1. INTRODUCTION

The laser-induced grating (LIG) technique is a versatile non-intrusive diagnostic method based on the generation of a spatially periodic modulation of the refractive index in the region of the fringe pattern created by strong electric fields of two crossed pulsed laser beams [1,2]. The technique gives access to a multitude of thermodynamic, fluid-dynamic, and molecular properties, starting from temperature [3–9], species concentration [4,9–15], pressure [16–18], and flow velocity [19–24], over speed of sound, thermal diffusivity, and bulk viscosity [25–27], and ending by the collisional energy transfer rates in the gas phase [28–31]. In more details, the research of the previous decades was reviewed in [32].

LIGs are formed by the interference of the two incident pump laser radiations, which creates the regions of high and low electric field strength [2,33,34]. The fringe spacing of the interference pattern is defined by the laser wavelength and the beam crossing angle. Due to electrostriction, a density modulation is generated in the medium [33,34]. This process can be regarded as instantaneous due to short duration of the pump pulses (typically nanosecond or smaller). The density modulation develops as two counter-propagating sound wave packets and decays while these sound waves diverge. The sound waves create the respective modulation of the complex refractive index. This grating is called a laser-induced electrostrictive grating (LIEG), and it is formed in any medium and at any wavelength of the pump beams.

If a probe laser beam passes through the region of the refractive index modulation at an appropriate angle to the optical axis (Bragg angle), it is efficiently diffracted [2,35]. The temporal evolution of the power of the diffracted continuous wave probe radiation (a LIG signal) gives straightforward access to the adiabatic sound velocity through the period of the characteristic oscillations of the signal intensity.

The sound velocity is a function of gas composition, temperature, and pressure. Therefore, two of the three parameters need to be determined separately for determination of the third, and that limits the diagnostic applicability of the technique [34,36,37].

For many diagnostic applications, a strong thermal contribution to a LIG has been used [2,5,8,38]. This thermal contribution to the grating (a laser-induced thermal grating, LITG) appears if absorption of pump laser energy occurs when the laser frequency matches a molecular resonance [2]. In this case, the LITG contribution to the signal may be a lot more intense than the LIEG one under otherwise identical experimental conditions due to strong enhancement of the induced gas density variations resulting from efficient collisional relaxation of the absorbed energy [30,39]. Diagnostics using LITGs employs either an accidental overlap of the wavelength of the available pump laser with a molecular absorption resonance or the appropriate adjustment of the pump laser wavelength. Use of a tunable laser complicates the experimental setup. However, it allows for us to record the spectrum of LITG excitation efficiency by scanning the pump wavelength, which yields additional complementary information [15,29,30,40–42]. Nevertheless, the LIG signal evaluation for diagnostics is predominantly aimed at determination of the easily accessible sound velocity.

Employment of a non-tunable powerful Nd:YAG laser at 1064 nm for pumping a molecular gas medium, as the simplest version of the LIG experiment, may result in efficient generation of not only LIEGs, but also LITGs, since many simple organic molecules or ${{\rm H}_2}{\rm O}$ weakly absorb this radiation in overtone and/or combination bands. This absorption may be, however, sufficient to provide noticeable LITG contributions to the LIG signals. If even a small thermal contribution is present, two parameters can be derived from the same LIG measurement. The temporal profile of the signal changes depending on the amount of energy, introduced to the medium as a result of the light absorption and the subsequent release of this internal energy as heat, and on the rate of this energy release [33,34]. If a LITG contribution, either due to a small absorption cross-section or a small mixture fraction of an absorbing species in a mixture, is observable in a LIEG signal profile, the ratio of the contribution strengths can be exploited for simultaneous multiparameter diagnostics without fitting the profile [4,9,13,14,37,43]. This peculiarity has been used in various LIG diagnostic experiments aimed to determine gas mixture composition and temperature [37,42].

In this work, we study the possibility of simultaneously determining the composition and temperature of a binary gas mixture in the case where weak absorption does not lead to the generation of a LITG contribution to a LIG signal but at the same time is sufficient to provide a contribution due to four-wave mixing of pulsed pump radiation and continuous-wave probe radiation.

2. EXPERIMENT

The setup employed for the laser-induced grating experiments is schematically shown in Fig. 1. The whole LIG setup was designed to be very compact and mobile to be used, for instance, both in the lab and in cramped test facilities. Therefore, it was decided to refuse wavelength tuning by, e.g., an OPO, and only the fundamentals and harmonics of commercially available Nd:YAG lasers have been used. The pump beams are supplied by a flash lamp pumped Q-switched Nd:YAG laser operating at a repetition rate of 10 Hz. The output at 1064 nm (${9395.4}\;{{\rm cm}^{- 1}}$, with a FWHM bandwidth of ${1}\;{{\rm cm}^{- 1}}$) has an energy of 42 mJ/pulse and a pulse duration of 8.4 ns at 1/e intensity level. The laser beam is passed through a telescope for collimation and beam diameter control. A half-wave plate and a Glan–Taylor polarizer guarantee polarization control and purity. For the studies of power dependence, they also serve as a pump power attenuator. A photodiode detects light scattered by the half-wave plate and provides trigger pulses for the experiment. A 50% beam splitter divides the laser beam into the two pump beams, which are guided onto a lens (750 mm focal length). For the polarization-dependent experiments, a half-wave plate is used in one of the pump beams to change its polarization. The pump beams are crossed inside the gas cell, with their pulses temporally overlapped, at an opening angle of 2.1° providing a fringe spacing $\Lambda \approx {29}\;{\unicode{x00B5}{\rm m}}$.

 

Fig. 1. Schematic setup for the laser-induced grating (LIG) experiments. The beams are distinguished by color: red, 1064 nm (pulsed); green, 532 nm (CW); A, aperture; BF, bandpass filter; BS, beam splitter; GP, Glan–Taylor polarizer; $\lambda /{2}$, half-wave plates; L, lenses; ND, neutral density filters; OSC, oscilloscope; OF, optical fiber; PD, trigger photodiode; PMT, photomultiplier tube.

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The probe beam is provided by a frequency-doubled continuous-wave Nd:YAG diode-pumped solid-state (DPSS) laser with an adjustable power range between 0 and 500 mW. Unless otherwise stated, the power was set to 50 mW as a compromise between the signal and the straylight levels. The output beam is guided through a telescope and a half-wave plate. It is focused into the cell by the same lens as the pump beams. At the common focal point, it is crossed with the pump beams at an angle of about 0.53° to the beam optical axis to satisfy the Bragg condition. Proper beam alignment is assured by use of aperture masks with the correct beam positions in front of and behind the cell. For routine alignment without the need to remove the cell, a fraction of the focused beams can be reflected from a glass slide onto a CCD camera (not shown in Fig. 1), which had been positioned at the correct location for the optimal spatial overlap of the beam waists. The probe volume diameter is about 300–380 µm, depending on the individual alignment of the beams, while its length is estimated, using the relations from [42], to be about 3 mm. These dimensions of the probe volume characterize the spatial resolution of the technique.

After passing through the cell, the pump and probe beams are blocked. The signal beam at 532 nm is collimated by a lens and directed through a series of apertures for the straylight suppression. Subsequently it is guided through a bandpass filter (FWHM 5 nm at 532 nm) and focused into an optical fiber for additional spatial filtering. The fiber delivers the signal radiation to a highly sensitive and fast-response photomultiplier tube (PMT), and its output voltage is recorded using a 20 GS/s digital oscilloscope. The linear response of the PMT ranges from about 0 to 1 V. Single-shot signal intensities are kept well below this threshold by use of neutral density filters. Note that the optical arrangement is stable enough to provide reasonable reproducibility of single-shot signal oscillation periods (which characterize the fringe spacing) and intensities during several hours of the experiments without realignment of the setup.

The gas mixtures are prepared by evacuating the cell and filling in gas 1 up to a defined partial pressure and then adding up gas 2 up to the desired measurement pressure. The mixture is allowed to equilibrate for about a few minutes.

Typically, 500 single-shot LIG signals are recorded and averaged. The fringe spacing is determined from the profile of the signal in pure ${{\rm N}_2}$ at 1 bar and 295 K, where the sound velocity is well known. The value of $\Lambda$ is assumed to be constant for subsequent measurements. The pressure measurement accuracy is estimated at $\pm {5}\;{\rm mbar}$, resulting in the mixture composition known to have an accuracy of better than $\pm {1}\%$. The electrically heated cell temperature was calibrated using a thermocouple type K. The accuracy of the temperature measurement in the LIG probe volume is $\pm {5}\;{\rm K}$.

3. RESULTS AND DISCUSSION

A. Laser-Induced Electrostrictive Gratings Signals in ${{\rm CO}_2}$

Figure 2 shows the LIG signal temporal profile in pure ${{\rm CO}_2}$ and ${{\rm N}_2}$ (plotted downward for visibility) at 2 bar and 295 K at the all-vertical (VVVV) polarization configuration of the ${{\rm pump}_{1,2}}$, probe, and signal beams. As a function of the time delay $t$ after the pump laser pulse, the profile represents a damped oscillation during $\approx {800}\;{\rm ns}$, which is characteristic of a LIEG signal resulting from the probe beam diffraction by a decaying standing acoustic wave. The difference in the oscillation periods ${T_{\rm{osc}}}({{\rm CO}_2}) \approx {55}\;{\rm ns}$ and ${T_{\rm{osc}}}({{\rm N}_2}) \approx {45}\;{\rm ns}$ is due to the inverse proportionality of ${T_{\rm{osc}}}$ to the gas sound velocity ${\upsilon _s}$. The dependence of the sound velocity of an ideal binary gas mixture on isentropic exponent $\gamma$, component mole fraction $x$, gas pressure $p$, density $\rho$, temperature $T$, and molar mass $M$ is defined by Eq. (1),

 

Fig. 2. Normalized LIG signal temporal profiles (500 averaged single-shot signals) in pure ${{\rm CO}_2}$ and ${{\rm N}_2},$ each at 2 bar and 295 K at the all-vertical (VVVV) polarization configuration (see text). The ${{\rm N}_2}$ signal is plotted downward for a clearer view. The profiles have been fitted using Eq. (2) (green and blue lines, respectively).

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An analytical description of the temporal evolution of the electrostrictive LIG contribution to the signal ${S_{\rm{LIEG}}}(t)$, starting from $t\; \approx {15}\;{\rm ns}$, can be given by Eq. (2) based on the linearized hydrodynamic equations and the assumption of weak acoustic damping [33],

It describes the result of probe light diffraction by a damped standing acoustic wave with the acoustic frequency ${\Omega _a} = {2}\pi {\upsilon _s}/\Lambda$, corresponding to the signal oscillation period ${T_{\rm{osc}}} = \pi /{\Omega _a}$. The parameter ${M_e}$ is the amplitude of the refractive index spatial modulation due to electrostriction [30], and ${S_0}$ is the scaling factor. The signal decay is governed by the acoustic wave damping time ${\tau _a}$, defined by sound absorption, and the transit time ${\tau _{{tr}}}$, dependent on the probe beam waist and the sound velocity, due to the running of the space-limited acoustic waves away from the probe volume [30,33].

The temporal evolution of the LIG signals in pure ${{\rm CO}_2}$ and ${{\rm N}_2}$ in Fig. 2 has been fitted, starting from the delay $t = {15}\;{\rm ns}$, by using Eq. (2). The fitting curve, also represented in the figure, reasonably reproduces the experimental profile. However, in both signals, there appears a background contribution between 100 and 500 ns, which leads to the oscillations above the baseline and is not reproduced by the fit. Despite the best efforts in the beam alignment, this slowly varying background likely arises from incomplete interference between the induced counter-propagating acoustic wave packets and non-ideal overlap of the pump beams [43–45]. Furthermore, the probe beam focal diameter larger than that of the pump beam overlap region can lead to diffraction from the periphery of this region resulting in an additional non-oscillating contribution to the signal [43,45]. Despite this background, Eq. (2) can describe the signal temporal evolution and the acoustic frequency can be readily obtained as ${\Omega _a}/{2}\pi = ({9.27} \pm {0.004})\;{\rm MHz}$ in ${{\rm CO}_2}$ and ${\Omega _a}/{2}\pi = ({11.57} \pm {0.003})\;{\rm MHz}$ in ${{\rm N}_2}$. To fit the LIG signal profile by Eq. (2), the first guess for the acoustic frequency can be obtained from the Fourier transformation of the signal. The initial values of the fitting parameters are calculated based on the relations in [30] and molecular data from [46]. In ${{\rm CO}_2}$ and ${{\rm N}_2}$, the derived transit times of the sound waves out of the probe volume ${\tau _{{tr}}}({{\rm CO}_2}) = ({404} \pm {4})\;{\rm ns}$ and ${\tau _{{tr}}}({{\rm N}_2}) = ({353} \pm {2})\;{\rm ns}$ are much smaller than the respective ${\tau _a}({{\rm CO}_2}) = {3.24}\;{\unicode{x00B5} \rm s}$ and ${\tau _a}({{\rm N}_2}) = {3.20}\;{\unicode{x00B5} \rm s}$ as calculated using the tabulated data, and they can be directly related to the pump beam waist radius ${w_0} \approx {150}\;{\unicode{x00B5}{\rm m}}$, which well corresponds to an ideal focused Gaussian beam waist radius under the given experimental conditions.

The rise of the LIG signal in ${{\rm CO}_2}$ is preceded by a narrow peak, exhibiting its maximum at $t \approx {0}$, i.e., at the temporal overlap of the pump pulses. Due to its specificity to ${{\rm CO}_2}$, the narrow peak may be applied for ${{\rm CO}_2}$ diagnostics. To elucidate the nature of the peak and to explore its potential diagnostic applicability, further analysis is done based on the behavior of this signal contribution relative to the known behavior of that of LIGs while changing laser polarizations and powers, mixture pressure and composition, and gas temperature.

B. Two-Color Four-Wave Mixing Signals in ${{\rm CO}_2}$

Figure 3 shows in detail the initial part of the signals in ${{\rm CO}_2}$ at different pump and probe beam polarization configurations, with vertical (V) or horizontal (H) polarizations. The profiles with the VVVV and VVHH configurations were normalized by the respective fitted amplitude of the first LIG oscillation peak [see Eq. (3)]. The visible slight shift of the oscillation peaks in the VVVV and VVHH configurations is related to the small difference in the fringe spacings due to required beam realignment after changing the polarizations. Despite the realignment, the values of the normalizing factors differed by less than 5%. Thus, the signal at the HVVH configuration has been normalized by the mean factor for the VVVV and VVHH configurations.

 

Fig. 3. Details of the initial part of the normalized signals in ${{\rm CO}_2}$ at different polarization configurations of the two pump, probe, and signal beams: black circles, VVVV; red circles, VVHH; green circles, HVVH. The signals have been fitted using Eq. (3) (blue lines). Note that realignment led to a different fringe spacing and resultingly different acoustic frequency between the VVVV and VVHH configurations.

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The narrow peak clearly observed in the VVVV and VVHH polarization configurations of the ${{\rm pump}_{1,2}}$, probe, and signal beams is far more intense in ${{\rm CO}_2}$ than in ${{\rm N}_2}$, and it is absent in vacuum and with the pump beams blocked in front of the cell. Therefore, it is unlikely to be the straylight bypassing the bandpass filter. Because of the “instantaneous” nature of the peak, it is assumed to be the result of two-color four-wave mixing (TCFWM) of the pulsed pump and continuous-wave (CW) probe radiations, with a non-resonant contribution to the signal in ${{\rm N}_2}$ and both non-resonant and resonant ones in ${{\rm CO}_2}$. With the pump beam polarizations crossed (the HVVH configuration), the pump beam interference vanishes, and hence the LIG contribution to the signal disappears, whereas the narrow peak is still visible (see the green trace in Fig. 3). This further corroborates the assumption of observing the TCFWM contribution, which is expected to decrease in intensity at orthogonal pump beam polarizations but not vanish entirely.

A similar peak, with a subsequent LIG contribution, had been observed by Danehy et al. [39] in UV DFWM-LIG experiments in a low-pressure ${\rm NO}/{{\rm CO}_2}$ mixture (FWM contribution from NO molecules), by Buntine et al. [47] and Boose et al. [48] in visible range DFWM-LIG experiments in a low-pressure ${{\rm H}_2}{\rm O}$ vapor/air mixture (FWM signal in ${{\rm H}_2}{\rm O}$ gas), and by Sahlberg et al. [40] in a mid-IR LIG experiment in an ethylene/air flame, where it was attributed to an FWM contribution from ${{\rm C}_2}$-radicals, although no diagnostic application was discussed.

To quantify the presumably TCFWM narrow peak electric field amplitude, relative to that of the first oscillation of the LIG signal at the VVVV or VVHH configurations, a simplified model of the temporal evolution of the initial part of the signal, including the peak, has been proposed as represented by Eq. (3),

Here, the signal temporal profile $S(t)$ is described as the square modulus of the sum of the electric fields: a Gaussian function with ${1/}e$ full width $\tau$ (about the ${1/}e$–level pump laser pulse width) and a phase-shifted sine wave from Eq. (2), with the acoustic period ${T_a} = {2}\pi /{\Omega _a}$ and a phase shift $\varphi$. The ratio of the peak and the LIG-diffracted radiation field amplitudes is represented by the parameter $r$, and $A$ is the signal intensity scaling factor, equivalent to ${S_0} \cdot M_e^2$ in Eq. (2) in case of $r = {0}$. The expression is assumed to be applicable at $t \le {T_a}/{2}$.

Figure 3 shows the fits of Eq. (3) to the experimental data. Because of the quasi-instantaneous response of the FWM, the peak maximum has been used as a reference for $t = {0}\;{\rm ns}$. For both polarization configurations, VVVV and VVHH, the model can properly describe the signal profiles, with both the narrow peak and the first LIG signal oscillation. The value of $\tau$ is found to be ${\approx} {9.1}\;{\rm ns}$. The parameter $r$ as a measure of the peak and the LIG-diffracted field amplitude ratio is ${r_{{\rm VVVV}}} = {1.180} \pm {0.004}$ for the VVVV polarization configuration. When the polarization configuration is changed to VVHH, the amplitude ratio is only ${r_{{\rm VVHH}}} = {0.47} \pm {0.01}$.

The phase shift $\varphi$-dependent cross term defines the observed signal intensity in the dip between the peak and the oscillation. Since the fall time of the PMT, evaluated as 2.4 ns from the single peak shape at the HVVH configuration (see Fig. 3), is on the order of the half-width of the FWM peak, the measured signal profile in the dip may be distorted, which is not accounted for in Eq. (3). Indeed, the respective relatively small discrepancy between the experimental and the fitted profiles is noticeable, especially in the case of the signal at the VVVV configuration, when the intense narrow peak sharply decreases in magnitude. Therefore, the variation of $\varphi$ aimed to improve the fitting is unjustified, and its value can be set from physical considerations: equal to $\pi /{2}$ if the FWM contribution is considered to be resonant and equal to 0 if the contribution is non-resonant.

The incident beam polarization dependence of the four-wave mixing signal intensity is governed by that of the square modulus of the electric field ${E_{\rm{FWM}}}$ created by the induced non-linear polarization, which is a function of the incident electric fields, and the third-order non-linear susceptibility tensor ${\chi ^{(3)}}$. In particular, the FWM signal intensity ${S_{\rm{FWM}}}$ at parallel pump polarizations can be shown to vary with the angle ${\Theta _{\rm{probe}}}$ between the pump and the probe fields as expressed by Eq. (4), where $\chi _{{ijkl}}^{(3)}$ are the components of the tensor ${\chi ^{(3)}}$ [49–51],

The ratio of the non-resonant components of ${\chi ^{(3)}}$ for the two polarization configurations is $\chi _{1111}^{\rm{NR}}/\chi _{2112}^{\rm{NR}} \approx {3}$, while the ratio of the resonant components equals to $\chi _{1111}^R/\chi _{2112}^R \approx {2}$ for most linear molecules [49]. The ratio of $r$ for the two polarization configurations, ${r_{{\rm VVVV}}}/{r_{{\rm VVHH}}}$, is found from the experimental data to be about 2.5 in neat ${{\rm CO}_2}$. Therefore, assuming that the LIG-diffracted electric field amplitude is independent of the polarization configuration (VVVV or VVHH), the peak in ${{\rm CO}_2}$ can rather be attributed to a resonant signal. This seems to be contradictory to the virtual absence of a resonant, thermal contribution to the LIG signal in Fig. 2, which is reasonably described by Eq. (2) as a purely electrostrictive one. A resonant FWM process of two pump waves and one probe radiation wave, which does not lead to the formation of an observable thermal grating, could either hint at the involvement of small absorption cross-section dipole IR-pump resonant transition or of a pump–probe Raman-resonant transition with negligible Raman pumping—as in the coherent anti-Stokes Raman scattering (CARS) process. The third option is the non-resonant pump and resonant probe (NRP) FWM process, similar to that observed by Sun et al. while tuning the probe wavelength in an FWM experiment to an electronic resonance in OH radicals [52]. Due to the incidence of the probe beam at the Bragg angle and the relation between the frequencies of the pump and probe laser radiations (${2}{v_{\rm{pump}}} = {v_{\rm{probe}}}$), all the three processes are intrinsically phase-matched and fulfill the energy conservation condition, yielding the signal at 532 nm (transmitted by the bandpass filter).

The NRPFWM processes would involve a (near-) resonance around 532 nm (${\approx} {18}{,}{800}\;{{\rm cm}^{- 1}}$). The HITRAN database covers this spectral range, albeit no transitions are listed [53], while the lowest electronic resonance A←X resides at ${46}{,}{000}\;{{\rm cm}^{- 1}}$ [54]. Therefore, the NRPFWM process can be excluded. The CARS-type process would employ the 532 nm beam as the pump and the 1064 nm beams as the Stokes and the probe ones of the process. As given by calculations based on the results of Majcherova et al. [55], there exists one Raman-active transition near ${9395}\;{{\rm cm}^{- 1}}$ within the pump laser linewidth (${\approx} {1}\;{{\rm cm}^{- 1}}$), corresponding to the 00001-70004 combination band S(14) resonance at ${9395.7} {{\rm cm}^{- 1}}$, with a low Raman cross-section. The IR-resonant process has both 1064 nm pump beams interacting to generate the coherence and the CW 532 nm beam to probe this coherence. The database [53] gives the relevant dipole-active transition that corresponds to the 00001-20033 combination band $R({8})$ resonance at ${9395.3}\;{{\rm cm}^{- 1}}$, with the FWHM of about ${0.44}\;{{\rm cm}^{- 1}}$ at 2 bar of ${{\rm CO}_2}$. Note that the neighboring transitions are separated by about ${1.2 {-} 1.3}\;{{\rm cm}^{- 1}}$. The small absorption line strength of only ${1.5} {\cdot} {{10}^{- 26}}\;{{\rm cm}^{- 1}} \cdot {{\rm cm}^2}/{\rm mol}$ and relatively large laser linewidth would result in negligible excitation of the upper rovibrational state and, hence, in a non-observable thermal contribution to the LIG signal.

The origin of the narrow peak can be characterized by its dependence on pump and probe laser intensity. Both TCFWM and LIG processes are expected to depend linearly on the probe and quadratically on the pump intensities, which is typical for any four-wave mixing interaction. As a result, the parameter $r$ is expected to be constant when either the pump laser pulse energy or the probe laser power is varied. Figure 4 shows these dependences of the parameter ${r_{{\rm VVVV}}}$ for two ${{\rm CO}_2}$ concentrations $x$ in ${{\rm N}_2}$: $x = {1}$ and $x = {0.5}$. As expected, there is no visible trend with the variation of either the probe or the pump intensity, and the corresponding average values are ${r_{{\rm VVVV}}}({1}) = {1.14} \pm {0.04}$ and ${r_{{\rm VVVV}}}({0.5}) = {0.85} \pm {0.03}$. This corroborates the origin of the narrow peak as a result of the TCFWM process. Furthermore, the determination of $r$ is, therefore, independent of the actual laser power, which is important for general use of this parameter for diagnostics.

 

Fig. 4. Parameter ${r_{\rm{VVVV}}}$ as a function of the pump pulse energy or the probe power for 100% ${{\rm CO}_2}$ and 50% ${{\rm CO}_2}$ in ${{\rm N}_2}$ at 295 K and 2 bar. The dashed blue lines represent the respective average of $r$.

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Figure 5 shows the dependence of the parameter ${r_{{\rm VVVV}}}$ and of the LIG contribution scaling factor $A$ from Eq. (3) (normalized by its value at 1 bar) on ${{\rm CO}_2}$ pressure $p$ in the range of 0.4–2.7 bar. As expected from the linear dependence of ${M_e}$ on gas density [30,32], the LIG signal intensity (${\sim}M_e^2$) increases quadratically with pressure as can be seen from the fit of the $A$ values by a parabola $a \cdot {p^2}$. The ratio of the narrow peak and the LIG-diffracted field amplitudes ${r_{{\rm VVVV}}}$ is practically independent on pressure (${r_{{\rm VVVV}}} = {1.12} \pm {0.02}$), indicating the similar quadratic increase for the TCFWM signal contribution. All this information combined, the narrow peak can be safely assumed to be the result of the four-wave mixing process.

 

Fig. 5. Normalized LIG signal scaling factor $A$ (blue *) and parameter ${r_{\rm{VVVV}}}$ (black $\circ$) as a function of ${{\rm CO}_2}$ pressure.The solid red line shows the result of fitting of the $A$ values by a parabola $a \cdot {p^2}$; the dashed black line represents the average of $r$.

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C. Application of the TCFWM Signal to Determination of ${{\rm CO}_2} {\text -} {{\rm N}_2}$ Concentration

Examples of LIG signal temporal profiles (at the VVVV configuration) in the ${{\rm CO}_2} {\text -} {{\rm N}_2}$ mixture at 2 bar total pressure and 74% and 34% ${{\rm CO}_2}$ mole fractions are depicted in Fig. 6. The narrow peak is clearly visible in both profiles. The experimental traces have been fitted using Eqs. (2) and (3) with ${\Omega _a}/{2}\pi = ({9.517} \pm {0.003})\;{\rm MHz}$ and ${\tau _{{tr}}} = ({394} \pm {1})\;{\rm ns}$ for 74% ${{\rm CO}_2}$ and ${\Omega _a}/{2}\pi = ({10.710} \pm {0.004})\;{\rm MHz}$ and ${\tau _{{tr}}} = ({346} \pm {1})\;{\rm ns}$ for 34% ${{\rm CO}_2}$ close to those found for the signal in pure ${{\rm N}_2}$ in Fig. 2. The amplitude ratio is smaller than that in pure ${{\rm CO}_2}$ [${r_{{\rm VVVV}}}({{\rm CO}_2}) = {1.18}$, ${r_{{\rm VVVV}}}({0.74}) = {0.92}$, ${r_{{\rm VVVV}}}({0.34}) = {0.57}$], which is likely due to dilution of ${{\rm CO}_2}$ in ${{\rm N}_2}$. The parameter $r$ can, therefore, be used for ${{\rm CO}_2}$ concentration determination, which is discussed in the following.

 

Fig. 6. Details of the initial part of the normalized LIG signal profiles in the ${{\rm N}_2} {\text -} {{\rm CO}_2}$ mixture at 295 K and 2 bar total pressure with 74% and 34% ${{\rm CO}_2}$ mole fractions (black and blue points) at the all-vertical (VVVV) polarization configuration. The experimental traces, fitted according to Eq. (3), are plotted using green and red lines, respectively.

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The ${{\rm CO}_2}$ concentration dependence of the FWM and the LIG-diffracted field amplitude ratio $r$ is further determined by deriving its value for an arbitrary ${{\rm CO}_2}$ concentration in ${{\rm N}_2}$ at room temperature. The FWM signal intensity is defined by the real non-resonant (${\chi _{{\rm N}_2}^{(3){\rm NR}}}$, ${\chi _{{\rm CO}_2}^{(3){\rm NR}}}$) and resonant (${\chi _{{\rm CO}_2}^{(3){\rm R}}}$, with the real and imaginary parts ${\chi ^{{{(3)}\prime}}}$ and ${\chi ^{(3)\prime\prime}}$, respectively) contributions to the third-order non-linear susceptibility ${\chi ^{(3)}}$ from ${{\rm N}_2}$ and ${{\rm CO}_2}$, as well as by the phase shift $\varphi$ between the FWM and the LIG-diffracted fields. The resonant contribution from ${{\rm N}_2}$ equals to zero due to the absence of the dipole- or Raman-active transitions. At the VVVV and VVHH configuration, the relation between the electric fields can be expressed by Eq. (5),

After calculating the FWM signal intensity as ${E_{\rm{FWM}}} \times {E_{\rm{FWM}}^*} = {[r(x){\cdot}{E_{\rm{LIEG}}}(x)]^2}$ and grouping the terms by powers of $x$, one obtains Eq. (6),

To obtain the final dependence of $r$ on $x$, the $x$-dependent terms of the LIEG electric field strength ${E_{\rm{LIEG}}}$ need to be considered. These terms are written explicitly in Eq. (7),

The mole fraction dependence of the gas density can be expressed as linear, defined by the ratio of the molar masses ${M_{\rm CO2}}$ and ${M_{\rm N2}}$ [Eq. (8)],

The $x$-dependence of the acoustic period ${T_a}$ can be expressed, according to Eq. (1), as the inverse of the adiabatic sound velocity, using the parameters of the mixture components,

The value of ($\partial n/\partial \rho$) for a neat gas $i$ can be estimated as $({n_{0i}} - {1}){\rm /}{\rho _{0i}}$, where index 0 refers to the values at the STP conditions. Then, for the ${{\rm CO}_2} {\text -} {{\rm N}_2}$ mixture, the dependence of ($\partial n/\partial \rho$) on $x$ can be linearly interpolated as shown in Eq. (10),

Figure 7 shows the FWM-LIG field amplitude ratio $r$ in ${{\rm CO}_2} {\text -} {{\rm N}_2}$ mixtures as a function of ${{\rm CO}_2}$ mole fraction $x$ at 2 bar total pressure for 0°, 30°, 60°, and 90° probe polarization angles ${\Theta _{\rm{probe}}}$. The values exhibit a non-linear composition dependence, as well as a strong dependence on ${\Theta _{\rm{probe}}}$. The data for 0° and 90° have been fitted using Eq. (6), with the known values of $\chi _{1111}^{\rm{NR}}$ for ${{\rm N}_2}$ and ${{\rm CO}_2}$ [49] and with $\chi _{2112}^{\rm{NR}}$ assumed to be equal to $\chi _{1111}^{\rm{NR}}/{3}$. The phase $\psi$ was defined from the data set with the highest signal intensity (i.e., at the VVVV configuration), which is considered to have the best precision, and was kept fixed for the other configurations. The value of $\psi$ is expected to be $\pi /{2}$ in case of a resonance and to approach 0 far from it. The fitted value of $\psi = ({0.4}\pm{0.2}) \cdot \pi {\rm /2}$ points to a larger-than-expected, but acceptable, shift of the laser line and the resonance frequencies. Note that the fitting curves closely represent the experimental values. For each trace, a separate set of parameters ${r_0}$, $a$, and $b$ was obtained, and the values $\chi _{1111,\rm CO2}^R$ and $\chi _{2112,\rm CO2}^R$ have been calculated. The values employed for fitting, as well as the obtained $\chi _{\rm CO2}^R$, are summarized in Table 1.

 

Fig. 7. Plot of the FWM-LIG amplitude ratio $r$ as a function of ${{\rm CO}_2}$ mole fraction $x$ for the ${{\rm CO}_2} {\text -} {{\rm N}_2}$ mixture at 295 K and 2 bar with 0° (= vertical, black *), 30° (magenta $\circ$), 60° (blue $\square$), and 90° (= horizontal, red ×) probe polarization angle ${\Theta _{\rm{probe}}}$. All the data sets were fitted using Eq. (11) (solid lines) (for 30° and 60°—under the limitations outlined in the text). Dotted lines represent the curves for 30° and 60° calculated using Eq. (11), where ${r_{{\rm VVVV}}}(x)$ and ${r_{{\rm VVHH}}}(x)$ are the values from the fitted curves at 0° and 90°, respectively.

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Table 1. Values Used for Fitting of Eq. (6) to the Experimental Data and the Results of the Fit

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In pure ${{\rm N}_2}$, at $x = {0}$, the narrow peak was weakly observable at the VVVV configuration (${r_{0,{\rm VVVV}}} = {0.375}\pm{0.006}$). Since the FWM signal is assumed to be a non-resonant one, the ratio is expected to decrease to about a third at the VVHH configuration, as discussed above and expressed by Eq. (4), which is at the noise level of the absolute FWM intensity. Nevertheless, we were able to obtain the value of ${r_{0,{\rm VVHH}}} = {0.116}\pm{0.005}$ corresponding to the ${r_{{\rm VVVV}}}/{r_{{\rm VVHH}}}$ ratio of 3.23, which is in a good agreement with the expected value of 3 [49], if the assumption that LIG diffraction efficiency does not depend on ${\Theta _{\rm{probe}}}$ is valid. In ${{\rm CO}_2}$, the derived significant values of $\chi _{\rm CO2}^R$, with the ratios $|\chi _{1111}^R|/\chi _{1111}^{\rm{NR}} = {4.6}$ and $|\chi _{2112}^R|/\chi _{2112}^{\rm{NR}} = {6.2}$, respectively, indicate the presence of a resonant contribution to the FWM peak. In this case, the ratio $|\chi _{1111, \rm CO2}^R|/|\chi _{2112 , \rm CO2}^R|$ of 2.2 is in a good agreement with an expected value of 2 for a dominant isotropic contribution to the susceptibility [49,51]. Note that, since the pump laser linewidth is significantly larger than that of the transition, the obtained susceptibilities $\chi _{\rm CO2}^{{R^\prime}}$ and $\chi _{\rm CO2}^{R^{\prime \prime}}$, in fact “integrated” over the resonance curve within the laser line, are the effective values as related to the values ${\chi ^{\rm{NR}}}$.

The data for 30° and 60° in Fig. 6, ${\rm r}(x,{\Theta _{\rm{probe}}})$, can be estimated based on the derived values and Eq. (4), assuming that LIG diffraction efficiency does not depend on ${\Theta _{\rm{probe}}}$. The resulting relation is given by Eq. (11),

The curves $r(x,{\Theta _{\rm{probe}}})$ calculated based on Eq. (11) are represented by the dotted lines in Fig. 7. The trend of the experimental $r(x)$ dependences is reasonably well reproduced for both probe polarizations; however, the absolute values are systematically slightly overestimated. This may be due to a small ${\Theta _{\rm{probe}}}$-dependence of the LIG diffraction efficiency that has not been considered when fitting the data with Eq. (6). Yet, the curves, as well as those calculated for any other ${\Theta _{\rm{probe}}}$, can be used if the ratio $r$ needs to be optimized, depending on the system under study, to avoid, e.g., detector saturation by potential intense FWM and weak LIG signal contributions in different gases.

Note that the curves, fitted to the data for 30° or 60° using Eq. (6) by variation of the parameters ${r_0}$, $a$, and $b$, can accurately describe the experimental data, as shown by the solid lines in Fig. 6, and may serve as calibration curves for determination of ${{\rm CO}_2}$ concentration. In this case, the derived parameters should be considered as the effective values.

Thus, we can define $x$ based on $r$, either using the experimental calibration dependences or calculations if the appropriate values are available. For ${{\rm CO}_2} {\text -} {{\rm N}_2}$ mixtures, the slope, and thereby sensitivity, is maximum in the VVVV configuration. The detection limits for trace amounts of ${{\rm CO}_2}$ in ${{\rm N}_2}$ and ${{\rm N}_2}$ in ${{\rm CO}_2}$ are estimated from the slope of the $r(x)$ dependence over a concentration range of less than 5% for each species (see Fig. 7) and the standard deviation of the $r$-values (see Fig. 5) to be $\approx {1}\%$ for ${{\rm CO}_2}$ and $\approx {4}\%$ for ${{\rm N}_2}$.

D. Temperature Dependencies

To simultaneously determine gas composition from the FWM-LIG field amplitude ratio $r$ and temperature from the sound velocity ${\upsilon _s}$ via the oscillation period, the temperature dependence of $r$ needs to be known. Figure 8(a) shows the LIG signal profiles at different temperatures between 295 K and 515 K in pure ${{\rm CO}_2}$ at 2 bar normalized by the amplitude of the first LIG signal oscillation peak at 295 K from a fit of Eq. (2). As can be seen from the figure, the absolute signal intensity of both TCFWM and LIG contributions rapidly decreases. This is first of all due to the proportionality of the FWM electric field amplitude, as well as that of the LIG-diffracted radiation, to gas density, which scales as ${1/}T$ at isobaric conditions. It means that proper fitting of the data is more challenging at elevated temperatures (and small concentrations), and the uncertainties are generally higher under these conditions. The fit of the profiles using Eq. (2) reveals a variation of the oscillation frequency from ${\Omega _a}/{2}\pi = {8.88}\;{\rm MHz}$ to 11.96 MHz. Figure 8(b) shows the same profiles but normalized by the amplitude of their first signal oscillation peak amplitude $A$ from the fit of Eq. (3) to the data. Apparently, the FWM and LIG contributions have different temperature dependences: The amplitude of the FWM peak decreases faster with temperature than that of the LIG oscillation peak. This temperature dependence needs to be known and, ideally, analytically described to allow independent determination of the two parameters.

 

Fig. 8. Experimental LIG signal profiles in pure ${{\rm CO}_2}$, at 2 bar and 295 K (gray profile), and 335 K to 515 K (in steps of 20 K) and 30° probe angle. (a) The intensities are normalized by the amplitude of the first LIG signal oscillation peak at 295 K as fitted using Eq. (2); the fitting curves are not shown for visibility but represent the data reasonably well. (b) First 40 ns of the same profiles, with the identical colors, each normalized by their first signal oscillation peak amplitude $A$ from a fit of Eq. (3) to the data.

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The temperature dependence of the FWM field amplitude at resonance is strongly influenced by the population difference $\Delta N(T)$ for the respective transition. As assumed above, the relevant rovibrational transition in ${{\rm CO}_2}$ responsible for the observed FWM peak is the 00001-20033 $R({8})$ transition. From Boltzmann’s relation, it becomes apparent that the population of the upper (20033) vibrational state is negligible in the studied temperature range due to its high energy. Therefore, the population difference is governed by the molecular number density in the lower rovibrational state. The energy ${E_J}$ of the lower rovibrational $J$-level can be calculated as ${E_J} = {2} \cdot B \cdot J(J + {1})= {28.1}\;{{\rm cm}^{- 1}}$, where $B = {0.39021}\;{{\rm cm}^{- 1}}$ is the rotational constant in the ground vibrational state (00001) from [56]. The canonical partition function ${Z_r}(T)$ is approximated by the linear temperature dependence ${Z_r}(T)\; \approx \;{k_B}T/B$ for the studied temperature range. The temperature dependence of the total number density $N(T)$ can be approximated as ${1/}T$. The temperature-dependent transition linewidths $\Gamma ({\rm FWHM}) = {0.22}\;{{\rm cm}^{- 1}}$ at 300 K [57] are narrower than the pump laser linewidth and, therefore, are not taken into account. All the mixture composition-dependent coefficients are combined into the proportionality factor. Summarizing, the temperature dependence of the FWM field amplitude can be defined from Eq. (12),

The LIG diffracted field amplitude ${M_e}$ is proportional to ${T^{- 3/2}}$ due to its proportionality to gas number density $N$ and inverse proportionality to the sound velocity ${\upsilon _s}$ [33,42]. The resulting relation for the temperature dependence of $r$ can be expressed by Eq. (13),

The experimentally derived values of $r$ as a function of temperature for different mixture compositions are shown in Fig. 9. Also plotted are the curves of Eq. (13), when $r(x,{T_0})$ are calculated using Eqs. (6) and (11) and the parameters in Table 1 (dotted lines), or are taken from the calibration curve for 30° probe angle in Fig. 7 (dashed lines). The model corresponds to the trend observed in the data, i.e., $r$ decreases non-linearly with $T$. Both methods for the determination of $r(x,{T_0})$ yield curves that lie within the scatter of the experimental data points. The slope of $r(T)$ is steeper the larger ${{\rm CO}_2}$ mole fraction is. Therefore, the combination of Eqs. (6) and (13) can be used as a complete description of $r(x,T)$.

 

Fig. 9. Experimental and calculated values of FWM/LIG ratio $r$ at 30° probe angle as a function of temperature for different selected mole fractions of ${{\rm CO}_2}$ in ${{\rm CO}_2} {\text -} {{\rm N}_2}$ mixtures. Dashed lines represent plots of Eq. (13), where $r(x,{T_0},{30}^\circ)$ were calculated using Eqs. (6) and (11) and the parameters in Table 1. Dotted lines are calcualted using in Eq. (13) with $r(x,{T_0},{30}^\circ)$ values as per the calbration curve for ${\Theta _{\rm{probe}}} = {30}^\circ$ in Fig. 7.

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The relations for the temperature and species concentrations can be readily used for various molecules provided TCFWM and LIEG signals can be simultaneously observed and thermal data, susceptibilities, and molecular constants are known.

If both parameters are to be determined from LIG experiments of a mixture of unknown ${{\rm CO}_2}$ concentration and temperature, the acoustic frequency ${\Omega _a}/{2}\pi$ and the ratio $r$ need to be determined. The acoustic frequency, e.g., obtained from a Fourier-transform of the signal profile, can be related to the local sound velocity by ${\upsilon _s} = {\Omega _a} \cdot \Lambda /{2}\pi$. By using Eq. (1) and assuming $T$-independence of the heat capacities ${c_p}$ and ${c_v}$, a relation for $T({\Omega _a},\; x)$ [Eq. (14)] can be formed,

By plugging Eq. (14) into Eq. (13) and combining with Eq. (6), a relation for $r(x,{\Omega _a})$ is formed that can be numerically solved for $x$ for known $r$ and ${\Omega _a}$. Once $x$ is known, Eq. (14) can be used to obtain the temperature.

4. SUMMARY

We report the observation of a narrow peak preceding laser-induced electrostrictive grating signals in ${{\rm CO}_2}$ and ${{\rm CO}_2} {\text -} {{\rm N}_2}$ mixtures. This peak is attributed to the result of resonant two-color four-wave mixing in ${{\rm CO}_2}$ of the pulsed pump and CW probe radiations. It is shown that, while the TCFWM signal exhibits gas density and laser power dependencies identical with those of the LIEG signal, the variation of these signals with temperatures and species concentrations differs. Therefore, the formulation of analytical relations of the temperature and concentration dependencies of the relative FWM and LIEG signal amplitudes enables its application, in conjunction with the LIEG signal profile, to simultaneous determination of ${{\rm CO}_2}$ species concentration and gas temperature in ${{\rm CO}_2} {\text -} {{\rm N}_2}$ mixtures. Through the analytical relations, a calibration-free extension of the technique to other molecules exhibiting simultaneous TCFWM and LIEG signals is possible.