Stimulated Scattering
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.
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 (ωP)
interacts with the vibrational fluctuations, losing a photon which is down shifted in frequency by
the vibrational frequency () to produce a Stokes wave (ωS,) 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.


 
 
 
1
ET (r , t )  eˆ E P e i ( k P r Pt )  c.c.  E S e i ( kS r S t )  c.c.
2 Pump (laser) field Stokes field,S  P  v
 ijn
 L
 ij  polarizability tensor  αij  q n
q n



p NL (S )  q
q
1  n
qn 
m qn

(1)
(1)

(

)

(

)
E
(P );
q 0
S
P

qn 0 

iin
qn


p NL (P )  q
q
2
2
is

eal
for



qn  0
mg
p

(1)
(1)

(

)

(

)
E
(S )
q 0
P
S
E P E*S
E*P E S
i ( P S )t
i (S  P )t

(

)

(

)

{
e

e
}
qn  0
P
S
*
D( P   S )
D ( P   S )
(1)
(1)




 n
1
i ( k S r  S t )
i ( k P r  P t ) 
NL
(1)
(1)
p
 qn
 EPe

qn  0  ( P )  (S )  E S e
qn
2

PSNL e i ( kS z S t ) 
 n
4m D * ( P   S ) q n
PPNL e i ( k P z Pt ) 
 n
4m D( P   S ) q n
N
N
[
[
drives E S
drives E P
i ( k S z S t )
2 (1)
(1)
2
2
qn 0 ] [  ( P )  (S )] | E P | E S e
i ( k P z  Pt )
2 (1)
(1)
2
2
]
[

(

)

(

)]
|
E
|
E
e
qn  0
P
S
S
P
VNB: both polarizations, PSNL and PPNL 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

NL
also occurs for non-collinear Stokes beams since PS is independent of k P .
d
NS

ES  i
[
dz
8m nS  0cD* (P  S ) q
N S
d

 I S ( z) 
[
dz
m nS nP c 2 02 q
2 ( 3)
2
q  0 ]  | E P | ES ;
q 0 ]

2 (3)
d
NP

EP  i
[
dz
8m nP 0cD (P  S ) q
2 ( 3)
q 0 ] 
| ES |2 EP
 v-1( P   S )
I P ( z)I S ( z) 2
( v  [ P   S ]2 ) 2  4 v-2 [ P   S ]2
d
Optical loss added
I S ( z )  g R I S ( z ) I P ( z )   S I S ( z ) phenomenogically
dz
N S
( P   S ) v1

2 (3)
gR 
[
] 
(Raman Gain coefficient)
2 2 q q 0
2
2 2
2
2
m nS n P c  0
( v  [ P   S ] )  4 v [ P   S ]

P
P


 I S ( L)  I S (0)e[ g R I P (0) S ] z → For gRI(p )>S, exponential growth of Stokes
I ( z ) I ( 0)
Phase of Raman signal independent of laser phase,
i.e.  g R | E P |2 ! But if temporal coherence of
laser is very bad, P may be larger than  v-1 →
must average over P to get net gain
1
E P ei ( k P z  P t )  c.c. can also have gain for Stimulated Stokes in the backward
2
direction! Get the same g R but boundary conditions at
1
z=0, L different!
E S ei (  k S z  S t )  c.c.
2
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
Recall
d
N P

2 (3)
EP  i
[
| E S |2 E P
q 0 ] 
dz
8m nP c 0 D( P  S ) q
d
P
 I P ( z)   g R
I S ( z)I P ( z)   P I P ( z) 
dz
S
1 d
1 d
I P ( z)  
I S ( z)
P dz
S dz
Optimum conversion: I P ( L)  0 and I S (0)  0

I P (0)
P

I S ( L)
S
When I S (z ) grows by one photon, I P (z ) decreases by one photon and
 ( P   S ) of energy is lost to the vibrational mode, and eventually heat
Raman Amplification – Attenuation, Saturation, Pump Depletion, Threshold
No pump depletion (small signal gain) but with attenuation loss
d
d
I P   P I P  I P ( z )  I P (0)e  P z  I S ( z )  g R I S ( z ) I P (0)e  P z   S I S ( z )
dz
dz
1  exp(  P L)
 I S ( z )  I S (0)e g R I P ( 0) Leff  S L with Leff 
P
Define unsaturated (no pump depletion) amplifier gain as G A  exp[ g R I P (0) Leff ]
Assume P = S =  (reasonable approximation)
Saturation in amplifier gain occurs due to pump
depletion.
d
I S ( z )  g R I S ( z ) I P ( z )  I S ( z )
dz
d

I P ( z )   g R P I S ( z ) I P ( z )  I P ( z )
dz
S
Saturated Gain :  GS 
with r0 
(1  r0 )e L
r0  G A(1 r0 )
 P I S ( 0)
(input condition)
 S I P ( 0)
Note that the higher the input
power, the faster the saturation
occurs, as expected.
Starting from noise, the Stokes seed intensity ( I Seff (0) ) 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:
1 / 2
I Seff (0)  Aeff  S
2
2gR
I P (0) Leff  2
 S
Aeff is the effective nonlinear core area
The stimulated Raman “threshold” pump intensity I Pth (0) is defined approximately as the input
pump intensity for which the output pump intensity equals the Stokes output intensity, i.e.
I S ( L)  I Seff (0) exp[ g R I P (0) Leff ]  I P ( L)  I Pth (0)e  P L
For backwards propagating Stokes
 g R I Pth (0) Leff  16
I Pth (0)e  P L  I S (0)
 I Seff ( L) exp[ g R I P (0) Leff ]  I P ( L)
where I Seff ( L)  Aeff  S

g R I Pth (0) Leff
 20
2
 gR
2
I P (0) Leff  2
1 / 2
 S
This threshold is higher than for forward
propagating Stokes. Therefore, forward
propagating Stokes goes stimulated first and
typically grows so fast that it depletes the pump
so that that backwards Stokes never really grows
glass
Raman Amplification – Pulse Walk-off
Stokes and pump beams propagate with different
group velocities vg (S) and vg(P). The
interaction efficiency is greatly reduced when
walk-off time  pump pulse width t. As a result
the Stokes signal spreads in time and space
For backward propagating Stokes, the pulse
overlap is small and the amplification is weak.
Raman Laser
[ g Rmax I P  S ] L
Threshold condition: Re
g Rmax 
N S
4m nS nP c 2 02 v1v
[

q
q 0 ]
1

2 (3)
Frequently fibers used for gain. Why? Example silica has a small gR but also an ultra-low loss
allowing long growth distances. For L10m, PPth=1W 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
2
2
To this point we have focused on terms like | E P | E S and | E S | E P which corresponded to
 S   P  v . What about  P  v , i.e. Anti-Stokes generation? This requires tracking the
optical 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
Ωv
P
Ωv
A
Coherent Anti-Stokes Generation
 
 

1
i ( K r   t )
* 
 i ( K r   t )
Again we write : q  {Q ( K , )e
 Q ( K , )e
}
2
E P E*S
E*P E S
1 
i ( P  S )t
i ( S  P )t
(1)
(1)


(

)

(

)
{
e

e
}
q 0
P
S
*
4m q
D( P  S )
D ( P  S )


*
Q ( K , )
Q ( K , )
 
1
i
(
k
 S   P  v  Stimulated Stokes;  A   P  v Anti-Stokes
E A e A r  At )  c.c.
2
1 d
N 
(1)
(1)
*
I S  i
q  0  ( P )  (S )Q E P E S  c.c.
S dz
8 q
1 d
N 
(1)
(1)
*
(1)
* *
 ikz
IP  i
]  c.c.
q  0  ( P )[  ( S )Q E P E S   ( A )Q E P E Ae
 P dz
8 q

1 d
N 
(1)
(1)
* *
ikz
I A  i

(

)

(

)
Q
E
E
e
 c.c
q 0
P
A
P A
A dz
8 q

- k  0  dispersion in refractive index means the waves are not collinear
for the Anti-Stokes case, similar to the CARS case discussed previously
-Thus Anti-Stokes process requires phase-matching (not automatic)

1 d
1 d
1 d
I S ( z) 
I A ( z)  
I P ( z)
S dz
 A dz
 P dz
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 destroyed
What is missing in the conservation of energy is the flow of mechanical energy Emech (t) into the
optical phonon modes via the nonlinear mixing interaction, and its subsequent decay (into heat).
Detailed analysis 
1 d
1 d
1 d
{ E mech  2 v-1 E mech } 
IS 
IA
 dt
 S dz
 A dz
Vibrational energy grows with the Stokes energy, and
decreases with the creation of Anti-Stokes and by
decay into heat. If Stokes strong  2nd Stokes
 3rd Stokes etc.
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).
The generation of
Anti-Stokes lags
behind 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.
Decays to thermal
“bath”, i.e. heat
Stimulated Brillouin
“Noise” fluctuations
in optical fields and
sound wave fields
Brillouin scattered light
Optical phonon (sound
wave) excited
Brillouin Amplification
Stokes signal injected.
Grow in opposite
directions but still
“drive” each other
Decays to thermal
“bath”, i.e. heat
Light waves
 
1
E (r , t )  eˆ[E P ei ( k P z Pt )  E S e i ( kS z S t )  E Ae i ( k A z  At )  c.c.]
2
Freely propagating sound waves
 1  i ( Kz  S t )  * i ( Kz St )  i (  Kz St )  * i ( Kz St )
q  [Q e
 Q e
 Q e
 Q e
]
2
Forward travelling
Backwards travelling
vSound 
S

( 103 m / s )  c  ( 108 m / s ) 
K
k
 / k  S / K
For k  K   S and need kK for measurable S, since S0 as K 0
For Stokes need k P  K  k S and S  P  S
* e i ( Kz St )
 interactio n via E P ei (k P z  P t )Q
Backwards Stokes couples to
forwards travelling phonons
To get stimulated scattering, light and sound waves must be collinear → Backscattering → K 2k
P  S   S → phonon wave picks up energy and grows
along +z.  Stokes can grow along -z
For Anti- Stokes need k P  K  k A and
A  P  S
 interactio n with E P ei (k P z  P t )Q-ei ( Kz st )
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.  kBT, a very small number! (Stokes process generates
sound waves in opposite direction.) Anti-Stokes NOT stimulated!
Comparison between Stimulated Raman and Stimulated Brillouin
Stimulated Raman
Stimulated Brillouin
1. Molecular property
1. Acousto-optics uses bulk properties
 Local field corrections
 NO local field corrections
2. Normal modes do NOT propagate.
2. Acoustic waves propagate.
3. Normal mode frequency S  K
3. Normal mode frequency is fixed at v
4. Backward Scattering only
4. Both forwards and backwards scattering
Equation of Motion for Sound Waves
Light-sound coupling
gas/liquid : Pi
NL
Mass density
 
 



2
2 2 AO
(r , t )   0 (n  1)[  q ]Ei (r , t ) solid :  0 ni n j piijj [  q ] jj Ei (r , t )


2



2  

 q  2 Sq  vS 2 q  Fq
z
Acoustic damping constant
Sound velocity
Force due to mixing of light beams
 S  decay time of sound wave
Gas or Liquid
0
0  0
 0 0
         
vs
Only compressional wave (longitudinal acoustic
phonon) couples to backscattering of light
Substituting into driven wave equation  

qz  2 Sq z  vS2
2
z
2
q z  Fz for qz
1
d
SVEA  [(( P   S ) 2 Q  S2Q )  2i ( P   S )  SQ  2iK Q  c.c.]  Fq
2
dz
Vint
    k  k  K ,  
1
1
1
 * 
2
2
S
P
S
P
s
  0 (n  1)[  q ]E  E         0 (n  1) q z (E S .E P )e i[P S ]t
2
2
2 z
Fq  

1
d
Vint  [(( P   S ) 2 Q  S2Q )  2i ( P   S )  SQ  2iK Q  c.c.]
q z
2
dz
1
 * 
2
  0 (n  1) (E S .E P )e i[P S ]t  c.c.
4
z
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. dQ / dz  0.
2

(
n
 1)
1
*
 Q* ( z )  i 0
[
K
E
( z )E S ( z )]
P
2
2
2  ( P  S )  S  2i( P  S ) S
Mixing of optical beams drives the sound waves
Power Flow (Manley Rowe)
Acoustic phonons modulate pump beam to produce Stokes.
 NL 
  
 
 
 
2
Recall P (r , t )   0 (n  1)  qE (r , t ) E (r , t )  ES (r , t )  EP (r , t )
1
 PSNL  i 0 (n 2  1) KQ* E P
2
Q

  
  q  q z  iKq
z
Q*
i ( P S )t iS t
 e  i P t
Note that for PPNL, Q+ is linked to ES with Q E S  e

PPNL
1
  i 0 ( n 2  1) KQ E S
2
 S (n 2  1) KQ*
d
SVEA  
dz
ES ( z)  
4 nS c
E P ( z );
S propagates along –z
d
 P (n 2  1) K
E P (z) 
[Q E S (z)]
dz
4nP c
 S (n 2  1) K 0 *
d
  I S ( z)  
{Q E P ( z )E *S ( z )  c.c.}
dz
8
 P (n 2  1) K 0 *
d
I P ( z) 
{Q E P ( z )E *S ( z )  c.c.}
dz
8

P travels along +z
Travels and grows along -z
Travels and depletes along +z
1 d
1
d
I P ( z) 
I S ( z ) Pump beam supplies energy for the Stokes beam!
P dz
S d (  z )
Phonon Energy Flow (need acoustic SVEA)
S
 0 (n 2  1)
d
Acoustic SVEA 
Q ( z )  Q ( z )  
(E P E S* )
dz
2
4 vS2
Decay of sound waves “heats up” the lattice
S  2
S
vS
Mixing of optical beams drives sound waves
S 0 (n 2  1) K *
d
 I  (S , z )   S I  (S , z )  
{Q E P ( z )E S* (z)  c.c.}
dz
8

1 d
1 d
1 d
[ I  (S , z )   s I  (S , z )]  
I P ( z)  
I S ( z)
 dz
 P dz
 S dz
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
When the growth of the acoustic phonons is limited by their attenuation constant.
 0 (n 2  1)
1
*
Recall : Q ( z )  i
K
E
(
z
)
E
( z)
P
S
2
2
2
( P   S )  S  2i ( P   S ) S
Signature of exponential growth
 S (n 2  1) 2 S
 S2
d
 I S ( z)  
I S ( z ) I P ( z )  -g B I S ( z ) I P ( z )
2
2
2
2
2
dz
4 vS Sc n ( P   S  S )   S
 S (n 2  1) 2 S
 S2
 S (n 2  1) 2 S
max
 gB 
 gB 
2
2 2
2
2
4 vS Sc n ( P   S  S )   S
4 vS2 Sc 2 n 2
This leads to exponential growth of Stokes along -z!!
1 d
1 d
Also, from 
I P ( z)  
I S ( z) 
 P dz
 S dz
P
d
I P ( z )  -g B
I S ( z)I P ( z)
dz
S
The energy associated with [ P   S ] , i.e. the sound waves, eventually goes into heat.
What is happening to acoustic phonons ?
 0 (n 2  1)
d
d
S
 0 ( n 2  1)
*
*
 Q ( z )  0  Q ( z )  
(
E
E
)
Q ( z )  Q ( z )  i
(
iK
E
E
)
P
S
P
S
2
2
dz
dz
2
2 S v S
4 Kv S
2
2
 (n  1) S
d
d
Substituting this Q into I S ( z )  I S ( z )  S 2 2
I S ( z ) I P ( z ), i.e. g Bmax
dz
dz
4 vS Sc nL nS
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
Pump signal decays exponentially in
the forward direction as the Stokes
grows exponentially in the backward
direction
For amplifying a signal IS(L) inserted
at z=L, the growth of the signal is shown
for different signal intensities relative
to the pump intensity.
Relative Intensity
1.0
0.8
g B I P (0) L  10
0.6
I S ( L) / I P (0)  0.01
0.4
0.2
Assume an isotropic solid – the pertinent
elasto-optic coefficient is p12 so that
 NL 
  
 
4
P (r , t )   0 n p12 [  q (r , t )]E (r , t )
 0
Pump
0.0
Stokes
0.2
0.4
I S ( L) / I P (0)  0.001
0.6
0.8
Distance z/L
(typically 1  p12  0.1).
2
 S n 6 p12
S
 S2
 gB 
4 vS2 Sc 2 ( P   S  S ) 2   S2
g Bmax
2
 S n 6 p12
S

4 vS2 Sc 2
Can add loss phenomenologically
d
I S ( z )  -g B I S ( z ) I P ( z )   S I S ( z )
dz
P
d
I P ( z )  -g B
I S ( z)I P ( z)   P I P ( z)
dz
S
1.0
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
d
I S ( z )  g B I S ( z ) I P ( z )  I S ( z )
dz
 I P ( z )  I P (0)e Leff

d
I P ( z )  I P ( z )
dz
I S ( 0)  I S ( L ) e
g B I P ( 0) Leff L
Leff 
1  exp( P L)
P
Signal output
Brillouin threshold pump intensity defined as I Pth (0) for which I P (0)e  P L  I S (0)
with unsaturated gain & with the Lorentzian line-shape for gB:
g B I Pth (0) Leff  21
To solve analytically for saturation which occurs in the presence of pump depletion, must
assume =0, P  S and define GA  g B I P (0) L (unsaturat ed gain)
I S (0) e[(1b0 ) g B I P (0) L ]  b0
I ( 0)
saturated gain : G S 

with b0  S
,
I S ( L)
1  b0
I P ( 0)
Plot of gain saturation after a propagation distance
L versus the normalized unsaturated gain GA.
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, vS=6km/s,
S /2= 11GHz, 1/S   17 MHz → gB  5x10-11 m/W.
This value is 500x larger the gR! 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
vg(P)
tP
vg(S)
tS
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.