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Using a four-mode theoretical analysis we show that highly efficient anti-Stokes conversion in waveguides is more challenging to realize in practice than previously thought. By including the dynamics of conversion to 2nd Stokes via stimulated Raman scattering and four-wave mixing, models predict only narrow, unstable regions of highly efficient anti-Stokes conversion. Experimental results of single-pass Raman conversion in confined capillary waveguides validate these predictions. This places more stringent conditions on systems that require highly efficient single-pass anti-Stokes conversion.
©2010 Optical Society of America
1. Introduction
With the advent of hollow-core photonic crystal fibers (HC PCF) [1], there has been a renewed interest in Raman conversion in molecular gases [2–4]. This is primarily due to the capability for both high sustained intensities and an increased interaction length with the Raman medium via the tight confinement (~100 μm2) and low losses (<0.1 dB/m) of the HC PCF. In a series of papers, Benabid and associates have shown nanoJoule thresholds for Stokes conversion [2], Stokes conversion efficiencies of greater than 90%, the capability to create multiple Stokes and anti-Stokes orders across Terahertz bandwidths that may be self-coherent [3], and efficient single-pass Stokes conversion with cw lasers [4].
While efficient stimulated Raman scattering (SRS) conversion to the Stokes has been demonstrated, efficient anti-Stokes conversion through four-wave mixing (FWM) processes has historically been very poor. Because anti-Stokes generation is a parametric process, phase matching becomes critical to efficient conversion. In addition, as phase matching is achieved, it subsequently reduces the total Stokes gain due to interference between the SRS and FWM processes making it harder to reach threshold (so called gain suppression). Roos et al. described the process theoretically for a cw Raman resonator and showed that efficient anti-Stokes conversion was possible by dispersion compensating an ultra-high finesse cavity, which simultaneously phase matches the process and increases the interaction length to overcome the resulting gain suppression [5]. Theoretically, close to 50% quantum efficiency could be obtained under the appropriate conditions. Subsequently, Zaitsu et al. partially validated this theory and showed a ~2% intracavity ratio between the anti-Stokes and Stokes beams with a dispersion compensated Raman resonator [6].
It is a natural extension of the work of both Benabid and Roos to consider whether single-pass, highly efficient anti-Stokes conversion is possible by use of a dispersion compensated waveguide. In such a system, the low anomalous dispersion of the waveguide is balanced by varying the pressure of the Raman active gas. The gain suppression is overcome and threshold is reached by using the long interaction length of the waveguide structure (i.e. their large enhancement factors [7,8]). In a recent article, Nazarkin theoretically describes just such an approach based on HC PCF and predicts that single pass anti-Stokes conversion efficiencies could reach the theoretical maximum of 50% [9]. Figure 1(a) shows a simulation based on the three mode theory [5,9], which treats the pump, the Stokes and the anti-Stokes fields and considers both SRS and FWM. By using suitable interaction lengths, high anti-Stokes efficiencies appear to be feasible.
Fig. 1 (a) A plot of the normalized anti-Stokes power versus the phase-gain mismatch factor, Δk / g 0 = (2k p-k a-k s) / g 0 (where g 0 is the Raman gain coefficient times the peak power, which was assumed to be 10 kW), and L is the interaction length after Roos [5] and Nazarkin [9]. High efficiencies are predicted but require long interaction lengths. (b-d) Raman simulations with varying levels of complexity. (b) Raman conversion with FWM simulation including only pump (green), Stokes (red), and anti-Stokes (blue) fields. The effect of phase matching (SRS gain g 0 = 10, Δk / g 0 = 0.5 solid lines, Δk / g 0 =0.05 dashed lines) can be seen by the increased threshold and conversion efficiency to anti-Stokes. (c) Simulation including cascaded SRS to the 2nd Stokes field (black). (d) This simulation includes full FWM interaction between all four fields and shows rapid conversion into the 2nd Stokes and pump fields.
Based on the work presented in this paper, we have found that the analysis performed by Nazarkin [9], and the earlier theoretical and experimental work by Roos [5], does not include the effects of conversion to second (2nd) Stokes frequencies via both SRS and FWM (see Eq. (1), which are critical when the 2nd Stokes field is allowed to build. If not properly controlled, these physical effects will severely reduce the efficiency of anti-Stokes conversion. In addition, our work has shown that the net gain in higher order spatial modes must also be suppressed when the FWM phase matching condition is obtained in the fundamental mode. If not, Stokes conversion in higher order spatial modes will be preferred and can ultimately keep the phase matched fundamental spatial mode from reaching Stokes conversion threshold.
These results have broad implications for those considering the use of efficient anti-Stokes conversion for parametric generation of UV sources [3], for efficient three-color visible laser sources [6], or for high energy conversion of pulsed visible laser sources [7]. The effective control of these physical processes while maintaining ideal phase matching conditions across the pump, Stokes, and anti-Stokes will be a significant challenge.
2. Background and theory
Raman processes including SRS and Raman resonant FWM have been well studied using numerical and analytic methods. However, few studies have included multiple effects including cascaded Raman scattering (conversion to multiple Stokes and anti-Stokes lines), FWM, and possible competition between spatial modes [11]. In particular, studies focusing on efficient anti-Stokes generation through phase-matched FWM have ignored the possibility of cascaded Raman processes to higher order modes [5,6,9,10].
In order to more fully understand and predict the Raman conversion process, a 1D numerical model has been developed that includes four fields; the pump, Stokes, anti-Stokes, and the 2nd Stokes fields, with all Raman resonant and non-resonant FWM terms [12]. The complex wave equations for the four fields follow the form used by Bobbs and Warner [12].
where a, p, s, s2 refer to the anti-Stokes, pump, first Stokes, and second Stokes fields respectively, the k’s refer to the wavenumbers in the gas-filled capillary, and the χ’s are the third-order non-linear susceptibilities tensors, which collapse to complex scalars for the single polarization case studied here. As described in [12] the various terms (i.e. SRS and FWM) and the wavenumber-dependent phase factors (i.e. the ε's) can be readily identified as driving terms in the classical derivation of the Raman resonant process. In addition to the three field model phase mismatch factor, Δk =2k p-k a-k s, two more phase factors are added in this four field model to account for phase matching between the various fields. For simple dispersive propagation, the phase mismatch associated with ε2 is about twice that of ε and ε1.These equations are solved by numerical integration using Runge-Kutta methods (energy conservation was monitored [11]). As the above equations involve stimulated processes, a small amount of Stokes field is included in the initial conditions along with the strong pump field to seed the SRS and FWM processes. Several solution runs are shown in Fig. 1(b-d). For each plot two runs were made: the first is for a larger phase mismatch (solid) and the second is for a smaller phase mismatch (dashed). The succeeding panels progress from the well known three mode case described in [12] to the full four mode case of Eq. (1). When the 2nd Stokes field is not included in the model one can see that the effect of better phase matching on the final output fields (dashed lines) is to increase the threshold length due to gain suppression and increase the final anti-Stokes output power as shown in Fig. 1(b) [13]. The 2nd Stokes field can be included into the model in two steps. First, the 2nd Stokes is generated only through SRS from the Stokes field as shown in Fig. 1(c). In this case, the Stokes and anti-Stokes generation proceeds similarly, however, after the pump has been depleted, the 2nd Stokes field may reach threshold and the Stokes field is then depleted through SRS. In addition, the anti-Stokes mode may be depleted through SRS back to the pump mode if the anti-Stokes to pump SRS threshold is surpassed. In this case, significant conversion to the anti-Stokes mode can occur for certain propagation lengths and input powers. In the second modeling step, the 2nd Stokes field interacts through both SRS and FWM terms as shown in Fig. 1(d). When both effects are included the behavior changes dramatically. Once the Raman process reaches threshold, the Stokes and anti-Stokes fields are rapidly converted into 2nd Stokes and pump for both the phase matched (dashed) or poorly phase matched case (solid). Our physical interpretation of this process is that FWM seeds the 2nd Stokes field subsequently allowing efficient conversion through SRS.
3. Experimental results and analysis
In order to explore the Raman interactions of laser pulses with a gas confined in a waveguide, a doubled Nd:YAG 1064 nm pulse laser was constructed and utilized. 532 nm laser pulses with arbitrary pulse shape and duration (from 10 - 1000 ns) could be created with high pulse energies of up to 70 mJ. Figure 2(a) is a schematic of the experimental Raman testbed, including a simplified sketch of the custom laser system. Glass capillaries (3-mm outer diameter, 100-µm hollow-core diameter, 30-cm length) were chosen as the waveguides for two reasons. First, we wanted to explore the conversion of high-energy pulses (~50 mJ) that were far above the damage thresholds of commercially available HC PCF (~100 μJ) due to their small core sizes. Second, the anomalous dispersion of the HC PCF is too large (2 to 3 orders of magnitude larger than ~100 µm capillaries) to adequately control using reasonable pressures of the desired Raman active gases.
Fig. 2 (a) A simplified schematic of the experimental setup. The capillary was strapped to an aluminum block with a V-groove, (b) shows perspective view and (c) an end on view, and placed entirely inside a windowed gas cell (not shown). Cameras were used to look into the cell to image the input (d) and output (e) faces of the capillary to aid in coupling the light into the capillary.
The linearly polarized 532 nm pulses were coupled into the waveguide with the aid of cameras imaging both the front and back facets; see Fig. 2(d,e). The capillary was strapped to a V-groove in an aluminum block to force linearity minimizing bend losses in the waveguide, which was then placed in a windowed gas cell (not shown) filled with a Raman active gas. The two gases used were the traditional high gain H2 and CO2, which was chosen for its smaller Raman shift. The lowest loss guiding mode of the capillary is the zero order Bessel function [14]. Theoretically, coupling efficiencies of up to 98% can be expected with a Gaussian beam and the theoretical transmission is 73% for the fundamental mode of a capillary. The output was comprised of the Raman-shifted pulses and residual 532 nm pump light, which were then dispersed through an equilateral prism onto several calibrated photodetectors for examination. Figure 3(a) shows the temporal profiles for Raman converted pulses from the first vibrational frequency of CO2 at 80 psi (ω mol = 1388 cm−1). The 532 nm input pump (yellow) has a FWHM of 32 ns and was captured before the capillary. The remaining 532 nm light after passing through the waveguide (green) shows a substantial central reduction where the Raman conversion takes place. The Stokes pulse (red) at 574 nm, the 2nd Stokes pulse (black) at 624 nm, and the anti-Stokes pulse (blue) at 495 nm were observed. The Stokes pulse also shows a substantial central reduction in power that is well correlated with 2nd Stokes generation. At this pressure and peak power level, Δk / g 0 ≈0.1, which should have provided anti-Stokes efficiencies well in excess of 10% based on the three mode theory above as shown in Fig. 1(a). We measured an efficiency of 0.3%, a result that can be attributed to the more complex dynamics involved with conversion to the 2nd Stokes.
Fig. 3 The temporal profiles of Raman converted pulses for 80 psi of CO2. (a) An experimental plot showing the various Raman converted pulses with an input pulse peak power of 1.2 MW. (b) Various experimental traces in ascending input peak power levels showing the threshold for Stokes conversion, the subsequent threshold for 2nd Stokes conversion, and evidence of conversion to 3rd Stokes. (c) Numerical simulation of conditions in (a) using Eq. (1). Note the scale change on the pump as compared to (a).
The evolution of these pulses as a function of increasing pump power is shown in Fig. 3(b). As the traces ascend, the threshold for Stokes conversion is quickly followed by the threshold for 2nd Stokes conversion. At even higher powers, a central reduction in the 2nd Stokes pulse is observed, suggesting that the threshold for 3rd Stokes had been reached (this wavelength was not monitored in these experiments). This unimpeded conversion to higher Stokes orders must be properly mitigated if efficient anti-Stokes conversion is to be achieved.
To better understand our experimental Raman conversion results in the capillary, the theoretical model was expanded and specialized to the experiment by: 1) calculating the phase matching parameters, ε, ε 1, ε 2 from the pressure dependent dispersion of the Raman gas and the spatial mode-dependent waveguide propagation factors, 2) adjusting the nonlinear Raman coefficients for pressure, 3) adding spontaneous Raman scattering of the pump field into the forward propagating Stokes mode, and 4) incoherent time-domain behavior including the cumulative effects of molecular excitations and weak optical feedback. The theoretical background and rationale for these extensions will be discussed in a future publication. The result is shown in Fig. 3(c). Quantitatively, the predictions of the thresholds based on the output pulse shapes agree to within about a factor of two to the experimental results and provides confidence in the model’s predictions.
Another important parameter that governs the behavior of the Raman conversion process in capillaries is the spatial mode the Raman modes occupy. The fundamental EH11 spatial mode has the lowest propagation losses and couples well to a free-space Gaussian beam, making it the preferred mode for the Raman pump and for the Stokes radiation. It also has the lowest dispersion, allowing the mode to be easily phase matched with low-pressure gases. This can be seen in Fig. 4(a) , where the phase-gain mismatch factor is plotted as a function of H2 gas pressure. However, other higher order spatial modes, such as the TE01+EH21 spatial mode, have larger dispersion and subsequently have a larger value for their phase-gain mismatch. This can lead to spatial mode competition as the fundamental mode undergoes significant gain suppression, while other higher-order modes do not. This effect was observed in a separate experiment with H2 gas in the capillary. Here the spatial mode of the output Stokes field from the first vibrational mode (ω mol = 4155 cm−1) was monitored as a function of H2 gas pressure. At lower pressures (38 psi) a higher-order donut mode was observed as shown in the middle of Fig. 4. This donut mode transitioned to the fundamental EH11 mode as the pressure was further increased (near 80 psi).
Fig. 4 (a) The phase-gain mismatch factor as a function of pressure for H2 assuming a 10 kW peak power. Experimental spatial mode profiles are shown for 38 and 82 psi showing that higher order spatial modes can have lower thresholds than the fundamental as it becomes phase matched. (b) The conversion efficiencies for the various Raman modes using the four mode treatment of Eq. (1). A very narrow and therefore gain sensitive region of efficient anti-Stokes conversion is seen (solid blue) as compared to the three mode treatment (dashed blue).
4. Conclusions
Due to the possibility of converting to the 2nd Stokes mode, efficient anti-Stokes conversion becomes more difficult than just obtaining good phase matching. Shen indicated this difficulty pointing out that significant intensity in two or more Raman modes causes all Raman modes, ω= ω p ± mω mol, to become strongly coupled [13]. This allows for rapid redistribution of the optical power into other modes. While there are propagation lengths with significant conversion to Stokes and anti-Stokes, these regions are narrow and very sensitive to the threshold conditions. Simulations using the four-mode theory support this conclusion as shown in Fig. 4(b). Therefore, although phase matching may allow significant generation of anti-Stokes radiation, this carries with it the seeds of its own destruction. In addition to good phase matching, conversion to the 2nd Stokes and higher order Stokes modes must be suppressed to ensure efficient conversion to the anti-Stokes wavelength. The ability to suppress 2nd Stokes in high finesse cavities by tailoring the mirrors could be a distinct advantage for generation of anti-Stokes. In situations where significant conversion to 2nd Stokes occurs, the inclusion of these higher order Stokes modes will need to be included for realistic modeling. In addition to inclusion of higher order Stokes processes, careful analysis of the propagating spatial modes of the waveguide structure must be considered to account for possible spatial mode competition, when efficient anti-Stokes is desired.
Considering these two major effects, we can now place four stringent constraints on the waveguide needed to create efficient single-pass anti-Stokes conversion. First, the waveguide must have good transmission properties at the pump, Stokes and anti-Stokes wavelengths. Second, it must strongly suppress conversion to the 2nd-Stokes and other higher order-Stokes wavelengths through high losses and large phase mismatch. Third, the waveguide must have fairly low values of anomalous dispersion across the pump, Stokes and anti-Stokes wavelength band so that reasonable pressures of a Raman or buffer gas can lead to suitable phase matching. Finally, the gain in higher-order spatial modes must be strongly suppressed especially as the fundamental spatial mode becomes phase matched. Currently, this combination of properties is not commercially available either in HC PCF or glass capillaries.
Acknowledgements
We gratefully acknowledge support of the U.S. Navy through the DEPSCoR program under grant number N000140810753. Special thanks to Prof. John Carlsten for discussions on Raman conversion and the generous use of equipment. We would also like to acknowledge Prof. Fetah Benabid and Prof. Michael Raymer for their time and discussions.
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