1. Introduction

Generation of spatiotemporal coherent light at frequencies that are not covered by standard lasing transitions has been an ongoing field of research since the invention of the first lasers. Many different methods based on nonlinear effects such as self-phase modulation [1] or high harmonic generation [2], have been developed to extend the accessible spectral regions from the far-infrared to the extreme ultraviolet and even beyond [3]. In this direction, stimulated Raman scattering (SRS) offers a versatile way to access new spectral regimes by the inelastic scattering of photons from molecules. In particular Raman-active gases, in contrast to solid-state materials, offer wide transparency windows and allow the generation of mutually coherent and equidistant spectral sidebands spanning several octaves [4]. In addition, optical signals of arbitrarily low amplitude can be shifted by the Raman frequency using the technique of molecular modulation [5–7]. In this approach, two intense laser signals with a frequency offset equal to the Raman shift drive the molecules coherently, creating a moving “coherence wave” that spatially modulates the refractive index. Under the correct conditions this wave can be used to shift the frequency of a probe signal. This approach has recently been proposed as an efficient means of generating tunable THz radiation from infrared pulses [8]. When implemented in bulk gas cells or wide-bore capillaries, however, all these methods require relatively high peak powers in the pump fields.

Gas-filled hollow-core photonic crystal fiber (HC-PCF) offers the twin advantages of tight modal confinement and long light-matter interaction lengths (not accessible in simple glass capillaries), which greatly reduces the peak power required for the observation of nonlinear phenomena [9, 10]. This makes it an ideal candidate for, e.g., the generation of multiple Raman sidebands [11, 12]. A further advantage is that the dispersion and nonlinearity can be widely tuned simply by varying the gas pressure [13].

In this paper we present an efficient fiber-based method for selective frequency up-conversion of a broadband optical signal using Raman-induced molecular modulation. By means of the differing dispersive properties of higher order modes [14, 15], perfect phase-matching for this up-shifting process is achieved in hydrogen-filled HC-PCF. In contrast with other approaches [5–7], most of the power of the probe beam is selectively converted to just a single band. The conversion efficiency of the linear scattering mechanism can be optimized for any single wavelength over a broad range by adjusting the gas pressure in the fiber.

2. Experimental set-up

The set-up is sketched in Fig. 1(a).We used a 1064 nm Q-switched laser delivering 3.8 ns pulses. The linearly-polarized beam was divided at a beam splitter. One part was used to generate a 150 nm wide supercontinuum (SC) signal in a 4 m-long solid-core PCF (core diameter 5.0 µm); this was used as the mixing signal. The other part, which we refer to as the pump beam, was diverted through a free-space delay line and combined with the SC mixing signal at a second beam splitter. The signals were then coupled through a pressure cell into a hydrogen-filled kagome-style HC-PCF [in Fig. 1(b)] with a length of 1 m and an area preserving core radius [16] of a_AP = 19.3 µm. An optical spectrum analyzer was used to monitor the spectrum of the out-coupled light and a CCD camera imaged the near-field modal pattern.

 

Fig. 1 (a) Schematic of the experimental set-up. ND: neutral density filter; BP: band-pass filter; BS: beam-splitter. (b) Scanning electron micrograph of the fiber.

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3. Experimental results

3.1 Basic observations

When only the pump beam (ω_P/2π = 282 THz) was launched into the gas-filled fiber, combs of both rotational and vibrational Raman lines were generated from noise [9, 11, 12]. The corresponding Raman shifts in hydrogen are Ω_R/2π = 17.6 THz for the rotational S(1) Raman transition and Ω_V/2π = 125 THz for the vibrational Q(1) Raman transition [17]. The measured spectrum is shown in Fig. 2(a) for a pump energy of 30 µJ and a gas pressure of 18.7 bar.

 

Fig. 2 Photon count-rate spectra normalized to the rate of the pump (the peak is not shown to enhance the other spectral lines). (a) Pump signal only. (b) SC mixing signal only. (c) Both pump and SC mixing signals. (d)-(h) Normalized near-field intensity distributions recorded at the end-face of the fiber filled with a gas pressure of 10 bar. (d) Pump beam, mainly coupled into LP₀₁ mode; (e) mixing beam at 1080 nm; (f) first AS; (g) shifted SC; (h) pump with higher modal content in the LP₁₁ mode.

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Note that the first vibrational Stokes line [ω_S/2π = 157 THz, red solid line in Fig. 2(a,c)] was measured with an uncalibrated optical spectrum analyzer, so that its height does not scale with the rest of the spectrum. In contrast to the intense pump pulse, the spectrum of the relatively weak SC mixing beam, when launched in the absence of the pump, is not altered by propagating along the fiber [Fig. 2(b)]; this is because it is not powerful enough to initiate SRS. When, however, both pump and SC beams are simultaneously coupled into the fiber with an appropriate delay and modal content, two major effects can be observed [see Fig. 2(c)]. First, the SC spectrum seeds the first rotational Stokes line (ω_rS/2π = 264 THz) and a slight increase in the absolute strength of the rotational sidebands is observed due to SRS. Second, a broad spectrum appears around the first vibrational anti-Stokes (AS) line (ω_AS/2π = 406 THz). These new spectral components correspond to light that is frequency up-shifted relative to the mixing signal.

In the experiment, the up-shifted signal was strongest when the pump beam was mainly launched in an LP₀₁ mode [Fig. 2(d)] while the mixing beam was both delayed by 1.2 ns with respect to the pump and misaligned so as to excite a mixture of LP₀₁ and LP₁₁ modes [Fig. 2(e)]. Under these conditions, the first vibrational AS signal [Fig. 2(f)], and the broad up-shifted spectral background located around it [Fig. 2(g)], were found to appear predominantly in the LP₁₁ mode. Remarkably, the maximum photon count-rate in the shifted spectrum is of the same order of magnitude as that in the mixing beam, indicating that the process has high quantum efficiency.

3.2 Analysis of the results

The mechanism behind the generation of vibrational AS lines, and the up-shifted broad SC in the vicinity of the first AS band in Fig. 2(a,c), can be understood with the help of a dispersion diagram. This requires an accurate knowledge of the dispersion relation for the LP₀₁ and LP₁₁ modes of the kagome-PCF, which can be obtained from the expression [18]:

Figure 3(a) shows the dispersion curves for the LP₀₁ and the LP₁₁ modes of the evacuated kagome-PCF, plotted on a diagram of frequency versus wavevector difference $(k_0-{\beta }_{mq})$, where $k_0=2\pi /\lambda$, λ is the vacuum wavelength and ${\beta }_{mq}=k_0{n}_{mq}$ is the modal wavevector. Using Eq. (1), the phase index of the coherence wave created by Raman conversion from a signal at frequency ω to a signal at $(\omega -\Omega )$ may be written as $n_k^l(\omega )=c{\Omega }^{-1}\left[{\beta }_k(\omega )-{\beta }_l(\omega -\Omega )\right]$where the subscript k denotes the mode at frequency ω and the superscript l the mode at (ω – Ω). In this way, we can define the coherence wave via the four-vector ${\kappa }_k^l(\omega )=\left[\Omega c^{-1}{n}_k^l(\omega ),\Omega \right]$. To simplify the notation, we will from now on use Ω to refer to the vibrational transition.

 

Fig. 3 (a) Dispersion diagram for the LP₀₁ and LP₁₁ modes at zero pressure. The horizontal lines mark the frequencies of the pump (P), first Stokes (S) and first anti-Stokes (AS) bands. The spectral widths of the mixing and the shifted SC signals are indicated by gray-scale shading. The backward (forward) sloping arrows indicate the four-vectors of the intra- (inter-) modal coherence waves. (b) Phase indices of different coherence waves as a function of the upper of the two frequencies used to generate them.

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In the first stage of the experiment, an intramodal coherence wave was created by launching an LP₀₁ pump wave (frequency ω_P) into the fiber with power sufficient to initiate SRS; the four-vector ${\kappa }_{01}^{01}({\omega }_{\text{P}})$ of this coherence wave is indicated by the green arrow in Fig. 3(a). In the second stage, LP₁₁ light at frequency ω_P, already present in the fiber owing to launch misalignment, could be frequency up-shifted to the first vibrational AS within the LP₁₁ mode by means of the previously excited ${\kappa }_{01}^{01}({\omega }_{\text{P}})$ coherence wave. For this process to be efficient, phase-matching is essential, i.e., $n_{01}^{01}({\omega }_{\text{P}})=n_{11}^{11}({\omega }_{\text{AS}})$. In the experiment it turned out that the phase mismatch was small, resulting in generation of a band of LP₁₁ AS light [Fig. 2(a)]. Conversely, direct intramodal conversion to the LP₀₁ AS was strongly suppressed owing to a higher degree of dephasing, which could not be compensated at any available pressure resulting in generation of a clean LP₁₁ AS signal [Fig. 2(f)].

As we will show later, it is also possible to create a comparatively strong intermodal coherence wave ${\kappa }_{11}^{01}({\omega }_{\text{P}})$ [Fig. 3(a)] by launching a higher proportion of pump power in the LP₁₁ mode. This occurs through conversion of LP₁₁ pump light to the LP₀₁ Stokes band, seeded by light that is already present in that band. This coherence wave is then able to convert LP₀₁ pump light to an AS signal in the LP₁₁ mode [14, 15], if ${\kappa }_{11}^{01}({\omega }_{\text{P}})\approx {\kappa }_{11}^{01}({\omega }_{\text{AS}})$. Note that these up-shifting processes, mediated by pre-existing coherence waves, are not in general restricted to light at the original pump frequency. If the phase-mismatch is arranged to be sufficiently small, signals at other frequencies can be efficiently up-converted. Since the SC mixing signal was launched in a superposition of LP₀₁ and LP₁₁ modes [Fig. 2(e)], SC light in the LP₁₁ mode could also be up-shifted by the ${\kappa }_{01}^{01}({\omega }_{\text{P}})$ coherence wave, resulting in a broad spectral band around the AS frequency [Fig. 2(c)].

Possible phase-matched transitions are illustrated in Fig. 3(b) for a pressure of 18.7 bar. The dashed vertical lines mark coherence waves with identical phase indices. The green dots at the lower end of these lines indicate the indices of the coherence waves generated when SRS occurs at a pump frequency of 282 THz. These waves can be used for phase-matched transfer of light to an AS band in the vicinity of the upper green dots. In particular, the intermodal coherence wave with index $n_{11}^{01}({\omega }_P)$ can be used for efficient generation of light at ~430 THz, and the intramodal coherence wave with index $n_{01}^{01}({\omega }_P)$ permits efficient up-conversion to ~400 THz in the LP₁₁ mode. Note that for symmetry reasons a coherence wave generated by intermodal SRS cannot be used for intramodal conversion (dotted line marked with a red cross) [14].

In the experiments, rotational excitations are also present, leading to the formation of rotational sidebands [Fig. 2(a)]. These bands are however quite weak since both SC and pump signals are linearly polarized, favoring vibrational transitions. Moreover, in hydrogen the vibrational gain is much higher than the rotational [17]. Thus, although the scattering of photons by the rotational coherence wave cannot be excluded, its efficiency can be kept extremely low if a SC mixing beam with a relatively narrow spectral width [see Fig. 2(b)] is used, so as to avoid strong seeding of rotational SRS.

3.3 Linear dependence and conversion efficiency

Given the presence of a sufficiently strong phase-matched coherence wave, the conversion probability for photons belonging to a subsequent mixing signal is expected to be independent of the power of the mixing signal itself, provided it is sufficiently weak [5]. We have studied the linearity of this scattering process by attenuating the SC mixing beam using a set of neutral density filters. The intensities of both the mixing beam and the shifted SC were measured at the output of the fiber while keeping the pump energy constant at 30 µJ and the pressure at 14 bar. The proportionality between both shifted and mixing signals observed in Fig. 4(a) indicates that the mixing does not noticeably influence the strength of the coherence wave. We can therefore infer that this scattering process occurs without any threshold and can be used for shifting of arbitrarily weak signals.

 

Fig. 4 (a) Comparison of the intensities of shifted and mixing signals, normalized to their values in the absence of any ND filters. The black line is a linear fit of the experimental data, yielding a slope of 0.997 ± 0.013. (b) Conversion efficiency η of the up-shifting process (solid curve). The dashed curve shows the analytical conversion efficiency after fitting to Eq. (3).

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Calibrated spectra as those shown in Fig. 2, allow estimation of the absolute photon number conversion efficiency $\eta (\omega )=N_{\text{P,M}}(\omega )/N_{\text{M}}(\omega -\Omega )$ of the scattering process, where $N_{\text{P,M}}(\omega )$ is the photon count-rate recorded with both pump and mixing beams present in the fiber [Fig. 2(c)], and $N_{\text{M}}(\omega -\Omega )$ is the reference spectrum corresponding to the original mixing beam [Fig. 2(b)].

The resulting conversion efficiency η(ω) is plotted in Fig. 4(b) for p = 18 bar and a pump pulse energy of 18 µJ. The maximum value $\eta \approx 0.79$ is located at 399 THz, which corresponds to the frequency of perfect phase-matching for this specific pressure. Apart from this isolated peak, the average conversion efficiency turns out to be ~50%. Note that this is likely to be an underestimate since only the LP₁₁ content of the original mixing beam (which is a mixture of LP₀₁ and LP₁₁ modes) can be converted by the excited ${\kappa }_{01}^{01}({\omega }_{\text{P}})$ coherence wave because of the need for phase-matching [Fig. 3(a)].

The observed gain bandwidth of η(ω), shows that the up-shifting process can tolerate a certain amount of phase-mismatch. This can be explained by modeling the coupling of the mixing beam to the AS band. The evolution with z of the electric field amplitudes of the mixing $E_{\text{mi}}(\omega -\Omega ,z)$ and shifted $E_{\text{sh}}(\omega ,z)$ signals can be written as:

3.4 Pressure dependence of the conversion efficiency

Perfect phase-matching of the up-shifting process, which occurs when ${\kappa }_{01}^{01}({\omega }_{\text{P}})={\kappa }_{11}^{11}(\omega )$, depends on both frequency and pressure. In order to quantify this feature of the system, the hydrogen pressure inside the HC-PCF was tuned from 1 to 37.5 bar, and the conversion efficiency η in a given frequency range was measured at each step. The results are shown in Fig. 5(a).The spectral component for which η is maximum shifts smoothly from 395 THz to 420 THz with increasing pressure. For lower pressures (below ~10 bar) and hence lower molecular densities, η drops rapidly owing to decreasing Raman gain [20]. At higher pressures η fluctuates slightly on a linear scale up to 35 bar, with an average value of ~50% and some isolated peaks reaching ~80% at 18 and 30 bar. Remarkably, these measured conversion efficiencies to a single sideband are about a factor of two higher, and the pulse energies 100 times lower, than the values previously reported in experiments performed using nanosecond pump pulses [6].

 

Fig. 5 Pressure-dependence of the conversion efficiency at different frequencies. (a) Pump beam mainly in the LP₀₁ mode. (b) Pump beam with a high LP₁₁ modal content. The dashed and solid lines correspond to numerical curves of perfect phase-matching for intramodal and intermodal coherence waves. The dotted line in (a) indicates the measurement in Fig. 4(b)

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We have verified that the overall shape of the data in Fig. 5(a) follows the condition for perfect phase-matching of the intramodal coherence waves $n_{01}^{01}({\omega }_{\text{P}})=n_{11}^{11}(\omega )$, in terms both of frequency and pressure (dashed line).

3.5 Effects of intermodal coherence waves

To explore the role of the intermodal coherence waves, the pressure dependence of η(ω) was monitored while adjusting the coupling of the pump beam so as to increase the fraction of the total power in the LP₁₁ mode [a typical near-field profile is shown in Fig. 2(h)]. The measured conversion efficiencies are shown in Fig. 5(b) for a pump pulse energy of 36 µJ. The branch of high η(ω) mediated by the intramodal coherence wave [see Fig. 5(a)] is still visible in this measurement, although the absolute values of η(ω) are slightly lower. However, the most interesting feature here is the appearance of a second branch crossing the first one at the AS frequency. We attribute this new observation to the excitation of the intermodal coherence wave ${\kappa }_{11}^{01}({\omega }_{\text{P}})$. The phase-matching condition for this intermodal scattering process $n_{11}^{01}({\omega }_{\text{P}})=n_{11}^{01}(\omega )$ is plotted in Fig. 5(b), again displaying good agreement with the experimental results (yellow solid line). Interestingly, in the case of single-frequency AS generation both scattering branches are usually degenerate and so phase-matched at the same pressure [15]. This can be directly understood from the diagram in Fig. 5(b), as the generation of the first AS signal corresponds to the crossing point of both intramodal and intermodal phase-matching lines. In contrast, this degeneracy between both branches vanishes when the up-shifted photons do not correspond to the AS line. The use of a SC mixing beam allows, for first time, to unambiguously access the non-degenerate regime, so that the intramodal and intermodal scattering schemes are perfectly phase-matched at different pressures for a given frequency as shown in Fig. 5(b).

4. Conclusions

In conclusion, the generation of coherence waves in a hydrogen-filled HC-PCF provides a simple and versatile means of achieving efficient and selective frequency conversion of arbitrary signals without any threshold. Since the dispersion of the different fiber modes changes with the gas pressure, perfect phase-matching can be finely tuned over a broad frequency range. Conversion efficiencies of ~50% are achieved over broad spectral bandwidths despite using pump pulse energies that are two orders of magnitude lower than in previous experiments [6]. Similar to the demonstrated up-shifting process, down-shifting would in principle be possible in our experimental set-up using a mixing beam with a frequency close to the AS band. In fact, our approach allows for both up- and down-shifting of arbitrary signals provided that suitable parameters (such as core diameter, pressure, pump wavelength and low attenuation at all the relevant wavelengths) are available. The technique may find applications in the development of novel tunable laser sources and highly sensitive in-fiber coherent anti-Stokes Raman spectroscopy of trace gases [21].