stimulated scattering is a fascinating process which requires a strong coupling between light and...
TRANSCRIPT
Stimulated scattering is a fascinating process which requires a strong coupling between light andvibrational and rotational modes, concentrations of different species, spin, sound waves and ingeneral any property which can undergo fluctuations in its population and couples to light. Theoutput light is shifted down in frequency from the pump beam and the interaction leads to growthof 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 inducedfluctuations 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 bythe 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 theclassical 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 wavegrow together. If the generation rate of Stokes light exceeds the loss, stimulated emission occursand the Stokes beam grows exponentially.
It is the product of optical fields which excites coherently thephonon modes. Since the “noise” requires a quantummechanical treatment here we consider only the classicalsteady state case, i.e. both the pump and Stokes are classical fields, i.e. it is assumed that both fields are present.
....ˆ2
1),( )()( ccecceetrE trki
Strki
PTSSPP
EEPump (laser) field Stokes field, v PS
0 r lity tensopolarizabi
nqn
ijnn
Lijij q
qα
for eal is 220 pmgq
n
iinnq
)()()()( ;)()()()( )1()1(0
)1()1(0 SSPqP
NLPPSqS
NL Eq
qpEq
qp
})(
EE
)(
EE{ )()(
1 )(*
*)(
*
S)1()1(
0ti
SP
SPti
SP
SPPq
n
nn
PSSPn
eD
eDqm
q
)()(S
)1()1(0 2
1)()( trki
Ptrki
SPqn
nn
NL PPSSn
eeq
qp EE
PEdrives SEdrives
)(22S
)1()1(20
)(
)(22S
)1()1(20*
)(
||)]()([][)(4
||)]()([][)(4
tzkiPSPq
n
n
SP
tzkiNLP
tzkiSPPq
n
n
SP
tzkiNLS
PPn
PP
SSn
SS
eqDm
Ne
eqDm
Ne
EEP
EEP
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
also occurs for non-collinear Stokes beams since is independent of .
NLP
NLS PP and
NLSP Pk
PSqSPP
PPSPq
SPS
SS qcDnm
Ni
dz
d
qcDnm
Ni
dz
dEEEEEE 2)3(2
00
2)3(20*
0
||][)(8
;||][)(8
22-v
222v
-1v)3(2
020
2 ][4)][(
)()()(][)(
SPSP
SPSPq
PS
SS zIzI
qcnnm
NzI
dz
d
)()()()( zIzIzIgzIdz
dSSPSRS Optical loss added
phenomenogically
t)coefficienGain (Raman ][4)][(
)(][
22v
222v
1v)3(2
020
2SPSP
SPq
PS
SR qcnnm
Ng
zIgSS
IzI SPRPP eILI ])0([)0()( )0()( → For gRI(p )>S, exponential 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| PRg-1v
..2
1 )( cce tzkiP
PP E
..2
1 )( cce tzkiS
SS E
can also have gain for Stimulated Stokes in the backward direction! Get the same but boundary conditions at z=0, L different!
Rg
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
)()()()( zIzIzIgzIdz
dPPPS
S
PRP
PSqSPP
PP qDcnm
Ni
dz
dEEE 2)3(2
00
||][)(8
Recall
)(1
)(1
zIdz
dzI
dz
dS
SP
P
Optimum conversion: 0)0( and 0)( SP ILI
S
S
P
P LII
)()0(
When grows by one photon, decreases by one photon and of energy is lost to the vibrational mode, and eventually heat
)(zI S )(zIP)( SP
No pump depletion (small signal gain) but with attenuation loss
])0(exp[ as gainamplifier depletion) pump (no dunsaturate Define
)exp(1 with)0()(
eff
eff)0( eff
LIgG
LLeIzI
PRA
P
PLLIgSS
SPR
)()0()()( )0()( zIeIzIgzIdz
deIzIII
dz
dSS
zPSRS
zPPPPP
PP
Raman Amplification – Attenuation, Saturation, Pump Depletion, Threshold
Saturation in amplifier gain occurs due to pumpdepletion.
)()()()(
)()()()(
zIzIzIgzIdz
d
zIzIzIgzIdz
d
PPSS
PRP
SPSRS
Assume P = S = (reasonable approximation)
condition)(input )0(
)0( with
)1( :Gain Saturated
0
)1(0
00
P
S
S
P
rA
L
S
I
Ir
Gr
erG
Note that the higher the inputpower, the faster the saturationoccurs, 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:
LthPPPRSS
PeILILIgILI (0))(])0(exp[)0()( effeff
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(thPI
2/1
2
2
eff
eff
)0(
2)0(
S
R
PSeffS
g
LIAI
16)0( effth LIg PR
Aeff is the effective nonlinear core area
glass
)0()0(thS
LP IeI P For backwards propagating Stokes
2/1
2
2
eff
eff
effeff
)0(
2)( where
)(])0(exp[)(
S
R
PSeffS
PPRS
g
LIALI
LILIgLI
This threshold is higher than for forwardpropagating Stokes. Therefore, forward propagating Stokes goes stimulated first andtypically grows so fast that it depletes the pumpso that that backwards Stokes never really grows
20)0( effth LIg PR
)0(effSI
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 resultthe Stokes signal spreads in time and space
For backward propagating Stokes, the pulseoverlap is small and the amplification is weak.
Raman Laser
Threshold condition: 1Re ][ max
LIg SPR
)3(20
v1
v20
2max ][
4
q
PS
SR qcnnm
Ng
Frequently fibers used for gain. Why? Example silica has a small gR but also an ultra-low loss
allowing long growth distances. For L10m, PPth=1W for lasing.
Multiple Stokes and Anti-Stokes Generation
Fused silica fiber excitedwith 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
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.
PSSP E|E| and E|E| 22
.v PS vP
vΩ
SP
vΩ
P A
Coherent Anti-Stokes Generation
})(
EE
)(
EE{)()(
4
1
}),(),({2
1: writeAgain we
)(*
*)(
*
S)1()1(
0
) (*) (
ti
SP
SPti
SP
SPPq
trKitrKi
PSSP eD
eDqm
eKQeKQq
),( KQ
),(* KQ
v PS Stimulated Stokes; Anti-Stokes ..2
1 )( cce trkiA
AA Ev PA
..)()(8
1 *)1()1(0 ccQ
q
NiI
dz
dSPSPqS
S
EE
.)()(8
1
..])()()[(8
1
**)1()1(0
A
**)1(*)1()1(0
cceQq
NiI
dz
d
cceQQq
NiI
dz
d
kziAPAPqA
kziAPASPSPqP
P
EE
EEEE
- 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)
0k
)(1
)(1
)(1
zIdz
dzI
dz
dzI
dz
dP
PA
AS
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 phononis also created, and for every Anti-Stokes photon created an optical phonon is destroyedWhat 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).
Vibrational energy grows with the Stokes energy, anddecreases with the creation of Anti-Stokes and bydecay into heat. If Stokes strong 2nd Stokes 3rd Stokes etc.
AA
SS
Idz
dI
dz
d
dt
d
11
}E2E{1
analysis Detailed mech1-
vmech
Anti-Stokes is not automatically wavevector matched! SinceStokes is generated in all directions, Anti-Stokes generation“eats out” a cone in the Stokes generation (angles small).
The generation of Anti-Stokes lagsbehind the Stokes
Stimulated Brillouin ScatteringThe normal modes involved are acoustic phonons. In contrast to optical phonons, acoustic waves travel at the velocity of sound.
Light waves
.].[̂2
1),( )()()( cceeeetrE tzki
Atzki
Stzki
PAASSPP EEE
Freely propagating sound waves
][2
1 )(*)()(*)( SSS tKzitKzitKzitKzi eQeQeQeQq S
Forward travelling Backwards travelling
Stimulated Brillouin“Noise” fluctuationsin optical fields andsound 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 AmplificationStokes signal injected.
Kksmk
smK
// )/10( c )/10( v S83S
Sound
and For S Kk need kK for measurable S, since S0 as K 0
Backwards Stokes couples toforwards travelling phonons
For Stokes need S and PSSP kKk )(*)( n viainteractio StKzieQtzkieP PP
E
To get stimulated scattering, light and sound waves must be collinear → Backscattering → K 2k
→ phonon wave picks up energy and grows
along +z. Stokes can grow along -zSSP
For Anti- Stokes need SA and PAP kKk )()(n with interactio stKzie-QtzkieP PP E
Backwards Anti-Stokes couplesto backwards travelling phonons
backwards phonon wave gives up energy and one phonon is lost for everyanti-Stokes photon created. But the only backwards phonons available aredue to “noise”, i.e. kBT, a very small number! (Stokes process generates
sound waves in opposite direction.) Anti-Stokes NOT stimulated!
),(][:solid ),(])[1(),(:gas/liquid 220
20 trEqpnntrEqntrP ijj
AOiijjjii
NLi
Stimulated Raman1. Molecular property Local field corrections2. Normal modes do NOT propagate.
3. Normal mode frequency is fixed at v
4. Both forwards and backwards scattering
Stimulated Brillouin1. Acousto-optics uses bulk properties ® NO local field corrections2. Acoustic waves propagate.
3. Normal mode frequency S K4. Backward Scattering only
Light-sound coupling
Equation of Motion for Sound Waves
Only compressional wave (longitudinal acousticphonon) couples to backscattering of light
qFqz
2
22SS v2 Mass density
Acoustic damping constant Sound velocity
Force due to mixing of light beams
wavesound of decay timeS
vs
0
0
0
0
0
Gas or Liquid
Comparison between Stimulated Raman and Stimulated Brillouin
)E.E(2
1)1(
2
1])[1(
2
1V ][*2
0,2
0intti
PSzKkk SPsPSPS e
zqnEEqn
Substituting into driven wave equation for qz
zzzz Fqz
2
22SS v2
..)E.E()1(4
1
.].2)(2))([(2
1 V
][*20
S2S
2int
ccez
n
ccQdz
diKQiQQ
qF
tiPS
SPSPz
q
SP
qSPSP FccQdz
diKQiQQSVEA .].2)(2))([(
2
1S
2S
2
The damping of acoustic phonons at the frequencies typical of stimulated Brillouin (10’s GHz) frequencies is large with decay lengths less than 100m. This limits (saturates) the growth of the phonons. In this case the phonons are damped as fast as they are created , i.e. .
0/ dzdQ
Mixing of optical beams drives the sound waves
)]()([)(2)(
1
2
)1()( *
S2S
2
20* zzK
i
nizQ SP
SPSP
EE
Acoustic phonons modulate pump beam to produce Stokes.
PNLS KQni E )1(
2
1P *2
0
Power Flow (Manley Rowe)
),()1(),( Recall 20 trEqntrPNL
Note that for , Q+ is linked to ES with
tititiS
PSSP eeQ )(ENL
PP
SNLP KQni E)1(
2
1P 2
0
iKqqz
q z
Q
*Q
),( ),( ),( trEtrEtrE PS
S propagates along –z
)](z[4
)1()(z );(
4
)1()(SVEA
2*2
SP
PPP
S
SS Q
cn
Kn
dz
dz
cn
KQnz
dz
dEEEE
P travels along +z
.}.)()({8
)1()( **0
2
cczzQKn
zIdz
dSP
SS
EE
.}.)()({8
)1()( **0
2
cczzQKn
zIdz
dSP
PP
EE
)()(
1)(
1zI
zd
dzI
dz
dS
SP
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)
)(v4
)1()(
2)( SVEA Acoustic *
2S
20S
SPn
zQzQdz
dEE
S
SS v
2
Mixing of optical beams drives sound wavesDecay of sound waves “heats up” the lattice
.}.)(z)({8
)1(),(),( **
20S
SSS cczQKn
zIzIdz
dSP
EE
)(1
)(1
)],(),([1
SS zIdz
dzI
dz
dzIzI
dz
dS
SP
Ps
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.
)()()(2)(
1
2
)1()(:Recall *
S2S
2
20 zzK
i
nizQ SP
SPSP
EE
)()( )()()(v4
)1()(
2S
2S
2S
22S
2S
S22
zIzI-gzIzInc
nzI
dz
dPSBPS
SP
SS
Signature of exponential growth
22S
2S
S22
max2S
2S
2S
22S
2S
S22
v4
)1(
)(v4
)1(
nc
ng
nc
ng S
BSP
SB
The energy associated with , i.e. the sound waves, eventually goes into heat.][ SP
)()()( )(1
)(1
from Also, zIzI-gzIdz
dzI
dz
dzI
dz
dPS
S
PBPS
SP
P
This leads to exponential growth of Stokes along -z!!
)(v4
)1()(
2)( *
2S
20S
SPiKK
nizQzQ
dz
dEE
)(v2
)1()( 0)( *
2SS
20
SPn
zQzQdz
dEE
What is happening to acoustic phonons ?
max2
S2S
S22
i.e. ,)()(v4
)1()( )( into thisngSubstituti BPS
SL
SSS gzIzI
nnc
nzI
dz
dzI
dz
dQ
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 1.0Distance z/L
Rel
ativ
e In
tens
ity
0.2
0.4
0.6
0.8
1.0
0.0
Pump
Stokes 001.0)0(/)( PS ILI
01.0)0(/)( PS ILI
0
10)0(
LIg PB
For amplifying a signal IS(L) inserted at z=L, the growth of the signal is shown for different signal intensities relativeto the pump intensity.
Pump signal decays exponentially inthe forward direction as the Stokesgrows exponentially in the backwarddirection
Assume an isotropic solid – the pertinent
elasto-optic coefficient is p12 so that
(typically 1 p12 0.1).
),()],([),( 124
0 trEtrqpntrP NL
2S
2S
S212
6max
2S
2S
2S
2S
2S
S212
6
v4
)(v4 c
png
c
png S
BSP
SB
Can add loss phenomenologically
)()()()( )()()()( zIzIzI-gzIdz
dzIzIzI-gzI
dz
dPPPS
S
PBPSSPSBS
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 SP= is an excellent
approximation. For no depletion of pump except for absorption
LLIgSS
LPP
PPSPSBS
PBeLIIeIzI
zIzIdz
dzIzIzIgzI
dz
d
effeff)0(
)()0( )0()(
)()( )()()()(
Signal output
P
P LL
)exp(1
eff
,)0(
)0( with
1
)(
)0( :gain saturated 0
0
0])0()1[( 0
P
SLIgb
S
SS I
Ib
b
be
LI
IG
PB
Brillouin threshold pump intensity defined as
with unsaturated gain & with the Lorentzian line-shape for gB:
)0()0(for which )0(thS
LPP IeII 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( LIgG PBA
21)0( effth LIg PB
Plot of gain saturation after a propagation distanceL 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 = 1.55m, 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 muchsmaller and requires stable single frequency input toutilize the larger gain – hence no advantage to stimulated Brillouin for amplification.
Pulsed Pump Beam
tP
tS
vg(P)
vg(S)
Stokes and pump travel in opposite directions, the overlapwith a growing Stokes is very small and hence theStokes amplification is very small! The shorter the pumppulse, the less Stokes is generated, i.e. this is a very inefficient process! Stimulated Raman dominates forpulses when pulse width << Ln/c.