## 2 Theoretical Background  
As shown in Figure 1, theoretical analysis of the proposed fiber-based MOM assumes an apparatus based on commercially available lasers to pump and probe gaseous Raman-active molecules in the HC-PCF. The HC-PCF is a photonic-bandgap type with a \(<\)10 \(\upmu\)m diameter core, low-loss transmission at 800 nm and a \(\sim\)100-nm bandwidth. The pump beam is a 1-GHz mode-locked laser with an average power of a few watts (average energy of a few n)) to excite coherent molecular motion in the HC-PCF. A 852-nm cw beam from a standard external-cavity laser diode is used to probe the molecular Raman coherence, leading to sideband generation in the MOM process. The amplitude of coherent molecular motion excited by the ultrashort pulses with a duration less than one period of the molecular motion can be expressed as:  
\[Q(t)=Q_{0}\exp\left(-\frac{t}{T_{2}}\right)\sin(\Omega t), \tag{1}\]  
where \(T_{2}\) is the dephasing time of the molecular coherence and \(\Omega\) is the frequency of molecular motion. The initial amplitude, \(Q_{0}\) when molecules are excited by the pump pulse at \(t=0\) was determined by the fluence of the pump pulse energy \(F_{p}\):  
Figure 1: Experimental apparatus assumed in the theoretical analysis. \[Q_{0}=\frac{2\pi}{Mc\Omega}\left\{\frac{\partial\alpha}{\partial Q}\right\}F_{p}, \tag{2}\]  
where \(M\) is the effective molecular mass and \(\partial\alpha/\partial Q\) is the change in polarizability with respect to the molecular coordinates. In Equation (2), larger coherent molecular motion is obtained when the pump pulse fluence is larger, i.e., larger energy per a unit area [19]. When a probe beam given by \(\varepsilon_{i}(z,t)=(1/2)E_{i}\exp{i(\omega_{i}t-k_{i}z)}+c.c.\) propagates through the coherently excited molecules, the molecular motion affects its electromagnetic field via the nonlinear polarizability \(P_{R}=N(\partial\alpha/\partial Q)Q(t)E_{i}\), where \(N\) is the molecular density. From the solution of the wave equation including the nonlinear polarizability, the amplitude of the electromagnetic field of the probe beam \(E_{i}\) propagating through the coherently oscillating molecules at distance \(l\) is:  
\[E_{i}(zit)=E_{i}(0,t)\exp\left\{-i\beta l\sin(\Omega t)\right\}. \tag{3}\]  
It is assumed that the phase-mismatch caused by the difference between the pump and probe phase velocities is negligibly small. Equation (3) is valid when dephasing of the molecular coherence during the period between successive pump pulses is negligible (see Discussion). \(\beta\) is a parameter corresponding to the Raman gain in stimulated Raman scattering given by:  
\[\beta=\frac{2\pi\omega_{i}N}{c}\left\{\frac{\partial\alpha}{\partial Q}\right\} Q_{0} \tag{4}\]  
Equation (4) is consistent with increased Raman gain with the amplitude of molecular coherence. The spectrum of the probe beam after interaction with the molecules is obtained by the Fourier transform of Equation (4):  
\[|F_{i}(\delta\omega)|^{2}=\sum_{n=-\infty}^{\infty}|f_{i}(\delta\omega-nt \Omega)|^{2}\cdot J_{n}^{2}(\xi), \tag{5}\]  
where \(\xi=\beta l\), and \(J_{n}(\xi)\) is an \(n\)-th order Bessel function. Here, \(f_{i}(\delta\omega)=f_{i}(\omega-\omega_{i})\) is an initial spectrum that is an extremely narrow line at \(\delta\omega=0\) for a cw laser beam. Equation (5) reveals a comb-like spectrum with equally-spaced lines separated by \(\Omega\) that is symmetrical about the initial probe frequency. The intensities of the 0-th order component and the \(n\)-th order sidebands are determined by the \(n\)-th order Bessel function, \(J_{n}(\xi),n=\pm 0,1,2,\cdot\cdot\cdot\).