## 4 Discussion  
Two important issues are: (1) relaxation of the molecular coherence, and (2) phase-mismatching between the probe beam and the MOM-generated sidebands, as well as that between the probe and pump beams. The coherent motion of molecules is excited by a train of ultrashort pulses, and is probed with a cw laser. The coherence decays with a certain time constant until the next incoming pulse restored it. Hence, the time interval between pulses is a critical factor that determined the amplitude of the coherent molecular motion. By using a 1 GHz mode-locked pump source, the dephasing time of  
Figure 4: Contour curves for \(\xi=0.25,0.5,1\) as a function of hydrogen pressure, the hollow-core photonic crystal fiber (HC-PCF) length, and the pump power.  
Figure 3: (**a**) Evolution of \(\xi\) as a function of pump beam power for hydrogen pressures of 1, 5, 10 atm in a 10-m HC-PCF. (**b**) Dependence on the HC-PCF length of the product of the pump power and the hydrogen pressure needed to satisfy \(\xi=0.25,0.5,1\).  
the molecular coherence is assumed to be longer than 1 ns. The dephasing time \(T_{2}\) of the molecular coherence is given by:  
\[T_{2}=\frac{1}{\pi\Gamma}, \tag{7}\]  
where \(\Gamma\) is the Lorentzian linewidth of the Raman gain curve. This relationship is satisfied for rotational Raman scattering of hydrogen in a pressure range of 0.5 atm to several atm, and \(\Gamma\) has a linear dependence on pressure (or concentration). Previously [24], \(\Gamma\) was expressed as: \(\Gamma(\rho)=97.3\rho\) (MHz), where \(\rho\) was the hydrogen pressure (atm). This immediately results in \(T_{2}=3.2/\rho\) (ns), which indicates that the hydrogen pressure should be ideally \(\sim\)1 atm to maintain molecular coherence until the next pump pulse. There is a trade-off between repetition rate and pump energy, although increased energy contributes to increased excitation of coherent molecular motion.  
The phase-matching condition \(\Delta k=0\) leads to maximum optical modulation. Larger \(\Delta k\) dramatically reduces the MOM efficiency and directly suppresses the sidebands intensities. The MOM \(\Delta k\) is approximately \(\Delta k=Df\), where \(D\) is a group delay dispersion of the modulation medium and \(f\) is the modulation frequency [25]. Although \(\Delta k\) can be negligible for \(f\) in the GHz range, an extremely high modulation frequency (larger than 10 THz) causes significant effects [10]. To satisfy \(\Delta k=0\), the elimination of the medium dispersion, i.e., \(D=0\), is required. This is difficult because gaseous media at a few atm usually produce dispersions at optical frequencies that are non-negligible for optical modulation greater than 10 THz. In addition, the medium dispersion is proportional to HC-PCF length. Thus, a longer HC-PCF leads to a larger dispersion, i.e., a larger \(\Delta k\) in the MOM process. This problem could be solved by the control of the total dispersion of the medium-filled HC-PCF that is determined by the sum of the contributions from the medium and the waveguide. Typical HC-PCFs have a transmission band with a negative waveguide dispersion [26]. Thus, the sign of HC-FCF dispersion is opposite to that of the medium filling the HC-PCF. The positive dispersion of the medium can be offset by the negative dispersion of the HC-PCF. This would allow the total dispersion to be virtually zero at a frequency of the probe beam for MOM, by using an optimized medium pressure. Kagome-type HC-PCFs may benefit from less frequency-dependent dispersion compared with photonic-bandgap HC-PCFs [27]. Under those conditions, phase matching of the optical modulation would be satisfied, leading to efficient modulation of the probe beam at a frequency greater then 10 THz.