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Multi-wave Mixing
In this lecture a selection of phenomena based on the mixing of two or more waves to produce anew wave with a different frequency, direction or polarization are discussed. This includesinteractions with non-optical normal modes in matter such as molecular vibrations which underthe appropriate conditions can be excited optically via nonlinear optics.In nonlinear optics “degenerate” means that all the beams are at the same frequency and“non-degenerate” identifies interactions between waves of different frequency. Since theinteractions occur usually between coherent waves, the key issue is wavevector matching. Because there is dispersion in refractive index with frequency, collinear wave-vector matched non-degenerate interactions are not trivial to achieve, especially in bulk media. Of course, thewaves can be non-collinear to achieve wavevector conservation in which case beam overlapreduces interaction efficiency.
Degenerate Four Wave Mixing (D4WM)
Beam Geometry and Nonlinear Polarization
“P1” and “p2” are counter-propagating pump waves, “s” is the input signal, “c” is the conjugate
0cs2p1p kkkk
cs kk
1p2p kk
)()(E)-()(E)()(E)-()(E
)()(E)()(E)()(E)-()(E
2
1)(
(c)*(c)(s)*(s)
*(p2)(p2)*(p1)(p1)
iiii
iiiiiE
321321321321)3(
0)3( )()()()(),,;(ˆ)( dddδEEEP lkjijkli
Each is the total field!Each total field is given by
General case for third order polarization, in frequency domain
Full 4-wave mixing process will contain 8x8x8 terms!But, will need wavevector conservation in interaction → reduces # of terms
1. Assume beams are co-polarized along the x-axis and treat as scalar problem
2. E(p1) and E(p2) (pump beams) are strong, and E(s) and E(c) are weak beams3. For the nonlinear polarization, only products containing both pump beams and either the signal or conjugate beam are important
4. Nonlinear polarization (products of 3 fields) need p1, p2 and c or s
rki*x
(xxxxxxxxxx
xxxxxxxx
xxxxxxxxx
e
s(c))2p)1(p1p2pcs
)3(2p1pcs
)3(
c1p2ps)3(
c2p1ps)3(
1pc2ps)3(
2pc1ps)3(
0s(3)
)},,;(ˆ),,;(ˆ
),,;(ˆ),,;(ˆ
),,;(ˆ),,;(ˆ{4
1)(P e.g.
EEE
).(Pfor resultsSimilar c(3) x
.].}2222
)}{()({}22
22)}{()([{4
1
)((c)*(p2))((s)*(p2))((c)*(p1))((s)*(p1)
21)((c)(p2))((s)(p2)
)((c)(p1))((s)(p1)21
spspspsp
spsp
spsp
ccee
ee
rkkirkkirkkirkki
rkkirkki
rkkirkki
EEEEEEEE
EEEE
EEEE
5. “grating model”: products of two fields create a grating
in space, and a third wave is deflected by the grating to form beams s or c. Initially the only 2
field products of interest are (p1 or p2) x (c or s).
)()( 21)3()1(
effective, xxxxxxxx EE
There are two kinds of time dependences present here correspondingto the first two inputs in 1. oscillates at 2 (requires electronic nonlinearity)2. DC in time
..)()( 21 cc ..)()( 21 cc
),,;(ˆ 321)3( xxxx),( 21
- Now form E(1)E(2) x E(3) subject to the following restrictions
1. Terms can only multiply terms with so that the output frequency is
2. The product of three different beams is required3. Because the pump beams are the “strong” beams, only products with two pump beams are
kept which generate signal or conjugate beams. Assume Kleinman limit.
)()( 21 )( 3
First term generates the conjugate and the second term the signal.
}{),,;(4
6)(P)()()( ss (c)*(s)(p2)(p1))3(
0(3)
321rki
xrki*
xxxxxxxx eeEEE EEEE
}{);(-2)(P ss (c)(s(p2)(p1)||2
20
2(3) rki*x
rki)*xxxx eencn
EEEE
D4WM Field Solutions
),'()()(),'('
),'()()(),'('
(c)*(p2)(p1)20
(s)
(s)*(p2)(p1)20
(c)
znnizdz
d
znnizdz
d
EEEE
EEEE
Using the SVEA in undepletedpump beams approximation
Simplifying in the undepleted pump approximation
)'(1
)'('
)()(1
)'(1
)'('
(c)*
4
(s)
(p2)(p1)20
4
(s)*
4
(c)
zizdz
d
nnzizdz
d
WM
WMWM
EE
EEEE
Applying the boundary conditions: 0),0( 0),'( (s)(c) EE L
Output of conjugate beam:
WM
Li
4
(s)*(c) ' tan),0(),0(
EE
R>1 ? YES! Photons come out of pump beams!Need to include pump depletion.
WM
L
4
22(s)
2(c) 'tan
|),0(|
|),0(|ty"Reflectivi"
E
E
WM
LL
4
22(s)
2(s) 'sec
|),0(|
|),'(|vity"Transmissi"
E
E
Can get gain on both beams!
T and R and n"oscillatio" 2
4
WM
L'
Unphysical, need pump depletion
Manley Rowe Relation
)'((s)* )'((c)*
4
1 )'((s)
')'((s)*
)'((c)* )'((s)*
4
1)'((c)
')'((c)*
zxzWM
izdz
dxz
zxzWM
izdz
dxz
EEEE
EEEE
)'('
)'('
)()( zIdz
dzI
dz
d cs Since signal travels along +z, and the conjugate travelsalong –z, both beams grow together at expense of the pump beams.
Can be shown easily when z z and allowing pump beam depletion
),(),(),(),(
),(),(),(),(
(p1)*(c)(s)20
(p2)
(p2)*(c)(s)20
(p1)
zzznnizdz
d
zzznnizdz
d
EEEE
EEEE
Note that the p1 and s, and the p2 and c beams travel in the same direction
)()( and )()( )c()2p()s()1p( zIdz
dzI
dz
dzI
dz
dzI
dz
d
Pump beam #1 depletes Pump beam #2 depletesSignal beam grows Conjugate beam grows
Wavevector Mismatch What if pump beams are misaligned, i.e. not exactly parallel?
cs2p1p kkkkk
Assume that z and z are essentially coincident
kzi
WM
kzi
WMeziz
dz
deziz
dz
d )(1
)( )(1
)( (c)*
4
(s)(s)*
4
(c) EEEE
Form of solutions, subject to the usual boundary conditions 24
22 4/ with WMk 2
4max
242
2 2)2/(R
]2
[)(cos
)(sinR
WMWM kL
kL
L
Linear Absorption absorption of all 4 beams, no pump depletion approximation to signal and conjugate used
)(2(p2)(p2)2(p1)(p1) )0()( ;(0)e(z)
Lzzez
EEEE
)(2
)()( );(2
)()( (c)(s)*
4
(c)(s)(c)*
4
(s) zzi
zdz
dzz
iz
dz
d
WMWMEEEEEE
224
2
2(p2)(p1)20
4WM
]2/[
)()0( 1
Redefine
WM
L
eLnn
EE
224
2
)]sin(}2
{)cos([
)(sinR
LL
L
WM
)()0(}4
);();({)(4R )2p()1p(
2vac
2||22
||22
vac LIIk
nLk
Complex )3(̂ But is in general a complex quantity, i.e. )3()3( i
index change absorption change
)3(̂
Both the real (n2) and imaginary (2) parts of contribute to D4WM signal)3(̂
Three Wave Mixing
Assume a thin isotropic medium. Co-polarized beams, x-polarized with small angles between input beams
)cos,(sin );cos,sin(),( 2p1p kkkkkk zy z
y
x
)1*(p2)2(p)2*(p2)1(p E][E and E][ELook at terms
rkki
rkki
ecnnz
ecnnz
)2()2(p*2)1(p||2
20
(s2)
)2()1(p*2)2(p||2
20
(s1)
2p1p
1p2p
])[;(-][)(P
])[;(-][)(P
EE
EE
zkkkiss
ss
ssssppppe
ki
zSVEA
)(2)2,1(
2,1
20
)2,1(2,12,11,2
2
(z)
PE
)cos,sin3(2p1p2 ][2
)(
)cos,sin3(1p2p2 ][2
)(
)2(p*2)1(p202s
)2(2s
(s2)
)1(p*2)2(p201s
)2(1s
(s1)
2s21p
1s1p2p
kkknn
eizdz
d
kkknn
eizdz
d
rkkki
rkkki
p
EEE
EEE
Assuming that the beams are much wider in the x-y plane than a wavelength, wavevector isconserved in the x-y plane. For the signal field which must be a solution to the wave equation,
)2
91()sin
2
91( ]sin91[sin9 2222222222 kkkkkkkkk zyz
222,2,1,1,1,2, 4 )
2
91(cos2 kkkkkkkkk szpzpzszpzpzz
).0(])0(2
)sinc;([)(
),0(])0(2
)sinc;([)( :equationsSVEA thegIntegratin
)2p(2)1p(2||2vac
)2s(
)1p(2)2p(2||2vac
)1s(
IILk
LnkLI
IILk
LnkLI
z
z
(s2)(s1) E and E can interact with the pump beams
again to produce more output beams etc.
(p2)(p1) E and E
Called the Raman Nath limit of the interaction
Non-degenerate Wave Mixing
0}|)(|
)()(
|)(|
)()(
|)(|
)()(
|)(|
)()({
1)()()()(
4
444
3
333
2
222
1
1114321
k
kn
k
kn
k
kn
k
kn
ckkkk
In the most general non-degenerate case with frequency inputs, in which and are the pump beams, then for frequency and wavevector conservation,
4321 and ,, 12 4321
A frequent case is one pump beam from which two photons at a time are used to generate twosignal beams at frequencies above and below the pump frequency which, for efficiency, requires
0 )()()(2 ;2 443311431 nnnk
..)(2
1 ; ..);(
2
1 ; ..)(
2
1 )(3
)4()(3
)3()(1
)1( 443311 cceEccerEcceE tzkitzkitzki EEE
Assuming(1) co-polarized beams, (2) the Kerr effect in the non-resonant regime,(3) cross-NLR due to the pump beam only,(4) and a weak signal (3) input,
the signal and idler (conjugate) beamnonlinear polarizations are
The fields are written as:
,);()(
);()(2 );()(2 :Defining
)(2111||2
4,3
14,3vac4,3
111||21vac111||24,3
14,3vac4,3
1
ieInn
nk
InkkInn
nk
ziezizizdz
d ),(),(),( :SVEA theinto ngsubstituti and 3,4*
4,34,34,34,3 EEE
})()]([),,;(~4
3
)(),,;(~4
62)(
])();(2[4,3
*213,4114,3
)3(0
3,414,3114,3)3(
14,3
)(
11vac11||2 zIknkixxxx
xxxxNL
e
Icn
EE
EP
, )(),( )(),( :onssubstituti theUsing 434433
zizi ezBzezBz EE
44344342
2
34334332
2
)]([ ;)]([ Bdz
diBB
dz
dB
dz
diBB
dz
d
.2
)( 4)]([
2
1
2
)( 4343
243
43 gii
zz eBeBB 4,34,34,3 :form theof solutionsFor
The signal and idler grow exponentially for when g is real!!For imaginary g, the solutions are oscillatory
24343 )]([4
0)0,( ,0)0,( conditionsboundary For the 34 EE
.)sinh()0,(),(
,)]sinh(2
)())[cosh(0,(),(
2
)(
34
4
2
)(43
33
43
43
zi
zi
egzg
iz
egzg
izgz
*EE
EE
A number of simplifications can be made which give insights into the conditions for gain.Expanding Δk around for small . The sign of Δk is negative in the normal dispersion region and positivein the anomalous dispersion region. The condition for gain can now be written as
2124311 )()()()(2)( kkkkk
3114
.0]2
1);()(2[]
2
1);()(2[ 2
2111||21vac2
2111||21vac kInkkInk
0)]}()()(2[);({
]);()()(2[
23vac
4
13vac
3
11vac111||2
2111||24vac3vac
43
1
kn
nk
n
nkInk
Inkknn
n
After some tedious manipulations valid for small Δ, the condition for gain becomes
Gain occurs for both signs of the GVD and the nonlinearity provided that the intensity exceeds the threshold value
.||2
1|||);(|)(2 2
2111||21vac kkInk
This means that the cross-phase nonlinear refraction due to the pump beam must exceed theindex detuning from the resonance for gain to occur.
Nonlinear Raman Spectroscopy
Usually refers to the nonlinear optical excitation of vibrational or rotational modes. A minimum of two unique input beams are mixed together to produce the normal mode at the sum or difference frequency. Although any Raman-active mode will work, vibrational modes typically are very active in modulating the polarizability.
Note: Must include dissipative loss of the normal modes in Manley-Rowe relations
a
a
Degenerate Two Photon Vibrational Resonance
Optical coupling between two vibrational levels (inside the vibrationalmanifold of the electronic ground state)
0 r lity tensopolarizabi
kqk
ijkk
Lijij q
qα
Vibrational amplitude
e.g. the symmetric breathing mode in a methane
molecule (CH4)
CH
H
H
H CH
H
H
H
..)(2
1),( )( ccetrE taraki
ainc
E
)()()(sdiscussion previous From )1()1(0 aaNLq
NL Eq
qp
EEqm
qqq aq
2)1(
02v
1v )]([
2
12 classical mechanics
)2(0
2)(2
0
2.].
)0(4
|)(|
)2(4
)([
2
1),(
ccqDm
eqDm
trq qatrki
qa
a aa
EE
Note: The molecular vibrations are not only drivenin time due to the field mixing, but also into aspatial pattern for the first term
qz
/ka
Real part gives 2 photon absorption; Imaginary part gives index change
21-v
2v
)3(2
0220
2
1-v
)()2(
)()(
32)(resonancenear 2PA For
a
aaq
aa
I
qcmn
N
dz
d EE
)3(2
01-v
220
2max||22
||216
);( )();()(
qa
aaaaaa qcmn
NII
dz
d
)()()()2(
2
32)( :);( associatedFor
21-v
2v
v)3(2
0220
2||2 aaa
aq
aa I
qcmn
Ni
dz
dn
EE
21-v
2v
v)3(2
020
2||2
2vac
)()2(
2
32);(
)()();()(
a
aq
aaaa
aaaaa
qcmn
Nn
I-nikdz
dEE
0);( 2 0);( 2 ||2v||2v nn aa
1-v
22v
2)3(
2
0NL
44
)(|)(|
8)(absorptionphoton woresonant tFor
aa
aaqa
iqm
N
EE
P
D(2a))2()]([ 2 )1(3)1()3(
aaaNL fff
21-v
2v
1-vv
2)3(
2
00 )()2(
2
4
)(|)(|
16)( SVEA
a
a
a
aaq
a
aa
i
qcmn
Ni
dz
d EEE
II
field strong - )(E
field strong - )(E
a
b
New fields generated at 2a-b and 2b-a
(only one can be phase-matched at a time)
CARS – Coherent Anti-Stokes Raman Spectroscopy
Nonlinear Raman Spectroscopy
Nonlinear process drives the vibration at the difference frequency a-b between input fields
..)(..)(
2
1 )()(inc ccecceE tbrbki
btaraki
a
EE
I
field weak polarized- )(E
field strong polarized- )(E
jy
ix
b
a
Makes medium birefringent for beam “b” and changestransmission of medium “b”
RIKES – Raman Induced Kerr Effect Spectroscopy ( Raman Induced Birefringence)
RIKES jbNLnqn
ijnNLijq
n
ijnn
Liji Eq
qNPE
qqp
nn)()( )1()1(
0,loc0
)()()()(1
2:mechanics classical )1()1(0
2v
1v bjaiabq
n
ijnnnn EE
qmqqq
n
..)(
)()()()(
4
1),( ])()[(
*
*)1()1(
0 cceDqm
trq trkki
ba
bjaibaq
n
ijnn
ababn
EE
cceDq
xqm
NtrP trki
ba
bjaiq
n
ijnq
n
jinNLj
bbabNL
bbnn
.)(
)(|)(|
8),(
is at on polarizatinonlinear the,)]()([ define , Since
][*
2)3(
0
*
0
2)1()1()3(
EE
1. Index change produced at frequency b by beam of frequency a
2. Nonlinear gain or loss induced in beam “b” by beam “a”3. One photon from beam “a” breaks up into a “b” photon and an optical phonon4. Propagation direction of beam “a” is arbitrary, only polarization important!
)()()(])[(
][][
8)(
21-v
2v
1-vv)3(
00v
220
bjaiba
baq
n
ijnq
n
jin
ab
bbj I
i
qqcmnn
Ni
dz
d
SVEA
nn
EE
Imaginary part contribution → to nonlinear refractive index coefficient
21-v
2v
v)3(00
v20
2
b2
)(])[(
][][
8);(
)()();(-)(
ba
baq
n
ijnq
n
jin
abab
bjaiab
bj
nn qqcmnn
Nn
Inc
idz
dEE
21-v
2v
)3(20
v22
0
1-v
2
21-v
2v
)3(00
v22
0
-1v
)(])[(
1][
4);(
)(])[(
)()(][
8)(
baq
n
jin
ab
bab
ba
bjaiq
n
ijnq
n
jin
ab
bbj
n
nn
qcmnn
N
I
qqcmnn
N
dz
d EE
Real part → contribution to nonlinear gain (or loss)
LII
ILI
LIeI
LI
aabb
bb
aabLI
b
b aab
)();()0,(
)0,(),(
)();(1)0,(
),(Solving
2
2)();(2
\Modulating the intensity of beam “a” modulates the transmission of beam “b”. Varying a - b
through gives a resonance in the transmission! Assumed was a crystal. If medium is random, need to work in both lab and molecule frames of reference and then average over all orientations.
v
tiebrabrQabr ab )2(),E(),()2,(NLPfor Looking
CARS
For simplicity, assume two input co-polarized beams, b>a.
Coherent Anti-Stokes Raman Scattering (Spectroscopy)
rki
ab
aibiabbaq
n
iin
c
cabi
ab
aibiq
n
iinab
NLi
eDqmcn
Ni
dz
d
Dqm
N
n
n
)(
)()()2()]()[(||
)(8)2(
)(
)()(||
4)2(
*2)1(2)1()1(2
00
*2)3(2
0
EEE
EEP
Field at is written as abc 2 ..)(2
1 ])([ cce tczckici E
This process requires wavevector matching to be efficient,
)( bk
)( ak
)( ck
)( bk
Cannot get collinear wavevector matchingbecause of index dispersion in the visible
cab kkkk
2
since c >> b - a then angles are small relative to z-axis
)(2)()( 2 c babca kkknnn
21-v
2v
22
2)3(402
v24
042
222
)()]([
)()()2
(sinc][||
64 ),(
ab
ab
qn
iin
abc
cc
IIL
k
qmcnnn
LNLI
n
Differences between RIKES and CARS
Automatically wave-vector matched inisotropic medium. No new wave appears.
Have resonance at ||v ba
Requireswave-vector matching
Can also have CSRS (different wavevector matching conditions)Signal appears at )2( ba
When is tuned through , resonantenhancement in the signal occurs. Monolayersensitivity has been demonstrated. There is also a
)( ab
“background” contribution due to electronic
transitions via:
v
),,];2[()3(bababxxxx
For comparable contributions ofbackground and resonance terms