**2.2.1 Classical description of CARS**

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 along Q is [53]:

$$\frac{d^2Q(t)}{dt^2} + 2\gamma \frac{dQ(t)}{dt} + \alpha\_\nu^2 Q(t) = \frac{F(t)}{m},\tag{2.10}$$

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 provided by the incident pump and Stokes fields:

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

quadratically with the third-order susceptibility of the medium.

frequency of a vibrational mode, ωP−ωS=ων,

mechanical description of the CARS process.

cn(t), and \* 

*nm n m t c tc t* .

and with the relation <sup>3</sup> 2 \* 3 

**2.2.2 Quantum mechanical description of CARS** 

system is usually expressed in terms of the density operator:

 

The expectation value for the electric dipole moment is then given by:

 *<sup>r</sup> N t EEP S* :

 3 *r*

Spectroscopy and Microscopy with Photonic Crystal Fiber Generated Supercontinuum 177

It is an inherent property of medium that describes the medium's response to the incident optical fields. When the frequency difference of incident optical fields matches with

> 

The intensity of CARS signal is proportional to the square modulus of the total polarization:

 2 2 3 3 <sup>2</sup> <sup>9</sup> *CARS AS r AS P S I P*

It scales quadratically with the pump intensity, linearly with the Stokes intensity, and

Although the classical description of CARS can provide a picture of CARS process and a simplified relationships among the medium, intensities of incident optical fields and CARS signals, it is unable to account for the interaction of the fields with the quantized states of the molecule. More accurate numerical estimates can only be achieved with a quantum

The quantum mechanism of CARS process can be effectively described by the timedependent third-order perturbation theory. In the quantum mechanical description, the

where the wave functions are expanded in a basis set *n* with time-dependent coefficients

 .

 

The third-order nonlinear susceptibility for the CARS process is found by calculating the third-order correction to the density operator through time-dependent perturbation theory

*n m*

*N A V i*

where Γν is the vibrational decay rate that is associated with the line width of the Raman

 2 4 0

where nP and nS are the refractive index at the pump and Stokes frequency, ρgg and ρνν is the element of the density matrix of the ground state and vibrationally excited state,

 

mode R. The amplitude Aν can be related to the differential scattering cross-section:

*PSS c n A R n*

 

  *mn nm*

<sup>3</sup> , *<sup>P</sup> gg diff*

 

*P S*

, (2.20)

, (2.21)

*nm*

 

*t t t tnm* , (2.18)

*t t* . (2.19)

*nm*

 

<sup>3</sup>

*r AS* will maximize.

*I I* . (2.17)

$$F(t) = \left(\frac{\partial \alpha}{\partial \mathbf{Q}}\right)\_0 E\_P E\_S^\* e^{-i(a\boldsymbol{\wp} - a\boldsymbol{\wp}\_\circ)t} \,\,\,\,\,\tag{2.11}$$

where the time-varying driven force oscillates at the beat frequency of the incident optical fields. A solution to equation (2.10) can be written as:

$$Q = Q(\alpha\_{\mathcal{P}} - \alpha\_{\mathcal{S}}) \left( e^{-i(\alpha\_{\mathcal{P}} - \alpha\_{\mathcal{S}})t} + e^{i(\alpha\_{\mathcal{P}} + \alpha\_{\mathcal{S}})t} \right) \dots$$

where Q(ωP-ωS) is the amplitude of the molecular vibration. Then, from the equation (2.10), it can be worked out:

$$Q\left(\alpha\_{\rm P} - \alpha\_{\rm S}\right) = \frac{\left(1/m\right)\left[\left\|\alpha/\partial Q\right\|\_{0}\right]E\_{\rm P}E\_{\rm S}^{\rm e}}{\alpha\_{\rm V}^{2} - \left(\alpha\_{\rm P} - \alpha\_{\rm S}\right)^{2} - 2i\left(\alpha\_{\rm P} - \alpha\_{\rm S}\right)\chi}.\tag{2.12}$$

From the equation (2.12), we know that the amplitude of molecular vibration is proportional to the product of the amplitudes of the driving fields and the polarizability change. When the frequency difference of the pump and the Stokes fields equals to the resonant frequency ων, the molecular vibration of active Raman mode will be resonantly enhanced. When a probe field with frequency of ωPr passes through the medium, it will be modulated by the resonant enhanced molecular vibrational mode, resulting in a component at the anti-Stokes frequency, ωPr+ωP−ωS.

The total nonlinear polarization is the summation of all N dipoles:

$$P(t) = N\mu(t) = N\left[\left(\frac{\partial\alpha}{\partial Q}\right)\_0 Q\right] E\_{\text{Pr}}(t) \,. \tag{2.13}$$

In order to simplify the experimental system, the pump field provides the probe field. The frequency of generated anti-Stokes signal is ωAS=2ωP−ωS. The total nonlinear polarization can be written as:

$$P(t) = P(o\_{AS})e^{-i(o\_{AS})t} \,. \tag{2.14}$$

With (2.12), (2.13), and (2.14), we can deduce the amplitude of total nonlinear polarization:

$$P\left(o\_{\rm AS}\right) = \frac{\left(N/m\right)\left[\partial\alpha/\partial Q\right]\_0^2}{\alpha\_\text{\nu}^2 - \left(o\_\text{P} - o\_\text{S}\right)^2 - 2i\left(o\_\text{P} - o\_\text{S}\right)\gamma} E\_P^2 E\_\text{S}^\* = 3\,\chi\_r^{(3)}\left(o\_{\rm AS}\right) E\_P^2 E\_\text{S}^\*.\tag{2.15}$$

From above discussion, we know that the amplitude of the total nonlinear polarization is proportional to the product of three incident optical fields. Here, we define the vibrational resonant third-order susceptibility <sup>3</sup> *r AS* :

$$\left(\mathcal{Z}\_r^{(3)}\right)\left(\mathcal{o}\_{AS}\right) = \frac{\left(N/\Im m\right)\left[\partial\mathcal{A}/\partial Q\right]\_0^2}{\left(\mathcal{o}\_\mathcal{V} - \left(\mathcal{o}\_\mathcal{P} - \mathcal{o}\_\mathcal{S}\right)^2 - 2i\left(\mathcal{o}\_\mathcal{P} - \mathcal{o}\_\mathcal{S}\right)\mathcal{Y}}.\tag{2.16}$$

It is an inherent property of medium that describes the medium's response to the incident optical fields. When the frequency difference of incident optical fields matches with frequency of a vibrational mode, ωP−ωS=ων, <sup>3</sup> *r AS* will maximize.

The intensity of CARS signal is proportional to the square modulus of the total polarization:

$$I\_{\rm CARS} \propto \left| P^{(3)} \left( \rho\_{\rm AS} \right) \right|^2 = 9 \left| \mathcal{Z}\_r^{(3)} \left( \rho\_{\rm AS} \right) \right|^2 I\_P^2 I\_S \,. \tag{2.17}$$

It scales quadratically with the pump intensity, linearly with the Stokes intensity, and quadratically with the third-order susceptibility of the medium.

Although the classical description of CARS can provide a picture of CARS process and a simplified relationships among the medium, intensities of incident optical fields and CARS signals, it is unable to account for the interaction of the fields with the quantized states of the molecule. More accurate numerical estimates can only be achieved with a quantum mechanical description of the CARS process.

#### **2.2.2 Quantum mechanical description of CARS**

176 Photonic Crystals – Innovative Systems, Lasers and Waveguides

 \* 0 *P S i t Ft EEe P S <sup>Q</sup>*

where the time-varying driven force oscillates at the beat frequency of the incident optical

 *PS PS i ti t QQ e e P S* 

where Q(ωP-ωS) is the amplitude of the molecular vibration. Then, from the equation (2.10),

2 2 1

From the equation (2.12), we know that the amplitude of molecular vibration is proportional to the product of the amplitudes of the driving fields and the polarizability change. When the frequency difference of the pump and the Stokes fields equals to the resonant frequency ων, the molecular vibration of active Raman mode will be resonantly enhanced. When a probe field with frequency of ωPr passes through the medium, it will be modulated by the resonant enhanced molecular vibrational mode, resulting in a component at the anti-Stokes

Pr

In order to simplify the experimental system, the pump field provides the probe field. The frequency of generated anti-Stokes signal is ωAS=2ωP−ωS. The total nonlinear polarization

*AS i t Pt P e AS*

With (2.12), (2.13), and (2.14), we can deduce the amplitude of total nonlinear polarization:

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

 

2

From above discussion, we know that the amplitude of the total nonlinear polarization is proportional to the product of three incident optical fields. Here, we define the vibrational

*Nm Q <sup>P</sup> E E E E i*

*AS P S r AS P S*

<sup>3</sup>

3 0 2 2 3

 

 *r AS* :

 *PS PS*

*Pt N t N QE t <sup>Q</sup>*

0

*Nm Q*

2

*i*

 

*PS PS*

 

,

 

2

*i*

 

*PS PS*

0

 

\*

. (2.12)

. (2.14)

. (2.16)

0 2 \* 3 2 \*

. (2.15)

2

 

. (2.13)

*P S*

, (2.11)

 

*m Q EE <sup>Q</sup>*

 

*P S*

The total nonlinear polarization is the summation of all N dipoles:

 

*r AS*

 

resonant third-order susceptibility

 

fields. A solution to equation (2.10) can be written as:

it can be worked out:

frequency, ωPr+ωP−ωS.

can be written as:

The quantum mechanism of CARS process can be effectively described by the timedependent third-order perturbation theory. In the quantum mechanical description, the system is usually expressed in terms of the density operator:

$$\rho\left(t\right) \equiv \left|\psi\left(t\right)\right\rangle \left|\psi\left(t\right)\right| = \sum\_{nm} \rho\_{nm}\left(t\right) \left|n\right\rangle \left\langle m \right|,\tag{2.18}$$

where the wave functions are expanded in a basis set *n* with time-dependent coefficients cn(t), and \* *nm n m t c tc t* .

The expectation value for the electric dipole moment is then given by:

$$\left\langle \mu \left( t \right) \right\rangle = \sum\_{n.m} \mu\_{mn} \rho\_{nm} \left( t \right) \,. \tag{2.19}$$

The third-order nonlinear susceptibility for the CARS process is found by calculating the third-order correction to the density operator through time-dependent perturbation theory and with the relation <sup>3</sup> 2 \* 3 *<sup>r</sup> N t EEP S* :

$$\mathcal{X}\_r^{(\mathfrak{z})} = \frac{N}{V} \sum\_{\nu} \frac{A\_{\nu}}{a\_{\nu} - \left(a\_{\mathcal{P}} - a\_{\mathcal{S}}\right) - i\Gamma\_{\nu}} \, \, \, \tag{2.20}$$

where Γν is the vibrational decay rate that is associated with the line width of the Raman mode R. The amplitude Aν can be related to the differential scattering cross-section:

$$A\_{\nu} \propto \frac{\left(\pi \varepsilon\_{0}\right)^{2} \varepsilon^{4} n\_{P}}{\hbar o\_{P} o\_{\rm S}^{3} n\_{S}} \left(\rho\_{\rm g\rm g} - \rho\_{\rm \nu \nu}\right) \sigma\_{\rm diff} \, ^{\prime} \mathsf{R} \, \, ^{\prime} \mathsf{R} \, \, ^{\prime} \tag{2.21}$$

where nP and nS are the refractive index at the pump and Stokes frequency, ρgg and ρνν is the element of the density matrix of the ground state and vibrationally excited state,

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

for practical applications.

**Epi-detection [54, 55]**

**2.3.2 Suppression of NRB noise** 

tissues, this method will not work. **Polarization-sensitive detection** 

nonresonant field is (3) (3)

resonant field is (3) (3)

1221 1111 *r r*

 

function of the angle between the pump and Stokes fields:

Spectroscopy and Microscopy with Photonic Crystal Fiber Generated Supercontinuum 179

polarization at 2ωP-ωS is established via field interactions with virtual levels. When 2ωP is close to the frequency of a real electronic state, it is a nonresonant two-photon enhanced electronic contribution, as shown in figure 2 (c). The nonresonant contribution is a source of background that limits the sensitivity of CARS microscopy. For weak vibrational resonances, the nonresonant background may overwhelm the resonant information. In biological samples, the concentration of interested molecules usually is low, while the nonresonant background from the aqueous surrounding is generally ubiquitous. The mixing of the nonresonant field with the resonant field gives rise to the broadened and distorted spectral line shapes. Therefore, the suppression of the nonresonant contribution is essential

Several effective methods have been developed in order to suppress the NRB noise. Here,

In samples, every object will be the source of NRB noise. The aqueous environment produces an extensive NRB noise that may be stronger than the resonant CARS signal from a small object in focus. Because the epi-CARS (E-CARS) has the size-selective mechanism, the NRB noise from the aqueous surrounding can be suppressed while the signal from small objects will be retained. It should be noted that the NRB noise can not be directly reduced in E-CARS. When samples have comparative sizes or in a highly scattering media, such as

The polarization-sensitive detection CARS (P-CARS) is based on the different polarization properties of the resonant CARS and nonresonant signals to effectively suppress the NRB noise [56-58]. According to the Kleinman's symmetry, the depolarization ratio of the

vary from 1/3. The nonlinear polarization, polarized at an angle θ, can be written as a

2 \*

where i is either the resonant or nonresonant component. The nonresonant field is linearly polarized along an angle θnr= tan-1(tan(φ)/3). When detecting the signal at an angle

2 \*

When φ=71.6° and θnr=45°, the ratio of resonant and nonresonant signals reaches the maximum. Under this condition, the nonresonant background can be negligible. The P-

<sup>3</sup> cos sin 1 3

 

*i ii P <sup>x</sup> x PS*

 

orthogonal to the linearly polarized nonresonant background, the resonant signal is:

*rr r P*

cos tan

 

*nr* [59]. However, the depolarization ratio of

 *nr E EP S* . (2.25)

*e e EE* , (2.24)

*<sup>r</sup>* , which depends on the symmetry of the molecule and may

1221 1111 1 3 *nr nr*

1111

1111

 

3

4

4

 

 

we briefly discuss several widely used techniques for suppressing the NRB noise.

respectively. The CARS signal intensity is again estimated by substituting equation (2.20) into (2.7).

The quantum mechanical description of CARS process can be qualitatively presented by considering the time-ordered action of each laser field on the density matrix ρnm(t). Each electric field interaction establishes a coupling between two quantum mechanical states of the molecule, changing the state of the system as described by the density matrix. Before interaction with the laser fields, the system resides in the ground state ρgg. An interaction with the pump field changes the system to ρjg. Then the system is converted into ρνg by the following Stokes field. The density matrix now oscillates at frequency ωvg=ωjg−ωvj that is a coherent vibration. When the third incident optical field interact with medium, the coherent vibration can be converted into a radiating polarization ρkg, which propagates at ωkg=ωjg+ωvg. After emission of the radiation, the system is brought back to the ground state.

As a coherent Raman process, the intensity of CARS signal is more than five orders of magnitude greater than that of spontaneous Raman scattering process. Because the radiating polarization is a coherent summation, the intensity of CARS signal is quadratic in the number of Raman scattering. Because of the coherence, the CARS signal is in certain direction that allows a much more efficient signal collection than Raman scattering. CARS signal is blue-shifted from incident beams, which avoids the influence from any one-photon excited fluorescence.

#### **2.3 Resonant and nonresonant signals in CARS**

#### **2.3.1 Source of nonresonant background signals**

From the theory of the CARS process, we can know that CARS signal comes from the thirdorder nonlinear susceptibility. The total CARS signal is proportional to the square modulus of the nonlinear susceptibility [46]:

$$I\left(\boldsymbol{\alpha}\_{\rm AS}\right) \propto \left|\boldsymbol{\chi}^{\rm (3)}\left(\boldsymbol{\alpha}\_{\rm AS}\right)\right|^2 = \left|\boldsymbol{\chi}\_r^{\rm (3)}\left(\boldsymbol{\alpha}\_{\rm AS}\right)\right|^2 + \left|\boldsymbol{\chi}\_{\rm nr}^{\rm (3)}\right|^2 + 2\left|\boldsymbol{\chi}\_{\rm nr}^{\rm (3)}\right| \operatorname{Re}\left(\boldsymbol{\chi}\_r^{\rm (3)}\left(\boldsymbol{\alpha}\_{\rm AS}\right)\right). \tag{2.22}$$

The total third-order nonlinear susceptibility is composed of a resonant ( <sup>3</sup> *<sup>r</sup>* ) and a nonresonant ( <sup>3</sup> *nr* ) part:

$$\mathcal{X}\_{(3)} = \mathcal{X}\_{(3)}^{\prime} + \mathcal{X}\_{(3)}^{\prime\prime},\tag{2.23}$$

where resonance 3 *<sup>r</sup>* is a complex quantity, 33 3 ' '' *rr r i* and represents the Raman response of the molecules. When the frequency difference between the pump and Stokes fields equals to the vibrational frequency of an active Raman mode, a strong CARS signal is induced. It provides the inherent vibrational contrast mechanism of CARS microscopy. However, it is not the only components in the total anti-Stokes radiations. In the absence of active Raman modes, the electron cloud still has oscillating components, at the anti-Stokes frequency ωAS =2ωP-ωS, coupling with the radiation field. It is the purely electronic nonresonant contribution from 3 *nr* that is frequency-independent and a real quality. Two energy level diagrams of nonresonant contribution are depicted in figure 2 (b) and (c), when all three incident optical fields overlap in time. As shown in figure 2 (b), a radiating polarization at 2ωP-ωS is established via field interactions with virtual levels. When 2ωP is close to the frequency of a real electronic state, it is a nonresonant two-photon enhanced electronic contribution, as shown in figure 2 (c). The nonresonant contribution is a source of background that limits the sensitivity of CARS microscopy. For weak vibrational resonances, the nonresonant background may overwhelm the resonant information. In biological samples, the concentration of interested molecules usually is low, while the nonresonant background from the aqueous surrounding is generally ubiquitous. The mixing of the nonresonant field with the resonant field gives rise to the broadened and distorted spectral line shapes. Therefore, the suppression of the nonresonant contribution is essential for practical applications.
