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C. Cascaded Fifth-Order Electronically Nonresonant Scattering

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



Copyright © 2001 by Taylor & Francis Group, LLC



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



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



448



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



Copyright © 2001 by Taylor & Francis Group, LLC



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



Copyright © 2001 by Taylor & Francis Group, LLC



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