)+f_{\tau}^{2}{\cal Q}_{1}(L)\Big{]}d\tau,$ (23)

where the effective scattering kernel is given by

$\displaystyle{\cal Q}_{g}(L)=$ $\displaystyle\exp\Big{(}-\frac{\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm
m}L}{2}\Big{)}\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\exp\Big{(}-\frac{|{{\itbf
s}}|^{2}}{2}\Big{)}$ $\displaystyle\times\Big{[}\exp\Big{(}\frac{\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm m}Lg}{2}\int_{0}^{1}C\big{(}\frac{L{{\itbf
s}}s^{\prime}}{k_{o}\ell_{\rm m}\ell_{\rm
s}}\big{)}ds^{\prime}\Big{)}-1\Big{]}d{{\itbf s}}.$ (24)

When $\frac{\lambda_{o}L}{\ell_{\rm m}\ell_{\rm s}}\ll 1$ the expression is
simpler:

${\cal Q}_{g}(L)=\exp\Big{(}-\frac{\sigma_{\rm m}^{2}k_{o}^{2}\ell_{\rm
m}L}{2}(1-g)\Big{)}-\exp\Big{(}-\frac{\sigma_{\rm m}^{2}k_{o}^{2}\ell_{\rm
m}L}{2}\Big{)}.$

The kernel (24) depends on the two-point statistics of the random medium
fluctuations and reflects cumulative scattering effects over the propagation
distance $L$. Note that $g=0$ corresponds to the situation with intensities of
wave fields having propagated through uncorrelated random media and that
indeed ${\cal Q}_{0}(L)=0$.

The first term in the square brackets in (23) corresponds to the scintillation
contribution from the fluctuations of the source and this contribution is
damped by temporal decorrelation of the medium fluctuations as well as
temporal averaging at the detector. The second term in the square brackets is
the scintillation contribution produced by the random medium fluctuations and
is again damped by temporal decorrelation of the random medium fluctuations.
The last term in the square brackets is a cross term reflecting the
scintillation contribution from the combined effect of medium and source
fluctuations.

We next discuss the behavior of the scintillation index in various special
cases.

  * •

Note first that a rapid decay of $f_{\tau}$ (short coherence time) corresponds
to rapid decorrelation of the source. Such a rapid decorrelation serves to
reduce the scintillation index due to averaging by the photodetector.
Similarly rapid decay of $g_{\tau}$ corresponds to rapid decorrelation in the
medium fluctuations and reduced scintillation due to averaging over incoherent
scattering contributions. We find from (23) that when $T$ becomes much larger
than the coherence times of the source and of the medium, and assuming that
$\tau\mapsto f_{\tau}\in L^{2}$ (i.e., is square-integrable) and that
$g_{\tau}$ goes to zero at infinity fast enough so that $\tau\mapsto
Q_{g_{\tau}}(L)\in L^{1}$ (i.e., is integrable), then we have

${\cal S}\stackrel{{\scriptstyle T\to+\infty}}{{\longrightarrow}}0$

for any propagation distance.

  * •

An interesting situation corresponds to $g_{\tau}\equiv 1$, which means that
the medium is frozen. We have

$\frac{\mu_{4}(L,{\bf 0},{\bf 0},{\bf 0},{\bf 0};\tau)-\mu_{2}(L,{\bf 0},{\bf
0})^{2}}{\mu_{2}(L,{\bf 0},{\bf 0})^{2}}=f_{\tau}^{2}+(1+f_{\tau}^{2}){\cal
Q}_{1}(L).$

This means that, even if $\tau$ is so large that the initial fields are
independent ($f_{\tau}=0$) the intensities of the transmitted fields at the
photodetector at different times are correlated, in fact the covariation
degree is zero at $L=0$, is not zero for positive $L$ and goes to zero as
$L\to+\infty$. The scintillation index is given by

$\displaystyle{\cal S}=$
$\displaystyle\frac{2}{T}\int_{0}^{T}\big{(}1-\frac{\tau}{T}\big{)}\Big{[}f_{\tau}^{2}+(1+f_{\tau}^{2}){\cal
Q}_{1}(L)\Big{]}d\tau,$ (25)

so that, when $T$ becomes much larger than the coherence time of the source,
and assuming that $f_{\tau}\in L^{2}$, then we have

${\cal S}\stackrel{{\scriptstyle T\to+\infty}}{{\longrightarrow}}{\cal
Q}_{1}(L).$

This shows that the scintillation index corresponding to averaging of the
initial incoherent intensity is zero, while the one corresponding to the
transmitted field is not.

  * •

For $\tau$ smaller than both the coherence times of the source and the medium,
we have $f_{\tau}=g_{\tau}=1$ and then for $T$ similarly small

${\cal S}=\frac{\mu_{4}(L,{\bf 0},{\bf 0},{\bf 0},{\bf 0};\tau)-\mu_{2}(L,{\bf
0},{\bf 0};0)^{2}}{\mu_{2}(L,{\bf 0},{\bf 0};0)^{2}}=1+2{\cal Q}_{1}(L).$ (26)

Thus, initially the scintillation index is one (because the source has
Gaussian distribution), then it reaches beyond one in a mixing region and
returns to one for large propagation distances. Indeed, the fluctuations of
the initial field happen on a spatial scale that is large relative to the
scale of the field variations that are imposed by the random medium
fluctuations resulting in a non-Gaussian mixture situation with scintillation
index beyond one.

  * •

Consider the strongly scattering regime so that

$\alpha_{L}:=\frac{\sigma_{\rm m}^{2}k_{o}^{2}\ell_{\rm m}L}{2}\gg 1.$ (27)

Note that then the propagation distance is larger than the scattering mean
free path since $\alpha_{L}=4L/\ell_{\rm mfp}$, so that in the case of
coherent sources most of the wave energy has been transferred to incoherent
wave energy due to scattering. We will in the context of the strongly
scattering regime (27) assume that $C$ is smooth and isotropic so that we have
(remember $C({\bf 0})=1$):

$C(\boldsymbol{\chi})=1-c_{2}|\boldsymbol{\chi}|^{2}+o(|\boldsymbol{\chi}|^{2}).$
(28)

Then we we can compute a simplified expression for ${\cal Q}_{1}(L)$ and find

$\displaystyle{\cal Q}_{1}(L)\stackrel{{\scriptstyle\alpha_{L}\gg
1}}{{\simeq}}\frac{1}{1+\frac{c_{2}\sigma_{\rm m}^{2}L^{3}}{3\ell_{\rm
m}\ell_{\rm s}^{2}}},$

so that ${\cal Q}_{1}(L)$ goes to zero when the propagation distance $L$
becomes large.

In Figure 2 we illustrate the behavior of the scintillation in the regime
$\ell_{\rm s}\gg\ell_{\rm m}$. The figure shows how the scintillation index in
(26) depends on the propagation distance. We introduce the length parameter

$\displaystyle\ell=\frac{\ell_{\rm m}\ell_{\rm s}}{\lambda_{0}}.$

Then we show the scintillation index as function of $L/\ell$ for three
different values of the medium fluctuation strength parameter

$\displaystyle\alpha_{\ell}=\frac{\sigma_{\rm m}^{2}k_{o}^{2}\ell_{\rm
m}\ell}{2},$

when $C(\boldsymbol{\chi})=\exp(-|\boldsymbol{\chi}|^{2}/2)$. Note that with
stronger medium fluctuations the maximum value for the scintillation index is
larger and happens for shorter propagation distances.

Figure 2: Scintillation index as a function of the relative propagation
distance $L/\ell$ for three values of the medium fluctuation strength
parameter $\alpha_{\ell}$. The figure corresponds to the regime $\ell_{\rm
s}\gg\ell_{\rm m}$ for small averaging times at the detector so that the
scintillation index is given by (26).

###  4.2 Source with Intermediate Correlation Radius

We consider next the case when the correlation radius of the source is of the
same order as the correlation radius of the medium so that
$\frac{\lambda_{o}L}{\ell_{\rm m}^{2}}\gg 1$ and
$\frac{\lambda_{o}L}{\ell_{\rm m}\ell_{\rm s}}\gg 1$. We carry out the
analysis in Appendix B.2 where we derive the following expression for the
scintillation index (22):

$\displaystyle{\cal S}=$
$\displaystyle\frac{2}{T}\int_{0}^{T}\big{(}1-\frac{\tau}{T}\big{)}\Big{[}f_{\tau}^{2}\exp\Big{(}-\frac{(1-g_{\tau})\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm m}L}{2}\Big{)}\Big{]}d\tau.$ (29)

As above the term in the square brackets corresponds to scintillation
contribution from the fluctuations in the source and this contribution is
damped by fast temporal decorrelation of the medium fluctuations (small
$g_{\tau}$), moreover, by smoothing at the detector. Note that this term
corresponds to the first term in the square brackets in (23). Indeed, the last
two terms in the square brackets in (23) becomes small when $\ell_{\rm s}$ is
reduced so that $\frac{\lambda_{o}L}{\ell_{\rm m}\ell_{\rm s}}\gg 1$. The
situation with $\ell_{\rm s}\sim\ell_{\rm m}$ is the regime considered here.
If we further reduce $\ell_{\rm s}$ so that $\ell_{\rm s}\ll\ell_{\rm m}$ then
we may transition towards the regime considered in the next section. In the
regime $\ell_{\rm s}\sim\ell_{\rm m}$ we can observe the following behaviors.

  * •

In the case when $T$ becomes much larger than the coherence time of the
source, and assuming that $f_{\tau}\in L^{2}$, we have

${\cal S}\stackrel{{\scriptstyle T\to+\infty}}{{\longrightarrow}}0$

for any propagation distance due to averaging at the photodetector.

  * •

For $\tau$ smaller than the coherence times of both the source and the medium,
we have $f_{\tau}=g_{\tau}=1$ and then with $T$ similarly small

${\cal S}=1.$

Here the field behaves as a complex Gaussian field, because the fluctuations
in the source and in the medium happen at the same scale.

In Figures 3 and 4 we illustrate the behavior of the scintillation index in
the regime $\ell_{\rm s}\sim\ell_{\rm m}$. The figures show how the
scintillation index depends on the magnitudes of the coherence times of the
source and of the medium relative to the integration time at the detector. The
two figures correspond to two different values for the strength of the medium
fluctuations parameter $\alpha_{L}$. The time coherence functions $f_{\tau}$
and $g_{\tau}$ are chosen to be Gaussian

$f_{\tau}=\exp\left(-\frac{\tau^{2}}{2\tau_{\rm s}^{2}}\right),\quad
g_{\tau}=\exp\left(-\frac{\tau^{2}}{2\tau_{\rm m}^{2}}\right).$ (30)

Note how strong medium fluctuations serve to reduce the scintillation index in
the case with a time-dependent random medium and temporal averaging at the
detector, moreover, how also long duration detector temporal averaging serves
to reduce the scintillation index.

$0$$T/\tau_{\rm s}=20$$T/\tau_{\rm m}=80$

Figure 3: Scintillation index ${\cal S}$ as a function of the coherence times
of the source and of the random medium, respectively $\tau_{\rm s}$ and
$\tau_{\rm m}$, relative to $T$, the averaging time at the detector, in the
regime $\ell_{\rm s}\sim\ell_{\rm m}$. The effective strength parameter for
the magnitude of the medium fluctuations is $\alpha_{L}=\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm m}L/2=3$. The two solid black lines correspond to the
contour levels ${\cal S}=.8$ and $.2$ respectively. Note that here and below
we adapt the color scale to the particular distribution of scintillation index
values.

$0$$T/\tau_{\rm s}=20$$T/\tau_{\rm m}=80$

Figure 4: As in Figure 3, however with the medium fluctuation strength
parameter $\alpha_{L}=\sigma_{\rm m}^{2}k_{o}^{2}\ell_{\rm m}L/2=1/3$.

###  4.3 Source with Small Correlation Radius

We finally consider the regime in which the correlation radius of the source
is smaller than the correlation radius of the medium and we have
$\frac{\lambda_{o}L}{\ell_{\rm s}^{2}}\lesssim 1$. A similar regime (called
spot-dancing regime) has already been considered in the literature to study
coherent and narrow beam propagation: the beam propagates with the same
transverse profile as in a homogeneous medium but its center randomly wanders,
more exactly, its center is a random process whose standard deviation
increases with propagation distance [19, 16]. Here, we also assume that ${C}$
is smooth and isotropic with an expansion as in (28). We carry out the
analysis in Appendix B.3 where we derive the following expression for the
scintillation index (22):

$\displaystyle{\cal S}=$
$\displaystyle\frac{2}{T}\int_{0}^{T}\big{(}1-\frac{\tau}{T}\big{)}\bigg{[}\frac{f_{\tau}^{2}}{1+\frac{(1-g_{\tau})c_{2}\sigma_{\rm
m}^{2}L^{3}}{3\ell_{\rm m}\ell_{\rm s}^{2}}}\bigg{]}d\tau.$ (31)

The numerator within the square brackets corresponds to the scintillation
contribution of the Gaussian source, while the denominator corresponds to
damping of the scintillation index due to temporal decorrelation in the random
medium. We can moreover make the following observations.

  * •

When $T$ becomes much larger than the coherence time of the source, and
assuming that $f_{\tau}\in L^{2}$, then we have

${\cal S}\stackrel{{\scriptstyle T\to+\infty}}{{\longrightarrow}}0$

for any propagation distance. Note that the scintillation index is small even
if the medium is frozen since the medium fluctuations do not strongly
contribute to the intensity correlations with the very small source
correlation radius.

  * •

For $\tau$ smaller than the coherence times of the source and of the medium,
we have $f_{\tau}=g_{\tau}=1$ and it follows that with $T$ similarly small

${\cal S}=1.$

This corresponds to a Gaussian situation since the random medium fluctuations
again does not strongly affect the correlations in this case with a source
with rapid stationary spatial fluctuations. This is in contrast to the
situation with a deterministic beam source when the spot dancing property
produces a heavy-tailed intensity distribution and large scintillation index
(a non-central chi-square distribution with two degrees of freedom, also known
as the Rice-Nakagami distribution [19]).

In Figures 5 and 6 we illustrate the behavior of the scintillation index in
the regime $\ell_{\rm s}\ll\ell_{\rm m}$. The figure shows how the
scintillation index depends on the magnitudes of the coherence times of the
source and of the medium relative to the integration time at the detector. The
two figures correspond to two different values for the effective strength of
the medium fluctuations

$\displaystyle\frac{c_{2}\sigma_{\rm m}^{2}L^{3}}{3\ell_{\rm m}\ell_{\rm
s}^{2}}.$

and we again assume a Gaussian time coherence function $f_{\tau}$ as in (30).
Note that in this case the strength parameter does not depend on the central
wavelength $\lambda_{0}$. As above note how strong medium fluctuations serve
to reduce the scintillation index in the case with a time-dependent random
medium and temporal averaging at the detector, moreover, how again long
detector temporal averaging serves to reduce the scintillation index.

$0$$T/\tau_{\rm s}=20$$T/\tau_{\rm m}=80$

Figure 5: Scintillation index ${\cal S}$ as a function of the coherence times
of the source and of the random medium, respectively $\tau_{\rm s}$ and
$\tau_{\rm m}$, relative to $T$, the averaging time at the detector, in the
regime $\ell_{\rm s}\ll\ell_{\rm m}$. The effective strength parameter for the
magnitude of the medium fluctuations is $c_{2}\sigma_{\rm
m}^{2}L^{3}/(3\ell_{\rm m}\ell_{\rm s}^{2})=3$. The two solid black lines
correspond to the contour levels ${\cal S}=.8$ and $.2$ respectively.

$0$$T/\tau_{\rm s}=20$$T/\tau_{\rm m}=80$

Figure 6: As in Figure 5, however with medium fluctuation strength parameter
$c_{2}\sigma_{\rm m}^{2}L^{3}/(3\ell_{\rm m}\ell_{\rm s}^{2})=1/3$.

##  5 Example with Experimental Data

We discuss an example with real data taken from the paper [29] by Nelson et
al. The experiment in [29] involves an over-the-water laser beam link at the
United States Naval Academy. The source is partially coherent (Multi-Gaussian
Schell Model) and realized via a SLM. The measurement procedure at the CDD
camera corresponds to an averaging interval of $T=60$sec. The experiment is
carried out for various values for the source coherence time $\tau_{\rm s}$
realized via varying the SLM cycling rate. The field trials were conducted in
July and were performed during the night in calm weather conditions over a
maritime link of 323 meters. We refer to the paper [29] for a more detailed
description of the experimental setup. Assuming a frozen medium in view of the
calm weather we can then model the observed scintillation index as in (25)

${\cal
S}=\frac{2}{T}\int_{0}^{T}\big{(}1-\frac{\tau}{T}\big{)}\Big{[}f_{\tau}^{2}+(1+f_{\tau}^{2}){\cal
Q}_{1}(L)\Big{]}d\tau\stackrel{{\scriptstyle\tau_{\rm s}\ll
T}}{{\longrightarrow}}\frac{c_{1}}{\tau_{\rm s}^{-1}}+c_{2},$ (32)

and we can fit the parameters $c_{1},c_{2}$ via least squares. The results are
shown in Figure 7 and we can see an excellent fit in between model and data.

Figure 7: Measurements of scintillation index as function of the SLM cycling
rate (red stars). The observations conform well with the theoretical
predictions (dashed line) assuming a frozen medium.

##  6 Conclusions

We have considered the scintillation of wave field that is observed after
propagation through a time-dependent random medium. The source is partially
coherent in time and space and constitutes a random field in lateral space and
time variables. We consider a high-frequency and far-field regime. We give
here precise characterizations of the scaling regimes leading to the different
canonical forms of scintillation. The central scaling parameters are the
temporal and spatial statistical coherence lengths of the source and of the
random medium, in addition to the propagation range and the strength of the
random medium fluctuations, and the response time of the photodetector. In the
high-frequency and far-field regime, three scaling regimes are identified
depending on the magnitude of the spatial correlation radius of the source
relative to that of the medium. We identify general formulas for the
scintillation index in each regime and discuss special cases corresponding to
an effective Gaussian situation with scintillation index being equal to one, a
non-Gaussian mixture situation with scintillation index reaching beyond one,
and situations with small scintillation index corresponding to a desirable
high signal-to-noise ratio for the measured intensity. In particular temporal
averaging creates situations with a low scintillation index. In the context of
for instance communication, however, long averaging times are not in general
desirable and our analysis presents quantitative insights about appropriate
tradeoffs that can be made for optimal system performance. Such particular
system optimization challenges are left for future work. We also remark that
we have considered the case when the source has infinite lateral spatial
extent and is a stationary stochastic process in lateral space coordinates and
time. The case when the source is modulated by a finite source aperture and
the associated challenge of identifying the spreading of the wave field and
the evolution of speckle statistics can be analyzed via similar theoretical
frameworks as those presented here, but is also left for future work.

##  Appendix A Analysis of Fourth-order Moment Equations

The main equation underlying the above results is a simplified equation
deriving from (19) and from which the expressions of the special fourth
moments $\mu_{4}(L,{\bf 0},{\bf 0},{\bf 0},{\bf 0};\tau)$ follow. We deduce
this equation here and analyze it in the specific scintillation regimes in
Appendix B.

Consider the general moments in (18) satisfying (19) with initial condition
(21). It will be convenient to parameterize the four points ${{\itbf
x}}_{1},{{\itbf x}}_{2},{{\itbf y}}_{1},{{\itbf y}}_{2}$ in (18) in the
special way:

$\displaystyle{{\itbf x}}_{1}=\frac{{{\itbf r}}_{1}+{{\itbf r}}_{2}+{{\itbf
q}}_{1}+{{\itbf q}}_{2}}{2},\quad\quad{{\itbf y}}_{1}=\frac{{{\itbf
r}}_{1}+{{\itbf r}}_{2}-{{\itbf q}}_{1}-{{\itbf q}}_{2}}{2},$ (33)
$\displaystyle{{\itbf x}}_{2}=\frac{{{\itbf r}}_{1}-{{\itbf r}}_{2}+{{\itbf
q}}_{1}-{{\itbf q}}_{2}}{2},\quad\quad{{\itbf y}}_{2}=\frac{{{\itbf
r}}_{1}-{{\itbf r}}_{2}-{{\itbf q}}_{1}+{{\itbf q}}_{2}}{2}.$ (34)

In particular ${{\itbf r}}_{1}/2$ is the barycenter of the four points
${{\itbf x}}_{1},{{\itbf x}}_{2},{{\itbf y}}_{1},{{\itbf y}}_{2}$:

$\displaystyle{{\itbf r}}_{1}=\frac{{{\itbf x}}_{1}+{{\itbf x}}_{2}+{{\itbf
y}}_{1}+{{\itbf y}}_{2}}{2},\quad\quad{{\itbf q}}_{1}=\frac{{{\itbf
x}}_{1}+{{\itbf x}}_{2}-{{\itbf y}}_{1}-{{\itbf y}}_{2}}{2},$
$\displaystyle{{\itbf r}}_{2}=\frac{{{\itbf x}}_{1}-{{\itbf x}}_{2}+{{\itbf
y}}_{1}-{{\itbf y}}_{2}}{2},\quad\quad{{\itbf q}}_{2}=\frac{{{\itbf
x}}_{1}-{{\itbf x}}_{2}-{{\itbf y}}_{1}+{{\itbf y}}_{2}}{2}.$

We denote by $\mu$ the fourth-order moment in these new variables:

$\mu(z,{{\itbf q}}_{1},{{\itbf q}}_{2},{{\itbf r}}_{1},{{\itbf
r}}_{2};\tau)=\mu_{4}(z,{{\itbf x}}_{1},{{\itbf x}}_{2},{{\itbf
y}}_{1},{{\itbf y}}_{2};\tau)$ (35)

with ${{\itbf x}}_{1},{{\itbf x}}_{2},{{\itbf y}}_{1},{{\itbf y}}_{2}$ given
by (33-34) in terms of ${{\itbf q}}_{1},{{\itbf q}}_{2},{{\itbf
r}}_{1},{{\itbf r}}_{2}$.

In the variables $({{\itbf q}}_{1},{{\itbf q}}_{2},{{\itbf r}}_{1},{{\itbf
r}}_{2})$ the function ${\mu}$ satisfies the system:

$\frac{\partial{\mu}}{\partial z}=\frac{i}{k_{o}}\big{(}\nabla_{{{\itbf
r}}_{1}}\cdot\nabla_{{{\itbf q}}_{1}}+\nabla_{{{\itbf
r}}_{2}}\cdot\nabla_{{{\itbf q}}_{2}}\big{)}{\mu}+\frac{\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm m}}{4}{U}({{\itbf q}}_{1},{{\itbf q}}_{2},{{\itbf
r}}_{1},{{\itbf r}}_{2};\tau){\mu},$ (36)

with the generalized potential

$\displaystyle{U}({{\itbf q}}_{1},{{\itbf q}}_{2},{{\itbf r}}_{1},{{\itbf
r}}_{2};\tau)={C}\left(\frac{{{\itbf q}}_{2}+{{\itbf q}}_{1}}{\ell_{\rm
m}}\right)+{C}\left(\frac{{{\itbf q}}_{2}-{{\itbf q}}_{1}}{\ell_{\rm
m}}\right)+g_{\tau}{C}\left(\frac{{{\itbf r}}_{2}+{{\itbf q}}_{1}}{\ell_{\rm
m}}\right)$ $\displaystyle\hbox{}+g_{\tau}{C}\left(\frac{{{\itbf
r}}_{2}-{{\itbf q}}_{1}}{\ell_{\rm m}}\right)-g_{\tau}{C}\left(\frac{{{\itbf
q}}_{2}+{{\itbf r}}_{2}}{\ell_{\rm m}}\right)-g_{\tau}{C}\left(\frac{{{\itbf
q}}_{2}-{{\itbf r}}_{2}}{\ell_{\rm m}}\right)-2.$ (37)

The Fourier transform (in ${{\itbf q}}_{1}$, ${{\itbf q}}_{2}$, ${{\itbf
r}}_{1}$, and ${{\itbf r}}_{2}$) of the fourth-order moment is defined by:

$\displaystyle\hat{\mu}(z,{\boldsymbol{\xi}}_{1},{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{1},{\boldsymbol{\zeta}}_{2};\tau)=$
$\displaystyle\iint_{\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathbb{R}^{2}}{\mu}(z,{{\itbf
q}}_{1},{{\itbf q}}_{2},{{\itbf r}}_{1},{{\itbf r}}_{2};\tau)$
$\displaystyle\times\exp\big{(}-i{{\itbf
q}}_{1}\cdot{\boldsymbol{\xi}}_{1}-i{{\itbf
q}}_{2}\cdot{\boldsymbol{\xi}}_{2}-i{{\itbf
r}}_{1}\cdot{\boldsymbol{\zeta}}_{1}-i{{\itbf
r}}_{2}\cdot{\boldsymbol{\zeta}}_{2}\big{)}d{{\itbf q}}_{1}d{{\itbf
q}}_{2}d{{\itbf r}}_{1}d{{\itbf r}}_{2}.\hskip 21.68121pt$ (38)

It satisfies

$\displaystyle\frac{\partial\hat{\mu}}{\partial
z}+\frac{i}{k_{o}}\big{(}{\boldsymbol{\xi}}_{1}\cdot{\boldsymbol{\zeta}}_{1}+{\boldsymbol{\xi}}_{2}\cdot{\boldsymbol{\zeta}}_{2}\big{)}\hat{\mu}=\frac{\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm
m}^{3}}{4(2\pi)^{2}}\int_{\mathbb{R}^{2}}\hat{C}({{\itbf k}}\ell_{\rm
m})\bigg{[}\hat{\mu}({\boldsymbol{\xi}}_{1}-{{\itbf
k}},{\boldsymbol{\xi}}_{2}-{{\itbf
k}},{\boldsymbol{\zeta}}_{1},{\boldsymbol{\zeta}}_{2})$
$\displaystyle\quad+\hat{\mu}({\boldsymbol{\xi}}_{1}+{{\itbf
k}},{\boldsymbol{\xi}}_{2}-{{\itbf
k}},{\boldsymbol{\zeta}}_{1},{\boldsymbol{\zeta}}_{2})-2\hat{\mu}({\boldsymbol{\xi}}_{1},{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{1},{\boldsymbol{\zeta}}_{2})$
$\displaystyle\quad+g_{\tau}\hat{\mu}({\boldsymbol{\xi}}_{1}+{{\itbf
k}},{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{1},{\boldsymbol{\zeta}}_{2}-{{\itbf
k}})+g_{\tau}\hat{\mu}({\boldsymbol{\xi}}_{1}-{{\itbf
k}},{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{1},{\boldsymbol{\zeta}}_{2}-{{\itbf
k}})$ $\displaystyle\quad-
g_{\tau}\hat{\mu}({\boldsymbol{\xi}}_{1},{\boldsymbol{\xi}}_{2}-{{\itbf
k}},{\boldsymbol{\zeta}}_{1},{\boldsymbol{\zeta}}_{2}-{{\itbf
k}})-g_{\tau}\hat{\mu}({\boldsymbol{\xi}}_{1},{\boldsymbol{\xi}}_{2}+{{\itbf
k}},{\boldsymbol{\zeta}}_{1},{\boldsymbol{\zeta}}_{2}-{{\itbf
k}})\bigg{]}d{{\itbf k}},$ (39)

starting from

$\displaystyle\hat{\mu}(z=0,{\boldsymbol{\xi}}_{1},{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{1},{\boldsymbol{\zeta}}_{2};\tau)=$
$\displaystyle(2\pi)^{8}\phi_{\ell_{\rm
s}^{-1}}({\boldsymbol{\xi}}_{1})\phi_{\ell_{\rm
s}^{-1}}({\boldsymbol{\xi}}_{2})\delta({\boldsymbol{\zeta}}_{1})\delta({\boldsymbol{\zeta}}_{2})$
$\displaystyle+(2\pi)^{8}f_{\tau}^{2}\phi_{\ell_{\rm
s}^{-1}}({\boldsymbol{\xi}}_{1})\phi_{\ell_{\rm
s}^{-1}}({\boldsymbol{\zeta}}_{2})\delta({\boldsymbol{\zeta}}_{1})\delta({\boldsymbol{\xi}}_{2}).$
(40)

Here $\hat{C}({{\itbf
q}})=\int_{\mathbb{R}^{2}}C(\boldsymbol{\chi})\exp(i{{\itbf
q}}\cdot\boldsymbol{\chi})d\boldsymbol{\chi}$ is the Fourier transform of $C$,

$\phi_{\kappa}({\boldsymbol{\xi}})=\frac{1}{2\pi\kappa^{2}}\exp\Big{(}-\frac{|{\boldsymbol{\xi}}|^{2}}{2\kappa^{2}}\Big{)},$
(41)

is the $2$-dimensional centered isotropic Gaussian density with standard
deviation $\kappa$, $\delta$ is the Dirac delta distribution, and
$f_{\tau},g_{\tau}$ are defined in (17). We now seek to characterize

$\displaystyle\mu_{4}(L,{\bf 0},{\bf 0},{\bf 0},{\bf 0};\tau)=$
$\displaystyle\mu(L,{\bf 0},{\bf 0},{\bf 0},{\bf 0};\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{8}}\iint_{\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathbb{R}^{2}}\hat{\mu}(L,{\boldsymbol{\xi}}_{1},{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{1},{\boldsymbol{\zeta}}_{2};\tau)d{\boldsymbol{\xi}}_{1}d{\boldsymbol{\xi}}_{2}d{\boldsymbol{\zeta}}_{1}d{\boldsymbol{\zeta}}_{2}.$
(42)

Thanks to the special initial condition that is proportional to
$\delta({\boldsymbol{\zeta}}_{1})$, the solution $\hat{\mu}$ to (39) is itself
proportional to $\delta({\boldsymbol{\zeta}}_{1})$, and we can therefore
reduce the problem (39) to the analysis of

$\hat{\eta}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)=\frac{1}{(2\pi)^{4}}\iint_{\mathbb{R}^{2}\times\mathbb{R}^{2}}\hat{\mu}(z,{\boldsymbol{\xi}}_{1},{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{1},{\boldsymbol{\zeta}}_{2};\tau)d{\boldsymbol{\xi}}_{1}d{\boldsymbol{\zeta}}_{1}.$
(43)

The quantity of interest is then

$\mu_{4}(L,{\bf 0},{\bf 0},{\bf 0},{\bf
0};\tau)=\frac{1}{(2\pi)^{4}}\iint_{\mathbb{R}^{2}\times\mathbb{R}^{2}}\hat{\eta}(L,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)d{\boldsymbol{\xi}}_{2}d{\boldsymbol{\zeta}}_{2}.$
(44)

The function $\hat{\eta}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2})$
is solution of the characteristic system

$\displaystyle\frac{\partial\hat{\eta}}{\partial
z}+\frac{i}{k_{o}}{\boldsymbol{\xi}}_{2}\cdot{\boldsymbol{\zeta}}_{2}\hat{\eta}=\frac{\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm
m}^{3}}{4(2\pi)^{2}}\int_{\mathbb{R}^{2}}\hat{C}({{\itbf k}}\ell_{\rm
m})\bigg{[}-2\hat{\eta}({\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2})+2\hat{\eta}({\boldsymbol{\xi}}_{2}-{{\itbf
k}},{\boldsymbol{\zeta}}_{2})$
$\displaystyle+2g_{\tau}\hat{\eta}({\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2}-{{\itbf
k}})-g_{\tau}\hat{\eta}({\boldsymbol{\xi}}_{2}-{{\itbf
k}},{\boldsymbol{\zeta}}_{2}-{{\itbf
k}})-g_{\tau}\hat{\eta}({\boldsymbol{\xi}}_{2}+{{\itbf
k}},{\boldsymbol{\zeta}}_{2}-{{\itbf k}})\bigg{]}d{{\itbf k}},$ (45)

starting from

$\hat{\eta}(z=0,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)=(2\pi)^{4}\phi_{\ell_{\rm
s}^{-1}}({\boldsymbol{\xi}}_{2})\delta({\boldsymbol{\zeta}}_{2})+(2\pi)^{4}f_{\tau}^{2}\phi_{\ell_{\rm
s}^{-1}}({\boldsymbol{\zeta}}_{2})\delta({\boldsymbol{\xi}}_{2}).$ (46)

This simplified system (45-46) underlies the scintillation results presented
above. Note that the fourth moment problem has been reduced to a problem
defined relative to two, rather than four, copies of the lateral spatial
variables. We derive explicit solutions of this system in different scaling
regimes in the next Appendix B.

##  Appendix B Derivation of Scintillation Results

As mentioned in Section 2.2, the Itô-Schrödinger equation is valid in the
white-noise paraxial regime, when the wavelength is much smaller than the
correlation radii of the source and of the medium, which are themselves much
smaller than the propagation distance. By the Itô-Schrödinger equation, the
fourth-order moment (18) satisfies a closed equation (19). In this section, we
derive closed form expressions of the solution of Eq. (19) in three special
white-noise paraxial regimes, depending on the ratio of the correlation radii
of the source and of the medium.

###  B.1 Scintillation Regime with a Large Correlation Radius of the Source

We consider the white-noise paraxial regime in which, additionally, the
correlation radius of the source is larger than the correlation radius of the
medium $\ell_{\rm s}\gg\ell_{\rm m}$ and derive the results presented in
Section 4.1. More exactly, we here deal with the following scaled regime:

$\frac{\ell_{\rm m}}{\ell_{\rm
s}}\sim{\varepsilon}\,,\quad\quad\frac{L}{\ell_{\rm
s}}\sim\alpha^{-1}\,,\quad\quad\frac{\lambda_{o}}{\ell_{\rm
s}}\sim\alpha{\varepsilon}\,,\quad\quad\sigma_{\rm
m}^{2}\sim\alpha^{3}{\varepsilon}\,,$ (47)

and we assume $\alpha\ll{\varepsilon}\ll 1$ [Note that $L/\ell_{\rm mfp}\sim
1$ and $\lambda_{o}L/\ell_{\rm m}^{2}\sim{\varepsilon}^{-1}$]. This means that
the paraxial white-noise limit $\alpha\to 0$ is taken first (and we get an
${\varepsilon}$-dependent Itô-Schrödinger equation), and then we want to apply
the limit ${\varepsilon}\to 0$ in the fourth-moment equation (45). In view of
(47) it is natural to introduce the rescaled function

$\tilde{\eta}^{\varepsilon}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)=\hat{\eta}\Big{(}\frac{z}{{\varepsilon}},{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau\Big{)}\exp\Big{(}i\frac{z}{{\varepsilon}k_{o}}{\boldsymbol{\xi}}_{2}\cdot{\boldsymbol{\zeta}}_{2}\Big{)}.$
(48)

In the regime (47) the rescaled function $\tilde{\eta}^{\varepsilon}$
satisfies the equation with fast phases

$\frac{\partial\tilde{\eta}^{\varepsilon}}{\partial z}={\cal
L}^{\varepsilon}_{z}\tilde{\eta}^{\varepsilon},$ (49)

where

$\displaystyle{\cal
L}^{\varepsilon}_{z}\tilde{\eta}({\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2})=\frac{\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm
m}^{3}}{4(2\pi)^{2}}\int_{\mathbb{R}^{2}}\hat{C}({{\itbf k}}\ell_{\rm
m})\bigg{[}-2\tilde{\eta}({\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2})+2\tilde{\eta}({\boldsymbol{\xi}}_{2}-{{\itbf
k}},{\boldsymbol{\zeta}}_{2})e^{i\frac{z}{{\varepsilon}k_{o}}{{\itbf
k}}\cdot{\boldsymbol{\zeta}}_{2}}$
$\displaystyle\hbox{}+2g_{\tau}\tilde{\eta}({\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2}-{{\itbf
k}})e^{i\frac{z}{{\varepsilon}k_{o}}{{\itbf
k}}\cdot{\boldsymbol{\xi}}_{2}}-g_{\tau}\tilde{\eta}({\boldsymbol{\xi}}_{2}-{{\itbf
k}},{\boldsymbol{\zeta}}_{2}-{{\itbf
k}})e^{i\frac{z}{{\varepsilon}k_{o}}({{\itbf
k}}\cdot({\boldsymbol{\zeta}}_{2}+{\boldsymbol{\xi}}_{2})-|{{\itbf k}}|^{2})}$
$\displaystyle\hbox{}-g_{\tau}\tilde{\eta}({\boldsymbol{\xi}}_{2}-{{\itbf
k}},{\boldsymbol{\zeta}}_{2}+{{\itbf
k}})e^{i\frac{z}{{\varepsilon}k_{o}}({{\itbf
k}}\cdot({\boldsymbol{\zeta}}_{2}-{\boldsymbol{\xi}}_{2})+|{{\itbf
k}}|^{2})}\bigg{]}d{{\itbf k}},\hskip 14.45377pt$ (50)

and the initial condition is

$\tilde{\eta}^{\varepsilon}(z=0,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)=(2\pi)^{4}\phi_{{\varepsilon}/\ell_{\rm
s}}({\boldsymbol{\xi}}_{2})\delta({\boldsymbol{\zeta}}_{2})+(2\pi)^{4}f_{\tau}^{2}\phi_{{\varepsilon}/\ell_{\rm
s}}({\boldsymbol{\zeta}}_{2})\delta({\boldsymbol{\xi}}_{2}).$ (51)

Note that $\phi_{\kappa}$ belongs to $L^{1}$ and has a $L^{1}$-norm equal to
one. The asymptotic behavior as ${\varepsilon}\to 0$ of the moments is
therefore determined by the solutions of partial differential equations with
rapid phase terms. We can now proceed as in [20] and we obtain the following
proposition.

###### Proposition B.1

In the regime (47), the function
$\tilde{\eta}^{\varepsilon}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)$
has the form

$\displaystyle\tilde{\eta}^{\varepsilon}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)=K(z)\phi_{{\varepsilon}/\ell_{\rm
s}}({\boldsymbol{\xi}}_{2})\delta({\boldsymbol{\zeta}}_{2})+K(z)A_{1}(z,{\boldsymbol{\xi}}_{2},{\bf
0})\delta({\boldsymbol{\zeta}}_{2})$ (52)
$\displaystyle\hbox{}+K(z)A_{g_{\tau}}\big{(}z,{\boldsymbol{\zeta}}_{2},\frac{{\boldsymbol{\xi}}_{2}}{{\varepsilon}}\big{)}\phi_{{\varepsilon}/\ell_{\rm
s}}({\boldsymbol{\xi}}_{2})+f_{\tau}^{2}K(z)\delta({\boldsymbol{\xi}}_{2})\phi_{{\varepsilon}/\ell_{\rm
s}}({\boldsymbol{\zeta}}_{2})$
$\displaystyle\hbox{}+f_{\tau}^{2}K(z)A_{g_{\tau}}(z,{\boldsymbol{\zeta}}_{2},{\bf
0})\delta({\boldsymbol{\xi}}_{2})+f_{\tau}^{2}K(z)A_{1}\big{(}z,{\boldsymbol{\xi}}_{2},\frac{{\boldsymbol{\zeta}}_{2}}{{\varepsilon}}\big{)}\phi_{{\varepsilon}/\ell_{\rm
s}}({\boldsymbol{\zeta}}_{2})$
$\displaystyle\hbox{}+R^{\varepsilon}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau),$

where the functions $K$ and $A_{g}$ are defined by

$\displaystyle K(z)=$ $\displaystyle(2\pi)^{4}\exp\Big{(}-\frac{\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm m}z}{2}\Big{)},$ (53) $\displaystyle
A_{g}(z,{\boldsymbol{\xi}},{\boldsymbol{\zeta}})=$
$\displaystyle\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}\Big{[}\exp\Big{(}\frac{\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm m}g}{2}\int_{0}^{z}C\Big{(}\frac{{{\itbf
x}}}{\ell_{\rm m}}+\frac{{\boldsymbol{\zeta}}z^{\prime}}{k_{o}\ell_{\rm
m}}\big{)}dz^{\prime}\Big{)}-1\Big{]}$
$\displaystyle\times\exp\big{(}-i{\boldsymbol{\xi}}\cdot{{\itbf
x}}\big{)}d{{\itbf x}},$ (54)

and the function $R^{\varepsilon}$ satisfies
$\sup_{z\in[0,L]}\|R^{\varepsilon}(z,\cdot,\cdot;\tau)\|_{L^{1}(\mathbb{R}^{2}\times\mathbb{R}^{2})}\stackrel{{\scriptstyle{\varepsilon}\to
0}}{{\longrightarrow}}0$.

We remark that

$\frac{K^{1/4}(z)}{2\pi}=\exp\Big{(}-\frac{\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm m}z}{8}\Big{)}$ (55)

represents damping of the mean wave field due to scattering and transfer of
coherent energy to incoherent wave energy in the case with frozen medium and
deterministic sources. We remark moreover that the factor $A_{g}$ depends on
the two point statistics of the random medium at lateral offsets and captures
effects of lateral scattering of wave field energy. As a result, the quantity
of interest in (44) is then

$\mu_{4}^{\varepsilon}(L,{\bf 0},{\bf 0},{\bf 0},{\bf
0};\tau)=1+f_{\tau}^{2}\exp\Big{(}-\frac{\sigma_{\rm
m}^{2}(1-g_{\tau})k_{o}^{2}\ell_{\rm m}L}{2}\Big{)}+{\cal
Q}_{g_{\tau}}(L)+f_{\tau}^{2}{\cal Q}_{1}(L)$ (56)

with

$\displaystyle{\cal Q}_{g}(L)=$ $\displaystyle\exp\Big{(}-\frac{\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm m}L}{2}\Big{)}\int_{\mathbb{R}^{2}}\phi_{\ell_{\rm
s}^{-1}}({\boldsymbol{\zeta}})$
$\displaystyle\times\Big{[}\exp\Big{(}\frac{\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm
m}g}{2}\int_{0}^{L}C\big{(}\frac{{\boldsymbol{\zeta}}z}{k_{o}\ell_{\rm
m}}\big{)}dz\Big{)}-1\Big{]}d{\boldsymbol{\zeta}}.$ (57)

Therefore the relative covariance of the intensities at time zero and time
$\tau$ is

$\displaystyle\frac{\mu_{4}^{\varepsilon}(L,{\bf 0},{\bf 0},{\bf 0},{\bf
0};\tau)-\mu_{2}^{\varepsilon}(L,{\bf 0},{\bf
0};0)^{2}}{\mu_{2}^{\varepsilon}(L,{\bf 0},{\bf 0};0)^{2}}$ $\displaystyle=$
$\displaystyle f_{\tau}^{2}\exp\Big{(}-\frac{\sigma_{\rm
m}^{2}(1-g_{\tau})k_{o}^{2}\ell_{\rm m}L}{2}\Big{)}$
$\displaystyle\hbox{}+{\cal Q}_{g_{\tau}}(L)+f_{\tau}^{2}{\cal Q}_{1}(L).$

This gives the result (23) for the scintillation index in the regime
$\ell_{\rm s}\gg\ell_{\rm m}$.

###  B.2 Scintillation Regime with an Intermediate Correlation Radius of the
Source

We consider the white-noise paraxial regime in which, additionally, the
correlation radius of the source is of the same order as the correlation
radius of the medium $\ell_{\rm s}\sim\ell_{\rm m}$. This is the regime when
the source lateral spatial fluctuations takes place on the same scale of
variation as that of the random microstructure fluctuations, rather than being
large relative to this scale as in the previous Section B.1. More exactly, we
here deal with the following scaled regime:

$\frac{\ell_{\rm m}}{\ell_{\rm s}}\sim 1\,,\quad\quad\frac{L}{\ell_{\rm
s}}\sim\alpha^{-1}{\varepsilon}^{-1}\,,\quad\quad\frac{\lambda_{o}}{\ell_{\rm
s}}\sim\alpha\,,\quad\quad\sigma_{\rm m}^{2}\sim\alpha^{3}{\varepsilon}\,,$
(58)

and we assume $\alpha\ll{\varepsilon}\ll 1$ [Note that $L/\ell_{\rm mfp}\sim
1$ and $\lambda_{o}L/\ell_{\rm m}^{2}\sim{\varepsilon}^{-1}$]. This means that
the paraxial white-noise limit $\alpha\to 0$ is taken first, and then we want
to apply the limit ${\varepsilon}\to 0$ in the fourth-moment equation (45). As
above we introduce the rescaled function

$\tilde{\eta}^{\varepsilon}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)=\hat{\eta}\Big{(}\frac{z}{{\varepsilon}},{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau\Big{)}\exp\Big{(}i\frac{z}{k_{o}{\varepsilon}}{\boldsymbol{\xi}}_{2}\cdot{\boldsymbol{\zeta}}_{2}\Big{)}.$
(59)

In the regime (58) the rescaled function $\tilde{\eta}^{\varepsilon}$
satisfies again the equation with fast phases (49-50), here with initial
condition given by (46). The asymptotic behavior as ${\varepsilon}\to 0$ of
the moments is therefore determined by the solutions of partial differential
equations with rapid phase terms. We can again proceed similarly as in [20]
and we obtain the following proposition.

###### Proposition B.2

In the scintillation regime (58), the function
$\tilde{\eta}^{\varepsilon}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)$
has the form

$\tilde{\eta}^{\varepsilon}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)=(2\pi)^{4}B_{1}(z,{\boldsymbol{\xi}}_{2})\delta({\boldsymbol{\zeta}}_{2})+(2\pi)^{4}f_{\tau}^{2}B_{g_{\tau}}(z,{\boldsymbol{\zeta}}_{2})\delta({\boldsymbol{\xi}}_{2})+R^{\varepsilon}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau),$
(60)

with

$B_{g}(z,{\boldsymbol{\xi}})=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}\exp\Big{(}-i{\boldsymbol{\xi}}\cdot{{\itbf
x}}-\frac{|{{\itbf x}}|^{2}}{2\ell_{\rm s}^{2}}-\frac{\sigma_{\rm
m}^{2}k_{o}^{2}\ell_{\rm m}z}{2}\big{[}1-gC(\frac{{{\itbf x}}}{\ell_{\rm
m}})\big{]}\Big{)}d{{\itbf x}},$ (61)

and the function $R^{\varepsilon}$ satisfies
$\sup_{z\in[0,L]}\|R^{\varepsilon}(z,\cdot,\cdot;\tau)\|_{L^{1}(\mathbb{R}^{2}\times\mathbb{R}^{2})}\stackrel{{\scriptstyle{\varepsilon}\to
0}}{{\longrightarrow}}0$.

As a result, the quantity of interest (44) is

$\mu_{4}^{\varepsilon}(L,{\bf 0},{\bf 0},{\bf 0},{\bf
0};\tau)=1+f_{\tau}^{2}\exp\Big{(}-\frac{\sigma_{\rm
m}^{2}(1-g_{\tau})k_{o}^{2}\ell_{\rm m}L}{2}\Big{)}$ (62)

and

$\frac{\mu_{4}^{\varepsilon}(L,{\bf 0},{\bf 0},{\bf 0},{\bf
0};\tau)-\mu_{2}^{\varepsilon}(L,{\bf 0},{\bf
0};0)^{2}}{\mu_{2}^{\varepsilon}(L,{\bf 0},{\bf
0};0)^{2}}=f_{\tau}^{2}\exp\Big{(}-\frac{\sigma_{\rm
m}^{2}(1-g_{\tau})k_{o}^{2}\ell_{\rm m}L}{2}\Big{)}.$ (63)

This then gives the result (29) for the scintillation index.

###  B.3 Scintillation Regime with a Small Correlation Radius of the Source

We finally consider the white-noise paraxial regime in which, additionally,
the correlation radius of the source is smaller than the correlation radius of
the medium $\ell_{\rm s}\ll\ell_{\rm m}$ and derive the result (31) for the
scintillation index in this regime. More exactly, we here deal with the
following scaled regime:

$\frac{\ell_{\rm m}}{\ell_{\rm
s}}\sim{\varepsilon}^{-1}\,,\quad\quad\frac{L}{\ell_{\rm
s}}\sim\alpha^{-1}\,,\quad\quad\frac{\lambda_{o}}{\ell_{\rm
s}}\sim\alpha\,,\quad\quad\sigma_{\rm
m}^{2}\sim\alpha^{3}{\varepsilon}^{-1}\,,$ (64)

and we assume $\alpha\ll{\varepsilon}\ll 1$ [Note that $L/\ell_{\rm
mfp}\sim{\varepsilon}^{-2}$ and $\lambda_{o}L/\ell_{\rm s}^{2}\sim 1$]. This
means that the paraxial white-noise limit $\alpha\to 0$ is taken first, and
then we want to apply the limit ${\varepsilon}\to 0$ in the fourth-moment
equation (45). We also assume that $C$ is smooth and isotropic, so that we
have (28), and also $\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}\hat{C}({{\itbf
q}}){{\itbf q}}\otimes{{\itbf q}}d{{\itbf q}}=2c_{2}{\bf I}$. We denote by
$\hat{\eta}^{\varepsilon}$ the function (43) in the regime (64). Then we find
that in the regime of small ${\varepsilon}$ the function
$\hat{\eta}^{\varepsilon}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)$
is solution to the system

$\frac{\partial\hat{\eta}^{\varepsilon}}{\partial
z}+\frac{i}{k_{o}}{\boldsymbol{\xi}}_{2}\cdot{\boldsymbol{\zeta}}_{2}\hat{\eta}^{\varepsilon}=\frac{\sigma_{\rm
m}^{2}k_{o}^{2}c_{2}(1-g_{\tau})}{2\ell_{\rm
m}}\Delta_{{\boldsymbol{\xi}}_{2}}\hat{\eta}^{\varepsilon},$ (65)

with initial condition given by (46). We can easily solve (65) via a Fourier
transform and find

###### Proposition B.3

In the scintillation regime (64), the function
$\hat{\eta}^{\varepsilon}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)$
has the form

$\hat{\eta}^{\varepsilon}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)=(2\pi)^{4}G_{1}(z,{\boldsymbol{\xi}}_{2};\tau)\delta({\boldsymbol{\zeta}}_{2})+(2\pi)^{4}f_{\tau}^{2}G_{2}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)\phi_{1/\ell_{\rm
s}}({\boldsymbol{\zeta}}_{2}),$ (66)

with

$\displaystyle G_{1}(z,{\boldsymbol{\xi}}_{2};\tau)=$
$\displaystyle\frac{\ell_{\rm s}^{2}}{2\pi}\frac{1}{1+\frac{\sigma_{\rm
m}^{2}(1-g_{\tau})c_{2}k_{o}^{2}\ell_{\rm s}^{2}L}{\ell_{\rm
m}}}\exp\Big{(}-\frac{\ell_{\rm
s}^{2}|{\boldsymbol{\xi}}_{2}|^{2}}{2\big{(}1+\frac{\sigma_{\rm
m}^{2}(1-g_{\tau})c_{2}k_{o}^{2}\ell_{\rm s}^{2}L}{\ell_{\rm
m}}\big{)}}\Big{)},$ (67) $\displaystyle
G_{2}(z,{\boldsymbol{\xi}}_{2},{\boldsymbol{\zeta}}_{2};\tau)=$
$\displaystyle\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}\exp\Big{(}-\frac{c_{2}(1-g_{\tau})\sigma_{\rm
m}^{2}k_{o}^{2}}{2\ell_{\rm m}}\int_{0}^{L}\big{|}{{\itbf
x}}-\frac{{\boldsymbol{\zeta}}_{2}z}{k_{o}}\big{|}^{2}dz-i{\boldsymbol{\xi}}_{2}\cdot{{\itbf
x}}\Big{)}d{{\itbf x}}$ $\displaystyle=$ $\displaystyle\frac{\ell_{\rm
m}}{2\pi c_{2}(1-g_{\tau})\sigma_{\rm
m}^{2}k_{o}^{2}L}\exp\Big{(}-\frac{iL}{2k_{o}}{\boldsymbol{\xi}}_{2}\cdot{\boldsymbol{\zeta}}_{2}\Big{)}$
$\displaystyle\times\exp\Big{(}-\frac{c_{2}(1-g_{\tau})\sigma_{\rm
m}^{2}L^{3}}{24\ell_{\rm m}}|{\boldsymbol{\zeta}}_{2}|^{2}-\frac{\ell_{\rm
m}}{2c_{2}(1-g_{\tau})\sigma_{\rm
m}^{2}k_{o}^{2}L}|{\boldsymbol{\xi}}_{2}|^{2}\Big{)}.$ (68)

As a result

$\mu_{4}^{\varepsilon}(L,{\bf 0},{\bf 0},{\bf 0},{\bf
0};\tau)=1+\frac{f_{\tau}^{2}}{1+\frac{c_{2}(1-g_{\tau})\sigma_{\rm
m}^{2}L^{3}}{3\ell_{\rm m}\ell_{\rm s}^{2}}}$ (69)

and

$\frac{\mu_{4}^{\varepsilon}(L,{\bf 0},{\bf 0},{\bf 0},{\bf
0};\tau)-\mu_{2}^{\varepsilon}(L,{\bf 0},{\bf
0};0)^{2}}{\mu_{2}^{\varepsilon}(L,{\bf 0},{\bf
0};0)^{2}}=\frac{f_{\tau}^{2}}{1+\frac{c_{2}(1-g_{\tau})\sigma_{\rm
m}^{2}L^{3}}{3\ell_{\rm m}\ell_{\rm s}^{2}}}.$ (70)

This then gives (31) for the scintillation index when $\ell_{\rm
s}\ll\ell_{\rm m}$.

## Funding

JG was supported by the Agence Nationale pour la Recherche under Grant No.
ANR-19-CE46-0007 (project ICCI), and Air Force Office of Scientific Research
under grant FA9550-18-1-0217.
KS was supported by the Air Force Office of Scientific Research under grant
FA9550-18-1-0217, and the National Science Foundation under grant DMS-2010046.

## Acknowledgments

We thank C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth and R. Malek-
Madani for allowing us to use their data from [29] to generate Figure 7.

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