Coherence waves (${\mathrm{C}}_{\mathrm{w}}\mathrm{s}$) of collective molecular oscillation, created by stimulated Raman scattering (SRS) [1–5], are useful in the synthesis of ultrashort pulses [6] and for efficient frequency conversion [3–5]. They arise through the beating between two intense quasi-monochromatic laser beams whose frequencies differ by the Raman transition frequency ${\mathrm{\Omega }}_{\mathrm{R}}$. The resulting ${\mathrm{C}}_{\mathrm{w}}$ takes the form of a traveling refractive index grating. Under phase-matched conditions, an arbitrarily weak mixing signal at a different wavelength can be scattered off this ${\mathrm{C}}_{\mathrm{w}}$, which is coherently amplified when the mixing signal is downshifted by ${\mathrm{\Omega }}_{\mathrm{R}}$ and absorbed when the mixing signal is upshifted by ${\mathrm{\Omega }}_{\mathrm{R}}$ [2–5].

Phase matching can be achieved using noncollinear beams in bulk materials [7,8] or higher-order guided modes in optical fibers [5,9]. Broadband-guiding kagomé-style hollow-core photonic crystal fiber (kagomé-PCF) is particularly attractive since the dispersion can be precisely controlled by varying the gas pressure [10]. In addition, tight modal confinement and long light–gas interaction lengths dramatically reduce the peak laser intensities required for the efficient excitation of ${\mathrm{C}}_{\mathrm{w}}\mathrm{s}$.

In this Letter, we report how a ${\mathrm{C}}_{\mathrm{w}}$, excited by an ${\mathrm{LP}}_{01}$ pump pulse in a hydrogen-filled kagomé-PCF, can be used, within its coherence lifetime, for the efficient phase-matched frequency-shifting of an ${\mathrm{LP}}_{01}$ mixing pulse at an entirely different wavelength. The process is made possible by the unique S-shaped dispersion curve that forms around the pressure-tunable zero dispersion point (ZDP) in a kagomé-PCF [10].

To illustrate the concept, the dispersion curve in the vicinity of a ZDP at wave vector ${\beta }_0$ and frequency ${\omega }_0$ is drawn schematically in Fig. 1, assuming no dispersion terms higher than third order. A ${\mathrm{C}}_{\mathrm{w}}$ is first “written” by pump and Stokes waves, labeled by $W_0$ and $W_{-1}$, respectively, on the diagram. The $\omega -\beta$ “four vector” of the resulting ${\mathrm{C}}_{\mathrm{w}}$ is marked by the blue dashed line. Once created, this ${\mathrm{C}}_{\mathrm{w}}$ can be used for fully phase-matched frequency upconversion to point $M_1$ of an optical mixing signal at point $M_0$ on the opposite side of the ZDP. It may also be used to seed frequency downconversion to $M_0$ of a signal at $M_1$, during which process it will be amplified. This procedure will of course also work the other way around, i.e., with writing signals on the low-frequency side of the ZDP and mixing signals on the high-frequency side.

 

Fig. 1. (a) Sketch of the dispersion of a gas-filled kagomé-PCF in the vicinity of the ZDP (the gray dot at $\omega ={\omega }_0$ and $\beta ={\beta }_0$), assuming no dispersion terms higher than third order. The blue dashed lines indicate the coherence wave ${\mathrm{C}}_{\mathrm{w}}$ excited by the writing signals $W_0$ and $W_{-1}$. ${\mathrm{C}}_{\mathrm{w}}$ is a four vector with frequency ${\mathrm{\Omega }}_R$ and wave vector $\mathrm{\Delta }\beta$ given by the difference between the wave vectors of the writing signals. It can be used either to upshift the frequency of a mixing signal $M_0$ placed at the position symmetric to $W_0$ on the opposite side of the ZDP or to seed the downconversion of a signal placed at $M_1$.

Download Full Size | PDF

In general, higher-order dispersion will increasingly distort the perfect S-shape as one moves further away from the ZDP. Nevertheless, as we shall see, it is always possible to find two distinct pairs of points, albeit asymmetrically placed about the ZDP, that are connected by the identical ${\mathrm{C}}_{\mathrm{w}}$. If the frequency of the mixing beam is not precisely correct, however, the upconversion process will be dephased at a rate given by

 

Fig. 2. (a) Dispersion curves of the ${\mathrm{LP}}_{01}$ mode for pressures of 3, 12, and 30 bar (see text). The notation is the same as in Fig. 1, and for clarity, only the 30 bar case is fully labeled. (b) Phase-matched frequency pairs (solid lines) and ZDPs (dotted line) as a function of pressure for a pump frequency ($W_0$) of 563 THz (532 nm). The ZDPs for the three pressures in (a) are marked with gray dots.

Download Full Size | PDF

The wavelength of the ZDP shifts from $\sim 400\mathrm{nm}$ to $\sim 1.1\mathrm{\mu m}$ as the pressure increases from 1 to 100 bar [Fig. 2(b)].

The mixing frequencies of perfectly phase-matched pairs can be accurately calculated by solving Eqs. (1) and (2). Figure 2(a) shows the ${\mathrm{C}}_{\mathrm{w}}\mathrm{s}$ created at three different pressures by beating a pump signal $W_0$ at 532 nm (563 THz) with a Stokes signal $W_{-1}$ at 685 nm (438 THz). The presence of higher-order dispersion shifts the positions of the two perfectly phase-matched pairs asymmetrically about the ZDP. At 30 bar, both the pump ($W_0$) and Stokes ($W_{-1}$) signals lie in the normal dispersion region ($\omega >{\omega }_0$), and the phase-matched mixing pair ($M_1$, $M_0$) is at (322, 197) THz in the anomalous dispersion region. At 3 bar, on the other hand, the ${\mathrm{C}}_{\mathrm{w}}$ is excited in the anomalous dispersion region and the mixing pair is phase matched at (749, 624) THz in the normal dispersion region.

A special situation occurs at 12 bar, when the lower frequency of the upper pair and the upper frequency of the lower pair coincide; this causes a strong second Stokes ($W_{-2}$) signal to appear when pumping with $W_0$ (see below). Overall, Fig. 2 shows that for a fixed $W_0$ frequency of 563 THz, the frequency of $M_1$ for perfect phase matching can be tuned from the UV to the near IR simply by changing the pressure.

To confirm these predictions experimentally, we developed a setup for exciting and probing ${\mathrm{C}}_{\mathrm{w}}\mathrm{s}$ under different conditions (see Fig. 3). The output of a linearly polarized 1064 nm laser, delivering nanosecond pulses, was split into two parts. The first part was used to generate the $W_0$ signal by frequency-doubling to 532 nm in a KTP crystal. The second was spectrally broadened to $\sim 350\mathrm{nm}$ in a 4 m long solid-core PCF and used as the $M_0$ mixing signal. The pulse durations were 3.2 ns for $W_0$ and 2.0 ns for $M_0$. Both pulses were then recombined and launched into the ${\mathrm{LP}}_{01}$ mode of a 1 m long gas-filled kagomé-PCF. The signals exiting the fiber endface were monitored using an optical spectrum analyzer and imaged using a CCD camera. The intensity of the mixing beam was kept low so as to avoid the onset of SRS, which would influence the ${\mathrm{C}}_{\mathrm{w}}$. The interaction was found to be strongest when the mixing pulse was delayed by $\sim 1\mathrm{ns}$ from the pump pulse.

 

Fig. 3. Schematic of the experimental setup. BS, beam splitter; DBS, dichroic BS; OAP, off-axis parabolic mirror; SC-PCF, solid-core photonic crystal fiber; DL, delay line; OSA, optical spectrum analyzer.

Download Full Size | PDF

The vibrational Raman gain of hydrogen saturates for pressures above $\sim 10\mathrm{bar}$ [12]. In this regime, the behavior of the system is dominated by changes in the phase-matching conditions. Below 10 bar, both the phase-matching conditions and the Raman gain are pressure-dependent and play a role.

We recorded the spectra generated when scanning the pressure from 5 to 40 bar, for two pulse energies, $E_P=20$ and 30 μJ (Fig. 4). The photon rates in the $W_0$, $W_{-1}$, and $W_{-2}$ bands are plotted in Figs. 4(a) and 4(c) and those in the mixing signals are in Figs. 4(b) and 4(d). Note that, for clarity, the sum of the rates in the two upshifted sidebands $M_1$ at 730 nm (411 THz) and $M_2$ at 560 nm (535 THz) is plotted, along with the signal $M_0$ at 1048 nm (286 THz); the full experimental data are available in Fig. S2 of Supplement 1.

 

Fig. 4. Experimental and theoretical normalized photon rates of the various signals, plotted against increasing pressure. (a) Pump ($W_0$), first ($W_{-1}$), and second ($W_{-2}$) Stokes waves for $E_P=20\mathrm{\mu J}$. (b) Mixing ($M_0$) and upshifted anti-Stokes ($M_1+M_2$) signals, coupled by the ${\mathrm{C}}_{\mathrm{w}}\mathrm{s}$ created in (a). (c) Pump ($W_0$), first ($W_{-1}$), and second ($W_{-2}$) Stokes waves for $E_P=30\mathrm{\mu J}$. (d) Mixing ($M_0$) and upshifted anti-Stokes ($M_1+M_2$) signals, coupled by the ${\mathrm{C}}_{\mathrm{w}}\mathrm{s}$ created in (c). The green solid lines show the pressure dependence of the phase-match parameter $\vartheta$. Note that the ${\mathrm{C}}_{\mathrm{w}}$ generated by the $W_{-1}\to W_{-2}$ conversion creates a second phase-matching pressure at point B. Upshifting to $M_1$ is most efficient at the phase-matching pressures (points A and B).

Download Full Size | PDF

In Fig. 4(a), the pump pulse energy was $E_P=20\mathrm{\mu J}$ (injected into the kagomé-PCF with $\sim 40\%$ coupling efficiency). As the pressure increases, $W_0$ converts to $W_{-1}$ via SRS, giving rise to a ${\mathrm{C}}_{\mathrm{w}}$ and becoming significantly depleted. At 12 bar, $W_{-1}$ is itself depleted and light appears at the $W_{-2}$ frequency (i.e., the second Stokes). This is because at this pressure both the $W_0\to W_{-1}$ and $W_{-1}\to W_{-2}$ processes are phase-matched to the same ${\mathrm{C}}_{\mathrm{w}}$. At higher pressures, the $W_{-1}\to W_{-2}$ phase mismatch increases and the $W_{-2}$ signal drops again.

At 14.2 bar, the ${\mathrm{C}}_{\mathrm{w}}$ created in the $W_0\to W_{-1}$ process is able to phase-match the $M_0\to M_1$ conversion from 1048 to 730 nm, with photon number conversion efficiencies (based on the depletion of the $M_0$ signal) well above 50% at the peak [Fig. 4(b)].

At 30 μJ pump pulse energy, a strong $W_{-2}$ signal appears for pressures above 15 bar [Fig. 4(c)]. This is because the $W_{-1}$ signal is strong enough to reach the threshold for SRS, generating a strong $W_{-2}$ signal and an independent ${\mathrm{C}}_{\mathrm{w}}$. This new ${\mathrm{C}}_{\mathrm{w}}$ causes a new phase-matching point to appear at a pressure of $\sim 27\mathrm{bar}$, resulting in a second peak of $M_0\to M_1$ conversion [Fig. 4(d)]. At the same time, the first conversion peak reaches conversion efficiency above 70%.

As shown in the right-hand column of Fig. 4, these measurements are in good overall agreement with numerical solutions of a set of coupled spatiotemporal Maxwell–Bloch equations [13]. Note that the model is only valid for pressures above 10 bar, i.e., outside the shaded region in Fig. 4, when the Raman gain is independent of pressure (see Supplement 1 for details).

The simulations also allow us to study the behavior of the ${\mathrm{C}}_{\mathrm{w}}$ amplitude $|Q|$, a quantity that is not accessible in the experiment. Figure 5 plots the evolution of both $|Q|$ and the ($M_1+M_2$) photon rate along the fiber at gas pressures of 12 and 27 bar for $E_P=30\mathrm{\mu J}$. At 12 bar, $|Q|$ grows through a SRS-related exponential gain in the $W_{-1}$ signal (upper left-hand panel in Fig. 5). It peaks at $\sim 40\mathrm{cm}$, decreasing thereafter because by that point the majority of pump photons have been converted to the $W_{-1}$ band. Note that the temporal peak of $|Q|$ is delayed by $\sim 1\mathrm{ns}$ relative to the center of the pump pulse, as expected in transient Raman scattering. For this special pressure (as explained above in Fig. 2), the same ${\mathrm{C}}_{\mathrm{w}}$ is simultaneously able to seed the downconversion from the $W_{-1}$ to the $W_{-2}$ band (and thus be amplified), giving rise to the strong $W_{-2}$ signals observed in Figs. 4(a) and 4(c). As expected, upconversion from the $M_0$ to the $M_1$ band is highest when $|Q|$ is strongest (upper right-hand panel in Fig. 5).

 

Fig. 5. Simulated spatiotemporal evolution of the normalized ${\mathrm{C}}_{\mathrm{w}}$ amplitude $|Q|$ and the normalized ($M_1+M_2$) photon rate for $E_P=30\mathrm{\mu J}$ at pressures of 12 bar (upper) and 27 bar (lower). The ${\mathrm{C}}_{\mathrm{w}}\mathrm{s}$ created in the $W_0\to W_{-1}$ process are marked with (I) and those created by the $W_{-1}\to W_{-2}$ process with (II). The inset in the upper right panel shows the normalized temporal profiles of the $W_0$ and $M_0$ pulses at the entrance to the fiber. Time is relative to a frame traveling at the group velocity of the $W_0$ pulse. The origin of the periodic oscillations in the lower left panel is discussed in Supplement 1.

Download Full Size | PDF

This special dual phase-matching condition is no longer valid at 27 bar, because the $W_0\to W_{-1}$ and $W_{-1}\to W_{-2}$ conversion processes produce different ${\mathrm{C}}_{\mathrm{w}}\mathrm{s}$. As a result, the first peak in $|Q|$ at $\sim 25\mathrm{cm}$ originates from $W_0\to W_{-1}$, whereas the second peak at $\sim 60\mathrm{cm}$ is created by the $W_{-1}\to W_{-2}$ conversion and appears at a later time (lower left-hand panel in Fig. 5). For the mixing beams, although the $M_0\to M_1$ conversion is very weak at the 25 cm point due to strong dephasing ($\vartheta =1.6{\mathrm{cm}}^{-1}$), it is strong at 60 cm because of phase matching with the second ${\mathrm{C}}_{\mathrm{w}}$ (lower right-hand panel in Fig. 5). This illustrates the supreme importance of phase matching for efficient frequency conversion.

From coupled mode theory, the bandwidth $\mathrm{\Delta }\nu$ of the upconversion process covers the range $-\pi <\vartheta L_c<\pi$, where $L_c$ is the length over which the ${\mathrm{C}}_{\mathrm{w}}$ amplitude is significant ($\sim 10\mathrm{cm}$ in the experiments). $\mathrm{\Delta }\nu$ depends on the curvature of the dispersion curve for the $M_0\to M_1$ process and ranges from $\sim 250\mathrm{THz}$ for $M_0$ at 800 THz, to $\sim 20\mathrm{THz}$ for $M_0$ at 300 THz. These large bandwidths make it possible to frequency-shift and replicate broadband signals, as shown in Fig. 6 where the full recorded spectrum at 14.5 and 27.6 bar is plotted for $E_P=30\mathrm{\mu J}$. Cascaded upshifting to the $M_2$ band is possible in the case of 14.5 bar (upper panel in Fig. 6) because of the relatively flat dispersion curve from $\sim 300$ to $\sim 600\mathrm{THz}$ [see for instance the 12 bar curve in Fig. 2(a)]. In contrast, at 27.6 bar, efficient conversion is only possible to the first ($M_1$) sideband (lower panel in Fig. 6), owing to the strong curvature of the dispersion over the whole frequency range [see for instance the 30 bar curve in Fig. 2(a)].

 

Fig. 6. Demonstration of broadband frequency shifting for 14.5 bar (upper panel) and 27.6 bar (lower panel). The broadband mixing signal $M_0$ is frequency-upshifted to the first ($M_1$) and second ($M_2$) anti-Stokes bands. The original spectrum $M_0^0$ (pump switched off) is depleted accordingly. The photon rates are normalized to the peak of the broadband mixing $M_0^0$ signal at 1064 nm.

Download Full Size | PDF

Since $\vartheta$ is also a function of pressure, it determines the width of the conversion peaks as shown in Fig. 4(d), where $\vartheta$ is plotted for the two ${\mathrm{C}}_{\mathrm{w}}\mathrm{s}$ excited in that case. It is clear that a weaker pressure dependence of $\vartheta$ translates into broader conversion peaks.

In this Letter, we have concentrated on frequency upshifting. As mentioned in the introduction, it is also possible to use a ${\mathrm{C}}_{\mathrm{w}}$ to frequency downshift a mixing signal. In this case, the ${\mathrm{C}}_{\mathrm{w}}$ acts as a (potentially very strong) seed, being amplified in the conversion process. In Fig. 7, we show the result of an experiment with the same setup where a broadband signal at $\sim 325\mathrm{THz}$ (920 nm) is downshifted. The maximum conversion efficiency achieved is $\sim 25\%$. Note also that since the phase-matching frequency shifts with pressure, so does the peak of the downshifted frequency band. The experimental trace nicely follows the analytical curve for perfect phase matching (the dashed curve in Fig. 7). Taken together with all-${\mathrm{LP}}_{01}$ operation, this further demonstrates the wide usable frequency range and versatility of this unique wavelength convertor.

 

Fig. 7. Frequency-downshifted signal versus pressure. The dashed line represents the theoretical perfectly phase-matched frequencies.

Download Full Size | PDF

In conclusion, the S-shaped dispersion curve in the vicinity of the ZDP makes the hydrogen-filled kagomé-PCF an ideal vehicle for highly efficient wavelength conversion of broadband optical signals. A ${\mathrm{C}}_{\mathrm{w}}$ (which behaves like a traveling refractive index grating) is first excited by SRS on one side of the ZDP. It is then used for the phase-matched conversion of a signal on the opposite side of the ZDP. The system is widely pressure-tunable, permitting highly efficient up- and downconversion, by 125 THz, of ${\mathrm{LP}}_{01}$ signals from the UV to the near IR. The system offers a new route for the frequency conversion of ultrashort laser pulses from mode-locked lasers. Further flexibility is possible by changing the fiber design, working with different Raman active gases, adding buffer gases, or working with a different pump wavelength. This uniquely flexible system has many potential applications in ultrasensitive Raman spectroscopy and laser science.