**3.3 Supercontinuum generation with photonic crystal fiber**

184 Photonic Crystals – Innovative Systems, Lasers and Waveguides

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

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

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

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

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

( , ) ( ) exp( ) *XFROG gate I*

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

*<sup>i</sup> <sup>i</sup> A z t R T A z T T dT i z T*

 *E t E t i t dt* 

2

*R*

2

 

, (3.2)

. (3.1)

was simulated with the split-step Fourier method (SSFM) to solve GNLSE [94].

 

1

neglected, ΓR=0. It has more detailed explanation in the paper [94].

 

*k k k k*

2

*A iA <sup>A</sup> z k T*

2 !

*k*

*T*

 

0

and the following equation:

that is mainly divided into three steps.

vacuum, and *A*eff the effective core area.

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 lengths are described in figure 8.

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

Fig. 7. Time (a) and spectrum (b) evolution of SC along the entire length of the PCF with the input pulse width 30 fs and peak power 10 kW [98].

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

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

**4. Broadband CARS spectroscopy and microscopy** 

**4.1 Introduction to broadband CARS** 

diagram of M-CARS shown in figure 10.

Fig. 10. Energy level diagram of M-CARS.

*P*

j

vi

g

*S*

ki

*P*

In the previous works, a narrowband and a broadband dye laser was used for the pump/probe and the Stokes beams respectively [36-38]. The recent progress in wavelengthtunable ultra-short pulse laser has been giving a powerful momentum to the development of M-CARS. The M-CARS micro-spectroscopy has been developed for fast spectral characterization of microscopic samples [35, 99, 100]. But because of the used laser limitation to the line-width, M-CARS is still unable to simultaneously obtain wider molecular

CARS spectroscopy and microscopy can be achieved with an optimized PCF.

Spectroscopy and Microscopy with Photonic Crystal Fiber Generated Supercontinuum 187

description of SC generation, and theoretical guides for experimental instruments. Dispersion and nonlinearity of a PCF can be modified and optimized by adjusting the airhole structure of a PCF. Under specifically experimental conditions, a perfect SC source for

In a traditional CARS microscopy, two or three ultra-short laser pulses with narrow linewidth and different frequencies are used as excitation beams. It permits high-sensitivity imaging based on a particular molecular bond, called single-frequency CARS. But for a mixture with various or unknown components, it is not adequate to distinguish the interested molecules from a complex based on the signal of a single active Raman bond. The broadband even complete molecular vibrational spectra will be beneficial for obtaining the accurate information of various chemical compositions. Although it can be achieved by sequentially tuning the frequency of Stokes beam, it is time-consuming and unpractical for some applications. This problem can be circumvented by using the multiplex CARS (M-

CARS) or broadband CARS spectroscopy with simultaneously detected wider band.

The M-CARS spectroscopy was first demonstrated by Akhamnov et al., a part of CARS spectra of a sample can be simultaneously obtained [34]. In M-CARS, a broadband laser beam is used as the Stokes beam for providing a required spectral range. A narrow line-width laser beam is used as the pump and probe beam that determines the spectral resolution of the system. The multiplex molecular vibrational modes of a sample can be resonantly enhanced, the corresponding CARS signals can be detected simultaneously, the energy

> *ASi*

*i*

simpler way to generate favorable SC for CARS applications. The simulation results are shown in figure 9, where we can see that the SC generated by a PCF with two ZDW is quite good for CARS applications when the laser pulse width is 300fs, as shown in figure 9 (c).

Fig. 8. Temporal-spectral distribution of SC when PCFs with lengthes of 10 cm (a), 20 cm (b), 25 cm (c), and 50 cm (d) pumped by laser pulse with pulse-width 30 fs, wavelength 780 nm, and peak power 10 kW [98].

Fig. 9. Temporal-spectral distribution of the SC when using the femtosecond laser pulse with a central wavelength of 780 nm, peak power of 10 kW and pulse-width of 50 fs (a), 100 fs (b), 300 fs (c) and 500 fs (b) as seed pulse to pump a PCF with length of 10cm.

By numerical simulations, we clearly understood the effects of the parameters of PCF and pumping laser pulse on the generation of SC. All simulation results provide us an intuitive description of SC generation, and theoretical guides for experimental instruments. Dispersion and nonlinearity of a PCF can be modified and optimized by adjusting the airhole structure of a PCF. Under specifically experimental conditions, a perfect SC source for CARS spectroscopy and microscopy can be achieved with an optimized PCF.
