Lecture 12

dinnerworkableUrban and Civil

Nov 16, 2013 (3 years and 8 months ago)

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Stimulated scattering is a fascinating process which requires a strong coupling between light
and

vibrational
and rotational modes,
concentrations
of different species, spin, sound waves and
in

general
any property which can
undergo
fluctuations in its population and couples to light.
The

output
light is
shifted down
in frequency from
the
pump beam
and the
interaction leads to
growth

of
the shifted light
intensity. This
leads to exponential growth of the signal
before saturation

occurs due
to pump beam depletion. Furthermore,
the matter
modes also experience gain.

Stimulated Scattering

The
Stimulated Raman Scattering (SRS)
process is initiated by noise, thermally
induced

fluctuations
in the optical fields and Raman active vibrational modes. An incident pump field (
ω
P
)

interacts
with the vibrational fluctuations,
losing
a photon which is down shifted
in frequency by

the
vibrational frequency
(

)
to produce a Stokes wave (
ω
S
,
)
and
also an
optical
phonon

(
quantum of vibrational energy
).
These
stimulate further break
-
up
of pump photons in
the

classical
exponential population dynamics process in
which “
the more you have, the more you get”.

The
pump decays with propagation distance
and both
the phonon population and Stokes
wave

grow
together. If the generation rate of Stokes
light exceeds
the loss, stimulated emission
occurs

and
the Stokes beam grows exponentially.

It is the product of optical fields which excites coherently the

phonon modes. Since the “noise” requires a quantum

mechanical treatment here we consider only the classical

steady state case, i.e. both the pump and Stokes are

classical fields, i.e. it is assumed that both fields are present.



.
.
.
.
ˆ
2
1
)
,
(
)
(
)
(
c
c
e
c
c
e
e
t
r
E
t
r
k
i
S
t
r
k
i
P
T
S
S
P
P
















E
E
Pump (laser) field

Stokes field,

v



P
S














0

r
lity tenso
polarizabi

n
q
n
ijn
n
L
ij
ij
q
q
α



for

eal

is

2
2
0
p
mg
q
n
iin
n
q








)
(
)
(
)
(
)
(

;
)
(
)
(
)
(
)
(
)
1
(
)
1
(
0
)
1
(
)
1
(
0
S
S
P
q
P
NL
P
P
S
q
S
NL
E
q
q
p
E
q
q
p


























}
)
(
E
E
)
(
E
E
{

)
(
)
(
1
)
(
*
*
)
(
*
S
)
1
(
)
1
(
0
t
i
S
P
S
P
t
i
S
P
S
P
P
q
n
n
n
P
S
S
P
n
e
D
e
D
q
m
q











































)
(
)
(
S
)
1
(
)
1
(
0
2
1
)
(
)
(
t
r
k
i
P
t
r
k
i
S
P
q
n
n
n
NL
P
P
S
S
n
e
e
q
q
p









E
E
P
E
drives

S
E
drives

)
(
2
2
S
)
1
(
)
1
(
2
0
)
(
)
(
2
2
S
)
1
(
)
1
(
2
0
*
)
(
|
|
)]
(
)
(
[
]
[
)
(
4
|
|
)]
(
)
(
[
]
[
)
(
4
t
z
k
i
P
S
P
q
n
n
S
P
t
z
k
i
NL
P
t
z
k
i
S
P
P
q
n
n
S
P
t
z
k
i
NL
S
P
P
n
P
P
S
S
n
S
S
e
q
D
m
N
e
e
q
D
m
N
e
































E
E
P
E
E
P
VNB: both polarizations,

have
exactly the correct
wavevector

for

phase
-
matching to the Stokes and pump fields respectively. Also, for simplicity in the

analysis, assume that the laser and Stokes beams are collinear.
However, stimulated
Raman

a
lso occurs
for non
-
collinear
Stokes beams
since
is
independent of
.

NL
P
NL
S
P
P

and

NL
S
P
P
k

P
S
q
S
P
P
P
P
S
P
q
S
P
S
S
S
q
cD
n
m
N
i
dz
d
q
cD
n
m
N
i
dz
d
E
E
E
E
E
E
2
)
3
(
2
0
0
2
)
3
(
2
0
*
0
|
|
]
[
)
(
8

;
|
|
]
[
)
(
8






















2
2
-
v
2
2
2
v
-1
v
)
3
(
2
0
2
0
2
]
[
4
)
]
[
(
)
(
)
(
)
(
]
[
)
(
S
P
S
P
S
P
S
P
q
P
S
S
S
z
I
z
I
q
c
n
n
m
N
z
I
dz
d























)
(
)
(
)
(
)
(
z
I
z
I
z
I
g
z
I
dz
d
S
S
P
S
R
S




Optical loss added

phenomenogically

t)
coefficien
Gain
(Raman

]
[
4
)
]
[
(
)
(
]
[
2
2
v
2
2
2
v
1
v
)
3
(
2
0
2
0
2
S
P
S
P
S
P
q
P
S
S
R
q
c
n
n
m
N
g
























z
I
g
S
S
I
z
I
S
P
R
P
P
e
I
L
I
]
)
0
(
[
)
0
(
)
(
)
0
(
)
(











For
g
R
I
(

p

)>

S
,

e
xponential
growth of Stokes

Phase
of Raman signal
independent of laser phase,

i.e
.

! But
if
temporal coherence of

laser is very
bad,

P

may
be larger
than


must average
over



P

to get
net gain

2
|
E
|
P
R
g

-1
v

.
.
2
1
)
(
c
c
e
t
z
k
i
P
P
P



E
.
.
2
1
)
(
c
c
e
t
z
k
i
S
S
S




E
can also have
gain for Stimulated Stokes in the backward
direction! Get the same but boundary conditions at
z
=0,
L

different!

R
g
In fact Stokes beam can go
in any direction
, however if the two beams are
not collinear
then
the net gain is small with finite
width beams

Raman
Amplification

)
(
)
(
)
(
)
(
z
I
z
I
z
I
g
z
I
dz
d
P
P
P
S
S
P
R
P







P
S
q
S
P
P
P
P
q
D
c
n
m
N
i
dz
d
E
E
E
2
)
3
(
2
0
0
|
|
]
[
)
(
8

Recall











)
(
1
)
(
1

z
I
dz
d
z
I
dz
d
S
S
P
P





Optimum conversion:

0
)
0
(

and

0
)
(


S
P
I
L
I
S
S
P
P
L
I
I


)
(
)
0
(


When grows by one photon,

decreases
by one photon
and


of
energy
is lost
to the vibrational mode, and eventually
heat

)
(
z
I
S
)
(
z
I
P
)
(
S
P




No pump depletion
(small signal gain) but with attenuation loss

]
)
0
(
exp[

as

gain
amplifier

depletion)

pump

(no

d
unsaturate

Define
)
exp(
1

with
)
0
(
)
(

eff
eff
)
0
(
eff
L
I
g
G
L
L
e
I
z
I
P
R
A
P
P
L
L
I
g
S
S
S
P
R










)
(
)
0
(
)
(
)
(

)
0
(
)
(
z
I
e
I
z
I
g
z
I
dz
d
e
I
z
I
I
I
dz
d
S
S
z
P
S
R
S
z
P
P
P
P
P
P
P













Raman
Amplification


Attenuation, Saturation, Pump Depletion, Threshold

Saturation in
amplifier
gain occurs due to
pump

depletion
.

)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
z
I
z
I
z
I
g
z
I
dz
d
z
I
z
I
z
I
g
z
I
dz
d
P
P
S
S
P
R
P
S
P
S
R
S









Assume

P

=

S

=


(reasonable approximation)

condition)
(input

)
0
(
)
0
(
with
)
1
(


:
Gain

Saturated
0
)
1
(
0
0
0
P
S
S
P
r
A
L
S
I
I
r
G
r
e
r
G











Note that the higher
the input

power
, the
faster the saturation

occurs, as expected.

Starting from noise, the Stokes seed intensity ( ) is a single “noise” photon


the Stokes
frequency bandwidth of the unsaturated gain profile, assumed to be
Lorentzian
.

Mathematically for the most important case of a single mode fiber:






L
th
P
P
P
R
S
S
P
e
I
L
I
L
I
g
I
L
I





(0)
)
(
]
)
0
(
exp[
)
0
(
)
(
eff
eff
The stimulated Raman
“threshold”
pump intensity is defined approximately as
the input
pump intensity for which the output pump intensity equals the Stokes output intensity,
i.e.

)
0
(
th
P
I
2
/
1
2
2
eff
eff
)
0
(
2
)
0
(





S
R
P
S
eff
S
g
L
I
A
I






16
)
0
(

eff
th


L
I
g
P
R
A
eff


is the effective nonlinear core
area


glass

)
0
(
)
0
(
th
S
L
P
I
e
I
P



For backwards propagating Stokes

2
/
1
2
2
eff
eff
eff
eff
)
0
(
2
)
(

where
)
(
]
)
0
(
exp[
)
(







S
R
P
S
eff
S
P
P
R
S
g
L
I
A
L
I
L
I
L
I
g
L
I






This threshold is higher than for forward

propagating Stokes. Therefore, forward

propagating Stokes goes stimulated first and

t
ypically grows so fast that it depletes the pump

so that that backwards Stokes never really grows

20
)
0
(

eff
th


L
I
g
P
R
)
0
(
eff
S
I
Raman
Amplification


Pulse Walk
-
off

Stokes and pump beams propagate with different
group velocities v
g

(

S
)

and v
g
(

P
). The
i
nteraction
efficiency
is greatly reduced
when

walk
-
off
time


pump pulse
width

t
. As a result

t
he Stokes signal spreads in time

and space

For backward propagating Stokes, the pulse

overlap is small and the amplification is weak.

Raman Laser

Threshold
condition:

1
Re
]
[
max


L
I
g
S
P
R

)
3
(
2
0
v
1
v
2
0
2
max
]
[
4







q
P
S
S
R
q
c
n
n
m
N
g




Frequently
fibers used for gain.
Why? Example silica has a small
g
R

but
also an ultra
-
low loss
allowing
long growth distances
.
For
L

10
m
,
P
P
th
=1
W

for
lasing.

Multiple Stokes and
Anti
-
Stokes

Generation

Fused silica fiber excited

with doubled
Nd:YAG

laser


=514nm.

Spectrally resolved multiple Stokes
beams

Spectrally resolved multiple
Anti
-
Stokes

beams

To this point we have focused on terms like

which corresponded
to


What
about
,
i.e. Anti
-
Stokes generation? This requires tracking the

o
ptical phonon
population since a phonon must be destroyed to upshift the
frequency. Therefore

Anti
-
Stokes generation
follows

Stokes generation which involves the generation of the phonons.

P
S
S
P
E
|
E
|

and

E
|
E
|
2
2
.
v



P
S


v


P

v
Ω

S


P

v
Ω

P


A

Coherent
Anti
-
Stokes Generation

}
)
(
E
E
)
(
E
E
{
)
(
)
(
4
1
}
)
,
(
)
,
(
{
2
1
:
write
Again we
)
(
*
*
)
(
*
S
)
1
(
)
1
(
0
)

(
*
)

(
t
i
S
P
S
P
t
i
S
P
S
P
P
q
t
r
K
i
t
r
K
i
P
S
S
P
e
D
e
D
q
m
e
K
Q
e
K
Q
q









































)
,
(

K
Q

)
,
(
*

K
Q

v



P
S




Stimulated
Stokes;


Anti
-
Stokes

.
.
2
1
)
(
c
c
e
t
r
k
i
A
A
A






E
v



P
A



.
.
)
(
)
(
8
1
*
)
1
(
)
1
(
0
c
c
Q
q
N
i
I
dz
d
S
P
S
P
q
S
S









E
E





.
)
(
)
(
8
1
.
.
]
)
(
)
(
)[
(
8
1
*
*
)
1
(
)
1
(
0
A
*
*
)
1
(
*
)
1
(
)
1
(
0
c
c
e
Q
q
N
i
I
dz
d
c
c
e
Q
Q
q
N
i
I
dz
d
kz
i
A
P
A
P
q
A
kz
i
A
P
A
S
P
S
P
q
P
P





















E
E
E
E
E
E









-



dispersion in refractive index means the waves are not collinear


for the Anti
-
Stokes case, similar to the CARS case discussed
previously

-
Thus
A
nti
-
Stokes
process requires phase
-
matching (not automatic
)

0


k

)
(
1
)
(
1
)
(
1


z
I
dz
d
z
I
dz
d
z
I
dz
d
P
P
A
A
S
S







For every Stokes photon created, one pump photon is
destroyed AND
for every Anti
-
Stokes photon
created another pump photon is
destroyed.
Also
, for every Stokes photon created an optical
phonon

is
also created,
and for
every Anti
-
Stokes photon created an optical phonon is destroy
ed

What is missing in the conservation of energy is the flow of mechanical energy
E
mech

(t) into the

optical phonon modes via the nonlinear mixing interaction, and its subsequent decay (into heat).

Vibrational energy grows with the Stokes energy, and

decreases with the creation of Anti
-
Stokes and by

decay into heat.
If Stokes strong


2
nd

Stokes




3
rd

Stokes etc
.

A
A
S
S
I
dz
d
I
dz
d
dt
d



1
1
}
E
2
E
{
1


analysis

Detailed
mech
1
-
v
mech





Anti
-
Stokes is
not

automatically
wavevector

matched!
Since

Stokes
is generated in all directions
, Anti
-
Stokes generation


eats out” a
cone in
the Stokes generation (angles small
).

T
he generation of

Anti
-
Stokes lags

b
ehind the Stokes

Stimulated
Brillouin

Scattering

The normal modes involved are acoustic phonons. In contrast to optical phonons,

acoustic waves
travel
at the velocity of sound
.

Light waves

.]
.
[
ˆ
2
1
)
,
(
)
(
)
(
)
(
c
c
e
e
e
e
t
r
E
t
z
k
i
A
t
z
k
i
S
t
z
k
i
P
A
A
S
S
P
P












E
E
E


Freely propagating sound
waves

]
[
2
1
)
(
*
)
(
)
(
*
)
(
S
S
S
t
Kz
i
t
Kz
i
t
Kz
i
t
Kz
i
e
Q
e
Q
e
Q
e
Q
q
S























Forward travelling

Backwards travelling

Stimulated
Brillouin

“Noise” fluctuations

in optical fields and

sound wave fields

Brillouin

scattered light

Optical
phonon (sound


wave)
excited

Grow

in
opposite


directions but still


“drive” each
other

Decays to
thermal


“bath”, i.e. heat

Decays to thermal


bath
”, i.e
. heat

Brillouin

Amplification

Stokes signal injected.

K
k
s
m
k
s
m
K
/
/


)
/
10
(

c


)
/
10
(

v
S
8
3
S
Sound











and


For
S




K
k
need
k

K

for measurable

S
, since

S

0 as
K


0


Backwards Stokes couples to

forwards travelling phonons

For
Stokes
need

S

and







P
S
S
P
k
K
k


)
(
*
)
(

n via
interactio
S
t
Kz
i
e
Q
t
z
k
i
e
P
P
P







E
To get stimulated scattering, light and sound waves
must

be
collinear

Backscattering


K


2
k


→ phonon wave picks

up energy and grows
along +
z
.


Stokes can grow along
-
z

S
S
P





For
Anti
-

Stokes
need

S
A

and







P
A
P
k
K
k


)
(
)
(
n with
interactio
s
t
Kz
i
e
-
Q
t
z
k
i
e
P
P
P






E
Backwards Anti
-
Stokes couples

to backwards travelling phonons


backwards

phonon
wave gives
up energy

and one phonon is lost for every

anti
-
Stokes photon created. But the
only
backwards
phonons
available
are

due
to “noise

, i.e
.


k
B
T
, a very small number
! (Stokes process generates

s
ound waves in opposite direction.)
A
nti
-
Stokes NOT stimulated!

)
,
(
]
[
:
solid

)
,
(
]
)[
1
(
)
,
(
:
gas/liquid
2
2
0
2
0
t
r
E
q
p
n
n
t
r
E
q
n
t
r
P
i
jj
AO
iijj
j
i
i
NL
i


















Stimulated Raman

1.
Molecular property



䱯捡L晩f汤l捯c散瑩潮



乯牭慬浯摥猠摯⁎佔灲潰慧慴攮


乯牭慬浯摥m晲敱略湣e楳⁦楸i搠慴

v

4.
Both forwards and backwards scattering

Stimulated
Brillouin

1.
Acousto
-
optics uses bulk properties


NO local field corrections

2.

Acoustic waves propagate.

3.
Normal mode frequency

S



K

4.

Backward Scattering only


Light
-
sound coupling

Equation
of
Motion
for
Sound Waves

Only
compressional

wave (
longitudinal acoustic

phonon
) couples to
backscattering
of light



q
F
q
z
q
q












2
2
2
S
S
v
2



Mass density

Acoustic
damping constant

Sound velocity

Force due to
mixing of
light beams

wave
sound

of

decay time
S


v
s





0





0





0





0





0

Gas or Liquid

Comparison between Stimulated Raman and Stimulated
Brillouin


)
E
.
E
(
2
1
)
1
(
2
1
]
)[
1
(
2
1
V
]
[
*
2
0
,
2
0
int
t
i
P
S
z
K
k
k
S
P
s
P
S
P
S
e
z
q
n
E
E
q
n





































Substituting into driven wave equation for
q
z




z
z
z
z
F
q
z
q
q





2
2
2
S
S
v
2






.
.
)
E
.
E
(
)
1
(
4
1




.]
.
2
)
(
2
)
)
(
[(
2
1

V
]
[
*
2
0
S
2
S
2
int
c
c
e
z
n
c
c
Q
dz
d
iK
Q
i
Q
Q
q
F
t
i
P
S
S
P
S
P
z
q
S
P




































q
S
P
S
P
F
c
c
Q
dz
d
iK
Q
i
Q
Q
SVEA














.]
.
2
)
(
2
)
)
(
[(
2
1
S
2
S
2







The damping of acoustic phonons at the frequencies typical of stimulated
Brillouin

(10’s GHz)
frequencies is large with decay lengths less than 100

m. This limits (saturates) the growth of
the phonons. In this case the phonons are damped as fast as they are created , i.e. .

0
/


dz
dQ
Mixing of optical
beams drives
the sound waves

)]
(
)
(
[
)
(
2
)
(
1
2
)
1
(
)
(

*
S
2
S
2
2
0
*
z
z
K
i
n
i
z
Q
S
P
S
P
S
P
E
E
















Acoustic phonons
modulate
pump beam
to produce
Stokes
.

P
NL
S
KQ
n
i
E

)
1
(
2
1
P
*
2
0





Power Flow (Manley Rowe)

)
,
(
)
1
(
)
,
(

Recall
2
0
t
r
E
q
n
t
r
P
NL












Note that for ,
Q
+

is linked to E
S

with

t
i
t
i
t
i
S
P
S
S
P
e
e
Q











)
(
E
NL
P
P
S
NL
P
KQ
n
i
E
)
1
(
2
1
P
2
0






iKq
q
z
q
z










Q
*

Q

)
,
(


)
,
(


)
,
(
t
r
E
t
r
E
t
r
E
P
S









S

propagates
along

z

)]
(z
[
4
)
1
(
)
(z

);
(
4
)
1
(
)
(
SVEA
2
*
2
S
P
P
P
P
S
S
S
Q
c
n
K
n
dz
d
z
c
n
KQ
n
z
dz
d
E
E
E
E












P

travels along +
z

.}
.
)
(
)
(
{
8
)
1
(
)
(
*
*
0
2
c
c
z
z
Q
K
n
z
I
dz
d
S
P
S
S







E
E


.}
.
)
(
)
(
{
8
)
1
(
)
(

*
*
0
2
c
c
z
z
Q
K
n
z
I
dz
d
S
P
P
P




E
E


)
(
)
(
1
)
(
1
z
I
z
d
d
z
I
dz
d
S
S
P
P





Travels
and
depletes along
+
z

Travels and
grows
along
-
z

Pump beam
supplies energy for
the
Stokes
beam!

Phonon Energy
Flow

(need acoustic SVEA)

)
(
v
4
)
1
(
)
(
2
)
(

SVEA

Acoustic
*
2
S
2
0
S
S
P
n
z
Q
z
Q
dz
d
E
E










S
S
S
v
2



Mixing of optical
beams drives sound waves

Decay of sound
waves “
heats up” the lattice

.}
.
)
(z
)
(
{
8
)
1
(
)
,
(
)
,
(
*
*
2
0
S
S
S
S
c
c
z
Q
K
n
z
I
z
I
dz
d
S
P












E
E


)
(
1
)
(
1
)]
,
(
)
,
(
[
1
S
S
z
I
dz
d
z
I
dz
d
z
I
z
I
dz
d
S
S
P
P
s














Phonon beam grows in forward direction by picking up energy from the
pump beam
. The
Stokes grows in the backwards direction because it also
picks up
energy from the pump.

Exponential
Growth

W
hen the growth of the acoustic phonons is limited by their attenuation constant.

)
(
)
(
)
(
2
)
(
1
2
)
1
(
)
(
:
Recall
*
S
2
S
2
2
0
z
z
K
i
n
i
z
Q
S
P
S
P
S
P
E
E
















)
(
)
(

)
(
)
(
)
(
v
4
)
1
(
)
(
2
S
2
S
2
S
2
2
S
2
S
S
2
2
z
I
z
I
-g
z
I
z
I
n
c
n
z
I
dz
d
P
S
B
P
S
S
P
S
S

















Signature of exponential growth

2
2
S
2
S
S
2
2
max
2
S
2
S
2
S
2
2
S
2
S
S
2
2
v
4
)
1
(


)
(
v
4
)
1
(
n
c
n
g
n
c
n
g
S
B
S
P
S
B






















The energy associated with

, i.e. the sound waves, eventually goes
into
heat.

]
[
S
P




)
(
)
(
)
(


)
(
1
)
(
1

from

Also,
z
I
z
I
-g
z
I
dz
d
z
I
dz
d
z
I
dz
d
P
S
S
P
B
P
S
S
P
P









This leads to
exponential growth
of Stokes along
-
z
!!

)
(
v
4
)
1
(
)
(
2
)
(
*
2
S
2
0
S
S
P
iK
K
n
i
z
Q
z
Q
dz
d
E
E








)
(
v
2
)
1
(
)
(


0
)
(
*
2
S
S
2
0
S
P
n
z
Q
z
Q
dz
d
E
E











What is happening to acoustic phonons ?

max
2
S
2
S
S
2
2

i.e.

,
)
(
)
(
v
4
)
1
(
)
(


)
(

into


this
ng
Substituti
B
P
S
S
L
S
S
S
g
z
I
z
I
n
n
c
n
z
I
dz
d
z
I
dz
d
Q








Therefore
,
acoustic damping leads to saturation of the phonon flux and exponential gain of the
Stokes beam!

→In the
undepleted

pump
approximation get exponential
gain for
backwards Stokes

0.2

0.4

0.6

0.8

1.0

Distance
z/L

Relative Intensity

0.2

0.4

0.6

0.8

1.0

0.0

Pump


Stokes

001
.
0
)
0
(
/
)
(

P
S
I
L
I
01
.
0
)
0
(
/
)
(

P
S
I
L
I
0

10
)
0
(



L
I
g
P
B
For amplifying a signal
I
S
(
L
) inserted

at
z
=
L,
the growth of the signal is shown

for different signal intensities relative

to the pump intensity.

Pump signal decays
exponentially in

the forward direction as the Stokes

g
rows exponentially in the backward

direction

Assume an
isotropic solid


the
pertinent

elasto
-
optic
coefficient is
p
12

so
that




(typically 1


p
12


0.1)
.

)
,
(
)]
,
(
[
)
,
(
12
4
0
t
r
E
t
r
q
p
n
t
r
P
NL












2
S
2
S
S
2
12
6
max
2
S
2
S
2
S
2
S
2
S
S
2
12
6
v
4

)
(
v
4
c
p
n
g
c
p
n
g
S
B
S
P
S
B



















Can add
loss
phenomenologically

)
(
)
(
)
(
)
(

)
(
)
(
)
(
)
(
z
I
z
I
z
I
-g
z
I
dz
d
z
I
z
I
z
I
-g
z
I
dz
d
P
P
P
S
S
P
B
P
S
S
P
S
B
S








Pump
Depletion and Threshold

The analysis for no pump depletion, threshold and saturation effects is
similar to
that
discussed previously for Raman gain
effects Since

S
,

P
>>

S

then

S

P
=


is an excellent
approximation. For no depletion of pump
except
for absorption

L
L
I
g
S
S
L
P
P
P
P
S
P
S
B
S
P
B
e
L
I
I
e
I
z
I
z
I
z
I
dz
d
z
I
z
I
z
I
g
z
I
dz
d














eff
eff
)
0
(
)
(
)
0
(


)
0
(
)
(
)
(
)
(

)
(
)
(
)
(
)
(
Signal output

P
P
L
L


)
exp(
1
eff




,
)
0
(
)
0
(
with
1

)
(
)
0
(

:
gain

saturated


0
0
0
]
)
0
(
)
1
[(
0
P
S
L
I
g
b
S
S
S
I
I
b
b
b
e
L
I
I
G
P
B






Brillouin

threshold
pump intensity defined as

with unsaturated
gain
& with
the
Lorentzian

line
-
shape for
g
B
:


)
0
(
)
0
(
for which

)
0
(
th
S
L
P
P
I
e
I
I
P



To solve
analytically

for saturation which occurs in the presence of pump depletion, must
assume

=
0,

P




S
and define


gain)

ed
(unsaturat

)
0
(
L
I
g
G
P
B
A

21
)
0
(

eff
th

L
I
g
P
B
Plot of gain saturation after a propagation distance

L
versus the normalized unsaturated gain
G
A
.

The higher the gain, the faster it saturates.

Stimulated
Brillouin

has been seen in fibers at
mW


power levels for
cw

single frequency inputs.

It is the dominant nonlinear effect for
cw

beams.

e.g. fused silica :

P

=
1.55

m
,
n
=1.45,
v
S
=6
km
/
s
,


S

/2

=
11GHz
,
1/

S




17 MHz



g
B



5x10
-
11

m
/
W.

This value is

500x larger the
g
R
!
But
, 1/

S

is much

smaller and requires stable single frequency input to

utilize the larger gain


hence no advantage to stimulated
Brillouin

for amplification.

Pulsed Pump Beam


t
P


t
S

v
g
(

P
)

v
g
(

S
)

Stokes and pump travel in opposite directions, the overlap

with a growing Stokes is very small and hence the

Stokes amplification is very small! The shorter the pump

pulse, the less Stokes is generated, i.e. this is a very

inefficient process! Stimulated Raman dominates for

pulses when pulse width <<
Ln
/c.