C. Cascaded Fifth-Order Electronically Nonresonant Scattering
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446
Fleming et al.
Figure 2 Representative ladder diagrams for the possible cascading ﬁfth-order
pathways. (a) The sequential cascade pathway. (b) The parallel cascade pathways.
k2 k3 , ωs1 D ω1 ω2 ω3 D ω0 , (b) ks2 D k1 k2 C k4 , ωs2 D ω1
ω2 C ω4 D ω0 ; the two parallel cascade intermediates are (a) kp1 D k1
k2 C k5 , ωp1 D ω1 ω2 C ω5 D ω0 , and (b) kp2 D k3 C k4 C k5 , ωp2 D
ω3 C ω4 C ω5 D ω0 . Using Equation (8), the corresponding third-order
induced polarizations are expressed as
3
3
Ps1 t D Ps2 t D Nε3 R 3 t υ t
3
Pp1
3
Pp2
t D Nε R
3
tυt
t D Nε R
3
t
3
3
2
2
4
υt
2
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2
4
22
Fifth-Order 2D Raman Spectroscopy
447
where the third-order Raman response function is
i
23
R 3 t D h[˛ˆ t , ˛ˆ 0 ] eq i
h¯
The spatial amplitude of the electric ﬁelds generated by these third-order
induced polarizations can be obtained using Equation (10). The four electric
ﬁelds are then expressed as
3
εs13 t D iAs1 Fs13 Ps1 t D iAs1 Fs13 Nε3 R 3 t υ t
εs23
εp13
εp23
t D
t D
t D
3
iAs2 Fs23 Ps2
3
iAp1 Fp13 Pp1
3
iAp2 Fp23 Pp2
t
t
D iAs2 Fs23 Nε3 R 3
D iAp1 Fp13 Nε3 R 3
t D
iAp2 Fp23
3
Nε R
3
2
tυt
2
tυt
2
4
υt
2
t
2
4
24
with Axj and Fxj deﬁned in Equations (11) and (12).
In addition to the ﬁve external ﬁelds, the molecular system can interact
with these four internal ﬁelds generated by the third-order NLO scattering
processes. The total electric ﬁeld is therefore
E r,t D Eex r,t C E 3 r,t C Ð Ð Ð
25
where the external ﬁeld, Eex r,t , was given in Equation (1), and the
third-order internal ﬁeld, which is involved in the cascading processes
contributing to the ﬁfth-order signal, is given as
4
εsj3 t exp iksj Ð r
E 3 r, t D
iωsj t C c.c.
26
jD1
If one of these internal ﬁelds, in place of one of the external ﬁelds,
participates in one of the three ﬁeld-matter interactions of another thirdorder NLO process, the cascading coherent ﬁeld will be generated. The
corresponding four ﬁfth-order cascaded polarization amplitudes are then
expressed as
5
Pseq1 t D iAs1 Fs13 N2 ε5 R 3 t
5
Pseq2
5
Ppar1
5
Ppar2
t D iAs2 Fs23 N2 ε5 R 3 t
t D
iAp1 Fp13
N 2 ε5 R 3 t
t D
iAp2 Fp23
N ε R
2 5
3
t
2
R3
2
υt
2
4
2
R3
2
υt
2
4
2
R3 t υ t
2
4
tυt
2
4
2
R
3
27
As can be seen from Equation (27), the sequential cascades result in
responses that are symmetrical along the two adjustable time variables,
Copyright © 2001 by Taylor & Francis Group, LLC
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Fleming et al.
while the parallel cascades result in responses that are asymmetrical along
the two time variables. Adding together all four cascaded contributions to
obtain the total ﬁfth-order cascaded polarization,
5
Pcas t D iAN2 ε5 Fs13 C Fs23 R 3 t
C iAN ε
2 5
Fp13
C
Fp23
R
3
2
t
R3
2
R
2
3
υt
2
tυt
4
2
4
28
where A D As1 D As2 D Ap1 D Ap2 . Using Equation (10) the ﬁeld associated with the total cascaded response is expressed as
5
5
t D iAF 5 Pcas t D
εcas
ðR3 t
2
A F
3
2 R
5 2 5
N ε Fp13
2
A2 F 5 N2 ε5 Fs13 C Fs23
υt
2
C Fp23 R 3 t
4
2
R3 t υ t
2
29
4
D. The Total Nonresonant Fifth-Order Raman Signal
Since the direct and cascaded responses satisfy the same phase matching
condition, the total nonresonant ﬁfth-order Raman signal will contain both
contributions. Using Equations (11), (14), and (28), the ratio between the
absolute values of the direct and cascaded contributions is
5
jPcas
5
jPdir
tj
tj
D
ω0 lN
nc
jf Fs13 C Fs23 R 3 2
C Fp13 C Fp23 R 3 2 C
jR 5 2 , 4 j
4
gR 3
4
j
30
Thus, the ratio is dependent on experimental parameters such as the optical
path length, sample number density, and the phase matching conditions
for the intermediate third-order processes, as well as the ratio of the thirdand ﬁfth-order response functions. The ratio of the response functions is
directly related to the magnitude of the nonlinearity in the system, which
is reﬂected by the magnitude of the potential anharmonicity, g 3 , and the
nonlinearity in the polarizability, ˛ 2 . For example, let us consider only
the NP contribution to the direct ﬁfth-order response [Equation (21)]. For
simplicity we will consider a system represented by a single mode, in other
words the response is isotropic. If we express the third-order response
functions in term of the coordinate [Equation (17)] and ignore all higher
order terms,
R 3 t 1˛ 1 ˛ 1 C t
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31
Fifth-Order 2D Raman Spectroscopy
449
Then the ratio between the cascaded and direct response functions becomes
5
Rcascade
5
RNP
1
[˛ 1 ]4
[˛ 1 ]2
D
1
2
2
[˛ ] ˛
˛2
32
Thus, in the NP case, the ratio on the right-hand side of Equation (30) is
proportional to the ratio of the square of the linear coefﬁcient of the polarizability to the quadratic anharmonicity in the polarizability with respect to
the coordinate.
In the experimental section of this chapter we will present spectra
obtained using both homodyne and heterodyne detection methods. The total
homodyne detected signal is expressed as
Shomo
2, 4
ns c
8
ns c
D
8
D
1
1
1
1
5
5
dt jεcas
t C εdir
t j2
5
5
dt jiAF 5 Pcas t C iAF 5 Pdir t j2
33
and will inherently contain both the cascaded and direct contributions to the
ﬁfth-order signal. In contrast to homodyne detection, heterodyne detection
is sensitive to the phase of the electric ﬁelds. Bearing this aspect in mind,
consider the two ﬁfth-order ﬁelds associated with the cascaded and direct
processes [Equations (29) and (15)]. If the phase-matching conditions for
both third- and ﬁfth-order processes are assumed to be perfect so that Fsj3 D
5
t , is
F5 ¾
D 1, the amplitude of the cascaded ﬁfth-order scattering ﬁeld, εcas
5
purely real, whereas that of the direct process, εdir t , is purely imaginary.
Therefore, if a heterodyne-detection technique based on injecting a phasecontrolled local oscillator ﬁeld is used, one may be able to separately detect
the cascaded and direct contributions. That is (35),
Shetero
2,
4
D
ns c
4
C
1
dt RefεŁLO t,
1
RefεŁLO
t,
5
Ð εdir
tg
5
Ð εcas
tg
34
This result suggests that controlling the phase factor, , to make εŁLO t,
imaginary, one can selectively measure the direct component alone. In this
case, the optical phase of the local oscillator ﬁeld should be controlled
with respect to that of the ﬁnal laser ﬁeld. However, experimentally this
is quite challenging to implement. The inability to have perfect phase
matching in an experiment with ﬁnite laser pulse dimensions results in
phase shifting of the signals, thus lessening the discrimination against the
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