1. Introduction

There has been widespread interest in femtosecond coherent anti-Stokes Raman scattering (CARS) spectroscopy due to the increased data rates [1–10], increased signal strength enabling line or even planar measurements [11–14], increased signal-to-noise ratios due to the shot-to-shot spectral stability of the ultrafast pulses, and the capability of performing collision-free measurements. Extension of the measurements to high-pressure gas-phase media is of great interest for measurements in practical combustion systems. Some recent fs CARS experiments in high-pressure gases have been reported [1,5,15–20], and in these studies the authors have claimed that the effects of molecular collisions were negligible within moderately high pressure range and within short probe delays [1,5,15–17].

We have performed chirped-probe-pulse (CPP) fs CARS experiments in a high-pressure cell to investigate the physics of the process under well-controlled conditions. In our initial room-temperature measurements of the CPP fs CARS spectrum of nitrogen (N₂), we observed drastic changes in the spectrum as the pressure was increased from 1 to 10 bars. Given that the characteristic collisional time for N₂ at room temperature is approximately 10 ps at 10 bar pressure [1,15], and the pulse length for our chirped probe pulse was approximately 2-3 ps, this drastic change in the spectrum was quite surprising. Upon measuring the spectra of the pump, Stokes, and probe beams before and after the gas cell, it became apparent that the spectra of the pump and Stokes beams were significantly altered by self-phase modulation (SPM) as they propagated through the cell.

In this paper, we describe the results of a study of SPM and its effects on CPP fs CARS spectra in a room-temperature gas cell with pressures between 1 and 10 bar. We investigated SPM effects due to the propagation of the pump and Stokes beams through a gas cell filled with N₂, CO₂, O₂, or CH₄. We discuss the physics of the SPM process and the effects of SPM on CPP fs CARS spectroscopy, as well as the implications of SPM for fs CARS measurements in high-pressure media.

Self-phase-modulation (SPM) is a nonlinear Kerr-like phenomenon in which an ultrashort pulse accumulates temporal phase as it propagates through a gas medium [21–23]. This nonlinear phase accumulation depends on the intensity profile of the laser pulse and the nonlinear refractive index of the gas. In the study of Nibbering et al. [24], the nonlinear refractive index was measured for N₂, O₂ and Ar at pressures up to 1.6 bar by recording the power spectrum of the transmitted pulses. Nibbering et al. [24] used a well-collimated, unfocused 120-fs, 800-nm beam directed through a 2.5-m long gas cell to measure self-modulation effects. At the exit of the cell, they selected the central 1-mm-diameter region of the beam for spectral analysis. In more recent studies by Langevin et al. [25] and by Wahlstrand et al. [26], the nonlinear refractive index of N₂ was measured for several different pulse durations because variations in the pulse duration can affect the accuracy of retrieving the nonlinear refractive index.

The theoretical framework for the self-phase modulation effect was described by the following equations:

The pure rotational Raman contribution to the nonlinear refractive index is given by [25,28]

 

Fig. 1. Numerical simulation of (a) the laser intensity profile in the time domain (b) the electronic and rotational Raman contributions to the accumulated nonlinear phase, (c) and the electronic and rotational Raman contributions to the instantaneous frequency shift. In this simulation, we assumed the original temporal pulse profile to be Gaussian with a full width half maximum (FWHM) pulse duration of 55 fs. The averaged laser intensity was set to 2.0 TW/cm² and the optical path length was set to about 0.6 m. The magnitude of the rotational and electronic contributions of nonlinear refractive index were extracted from values quoted in the literature [25,26], i.e., ${n_{2,elec}} = 0.74 \times {10^{ - 23}}{m^2}/W$ and ${n_{2,rot}} = 1.1 \times {10^{ - 23}}{m^2}/W$. The values for the characteristic frequency ${\omega _R}$=16 THz, and the dephasing rate, $\Gamma $=7 THz, are also extracted from the literature [25].

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Fig. 2. Simulated power spectrum for the laser pulses by considering SPM. The comparisons were made among unmodulated power spectrum (black solid), modulated power spectrum as a result of electronic nonlinearity (blue dash) and the modulated power spectrum as a result of pure rotational nonlinearity (red point-dash). The simulation parameters are the same as for the calculations shown in Fig. 1.

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Notice also that in Eqs. (3) and (4) the nonlinear phases are written in their differential forms. The overall nonlinear phase accumulation requires spatial integration over the optical path inside the gas cell. For a collimated beam, this spatial integration will be straightforward as only the laser electric field characterization at $z = 0$ is needed (i.e., $I({t,0} )$). For the simulation results shown in Figs. 1 and 2, a collimated laser beam was assumed.

For our CPP fs CARS experiments, however, the laser beams were focused at the probe volume in order to generate the higher signals and to achieve the desired high spatial resolution for the measurments. Consequently, a more complicated spatial integration is required to take into account the laser intensity profile variation along the direction of propagation $\left( {{\phi_{NL}}({t,z} )\propto \int_0^z {I({t,z^{\prime}} )dz^{\prime}} } \right)$. We assume in this analysis that the laser beams are ideal Gaussian beams. When the frequency dispersion inside the gas medium is negligible (as it is in our case), the time and the spatial laser intensity characterization can be separated [30],

While most of the studies reported the measurements of nonlinear refractive index near atmospheric pressure [24–26,28], the pressure dependence of the nonlinear refractive index was measured by Börzsönyi et al. [31] for pressures up to 1 bar. It was found that, for these pressures, the nonlinear refractive index was approximately proportional to pressure for several inert gases such as N₂ and Ar. Consequently, we will assume in our numerical analysis that the nonlinear refractive index coefficients are proportional to pressure.

The calculations of the accumulated nonlinear phase in the direction of the laser propagation are shown in Fig. 3. We conclude that while at atmospheric pressure, the nonlinear phase accumulation due to glass windows is not negligible (though it is still small as compared to the accumulated nonlinear phase in N₂ gas medium), for high pressure conditions where SPM becomes significant, almost all of the accumulated nonlinear phase is due to the N₂ gas medium.

 

Fig. 3. Accumulated nonlinear phase as the pulse propagates through the gas cell, the simulation was evaluated at 10 bar and 1 bar pressure conditions. The top schematic drawing denotes the focusing geometry of the laser pulse. For the plot of nonlinear phase, the red line at both ends of the laser propagation is because of the nonlinear phase accumulated on the cell windows, and the nonlinear refractive index information for fused silica windows was taken from literature [16].

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2. Experimental system

A titanium:sapphire (Ti:S) laser source generated 55-fs (center wavelength at 798 nm; full-width half-maximum bandwidth = 270 cm⁻¹) pulses at 5 kHz with a pulse energy of 2 mJ (Legend Elite Duo, Coherent Inc). Most of the of the amplifier output, 80%, was directed into the optical parametric amplifier (OPA) to generate the 675 nm pump pulse. The rest of the 798-nm amplifier output was split further in a 20%-80% ratio to serve as the Stokes and probe beams, respectivley. The probe beam was chirped extensively from less about 55 fs to about 3 ps to map the Raman coherences into the frequency domain for single shot measurements. The maximum pulse energies for the pump, Stokes, and probe beams, attainable at the probe volume, were approximate 80 µJ, 80 µJ, and 320 µJ respectively

The pump and Stokes pulses were focused at the probe volume through a 500 mm plano-convex lens, the beam diameter on the focusing lens was about 11 mm. A profiling camera was used to characterize the laser beam profile near the probe volume (DCC1545M, Thorlabs). A half-wave plate and a linear polarizer were used to control the laser energy for pump and Stokes pulses.

A folded BOXCARS geometry [32] was used to spatially separate the CARS signal from the input beams. The CARS signal was directed into a 0.5 m, Czerny-Turner-type spectrometer and detected using an electron-multiplying charge-coupled device (EMCCD) camera system which was synchronized with the laser output.

A high-pressure test cell was designed and built to provide controlled conditions up to 25 bar pressure and 1000 K temperature. The room-temperature experiments described in this paper were conducted at a maximum pressure of 10 bar because the effects of SPM were already very significant. Two pieces of anti-reflection-coated 13-mm-inch thick fused silica windows were placed at the ends of the gas cell to allow for optical access. The path length inside the cell between the inner window surfaces is 660 mm.

3. Results and discussions

SPM in pure N₂ was evaluated by measuring the optical spectra of the 800 nm Stokes pulse before and after transmission through the high-pressure cell with different cell pressures. The optical spectra for two laser energy settings are displayed for comparison in Fig. 4. For the low laser energy case (10 µJ per pulse), the optical spectra measured after the cell were relatively unchanged throughout the pressure range from 1 to 10 bar. For the high laser energy case (60 µJ per pulse), the spectral components of the 800-nm pulse tended to shift toward lower frequencies (red shift). At higher pressures and the spectra appear to be much more asymmetric due to significant SPM effects. We note that the presence of a second beam did not influence SPM induced in the first beam (for example, the crossing of the pump beam didn’t affect the SPM pattern of the Stokes beam), which indicates the cross phase modulation is not a major concern in the context of this work.

 

Fig. 4. (a) Optical spectra of the 800-nm Stokes pulse after the high pressure cell, recorded at various pressures with a) 60 µJ per pulse and (b) 10 µJ per pulse. The reference spectra (dashed green line) were measured before the high-pressure cell.

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Similarly, we measured the self-phase-modulated spectra for the 800-nm beam after transmission through pure CO₂, O₂ and CH₄. This is of practical interest because for many high-pressure combustion applications, these species may be present in high concentrations, especially for oxy-fuel flames. In terms of SPM effects, O₂ has both higher rotational and electronic contributions the nonlinear refractive index than N₂ [24], and the stronger SPM effects are obvious in Fig. 5. Carbon dioxide (CO₂) has a much stronger rotational Raman cross section than either N₂ and O₂ [33,34], and exhibits an especially strong red shift in Fig. 5. Because CH₄ does not have any rotational Raman transitions, the SPM is solely due to the electronic contribution of the nonlinear refractive index. As shown in Fig. 5, the self-modulated optical spectra of CH₄ exhibit homogeneous broadening with no obvious red shift. However, the shape of the spectrum does appear to change in a complex fashion as SPM effects become stronger

 

Fig. 5. Self-modulated optical spectra in different gaseous species. The Stokes optical spectra was measured after the high pressure cell, laser energy was 40 µJ per pulse. Notice the large and negligible red-shift in the CO₂ and CH₄ spectra, respectively.

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One way to quantify SPM effects is to calculate the least squares difference between the modulated spectrum $S(\omega )$ of the 800-nm beam and a spectrum ${S^0}(\omega )$ recorded at low enough pressure and laser intensities that SPM was negligible. The parameter $\Delta SPM$ is given by

 

Fig. 6. Evaluation of effect of SPM extent on the modulated optical spectra of different gaseous species.

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The time domain electric field function $E({t,0} )$ can be obtained from the inverse Fourier transform of $\sqrt {S({\omega ,0} )}$ if the spectral phase is specified. According to Eq. (1), we can even simulate the post-gas-cell optical spectrum $S({\omega ,L} )$ if the nonlinear phase ${\phi _{NL}}({t,L} )$ is known. We can also determine the SPM-induced nonlinear phase ${\phi _{NL}}({t,L} )$ from theoretical fits to the experimentally measured spectra $S({\omega ,L} )$ and $S({\omega ,0} )$.

In practice, we allowed ${n_{2,elec}}$, ${n_{2,rot}}$ and the quadratic spectral phase to be varied to obtain the best theoretical fit to the SPM spectrum $S({\omega ,L} )$ based on the unmodulated power spectrum $S({\omega ,0} )$. The linear and cubic spectral phase were not included since the former only corresponds to a temporal shift and the latter is negligible for pump and Stokes beams. A summary of the fitting results is as shown in Fig. 7. The fitted power spectra match satisfactorily with the experimental data for both the pump and Stokes pulses. The magnitude of ${n_{2,elec}}$ and ${n_{2,rot}}$ obtained from the Stokes pulse spectral fitting at different pressures are as shown in Fig. 8. For the electronic contribution of nonlinear refractive index, the fitted value evaluated at room temperature, 10 bar pressure was $4.1 \times {10^{ - 23}}\,\,{{\,{m^2}} \mathord{\left/ {\vphantom {{\,{m^2}} W}} \right.} W}$. If we extrapolate the values reported by Wahlstrand et al. [26] and by Nibbering et al. [24] assuming a linear correlation between the nonlinear refractive index and the pressure, our fitted values are low by a factor of 1.8 times as compared to the value reported by Wahlstrand. and low by a factor of 5.6 as compared to Nibbering’s findings. For the pure rotational contribution of the nonlinear refractive index our fitted value is $1.28 \times {10^{ - 22}}\,\,{{{m^2}} \mathord{\left/ {\vphantom {{{m^2}} W}} \right.} W}$, which is low by a factor or 2 compared to the value reported by Sprangle et al. [28]. However, it is very close to the value as reported by Langevin et al. [25]. Despite the fact that the literature reported nonlinear refractive index varies among the different studies [24–26,28], the magnitude of the nonlinear refractive index extracted in this study agrees to better than an order of magnitude with all these reported.

 

Fig. 7. Fitted optical power spectra from the SPM simulation compared to the experimental spectra. (a) For pump pulse with 60 µJ. (b) The calculated electronic and pure rotational nonlinear phases for the pump beam. (c) For Stokes pulse with 60 µJ. (d) The calculated electronic and pure rotational nonlinear phases for the Stokes beam.

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Fig. 8. (a) Fitted nonlinear coefficients for the electronic contributions to the nonlinear refractive index of N₂ (b) Pure rotational contribution of the nonlinear refractive index. (c) The 800 nm Stokes beam profile at the focal plane. Stokes pulse energy was 60 µJ.

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Our reported values for the nonlinear refractive index cofficients are lower than the reported values. This systematic discrepancy is attributed to our assumption that the laser beams are ideal Gaussian beams and that they focus like ideal Gaussian beams. We measured the beam profile at the focal plane using a profiling camera (DCC1545M, Thorlabs) with a pixel size of 5.2 µm. The focus of the 798-nm beam is shown in Fig. 8(c). We calculated the laser beam area where the laser intensity drops to about ${e^{ - 2}}$ of the peak intensity. The effective beam diameter ($2{w_0}$) of the laser spot area is about 120 µm, which is approximately two times greater than the calculated ideal Gaussian beam focal diameter of 50 µm. Correspondingly, our assumption of an ideal Gaussian beam focus will result an overestimation of the laser intensity at the focus by a factor of approximately four. Consequently the nonlinear refractive indices extracted from our analysis are lower compared to reported values from studies where collimated beam were used [24,26].

CPP fs CARS signals for gaseous N₂ at different pressures were obtained using both high and low laser energy settings. As shown in Fig. 9, for the high laser energy setting (60 µJ per pulse), the CARS spectral variation is much more evident than for the low energy case (10 µJ per pulse). Spectral distortion of the signal, is evident, especially between 14600 cm⁻¹ to 14800 cm⁻¹, at 10 bar pressure. This spectral distortion as can be correlated with the SPM effects in N₂ shown in Fig. 4. One can also observe slight spectral modulation of the CARS signal for the 10 µJ per pulse energy case. This is attributed to the wavelength dependence of the linear refractive index of N₂. According to Lorenz-Lorenz equation, the linear refractive index difference of N₂ at 675 nm and 800 nm can be approximated to scale linearly from 1 to 10 bar [35](higher refractive index at 675 nm). The optical path length of the pump and Stokes pulses inside the high pressure cell (from the first cell windows to the probe volume) is about 330 mm, which leads to Stokes pulse being delayed by as much as 10 fs, relative to the pump beam, as the pressure was increased from 1 to 10 bar.

 

Fig. 9. (a) Experimental CPP fs CARS spectra with high laser energy setting (pump and Stokes both with 60 µJ per pulse energy), the spatially averaged peak intensity near the probe volume is about 5 × 10¹⁷ W/m² (b) Low laser energy setting (10 µJ per pulse energy), peak intensity near probe volume is about 9 × 10¹⁶ W/m². The probe pulse delay is about 1 ps.

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Using the optical spectra of the pump and Stokes pulses recorded at 1 bar, we can fit the CPP fs CARS spectrum. A comparison of the fitted CARS spectrum and experimental data taken at 10 bar at lower energy is shown in Fig. 10. The embedded figure shows the change of the fitted Stokes time delay for pressure increasing from 1 to 10 bar. In general, the theoretical spectrum is in good agreement with the experimental spectrum. The trend of the linear delay of Stokes beam, with increasing pressures, matches with the fitting code prediction. However, the fitting code exhibits a best fit of value for the Stokes delay of 20 fs from 1 to 10 bar rather than the 10 fs calculated from the dispersion of the refractive index. This may indicate slight residual SPM effects even for low laser energies.

 

Fig. 10. Fitted CPP fs CARS spectra at 10 bar condition as compared to the experimental data when 10 µJ per pulse energy was used for both pump and Stokes pulses. The fitting residual was displayed below the CPP fs CARS spectra. Embed: Fitted value of the Stokes delay at various pressures, relative Stokes delay reduces from about 25 fs to about 5 fs from 1 to 10 bar pressure. In the spectral fitting routine, all the laser parameters were kept identical except for the Stokes delay, resonant scaling factor and the relative phase angle were allowed to vary.

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Efforts were also made to fit the CPP fs CARS signal for the high laser energy setting by using the pump and Stokes spectra recorded either before or after the high pressure cell. As shown in Figs. 11(a) and 11(b), the fitting accuracy is much worse as compared to low energy case. This is because when SPM is significant, neither the pump/Stokes spectrum recorded before nor after the test cell gives a satisfactory approximation for the pump/Stokes spectrum at the probe volume. For the pure N₂ measurements, better fitting accuracy was obtained by simulating the SPM inside the high pressure cell as discussed in previous section and as shown in Fig. 7. For example, once the overall nonlinear phase ${\phi _{NL}}({t,z} )$ was determined, we assume that ${\phi _{NL}}({t,z/2} )\approx {\phi _{NL}}({t,z} )/2$ at the probe volume. This is a reasonable assumption when we consider the symmetric focusing geometry. The corresponding pump and Stokes spectra at the focus can then be calculated using Eq. (1). The CARS fitting result obtained from these “interpolated” power spectra is shown in Fig. 11(c). The fitting accuracy is obviously better than the CARS fitting as shown in Figs. 11(a) and 11(b).

 

Fig. 11. With 60 µJ per pulse energy, fitted CPP fs CARS spectra are compared to the experimental measurements at 10 bar. (a) Using the pump and Stokes spectra recorded before the gas cell. (b) Using the pump and Stokes spectra recorded after the gas cell. (c) Using the power spectra for the pump and Stokes pulses retrieved from the interpolated nonlinear phase. The laser parameters were the same for all three cases.

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The complex behavior and the species dependent features of SPM indicate that modeling and accounting for SPM effects in practical combustion systems may be very difficult. Thus suppression of SPM by limiting laser pulse energies is desirable, especially, when multiple gaseous species are expected at the probe volume. SPM effects will increase in severity as the density of the gas medium increases. Fortunately, the signal for fs CARS will increase approximately as the square of the gas density, so in many cases it is likely that the beam energies can be reduced to eliminate SPM effects while still maintaining acceptable fs CARS signal levels.

4. Summary

In this paper, we investigated the effects of SPM, at high pressures and with varying laser energies, on the power spectrum of pump and Stokes pulses employed in CPP-fs-CARS measurements. We found that with sufficiently high laser energy, the SPM can become significant and result in modulation of N₂ CPP fs CARS spectra even at moderate pressures. We also measured the SPM effects for several different gaseous species including CO₂, O₂ and CH₄. It was found that N₂ exhibits the lowest level of SPM while CO₂ has strongest SPM among those molecules being studied. The nonlinear refractive coefficients that we extract from our modeling of the modulated pump and Stokes spectra are in rough agreement with literature values despite the uncertainties associated with our assumption of an ideal Gaussian beam focusing for these beams. Thus, our results can be used to estimate SPM effects for other experiments where such as hybrid fs/ps CARS and other spectroscopic studies involving focused ultrafast beams in high pressure gas-phase media. SPM effects on the fs pump and Stokes pulses can significantly affect the Raman excitation profile, and this will affect the temperature accuracy of both CPP fs CARS and hybrid fs/ps CARS.

Also, we conclude that it is preferable to eliminate SPM effects by reducing the laser intensity rather than to try to incorporate these effects into the CARS modeling especially when several gaseous species are involved. Under most conditions, it appears that the beam energies can be reduced to eliminate SPM effects while still maintaining acceptable CPP fs CARS signal levels, and this will be explored in more detail in future experiments. So far, our measurements have been conducted at room temperature and at moderately high pressures (1-10 bar). In future work, we will extend our investigation of SPM to higher pressures and higher temperatures.