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

178 Photonic Crystals – Innovative Systems, Lasers and Waveguides

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

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

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

2 22 3 3 3 33 2 Re *AS AS r AS nr nr r AS <sup>I</sup>*

333

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

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

 

*<sup>r</sup>* is a complex quantity, 33 3 ' '' *rr r*

 

 

 

 

*nr* that is frequency-independent and a real quality. Two

. (2.22)

 

*r nr* , (2.23)

*i* and represents the Raman

*<sup>r</sup>* ) and a

 

The total third-order nonlinear susceptibility is composed of a resonant ( <sup>3</sup>

into (2.7).

excited fluorescence.

nonresonant ( <sup>3</sup>

where resonance 3

of the nonlinear susceptibility [46]:

*nr* ) part:

nonresonant contribution from 3

**2.3 Resonant and nonresonant signals in CARS 2.3.1 Source of nonresonant background signals** 

> 

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 tissues, this method will not work.

#### **Polarization-sensitive detection**

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 nonresonant field is (3) (3) 1221 1111 1 3 *nr nr nr* [59]. However, the depolarization ratio of resonant field is (3) (3) 1221 1111 *r r <sup>r</sup>* , which depends on the symmetry of the molecule and may vary from 1/3. The nonlinear polarization, polarized at an angle θ, can be written as a function of the angle between the pump and Stokes fields:

$$P^{i}\left(\theta\right) = \frac{3}{4} \mathcal{Z}\_{1111}{}^{i} \cos\phi \left\{\overline{e}\_{\text{x}} + \rho^{i} \tan\phi \overline{e}\_{\text{x}}\right\} E\_{P}^{2} E\_{S}^{\*}\,. \tag{2.24}$$

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

$$P^{r}\left(\theta\right) = \frac{3}{4} \mathcal{X}\_{1111}{}^{r} \cos\phi \sin\theta\_{nr} \left(1 - 3\rho^{r}\right) \mathcal{E}\_{P}^{2} \mathcal{E}\_{S}^{\*}\,. \tag{2.25}$$

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-

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

**2.4 Condition of momentum conservation: Phase-matching** 

phase conditions of focused fields.

ωP ωS

smaller than the interaction length.

P

Spectroscopy and Microscopy with Photonic Crystal Fiber Generated Supercontinuum 181

Unlike fluorescence or spontaneous Raman microscopy, the CARS process is a parametric process, in which the conditions of energy and momentum must conserve. The generation of CARS signals thus relies on not only the intensity of focused incident optical fields, but the

 l<<lC=π/|Δk|, (2.26) where l is the effective interaction length, lC is the coherent length, the wave-vector mismatch Δk=kAS-(kP+kP'-kS), kP,kP',kS and kAS is the wave-vector of pump, probe, Stokes and anti-Stokes field respectively. Under the tight focusing condition, in the CARS microscopy with collinear geometry, a small excitation volume and a large cone angle of wave-vectors compensate the wave-vector mismatch induced by the spectral dispersion of the refractive index of the sample, and the phase matching condition can easily be fulfilled [73, 74]. Therefore, the collinear geometry is the best configuration choice of CARS microscopy.

Detector

Fig. 4. Schematics of three typical CARS microscopy. (a) forward-detection CARS (F-CARS), (b) epi-CARS (E-CARS), and (c) counterpropagating CARS (C-CARS) microscopy. P,

Three typical geometries of CARS microscopes are shown in figure 4. In figure 4(a) and (b), both the pump and Stokes beams collinearly propagate, and the anti-Stokes signal is detected in the forward direction (a) and backward direction (b). For the forward-detection CARS (F-CARS), the phase matching condition can be easily fulfilled by using an objective with high numerical aperture (NA). For the epi-detection CARS (E-CARS) and counterpropagating CARS (C-CARS) (c) with collinearly propagating geometry, large wavevector mismatch is introduced and is |Δk| = 2|kAS| = 4nπ/λAS, and |Δk| = 2|kS| =4nπ/λS. Here, n is the refractive index of medium assumed to be independent of frequency. Therefore, the latter two CARS microscopes have higher sensitivity for object of much

(b) E-CARS

ωP ωS

OL

polarizer; D, dichroic beam splitter; OL, objective lens; S, specimen; F, filter.

S

OL

F

ωAS

Detector

(a) F-CARS

OL

S

OL

F

D

ωAS

Detector

(c) C-CARS

ωS

OL

F

D

ωAS

S

ωP

CARS has been successfully applied in spectroscopy and microscopy [60]. A schematic of a typical P-CARS system is shown in figure 3.

Fig. 3. Schematic of P-CARS. P, polarizer; HW, half-wave plate; D, dichroic beam splitter; OL, objective lens; S, specimen; F, filter.

In a P-CARS system, an analyzer in front of the detector is used to block the nonresonant signal, while the portion of the differently polarized resonant signal passes through the analyzer. Although the P-CARS can effectively suppress the NRB noise, the acquisition time is longer because of the loss of resonant signals

#### **Time-resolved CARS detection**

In the time-resolved CARS detection (T-CARS), the ultra-short laser pulse is used as excitation laser. The resonant and nonresonant contributions are separated in the time domain due to the different temporal response characteristics [61, 62]. Because of the instantaneous dephasing characteristics of nonresonant signal, it exists only when three laser pulses temporally overlap. In T-CARS, a pair of temporally overlapped laser pulses is used as the pump and Stokes pulse to resonantly enhance the molecular vibration. A laser pulse with time delay is used as the probe pulse. The resonant CARS signal decays in the finite dephasing time of the vibrational mode. The dephasing time is related to the width of spectral line of the corresponding Raman band and is typically several hundred femtoseconds (in solid) to a few picoseconds (in gas or liquid) [63]. Therefore, the NRB noise can be eliminated by introducing a suitable time delay between the pump/Stokes and probe pulses [64]. The detail discussions will be given in the next section.

#### **Phase control**

In the phase control method, a phase-mismatched coherent addition of nonresonant spectral components is introduced with phase shaping of the femtosecond laser pulses to suppress the nonresonant signal [65-67]. For CARS imaging with picosecond pulses, the phase control can be achieved by heterodyning the signal with a reference beam at the anti-Stokes wavelength [68, 69]. With the heterodyne CARS interferometry, the imaginary part of the third-order nonlinear susceptibility (Im{χ<sup>r</sup> (3)}) can be separated to suppress NRB noise [70-72].

#### **2.4 Condition of momentum conservation: Phase-matching**

180 Photonic Crystals – Innovative Systems, Lasers and Waveguides

CARS has been successfully applied in spectroscopy and microscopy [60]. A schematic of a

Fig. 3. Schematic of P-CARS. P, polarizer; HW, half-wave plate; D, dichroic beam splitter;

In a P-CARS system, an analyzer in front of the detector is used to block the nonresonant signal, while the portion of the differently polarized resonant signal passes through the analyzer. Although the P-CARS can effectively suppress the NRB noise, the acquisition time

In the time-resolved CARS detection (T-CARS), the ultra-short laser pulse is used as excitation laser. The resonant and nonresonant contributions are separated in the time domain due to the different temporal response characteristics [61, 62]. Because of the instantaneous dephasing characteristics of nonresonant signal, it exists only when three laser pulses temporally overlap. In T-CARS, a pair of temporally overlapped laser pulses is used as the pump and Stokes pulse to resonantly enhance the molecular vibration. A laser pulse with time delay is used as the probe pulse. The resonant CARS signal decays in the finite dephasing time of the vibrational mode. The dephasing time is related to the width of spectral line of the corresponding Raman band and is typically several hundred femtoseconds (in solid) to a few picoseconds (in gas or liquid) [63]. Therefore, the NRB noise can be eliminated by introducing a suitable time delay between the pump/Stokes and probe

In the phase control method, a phase-mismatched coherent addition of nonresonant spectral components is introduced with phase shaping of the femtosecond laser pulses to suppress the nonresonant signal [65-67]. For CARS imaging with picosecond pulses, the phase control can be achieved by heterodyning the signal with a reference beam at the anti-Stokes wavelength [68, 69]. With the heterodyne CARS interferometry, the imaginary

(3)}) can be separated to suppress

typical P-CARS system is shown in figure 3.

OL, objective lens; S, specimen; F, filter.

**Time-resolved CARS detection** 

**Phase control** 

NRB noise [70-72].

is longer because of the loss of resonant signals

pulses [64]. The detail discussions will be given in the next section.

part of the third-order nonlinear susceptibility (Im{χ<sup>r</sup>

Unlike fluorescence or spontaneous Raman microscopy, the CARS process is a parametric process, in which the conditions of energy and momentum must conserve. The generation of CARS signals thus relies on not only the intensity of focused incident optical fields, but the phase conditions of focused fields.

$$1 \lhd \mathfrak{l}\_{\mathbb{C}} = \mathfrak{n} / \left| \Delta \mathbf{k} \right| , \tag{2.26}$$

where l is the effective interaction length, lC is the coherent length, the wave-vector mismatch Δk=kAS-(kP+kP'-kS), kP,kP',kS and kAS is the wave-vector of pump, probe, Stokes and anti-Stokes field respectively. Under the tight focusing condition, in the CARS microscopy with collinear geometry, a small excitation volume and a large cone angle of wave-vectors compensate the wave-vector mismatch induced by the spectral dispersion of the refractive index of the sample, and the phase matching condition can easily be fulfilled [73, 74]. Therefore, the collinear geometry is the best configuration choice of CARS microscopy.

Fig. 4. Schematics of three typical CARS microscopy. (a) forward-detection CARS (F-CARS), (b) epi-CARS (E-CARS), and (c) counterpropagating CARS (C-CARS) microscopy. P, polarizer; D, dichroic beam splitter; OL, objective lens; S, specimen; F, filter.

Three typical geometries of CARS microscopes are shown in figure 4. In figure 4(a) and (b), both the pump and Stokes beams collinearly propagate, and the anti-Stokes signal is detected in the forward direction (a) and backward direction (b). For the forward-detection CARS (F-CARS), the phase matching condition can be easily fulfilled by using an objective with high numerical aperture (NA). For the epi-detection CARS (E-CARS) and counterpropagating CARS (C-CARS) (c) with collinearly propagating geometry, large wavevector mismatch is introduced and is |Δk| = 2|kAS| = 4nπ/λAS, and |Δk| = 2|kS| =4nπ/λS. Here, n is the refractive index of medium assumed to be independent of frequency. Therefore, the latter two CARS microscopes have higher sensitivity for object of much smaller than the interaction length.

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

conditions for realizing an ideal SC source are discussed.

controllable dispersion.

microscope.

**3.2 Numerical modeling** 

**3.1 Photonic crystal fiber used for supercontinuum generation** 

because of its high nonlinearity and group velocity dispersion effects.

Spectroscopy and Microscopy with Photonic Crystal Fiber Generated Supercontinuum 183

supercontinuum (SC) generation, was first demonstrated in the early 1970s [85-87]. With the advent of a new kind of optical waveguides in the late 1990s, photonic crystal fiber (PCF) has led to a great revolution in the generation of SC with ultra-broad spectral range and high brightness [39, 88, 89]. In this section, we will introduce the SC generation with PCFs by theoretical analysis and modeling. Based on the requirements of CARS, the method and

SC generation involves many nonlinear optical effects, such as self- and cross-phase modulation, four-wave mixing (FWM), stimulated Raman scattering (SRS) and solitonic phenomena, which add up to produce a output with an ultra-broadband spectra, sometimes spanning over a couple of octaves. With the developments of theories and techniques of modern nonlinear optics, various optical materials are realized and widely used in various fields. A photonic crystal fiber (PCF) (also called holey fiber (HF) or microstructure fiber (MF)) [90-92], based on the properties of two-dimension photonic crystal, is a kind of special optical fiber, which can confine the incident light passing through the entire length of fiber with its tiny and closely spaced air holes. Different arrangements of the air holes make PCFs with various optical characters, such as the single-mode propagation, high nonlinearity, and

According to different guiding mechanism, there are mainly two categories of PCF, photonic bandgap (PBG) PCF [93] and the total internal reflection (TIR) PCF, as shown in figure 5. The PBG PCF is usually used for transmission of high-energy laser pulses and optical signals. The major energy propagates through the hollow core of a PBG PCF with low loss, dispersion and nonlinear effects. The TIR PCF is used for the SC generating with wide spectral range. When high-intensity laser pulses with narrow line-width propagate in a TIR PCFs, the SC, sometimes spanning over a couple of octaves, could be generalized

Fig. 5. Structures of a typical PBG PCF (a) and TIR PCF (b) obtained with scanning electron

The process of SC generation is a synthesis result of a variety of the nonlinear optical effects, when ultra-short laser pulses with high intensity propagate in a PCF [94, 95]. Used as the source for CARS, we have mostly concerned about, however, the single-mode propagation
