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A. General: Nonresonant Nonlinear Optical Response

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



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





1



dt1 E2 r, t

0



Copyright © 2001 by Taylor & Francis Group, LLC



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





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 ikl/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



Copyright © 2001 by Taylor & Francis Group, LLC



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,



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



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