Transient stimulated Raman scattering is investigated in methane-filled hollow-core photonic crystal fiber. Using frequency-chirped ps-pulses at 1.06 μm as pump and tunable CW-radiation as Stokes seed, the vibrational excitation of the CH4 molecules can be controlled on the sub T 2 time-scale. In this way the generated Stokes pulse can be phase-locked to the pump pulse and its spectrum manipulated.
©2009 Optical Society of America
In recent years, nonlinear phenomena based on stimulated Raman scattering (SRS) have been the focus of many studies. A fundamental problem that arises is how to achieve efficient frequency conversion while keeping the phases of Stokes and anti-Stokes components mutually coherent, i.e., phase-locked. In the short (ps and fs) pulse regime, phase-control of the fields is particularly difficult, because in addition to the transient character of the response, the Raman process is often complicated by self-phase modulation, self-focusing, and ionization as a result of high pulse intensities [1-4]. Various techniques have been developed to improve the coherence of the process [5-8]. In one of these [7-8], the pump pulse is first stretched (i.e., frequency-chirped) to keep its intensity below the threshold for self-phase modulation and self-focusing. The Stokes field generated by the chirped pulse is itself frequency chirped and thus can be compressed to a shorter duration . However, the spectrum of the Stokes field (which develops from spontaneous noise) has an irregular character and is not reproduced from pulse to pulse, limiting potential applications of this technique .
In this letter, we show that the use of gas-filled hollow-core photonic crystal fiber (HC-PCF)  for ps-pulse SRS offers new possibilities for controlling the generated Raman spectrum. Recent papers have reported the excellent performance that gas-filled HC-PCF offers as a Raman shifter, extremely low threshold powers being reported for Stokes and anti-Stokes SRS in the quasi-CW regime (with ns-pulses) [10-11]. Here we discuss SRS in HC-PCF with pulses shorter than the phase relaxation time T 2 of the Raman medium (i.e., the coherent interaction regime). In contrast to ns-pulse SRS, the coherence state of the molecular motion is determined by the whole pre-history of the interaction. In particular, once excited, the Raman coherence “lives” for a time comparable to T 2, which makes possible coherent Stokes and anti-Stokes scattering of a temporally delayed optical field. We show that this property can be used to control the spectrum of the Stokes (or anti-Stokes) radiation. Using an intense frequency-chirped pulse as a pump and tunable CW-radiation as Stokes seed, coherent Stokes-shifted radiation is generated, locked to the phases of the two fields. The form of the spectrum can be controlled simply by tuning the frequency of the seed CW-laser, which changes the position of the low (or high) frequency wing. Since the Stokes field results from scattering of the pump field by the resonantly excited coherence, its chirp replicates the chirp of the pump field and thus can be compressed to a short pulse duration.
2. Idea behind controlling short-pulse SRS
The principle behind coherent control of ps-pulse SRS is illustrated in Fig. 1, which summarizes the interaction of a broadband frequency chirped pump pulse and a CW-Stokes seed with a Raman transition. The interaction starts at time t *, when the difference between the instantaneous frequency of the pump field ωp(t) and the frequency of the seed field ω seed is equal to the frequency Ωv of the Raman transition (for time t < t * the interaction is negligibly small due to the large detuning Δω = ω p(t) - ω seed - ΩV). During the resonant interaction, which lasts approximately t int ≈ ∣α∣-1/2 (where α is the chirp parameter of the time dependent pump pulse frequency, ω p(t) = ω 0+αt), the ground state population is partly transferred to the upper state. This creates a Raman coherence ρ12(t) which “lives” within the vibrational dephasing time T 2 of the transition. Scattering of the remainder of the pump field at time t > t * by the Raman coherence produces a Stokes signal that is locked in phase to the pump field. For this to work it is important that the dephasing time should be long compared to the pump pulse duration, T 2 > τp, so that all pump field spectral components arriving at times t > t * can be converted to the Stokes band. By tuning the seed frequency ω seed, one changes the starting time of the interaction and, as a result, the position of the lowest or the highest frequencies of the generated Stokes spectrum (Fig. 1), depending on the sign (positive or negative) of the chirp. Note that if phases of the fields are properly matched, this approach can also provide efficient tailoring of the anti-Stokes spectrum.
3. Experimental results
To explore this effect experimentally, we used the set-up illustrated in Fig. 2. As pump we used 40 ps pulses from a Fianium fiber laser operating at wavelength λ = 1.06 μm. The ouput spectrum of the laser is broadened to 4 nm by self-phase modulation in the amplifier and can be compressed to 1.5 ps using an external grating compressor. The maximum energy of the pulses was ~5 μJ. A 10 mW CW-laser (TSL-510 Santec Inc.), tunable in the range 1510-1630 nm, was used to seed the Stokes. The frequency difference between the two lasers could be finely tuned near the ν1 vibrational transition of the CH4 molecules (due to its high symmetry the CH4-molecule has no rotational spectrum ). The radiation from the two lasers was launched into a 25 cm long kagome-lattice HC-PCF filled with CH4. The attenuation loss of the fiber was ~1 dB/m and the core diameter was 26 μm. The intensities of the pump and the seed radiation inside the HC-PCF were chosen so that the influence of competing SRS from spontaneous noise was negligibly small. At the gas pressure used in the experiments (10 bar), the phase relaxation time (T 2) of the Raman transition is slightly longer than the pump pulse duration, meaning that the conditions necessary for observing coherent laser-molecule interactions are fulfilled.
In the first series of experiments we seeded the SRS while pumping with a positively chirped pulse (pulse spectrum shown in Fig. 3(b)). The Stokes spectra obtained at different seed frequencies are presented in Fig. 3(a). Note that two distinct spectral regions can be identified. The first contains a narrow peak that comes from the CW seed laser, while the second broad spectrum one is caused by the seeded SRS.
The most striking feature of the spectra in Fig. 3(a) is the asymmetry, i.e., the absence of signal below the seed frequency. In fact, it is the seed frequency that determines the position of the low frequency wing of each spectrum. This means that one can effectively control the width and the position of the generated spectrum by tuning the seed frequency. A further interesting feature is that the generated spectrum extends far beyond the seeded Stokes frequency, with a width that is much larger than the Raman linewidth ΔνR=2/T 2 (~1.3 cm-1 in these experiments). Note that the observed broad Stokes spectrum cannot be attributed to a competing noise-seeded signal, because under the conditions of the experiment this signal is several orders of magnitude smaller. As can be seen, the Stokes spectrum reproduces the structure of the pump spectrum in considerable detail, suggesting that phase modulation of the pump is transferred to the Stokes field.
Additional insight into the physics comes from a series of measurements in which the system was pumped with pulses having the opposite sign of chirp. The Stokes spectra generated by a negatively-chirped pulse are shown in Fig. 4(a) for different seed frequencies. They suggest that the higher frequency wing of the Stokes spectrum can be controlled simply by reversing the direction of frequency chirp.
The dynamics of the interaction can be further clarified by studying seeded SRS with a chirp-compensated pump pulse. Using a pair of gratings (see Fig. 2) the output of the Fianium laser was compressed to a duration of about 1.5 ps. As seen in Fig. 4(b), tuning the seed frequency has no influence on the shape of Stokes spectra produced by the chirp-compensated pump pulse.
4. Discussion and analysis
The physical picture described in Section 2 provides a qualitative explanation for the experimental results. The fact that the cut-off frequency of the spectra in Fig. 3 is determined by the seed frequency is a consequence of chirping of the pump pulse and the presence of coherent “memory” in the system. The broad Stokes spectrum is the result of scattering of the pump pulse by the long-lived Raman coherence produced by resonant excitation of the molecules. It contains the Stokes-shifted frequencies of the pump pulse that arrive after the moment of resonant crossing t * (see Fig. 1). This suggests that the form of the Stokes spectrum in Fig. 3 is directly related to the long phase-relaxation time of molecular vibrations T 2 ≈ 48 ps, which in our experiments exceeds the pump pulse duration τp = 40 ps. The scattering regime is quite different when the pump pulse chirp is removed using a grating compressor. In that case all the spectral components of the short pump pulse arrive at the same moment and interact with the molecules simultaneously. Under these circumstances, variations in seed frequency have no significant effect on the shape of the Stokes spectrum generated, provided the seed frequency lies within the bandwidth of the pump.
To provide quantitative support for these conclusions we now present the results of a theoretical study of chirped pulse SRS. The SRS process can be described semi-classically by density-matrix equations for the Raman transition :
and Maxwell’s equations for the fields:
Here E p,s(z,t) are the complex amplitudes of the pump and the Stokes fields propagating in the fundamental mode of the HC-PCF in the +z direction, and ω p,s,/β p,s and ν p,s = ∂ω/∂β p,s are the related frequencies, propagation constants and group velocities. The functions n = ρ 22-ρ 11 and ρ 12 describe the population difference (∣ρ 11∣2 + ∣ρ 22∣2 = 1) and the coherence on the vibrational levels of CH4, T 1 and T 2 being respectively the energy and phase relaxation times. The parameter r 12 is the two photon matrix element of the transition . Before the interaction with the fields the population difference equals its equilibrium value n(-∞) = n 0 and ρ 12= 0.
Before presenting numerical results, it is helpful to analyze the initial stage of the SRS process when relative changes of the fields are still small. The response of the Raman transition can be calculated using the input values of the field amplitudes, i.e., by setting:
where τ = t-z/ν is the retarded time and νp ≈ νs = ν. The offset Δω of the tunable seed frequency ῶs from the central frequency ω s of the seed laser is defined by Δω = ῶs-ωs. In the experiment, the chirp parameter α of the pump pulse is such that q=ατp 2 ≫1, where τp is the pump pulse duration, δωp ≈ ατp is the chirp-related spectral broadening and τp -1 is the spectral width of a transform-limited (unchirped) pulse of the same duration. In our experiments q ≈ 30. This means that during the Raman interaction the phase of the pump pulse changes much faster than its amplitude. Assuming that the duration of the pump pulse τp is short compared to T 2 and that the Raman transition is only weakly saturated (n ≈ n 0 = -1), the off-diagonal density matrix element turns out to be:
where Φ(τ) = 0.5(1+ erf (θ)), and
Owing to the rapid change of θ on the time-scale of the pump pulse (due to q ≫ 1) and the asymptotic behavior of erf(x) at ∣x∣→∞, the function Φ(τ) is close to a step-like unit function U(τ) centered at the time of resonant crossing τ *= Δω/α. By integrating Eq. (3) with help of Eqs. (5) and (6), we find that increase in the Stokes field at distance z is
where ρ̄12 is the asymptotic value of the off-diagonal density matrix element (Eq. (5)) for θ ≫ 1. According to Eq. (7), before the resonant crossing point (τ < τ *), there is no Stokes generation at all, while after the resonant crossing (τ > τ *), the Stokes field grows in proportion to the complex amplitude of the pump field. This makes possible transfer of energy from the pump band to the Stokes band and manipulation of the spectrum simply by tuning the seed frequency.
In our numerical calculations, we studied two different situations. In the first, the duration of the linearly chirped pump pulse was assumed to be slightly shorter than the phase relaxation time T 2 of the transition. The Raman process starts at the moment of resonant crossing. Due to the presence of coherent “memory” in the system, all the photons in the broad pump spectrum that arrive after the moment of resonant crossing are scattered by the Raman coherence (Fig. 5(a)). By varying the seed frequency, the low frequency edge of the Stokes spectrum can be shifted within the broad generated Stokes spectrum, in close agreement with the experimental observations. The modeling shows that during the initial stage of Stokes generation, when only a small part (a few %) of the pump energy is converted to the Stokes, the generated spectrum reproduces very well all the details of the pump spectrum. But with increase of the propagation length (and conversion), the Stokes spectrum can show significant differences from the pump spectrum.
This physical picture is qualitatively different in the opposite situation when the duration of the chirped pump pulse considerably exceeds the phase relaxation time T 2. In this incoherent interaction regime, we find that the transfer of the broadband pump spectrum to the Stokes band, and the control of its low and high frequency wings, appear to be impossible. Instead, Stokes generation is observed only within the Raman linewidth ΔνR = 2/T 2 (Fig. 5(b)), which is similar to the situation when a broadband pump interacts with a narrowband Stokes field. All these observations count in favor of a scenario where resonantly excited Raman coherence gives rise to scattering of the remainder of the pump field within a time interval determined by the phase relaxation time T 2. This is an intriguing example of a situation where treatment of the problem in terms of a Raman linewidth and a radiation spectral width can lead to the wrong conclusions, it being essential to analyze the full temporal behavior of the system.
Comparison of the Stokes spectra in Figs. 3(a) and 4(a) with the input pump spectrum in Fig. 3(b) shows that, although some features of the pump spectrum are well reproduced in the Stokes spectra, the process does not provide an exact replica of the pump spectrum. We explain this by the fact that, due to SPM in the pump laser, the pump pulse chirp is far from linear. Because of the S-like form of the chirp, the resonant crossing ω p(t *) - ω seed-Ωv = 0 takes place not at one, but at two different times t 1 * and t 2 * . The additional crossing appears on the wings of the pump pulse where field intensity drops. Therefore the contribution of this process to SRS is relatively small within the linear part of the pump chirp, i.e., near the center of the pump pulse spectrum. Indeed, in this case, high fidelity transfer of the pump spectrum to the Stokes region is observed experimentally. These observations are also supported by the results of modeling, which show that for an ideally linear chirp (see Fig. 5(a)), close correspondence between the pump and Stokes spectra is observed. This is true as long as only a small part (a few %) of the pump energy is converted to the Stokes field. Experimentally, conversion to the Stokes field was higher (about 10%), and therefore a noticeable nonlinear distortion of the pump pulse amplitude and phase due to interaction with the Stokes field could occur. We also note that the influence of group velocity (GV) walk-off of the pulses on the Stokes generation was quite small. The group velocity dispersion parameters due to dispersion of the HC-PCF and dispersion of the methane gas at 10 bar pressure were β 2 (PCF) ≈ -900 fs2m-1 and β 2 (gas) ≈ 200 fs2m-1, respectively, resulting in a walk-off between the pulses of about 100 fs , which is nearly three order of magnitude smaller than the pump pulse duration. This means that GV walk-off starts to be important for significantly shorter (sub-ps) pump pulses.
The long interaction lengths, and well-controlled environment, offered by hollow core PCF in the study of gas-laser interactions makes it possible to explore the coherent properties of the Raman response in unprecedented detail. CW seeding of chirped-pulse SRS can be used to control and manipulate broadband Stokes generation simply by tuning the frequency of the seed-laser within the spectral width of the pump pulse. This leads to a change of the position of the low (or high) frequency wing of the Stokes spectrum. Since the phase modulation of the Stokes field is close to the phase modulation of the frequency chirped pump pulse, the Stokes pulse could be compressed to a short pulse duration. Gas-filled HC-PCF offers the unique possibility to observe ps-pulse transient SRS phenomena at pump energies (a few μJ) at least four orders of magnitude lower than those reported so far in Raman experiments with ps pulses .
We thank Dr. M. S. Kang for useful discussions. A. C. gratefully acknowledges support from Alexander von Humboldt Foundation.
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