## 1 Introduction  
Advances in technologies to manipulate light have been reported for innovative uses in science and technology. Periodic or quasi-periodic light modulation plays important roles in spectroscopy, microscopy, optical information and communication technology, and ultrafast laser engineering [1, 2]. Optical modulation is commonly achieved with acoustic- or electro-optical effects in solid-state crystals [3]. Frequencies have been limited to those of acoustic or high-voltage drivers, which are typically \(\sim\)10\({}^{7}\) Hz or \(\sim\)10\({}^{10}\) Hz at most, respectively. Methods to expand the frequency range include ultrafast Kerr effects in a silicon-polymer hybrid medium [4], recollisions of excitons in a quantum well structure [5], and a plasma wakefield of intense lasers [6]. These approaches, however, cannot operate at optical frequencies (\(>\)10\({}^{13}\) Hz), and new methods are required for future applications such as high-speed broadband optical communication systems.  
Molecular optical modulation (MOM) is a promising way to realize operation at frequencies greater than 10 THz [7]. Molecules in a state of coherent motion can produce an ultrafast temporal variation of a refractive index in a macroscopic view [8]. The reflective index of the coherently excited molecular assembly oscillates at the same frequencies as the molecular motion and thus could modulate a light wave at the 10\({}^{12}\)-10\({}^{14}\) Hz range. MOM has been demonstrated in the pulsed regime. Intense optical pulses have been used to coherently excite molecules, which modulated relatively weak optical pulses before the coherence totally decayed. MOM of continuous-wave (cw) light would have many applications, particularly for optical communication. However, there are intrinsic difficulties in preparing the molecular coherence because of the relatively low intensities of cw lasers. A cw-MOM has been reported that used an optical cavity to enhance the intensities of the excitation beams [9], achieving MOM for a near-infrared cw laser with a efficiency greater than \(10^{-4}\)[10]. In these approaches, however, bulky optical cavities were used that might limit future applications that require robust, compact devices.  
A hollow-core photonics crystal fiber (HC-PCF) is a flexible component that is highly compatible with fiber-based optical systems that could improve information transduction capacity in current optical communication systems [11]. Gaseous media inside a fiber-core waveguide having a diameter of several to tens of \(\upmu\)m can interact with relatively low intensity light propagating through the core, and produce nonlinear optical phenomena such as Raman scattering [12], four-wave mixing [13], and two-photon absorption [14]. The use of HC-PCFs for MOM has been reported in theoretical studies [15; 16] and in experimental work [17; 18], including few-cycle pulse generation [15] and supercontinuum up-conversion [17]. These reports used optical pulses for probing the molecular coherence, rather than low-intensity cw beams.  
Here, a proposal is presented that takes advantage of HC-PCF as a waveguide for MOM of a cw laser based on Raman-active molecules in the core. A key idea is the use of a GHz pump beam train of ultrashort pulses to induce coherent motion of the molecules. The short interval time between successive pulses relative to the dephasing time of the molecular coherence would enable continuous excitation of the coherent motion. A theoretical analysis of the dependence of the molecular coherence amplitude on the pressure, the HC-PCF length, and the pump beam fluence indicates feasibility for 10-terahertz sideband generation for a cw probe beam under phase-matched conditions. This proposal can potentially lead to a fiber-based ultrafast optical modulator compatible with existing optical communication systems, and with the capacity to improve information transmission.