## 3 Results  
The evolution of the sidebands was estimated by the increase of \(\xi\) within a reasonably available range. Figure 2a plots the depletion of the fundamental component and the growth of the sidebands (\(n=\pm 1,\pm 2,\pm 3\)) as a function of \(\xi\) calculated by Equation (6). Thus, the power of the 0-th order fundamental component was converted into higher order sidebands sequentially with increasing \(\xi\). Figure 2b plots the spectrum of a probe beam optically modulated by coherently excited molecules for \(\xi=1\). The intensity of the 0-th order component was 0.586, whereas those of the \(\pm 1\)-st and \(\pm 2\)-nd-order sidebands were 0.194 and 0.013, respectively. The intensities of the \(\pm 2\)-nd-order sidebands for \(\xi=1\)were sufficient to provide a high signal-to-noise ratio for some applications, e.g., phase-locking of a multifrequency laser [20]. Figure 2b also plots the sideband intensities with the same absolute values for each order, e.g., \(n=+1\) and \(-1\), were equal. Actually, the dispersions of the waveguide and the gaseous medium produced a slight asymmetry between the Stokes and anti-Stokes sidebands because the anti-Stokes sideband generation depended on the phase mismatch between the fundamental and the sidebands. In this calculation, perfect phase-matching was assumed for the sideband generation. Hence, this result was valid only when the fundamental frequency had zero-dispersion in the HC-PCF [21]. The parameter \(\xi\) was estimated using Equation (6) for reasonable ranges of those parameters to achieve MOM with molecular hydrogen. The modulation was based on the rotational transition of _ortho_-hydrogen (\(S_{0}(1)\), \(J=1\to J=3\)), that produced a modulation frequency, \(\Omega\), of 17.6 THz. This was used previously [9; 10] for MOM of a cw laser in a high-finesse optical cavity filled with hydrogen. The 852-nm probe beam was a commercially available laser diode. The pump beam was an ultrashort 30-fs optical pulse that was less than one 57-fs rotational period of the hydrogen molecule, and had a \(\sim\)25-nm full-width at half-maximum spectral bandwidth. Thus, it created coherent molecular motion efficiently [22]. The 852-nm center wavelength of the pump beam was the same wavelength as that of the probe beam. However, The transverse modes of both the pump and probe beams had the lowest mode of a Gaussian shape, with beam diameters of 10 \(\upmu\)m at \(1/e^{2}\) intensity, which was similar to the core diameters of typical HC-PCFs. The specific parameter of a hydrogen molecule that determines its Raman polarizability, \(\partial\alpha/\partial Q\), was assumed to be 2.6 \(\times\) 10\({}^{-17}\), which was estimated from an experimental rotational Raman gain coefficient [23]. Figure 3a shows the dependence of \(\xi\) on the power of the pump pulse \(P\) (W) in the core of a 10-m-long HC-PCF for hydrogen pressures \(\rho\) of 1, 5, and 10 atm. The figure shows that the input pump power that equaled \(\xi\) decreased inversely proportional to the pressure. However, lower pressures were more desirable to suppress relaxation of the excited molecular coherence, as discussed below. \(\xi\) reached 1 for a 4.5-W input pump power when the pressure was 1 atm. Increasing the length of the HC-PCF contributed to the reduction of the pump power and the hydrogen pressure; Figure 3b plots the dependence of the product \(P\times\rho\) (W\(\cdot\)atm) on the length of the HC-PCF, \(l\) (m). For \(l=50\), \(\xi=1\) was attained by \(P\times\rho=0.91\), which meant that input pump beam less than 1 W was sufficient to produce sidebands with intensities shown in Figure 2b at a hydrogen pressure of 1 atm. A longer HC-PCF would be a much more effective way to reduce the cost of the apparatus than increasing the pump beam power. However, a longer HC-PCF would make it more difficult to satisfy the MOM phase-matching conditions. This problem is addressed below in the Discussion section.