When a laser pump beam of sufficient intensity is incident on a Raman-active medium such as hydrogen gas, a strong Stokes signal, redshifted by the Raman transition frequency , is generated. This is accompanied by the creation of a “coherence wave” of synchronized molecular oscillations with wave vector determined by the optical dispersion. Within its lifetime, this coherence wave can be used to shift by the frequency of a third “mixing” signal, provided phase matching is satisfied, i.e., is matched. Conventionally, this can be arranged using noncollinear beams or higher-order waveguide modes. Here we report the collinear phase-matched frequency shifting of an arbitrary mixing signal using only the fundamental modes of a hydrogen-filled hollow-core photonic crystal fiber. This is made possible by the S-shaped dispersion curve that occurs around the pressure-tunable zero dispersion point. Phase-matched frequency shifting by 125 THz is possible from the UV to the near IR. Long interaction lengths and tight modal confinement reduce the peak intensities required, allowing conversion efficiencies in excess of 70%. The system is of great interest in coherent anti-Stokes Raman spectroscopy and for wavelength conversion of broadband laser sources.
© 2015 Optical Society of America
Coherence waves () of collective molecular oscillation, created by stimulated Raman scattering (SRS) [1–5], are useful in the synthesis of ultrashort pulses  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 . The resulting 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 , which is coherently amplified when the mixing signal is downshifted by and absorbed when the mixing signal is upshifted by [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 . In addition, tight modal confinement and long light–gas interaction lengths dramatically reduce the peak laser intensities required for the efficient excitation of .
In this Letter, we report how a , excited by an pump pulse in a hydrogen-filled kagomé-PCF, can be used, within its coherence lifetime, for the efficient phase-matched frequency-shifting of an 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 .
To illustrate the concept, the dispersion curve in the vicinity of a ZDP at wave vector and frequency is drawn schematically in Fig. 1, assuming no dispersion terms higher than third order. A is first “written” by pump and Stokes waves, labeled by and , respectively, on the diagram. The “four vector” of the resulting is marked by the blue dashed line. Once created, this can be used for fully phase-matched frequency upconversion to point of an optical mixing signal at point on the opposite side of the ZDP. It may also be used to seed frequency downconversion to of a signal at , 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.
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 . If the frequency of the mixing beam is not precisely correct, however, the upconversion process will be dephased at a rate given by11]. This dispersion relation is plotted in Fig. 2(a) at three different pressures for the fiber used in the experiments ( and ). In order to magnify the very flat S-shape, we have subtracted off the linear dispersion by plotting frequency versus (), where is a pressure-dependent linear function of frequency chosen such that () is 0 at 800 THz.
The wavelength of the ZDP shifts from to 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 created at three different pressures by beating a pump signal at 532 nm (563 THz) with a Stokes signal 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 () and Stokes () signals lie in the normal dispersion region (), and the phase-matched mixing pair (, ) is at (322, 197) THz in the anomalous dispersion region. At 3 bar, on the other hand, the 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 () signal to appear when pumping with (see below). Overall, Fig. 2 shows that for a fixed frequency of 563 THz, the frequency of 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 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 signal by frequency-doubling to 532 nm in a KTP crystal. The second was spectrally broadened to in a 4 m long solid-core PCF and used as the mixing signal. The pulse durations were 3.2 ns for and 2.0 ns for . Both pulses were then recombined and launched into the 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 . The interaction was found to be strongest when the mixing pulse was delayed by from the pump pulse.
The vibrational Raman gain of hydrogen saturates for pressures above . 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, and 30 μJ (Fig. 4). The photon rates in the , , and 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 at 730 nm (411 THz) and at 560 nm (535 THz) is plotted, along with the signal at 1048 nm (286 THz); the full experimental data are available in Fig. S2 of Supplement 1.
In Fig. 4(a), the pump pulse energy was (injected into the kagomé-PCF with coupling efficiency). As the pressure increases, converts to via SRS, giving rise to a and becoming significantly depleted. At 12 bar, is itself depleted and light appears at the frequency (i.e., the second Stokes). This is because at this pressure both the and processes are phase-matched to the same . At higher pressures, the phase mismatch increases and the signal drops again.
At 14.2 bar, the created in the process is able to phase-match the conversion from 1048 to 730 nm, with photon number conversion efficiencies (based on the depletion of the signal) well above 50% at the peak [Fig. 4(b)].
At 30 μJ pump pulse energy, a strong signal appears for pressures above 15 bar [Fig. 4(c)]. This is because the signal is strong enough to reach the threshold for SRS, generating a strong signal and an independent . This new causes a new phase-matching point to appear at a pressure of , resulting in a second peak of 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 . 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 amplitude , a quantity that is not accessible in the experiment. Figure 5 plots the evolution of both and the () photon rate along the fiber at gas pressures of 12 and 27 bar for . At 12 bar, grows through a SRS-related exponential gain in the signal (upper left-hand panel in Fig. 5). It peaks at , decreasing thereafter because by that point the majority of pump photons have been converted to the band. Note that the temporal peak of is delayed by 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 is simultaneously able to seed the downconversion from the to the band (and thus be amplified), giving rise to the strong signals observed in Figs. 4(a) and 4(c). As expected, upconversion from the to the band is highest when is strongest (upper right-hand panel in Fig. 5).
This special dual phase-matching condition is no longer valid at 27 bar, because the and conversion processes produce different . As a result, the first peak in at originates from , whereas the second peak at is created by the conversion and appears at a later time (lower left-hand panel in Fig. 5). For the mixing beams, although the conversion is very weak at the 25 cm point due to strong dephasing (), it is strong at 60 cm because of phase matching with the second (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 of the upconversion process covers the range , where is the length over which the amplitude is significant ( in the experiments). depends on the curvature of the dispersion curve for the process and ranges from for at 800 THz, to for 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 . Cascaded upshifting to the band is possible in the case of 14.5 bar (upper panel in Fig. 6) because of the relatively flat dispersion curve from to [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 () 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)].
Since is also a function of pressure, it determines the width of the conversion peaks as shown in Fig. 4(d), where is plotted for the two excited in that case. It is clear that a weaker pressure dependence of 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 to frequency downshift a mixing signal. In this case, the 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 (920 nm) is downshifted. The maximum conversion efficiency achieved is . 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- operation, this further demonstrates the wide usable frequency range and versatility of this unique wavelength convertor.
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 (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 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.
See Supplement 1 for supporting content.
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