1. Introduction

Coherent anti-Stokes Raman scattering (CARS) is one of the most typical nonlinear optical effects which has extensively been used during the last four decades as a high-sensitive and high-precision spectroscopy tool [1]. Recent advances in CARS include time-resolved measurements in vibrational dynamics, plasma and combustion diagnostics, and bio-molecular imaging that use micro objectives as well as near-field optics. Another hot topic in the CARS application is broadband radiation associated with high-order stimulated Raman scattering, which enables generation of attosecond optical pulses, in molecular gases as well as crystalline solids [2–4].

Diamond is ideally suited as a host for efficient CARS emission because of its strong electron-lattice coupling and, therefore, high Raman susceptibility. Since the first demonstration of CARS in diamond at the early stage of nonlinear optics [5], various nonlinear optical studies have been made, such as Raman-induced Kerr effects (RIKES) [6], multiple stimulated Raman generation [7], and coherent phonon generation using sub-10 fs pulses [8]. All of these works, however, examined the Γ(²⁵⁺) optical phonon at the Brillouin zone center, the only mode which is Raman-active in the first order. More recently, high-order CARS has been demonstrated using amplified fs pulses, thus the phenomenon appeared at high excitation regime, possibly above the critical pumping power for the parametric effect [9].

In this paper, we report on the observation of a new CARS signal which originates from the second-order Raman process at purely third-order nonlinear optical regime. It is known that the spontaneous Raman spectrum of diamond shows multiple broad peaks together with a sharp unique line of the Γ(²⁵⁺) phonon [10]. The broad component is due to scattering by two phonons, which were originally Raman-inactive, but have come to be Raman-active through the second-order electron-phonon interactions. Here, we confirm resonance enhancement in CARS at excitation energy corresponding to these two-phonon complexes. CARS generation associated with the two-phonon resonance suggests potential application to continuous spectral spanning of broadband radiation [11] with avoiding phase-matching problems inherent in the multi-order nonlinear process [12].

2. Experimental details

The experiment was performed in a high-purity single crystal of diamond (categorized as type IIa), which had been synthesized at high pressure and high temperature. It was cut to a small plate of 0.3 mm thickness with an optically-smoothed surface perpendicular to the [111] crystallographic axis.

We used a pair of mode-locked Ti-sapphire lasers for excitation. Each laser produced frequency-tunable ps pulses with a repetition rate of 76 MHz. Hereafter, ω ₁(2) denotes the angular frequency of excitation pulses, and ω ₁>ω ₂ [see Fig. 1(a)]. The pulse energy and the temporal duration of ω ₁ (ω ₂) pulses were characterized to be 4 ps and 20 nJ (2 ps and 5 nJ), respectively. The two pulse sequences were precisely synchronized to each other by means of a novel phase-locking device (Coherent, Syncrolock) achieving a timing jitter of less than 500 fs.

The excitation beams were incident on the sample with a crossing angle of a few degrees, roughly normal to the sample surface. The focusing diameter was in the range of 50–80 µm. With a defined wave vector of ω ₁₍₂₎ photons as k ₁₍₂₎, CARS signals were generated in a phase-matching direction of 2k ₁-k ₂ [Fig. 1(b)]. The emitted signal was selected by an iris, passed through a high-pass spectral filter, then fed into a grating spectrometer equipped with a cooled charge coupled device (CCD) detector.

 

Fig. 1. (a) Transition scheme and (b) spatial arrangement for the CARS generation.

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To observe variation in CARS as a function of frequency shift, we changed the wavelength of ω ₂ pulses, while keeping that of ω ₁ pulses constant. Great care was taken to eliminate signal fluctuation with tuning wavelength. For this purpose, we split the excitation beam path into two sets, and measured CARS, and sum-frequency generation in a LiIO₃ crystal as a reference. Then, the CARS spectrum was divided by the reference spectrum, removing potential fluctuations due to the frequency tuning. All experiments were performed at room temperature.

3. Results and discussion

The spontaneous Raman (SR) spectrum of the present sample is shown in Fig. 2. Similar SR spectra have been reported in the literature [10, 13]. The sharp prominent line at 1332 cm^-1 was due to the triply degenerate zone-center phonon with Γ(²⁵⁺) symmetry, which is the only Raman-active mode in a cubic structure with two atoms per unit cell. The second-order Raman signal manifests itself in the range between 2100 cm^-1 and 2700 cm^-1. Note that the second-order Raman process originates from two phonons which must have equal and opposite momentum. Such a condition is satisfied by many sets of phonons throughout the Brillouin zone. Thus, the spectrum becomes a broad continuum, and its peak structures appear due to the joint density of states of the two phonons, which becomes maximum at several critical points in the dispersion curve.

Assignments for the two-phonon peaks have been made through comparisons with inelastic neutron and X-ray results [13]. The relatively sharp peak at 2668 cm^-1 is the overtone Γ(²⁵⁺) phonon. The energy of this peak is slightly higher than twice that of the Γ(²⁵⁺) phonon, reflecting the overbending of the optical phonon branch [14]. The highest peak at 2460 cm^-1, which will be the focus of this study, is assigned to the overtone ∑⁽¹⁾ phonon branch at the K point. Some of the other assignments are labeled by the arrows in Fig. 2.

 

Fig. 2. Spontaneous Raman spectrum in the present sample of diamond. It was excited by the 512.45 nm line of an Ar⁺ laser. The blue line is a ×200 magnified view of the red line. The dotted line shows background noise levels.

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Fig. 3. Comparison between the spectra of the excitation pulses and those of the CARS signals which were resonant with the Raman-active Γ(²⁵⁺) phonon. (a) ω ₂ pulse, (b) ω ₁ pulse, (c) first-order CARS, and (d) second-order CARS. A spontaneous Raman spectrum pumped by ω ₁ pulse is also shown by the dotted line in (a).

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A comparison between the spectra of excitation pulses and those of the CARS signal is summarized in Fig. 3. Here, we analyzed CARS which was resonant with the fundamental Γ(²⁵⁺) phonon, whose energy will be denoted by Ω_Γ. Figures 3(a) and 3(b) present the spectra of ω ₂ and ω ₁ pulses, respectively, showing the former being narrower than the latter, reflecting different Q values for each laser cavity. Figure 3(a) also shows the Stokes-shifted SR spectrum pumped by ω ₁ pulses. Overlapping of the ω ₂ spectrum and the SR spectrum assured that ω ₁-ω ₂=ΩΓ. Within this condition, we could observe CARS signals which were emitted in a phasematching direction of 2k ₁-k ₂, as shown in Fig. 3(c). The emission spectrum was centered at ω ₁+Ω_Γ, with a spectral width similar to that of ω ₁ pulses.

In addition to the first-order CARS signal, we were able to analyze the second-order CARS, which was generated in a direction of 3k ₁-2k ₂. The spectrum in this case appeared at ω ₁+2Ω_Γ in center energy, as shown in Fig. 3(d). The intensity of the second-order CARS was approximately six orders of magnitude lower than that of the first-order CARS, which was close to our instrumental sensitivity. Such a low signal intensity suggests that the second-order CARS arose from cascade scattering, i.e., anti-Stokes scattering of the first-order CARS by coherently generated phonons. Moreover, the scattering amplitude depended roughly on the fifth power of the excitation power, that was simply expected by the perturbation theory. Parametric effects may appear for much higher excitation, which would produce multiple phonon sidebands through the stimulation effect [9].

 

Fig. 4. CARS intensity as a function of energy shift between the two excitation pulses, ω ₁-ω ₂ (blue dots). Spontaneous Raman spectrum excited by the ω1 pulse is also plotted (red line).

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Figure 4 shows the dependence of the first-order CARS intensity on frequency shift in the vicinity of the Ω_Γ resonance. Resonance enhancement in CARS was actually found at ω ₁-ω ₂~Ω_Γ. Moreover, the excitation spectrum of CARS exactly followed the SR spectrum by ω1 pulses. This confirms that strict resonance between the pump energy and the sharp phonon energy is absolutely required for efficient CARS generation. Such a strict condition, however, should be loosened for CARS with the second-order Raman process, because of its broad spectral characteristics, as will be discussed below.

In order to demonstrate the two-phonon assisted CARS generation, we tuned a frequency shift between the excitation lasers to the highest peak in the two-phonon SR spectrum, which was at 2460 cm^-1. This peak was assigned to the overtone ∑⁽¹⁾ phonons (denoted by 2Ω_∑). With this pumping condition, even for off-resonance to Raman-active phonons, we were able to observe a CARS signal with an intensity as low as 10^-3 of that for the Raman-active phonon. Such a signal reduction is reasonable because CARS intensity should be proportional to the square of a Raman cross-section, which was two orders magnitude lower for the 2Ω_∑ mode than for the Raman active Ω_Γ mode, according to the SR spectrum in Fig. 2.

Dependence of the CARS intensity on frequency shift is plotted in Fig. 5. It shows significant peak and dip structures. We note that such a peculiar dependence was reproducible in several experiment runs, thus it was not an artifact of the experimental system or associated noise. This frequency variation is explained by the copresence of the resonant and nonresonant scattering processes.

The third-order nonlinear susceptibility which is responsible for the CARS process can be written in the form [1, 15, 16],

where Δ=ω ₁-ω ₂-Ω is a frequency detuning, and Γ is a Lorentzian width of the Raman resonance. Two terms followed by χ ⁽³⁾ _NR and χ ⁽³⁾ _R represent the nonresonant and resonant parts, respectively, in CARS generation. If the frequencies of ω ₁, ω ₂, and ω ₁+ω ₂ are far below the electronic resonance, we can assume χ ⁽³⁾ _NR and χ ⁽³⁾ _R to be real constants without loss of generality. Then, CARS spectrum is proportional to, and given by |χ ⁽³⁾ _CARS|².

 

Fig. 5. CARS intensity as a function of excitation energy shift ω ₁-ω ₂, close to the resonance of 2Ω_∑. Model function is also plotted by the red line, where χ ⁽³⁾ _R/χ ⁽³⁾ _NR=0.25, Γ= 3.5 cm^-1, and Ω=2460 cm^-1 in Eq. (1).

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The best fit of this model to the frequency-dependent data is plotted by the red line in Fig. 5. A large asymmetric feature, with a positive peak at a frequency lower than 2Ω_∑ and a dip at a frequency higher than 2Ω_∑, is reproduced by this model. The discrepancy between this model and the CARS spectrum is due to several reasons including (1) the effect of multiple Raman resonances (only single resonance was taken into account in this model, although the SR spectrum consisted of broad and multiple components), and (2) Lorentzian spectral response was assumed in this model, while the actual two-phonon spectrum reflects the density-of-states singularity, which is different from a Lorentzian.

The above analysis suggests the contribution of the nonresonant four-wave mixing (FWM) and the resonant CARS processes with similar magnitude. This argument has also been verified by measuring FWM signal at ω ₁-ω ₂=2000 cm^-1, which was completely off-resonant to both the Raman-active line and the two-phonon peaks. Nevertheless, we were able to confirm the nonlinear scattering signal with its amplitude as high as that of two-phonon resonant CARS within an experimental accuracy of a factor of five. We can therefore conclude that there was a copresence of the nonresonant FWM and the resonant CARS at the two-phonon resonance region, and that the CARS spectrum was remarkably influenced by the interference between these two components.

Finally, we would like to point out the importance of our findings. First, to our knowledge, this is the first observation of CARS induced by two phonons, each of which is inactive in the first-order Raman process, that was proposed theoretically in [17]. This is in contrast to the two-phonon RIKES measurement previously made [6], which dealt with Raman resonance in the vicinity of twice the frequency of Raman-active phonons. CARS therefore allows the high-sensitivity probing of the two-phonon spectra and dynamics.

Second, two-phonon CARS has a potential for application to new frequency-shifting devices, whose spectral shift is widely tunable, and can be higher than the fundamental phonon mode. Note that a frequency shift by 2Ω is possible with a conventional Raman shifter in a parametric regime, while the relevant process is, in principle, the fifth-order process, suggesting a low scattering efficiency. This was revealed in Fig. 3(d). In contrast, the two-phonon CARS produces 2Ω-shifted scattering in the third-order process, possibly leading to higher conversion efficiency.

Third, the two-phonon CARS has great advantage to produce high-order Raman sidebands. Multiple CARS generation up to the 22nd-order was reported in TiO₂ above the critical pumping power [3], which was explained in terms of highly-populated phonons at the Brillouinzone edge, achieving a standing-wave phonon grating, and high-order Raman-Nath type scattering [12]. Such a new two-phonon scattering mechanism avoids the phase-matching problem inherent in the high-order process, enabling broadband coherent radiation in solid states.

4. Conclusion

We observed coherent anti-Stokes Raman scattering in diamond, which was in resonance with the Raman-active phonon, as well as the two-phonon complexes which were originally Ramaninactive, but became Raman-active through the second-order Raman process. The interference nature in the two-phonon CARS process was analyzed. The present study reveals a high potential of coherent broadband radiation based on multi-phonon nonlinear process in diamond.