1. Introduction

Using material coherence excited by external radiation field absorption and emission at the resonant frequency can be suppressed [1, 2]. The dark state (coupled upper state) in a lambda-type three-level system is given by superposition of two lower states; absorption and stimulated emission form the dark state are balanced and are canceled out. The quantum interference effect, the electro-magnetically induced transparence (EIT), is one of remarkable situations for the lasing without inversion, slow-light, and the self-induced phase matching in anti-Stokes generation [3].

In this paper we have applied quantum interference to stimulated Raman scattering with different phonon modes. Our scheme is shown in Fig. 1.Two phonon modes are differentially coupled with external optical fields. Differential frequency between two external field ${\omega }_{ex1}$ and ${\omega }_{ex2}$ have been tuned to differential frequency of the two phonon modes at the first BZ (Brillouin Zone) boundary, i.e. ${\Omega }_{diff}={\omega }_{ex1}-{\omega }_{ex2}={\omega }_{p1}-{\omega }_{p2}$ where ${\Omega }_{diff}$is frequency difference between a couple of phonon modes having a frequency of ${\omega }_{p1}$ and ${\omega }_{p2}$, respectively. Direction of the arrows showing optical fields in Fig. 1(a) are indicating direction of instantaneous population flow of the coherent coupling i.e. Rabi flopping. A couple of phonons are simultaneously excited by two-phonon processes at the BZ boundary [4]. The emissions at the Stokes frequency that are indicated by the red arrows in Fig. 1(a) are suppressed by the interference between the emission and excitation. As shown in Fig. 1(b), a wavenumber of phonons emitted at the BZ boundary (2.8 × 10⁷ cm⁻¹) is approximately thousand times larger than that of visible photons. So the differential number between the phonons can be taken for wide range of wavenumbers. In other words, in almost all cases the phase matching condition is satisfied between any external fields.

 

Fig. 1 (a) Energy and (b) momentum conservation of differential two-phonon coupling. These two phonon modes are differentially coupled with the two external fields simultaneously via two-phonon excitation at the BZ boundary. A couple of phonons are emitted in the opposite directions. The momentum difference between these phonons couples to that between the external fields.

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Because of the differential excitation, a line broadening of each phonon can be canceled out when the coupling fields are strong enough. Thus the system generates a narrow line-width sideband with the small (differential) frequency shift.

2. Experimental setup

For the differential two-phonon pumping, a Raman medium that has two-phonon activities at the BZ boundary is required. A single crystal diamond made by chemical vapor deposition was used for the Raman medium. The beam propagation is along (100) plane and input and output planes are (110). For the (ς00) or (00ς) wave vector, frequencies of TO and LO phonon at a point Γ are degenerated at 1,333 cm⁻¹ and these at a point X are 1200 cm⁻¹ and 1060 cm⁻¹, respectively. For (1ς0) wave vector, these at W are 1,030 cm⁻¹ and 1,200 cm⁻¹, respectively [5,6]. The frequency difference ${\Omega }_{diff}$ at the point W is ${\Omega }_{diff}={\omega }_{p1}-{\omega }_{p2}={\omega }_{\text{TO}}\text{(W)-}{\omega }_{\text{LO}}\text{(W)}$, ${\Omega }_{diff}/2\pi$ = 170 cm⁻¹. The instantaneous frequency difference between the pump pulses is tuned by the delay-time between frequency chirped pulses. A 3-ps positively chirped pump pulse around 800 nm having a frequency bandwidth of 246 cm⁻¹ is generated with a 50-fs Ti:Al₂O₃ laser system by closing grating separation of the pulse compressor. A delay time of 1.93-ps corresponds to the ${\Omega }_{diff}/2\pi$ = 170 cm⁻¹. The two pump beams are focused into a 6 mm wide, 3 mm long, and 2 mm thick diamond crystal with a f = 200 mm cylindrical lens. A crossing angle of beams in the air is 27 mrad. A pulse energy of approximately 0.5 mJ was used for the experiments. The maximum pulse energy is limited by the damage threshold of crystal.

Remarkable spectrum changes with the delay-time have been observed [7] as shown in Fig. 2.When the detuning between the external fields was zero, conventional Raman spectrum under single phonon excitation was observed. The first Stokes and 1st to 6th anti-Stokes lines were generated by the 1,333 cm⁻¹ phonon mode. In contrast, at the delay time of 1.96 ps i.e. at the detuning of 170 cm⁻¹, these lines by the single-phonon mode disappeared. In the Stokes-side, the 1st to the 11th sidebands separated by ${\Omega }_{diff}$were clearly observed as shown in Fig. 3.A remarkable feature is that a line-width of sideband $\Delta \Omega /2\pi$ = 40 cm⁻¹ is a six times smaller than the pump line-width of 246 cm⁻¹ as shown in Fig. 4.Generally sideband narrower than pump line-width cannot be observed in four-wave-mixing or stimulated Raman scattering. The narrow line-width comb-like spectra means a generation of a phase-locked pulse train. The pulse train has a pulse separation of $2\pi /{\Omega }_{diff}$ = 196 fs and an entire duration of $2\pi /\Delta \Omega$ = 830 fs. The entire duration is consistent with the temporal overlapping between pump pulses. An individual pulse width in the pulse-train is estimated to be 20 fs when we assume a transform-limited (TL) sech² pulse with an effective band-width of $3\times {\Omega }_{dif}=\text{15}\text{.3}\text{THz}$.

 

Fig. 2 Stokes suppression and super-continuum generation by the differential two-phonon coupling. The delay-time of 1.93 ps corresponds to the detuning of 170 cm⁻¹. The inset is the beam profile of the supercontinuum at the delay-time of 1.93 ps.

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Fig. 3 Differential-frequency side-bands generated by two-phonon coupling. The broken line is showing spectra by single phonon excitation at the zero delay.

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Fig. 4 A magnification of spectra in the Stokes side. The line-width of external fields was 246 cm⁻¹.

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In the anti-Stokes side, in contrast, no spectral line has been observed. Because anti-Stokes is generated by sum-frequency mixing of the pump and the phonon the interference pattern is washed out by frequency chirping of the pump field. Spectral broadening in the pedestal was observed on the anti-Stokes side even with the Stokes side.

The supercontinuum covering the visible region was probably generated by the cross-phase-modulation (XPM) [8] with the ultrafast phase-modulation mentioned above. The spectral shape is quite different from the cascaded multiple anti-Stokes generation with diamond [9, 10]. A bandwidth of the supercontinuum (4,000 cm⁻¹ FWHM) corresponds to a pulse width of 2.6 fs when we assume transform-limited sech² pulse shape. The beam profile shown in Fig. 2 was taken at the delay of 1.93 ps on a screen located behind the crystal with a c-MOS camera. The left most is showing the line-focused IR pump beams (saturated).

3. Summary

Super-continuum generation and Stokes suppression by the differential two-phonon excitation has been demonstrated. The phase-locked multi-line sidebands separated by the beat frequency between phonon modes have been observed. This scheme can be extended to frequency comb and/or ultrashort pulse generation by the use of frequency stabilized two-color pump source. In this case the supercontinuum possibly becomes a broadband comb and an ultrafast pulse train.