tates model, although exact, requires a detailed knowledge of many p generally available. Experience has shown that the simplest model fo ipole moments requires a minimum of two states, the ground state and o In fact, such a two level model has been effective in predicting no r molecules optimized for large second order nonlinearities. ons associated the electronic states of symmetric molecules (without have either even (gerade) or odd (ungerade) symmetry, with the groun n symmetry. In the electric dipole transition approximation, one pho imum of one odd symmetry excited state and two photon absorption req even symmetry excited state and one odd symmetry excited state. Hen mmetric molecules is three levels. states for the sum over states calculation in the two level model a Molecular Nonlinear Optics: Two Level Model for }. ) ˆ )( ˆ ( ) ( ) ˆ )( ˆ ( ) ( ) ˆ )( ˆ ( ) ( { ) , ]; [ ( ˆ 10 * 10 10 , 00 11 01 * 10 * 10 10 , 00 11 01 10 10 10 00, 11 01 ) 2 ( 2 0 ) 2 ( p q ,j i ,i ,k q p q ,i j ,j ,k p q p ,j k ,k ,i q p q p ijk μ μ μ μ μ μ μ μ μ μ μ μ N ) 2 ( ˆ ijk
Transcript
The Sum Over States model, although exact, requires a detailed knowledge of many parameterswhich are not generally available. Experience has shown that the simplest model for moleculeswith permanent dipole moments requires a minimum of two states, the ground state and oneexcited state. In fact, such a two level model has been effective in predicting nonlinearities incharge transfer molecules optimized for large second order nonlinearities.The wavefunctions associated the electronic states of symmetric molecules (without permanentdipole moments) have either even (gerade) or odd (ungerade) symmetry, with the ground statesexhibiting even symmetry. In the electric dipole transition approximation, one photon absorptionrequires a minimum of one odd symmetry excited state and two photon absorption requires aminimum of one even symmetry excited state and one odd symmetry excited state. Hence theminimum for symmetric molecules is three levels.The relevant states for the sum over states calculation in the two level model are
}.)ˆ)(ˆ(
)(
)ˆ)(ˆ(
)(
)ˆ)(ˆ(
)({),];[(ˆ
10*
10
10,001101*
10*
10
10,001101
1010
1000,1101)2(2
0
)2(
pq
,ji,i,k
qpq
,ij,j,k
pqp
,jk,k,iqpqpijk
μμμμμμμμ
μμμμN
Molecular Nonlinear Optics: Two Level Model for )2(ˆijk
Two Level Model: Examples of Second Order Susceptibilities
e.g. In PPLN the dominant second order nonlinearity lies along the z-axis. The generation of new frequencies is normally done away from the resonances and involves the real part of (2):
.)4)((
3)(||),;2(
}))((
1
)2)((
1
))(2(
1){(||),;2(ˆ
2210
2210
210
00112
10)2(
20
(2),
1010101010100011
210
)2(2
0
(2),
N
N
zzz
zzz
The dispersion with frequency in NLSHO model is proportional towhich is different from the SOS result.
12210
22210 )]4()([ -
e.g. Sum frequency generation in Periodically Poled Lithium Niobate
}))((
1
))((
1
))((
1
))((
1
))((
1
))((
1{
)(||),];[(),];[(
2101102101101210210
121011021012101101210
00112
0120
)2(2112
)2(,1212
)2(,
μμμN
zzzzzz
.]][][)([
)(E)(]E)(3[)(||)(P
22
210
21
210
221
210
21212
212
102
100011
201
)2(221
)2(
zz
z μμμN
)(E)(E)],];[(),];[([2
1)(P 211212
)2(,2112
)2(,021
)2( zzzzzzzzz
Two Level Model: Third Order Susceptibilities)3(ˆ ijk
}.)ˆ)(ˆ)(ˆ()ˆ)(ˆ)(ˆ(
)ˆ)(ˆ)(ˆ()ˆ)(ˆ)(ˆ(
{})ˆ)(ˆ)(ˆ(
))((
)ˆ)(ˆ)(ˆ(
))((
)ˆ)(ˆ)(ˆ(
))((
)ˆ)(ˆ)(ˆ(
))(({ ),,];[(ˆ
*10
*10
*10
10011001
101010
10011001
1010*
10
10011001
101010
10011001
)3(3
0*
10*
10*
10
,10,00,11,00,11,01
1010*
10
,10,00,11,00,11,01
10*
10*
10
,10,00,11,00,11,01
101010
,10,00,11,00,11,01)3(3
0
)3(
rqppr
,k,j,i,l
qp*
r*
,k,j,i,l
prq
,j,k,l,i
prrqp
,j,k,l ,i
rqppqp
illkkj
ppqr
jkkiil
rpqp
liikkj
ppqrqp
jkkllirqprqpijkl
μ μ μ μ
ωωωωωω
μ μ μ μ
μ μ μ μμ μμ μ
Nμμμμμμ
μμμμμμμμμμμμ
μμμμμμN
e.g. Third Harmonic Generation (z-polarized input)
}.)3ˆ)(ˆ)(ˆ(
||
ˆˆˆ
||
)ˆ)(ˆ)(ˆ(
||
)ˆ)(ˆ)(3ˆ(
||{
})3ˆ)(2ˆ)(ˆ(
||)(
)ˆ)(2ˆ)(ˆ(
||)(
)ˆ)(2ˆ)(ˆ(
||)(
)ˆ)(2ˆ)(3ˆ(
||)({),,;3(ˆ
*10
*10
*10
410
101010
410
1010*
10
410
101010
410)3(
30
*10
*10
*10
2,10
2,00,11
1010*
10
2,10
2,00,11
10*
10*
10
2,10
2,00,11
101010
2,10
2,00,11)3(
30
)3(
r
,z**
,z
,z,z
zzzzzz
zzzzzzzzzz
μ
ω)ωω)(ωω)(ω(
μ
μμN
μμμμμμ
μμμμμμN
Third harmonic resonance peaksoccur for and . The anharmonic oscillator modeldoes not have the resonance!
103 102
102
Two Level Model: for Nonlinear Refraction and Absorption )3(̂
The three different needed to evaluate nonlinear absorption and refraction.)3(̂
),,;(ˆ :I Case )3( xxxx
]}.)ˆ(
1
)ˆ()ˆ(
1
)ˆ()ˆ(
1
)ˆ(
1 [||
]ˆ)ˆ(
1
)ˆ(ˆ)ˆ(
1
)ˆ(ˆ)ˆ(
1ˆ)ˆ(
1[
)(|{|),,;(ˆ
3*1010
2*10
210
*10
310
401
*10
2*1010
*10
*101010
*1010
210
20011
210
)3(3
0
)3(
μμμN
xxxx
Case II: ),,;(ˆ )3( xxxx
]}, ))ˆ(
1
)ˆ(
1(
)ˆ(
1)
)ˆ(
1
)ˆ(
1(
)ˆ(
1[||
]ˆ)ˆ(
1
)ˆ(ˆ)ˆ(
1
)ˆ(ˆ)ˆ(
1
ˆ)ˆ(
1[)(|{|),,;(ˆ
*1010
22*10
*1010
2210
401
*10
22*1010
*10
*101010
*10
1022
10
20011
210
)3(3
0
)3(
μμμ
Nxxxx
Case III: ),,;(ˆ )3( xxxx
]}.))ˆ(
1
)ˆ(
1(
)ˆ(
1)
)ˆ(
1
)ˆ(
1(
)ˆ(
1[||
] ))ˆ(
1
)ˆ(
1(
)ˆ)(2ˆ(
1)
)ˆ(
1
)ˆ(
1(
)ˆ)(2ˆ(
1[)(|{|),,;(ˆ
*1010
22*10
*1010
2210
401
*1010
*10
*10
*1010
1010
20011
210
)3(3
0
)3(
μμμ
Nxxxx
This last case is the only one which gives rise to a two photon peak! Hence the termsproportional to are due to two photon transitions. Conversely, the termsproportional to are associated with one photon transitions.
20011
210 )(|| μμμ
401 ||
The typical frequency dependence of the different contributions to the total is shown below. Subscripts lyrespective componentsimaginary and real therefer to and
)3(̂
The blue curves are for the total of the one photon terms ( ) and the red curves are forthe total two photon terms ( ). The upper insets show the dispersion ofthe two photon resonance terms on a linear scale. assumed.The key results here are that changes sign at the two photon resonance and that the imaginarycomponent goes to zero as 0.
410 ||
20011
210 )(||
)3(
210
20011 ||)(
It is instructive here to examine approximate formulas which are valid in each of the fourfrequency regimes defined below. “On and near resonance”
The sign of the non-resonant term depends on . Molecules with largepermanent dipole moments will always have a positive non-resonant nonlinear realsusceptibility.
210
20011 ||)( μμμ
Two Level Model: First Order Effect on of Population Changes )3(̂
The assumption has implicitly been made that the excited state populations are very small and that the probability of exciting an electron to a higher lying state is independent of the excited state populations. Next it will be shown that this not the case by estimating the effect of the excited state population on the nonlinearity. The physics is simple: The probability of a linear one photon transition is proportional to the population difference between the ground state and the excited state and, when the two populations become comparable, saturation of the linear index change and absorption change occurs.
10
110
2)1(0
10
110
2)1(1 ))((][ ))((][
N
NNIBNdt
dNNNIBN
dt
d
The equations deal with the local intensity and hence the Maxwell intensity is enhanced by .2)1( ][
.)(][1
2)(][210
2)1(
0 statesteady
10
02)1(
IfB
NN
NNIBNN
dt
d Ndt
d
.)](/)(1[
]}{[)( Defining ;-sat
110
2)1(sat10
II
NNfBINNN
])][()[(
22||
)](/)(1[
1)();(ˆ
210
210
210
210
110
210
2210
102
10sat
)1(
0
)1(
i
IIf
Nxx
Linear susceptibility, including the first order saturation term is
])][()[(
22||]|)(|
)(21[)();(ˆ
210
210
210
210
110
210
2210
102
102
sat
0)1(
0
)1(
i
I
ncNf
Nxxx
E
In the limit )()(sat II
In general the total polarization implied by this equation can be expanded in the usual way as
].|)(|)(),,;(ˆ4
1)();(ˆ[)()( 2)3(
,)1(
0)3()1( xxeffxxxxxxxx EEEPP
])][()[(
2
)(
||)(4),,;(ˆ
210
210
210
210
110
210
2210
10sat
210)1()3(
,
i
I
ncNeffxxxx
The signs identify the sign of the nonlinearity. The vertical lines indicate sign changes.
Need to express B in terms of the imaginary (absorption) part of linear susceptibility
Nonlinear Absorption (NLA) and Refraction (NLR)
Single Input Beam NLA & NLR e.g. x-polarized along a symmetry axis
),(|),(|),,;(ˆ4
3),( 2(3)
0)3( zzz xxxxxxx EEP
Inserting into the SVEA, writing , and separating into real and imaginary
)()(),( zix
NLezz E
.absorption theon tocontributinonlinear a also is there
),()},,;(ˆmag{4
3)()()(2
NLA )()},,;(ˆmag{8
3)(
and shift, phasenonlinear additionalan is therei.e.
NLR, )()},,;(ˆeal{8
3)(
4)3(2
3)3(
2)3(
znc
zdz
dz
dz
dz
znc
zdz
d
znc
zdz
d
xxxx
xxxx
xxxxNL
Converting to the intensity I(z) and defining (also called ) by);(||2
)}.,,;(ˆmag{2
3);( )();()( )3(
20
2||22
||2
xxxxcn
ωzIzI
dz
d
)],,;(ˆ),,;(ˆ),,;(ˆ[3
1),,;(ˆwith )3()3()3()3( xxxxxxxxxxxxxxxx
Here the first term in identifies the beam undergoing absorption and the secondthe beam causing the nonlinear absorption. The identify the relative polarizations.
);(||2 and ||
2 photon resonance term
),,;(ˆ )3( xxxx
LII
LIT
LI
II(L)
)0();(1
1
)0(
)(
)0();(1
)0(
:equation aldifferenti thegIntegratin
||2||2
If linear loss is also present
.
)(]1[
)0();(1
)0( )();()()()(
1
)(
||2
)(2
||211
1
L
L
eI
eII(L)zIzIzI
dz
d
].)0();(1ln[)},,;(ˆeal{);(8
3)(
. )()},,;(ˆeal{8
3)(
2||2
)3(
||2
20
)3(
znc
z
zdznc
z
xxxxNL
zxxxx
NL
Integrating the equation for in the presence of nonlinear absorption)(zNL
.)0()},,;(ˆeal{8
3)( )0();(1 2)3(2
||2 znc
zz xxxxNL
Defining and adopting the same notation as for vacNLNLNL knz
dz
dk )( );(||2
)},,,;(ˆeal{ 8
3);( ;|),0(E|);( )3(
E||,22
E||,2 xxxxNL
nnnn
)}.,,;(ˆeal{ 4
3);( );0();( definingor )3(
02||2||2
xxxxNL
cnnInn
Leff
Molecular Nonlinear Optics: Three Level Model for )3(ˆ ijk
As discussed previously, a three level model is the minimum required to describe the thirdorder nonlinearity for the symmetric molecules (i.e. no permanent dipole moments).
}.)ˆ)(ˆ)(ˆ(ˆˆˆ
)ˆ)(ˆ)(ˆ()ˆ)(ˆ)(ˆ({
})ˆ)(ˆ)(ˆ()ˆ)(ˆ)(ˆ(
)ˆ)(ˆ)(ˆ()ˆ)(ˆ)(ˆ({
),,];[(ˆ
***
*'
,)3(
30
***,,,,
*,,,,
**,,,,,,,,
'
,,)3(
30
)3(
rqpngpmgrng
mg,kgm,jng,ign,l
qmgp*mgr
*ng
mg,kgm,jng,ign,l
pmgrngqmg
mg,jgm,kng,lgn,i
pmgrngrqpng
mg,jgm,kng,l gn,imn
rqpmgpqngpg
imglnmknjgv
pmgpqngrg
jmgknmivnlgv
rmgpqngpg
lmginmkvnjgv
pmgpqngrqpg
jmgknmlnigv
mnvrqprqpijkl
μ μ μ μ
)ωω)(ωω)(ωω(
μ μ μ μ
μ μ μ μμ μμ μN
μμμμμμμμ
μμμμμμμμx
N
Convention for labeling states,transition moments, lifetimes etc.Due to the even symmetries of the wave functions for the
two even symmetry states mAg and 1Ag, spontaneous decay to the ground state is not allowed from mAg and the statemAg can only decay to 1Bu via with subsequent decayto the ground state via .
2110
One photon terms
Two photon terms
- Total of the one photon terms (blue); total two photon terms (black, negative and red positive).- The upper curves show the dispersion of the two photon resonance terms on a linear scale.
}])([
)(4)(2||
]])[()2(2
)(]2))[(2)()((2[|{|
||),,;(ˆ),,;(ˆ),,;(ˆ:ResonancePhoton One
3210
210
210
110
3102
10
2210
210
2102020
210
2221020
121
11010
221010201020
22102
21
210
)3(3
0
)3()3()3(
iμ
ii
Nxxxxxxxxxxxx
Near & On-resonanceDominant one photon terms
)( PTS e.g. 2110
Dominant two photon terms
})(4
)12(128||]
)4]()2[(
)2)(4(
)4]()2[(
])2(2)2(64)[2([2|{|||
})(4
)12(16||]32
])2[(
)4)(4([|||{|
)4([
4),,;(ˆ),,;(ˆ),,;(ˆ
4220
210
220
21020101
102
103220
210
221
220
21020
220
210
121
3220
210
221
220
3102020
21020
1102
212
10
220
21010
220
210204
10221
220
210
210
220
22202
102
21
2220
21020
210)3(
30
)3()3()3(
μ
μμi
Nxxxxxxxxxxxx
Two Photon Resonance
Off-resonance
].)(
4)[()3(4||]}
)4()(
)2)(2(2
)4()(
)2)(8()27([4
)4()(
)84(34{
|||[|),,;(ˆ),,;(ˆ),,;(ˆ
42210
110
221022
10104
1022220
22210
2102010201
21
2220
3221020
102010202
10202
102011022
20222
1020
2101020
220
2220
210
221
210
)3(3
0
)3()3()3(
i
i
N
Non-resonant
]. ||||
[||
12),,;(),,;(),,;(10
210
20
221
210
30
210
)3()3()3()3(
μμμ
Nxxxxxxxxxxxx
10
210
20
221 ||||
μμ
Note that the sign of the non-resonant nonlinearity is determined by the sign of .
The one photon equations for NLA and NLR are the same as for thetwo level case. There is an interesting consequence to the two photonresonance absorption lineshape due to the two relaxation times needed.
1020 333.1
1021 100
1021 10
1021
001.02120 01.02120
Nonlinear Absorption with Two Input Beams
).(|)(|)},,;(ˆ{4
6)(
)(|)(|)},,;(ˆ{4
6)(
2)3(0
)3(
2)3(0
)3(
bxaxaabbxxxxbx
axbxbbaaxxxxax
EEP
EEP
Proceeding as outlined for the single beam case
).()();()();()(
)()();()();()(
||22
||2
||22
||2
zIzIzIzIdz
d
zIzIzIzIdz
d
ababbbbb
babaaaaa
)},,;(ˆmag{3);(
)},,;(ˆmag{3);(
)3(2
0||2
)3(2
0||2
aabbxxxxba
bab
bbaaxxxxba
aba
cnn
ω
cnn
ω
),,;(ˆfor similar and )],,,;(ˆ
),,;(ˆ),,;(ˆ),,;(ˆ
),,;(ˆ),,;(ˆ[6
1),,;(ˆ
)3()3(
)3()3()3(
)3()3()3(
aabbxxxxbabaxxxx
abbaxxxxabbaxxxxbabaxxxx
bbaaxxxxbbaaxxxxbbaaxxxx
Two Input Beams (Non-degenerate Case, two input frequencies: )ba ,
Two Input, Orthogonally Polarized Beams, Equal or Unequal Frequencies
)],,,;(ˆ),,;(ˆ),,;(ˆ),,;(ˆ
),,;(ˆ),,;(ˆ[6
1),,;(ˆ
)(|)(|),,;(ˆ4
6)(
)3()3()3()3(
)3()3()3(
2)3(0
)3(
ababxyxybaabxyyxbaabxyyxababxyxy
aabbxxyyaabbxxyyaabbxxyy
bxayaabbxxyybx
EEP
)].,,;(ˆ),,;(ˆ),,;(ˆ),,;(ˆ
),,;(ˆ),,;(ˆ[6
1),,;(ˆ
)(|)(|),,;(ˆ4
6)(
)3()3()3()3(
)3()3()3(
2)3(0
)3(
babayxyxabbayxxyabbayxxybabayxyx
bbaayyxxbbaayyxxbbaayyxx
aybxbbaayyxxay
EEP
)},,;(ˆmag{3);( )()();()(
)},,;(ˆmag{3);( );()();()(
)3(2
022
)3(2
022
bbaaxxyyba
abababaa
aabbyyxxba
babbaabb
cnn
ωzIzIzI
dz
d
cnn
ωzIzIzI
dz
d
).()();()();()(
)()();()();()(
22
||2
22
||2
zIzIzIzIdz
d
zIzIzIzIdz
d
ababbbbb
babaaaaa
Combining self and cross nonlinear absorption
Nonlinear Refraction with Two Input Beams
Two Co-polarized Input Beams (ab and a=b, but not co-directional) – three input “modes”
])(
)},,;(ˆal{)(
)},,;(ˆeal{2[4
3)(
])(
)},,;(ˆeal{)(
)},,;(ˆeal{2[4
3)(
)3()3(2
0
)3()3(2
0
b
bbbbbxxxx
a
aaabbxxxx
b
bNLb
a
aaaaaxxxx
b
bbbaaxxxx
a
aNLa
n
zIe
n
zI
cnz
dz
d
n
zI
n
zI
cnz
dz
d
)();()();( ,,,||2,,,||2, zInzInn bababaababbaNL
ba
)},,,;(ˆeal{4
6);( )},,,;(ˆeal{
4
6);(
)},,;(ˆeal{4
3);( )},,,;(ˆeal{
4
3);(
)3(
0||2
)3(
0||2
)3(
02||2
)3(
02||2
aabbxxxxba
abbbaaxxxxba
ba
bbbbxxxxb
bbaaaaxxxxa
aa
cnnn
cnnn
cnn
cnn
Note that in the Kleinman (non-resonant) limit all of the individual susceptibilities are the sameso that the ratios of the nonlinear index coefficients are determined by the number of nonlinearsusceptibilities (which depend on the number of unique permutations of the frequencies) thatcontribute to each effect.
),;( 2
1);(
2
1);();( ||2||2||2||2
0abbabbaa nnnn
Orthogonally Polarized Beams (equal or unequal frequencies)
