A. General: Nonresonant Nonlinear Optical Response
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Fifth-Order 2D Raman Spectroscopy
441
denoted as 2 and 4 , respectively (see Fig. 1). The external ﬁeld is thus
given by a sum of the ﬁve ﬁelds:
5
Eex r, t D
Ej r, t
1
jD1
where
Ej r,t D uj εj t exp ikj Ð r
iω0 t C c.c.
2
Here c.c. refers to the complex conjugate and εj t and uj are the temporal
envelope function and unit vector of the polarization of the jth electric ﬁeld.
The frequencies of the ﬁve ﬁelds are assumed to be identical, i.e., ωj D ω0
for j D 1, 2, 3, 4, and 5. This can be experimentally achieved by using
time-delayed pulses generated from a common laser oscillator. Although
the femtosecond pulses generated in the laboratory have a ﬁnite width, for
the sake of simplicity the laser pulses are assumed to be impulsive in this
section, i.e.:
ε1 t D ε2 t D ευ t
ε3 t D ε4 t D ευ t
ε5 t D ευ t
2
2
3
4
Even though the laser pulses are approximated as delta functions, the slowly
varying amplitude approximation can still be applied to pulses as wide as
tens of femtoseconds, where the time scales for the nuclear degrees of
freedom remain much slower than the pulse width (1).
When the external ﬁeld frequency is far off-resonant with respect to
the optical transition, the ﬁeld-matter interaction can be described by the
effective interaction Hamiltonian given by
Heff
l D
˛:E
ˆ 2 r, t
4
Then, the Liouville equation for the ground state density matrix,
given as
g
r,t , is
i
υ
[H0 C Heff
5
g r, t D
l , g r, t ]
υt
h¯
where H0 is the material Hamiltonian determining the nuclear dynamics
in the electronic ground state. The second-order density matrix, expanded
with respect to the ﬁeld-matter interactions, can be obtained as
2
g
r, t D
i
h¯
1
dt1 E2 r, t
0
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t1 e
iH0t1 l¯h
[˛,
ˆ
eq ]e
iH0t1 l¯h
6
442
Fleming et al.
where eq denotes the equilibrium density operator in the electronic ground
state. It should be noted that the effective interaction Hamiltonian deﬁned in
Equation (6) represents two ﬁeld-matter interactions. Thus, the ﬁrst-order
term with respect to the effective interaction Hamiltonian is essentially
identical to the second-order perturbation expansion term with respect to
the ﬁeld-matter interaction when the hyperpolarizability contribution is
ignored.
The 2n-order expanded density matrix, which is obtained by the
n-order perturbation expansion with respect to the effective interaction
Hamiltonian, is then given as
r, t D
2n
g
i
h¯
1
dt1 E2 r, t
1
n
dtn Ð Ð Ð
0
tn Ð Ð Ð E2 r, t
Ð Ð Ð t2
t1
0
ðe
ceiH0 t2 l¯h Ð Ð ÐceiH0 tn l¯h
7
By deﬁnition, the polarization is the expectation value of the dipole operator,
iH0 tn l¯h
b˛,
ˆ ÐÐÐe
iH0 t2 l¯h
P n r,t D N Tr[ ˆ ind r, t
D NE r, t Ð Tr[˛ˆ
b˛,
ˆ e
iH0 t1 l¯h
n 1
g
n 1
g
r, t ]
[˛,
ˆ
eq ]e
iH0 t2 l¯h
r, t ]
8
where the induced dipole operator was deﬁned as ˆ ind r, t D E r, t Ð ˛,
ˆ
and N denotes the number density.
The spatial amplitude of the electric ﬁeld generated by one of the norder induced polarizations should satisfy the following Maxwell equation:
r ð r ð fεsn r, t exp ik0s Ð r g
D
4 ωs2 n
P t exp iks Ð r
c2 s
ns2 ωs2 n
ε r, t exp ik0s Ð r
c2 s
9
Here we look for a solution of the form, E n r, t D ε n r, t exp ik0s Ð r
iωs t , where k s is different from ks , which is given by a combination of the
incoming wave vectors, due to the frequency dispersion of the refractive
index of the optical sample.
Within the slowly varying amplitude approximation, the generated
electric ﬁeld amplitude grows linearly with respect to the distance from the
front boundary, z, of the optical sample. Direct integration over z gives the
Copyright © 2001 by Taylor & Francis Group, LLC
Fifth-Order 2D Raman Spectroscopy
443
generated electric ﬁeld amplitudes of the n-order NLO processes, approximately given as
ε n t D iA F n P
n
t
10
where the constant A is deﬁned as
Asj Á
2 jωsj j
ns c
l
2
11
and the phase matching factor, F n , is deﬁned as
Fn D
sin[kl/2]
exp ikl/2
kl/2
12
with k D k k0 . Here ns is the refractive index of the sample at the
frequency of ωs , and l is the sample thickness. The factor of 2 in the
denominator inside the parenthesis of Equation (1) is introduced to take
the average amplitude of the generated ﬁeld within the sample — note
that the generated ﬁeld amplitude increases linearly from 0 at the front
boundary to a maximum value proportional to l at the rear boundary so
that the average ﬁeld amplitude within the optical sample is proportional
to l/2.
B. Direct Fifth-Order Electrically Nonresonant Scattering
The ﬁfth-order nonlinear polarization can be obtained by inserting the electric ﬁeld given in Equation (1) into Equation (8). In particular, we look for
a speciﬁc polarization component whose wave vector and frequency are
given as, for example, ks D k1 k2 k3 C k4 C k5 and ωs D ω1 ω2
ω3 C ω4 C ω5 D ω0 , respectively:
5
5
Pdir
r, t D Pdir t exp iks Ð r
iωs t
13
Then, within the impulsive limit, carrying out the double integration
involved in the calculation of g4 r, t and using Equation (4), we ﬁnd
the amplitude of the direct ﬁfth-order off-resonant scattering polarization:
5
Pdir t D Nε5 R 5
2, 4
υt
2
14
4
Using Equation 10, the electric ﬁeld generated from the direct ﬁfth-order
polarization can be expressed as
5
5
εdir
t D iAF 5 Pdir t D iAF 5 Nε5 R 5
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2, 4
υt
2
4
15
444
Fleming et al.
where the ﬁfth-order nonlinear response function is deﬁned as
R5
2, 4
1
h[[˛ˆ
h¯ 2
D
2
C
4
, ˛ˆ
2
], ˛ˆ 0 ]
eq i
16
This is the standard result originally obtained by Tanimura and Mukamel
(5). It should be noted that ˛ is a second rank tensor with respect to the
relative polarizations of the interacting ﬁelds, and thus R 5 is a sixth rank
tensor. However, for the sake of simplifying the notation, we have dropped
the tensoral indices.
The 2D response [Equation (16)] can be expressed in terms of the
coordinates via a Taylor expansion of the polarizability operator (12,16,27):
˛j 1 Qj t C
˛ t D ˛0 C
j
1
2
2
˛j,k
Qj t Qk t C Ð Ð Ð
17
j,k
where ˛ n is the nth derivative of the polarizability with respect to the coordinate(s). For third-order nonresonant Raman spectroscopy, it is common to
use the Placzek approximation and assume that all higher-order terms in the
Taylor expansion are negligible, thus truncating the expansion at the linear
term. However, the assumption of a linear dependence of the polarizability
on the coordinates within a harmonic ground state potential leads to a zero
value for the three-point correlation function in Equation (16). This demonstrates the intrinsic dependence of the ﬁfth-order signal on the microscopic
coupling in the system. There are two types of coupling that can generate
the ﬁfth-order signal: anharmonicity in the vibrational potential, AN, and
nonlinearity in the dependence of the polarizability on the vibrational coordinate, NP (5,12,28). The resulting response can be expressed as the sum
of the two individual contributions:
R5
2, 4
D R 5 ,AN
2, 4
C R 5 ,NP
2, 4
18
The anharmonic response can be obtained by inclusion of the cubic anharmonicity in the vibrational potential,
V¾
D
1
2
kj Q2j C
j
1
3!
gijk3 Qi Qj Qk
19
ijk
where g 3 is the third derivative of the ground state potential with respect
to the coordinates. The magnitude of the individual AN and NP responses
reﬂects the magnitude of the potential anharmonicity, g 3 , and the magnitude of the nonlinearity of the polarizability, ˛ 2 , respectively. The individual responses based on AN and NP coupling can be expressed in terms
Copyright © 2001 by Taylor & Francis Group, LLC