**2. Inverse Raman scattering: theory**

To describe the SRS and its origin, we begin by discussing the interaction of an optical field of frequency with a Raman‐active medium, restricted to the scalar approximation. A vibra‐ tional mode in a Raman‐active medium can be described as a simple harmonic oscillator with frequency . Due to vibrations, the optical polarizability of the molecules will change in time, depending on their reciprocal distance (internuclear separation). The periodic variation of the molecule polarizability will generate a modulation in the refractive index, which in turn will modify an incoming light beam of frequency . Specifically, the frequencies ± will be superimposed upon the transmitted light beam. The stimulated Raman scattering can be visually understood in terms of the interactions shown in **Figure 1**. **Figure 1a**

**Figure 1.** Schematic showing the stimulated Raman‐scattering process. The molecules in the Raman‐active medium are described by harmonic oscillators of frequency . The incoming laser beam has frequency . (a) Modulation of the refractive index and subsequent emission of Stokes and anti‐Stokes radiation and (b) beating of the Stokes frequency with the laser field reinforcing the vibrational oscillation of the molecule.

 shows the modulation of the refractive index due to the vibrational frequencies of the medium, thereby the transmitted light will carry the Stokes ( <sup>=</sup> − ) and anti‐Stokes ( <sup>=</sup> + ) frequencies besides the laser frequency . At the same time, one of the newly generated frequencies can beat with the incoming laser beam frequency, modulating the amplitude of the molecular vibrations (**Figure 1b**).

of matter. Among the third‐order nonlinear effects (e.g., third‐harmonic generation, optical Kerr effect), the four‐wave mixing (FWM) is the mostly explored since it generalizes all the third‐order nonlinearities. The FWM relies on the mixing of three input signals, which results in the generation of a fourth output field. When one of the input signals is resonant with the frequencies of the material, the FWM process can be enhanced and is called *stimulated Raman scattering* (SRS). Coupling this process with laser pulses delayed in time, namely using a pump‐ probe setup, it is possible to investigate the temporal behavior of the material and the evolution of its properties. Nonlinear Raman spectroscopy is an example of such combination between

In this chapter, we discuss one of the FWM processes which contributes to the stimulated Raman scattering, called inverse Raman scattering (IRS). The theory behind the IRS effect will be explained, resorting to Feynman dual‐time line (FDTL) diagrams [3], as well as its appli‐ cation as a spectroscopic tool. Furthermore, the connection held between the IRS and the femtosecond transient absorption (FTA) spectroscopy will be clarified, pointing out the important role the IRS effect plays on the temporal evolution of relaxation dynamics in FTA.

To describe the SRS and its origin, we begin by discussing the interaction of an optical field of frequency with a Raman‐active medium, restricted to the scalar approximation. A vibra‐ tional mode in a Raman‐active medium can be described as a simple harmonic oscillator with frequency . Due to vibrations, the optical polarizability of the molecules will change in time, depending on their reciprocal distance (internuclear separation). The periodic variation of the molecule polarizability will generate a modulation in the refractive index, which in turn will modify an incoming light beam of frequency . Specifically, the frequencies ± will be superimposed upon the transmitted light beam. The stimulated Raman scattering can be visually understood in terms of the interactions shown in **Figure 1**. **Figure 1a**

**Figure 1.** Schematic showing the stimulated Raman‐scattering process. The molecules in the Raman‐active medium are described by harmonic oscillators of frequency . The incoming laser beam has frequency . (a) Modulation of the refractive index and subsequent emission of Stokes and anti‐Stokes radiation and (b) beating of the Stokes frequency

with the laser field reinforcing the vibrational oscillation of the molecule.

third‐order nonlinear optical effect and pump‐probe technique [2].

**2. Inverse Raman scattering: theory**

270 Raman Spectroscopy and Applications

The two processes hereby depicted reinforce one another: the modulation of the vibrational frequencies interacts with the incoming laser beam frequency, leading to a Stokes field which increases the amplitude of the vibrational oscillation, and eventually strengthens the Stokes frequencies. This condition of amplification is called vibrational coherence, and it can be probed by a third laser beam, determining the stimulated Raman scattering [4]. The mathe‐ matical expressions of energy conservation and phase matching are given below:

$$\begin{aligned} \alpha\_{\text{SRS}} &= \left(\alpha\_L - \alpha\_S\right) \pm \alpha\_L = \alpha\_v \pm \alpha\_L\\ \mathbf{k}\_{\text{SRS}} &= \left(\mathbf{k}\_L - \mathbf{k}\_S\right) \pm \mathbf{k}\_L = \mathbf{k}\_v \pm \mathbf{k}\_L \end{aligned} \tag{1}$$

with SRS and SRS the frequency and the wave vector of the SRS signal, respectively. If the incoming optical field is a femtosecond pulse, and the probing beam is a broadband white‐ light continuum (WLC), then the process is called *femtosecond stimulated Raman scattering* (FSRS), and is one of the four‐wave‐mixing processes. Among all the possible interactions of the fields involved, these can be narrowed down to two main terms called stimulated Raman scattering, and inverse Raman scattering [2]. Hereby, we focus on the inverse Raman‐scattering effect.

The probability of annihilation of a photon with frequency and the simultaneous creation of another photon with frequency 1 = ± , according to the Kramers‐Heisenberg dispersion formula, is given by [5]

$$P = \frac{16\pi^4}{h^4} \int \left| \mu \right|^2 \rho\_0 \left( \rho\_1 + \frac{8\pi h \alpha\_1^3}{c^3} \right) d\alpha\_0,\tag{2}$$

where is the probability of the two‐photon process, 01 is associated to the IRS process, while 08ℎ1 3 takes into account the spontaneous Raman scattering. A generalization of Eq. (2) was proposed by Mukamel to describe the stimulated Raman‐scattering signal as [6]

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2 2 <sup>2</sup> <sup>2</sup> , , , 2 2 <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>3</sup> 2 3 , 4 , , . ww p e e w dw w w e e w dw w w e dw w e e e w ww e e w w w dw w ¢ ¢ ¢ ¢ ¢ = - - - - - + - - + - + - å % *SRS pu pr pu pr g g pu pu pr g g gg ef pu pr g g pr pr pu g g pr eg pr eg pr eg pu pr eg pr pu pr pu pr eg pr pu pu pr eg S N Pg T Pg T Pg T Pg T T T* (3)

Here, is the number of molecules involved in the scattering process, pu (pr) and pu (pr) are the amplitudes and frequencies, respectively, of the pump (probe) field. indicates the ground‐state population at equilibrium , while eg 1 is a matrix defining the scattering process, showing the order of the transition ( = 2 two‐photon process, = 3 three‐photon process, etc.), and the initial and final states involved. Eq. (3) can be explained as follows: the first term describes the loss of a photon with frequency pu, and the emission of a photon (gain process) with frequency pr. Conversely, the second term takes into account the emission of a pump‐beam photon after the annihilation of a photon of the probe beam (loss process). Both third and fourth terms describe the perturbation in the probe‐field absorption due to the coexistence of the pump field [7].

As stated in Eq. (3), the absorption of a photon with frequency pu, or the emission at frequency 1, depends on the population of the states involved in the transition, and specifically on the ratio between the number of molecules in the excited vibrational state and the number of molecules in the ground state. At room temperature, the excited states are scarcely populated; hence, a radiation is most likely emitted at Stokes frequencies, while absorbed at anti‐Stokes frequencies [8].

Even if there are other nonlinear processes involved, due to the intensity of the laser field typically used, the treatment of the SRS can be narrowed down to three relevant equations in the classical description, reporting on the laser excitation frequency pu, and the Stokes and anti‐Stokes frequencies , :

$$\frac{dA\_n \left(\alpha\_{\mu\nu}\right)}{d\varpi} = \frac{2\pi i \alpha\_{\mu\nu}}{n\_{\mu\alpha}c} \mathcal{X}\_{n\bar{\mu}d} A\_j \left(\alpha\_{\bar{s}}\right) A\_k \left(\alpha\_{\bar{a}\bar{s}}\right) A\_l^\* \left(\alpha\_{\mu\nu}\right) e^{-i\hbar kz} + \sum\_{\mu=L,S,a\bar{s}} \mathcal{X}\_{n\bar{\mu}d} A\_j \left(\alpha\_{\mu\nu}\right) A\_k \left(\alpha\_{\mu}\right) A\_l^\* \left(\alpha\_{\mu}\right), \tag{4}$$

$$\frac{dA\_k\left(o\_{\mathcal{S}}\right)}{dz} = \frac{2\pi i o\_{\mathcal{S}}}{n\_{\mathcal{S}}c} \chi\_{\mathbb{H}^{\text{int}}}\left(o\_{\mu\nu}\right) A\_n^\*\left(o\_{\omega\mathcal{S}}\right) A\_l\left(o\_{\mu\nu}\right) e^{i\Lambda\bar{z}} + \sum\_{\mu=1,\mu\not\le\bar{\mathcal{S}}} \chi\_{\mathbb{H}^{\text{int}}} A\_l\left(o\_{\mathcal{S}}\right) A\_n\left(o\_{\mu}\right) A\_{\mu}^\*\left(o\_{\mu}\right), \tag{5}$$

Inverse Raman Scattering in Femtosecond Broadband Transient Absorption Experiments http://dx.doi.org/10.5772/65479 273

$$\frac{d\boldsymbol{A}\_{j}^{\*}\left(\boldsymbol{\alpha}\_{\boldsymbol{\alpha}\boldsymbol{\Sigma}}\right)}{d\boldsymbol{z}} = -\frac{2\pi i\boldsymbol{\alpha}\_{\boldsymbol{\alpha}\boldsymbol{\Sigma}}}{n\_{\boldsymbol{\alpha}\boldsymbol{\Sigma}}}\boldsymbol{\chi}\_{j\boldsymbol{\text{lin}}}A\_{k}^{\*}\left(\boldsymbol{\alpha}\_{\boldsymbol{\mu}\boldsymbol{\text{a}}}\right)A\_{l}\left(\boldsymbol{\alpha}\_{\boldsymbol{\text{s}}}\right)A\_{m}^{\*}\left(\boldsymbol{\alpha}\_{\boldsymbol{\mu}\boldsymbol{\text{a}}}\right)e^{-i\boldsymbol{\alpha}\boldsymbol{\text{s}}} + \sum\_{\boldsymbol{\mu}=\boldsymbol{L},\boldsymbol{\alpha}\boldsymbol{\text{S}},\boldsymbol{S}}\boldsymbol{\chi}\_{j\boldsymbol{\text{lin}}}A\_{k}^{\*}\left(\boldsymbol{\alpha}\_{\boldsymbol{\text{s}}\boldsymbol{\text{s}}}\right)A\_{l}\left(\boldsymbol{\alpha}\_{\boldsymbol{\text{s}}}\right)A\_{m}^{\*}\left(\boldsymbol{\alpha}\_{\boldsymbol{\text{s}}}\right). \tag{6}$$

( ) ( ) ( ) ( ) ( )

 w dw

2 2 <sup>2</sup> <sup>2</sup>

*SRS pu pr pu pr g g pu pu pr g g*


( ) ( ) ( )

 e e


+ - -

 w w w dw

 dw

*pr eg pr eg*

e e

<sup>2</sup> <sup>2</sup> <sup>1</sup>

*Pg T T*

+ -

, , ,

*gg ef*

¢

*S N Pg T*

e e

*Pg T*

e

*Pg T*

e

e e

the ground‐state population at equilibrium , while

, 4

å %

 p

ww

272 Raman Spectroscopy and Applications

coexistence of the pump field [7].

anti‐Stokes frequencies , :

p w

p w c

 ww

cw

<sup>=</sup> <sup>+</sup> å *m pu pu i kz*

<sup>=</sup> <sup>+</sup> å *k S <sup>S</sup> i kz*

frequencies [8].

w

*dA i*

w 2 3

*T*

( ) ( ) ( ) ( )

*pu pr g g pr pr pu g g*

= - -

 w

*pr eg pu pr eg pr pu pr*

2 2 <sup>2</sup> <sup>2</sup>

 w

¢ ¢

( ) ( ) ( ) ( )

*pu pr eg pr pu pu pr eg*

Here, is the number of molecules involved in the scattering process, pu (pr) and pu (pr) are the amplitudes and frequencies, respectively, of the pump (probe) field. indicates

process, showing the order of the transition ( = 2 two‐photon process, = 3 three‐photon process, etc.), and the initial and final states involved. Eq. (3) can be explained as follows: the first term describes the loss of a photon with frequency pu, and the emission of a photon (gain process) with frequency pr. Conversely, the second term takes into account the emission of a pump‐beam photon after the annihilation of a photon of the probe beam (loss process). Both third and fourth terms describe the perturbation in the probe‐field absorption due to the

As stated in Eq. (3), the absorption of a photon with frequency pu, or the emission at frequency 1, depends on the population of the states involved in the transition, and specifically on the ratio between the number of molecules in the excited vibrational state and the number of molecules in the ground state. At room temperature, the excited states are scarcely populated; hence, a radiation is most likely emitted at Stokes frequencies, while absorbed at anti‐Stokes

Even if there are other nonlinear processes involved, due to the intensity of the laser field typically used, the treatment of the SRS can be narrowed down to three relevant equations in the classical description, reporting on the laser excitation frequency pu, and the Stokes and

( ) ()( ) ( ) ( ) ()() \* \*

( ) ( ) ( ) ( ) ( ) ()() \* \*

 w

*dA <sup>i</sup> A A Ae AA A*

*pu L S aS*

 ww

*S L aS S*

<sup>2</sup> ,

<sup>2</sup> ,

*kjml j pu m aS l pu klmn l S m n*

*dz n c* (5)

*mjkl j S k aS l pu mjkl j pu k l*

*AA A e A AA dz n c* (4)

, ,

, ,

cw


<sup>D</sup>

m

m

=

=

( ) ( )( )

<sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>3</sup>

 w ww

,

 w

 dw

¢ ¢

 w w  w w

1 is a matrix defining the scattering

m

 ww

m

cwww

 m

 m (3)

, .

eg

Here, are the nonlinear susceptibilities at the interacting frequencies, while is the refractive index modulated by the fields at their specific frequency. The wave vectors are related to one another by the phase‐matching condition = 2pu <sup>−</sup> <sup>−</sup> = 0. On one hand, the real part of the susceptibility describes the refractive index modulation due to the incoming field, and thus expresses the phase modulation. The imaginary part, on the other hand, accounts for the nonlinear Raman scattering and the two‐photon absorption. Resorting to Eqs. (4)–(6), it is possible to describe all the processes involved in the SRS effect. The equations needed to describe the IRS effect alone are fewer, though. To simplify the theoretical treatment, few assumptions can be done without loss of generality. In fact, it can be expected that the incoming light intensity is stronger than the scattered one, that is, pu ≫ , . Furthermore, the intensity of the IRS is stronger than the spontaneous Raman scattering if detected at the same frequency. Thus, in the scalar approximation for the susceptibility, Eqs. (4)–(6) can be rewritten as

$$\frac{dA\_s}{d\varpi} = \frac{2\pi i \alpha\_S}{n\_s c} \left( \mathcal{Z} \left| A\_{\mu u} \right|^2 A\_s + \mathcal{X} A\_{\mu u}^\* A\_{aS}^\* e^{i\Lambda kz} \right), \tag{7}$$

$$\frac{dA\_{\rm as}}{dz} = -\frac{2\pi i \alpha\_{\rm as}}{n\_{\rm as}c} \left( \mathcal{X} A\_{\rm pa}^\* A\_{\rm s}^\* e^{-i\Lambda kz} + \mathcal{X} \left| A\_{\rm pa} \right|^2 A\_{\rm as}^\* \right), \tag{8}$$

assuming that the incoming field intensity is constant, that is, pu/ = 0. It follows that the intensities of the Stokes and anti‐Stokes fields are

$$I\_S\left(z\right) = r\_1^2 \exp\left(2\left|\mathbf{g}\right|I\_{\mu u}z\right) + r\_2^2 \exp\left(-2\left|\mathbf{g}\right|I\_{\mu u}z\right) + 2r\_1r\_2\cos\left(\Delta kz + \delta\phi\right),\tag{9}$$

$$I\_{as}\left(z\right) = \alpha\_1 r\_1^2 \exp\left(2\left|\mathbf{g}\right|I\_{\mu n}z\right) + \alpha\_2 r\_2^2 \exp\left(-2\left|\mathbf{g}\right|I\_{\mu n}z\right) - 2r\_1 r\_2 \cos\left(\Delta k z + \delta \phi\right). \tag{10}$$

where = 2 ℑ. The terms and are defined by the initial conditions, and specifi‐ cally 1 ≤ 1, 2 ≥ 1; is the initial phase shift at the front of the sample. Depending on the ratio pu/ and hence on the Stokes‐anti‐Stokes coupling, three different solutions to the system can be found [8]. Here, the solution for zero coupling (pu < ) will be presented, due to the noncollinear geometry of the pump‐probe setup (for the other solutions, refer to Weigmann's paper [8]). The condition hereby described predicts an enhancement of the IRS signal, with the intensities at the Stokes and anti‐Stokes frequencies given by

$$I\_{\rm s} \left( z \right) = r\_{\rm l}^{2} \exp \left( 2 \left| \mathbf{g} \right| I\_{\mu \nu} z \right), \tag{11}$$

$$I\_{\alpha S} \left( z \right) = \alpha\_2 r\_2^2 \exp \left( -2 \left| \mathbf{g} \right| I\_{\mu} z \right). \tag{12}$$

It follows that the intensities at the anti‐Stokes frequencies are higher than the corresponding ones at the Stokes frequencies [7, 9, 10]. In the spontaneous Raman scattering, at room temperature the populations in the excited vibrational levels are negligible, as described by Boltzmann distribution, thus the anti‐Stokes peaks are weak. However, in the stimulated process (such as IRS) the Boltzmann distribution does not describe anymore the energy levels population, and a strong anti‐Stokes emission can be observed [11].

#### **3. Inverse Raman scattering: spectroscopy**

The IRS effect can be described by the Feynman dual‐time line (FDTL) diagrams in the Liouville space with the ket and bra evolution (**Figure 2**) [12]. Using the FDTL diagrams and the related energy level diagrams, it is possible to illustrate the temporal evolution of the density matrix in the four‐wave mixing process, which the IRS is based on. The temporal evolution goes from 3 to when the third‐order polarization induced in the medium is computed [3]. The arrows pointing into the time line represent the absorption from the ground state Ψ0 to the excited state Ψ and the related wave vector is + On the contrary, the arrows pointing far from the time line illustrate the stimulated emission from Ψ to Ψ0 and their wave vector is −. In the energy diagrams, pr, pu, and pu \* are the vectors of the fields involved in the interaction in the medium, having frequencies pr, pu, and pu respectively. In both IRS (I) and IRS (II) processes, the wave vectors of the incoming pump fields, that is, pu and pu \* , vanish, leading to a third‐order polarization having the same direction of the incoming pump pulse [12]. The difference between the IRS (I) and IRS (II) lays in the sequence with which pump and probe pulses reach the sample. In the IRS (I) (**Figure 2a**), the excitation process is started by the probe pulse alone at frequencies higher than those of pump pulse, that is, in the anti‐Stokes region. The vibrational coherence persists as long as the pump pulse overlaps with the probe pulse, then a photon with frequency pr is annihilated due to the interaction between the two pulses, and a loss in intensity in the probe spectrum is observed at the specified anti‐Stokes frequency [13]. In the IRS (II) (**Figure 2b**), the pump pulse is followed by the probe pulse, it induces the excitation in the sample and due to the interaction of the two fields, the vibrational coherence is achieved. A photon having frequency pr and wave vector (i.e., same direction of the Stokes radiation) is created, and it adds to the probe pulse, leading to an intensity gain in the probe spectrum at the specified frequency. According to the energy conservation, a loss in energy, equal to the gain obtained at the Stokes frequency, is observed in the pump pulse [4]. This is clearly shown in the bottom part of **Figure 2a** and **2b**, where the frequency of the outcoming beam is higher (lower) than the one of the pump fields for the anti‐Stokes (Stokes) signal.

Weigmann's paper [8]). The condition hereby described predicts an enhancement of the IRS

<sup>1</sup> exp 2 , *<sup>S</sup> pu I z r gI z* ; (11)


\* are the vectors of the fields involved in the interaction

\* , vanish, leading

( ) ( ) <sup>2</sup>

( ) ( ) <sup>2</sup> 2 2 exp 2 . *aS pu I z r gI z* ;

It follows that the intensities at the anti‐Stokes frequencies are higher than the corresponding ones at the Stokes frequencies [7, 9, 10]. In the spontaneous Raman scattering, at room temperature the populations in the excited vibrational levels are negligible, as described by Boltzmann distribution, thus the anti‐Stokes peaks are weak. However, in the stimulated process (such as IRS) the Boltzmann distribution does not describe anymore the energy levels

The IRS effect can be described by the Feynman dual‐time line (FDTL) diagrams in the Liouville space with the ket and bra evolution (**Figure 2**) [12]. Using the FDTL diagrams and the related energy level diagrams, it is possible to illustrate the temporal evolution of the density matrix in the four‐wave mixing process, which the IRS is based on. The temporal evolution goes from 3 to when the third‐order polarization induced in the medium is computed [3]. The arrows

pointing into the time line represent the absorption from the ground state Ψ0 to the excited state Ψ and the related wave vector is + On the contrary, the arrows pointing far from the

time line illustrate the stimulated emission from Ψ to Ψ0 and their wave vector is −.

in the medium, having frequencies pr, pu, and pu respectively. In both IRS (I) and IRS (II)

to a third‐order polarization having the same direction of the incoming pump pulse [12]. The difference between the IRS (I) and IRS (II) lays in the sequence with which pump and probe pulses reach the sample. In the IRS (I) (**Figure 2a**), the excitation process is started by the probe pulse alone at frequencies higher than those of pump pulse, that is, in the anti‐Stokes region. The vibrational coherence persists as long as the pump pulse overlaps with the probe pulse, then a photon with frequency pr is annihilated due to the interaction between the two pulses, and a loss in intensity in the probe spectrum is observed at the specified anti‐Stokes frequency [13]. In the IRS (II) (**Figure 2b**), the pump pulse is followed by the probe pulse, it induces the excitation in the sample and due to the interaction of the two fields, the vibrational coherence is achieved. A photon having frequency pr and wave vector (i.e., same direction of the

signal, with the intensities at the Stokes and anti‐Stokes frequencies given by

a

population, and a strong anti‐Stokes emission can be observed [11].

**3. Inverse Raman scattering: spectroscopy**

pr,

pu, and

pu

processes, the wave vectors of the incoming pump fields, that is, pu and pu

In the energy diagrams,

274 Raman Spectroscopy and Applications

**Figure 2.** Feynman dual‐time line (FDTL) diagrams (top) and energy‐level diagrams (bottom) describing the IRS effect. The wavy line is the field coming out from the stimulated Raman‐scattering process, and illustrates the third‐order po‐ larization of the involved energetic states. In the energy diagrams, the solid and the dashed lines represent the real and the virtual levels, respectively. (a) The pump pulse arrives after the probe pulse, and excites the sample. This originates the anti‐Stokes line since the outcoming frequency is higher than the one of the incoming pump fields. This is called IRS (I) process and (b) the probe pulse follows temporally the pump pulse and the Stokes line is generated because the outcoming frequency is smaller than the incoming one. This is the IRS (II) process.

According to Lee's papers [3, 14], it is possible to extrapolate the mathematical expression of the third‐order polarization from the FDTL diagrams. The time‐dependent third‐order polarization in the IRS effect in the probe direction (i.e., wave vector pr, is given by the overlap of the wave packet on the ket side of the time line and a ground vibrational state on the bra side. It depends on the transition dipole moment ge on the inhomogeneous broadening of the wave packet and it is integrated over the time intervals , 1, 2,, and 3 :

$$\begin{split} P\_{\text{Res}(t)}^{(3)}\left(t\right) &= \left(\frac{i}{\hbar}\right)^3 \Big[ \int\_0^{\tau\_1} \int\_0^{\tau\_1} \int\_0^{\tau\_2} \tau\_3 \, d\tau\_3 \\ &\times \exp\left[ -\frac{\mathcal{V}\_g t}{2\hbar} - \frac{\mathcal{V}\_s \left(t - \tau\_1\right)}{2\hbar} - \frac{\mathcal{V}\_d \left(\tau\_1 - \tau\_2\right)}{\hbar} - \frac{\mathcal{V}\_g \left(\tau\_1 - \tau\_2\right)}{2\hbar} - \frac{\mathcal{V}\_s \left(\tau\_2 - \tau\_3\right)}{2\hbar} - \frac{\mathcal{V}\_g \tau\_3}{2\hbar} \right] \\ &\times E\_{\mu\nu}\left(\tau\_1\right) E\_{\mu\nu}^\*\left(\tau\_2\right) E\_{\mu\nu}\left(\tau\_3\right) G\left(t - \tau\_1 + \tau\_2 - \tau\_3\right) I\left(t, \tau\_1, \tau\_2, \tau\_3\right) \end{split} \tag{13}$$

$$\begin{split} P\_{\text{RS}(H)}^{(j)}\left(t\right) &= \left(\frac{i}{\hbar}\right)^{j} \Big[ \int\_{0}^{\tau\_{1}} \int\_{0}^{\tau\_{2}} \int\_{0}^{\tau\_{3}} \pi\_{2} \right] d\tau\_{3} \\ &\times \exp\left[ -\frac{\mathcal{Y}\_{g}t}{2\hbar} - \frac{\mathcal{Y}\_{s}\left(t-\tau\_{1}\right)}{2\hbar} - \frac{\mathcal{Y}\_{d}\left(\tau\_{1}-\tau\_{2}\right)}{\hbar} - \frac{\mathcal{Y}\_{s}\left(\tau\_{1}-\tau\_{2}\right)}{2\hbar} - \frac{\mathcal{Y}\_{s}\left(\tau\_{2}-\tau\_{3}\right)}{2\hbar} - \frac{\mathcal{Y}\_{s}\tau\_{3}}{2\hbar} \right] \\ &\times E\_{\mu\tau}\left(\tau\_{1}\right) \mathcal{E}\_{\mu\upsilon}^{\prime}\left(\tau\_{2}\right) E\_{\mu\upsilon}\left(\tau\_{3}\right) G\left(t-\tau\_{1}+\tau\_{2}-\tau\_{3}\right) I\left(t,\tau\_{1},\tau\_{2},\tau\_{3}\right) \end{split} \tag{14}$$

Here, the four‐time correlation function , 1, 2, 3 is defined as follows:

$$\begin{split} &I\left(t,\tau\_{1},\tau\_{2},\tau\_{3}\right) \\ &= \left\langle \Psi\_{\boldsymbol{g}^{\text{o}}}\left(\boldsymbol{\varrho}\right) \middle| e^{\operatorname{\boldsymbol{\mathfrak{e}}}\_{\boldsymbol{\mathfrak{e}}^{\text{i}}}/\hbar} \mu\_{\boldsymbol{g}^{\text{o}}} e^{\operatorname{\boldsymbol{\mathfrak{e}}}\_{\boldsymbol{\mathfrak{e}}^{\text{i}}} \left(t-\boldsymbol{\varepsilon}\_{1}\right)/\hbar} \times \mu\_{\boldsymbol{\mathfrak{e}}} e^{\operatorname{\boldsymbol{\mathfrak{e}}}\_{\boldsymbol{\mathfrak{e}}} \left(\boldsymbol{\varepsilon}\_{1}-\boldsymbol{\varepsilon}\_{1}\right)/\hbar} \times \mu\_{\boldsymbol{\mathfrak{e}}^{\text{i}}} e^{\operatorname{\boldsymbol{\mathfrak{e}}}\_{\boldsymbol{\mathfrak{e}}^{\text{i}}} \left(\boldsymbol{\varepsilon}\_{1}-\boldsymbol{\varepsilon}\_{1}\right)/\hbar} \left| \Psi\_{\boldsymbol{g}^{\text{o}}}\left(\boldsymbol{\varrho}\right) \right\rangle \end{split} \tag{15}$$

where is the multidimensional vibrational coordinate, Ψ0 the initial ground vibrational state of the electronic ground state , and and the vibrational multidimensional Hamiltonians associated to the energetic levels involved in the transition and The terms and in Eqs. (13) and (14) are the line bandwidths of the two energetic levels, while is the time interval between 2 and 1, during which the vibrational coherence is lost (Raman vibrational dephasing time) [3].

Once defined the third‐order polarization for both effects IRS (I) and IRS (II), it is possible to show the variation in intensity of the stimulated Raman signal as function of the frequencies, to be given by

$$
\Delta I\_{\rm lBS} \left( \alpha \right) = -\frac{8\pi^2 lC}{3n} \alpha \Im \left\{ E\_{\mu\nu}^\* \left( \alpha \right) P\_{\rm lBS}^{(3)} \left( \alpha \right) \right\} \tag{16}
$$

where is the optical path length, the number of molecules per unit volume, and the refractive index of the Raman‐active medium. pr \* is the incoming probe field with fre‐ quency and IRS (3) is the Fourier transform of the third‐order polarization calculated using Eqs. (13) and (14). The variation in intensity IRS is directly depending on the imaginary part of the third‐order Raman polarization, and hence on the Raman susceptibility as already stated in the previous section [15]. <sup>ℑ</sup> (3) is negative at the Stokes frequencies ( <sup>=</sup> − ) and positive at anti‐Stokes frequencies ( <sup>=</sup> + ), as depicted in **Figure 3**, where the relation between the Stokes and anti‐Stokes susceptibilities is presented. It follows that the Raman spectrum will be positive at the Stokes frequencies, experiencing a gain in the probe‐beam intensity for each frequency which fulfills <sup>=</sup> − . On the contrary, IRS < 0 in the anti‐Stokes region. Here, the probe‐beam intensity will be decreased for all those frequencies which meet the condition <sup>=</sup> + .

( ) ( ) ( )

276 Raman Spectroscopy and Applications

( ) ( ) ( )

3

*IRS II*

æ ö <sup>=</sup> ç ÷ è ø

*<sup>i</sup> P t dd d*

*exp*

æ ö <sup>=</sup> ç ÷ è ø

( )

ttt

,,,

*I t*

123

vibrational dephasing time) [3].

to be given by

quency and IRS

*<sup>i</sup> P t dd d*

*exp*

3

*IRS I*

1 2 3

g

òòò <sup>h</sup>

*t*

t

*t*

t t

tt

\*

1 2 3

g

òòò <sup>h</sup>

t

t t

tt

\*

m

*pr pu pu*

*pu pu pr*

12 3 00 0

g

*t t*

 t

12 3 00 0

g

*t t*

 t  t

> t gt

´ -+ -

t  t

*E E E Gt I t*

Here, the four‐time correlation function , 1, 2, 3 is defined as follows:

 t

> t gt

´ -+ -

 t

*E E E Gt I t*

( ) ( ) ( ) ( )

hh h h h h

( ) ( ) ( ) ( )

hh h h h h

gt

t t


2 2 2 22

*g gg e d e*

gt

2 2 2 22

*g gg e d e*

1 12 1 2 2 3 3

 t

 ttt

1 12 1 2 2 3 3

 t

 ttt

,,,

 m

,,,

gt

gt

t

 t

> t

g t

g t (13)

(14)

() ( ) ()( )( )

() ( ) ()( )( )

1 2 3 1 2 3 123

 t t t

 t

( ) ( ) ( ) ( ) ( ) <sup>1</sup> 1 2 2 3 <sup>3</sup>

where is the multidimensional vibrational coordinate, Ψ0 the initial ground vibrational state of the electronic ground state , and and the vibrational multidimensional Hamiltonians associated to the energetic levels involved in the transition and The terms and in Eqs. (13) and (14) are the line bandwidths of the two energetic levels, while is the time interval between 2 and 1, during which the vibrational coherence is lost (Raman

Once defined the third‐order polarization for both effects IRS (I) and IRS (II), it is possible to show the variation in intensity of the stimulated Raman signal as function of the frequencies,

( ) ( ) ( ) { ( )} <sup>2</sup> <sup>8</sup> \* <sup>3</sup>

where is the optical path length, the number of molecules per unit volume, and the

 wwI *pr IRS*

pr

(3) is the Fourier transform of the third‐order polarization calculated using

*lC <sup>I</sup> E P <sup>n</sup>* (16)

\* is the incoming probe field with fre‐

IRS is directly depending on the imaginary

 w

3 p

 D =- *IRS* w

refractive index of the Raman‐active medium.

Eqs. (13) and (14). The variation in intensity

/ // / / 0 0

Ψ | |Ψ *g g e e <sup>g</sup> ii i i t <sup>i</sup> g ge eg ge eg g*

*ee e e e*

t t

mm

<sup>é</sup> - - - ù - ´ -- - - - - ê ú ë û

1 2 3 1 2 3 123

 t t t

 t

<sup>é</sup> - - - ù - ´ -- - - - - ê ú ë û

**Figure 3.** Relation between the Stokes and anti‐Stokes susceptibilities. The solid black line is the imaginary part of the Raman susceptibility; the red dashed line is the real part of the Raman susceptibility.

**Figure 4.** Extracted Raman spectrum of eumelanin dispersion in DMSO‐methanol mixture.

The aforementioned variation in intensity IRS will be analyzed for a case of study of a dispersion of eumelanin in dimethyl sulfoxide DMSO‐methanol mixture (1:20 ratio) in the following. In **Figure 4**, the extracted Raman spectrum of the dispersion is depicted. On the red side of the white‐light broadband spectrum (Stokes side), some frequencies resonate with the vibrational modes of the Raman‐active medium. Hence, the signal appears as gain features in the probe pulse at those specific frequencies. At the same time, on the blue side (anti‐Stokes side) a loss in the intensity is achieved at frequencies <sup>=</sup> + .
