Download Multiply scattered waves through a spatially random medium

Multiply scattered waves through a spatially random
medium : entropy production and depolarization
Dominique Bicout, C. Brosseau
To cite this version:
Dominique Bicout, C. Brosseau. Multiply scattered waves through a spatially random medium
: entropy production and depolarization. Journal de Physique I, EDP Sciences, 1992, 2 (11),
pp.2047-2063. <10.1051/jp1:1992266>. <jpa-00246685>
HAL Id: jpa-00246685
https://hal.archives-ouvertes.fr/jpa-00246685
Submitted on 1 Jan 1992
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J.
Phys.
f
France
(1992)
2
2047-2063
1992,
NOVEMBER
2047
PAGE
Classification
Physics
Abstracts
05.701
42.20G
05.40
Multiply
scattered
entropy
production
Dominique
(')
(2)
(Received
par
February
J2
Dans
interaction
d'entropie
Joseph
avec
incident,
in
Magnet
Hochfeld
ddduisons
nous
87,
CNRS,
Labor,
B-P-166,
les
forrne
July
de
Par
la
France
J992)
phdnom~nes de
polarisation
pur de
ddsordonn£.
la
3J
Cedex,
Saint-Martin-d'Hkres
38402
final form
d'£tat
diffusant
milieu
un
B-P.
considbre
on
planes
maximum,
Fourier,
accepted
Etude,
cette
:
France
J992,
d'ondes
faisceau
medium
random
(2)
Brosseau
Festkorperforschung,
fur
09,
Universit6
R4sum4.
d'un
Cedex
Christian
and
Institut
Grenoble
CERMO,
and
(I)
Bicout
Planck
Max
38042
through a spatially
depolarization
waves
des
d6polarisation
d'6tat
et
arguments
matdce
de
de
et
de
d£coh6rence
arbitraire
de
sym£trie
et
Mueller
cohdrence,
principe
un
caract6risant
Ie
milieu
qui est en accord
le calcul
explicite base sur I'£quation de Bethe-Salpeter trait6e
avec
dans
l'approximation de la diffusion. Le r£sultat principal exprime les degr6s de polarisation et de
coherence
spatiale en
fonction
du
nombre
de
diffusions.
Deux
forts
saillants
h
Le
sont
noter.
exponentieIIe'de la production d'entropie due h l'irrdversibilitd
du
premier exprime la d6croissance
diffusions.
indique que la
de
ddpolarisation,
fonction
du
nombre
de
Le
second
processus
en
d6polarisation complkte d'un
incident
lin6airement
faisceau
polarisd
diffuseurs
de type
par des
Rayleigh ndcessite davantage de diffusions
(facteur 2) que pour une polarisation
circulaire.
diffusant
This
Abstract.
of
state
paper deals
and of
polarization
changes
randomly
deduce
the
with
the
arbitrary
position. Using
depolarization
state
of
decoherence
and
by
coherence
effects
linear
a
of
an
scattering
incident
medium
pure
which
entropy principle we
is
scattering
medium
which
consistent
with the explicit computation done in the
of the Bethe-Salpeter equation handled
context
diffusion
in the
approximation. The main result
the output degree of polarization
and
expresses
degree of spatial
function
of
coherence
of
the
number
scattering
From
these
results,
events.
as
a
main
conclusions
be drawn.
The first is that the entropy
production per scatteRng due to
two
can
the
irreversible
of depolarization is an exponentially
decreasing function of the number of
process
scattering
second
result
obtained
is that full
depolarization of linearly polarized light
The
events.
by Rayleigh
requires more scattering events (typically a factor-of-2) than are required for
scatterers
circularly polarized lightwave.
a
1.
general
with
form
of
symmetry
Mueller
the
matrix
arguments
and
describing
a
maximum
the
Introduction.
Studies
of
localization
phere [3],
In
this
scattering
interest,
research
[1-2]),
and
respect,
of
in
waves
connection
propagation
numerous
the
optical
others
subject
of
by
with
of
inhomogeneous
condensed
electromagnetic
media
matter
waves
currently attracting a wide
are
physics (e.g. weak
Anderson
under
the
sea
or
in
the
atmos-
(e.g, image
reconstruction
information
[4],
retrieval
[5]).
polarization
correlations
if of pRmary
importance. Multiple
fields
2048
JOURNAL
PHYSIQUE
DE
N°
I
I1
inhomogeneities in optically dense
randomizes
media
the
of
scattering of light from
state
effect
evidenced
(both
polarization,
however
the polarization
that
has
been
recently
memory
confirmed
experimentally [8] soon
afterwards)
theoretically [6-7] and
that
suggests
some
of
polarization
incident
irretrievably
information
about the
of
the
beam
is
lost by
state
not
characterize
depolarization of optical
multiple scattering. A point of further
interest is to
the
propagating
through
random
multiple
scattering
medium.
propagating in such
A
a
wave
waves
rapidly
depolarized
the
characteristic
medium
but
knows
little
about
spatial
becomes
one
a
proceeds. This question is important as regards the validation of the
scale on which this process
theory [2] to describe the transport properties of multiply scattered light. This
scalar
diffusion
of depolarization
relation
mechanism
and its
scattering
raises
also the question of the
to the
depolarization, it is useful to consider two main approaches.
understand
medium.
In trying to
absorption of polarization states is the first approach [9]. The second approach
The
selective
induces
depolarization by
decorrelation
of the phases and amplitudes
total flux and
preserves
field
[9]
it is essentially
of the
electric
entropic effect arising from the
components
an
of the polarization
evolution
during scattering. The present paper is mostly
irreversible
state
latter
approach and we will argue that depolarization of light by multiple
devoted
this
to
scattering is connected to a process of entropy production which falls off exponentially with
The key
observation
underlying the work reported here is that
of scattering
the
number
events.
this
result
derived
be
can
from
maximum
a
investigate
principle.
entropy
scattering
of
not inquire into
spatially random
the
electromagnetic
of pure
of
own
state
waves
polarization and arbitrary state of
from a
coherence
medium
fluctuates
which
in
space.
Example of such
medium
would
be a
collection
of randomly
oriented
scattering particles
suspended in a liquid. The purpose of the present work is to calculate the dependence of the
polarization and coherence
characteristics
(Stokes vector, degree of polarization and degree of
spatial coherence) upon the number of scattering events.
Besides, we will briefly consider an
application of this
formalism
technique of photon-correlation
known
to a
spectroscopy
as
diffusing wave
(DWS) [10] which is relevant to probe the nature of dynamic
spectroscopy
within
correlations
dense
dielectric
random
media (e,g,
colloidal
suspensions).
Our
emphasize
We
propagation
on
al. [I I ],
To
this
on
here
is
of
of the
different
of
:
us
the
defined
field
intensity
following
than
that
the
is
both
simply
Von
the
the
Neumann
radiation
spectral
entropy
energy
of
trace
is
been
has
backscattering
derived
approach
Such
an
Wolf
[12-13].
transport
and
coherency
the
analytically
entropy
of
treated
[I].
scatterers
coherence.
the
light arising
by Wolf et
polarized light from
of
spectrum
in the
which
largely by
initiated
been
characterize
to
in the
Rayleigh
theory of partial
has
change
medium
localization
distributed
[9]. Note
nature
the
weak
the
use
processes
field
entropy
polarization
we
permits
matrix
a
randomly
problem,
approach
electromagnetic
will
we
effects
the
stochastic
of
radiation
that
light through
containing
treat
theory
to
here
of
neither
medium
a
is
interest
with
related
which
from
we
Planck's
based
This
the
entropy
matrix
on
the
coherency
of
while
the
the
degree
to
the
shall
be
concemed
formula
[14].
of
organized as follows.
Section 2 is
devoted
formulation
paper
to the
of the
outlines
problem and
the
elementary
of
which
the
problems of
concepts in
terms
depolarization and
decoherence
will be approached throughout the paper.
Section 3 outlines
the principle of our
entropic approach. Then,
section 4 investigates
the
production
entropy
associated
with multiple scattering of light using the Bethe-Salpeter equation and deals
with
dependence of polarization
the
characteristics
(I.e. degrees of polarization and of spatial
light-intensity
autocorrelation
function) with the number of scattering
coherence,
Some
events.
Freund's
about
prediction [5] that the Stokes
of
incident
of
comments
vector
state
an
pure
polarization can be fully
after
reconstructed
total
depolarization by multiple scattering are
discussed
section 5. Finally,
in
several
concluding remarks
presented in section 6.
are
The
remainder
of the
is
DEPOLARIZATION
N°
II
2.
Preliminary
Let
us
for
the
OF
LIGHT
MULTIPLE
BY
2049
SCATTERING
considerations.
begin with a brief
development of
outline
assumptions
main
of the
background
and
required
information
theory.
the
Ej(r,
orthogonal
of the
t) Q
two
components
x, y) the
plane perpendicular to the direction of propagation (of
unit
ez), r is a position vector of a typical point in space and t is the time. Then, the
vector
electric
writes : E
E~ e~ + E~ e~, where e~ and e~ are orthogonal unit vectors
transverse
vector
triad
in said plane (e~ ez
(i, j
0, e~ e~
x, y)) with (e~ e~, ez) forming a right-handed
~~
quasimonochromatic
of real
field
is
of
Suppose next that this plane-wave
vectors.
narrow
homogeneous
that the field is statistically
spectral range
centered
around
assume
wo. We also
spatial
We
characterize
the
second-order
and
stationary, at least in the wide
sense.
may
coherence
properties of the field by its cross-spectral density tensor W (also termed the
(E, (rj E~*(r~)) at two
coherency matrix) [12] of components : Wij(rj, r~)
r~)
W~j(rj
points whose location is specified by position vectors rj and r~ the asterisk denoting complex
conjugation.
the
taken
the
ensemble
that
The
angular
brackets
denote
average
over
and
characterizes
the
statistical
properties of the incident field. By definition W is Hermitian
recall
basic
polarization
properties
of
the
non-negative
definite.
For later
purpose,
we
some
light
is
given
by
field.
polarization
P
of
The degree of
the
:
2.I
THE
INCIDENT
electric
Let
FIELD.
of the
vector
incident
in
wave
=
a
=
=
=
=
=
=
P
as
function
a
from
of the
rotational
two
(S~)
with:
W
=
of W.
«~ is
two-level
The
and
wave
write
to
=
choose
we
the
of
axes
(lExl~+ lEyl~),
(Sol
(Sil
(S~)
=
four
These
discussed
parameters
form
length
references
at
interpretation
in
in
Z (Sk)
+
of
terms
similarities
and
coordinates
rather
spatial.
than
under
Note
3
Stokes
parameters
indicating
tion
of the
the
is
degree
Stokes
:
to
;
£
P
which
parameters
of
Barakat
that
an
(s~)
[16].
Stokes
the
altemative
then
is
the
It
also
Stokes
in
bear
should
E~* j
-
interesting
with
of P in
terms
E~
E~)
parameters
mind
be
parameters
formulation
(E~ Efl
are
physical
emphasized that
their
=
x,
y
whereas
comments
respect
of the
on
to
the
time
average
1/2
2
wavefield
given by
I
vector.
to
operation E~
[16] made
It
the
is
e~,
=
useful
intensities
~~0)
=
and
(S~)
Stokes
I
x
it is
conjugation
the
considers
one
k=1~
e~
,
combinations
when
along
E~ El )
+
of the 4
transformation.
differences
(E~ El
however
invariant
are
(2)
"k
system
components
linear
(So), (Sj), (S~), are
(53) changes sign by this
which
i
[12, 15]
the
matrices
polarized
(lExl~- lEyl~),
=
=
obtained
are
Pauli
~
"o
coordinate
Cartesian
our
the
:
~=
If
parameters
for
decomposition [9] of any partially
completely polarized
wave
a
j
j (Sol
W
(1)
'
Stokes
average
usual
notation
the
where
(«o being the 2 x 2 unit matrix). The
light into a completely
unpolarized
independent of each other allows us
1'2
(w))2
(tr
invariants
(W«~),
tr
(W )
dot
4
(1
=
ordered.
so-called
viewed
be
may
A
as
well-known
Poincar£
sphere
an
order
parameter
geometric
representa3( [9, 12] of radius
JOURNAL
2050
SO(3)
(I.e.
P
~~~~.
(So)
similarity
to
possesses
two
the
inside
the
the
and
Further,
of
one
:
=
the
case
This
derived
and
I, s(x)
OS
that
with
here
other
(52)
(q~(
( (So)
<P
extreme
w
+
~~~
(Si) ) ~~~
~
The
I.
~~~
,
extreme
I
q~
case
usually
is
W
a
=
when
case
spatial
represents
taken
be
to
Von
the
simple analytical expression [9]
q~
0
=
coherence
Neumann
when
tr
(W)
measure
I
which
is
=
:
(s(P )),
In
=
with
(x)
s
=
(I
i
x)~
+
x)~
(l
-x
(4)
decomposition and was
above-mentioned
two-level
decomposition theorem [9]. For x varying between 0
1/2 and I, s(x) is a bijective strictly increasing
values
between
function : its
is noted
Two
remarks
importance.
of
First,
the
in (4)
s~ '.
entropy
are
takes
exists
and
depends only
both
on
linear
(P
is
expression is
reminiscent
the
to
from equation (2) via the spectral
inverse
S
polarization
of
:
1+x
S(P
axes
the convexity
[9] bears
of
property
states
on
some
analysis of spin1/2 [17]. The
of
set
states
convex
2 corresponding to mixed
(partially polarized) located
states
(totally polarized) on the
rank I corresponding to pure
states
of the
coherence
the complex degree of spatial
components
the
It takes
considered
be
to
of
;
degree
the
with
space
components.
(W)).
In
where
~~/~~o~~i/2 (w~~(o))~'~
associated
(W
polarization
dimensional
three
ii
the
ensures
field
in
W~(r)
incoherence
entropy
tr
rank
=
(3)
in
parameter)
N°
I
based
in
expression
the
normalization
spatial
represents
between
the two
The
of
other
qxy(r)
S(W)
states
introduce
we
E~ and E~ [12] by
the
order
description
strata
El
ball
the
encountered
that
3(.
surface
of
set
geometrical
This
P.
is
this
of
symmetry
PHYSIQUE
DE
I
#
)
and
P
not
S
S
S
(P
S
the
on
detailed
state
polarizations.
circular
and
of
polarization
Second,
for
:
instance,
satisfies
it
it is the
the
same
for
inequalities:
0 ).
=
We
will
medium
is linear,
that
the
scattering
assume
occupying a finite volume il (of arbitrary shape and orientation)
fluctuations
in free
and that the spatial
dielectric
susceptibility q~~ (r) tensor are
of its
space,
statistically homogeneous and stationary (at least in the wide sense). We restrict our
treatment
considering a non-absorbing
medium.
The
effect due to a weak
absorption can be easily
to
formalism
introduced
into our
and will be
considered
in a subsequent work. Typical
realization
of
of such
medium
would be a
collection
pointlike scattering
discrete
sizes is
whose
centers
small
compared to the wavelength. We also
the
time
fluctuations
of
that
the
scatterers
assume
sufficiently slow relative to the period of the field
oscillations
that the scattering
medium
are
so
essentially
time-invariant
approximation). The usual boundary
behaves
(I,e, adiabatic
as it is
conditions
require continuity of the magnetic field H and tangential electric field at every
discontinuity surface.
From
assumptions,
characterize
dielectric
susceptibility of the
the
above
the
we
may
(r)
being
medium by q~~ (r)
the
Kronecker
symbol),
of
white-noise
and
zero
mean
q
(&~~
2.2
THE
SCATTERING
isotropic,
and
MEDIUM.
non-magnetic
&~~
=
correlation
function
:
(R (ri)
R
(r2))
=
u
3
(ri
r2)
0
when
rj
,
where
u
is
a
constant
and
3
(r)
the
Dirac
delta
function.
e
i2, r~
e
i2
~~~
otherwise
N°
DEPOLARIZATION
II
LIGHT
OF
assumption is made regarding the thermodynamic
disorder (far from the
Anderson
consider only weak
No
will
~
fm
path
free
(
is
larger
much
than
of
state
the
transition)
wavelength
the
2051
SCATTERING
MULTIPLE
BY
of
scattering
medium.
that
elastic
such
the
mean
kof
(I,e,
radiation
the
We
ml),
uko
ko
w
=
light
o/c is
in
vacuo
diffusion
number
wave
vacuum
:
process
of the
fluctuations
2.3
the
consequently,
[2]. Finally,
SYMMETRY
are
RELATIONSHIPS
frequency
the
c
being
the
speed
that
described
by a
may be
medium
fluctuations
of the
the
statistically
independent.
SCATTERING
OF
FOR
wo,
propagation
assume
we
field
incident
with
associated
wavefield
the
by
related
matrix
=
linear
a
the
and
input
The
LIGHT.
POLARIzED
medium S~ are
vector S~ and
output Stokes
vector
to the scattering
the
form : S~
MS,, where M is the 4 x 4 Mueller
real-valued
of
classical
Stokes
of
relation
characterizing
the
scattering. A number of
restrictions
the form of the
M-matrix
placed at the outset
are
on
depending upon the symmetry and reciprocity requirements. Perrin [19] was the first to give a
detailed study of the
of independent
number
of
matrix
elements
parameters (among the sixteen
M) which are
for
specifying
the
polarization
of
characteristics
light
scattered
by
necessary
an
arbitrary medium.
Following the analysis made by Perrin, the forward (or backward) axial
scattering by a symmetrical
medium
(e,g,
identical
particles having spherical symmetry)
involves
coefficients
only three
Mueller
matrix is diagonal. We note in passing
the
moreover
;
further
for a variety of symmetry
that
refinements
properties of the matrices M describing the
of polarized
radiation by a slab of randomly
reflection
and
transmission
oriented
particles have
Hovenier
[21]. In a statistically isotropic threebeen
treated by Van de Hulst [20] and later by
medium,
dimensional
dimensions,
in
3.
In
section,
the
symmetry
Consequently
proved by Freund
arguments
the
elements
the
presented in
general form
of
is
In
two-
not
diagonal
the
multiple
[22].
of
production
per
AS (n
S
=
section
and
the
/
'~~
scattering
proceed
0
0
reads
+
matrix
M
as
fact
that
as
view
In
multiple
written
be
can
follows.
of
scattering
:
~~
0
(6)
/~
0
(P (n
describing
matrix
We
Mueller
the
l
Mueller
entropy.
previous
the
0
entropy
the
maximum
of
argument
an
M=
The
direction.
matrix
Mueller
the
~.
as
propagation
the
around
symmetry
C~
to
diagonal
evaluate
by
dissipative,
non
reduced
principle.
we
medium
C~~
full
is
is
block
but
entropy
Maximum
this
there
symmetry
dimensions
two
scattering
is
the
as
:
i(n)
S(P (n))
))
I
(7)
=
,
degree of polarization after n + I scattering events, S(P ) being given
particular state of
indicates
that ( depends
the
by equation (4) and the superscript
on
production after n + 2 scatterings is given by :
polarization.
Then
the total
entropy
where
P (n )
denotes
the
'
S (P (n
AS
=
+
I
))
jj
S(P (0))
IQ
=
=
)~~ ~~)~~
In
~
j
The
this
function
is to
( (x)
expected
is
taken
from
to
the
be
a
theory
monotonically
of
irreversible
l~
(8)
+
=o
decreasing function from ( (0
thermodynamics [18]. It can
down
be
to
shown
zero
that
;
candidate
the
("(x)~0
where
postulate
((x)
(P (n
exp(-
~k
=
+
)
xx)
=
~~~~~
expression
equivalent
An
~~~
get
we
~ ~°
It
x.
work
which
the
is
necessary
to
function
the
out
polarized
linearly
is
11
condition
I-i)tf)~
~
~~~
to
to
requirements.
of
polarization
state
pure
equation (8),
respect
chosen
satisfy
and
L
space
with
have
we
above
the
meets
From
differentiation
N°
I
metric
the
to
((x):
of
incident
an
mjj (n)).
indicates
form
which
consider
we
I
prime
the
belong
also
must
functional
the
Next,
h(x)
function
PHYSIQUE
DE
JOURNAL
2052
~~~
is
n-1
mu(n)
where
B
have
we
I
=
i~~~
(lo)
,
=
~k) and
be
can
fully
B is
exp(-
A
B
that
the
to
s-
=
convenience
notational
for
set
important
fact
S(P (I ))/In (2) : consequently
attention
AJ
by
determined
exp(-x).
B
=
written
We
call
equation(7)
from
double
scattering.
require
that A
as
Moreover
i
when
maximum
entropy
(2) (1
exp(-
In
~k
=
(I,e, in the
achieved
is
x)).
The
final
result
mj j (n
This
equation
parameter
B.
It
allows
is
successive
the
worth
noting
circularly
method
us
what
((x)
polarized (I,e. P (n + I )
is quite general and may be
functions ( (x)
kind of trial
that
satisfies
physical
the
Light depolarization
approach
4.
=
for
used
are
be
to
we
~
=
~
i.e,
;
iteration
with
a
to
will
change
involved
more
used.
(l
~)
formula
same
m~~(n))
tx~),
-
(2~~~
of
orders
n
:
=
This
expressed
apply for a
be
in
the
value
can
of the
terms
pure
of B.
matrix,
Mueller
simulation
in
state
which
Note
that
but it does
incorporate
any
not
I
sole
is
this
tell
function
constraints.
by
decoherence
and
is
s~
the
that
limit
a
scattering
medium
:
the
Bethe.Salpeter
equation
calculation
of the
dependence of the degrees of
problem now
shifts
the explicit
to
polarization and of spatial coherence in function of the number of scattering events when the
fulfills
described
in
section 2 and the
medium
the
incident
wavefield
plane
is of the type
evaluate
exactly
the
Mueller
matrix
elements
Indeed,
assumptions stated
above.
can
one
approximation (see Appendix A).
handled in the ladder
mii (n) by the Bethe-Salpeter equation
entropic
approach developed in the above
justify
of
the
This will
the
to
use
serve
our
purpose
distances
derivation is valid
section. We emphasize that this
greater than the elastic
over
mean
by Stephen and Cwilich [23], the problem of evaluating the
free path. As
demonstrated
coherency matrix W reduces to a matrix eigenvalue problem. We leave the details of formulae
in the
Appendices A and B.
The
4, I
DEPOLARIZATION
(labeled I) pure
normally
on
the
TIME-INVARIANT
BY A
SCATTERING
MEDIUM.
We
consider
polarization and arbitrary degree of spatial coherence,
half-space z ~ 0 its Stokes
writes
follows :
vector
as
state
of
(So )
~
'
(Si I
(S~)
(S~)
=
E~ ~)
=
Ex 1~)
=
=
+
E~
~
Ey 1~)
(Efl E~ + E~ Ef)
I (Efl E~
E~ Ef)
an
of unit
incident
intensity
l
=
~~
DEPOLARIZATION
II
N°
limit
In the
by
ensemble
derivation
of
techniques
expression
scattering limit,
covariance
averaged
weak
of
the
the
correlation
has
been
for
the
the
discussed
(labeled o)
output
of the scattering
medium is
determined
Bethe-Salpeter equation (Eq. (Al )). The
Green's
functions
of the
field by
time-independent
[23, 30, 41-42]. Following this method, we obtain the
linear
Stokes
(S21 (
°
(S31 (
where
the
ignored,
with
respect
normalize
we
write
to
the
denote
G~~'s
components
S~ in the
to
vector
(tit
(Gt
Gxxl
:
Cartesian
respect
(lGxyl~)
(tit G~yl
+
(Gi
G~X
~~~~
)
'
G~yl )
G~ ~).
+
and
the
suffices
x, y
label
has
been
absorption
equations (A9)
Since
chosen.
system
~)
)
(Eq. (A6))
function
coordinate
to
(lG~yl~)
+
Gxxl
Green's
retarded
the
S~ with
form
response
the
(lGxxl~)
(Sil ((lGxxl~)
~
2053
SCATTERING
MULTIPLE
BY
satisfying
function
largely
LIGHT
OF
Then,
allow
us
:
~l~~~
~)~
~°
~~~~
'
(53)
~~~~~~~~
f(n)
with
=
2
~~~~~~~
g(n)=
and
(7/10)~
+
expressions
g(n)
2
n
(7/10)~
+
+
being
I
number
the
of
scattering
derived by Freund [6] (his Eq. (I I)),
albeit
equations (12, 14) that the Mueller matrix of
the scattering
medium is of the form given by equation (6). The fact that mjj
m~~ is further
substantiated
by the property of rotational
invariance
of the degree of polarization.
Having specified S~, it follows from equation (14) that the output degree of polarization is
given by :
These
events.
in
a
different
fashion.
very
similar
readily
verified
are
It is
those
to
from
=
~o
which
degree
involves
of
three
spatial
~
~~l) ~
f(~)
independent
coherence
~~~~~~~
5
~~3) ~
+
2n
f
Combining
parameters.
writes
equations (3)
~~3)
+1
(S~)
+
I
j~
l
(53)
=
For
f~~
0
0
it is
concreteness,
instance,
degree
,
of
an
worthwhile
input
polarization
specialize
polarization
formulae
to
linear
P~
and
(14),
the
output
=
f(n)
which
is
state
a
(Sj)~
l/2
~~~~
(Sj)~ ~f(n))~
l
they stand, equations (15-16) show that the output degree of
will, in general,
differ
from
input degree of
coherence)
the
coherence)
because
of the
effect of the scattering
medium.
These
the
fact
single scattering (I,e. n
0) by particles
with
that
the
and
degree of polarization [19-21].
state
preserves
For
(~~)
'
As
interest.
I
1/2
:
~~2)
~~~
~~2) ~
~
(E
=
polarization (I,e,
polarization (I.e.
equations are in
having spherical
of
of
accordance
symmetry
(15-16) to some special
e~) has for output Stokes
monotonically
decreasing
spatial
spatial
function
cases
of
vector
of the
JOURNAL
2054
number
of
scattering (see Fig. I)
and
PHYSIQUE
DE
degree of spatial
a
I
N°
equal
coherence
to
Similarly,
zero.
ii
for
I
input
an
right-handed
P~
g(n)
meanings.
=
Numerical
thjt
see
we
(q(~(
(Mf~(
and
=
~~~
polarization
circular
/
=
ie~)
(e~
gets
one
°
:
°
,
g
P~.
calculations
jjnctions
these
E
state
Then,
have
not
f and g have clear physical
Upon comparing f and g in figure I,
f(n)~g(n)).
decay (I,e.
The
parameter
functions
the
two
in
figure
shown
are
the
same
,
(n)
I.
~~~~
~
which
polarization has been
represents the ratio of the linear to circular
~~3)
the exponential
indicates
that as the degree of polarization
also displayed in figure I
increase
decreases, the polarization ellipse flattens
towards
This fact is also expressed
the major axis.
by saying that depolarization of linearly polarized light requires more scattering events than are
required for a circularly polarized light. Bourret [24] made a similar
in his study of
comment
propagation of light in a medium with a stochastic
refractive
index. In this insightful paper,
he
found by using a
random
perturbation
depolarization
characteristic
that the
length
treatment
associated
with multiple
scattering is for linearly polarized light exactly twice the value for
circularly polarized light. From the numerical expressions of the functions f and g we find this
v
=
ratio
by
found
For
the
be in
to
n
~
the
In
~~~~
(0.7)
of
depolarization
~~
range
(n
l.94
=
-
), 2, lo (low
tx~
n)j,
I,e,
very
close
to
the
value
Bourret.
l,
Poincar6
the
process
3(
sphere
~~ ~~
but
induces
be
cannot
symmetry
a
assimilated
breaking
to
(I,e.
an
the
isotropic
contraction
of
symmetry
SO(3)
of
is
i
loco
0. 8
j
~
j~~
V
(0)
100
~
0. 6
i
°.4
~,
'
o
4
I
>6
i~
~
'
0.2
'
,
'
0
'
4
0
8
16
12
n
Fig. I.- Degree of polarization
(15)) for an input pure state
n (Eq.
(dashed
line).
The
inset
depicts
the
of
of
scattered
linear
of
scatterings
function
of the
number
light as a
polarization (solid line), right circular polarization
"~~~
parallel
dependence
of
the
normalized
parameter
v
(0)
on
n.
N°
DEPOLARIZATION
Ii
LIGHT
OF
2055
SCATTERING
MULTIPLE
BY
significance of these results, we have also plotted (Fig. 2) the
variation
of polarization using equation (4). The
of the
states
production AS(n)
S(P(n + I))
S(P (n)) with the
number
of scattering
events
entropy
with ~k and
(Fig. 3) is well represented by an exponential decay AS(n)
~kexp(-xn),
consistent
with
depending on the particular state of polarization. This form is
results
x
obtained
from
equation (11). For example, mjj (resp. m~~) given by equation (11) was
numerically tested assuming that the exact value is given by f(n) (resp, g(n)). From such
calculations,
find that the fitting parameter B was equal to within 4 fb of the
value.
exact
we
This
irreversible
of
behavior
of the entropy
production already expresses in a way the
nature
radiation
of n (say,
lo), the entropy of
is
the
considered.
For large
values
process
nm
defines
of
further
the
maximum
unaffected
by
scattering,
it
steady
state
entropy
(S(P
0)
attainable
standard
of
In (2)),
by multiple scattering. This is of course
the
sort
transition
from
physics
about this
from
statistical
[14, 25]. Relatively litle is known
argument
broken).
entropy
discuss
To
associated
the
physical
with
different
=
=
=
=
~j~~
o.~
'
/
0.6
'
/
/
/
0.4
i
0.2
0
0
6
4
2
8
lo
n
Fig.
2.
Sarne
Entropy (Eqs. (4, 15))
symbols as in figure I.
plotted
radiation
of
as
function
a
of
the
number
of
scattering
events,
Asin)
10-~
,
,
,
,
,
10-~
'
,
,
,
,
,
,
10-~
'
,
,
,
10~?
0
6
4
2
8
lo
n
Fig.
3.
events.
Entropy production AS (n )
Same
symbols as in figure1.
=
S (P (n
+
I
))
S(P (n ))
as
a
function
of the
number
of
scattering
JOURNAL
2056
PHYSIQUE
DE
N°
I
ii
AS
considered
by several
0, but some problems of this type have been briefly
[26-27] (e,g, loading of a spring under the action of a gravitational force, charging of a
capacitor, compression of a perfect gas). They found that the irreversibility
involved
in such
physical processes in transforming the system from an initial state to a terminal
bears an
state
relationship to the number of discrete steps in which it is carried out [26, 27].
inverse
AS
0
~
and
=
authors
4.2
POLARIZATION
DEPENDENCE
autocorrelation
of the
OF
TIME
THE
temporal
of the
AUTOCORRELATION
in the
motion
of the scattering
termed
is
centres
technique is described in details in [lo, 29] and was
devised
to
light scattering to the multiple scattering regime for probing the
of dense
scattering media. Light scattering experiments
measure
the
correlation
field
as
defined
~~~~~ ~~ ~°~~
=
m
I +
correlation
intensity.
The
evidenced
was
gl'l (t
form
of
scattering
the
for
which
[30],
al.
Treating
time.
the
the
of
times,
short
the
of
memory
field
theorem
field
the
auto-
polarization
leads
to
correlation
of
function
is
state
of
incident
their
[30].
decay
a
:
of the
transport
of
state
path
diffusion
each
the
factorization
diffusive
the
of
functions
measurement
very
have
lost
waves
delay
a
Siegert
the
For
events.
the
the
describing
depends on
functions
auto-correlation
et
related
function
Green's
MacKintosh
depends on
by long paths
[10]
Wolf
is
auto-correlation
through
related
t
multiply-scattered light by
Spectroscopy (DWS). This
extend
the usual
quasielastic
dynamic
structural
properties
the
intensity autoscattered
of
,
the
process,
are
and
the
to
these
by
which
dominated
Maret
gl'~ (t)
function
As
intensity
where
~~§~~
(f(0))
=
Gaussian
and
(f(0))
[gl~~ (t)[~.
~~~~~
gl~l (t)
as
complex
zero-mean
a
gl~~ (t)
gl~~ (t)
function
Measurement
FUNCTION.
intensity
Diffusing Wave
fluctuations
polarization.
In the following, we
derive an expression of the degree of polarization for multiply
scattered
light from a half-space. For the incident pure state of polarization defined by equation (6), the
writes
(for t*
l) :
output degree of polarization
~
P~(t*)
((Sj)~P((t*)
P(t*)
(S~)~P((t*)
+
=
(S~)~P((t*))~~~,
+
(17)
~'~
,
/
0.4
'
/
/
/
/
/
/
0.2
1'
/
/
/
/
/
0
0.2
0
0.4
0.6
0.8
@
Fig.
4.
Polarization
dimensionless
(dashed line).
time
t*.
dependence
of
The
are
curves
the
linear
equation (17)
functions
parallel polarization (solid line), right
autocorrelation
as
a
function
circular
of
the
polarization
N°
DEPOLARIZATION
II
LIGHT
OF
2057
SCATTERING
MULTIPLE
BY
in the
respectively the
where
the P~(tl's Q
defined
Appendix B and denote
1, 2, 3) are
polarization at 45° and
circular
degree of polarization
for
linear
polarization,
linear
polarization,
while t* is the
dimensionless
time (t/r~ with r~ being the time required by a
optical wavelength).
to
scatterer
one
move
depolarized at very
times
As
displayed in figure4,
the
light is completely
short
*
diffusion
of
contributions
(t « I corresponds to long
paths). Because
different
due to long
the
paths, the P~ (t*) versus (t *)~'~ differ in slope according to the input state of polarization. As
before, we find that complete depolarization of circularly polarized incident light requires less
(short
scattering events than for the linearly polarized case. This
with longer times
contrasts
diffusion
paths).
=
5.
Comment
To
close
the
Stokes
vector
useful
in
[14]. As
investigate
to
Recently
[14]
Gudkov
method.
implications of the recently proposed method [5] of
some
complete
depolarization, to the question of irreversibility.
a
about
irreversibility in
Polarization
Optics which we found
comment
after
references
indicated
are
first
reconstruction
paper,
we
now
reconstruction
representative
Some
the
Freund's
on
historical
an
off
trade
the
been
has
using
able
should
it
note,
between
Planck's
mentioned
be
that
polarization
light
spectral
entropy,
law
calculate
to
[14]
Jones
was
irreversibility.
and
the
of
flux
total
spectral entropy of an unpolarized light across the surface of a scattering object. A similar
but generalized to partially polarized light,
treatment,
put forward by Callies [14]. This
was
derived a formula
for the local
author
entropy production rate by a single scattering using the
radiative
transfer
theory.
However
these
connected
approaches are not easily
the multiple
to
the
scattering
the
situation
which
in
sense
medium
and
interaction
a
if
and
irreversible.
a
unitary
operation [9, 14]. Typical
tions
which
states
(I.e.
Freund
random
diffusive
reconstructed
process
by
of
prediction of
entropic approach
The
We
show
that
compensation by
is
therefore
it
consequence,
Mueller
matrix
(e,g,
evolutions
degree
lead
irreversible.
filter
to
scattering).
by
Measurements
scattering
under
the
otherwise
optical
any
indicate
to
a
invariant
reversible,
is
that
prove
useful
between
it is
transformation
selective
reversible
a
transforma-
absorption
of
produces
polarization
of
to
loss
any
of
light from
complete
w
BaS04 coating
depolarization of
argument
It is
Stokes
imply
that
0.03).
Reference
information
about
of
parameters
the
process
is
(assumed
a
the
a
[5]
incident
the
he
beam
reversible
polarization
incident
says is very
of light by
in
as
a
any input totally
claims
show
that
to
state
state
can
similar
to
be
the
filtering [15].
contradiction
with
the
here.
speckle-pattem
produces
that
comprehensive
note
we
a
only with precise knowledge
circular
it is
constitutes
rotator)
compensator,
increase
the
depolarizing
entropy are
reflected
of the
would
developed
the
evolution
the
depolarization by multiple scattering,
termed
speckle-pattem filtering which
process
determination
Freund
say
easy
which
is
not
key point here is that the entropy
tenable.
The
induces a loss of intensity and
suggested by Freund
Stokes
and simple to analyze this by expanding the
light (of intensity (S~)) in
of
orthogonal
terms
filtering
instructive
corresponding to unpolarized
vector
polarization states (e,g, right and left
handed
observations,
these
irreversible.
diffusively
not
is
that
(e, g. multiple
a
does
a
shall
scatterers)
complete
after
we
also
(residual
transport
even
usual
point
of
light
polarized
and
filtering) are
[5]
considers
array
Given
here.
occurs,
of
irreversible
are
considered
direct
a
by
is
absorption
no
As
characterized
which
irreversibility is featured.
Consider a linear
interaction
plane-wave
field.
If and only if the
remains
entropy
circular
states
each
of
intensity (S~)/2).
Now
a
right-
entropy (- In (2 )) and intensity (50 fb) losses [9]. Finally
discussion
of the data produced by Freund [5] could
take place
of both the
determined
scattering
characteristics
experimentally
both
JOURNAL
2058
of
BaSo4
the
coating
and
PHYSIQUE
DE
procedure
normalization
the
N°
I
the
in
of
measurement
I I
Stokes
the
parameters.
Concluding
6.
We
remarks.
details
depolarization and
decoherence
Three
non-absorbing
scattering
medium.
pointlike scattering centers,
uncorrelated
disorder
and
considered
have
weakly
in
disordered
introduced
:
typical
realization
scattering particles
considerations
(a)
of
some
such
of
much
smaller
points
main
Maximum
would
size
of
two
medium
entropy
a
collection
the
wavelength.
from
this
paper
plane-wave field by a
assumptions
were
approximation.
diffusion
On
of
basis
the
A
spherical
entropic
non-interacting
of
:
depolarized
totally
output
an
dense
than
drawn
be
can
yields
be
a
main
and
light
beam
(large
vector
I
n)
which
is
similar
of
that
to
light
natural
Stokes
whose
vector
is of the
°
form
The
results
0
depend on the specific nature of the
but
scatterers,
the
properties of the scattering
rather
medium.
Now
returning to the question
symmetry
on
raised in the
Introduction,
found that the length scale
which the depolarization and loss
we
over
nf, with typical number of scattering events n being about ten.
of
take place is (
coherence
for
distances
Then
larger than (, one is justified in describing the transport properties of
multiply scattered light by a radiative
transfer
theory which treat the radiation field as a scalar
than a four
Stokes
rather
In
addition, we found that the entropy production
component
vector.
falls
off
exponentially with (.
derived
in
preceding
the
section
do
not
=
(b)
principle
The
of
maximum
allows
entropy
us
also
deduce
to
the
form
of
Mueller
the
describing the scattering medium.
The explicit
calculation
of these
elements
from the Bethe-Salpeter
derived
equation permitted to test the validity of this approach. In that
knowledge of the degree of polarization after a double scattering suffices to enable one
case
a
matrix.
the
Mueller
Specifically, it was
that full
depolarization of linearly
construct
to
seen
polarized light requires more scattering events (typically a factor-of-2) than are required for a
circularly polarized light. It is believed
that this
reflects
behavior
the specific
symmetry
properties of (S~)
the
details
thereof
however
have not been
completely disentangled. This
be put in parallel with the results of
MacKintosh
who investigated the effects of
et al. [31]
can
polarization on backscattering of a laser
radiation
from
monodisperse
polystyrene spheres in
These
authors
also
pointed out that due to the large
of
have
number
statistically
water.
the
independent
contributions,
scattered
field is
Gaussian
distributed
by virtue of the central
matrix
elements
limit
theorem.
distribution
The
of
Stokes
for
parameters
a
Gaussian
distributed
field
has
been
spatial [36]
fluctuations.
is also
Reference
examined
made to a
in
details
the
recent
paper of Cohen et al. [32] who have recently
some
statistical
distribution
of the polarization
of multiply
scattered
optical
The ratio of
state
waves.
semi-minor
axis
the
semi-major
axis
polarization ellipse is
the
of
the
denoted
by
to
[32]. They
found
that
the
distribution
function
of
ellipticity
p(e)
parameter
e
~the
~~~
defining
the
ellipse
of
polarization
is
given
by
p(e)
indicating that the least
:
e
characterized
for
both
temporal [16, 34, 35]
and
=
probable
obtained
There
issue
of
state
with
do of
how
completely
is
polarization
data [32].
circular
experimental
remain
course
these
results
uncorrelated
are
: a
number
a
(I,e.
of
changed
more
realistic
(e)
=
points
when
the
to
be
(I
0.307).
+
A
e
)
satisfactory
agreement
was
investigated, regarding the
fundamental
susceptibility of the medium is not
dielectric
assumption
would
be to
allow
for
a
finite
correlation
N°
DEPOLARIZATION
ii
length
Gaussian
I.e.
LIGHT
OF
distribution
(rj )
q
BY
=
(2
2059
j~
~
(r~ ))
q
SCATTERING
MULTIPLE
exp
~ ~'~
~~
gr«
j2
where
«~
2
,
positive
the analysis is
extended
radiations
when
with a broad
constants
to
or
possibility
of
generating
frequency
shifts
scattering.
by
Our
method
of
:
calculating the dependence of the Mueller
matrix
elements
the
number
of
scattering
events
on
gives also rise to interesting possibilities for exploring the behavior of the above in more
complicated systems (e,g, scattering from large particles for which the transport of light
becomes
ballistic
[30],
ideal
particles shapes which do not
extension
less
light
to
scatter
isotropically and involving non-diagonal
Mueller
matrices).
C
and
are
«
I,e,
bandwidth
Acknowledgments.
(C.B.) is indebted to Prof. Craig. F.
correspondence and for bringing some
pleased to acknowledge Prof. R. Maynard for
of
One
us
useful
for
of the
related
research
critical
reading
of the
Appendix
of this Appendix is to briefly
purpose
field that is
needed
in
section 4.
The
State
his
to
University
attention.
manuscript
and
for
We
G.
are
Maret
discussions.
useful
the
Pennsylvania
Bohren
earlier
A
outline
the
conveniently
The
following
calculations
are
more
point is the Bethe-Salpeter equation for the field-field
diagrammatic
expansion [23, 30, 37-42].
(Gim Gjsl
G,~ GjS
"
+
derivation
done
of the
in
correlation
of
Our starting
space.
derived
from
usual
momentum
correlation
function
function
Gim'Gjt'w>jm'n' Gm'm Gfl
(Al)
n
,
repeated indices is presumed, G* being the complex conjugate of
which
coherent
contribution
right side of (Al
represents the
goes to
is the
incoherent
contribution.
In the
while the other
term
zero
very rapidly and is neglected
incoherent
of
contributions
of light
weak
scattering situation, the observed intensity is an
sum
scattered
through all possible paths. For scalar
shows
that
within
the
diffusion
waves,
one
approximation (I,e,
incoherent
intensity can be
incoherent
diffusive
and
transport of light), the
written
[39, 43] :
as
summation
where
G.
first
The
over
in the
term
fin~ (r)
~/ i
=
£
W(q)
~
)
(iq
exP
r
~£
and
be
may
For
Green's
function
diffusion
the
successive
ladder
by summing
field, the intensity operator Wij~~
obtained
polarized
a
(r, n) of
F
~
wijmn(~)
(A2)
n
,
i
q
the
where
(r,
F
=
10
£ £
(ol"I
equation is
diagrams.
can
be
(Aa(~)Y
~
weight
the
written
as
of
diffusion
paths
:
(" lmnj,
(A3)
a
the
first
term
n
[a)
eigenvectors
O
(q~))
(A2)
have
and
0
=
been
(A3),
corresponding
associated
and
derived
one
to
finds
by
a
single
eigenvalues
the
number
of
~
(
:
(G~~ G()
=
~
authors
scattering
A~(q) (to
[23, 30] and
situation.
the
are
The
(ij la ) A~(r, n) (a [mn)
~
nine
ortho-normal
degree of approximation
reproduced here. From
not
(A4)
,
a
~~~~
(~ ~)
a
,
l
=
£
n
(q)Y
(A
a
q
exp
(iq
r),
(A5)
2060
JOURNAL
and
ii j ), [mn) denoting respectively
the
dependence
of
interested
evaluate
to
number,
write
we
the
the
propagation
of
direction
(G~X
Gj~)
(G~~
=
(G~ G()
F~~~
=
the
and
the
final
N°
Green's
function
as
F
G()
=
(r,
ez,
where
follows
It
the
have
we
at
(G,~ G~()
F~
=
~
lo
n, L~
for
~
(r,
F
j
~
once
are
set
(r,
n,
L,
=
j
~~ ~~
~~, L(
L(
=
(r,
F
3
carrying
given by the
that
~
formulae
scattered
the
by a diffusion
approximation,
diffusion
[10, 30, 43]
Appendix is
within
light
entire
space,
7
7
4
lo
)~
scattered
~
jnj
2
(A9c)
'
l
)~
jnj
2
(A9d)
B
of
calculation
the
that
(A9b)
'
l
~
one
(A9a)
~~
lo
lo
D~,
j
of
the
the
DWS.
shows,
in the
electric
field
degree
Brownian
For
weak
polarization
of
scattering
correlation
scatterers
limit
function
for
and
is
within
given by
:
~~~~~~~ ~~
n
exp
n)
scattering paths
denoting the
length nf,
of
(- 2 nt/rq )
dn
~~~~
~~~~~~~~
F
F(r,
the
over
,
IF (r,
where
,
n
(
2
+
4
coefficient
the
:
5
1
framework
the
(A7d)
,
(Ji8)
L)
equations (A7)
of
=
~
consider
to
L~ )j
:
Appendix
this
n,
~ ~~
L(
=
~
=
(r,
F
~2
4
n
(G~~ Gj~)
(G~G()
(A7a)
(A7b)
given by
medium
integration
the
out
Lj
n,
Lj
n,
2
eXp
and
49
=
=
characterized
are
,
infinite
7rL)
4
(G~G()
multiply
we
scattering
,
L~
n,
(r,
F
lo
~
an
~~'~ ~~~
of
the
(A6)
lo
2
4
=
purpose
a
Since
of
~
l + 2
2
l
=
~
(G~ G£)
The
function
ii
have
we
F~~~
=
=
~ijmn (~,
~
3/2
F
and
polarization.
of
states
intensity
mean
GJ)
along
is
(G~ Gj~)
with
initial
I
:
(Gim
As
PHYSIQUE
DE
function
Green's
and
To
ki
=
~
Dj
of
time
dn
equation
required by a
diffusion
the
is the
(r, n)
is
scatterer
number
the
to
move
of
one
N°
DEPOLARIZATION
II
OF
LIGHT
SCATTERING
MULTIPLE
BY
2061
normalized
optical wavelength. From inspection of equation (B I), we note that gl~ ~(r, t is the
of
the
number
Laplace transform of F (r, n ), giving then a one to one correspondence
between
and
dimensionless
t*
scattering
the
variable
t/ro.
events
n
situation
of reflected
We
consider
the special
light from a half-space. In this geometry,
now
scattering
medium
occupies
of space z ~ 0 and that the light is
that
the
the
region
assume
we
semi-infinite
semi-infinite
medium
For a
incident
normally on the
from the
tx~.
vacuum
z
from
the
medium,
diffusion
paths for scalar
be
obtained
above
the weight of
waves
may
of
images
infinite-space
by
the
method
[37].
case
=
=
a)
LINEAR
POLARIZATION.
Considering an
perpendicularly
polarized along e~ (labeled I ),
L) polarized output light [30] :
incident
field
(labeled
(xx
(n )
F
xx)
F
(G~X
=
=
GQ)
l
=
2
=
2
ii
(xx [F[ yy)
Fi(n)
Then,
obtain
we
a
where
/~
=
In
CIRCULAR
a
vector
~
(l
(B2a)
(n, Lj )
,
)~j
lo
and
F(n,
Lj).
(B2b)
;
=
b
+
t*
gi~~(t)
=
3
n
~~ ~~~
~~~ ~~
(t )
j
~/6
+
(84)
j)(t)
g
POLARIZATION.
similar
the
manner,
E,
is
F
lo
parallel
:
P
b)
(G~G()
=
=
j
~
2
+
respectively
get for
we
=
/
ie~)
(e~
of
case
and
polarization
circular
polarization
the
I+ +1
ll~l
i
"
ly~l
developed [30].
is
states
are
l~yl
+ I
The
incident
unit
:
lyyll
+
(85~)
'
passing,
conjugation
In
We
call
helicity.
In
we
would
=
like
operation.
F~ (resp. F_ )
like
fashion
)1[xxi
)
1+
to
the
as
the
remark
Green's
above
iYx)
=
F_
=
<+
<+
ixY)
I
permutation
function
have
we
F~
+ I
+
symmetry
corresponding
(B5b)
IYY)1
to
of
the
these
vectors
positive (resp,
((G~GQ)
+
the
negative)
:
+
F
+
(B6a)
+
,
F
+
(B6b)
+
,
)~ l=
under
(G~ G()
±
(G~G()
(± )
(G~ G()
2062
JOURNAL
PHYSIQUE
DE
I
N°
11
with
corresponding
The
temporal
correlation
functions
~~~~
~
:
~
~~~
~
~
are
~~~
~
~~,
)~)~~~)
(88)
=
~+
with
a,
b
(2).
In
2
the
aforementioned
by
iven
b'
(5/3)
d'=
+exp(-b')±-exp(-d')
~~~
corresponding
The
expressions and
of polarization
degree
~j~~~~~
P~(t)
=
Similarly,
for
gets
one
polarization
a
state
~
(t
=
gi
(2 t*
(2))
In
+
:
(89)
(t)
~
(t)
i
writes
45°
at
))~~~
P
=
j~~~~~~
(t)+g_
g+
~/(5/3)
d
+
~~
gi
i
(B lo)
(t)
,
with
~~~~~~~
ii(~(t)) _~~~~~
~~,
2+exp(-b')±~~exp(-c')
gi((t)
~/(49/23)
~~~~~~~~~
(Bll)
~/(49/23 ) In (10/7).
(10/7))
c'
+ In
indicate
compared
with
those
of
reference
[30] : for this purpose
These
results
we
may
/~
respectively
of
P~(t)
the
origin
equal
Pj(t)
and
taken
that the slopes
to
at
are
vs.
figure
4
also
and
)/2,
in
notations
of
reference
[30].
From
)/2
(y~
the
(yi
note
we
y_
yjj
discussed
in
section
4,I.
close to the value
that the ratio of these slopes is equal to 2.2, I.e.
and
c
(2 t*
=
=
be
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Scattering and
1990). For
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reference
SHENG
P.,
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P.
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two
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Y. N.,
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t
T
2, N'
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J. A.,
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