**2.1 Raman scattering process**

When a beam of light passes through a media, one can observe light scattering phenomenon besides light transmission and absorption. Most of elastically scattered photons (so called

Ultra-Broadband Time-Resolved Coherent Anti-Stokes Raman Scattering

 

is the basis of the selection rule of Raman spectroscopy.

intensity divided by the incident intensity:

summation of contributions from all N molecules:

process, a quantum mechanical description is necessary.

transition rate τ-1 can be presented as [50, 51]:

**2.1.2 Quantum mechanical description of raman scattering** 

can be rewritten as:

Spectroscopy and Microscopy with Photonic Crystal Fiber Generated Supercontinuum 173

where Q(t) is a simple harmonic oscillator, Q(t)=2Q0cos(ωRt). So the induced dipole moment

 0 0 *<sup>P</sup> P R i t i t t Er e ErQe <sup>Q</sup>*

On the right-hand side of equation (2.3), the first term corresponds to Rayleigh scattering with the same frequency of incident light. The second term describes the Raman frequency shift of ωP±ΩR. Because the Raman frequency shifting term depends on ∂α/∂Q, Raman scattering occurs only when the incident optical field induces a polarizibility change along the specific molecular vibrational mode. This specific mode is an active Raman mode, which

The differential scattering cross-section is one of key parameters to express the intensity of Raman scattering signal. In a solid angle ΔΩ, it can be defined as the amount of scattered

where IRaman is the intensity of Raman scattering light in ΔΩ, V is the volume of scattering medium, N is the molecular density, IP is the intensity of incident optical field. Therefore, the total intensity of the Raman scattering light in whole solid angle can be described as the

> *Raman P diff I NI d*

Obviously, the spontaneous Raman is a linear optical process, because the intensity of scattering signal has the linear relationship with the intensity of incident optical field and number of scattering molecules respectively. The classical description of Raman process only provides a qualitative relationship between the Raman scattering cross-section and intensities of the Raman signals. In order to achieve the quantitative study for the Raman

When the interaction between incident optical field and medium is studied with quantum mechanical method, the molecular system of medium should be quantized. The Raman scattering is a second-order process in which two interaction processes between incident optical field and medium are involved. The quantum mechanical explanation of the Raman scattering process is based on the estimation of the transition rate between the different molecular states. In quantum physics, the Fermi's golden rule is a common way to calculate

proportional to the square modulus of the transition dipole μvg. But in order to describe the Raman process, we need to calculate the second-order transition rate. The second-order

the first-order transition rate between an initial state *g* and a final state

*diff*

*Raman*

*I VNI*

*P*

. (2.3)

, (2.4)

. (2.5)

that is

Rayleigh scattering) from atom or molecule have the same energy (frequency) as the incident photons, as shown in figure 1 (a). However, a small part of the photons (approximately 1 in 10 million photons) are inelastically scattered with the frequencies different from the incident photons [45]. This inelastic scattering of light was theoretical predicted by A. Smekal in 1923 [46]. In 1928, Indian physicist Sir C. V. Raman first discovered this phenomenon when a monochromatic light with frequency of ωP was incident into a medium. He found that the scattered light components contained not only Rayleigh scattering with frequency of ωP, but also some weaker scattering components with frequencies of ωP±ΩR, which come from the inelastic scattering phenomenon of light named as Raman scattering or Raman effect [47, 48], as shown in figure 1 (b) and (c) in energy level diagram. Raman scattering originates from the inherent features of molecular vibration and rotation of individual or groups of chemical bonds. The obtained Raman spectra contain the inherent molecular structural information of the medium and can be used to identify molecules. Because of this significant feature, it has been widely used as a tool for analyzing the composition of liquids, gases, and solids.

Fig. 1. Energy level diagram of the elastic Raleigh scattering (a), the inelastic Stokes Raman scattering (b) and anti-Stoke Raman scattering (c), where ωP, ωS, ωAS and ΩR represents the frequency of incident light, Stokes scattering light, anti-Stoke scattering light and resonance respectively. The ground state level, the vibrational level and the virtual intermediate level is labeled with g, v, and j respectively.

#### **2.1.1 Classical description of raman scattering**

The Raman scattering phenomenon arises from the interactions between the incident photon and the electric dipole of molecules. In classical terms, the interaction can be viewed as fluctuation of the molecule under the influence of the applied optical field. With the used optical field of frequency ωP expressed as *<sup>P</sup> i t Et Ere* , the induced dipole moment of a molecule is:

$$
\mu(\mathbf{t}) = \mathbf{a}(\mathbf{t})\mathbf{E}(\mathbf{t})\tag{2.1}
$$

where α(t) is the polarizibility of the material. When the incident optical field interacts with the molecules, the polarizibility can be expressed as a function of the nuclear coordinate Q, and expanded to the first order in a Taylor series [49]:

$$a(t) = a\_0 + \frac{\partial a}{\partial Q} Q(t) \, \, \, \, \, \, \, \tag{2.2}$$

Rayleigh scattering) from atom or molecule have the same energy (frequency) as the incident photons, as shown in figure 1 (a). However, a small part of the photons (approximately 1 in 10 million photons) are inelastically scattered with the frequencies different from the incident photons [45]. This inelastic scattering of light was theoretical predicted by A. Smekal in 1923 [46]. In 1928, Indian physicist Sir C. V. Raman first discovered this phenomenon when a monochromatic light with frequency of ωP was incident into a medium. He found that the scattered light components contained not only Rayleigh scattering with frequency of ωP, but also some weaker scattering components with frequencies of ωP±ΩR, which come from the inelastic scattering phenomenon of light named as Raman scattering or Raman effect [47, 48], as shown in figure 1 (b) and (c) in energy level diagram. Raman scattering originates from the inherent features of molecular vibration and rotation of individual or groups of chemical bonds. The obtained Raman spectra contain the inherent molecular structural information of the medium and can be used to identify molecules. Because of this significant feature, it has been widely used as a tool for analyzing

ωP ωS

Fig. 1. Energy level diagram of the elastic Raleigh scattering (a), the inelastic Stokes Raman scattering (b) and anti-Stoke Raman scattering (c), where ωP, ωS, ωAS and ΩR represents the frequency of incident light, Stokes scattering light, anti-Stoke scattering light and resonance respectively. The ground state level, the vibrational level and the virtual intermediate level

The Raman scattering phenomenon arises from the interactions between the incident photon and the electric dipole of molecules. In classical terms, the interaction can be viewed as fluctuation of the molecule under the influence of the applied optical field. With the used

where α(t) is the polarizibility of the material. When the incident optical field interacts with the molecules, the polarizibility can be expressed as a function of the nuclear coordinate Q,

> *t Qt* <sup>0</sup> *<sup>Q</sup>*

 

 

g

(b) (c)

v

ωP ωAS

ΩR ΩR

g

, the induced dipole moment of a

, (2.2)

μ(t)=α(t)E(t), (2.1)

v

j

j

the composition of liquids, gases, and solids.

(a)

**2.1.1 Classical description of raman scattering** 

optical field of frequency ωP expressed as *<sup>P</sup> i t Et Ere*

and expanded to the first order in a Taylor series [49]:

g

v

j

ωP ωP

Energy

is labeled with g, v, and j respectively.

molecule is:

where Q(t) is a simple harmonic oscillator, Q(t)=2Q0cos(ωRt). So the induced dipole moment can be rewritten as:

$$
\mu \mu(t) = \alpha\_0 E(r) e^{-i\alpha\_0 t} + \frac{\partial \alpha}{\partial Q} E(r) Q\_0 e^{\pm i(\alpha\_0 \pm \Omega\_R)t} \,. \tag{2.3}
$$

On the right-hand side of equation (2.3), the first term corresponds to Rayleigh scattering with the same frequency of incident light. The second term describes the Raman frequency shift of ωP±ΩR. Because the Raman frequency shifting term depends on ∂α/∂Q, Raman scattering occurs only when the incident optical field induces a polarizibility change along the specific molecular vibrational mode. This specific mode is an active Raman mode, which is the basis of the selection rule of Raman spectroscopy.

The differential scattering cross-section is one of key parameters to express the intensity of Raman scattering signal. In a solid angle ΔΩ, it can be defined as the amount of scattered intensity divided by the incident intensity:

$$
\sigma\_{\text{diff}} = \frac{\partial \sigma}{\partial \Omega} = \frac{I\_{Ramam}}{V N I\_P \Delta \Omega} \,\,\,\,\tag{2.4}
$$

where IRaman is the intensity of Raman scattering light in ΔΩ, V is the volume of scattering medium, N is the molecular density, IP is the intensity of incident optical field. Therefore, the total intensity of the Raman scattering light in whole solid angle can be described as the summation of contributions from all N molecules:

$$I\_{Ramm} = \text{NI}\_P \int \sigma\_{diff} d\Omega \,. \tag{2.5}$$

Obviously, the spontaneous Raman is a linear optical process, because the intensity of scattering signal has the linear relationship with the intensity of incident optical field and number of scattering molecules respectively. The classical description of Raman process only provides a qualitative relationship between the Raman scattering cross-section and intensities of the Raman signals. In order to achieve the quantitative study for the Raman process, a quantum mechanical description is necessary.

#### **2.1.2 Quantum mechanical description of raman scattering**

When the interaction between incident optical field and medium is studied with quantum mechanical method, the molecular system of medium should be quantized. The Raman scattering is a second-order process in which two interaction processes between incident optical field and medium are involved. The quantum mechanical explanation of the Raman scattering process is based on the estimation of the transition rate between the different molecular states. In quantum physics, the Fermi's golden rule is a common way to calculate the first-order transition rate between an initial state *g* and a final state that is proportional to the square modulus of the transition dipole μvg. But in order to describe the Raman process, we need to calculate the second-order transition rate. The second-order transition rate τ-1 can be presented as [50, 51]:

Ultra-Broadband Time-Resolved Coherent Anti-Stokes Raman Scattering

**2.2 CARS process** 

domains [22-26].

third-order nonlinear susceptibility.

ωP ωS

**2.2.1 Classical description of CARS** 

g

along Q is [53]:

v

j

(a)

provided by the incident pump and Stokes fields:

k

ωP' ωAS

Spectroscopy and Microscopy with Photonic Crystal Fiber Generated Supercontinuum 175

The disadvantage of the Raman scattering is the low conversion efficiency due to the small scattering cross-section. Only 1 part out of 106 of the incident photons will be scattered into the Stokes frequency when propagating through 1cm of a typical Raman active medium. It makes Raman spectroscopy and microscopy more complex and costly that limits its broad applications. As one of nonlinear techniques with coherent nature, intensity of CARS signal is about 105 stronger than spontaneous Raman. Therefore, the CARS spectroscopy and microscopy have been widely used in physics, chemistry, biology and many other related

In the CARS process, three laser beams with frequencies of ωP, ωP' and ωS are used as pump, probe and Stokes, the energy level diagram of CARS is shown in figure 2. The primary difference between the CARS and Raman process is that the Stokes frequency stems from an applied laser field in the former. We can simply consider the joint action of the pump and Stokes fields as a source for driving the active Raman mode with the difference frequency ωP-ωS. Here, we will first describe CARS process with the classical model, after that a quantum mechanical explanation will be applied for finding the correct expression of the

ωP ωS

Fig. 2. Energy level diagram of CARS. (a) resonant CARS, (b) nonresonant electronic contribution and (c) electronically enhanced nonresonant contribution. Solid lines indicate

The classical description of an active vibrational mode driven by the incident optical field is a model of damping harmonic oscillator. The equation of motion for the molecular vibration

> <sup>2</sup> 2

where γ is the damping constant, m is the reduced nuclear mass, and F(t) is the external driving force of the oscillation from the incident optical fields. In the CARS process, F(t) is

 

<sup>2</sup> <sup>2</sup> *d Q t dQ t F t Q t dt dt <sup>m</sup>* 

k

ωP' ωAS

g

real states (g and ν); dashed lines denote virtual states (j and k).

v

j

ωP

ωP'

<sup>j</sup> ωAS

, (2.10)

g

(b) (c)

ΩR ΩR ΩR

v

ωS

$$\frac{1}{\tau} = \sum\_{\nu} \sum\_{R} \frac{\pi e^{4} \alpha\_{\text{P}} \alpha\_{\text{R}} \eta\_{\text{R}}}{2 \varepsilon\_{0}^{2} \hbar^{2} V^{2}} \left| \sum\_{j} \left\{ \frac{\mu\_{\text{v}} \mu\_{\text{jg}}}{\alpha\_{j} - \alpha\_{\text{P}}} + \frac{\mu\_{\text{v}} \mu\_{\text{jg}}}{\alpha\_{j} + \Omega\_{\text{R}}} \right\} \right|^{2} \delta \left( \alpha\_{\text{v}} + \alpha\_{\text{R}} - \alpha\_{\text{P}} \right), \tag{2.6}$$

where e is the electron charge, ε0 is the vacuum permittivity, is Planck's constant, nR is the refractive index at Raman frequency, and δ is the Dirac delta function. ωP is the frequency of incident optical field and ωR is the frequency of Raman scattering light. The frequencies ων and ωj are the transition frequencies from the ground state to the final state and intermediate state *j* , respectively.

In Raman scattering process, an incident optical field first converts the material system from the ground state *g* to an intermediate state *j* , which is an artificial virtual state. Then, the transition from the intermediate state *j* to the final state happens that is considered as an instantaneous process. A full description of Raman scattering thus incorporates a quantized field theory [52]. From the quantized field theory, we can find the number of photon modes at frequency of ωR in volume of medium V [52], and perform the summation over R in equation (2.6). From the argument of the Dirac delta function, the only nonzero contributions are the ones for which the emission frequency ωR=ωP-ων, the redshifted Stokes frequencies. With σdiff=d/dΩ(V/cnRτ), the expression for the transition rate can be directly presented with the differential scattering cross-section [51]:

$$\sigma\_{\rm diff} = \sum\_{\nu} \frac{e^4 \alpha\_{\rm P} \alpha\_{\rm R}^3}{16\pi^2 \varepsilon\_0^2 \hbar^2 \text{c}^4} \left| \sum\_{j} \left\{ \frac{\mu\_{\nu j} \mu\_{\rm jg}}{\alpha\_j - \alpha\_{\rm P}} + \frac{\mu\_{\nu j} \mu\_{\rm jg}}{\alpha\_j + \alpha\_{\rm R}} \right\} \right|^2. \tag{2.7}$$

This is the Kramers-Heisenberg formula, which is very important in quantum mechanical description of light scattering [50]. For Raman scattering process, the differential scattering cross-section for a particular vibrational state can be simplified to:

$$
\sigma\_{\rm diff} = \frac{\alpha\_{\rm P} \alpha\_{\rm R}^3}{16 \pi \varepsilon\_0^2 \hbar^2 \text{c}^4} \left| a\_{\rm R} \right|^2 \,, \tag{2.8}
$$

where the Raman transition polarizibility αR can be written as:

$$\alpha\_{\mathbb{R}} = \sum\_{j} \left\{ \frac{\mu\_{\nu^j} \mu\_{j\mathbb{\hat{S}}}}{\alpha\_j - \alpha\_{\mathbb{P}}} + \frac{\mu\_{\nu^j} \mu\_{j\mathbb{\hat{S}}}}{\alpha\_j + \alpha\_{\mathbb{R}}} \right\}. \tag{2.9}$$

From the above discussion, we can know that Spontaneous Raman scattering is a weak effect because the spontaneous interaction through the vacuum field occurs only rarely. Although Raman scattering is a second-order process, the intensity of Raman signal depends linearly on the intensity of the incident optical field. When the frequency approaches the frequency of a real electronic state of molecule, the Raman scattering is very strong, which is known as the resonant Raman and is one of effective methods to improve the efficiency of Raman scattering. If the spontaneous nature of the j→ν transition can be eliminated by using a second field of frequency ωR, the weak Raman scattering can also be enhanced, such as CARS process.

### **2.2 CARS process**

174 Photonic Crystals – Innovative Systems, Lasers and Waveguides

 

, (2.6)

 

 

*P RR j jg j jg*

where e is the electron charge, ε0 is the vacuum permittivity, is Planck's constant, nR is the refractive index at Raman frequency, and δ is the Dirac delta function. ωP is the frequency of incident optical field and ωR is the frequency of Raman scattering light. The frequencies ων

In Raman scattering process, an incident optical field first converts the material system from the ground state *g* to an intermediate state *j* , which is an artificial virtual state. Then,

considered as an instantaneous process. A full description of Raman scattering thus incorporates a quantized field theory [52]. From the quantized field theory, we can find the number of photon modes at frequency of ωR in volume of medium V [52], and perform the summation over R in equation (2.6). From the argument of the Dirac delta function, the only nonzero contributions are the ones for which the emission frequency ωR=ωP-ων, the redshifted Stokes frequencies. With σdiff=d/dΩ(V/cnRτ), the expression for the transition rate

*P R j jg j jg*

This is the Kramers-Heisenberg formula, which is very important in quantum mechanical description of light scattering [50]. For Raman scattering process, the differential scattering

3

224 <sup>0</sup> 16 c *P R diff R* 

 

*j jg j jg*

 

 

 

*j j P j R*

From the above discussion, we can know that Spontaneous Raman scattering is a weak effect because the spontaneous interaction through the vacuum field occurs only rarely. Although Raman scattering is a second-order process, the intensity of Raman signal depends linearly on the intensity of the incident optical field. When the frequency approaches the frequency of a real electronic state of molecule, the Raman scattering is very strong, which is known as the resonant Raman and is one of effective methods to improve the efficiency of Raman scattering. If the spontaneous nature of the j→ν transition can be eliminated by using a second field of frequency ωR, the weak Raman scattering can also be

*j jP jR*

2

 

can be simplified to:

 

 

 . (2.7)

*R j jP j R*

and ωj are the transition frequencies from the ground state to the final state

the transition from the intermediate state *j* to the final state

can be directly presented with the differential scattering cross-section [51]:

*e*

 

where the Raman transition polarizibility αR can be written as:

*R*

*diff*

cross-section for a particular vibrational state

enhanced, such as CARS process.

4 3 2224 <sup>0</sup> 16 c

 

4

2

 

1

intermediate state *j* , respectively.

22 2 0

*e n V*

*R P*

2

, (2.8)

. (2.9)

happens that is

and

2

 The disadvantage of the Raman scattering is the low conversion efficiency due to the small scattering cross-section. Only 1 part out of 106 of the incident photons will be scattered into the Stokes frequency when propagating through 1cm of a typical Raman active medium. It makes Raman spectroscopy and microscopy more complex and costly that limits its broad applications. As one of nonlinear techniques with coherent nature, intensity of CARS signal is about 105 stronger than spontaneous Raman. Therefore, the CARS spectroscopy and microscopy have been widely used in physics, chemistry, biology and many other related domains [22-26].

In the CARS process, three laser beams with frequencies of ωP, ωP' and ωS are used as pump, probe and Stokes, the energy level diagram of CARS is shown in figure 2. The primary difference between the CARS and Raman process is that the Stokes frequency stems from an applied laser field in the former. We can simply consider the joint action of the pump and Stokes fields as a source for driving the active Raman mode with the difference frequency ωP-ωS. Here, we will first describe CARS process with the classical model, after that a quantum mechanical explanation will be applied for finding the correct expression of the third-order nonlinear susceptibility.

Fig. 2. Energy level diagram of CARS. (a) resonant CARS, (b) nonresonant electronic contribution and (c) electronically enhanced nonresonant contribution. Solid lines indicate real states (g and ν); dashed lines denote virtual states (j and k).
