**3.2 Numerical modeling**

182 Photonic Crystals – Innovative Systems, Lasers and Waveguides

As one of noninvasive research tools with high sensitivity, specificity and resolution, CARS microscopy has attracted more and more attention and been widely used in physics, chemistry, biology, medicine and life science et al. The capabilities and availability of CARS microscopy has been further improved with the recent technique's advances. Many exciting

Because of the label-free characteristic of CARS microscopy, it has been regarded in the biological research, especially in the unstained cells. The first CARS microscopy was used to obtain the structural image of epidermal cells of onion immersed in D2O [21]. The water diffusion in live dictyostelium cells was researched with a broad vibrational resonance centered at 3300 cm-1, which could not be observed with fluorescence microscopy [75]. These early experimental results have proved that the CARS microscopy is an effective complementary method of fluorescence microscopy. Since many cellular processes take place on a subsecond timescale, high temporal resolution is required. By improving the temporal resolution, it is possible to image the chromosome distribution during mitosis using the symmetric stretching vibration of the DNA phosphate backbone [76]. Because of the good detectability of lipids, the structural and functional images of various living cells were obtained with CH bond of lipid [75, 77-79]. The sensitivity of CARS microscopy is high enough to detect lipid vesicles with sizes smaller than 300 nm in diameter [79]. Compared with fluorescence microscopy, CARS microscopy allows long-term investigations of cell without photobleaching. Therefore it can be used to long-term track biological molecules, such as lipid droplets, in living cells [80]. Nan and associates used the CARS microscopy to study the growth and transport of lipid droplets in live cells [79]. By tuning to the CH2 lipid vibration, Cheng and his colleagues observed the apoptosis, and identified different stages in the apoptotic process [76]. Potma and his associates visualized the intracellular hydrodynamics

On the basis of cell imaging, the CARS microscopy is used in the living animal's tissue imaging, in which the tissue's optical properties, such as absorbability and scattering, are of obvious concern. The method of epi-detection is a good solution in tissue imaging with CARS. CARS microscopy has been successfully used for imaging of nonstained axonal myelin in spinal tissues in vitro [82]. Both the forward and backward CARS signals from the tissue slab were detected. The lipid distributions in skin tissue of live animals have been observed [83]. These all preliminary experimental results show us a vast potential of CARS

As we have discussed in previous sections, in a CARS spectroscopy or microscopy, it is necessary that two ultra-short laser pulses with high peak power and different frequencies reach focus at the same time. In order to quickly distinguish different molecules in a complex system with the complete CARS spectra, such as various biological molecules in cells, it is required that the output of source must have not only a wide enough spectral range, but the spectral continuity and simultaneity of various spectral components [84]. Spectral broadening and the generation of new frequency components are inherent features of nonlinear optics. When ultra-short laser pulses propagate through a nonlinear medium, a dramatic spectral broadening will happen. This particular physical phenomenon, known as

**2.5 Applications of CARS spectroscopy and microscopy** 

with the CARS signal of the O-H stretching vibration of water [81].

microscopy in biomedical imaging and early diagnosis of diseases.

**3. Supercontinuum with photonic crystal fiber** 

results have been presented in many literatures.

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

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

**3.3 Supercontinuum generation with photonic crystal fiber** 

lengths are described in figure 8.

microscopy. Some representative results will be shown in the next section.

Fig. 6. Group velocity dispersion curve of the PCF with two ZWDs [98].

input pulse width 30 fs and peak power 10 kW [98].

Fig. 7. Time (a) and spectrum (b) evolution of SC along the entire length of the PCF with the

In figure 8(c), the spectral range of generated SC is 500nm by using a PCF with two ZDW under proper pumping conditions. In SC, the spectral continuity, simultaneity and intensity of red-shifted SC components are all good enough for a source of CARS. But for this purpose, an ultra-short pulse laser system with pulse width of 30fs is needed, which is not easily sustainable during practically experimental operations. Therefore, we have tried to seek a

Spectroscopy and Microscopy with Photonic Crystal Fiber Generated Supercontinuum 185

order to find out a way to achieve an ideal SC source for CARS spectroscopy and

Some of our computational results are shown here in order to account for the whole processing course clearly. We have simulated the SC generation by using a PCF with two zero dispersion wavelengths (ZWD) [98]. The calculated group velocity dispersion (GVD) curve is shown in figure 6. By solving GNLSE with SSFM, the temporal and spectral distributions of the SC generation along the whole length of PCF are shown in figure 7. With the XFROG trace, the results of temporal-spectral distributions of PCFs with different

and temporal distributions of various spectral components of SC, called temporal-spectral distribution [96]. For a PCF with a given structure, a number of numerical modeling and computational methods have been constructed and reported to obtain the entire properties of a PCF. Here, we carry out an entire analysis on the SC generation with a common method that is mainly divided into three steps.

Firstly, as one of most effective methods, the finite element method (FEM) can be used to obtain the coefficients of the chromatic dispersion (the effective propagation constant *β*eff) based on the structural parameters of PCFs (the diameter of air-hole, and the pitch between two holes). The dispersion coefficients *β*k (k≥2) can be derived by the Taylor series expansion at the central frequency *ω*0, and the nonlinear coefficient *γ* can be approximately calculated by *γ* =*n*2*ω*0/*cA*eff, with n2 the nonlinear-index coefficient for silica, *c* the speed of light in vacuum, and *A*eff the effective core area.

Secondly, a propagation equation is used to calculate the SC generation during the propagation of ultra-short laser pulses, although the generalized nonlinear Schrödinger Equation (GNLSE) is not the only way to realize it. The process of the pulses propagation was simulated with the split-step Fourier method (SSFM) to solve GNLSE [94].

$$\begin{split} \frac{\partial A}{\partial z} + \frac{\alpha}{2} A - \sum\_{k \ge 2} \frac{i^{k+1}}{k!} \beta\_k \frac{\partial^k A}{\partial T^k} &= \\ i\gamma \left( 1 + \frac{i}{\alpha\_0} \frac{\partial}{\partial T} \right) \Big| A \left( z, t \right) \Big|\_{-\alpha}^{+\alpha} \mathcal{R}(T') \times \left| A \left( z, T - T' \right) \right|^2 dT' + i\Gamma\_R \left( z, T \right) \Big) \end{split} \tag{3.1}$$

In equation (3.1), the linear propagation effects on the left-hand side and nonlinear effects on the right-hand side are given, where α and A are the loss coefficient and the spectral envelope with the new time frame T=t-β1z at the group velocity β1-1. R(T) presents the Raman response function. The noise ΓR, which affects the spontaneous Raman noise, is neglected, ΓR=0. It has more detailed explanation in the paper [94].

For a CARS spectroscopy or microscopy, the temporal-spectral distribution of SC is also an important factor. Therefore, thirdly, we have to figure it out in order to fully understand the temporal distribution of various spectral components in SC, although the spectral envelope of SC can be obtained in the second step. To obtain the temporal-spectral distribution of SC, cross-correlation frequency resolved optical gating method (XFROG) was applied for characteristic of SC and could be proved by an experimental instrument of XFROG [97]. The two-dimensional XFROG spectrogram can be plotted by using two electromagnetic fields and the following equation:

$$I\_{XFROG}(\alpha, \tau) = \left| \int\_{-\infty}^{+\infty} E(t) E\_{\text{gate}}(t - \tau) \exp(-i\alpha t) dt \right|^2,\tag{3.2}$$

where *E*(*t*) is the calculated envelope of the SC with the variable t, and *E*gate(*t*-*τ*) is the gating pulses with the delay time *τ* between the seed laser pulses and the SC. It can be concluded that XFROG measurement is a good way to characterize the temporal and spectral evolution of the SC generation and interpret the particular time and frequency domain information of the optical effects. With the above introduced method, we carried out simulation analysis in order to find out a way to achieve an ideal SC source for CARS spectroscopy and microscopy. Some representative results will be shown in the next section.
