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
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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):
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Two Level Model: Third Order Susceptibilities)3(ˆ ijk
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e.g. Third Harmonic Generation (z-polarized input)
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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
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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”
1 photon resonance: ( ) 2 photon resonance: “Off-resonance”“Non-resonant” (0)
5|| 1010 )5|2| ( 1010 5|2| and 5|| ( 10101010
Near & On-resonance
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210
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2102
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102
10
2101
10
210
210
210
2104
10210
2102
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410
4104
102
102
1010
20011
210
110
310
4103
102
10102
10
1020011
210
)3(3
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Dominant one photon terms
Dominant two photon terms
Off-resonance
})(
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42210
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22210
22101
102210
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00112
10
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All terms are important
Non-resonant
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),,;(~),,;(~),,;(~
210
200113
10
210)3(
30
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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 ))((][ ))((][
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210
210
210
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Linear susceptibility, including the first order saturation term is
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210
210
210
210
110
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2210
102
102
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In the limit )()(sat II
In general the total polarization implied by this equation can be expanded in the usual way as
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The signs identify the sign of the nonlinearity. The vertical lines indicate sign changes.
Kerr nonlinearity (dotted line) total nonlinearity (solid line) saturation contribution (dashed line)
210
20011 ||)( μμμ
210
20011 ||2.1)( μμμ
total nonlinearity(dash-dot line)
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22210
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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
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and shift, phasenonlinear additionalan is therei.e.
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2)3(
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20
2||22
||2
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);(||2 and ||
2 photon resonance term
),,;(ˆ )3( xxxx
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LIT
LI
II(L)
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1
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:equation aldifferenti thegIntegratin
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If linear loss is also present
.
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znc
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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
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30
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μ μ μ μ
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μ μ μ μ
μ μ μ μμ μμ μN
μμμμμμμμ
μμμμμμμμx
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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.
}])([
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3210
210
210
110
3102
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2102020
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11010
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22102
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Near & On-resonanceDominant one photon terms
)( PTS e.g. 2110
Dominant two photon terms
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4220
210
220
21020101
102
103220
210
221
220
21020
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210
121
3220
210
221
220
3102020
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1102
212
10
220
21010
220
210204
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220
210
210
220
22202
102
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Two Photon Resonance
Off-resonance
].)(
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42210
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221022
10104
1022220
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2102010201
21
2220
3221020
102010202
10202
102011022
20222
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2101020
220
2220
210
221
210
)3(3
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i
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Non-resonant
]. ||||
[||
12),,;(),,;(),,;(10
210
20
221
210
30
210
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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
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bbaaxxxxbbaaxxxxbbaaxxxx
Two Input Beams (Non-degenerate Case, two input frequencies: )ba ,
Two Input, Orthogonally Polarized Beams, Equal or Unequal Frequencies
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aabbxxyyaabbxxyyaabbxxyy
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EEP
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bbaayyxxbbaayyxxbbaayyxx
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)},,;(ˆmag{3);( )()();()(
)},,;(ˆmag{3);( );()();()(
)3(2
022
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022
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abababaa
aabbyyxxba
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22
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22
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Nonlinear Refraction with Two Input Beams
Two Co-polarized Input Beams (ab and a=b, but not co-directional) – three input “modes”
])(
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aabbxxxxba
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ba
bbbbxxxxb
bbaaaaxxxxa
aa
cnnn
cnnn
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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)
ba e.g.
)},,,;(ˆ),,;(ˆ),,;(ˆ),,;(ˆ
),,;(ˆ),,;(ˆ{6
1),,;(ˆ
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abbaxyyxabbaxyyxbabaxyxybabaxyxy
bbaaxxyybbaaxxyybbaaxxyy
)}.,,;(ˆeal{4
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e.g. b=a
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022
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aaaaxxyya
aa
IInn
ncn
cnn