九州大学学術情報リポジトリKyushu Control of quantum pathways for the generation of continuous-wave Raman sidebands

: The generation of a multifrequency continuous-wave laser through stimulated Raman scattering and phase-matched four-wave mixing in a medium-ﬁlled optical cavity is demonstrated. Three different quantum pathways for the four-wave mixing, two of them degenerate and one of them nondegenerate, can be excited independently by tuning the intracavity dispersion. The results suggest that phase-matched Raman sidebands were generated on the longer wavelength side as well as on the shorter wavelength side, which can be used for the Fourier synthesis of a train of ultrashort optical pulses.


Introduction
Nonlinear optical interactions rely on the phase-relationship between multiple laser fields coupled to each other through the strongly-driven nonlinear polarization of optical materials. A constant relationship between the phases of the nonlinear polarization and those of the laser fields substantially enhances the energy transfer from one beam to the others in the frequency conversion process; this is known as "phase-matching". The phase-mismatch, Δk, is defined as the difference between wavevectors of the interacting laser fields, and the efficiency of the nonlinear optical interaction is maximized for Δk = 0. Unfortunately, perfect phase-matching is generally prevented by the chromatic dispersion of nonlinear optical materials that disturbs the phase-relationship during the propagation of the laser fields. To solve this problem, sophisticated techniques based on the use of birefringent crystalline materials [1] and periodicallypoled structures [2] are widely used for efficient frequency conversions.
Meanwhile, for the use of isotropic nonlinear optical materials, such as gases, the anomalous dispersion of optical waveguides [3,4] and negative-dispersive optical cavities [5] can be utilized to improve the efficiency of the frequency conversion. In these methods, the phaseslip caused by the positive dispersion of nonlinear optical materials can be compensated for by the negative dispersion of an optical waveguide or an optical cavity, leading to the extension of the effective coherent length of the nonlinear optical interaction. The generation of deepultraviolet ultrashort pulses [3] and high-order harmonics [4] was successfully demonstrated in filled hollow optical waveguides, and a phase-matched nonlinear optical interaction pumped by a low-power continuous wave (cw) was demonstrated in a dispersion-compensated high-finesse optical cavity [5].
To generate broadband radiation through these frequency conversion processes, a largebandwidth dispersion-compensated region for the phase-matched interactions is required. In Phase-mismatch, Δk, for CARS shown in (a). k 0 , k 1 , and k −1 are wavevectors for ω 0 , ω 1 , and ω −1 , respectively. (c) Longitudinal modes responsible for the intracavity CARS.
the case of frequency conversion using solid materials, noncollinear beam geometries allow us to increase the bandwidth for the phase-matching [6][7][8][9]. For the use of isotropic gases as nonlinear optical materials, photonic bandgap crystal fibers are a promising tool for broadband frequency conversion [10]. The control of the dispersive properties of hollow-core waveguides by altering the photonic bandgap structure [11,12] has the potential to achieve broadband frequency conversion, but the bandwidth is still limited at this time. However, the progress in design schemes for negative dispersive mirrors associated with ultrafast laser technology pushes the limit of the tunability of the intracavity dispersion, leading to the generation of mode-locked ultrashort pulses with a bandwidth of greater than one octave [13][14][15]. Therefore, the dispersion-compensated optical cavity is a promising candidate for the realization of the broadband frequency conversion through the phase-matched processes even when pumped by low-power cw lasers. Such frequency conversions will allow for molecular optical modulators [16,17] or molecular mode-locked lasers [18] that operate at frequencies of more than 10 THz through the generation of a cw multifrequency laser. In this paper, the frequency conversion from a single frequency laser into a broadband (∼52.8 THz) multifrequency laser in a high-finesse optical cavity is demonstrated. Broadband compensation of the dispersion of an optical cavity enables us to drive two types of cw-based four-wave mixing, coherent anti-Stokes Raman scattering and nondegenerate four-wave mixing, which involve four single-frequency emission lines. We found that the control of total intracavity dispersion allows us to differentiate the phase-matching conditions for three different pathways in the intracavity four-wave mixing processes. The results suggest that there is phase-matched generation of Raman sidebands on both the shorter and longer wavelength sides, which allows for the Fourier synthesis of a cw-based optical pulse train with an arbitrary waveform [19].

Theoretical background
Nonlinear optical interactions in gas-phase media are mainly based on the third-order susceptibility (χ (3) ) because of its isotropic nature [20]. Herein, we focus on coherent anti-Stokes Raman scattering (CARS), which is a χ (3) effect, in a Raman-active gas-phase medium that has a Raman shift frequency of Ω R between three singe-frequency emissions at a pump emission, ω 0 , Wavelength (nm) a Stokes emission, ω 1 , and an anti-Stokes emission, ω −1 (see Fig. 1(a)). The phase-mismatch, Δk of CARS is defined by the difference between the wavevectors, i.e.
as shown in Fig. 1(b). The efficiency of the process is maximized for phase-matched conditions, i.e. Δk = 0. This Δk can be expressed using Ω R = ω 0 − ω 1 = ω −1 − ω 0 and even-order dispersion coefficients, β 2n [21]: Equation (1) indicates that the compensation of all the even-order dispersion of the medium satisfies the phase-matching condition for the CARS process. When the CARS process is driven in an optical cavity, three longitudinal modes spaced at the frequency of Ω R interact with each other through the χ (3) -response of the intracavity medium (see Fig. 1(c)). In this intracavity CARS, the phase-matching condition is satisfied when the separation frequencies between these three longitudinal modes are equal, i.e. Ω 1 = Ω −1 in Fig. 1(c). Unfortunately, the frequency intervals between adjacent longitudinal modes, i.e. the free spectral range (FSR), of a medium-filled optical cavity are not constant as a function of the pump frequency because of the dispersion of the intracavity medium [22]. The frequency dependence of FSR, δ Ω(ω), is calculated by the sum of the contributions of the dispersion given by the intracavity medium and the cavity mirrors: where c is the speed of light; n g is the group refractive index of the intracavity medium; L is the length of the optical cavity; and β mirror (ω) is the group delay given by one bounce of light on the surface of the cavity mirror. Using this δ Ω(ω), the frequency intervals between longitudinal modes responsible for the CARS (Ω j , j = −1, 1) can be calculated by the following equation: where ω N is the frequency of the Nth longitudinal mode and N j is the number of the longitudinal mode responsible for ω j , j = −1, 0, 1. The solid circles in Fig. 2(a) show the wavelength dependence of Ω j for an 8-cm-long Fabry-Perot-type optical cavity filled with hydrogen gas at a pressure of 750 kPa, calculated using Eq. (2) and (3) for β mirror (ω) = 0. These plots show that the large wavelength dependence of Ω j prevents the phase-matching in the CARS process. However, the use of cavity mirrors with negative dispersive properties enables us to compensate for the dependence of Ω j to satisfy the phase-matching condition of CARS in an optical cavity [5]. Figure 2(b) shows the wavelength dependence of group delay dispersion (GDD) for a mirror with a negative value of GDD from 740 nm to 950 nm, which was used in the following experiment. The solid squares in Fig. 2(a) represent the dependence of Ω j on the pump wavelength for the use of a pair of these negative dispersive mirrors (NDMs) as cavity components under the same conditions as for the solid circles in Fig. 2(a). These plots indicate that the compensation of the positive dispersion of the hydrogen gas by the negative dispersion of the cavity mirrors reduces the dependence of Ω j . Figure 2(c) shows the pressures that satisfy Ω 1 = Ω −1 , i.e. the phasematching condition, as a function of wavelength when the negative dispersive cavity mentioned above is used. This indicates that the phase-matching condition for CARS in the dispersioncompensated optical cavity can be found from 700 kPa to 900 kPa in the wavelength range from 785 nm to 800 nm.
In the following section, we discuss intracavity four-wave mixing (FWM) processes between the four longitudinal modes of the filled optical cavity shown in Fig. 3(a). In this case, three different quantum pathways of the FWM process shown in Figs. 3(b-d) can be allowed for the pumping frequency of ω 0 . The phase-matching conditions of these FWM processes are satisfied when two of the three frequency separations, Ω −1 , Ω 1 , and Ω 2 , are matched: Ω −1 = Ω 1 = Ω 2 ( Fig. 3(b)), Ω −1 = Ω 1 = Ω 2 (Fig. 3(c)), and Ω −1 = Ω 2 = Ω 1 (Fig. 3(d)). In the FWM processes shown in Figs. 3(b) and (c), ω 1 and ω 0 are degenerate, respectively, and are called CARS, as mentioned in the previous paragraph. On the other hand, Fig. 3(d) shows a nondegenerate FWM (NDFWM) that involves all four longitudinal modes. Figure 3(e) shows the dependence of Ω j , j = −1, 1, 2, on the frequency of ω 0 for the optical cavity consisting of a pair of the mirrors with negative dispersive properties ( Fig. 2(b)), and filled with hydrogen gas at a pressure of 750 kPa. The phase-matching conditions for the three pathways shown in Figs. 3(b-d) are satisfied under the wavelength given at the intersections of the three curves, A, B, and C, which correspond to the pathways shown in Figs. 3(b), (c), and (d), respectively. This predicts that the pathway of the intracavity FWM can be controlled by optimizing the wavelength and the medium pressure in the dispersion-compensated optical cavity when more than four emission lines are involved in the FWM process.

Experimental setup
The experimental setup is presented in Fig. 4, which consists of four main parts, a hydrogenfilled high-finesse optical cavity, a pumping laser, coupling optics, and measuring instruments. Negative dispersive mirrors that have the dispersive properties shown in Fig. 2(b) were used as the components of an optical cavity in the following experiment. These mirrors had a reflectivity of >99.96% in the wavelength range from 740 to 920 nm. The optical cavity, which consisted of a pair of these mirrors spaced at 8 cm, was installed in the same type of chamber used in the previous experiment [16,23]. The chamber was filled with hydrogen gas. The  gas was used as a nonlinear optical medium for the intracavity FWM, which was driven by a single-frequency cw Ti:sapphire laser (Coherent Inc., MBR110, linewidth, <100 kHz, power, 2 W). The laser beam was coupled to the optical cavity through mode-matching lenses to obtain an optimized resonant condition for the enhancement of the intensity of the coupled beam. The frequency of the beam was measured with a wavemeter (Burleigh Inc., WA-1500) and set at ∼800 nm. The hydrogen pressure in the cavity was finely adjusted to determine the phasematching conditions of the intracavity FWM such that the intracavity medium had a positive dispersion that precisely compensated for the negative dispersion of the cavity mirrors. The intracavity power was controlled by detuning the cavity length slightly from the point at which the intracavity power was maximized [24]. The spectrum of the output beam from the cavity was measured with a spectrometer (Ocean Optics Inc., HR4000) and the dependence of the intensities of the emission lines on the total output power was recorded with LabVIEW Software (National Instruments Inc.).  Fig. 4. Schematics of the experimental setup. The high-finesse optical cavity is installed in a stainless-steel chamber equipped with silica windows on both input and output side. The mode-matching lenses (ML) consist of a beam expander (×2.5) and a lens with f = 800 mm. The measuring system includes a beam profiler, a power meter, a spectrometer, and a photo detector. Operating parameters of the cw Ti:sapphire laser and properties of the negative dispersive mirrors are described in text.

Results and discussions
obtaining these spectra are shown in the inset of the figures. Figures 5(c) and (d) show the dependence of the intensities of ω −1 , ω 1 , and ω 2 as a function of the total output power measured for Figs. 5(a) and (b), respectively. Note that there is a difference between the thresholds for the generation of the emission lines. In Fig. 5(c), the threshold for the generation of ω −1 coincides with that of ω 1 at 15 mW. The threshold for the generation of ω 2 (30 mW) is larger than this value. However, in Fig. 5(d), the threshold for the generation of ω −1 corresponds to that of ω 2 at 20 mW. The threshold for the generation of ω 1 (8 mW) is substantially lower than this value. From Figs. 3(b-d), it can be seen that FWM is a process that generates two photons at the frequencies of the sidebands at the expense of two photons at the fundamental frequency (or frequencies). This leads to the identical thresholds for the generation of the sidebands through the FWM process. Therefore, the coincidence of the thresholds of ω −1 and ω 1 in Fig. 5(c) suggests that ω −1 was generated through a CARS process that satisfies the phase-matching condition, Δk 0 = 2k 0 − k 1 − k −1 = 0 (Fig. 2(c)). In this case, ω 2 was generated thorough cascaded SRS using ω 1 as a pump frequency. On the other hand, in the case of the FWM shown in Figs. 5(b) and (d), ω 1 was first generated through SRS from ω 0 at 8 mW. Then, the photons of ω −1 and ω 2 were generated at 20 mW at the expense of the photons of ω 0 and ω 1 because the phasematching condition of NDFWM, Δk 0 = k 0 + k 1 − k −1 − k 2 = 0, was satisfied at this fundamental wavelength (800.380 nm) and intracavity pressure (819 kPa). This is the first demonstration of phase-matched NDFWM in an optical cavity driven by a cw laser. In addition, we showed that the dispersion in the optical cavity determined the pathway for the frequency conversion based on χ (3) nonlinearity.
It was also found that the phase-matching for FWM affected the cascaded SRS for the generation of the high-order Stokes emission in the longer wavelength side. When the pump wavelength was set at 800.355 nm, different behaviors for the generation of ω 2 were observed at different intracavity pressures. Figures 6(a) and (b) show the spectra of the cavity output beam measured at intracavity pressures of 787 kPa and 800 kPa, respectively. Although these two spectra are similar, the thresholds for the generation of ω 2 shown in Figs. 6(c) and (d) are different. Fig. 6(c) shows the typical situation for cw-based cascade SRS in an optical cavity [24],