1. Introduction

The method of four-wave mixing laser spectroscopy using laser-induced gratings (LIGs) [1], i.e. spatially-periodic modulations of the complex refractive index, has been broadly developed and employed for fundamental studies and in applications for remote, non-intrusive, and local diagnostics of gaseous media during the last two decades. Measurements of parameters like the adiabatic sound velocity, temperature and composition of a gas, flow velocity, excited molecular states collisional deactivation rates, thermal and mass diffusion coefficients have been accomplished (see review [2] and the recent publications [3–9]).

When the probe laser beam, diffracted by LIGs, is not absorbed in the medium, the LIG signal strength is defined by the value (Δn)², where Δn is the induced variation of the refractive index, n, across the fringe of the grating. In the simplest experimental realization of the technique the radiation of a single pulsed pump laser, with a few nanoseconds pulse duration and an arbitrary wavelength, is employed for non-resonant excitation of LIGs due to electrostriction. In this situation the LIG signal strength is determined by the value of the electrostrictive constant [10] γ_e=ρ·(∂ε/∂ρ)_T=2n ·ρ·(∂n/∂ρ)_T, where ρ is the gas density and ε is the dielectric permittivity. Since in a gas the approximation ρ·(∂n/∂ρ)_T≈(n-1) is valid the parameter γ_e has a small value. Therefore the use of the electrostrictive LIGs, despite the advantages of universality and relative simplicity of the experiment and the data analysis, is characterized by low signal levels and limited sensitivity. Though by employing a non-resonant pumping in combination with one-photon resonance probing the sensitivity can be increased [11], in general a pump laser with high pulse energy is required anyway in order to excite electrostrictive LIGs. In addition, due to a uniqueness of the LIGs formation and evolution mechanism, the set of the measurable parameters of the medium is usually reduced to gas temperature and gas composition.

A source of pump radiation, which is resonant with a one-photon transition of any molecular species in a gas, allows one to enhance the efficiency of LIG excitation and hence the sensitivity of the technique, up to a few orders of magnitude. In particular, this is valid in cases of rapid local collisional exchange of the excited species internal energy with the environment. This exchange results in spatially-periodic modulations of temperature and in the corresponding variations of gas density providing large thermal contributions to the refractive index modulation (so-called thermal LIGs). The pump radiation wavelength may lie in the UV or visible spectral range, which is characteristic for electronic or combination vibrational molecular transitions. Up to now this approach has been employed for gases like NO₂ [3,5,12–16], NO [17,18], OH [18,19], O₂ [20–22], H₂O [23–26] and CH₄ [9]. In contrast, pump radiation wavelengths in the IR range correspond to the low-lying fundamental vibrational transitions which have been used in LIG experiments for the excitation of C₂H₄ [27,28], NH₃ [27], CH₃OH [28], and C₃H₈ [6,8]. Note that the significant enhancement of the LIG signal strength can be observed even if the mole fraction of the absorbing molecules is as small as 10^-3–10^-4, or the absorption is extremely weak. This is the case because large amounts of energy can be exchanged in a single collision. (Note that the excitation energy as well as the amount of energy exchanged with the environment in the course of collisional relaxation is, in case of rotational molecular states, about one to two orders of magnitude smaller than that related to deactivation of vibrational or electronic states.) In addition, a few mechanisms can be involved in the formation and decay of thermal LIGs. Hence, a larger number of characteristic parameters of the medium can be derived from the LIG signals with thermal contributions. However, the appropriate one-photon resonance pump laser source is not always available for a given species, thus conducting experiments may be difficult or even impossible. In order to overcome this problem multi-photon processes offer a possibility for the generation of thermal LIGs.

Two-photon stimulated Raman excitation of rovibrational molecular transitions, which are inactive in absorption, has been demonstrated for CO₂ molecules by employing a pair of frequency-shifted pump lasers [29,30]. The result was the efficient production of thermal LIGs, thereby enlarging the applicability of the LIG technique for gas phase diagnostics. However, the addition of the required second laser results in an obvious complication of the experiment. As an extension of this approach, the stimulated Raman excitation can be applied to pure rotational transitions of molecules. Since the transition frequencies are significantly smaller in this case, a single pump laser with spectrally broad output radiation, that covers the frequency range of the strongest Raman-active rotational molecular transitions, can be employed for the generation of thermal LIGs. In doing so, two-photon excitation of each transition is obtained by a number of different spectral components.

In this paper we report the first experiments on the generation of thermal contributions to LIGs via simultaneous stimulated Raman excitation of a number of rotational molecular transitions by using a single broadband pump laser. Rotational Raman excitation of LIGs is demonstrated in N₂, CO₂, and C₃H₈ under stationary conditions in a gas cell at pressures P=0.1-5 bar at room temperature (T=298 K). In view of the above-mentioned preference of employing thermal LIGs, it is of interest (1) to experimentally investigate the possibility of broadband stimulated Raman excitation of LIGs, (2) to evaluate the strengths of LIG signals obtained, and (3) to assess the efficiency of such excitation as compared to the non-resonant one.

2. Experimental

A schematic of the experimental setup is shown in Fig. 1. The broadband pump dye laser consists of a rectangular dye cell, DC, placed into a ~230 mm length plane resonator formed by a highly-reflecting mirror and an output coupler with 10% reflectivity. The dye cell is transversely pumped through a cylindrical lens, CL, by the frequency-doubled radiation of the Nd:YAG laser at 532 nm with the pulse energy of up to 220 mJ and pulse duration about 8 ns. A solution of DCM dye in methanol is employed providing a conversion efficiency of about 14% so that the pulse energies in each of the two pump beams after the 50 % beam splitter, BS, are about 15 mJ. The spectral profile of the dye laser output, centered at 638 nm, is close to a Gaussian one. At full pump power of the Nd:YAG laser it has the width Γ=17.4 nm (FWHM), or 430 cm^-1, while at the 1/e-level its width is 20.9 nm, or 514 cm^-1.

The delay line, DL, in one of the pump beam pathways is necessary to accurately adjust the difference of the optical path lengths of the two beams, Δy, to be close to zero. This is important for the spatial overlapping of intensity fringes at different wavelengths that provides efficient spatially-periodic two-photon excitation of thermal LIGs with a high contrast. Experimentally, if the optical path length was varied about ±30 µm by moving the prism of the delay line out of the optimal position, the signal disappeared. The upper limit of the mismatch, Δy_max, can be estimated as follows: for a good spatial overlap of the fringes the condition ΔkΔy_max≈2πΔλ/λ²₀ n Δy_max=π should be fulfilled taking into account the broad spectral range 2Δλ at the center wavelength λ ₀. At Raman excitation the number of the excited molecules and, hence, the amplitude of the refractive index modulation, Δ_n, is defined by the characteristic frequency width, FW, of the spectral profile of the squared pump intensity, I² (rather than of the profile of the pump intensity, I, like in absorption). In case of a Gaussian profile one obtains FW(I ²)=FW(I)/2^0.5. Substitution of FW(I) by the Γ value gives Δy_max≈33 µm. Note that small ajustments of the delay line are desirable if the species or gas pressure in the cell is changed as this exerts influence on the refractive index of the medium inside the cell.

 

Fig. 1. Scheme of the experimental setup: DC, dye cell; CL, cylindrical lens; BS, beam splitter;DL, delay line; FL, focusing lens; IF, interference filter; PMT, photomultiplier tube.

Download Full Size | PDF

The linearly polarized pump beams are focused into the pressurized gas cell by an f=1 m lens, FL, and their crossing angle provides the interference fringe spacing Λ≈28 µm. The temporal evolution of the LIG is recorded utilizing a beam of a cw air-cooled multi-line Ar⁺-laser (70 mW at 514.5 nm) in a 3D forward interaction geometry. The diffracted signal beam passes through a multi-mode optical fibre and an interference filter, IF, centred at the probe wavelength. Finally, the signal is detected by using a photomultiplier tube, PMT, connected to a 500 MHz digital oscilloscope which provides accumulation of typically 100 single-shot LIG signals and data storing.

The two-photon excitation of a given Raman transition at frequency Δν by the broadband laser radiation is defined by the sum of the products of the different spectral components’ intensities. Therefore it is worthwhile to estimate the pumping efficiency, η_p, in terms of Δν. For the case of a continuous Gaussian spectral profile the straightforward integration gives

where Γ, as above, is the FWHM of the pump laser spectral profile. Hence, all the transitions with frequencies Δν≪Γ/(2ln2)^0.5≈0.85 Γ, or Δν≪365 cm^-1 in our case, are pumped with approximately equal efficiency. Compare this value with the characteristic spectral parameters of linear molecules. Frequencies of Raman-active rotational transitions in these molecules (in O- and S-branches, in accordance with the selection rules ΔJ=±2) are defined by the values Δν_J≈4B·J, where B is the rotational constant of a molecule, and J is the total angular momentum quantum number. The typical values of J for the most populated rotational levels at a given temperature T satisfy the well-known relation J≤J_max=2·(kT/2Bhc)^0.5, where k and h are Boltzmann and Planck constants, respectively, and c is the speed of light. Thus, for the transition frequencies of interest the relation Δν_J≤Δν_max=8ν(BkT/2hc)^0.5 should be valid. For N₂, with B=2 cm^-1, J_max≈14 at ambient temperature, this means that the frequencies of the strongest transitions are less than Δν_max≈116 cm^-1, hence lie within the characteristic pump laser bandwidth of 365 cm^-1. For this reason, all the rotational transitions in N₂ that should contribute to the LIG signal are pumped with the same efficiency. For CO₂, with B=0.390 cm^-1, J_max≈34 and Δν_max≈53 cm^-1, this condition holds even better.

Note that for efficient Raman excitation of rotational transitions in the major part of molecules of interest it would be sufficient to provide a narrower pump laser spectral width (Γ~100–150 cm^-1) than in the presented experiment. In this case reduction of Γ at the comparable pulse energy level, e.g., by using a band-pass filter in the dye laser resonator, would enhance the signal strength. Moreover, it would make the adjustment of the two pump beams optical path lengths difference Δy less critical due to a corresponding increase of the admissible value of the mismatch Δy_max. It should also be noted that in fact the pump laser spectrum is not continuous, but consists of a large number of mode frequencies separated by about 0.02 cm^-1. Hence, it may happen that there is no pair of modes with the frequency difference exactly matching the frequency of a Raman transition. However, since under the experimental conditions of the present work the linewidths (FWHM) of the rotational transitions involved are usually a few times larger than the mode spacing (e.g. above 0.07 cm^-1 in N₂ and 0.18 cm^-1 in CO₂ at 1 bar), the pump energy is used quite efficiently.

3. Results and discussion

The examples of temporally resolved LIG signals obtained with the broadband pump laser in Ar and in N₂ at similar pressures of 3.8 and 3.4 bar, respectively, are presented in Fig. 2 at the same absolute vertical scale. The temporal shape of the signal in Ar, as an atomic species, is typical for that defined by only electrostriction. In contrast, the alternation of the oscillation peak heights distinctly observed in the temporal evolution of the LIG signal in N₂ is indicative of the interference of two contributions to the refractive index modulation: the electrostrictive and the thermal ones. An analysis shows that the latter is characteristic for an “instantaneous” energy exchange [22,31]. In N₂ it is assumed to result from rapid (as compared to the pump pulse duration) collisional R-T relaxation of spatially-periodic Raman-excited rotational molecular states. The magnitude of the peaks heights variation is defined by the ratio M_i/M_e of the amplitudes M_i and M_e of the thermal and electrostrictive contributions, respectively.

 

Fig. 2. Comparison of pure electrostrictive (in Ar at 3.8 bar) and combined electrostrictive and thermal (in N₂ at 3.4 bar) LIG signals. The amplitude of the signal in Ar is reduced by a factor of two for clarity. An offset of 0.034 µs from the pump laser pulse is introduced to the time scale. To the right, the twentyfold magnified weak tail of the signal in N₂ (the solid line, —) and the result of its fitting taking into account the electrostrictive and “instantaneous” energy exchange contributions (the short-dash line, - - -) are plotted.

Download Full Size | PDF

The value of M_e is defined by the refractive index, n, gas density, ρ, and adiabatic sound velocity, v_s, [22]:

Since in a gas (∂n/∂ρ)_T≈(n_STP - 1)/mN_L, where n_STP is the refractive index at standard conditions, m is the molecular mass, and N_L=2.68676·10¹⁹ cm^-3/Amagat is the Loschmidt number, M_e is proportional to gas pressure, P, and the value (n_STP - 1)²/(v_sm)∝M_e/P is the characteristic parameter of the strength of an electrostrictive LIG signal in a gas at 1 bar. At a given pressure the amplitudes of the electrostrictive contribution to LIGs in the set of the investigated molecular gases, i.e. the values of the coefficients M_e/P, normalized to that of Ar, grow from 1.4 and 2.7 for N₂ and CO₂, respectively, to 17.0 for C₃H₈. As a consequence, the absolute LIG signal strength in this row should increase.

The value of M_i is defined by the refractive index, the amount of energy per unit volume ΔE_i exchanged with the environment by Raman-excited molecules as a result of collisional R-T relaxation during the pump laser pulsẹ, and the specific heat, c_p, [22]:

The probability of the Raman pumping, η_J, of a transition between the rotational levels with J and J+2 of a linear molecule in the presence of two Raman-shifted radiation fields is defined by the relation (see, e.g., [32]):

that accounts for simultaneously occurring forward, J→J+2, and backward, J+2→J, rotational transitions. Here, g_J is the nuclear statistical weight of the lower level, Z_R is the rotational partition function, E_J and E_{J +2} are the energies of the lower and upper levels (with J and J+2), respectively, with E_J≈hcB·J·(J+1), and β is the polarizability anisotropy. The average amount of energy exchanged with the environment due to collisions after the excitation of one pair of rotational molecular states, Δε_i, is defined as

and, if Δν_J≪Γ/(2ln2)^0.5, is scaled by the value Δε_i≈4B·J_max in linear molecules. This energy is in fact essentially smaller than that related to excitation of a combination vibrational or an electronic state. The total energy density ΔE_i can then be expressed as ΔE_i∝B·J_max·β²·P·τ_L, where τ_L is the pump laser pulse duration. Note that similar estimates can be made for symmetric top molecules.

The noticeably different LIG signal oscillation periods in Fig. 2, at a given fringe spacing Λ, are defined by the respective adiabatic sound velocities v_s=322 m/s (Ar) and v_s=352 m/s (N₂). The signal decay time of about 0.5 µs at pressures below 5 bar is mainly determined by the time interval in which the acoustic waves traverse the waist of the pump and probe beams (the acoustic transit time). To a smaller extent, it is defined by the acoustic damping (caused by the gas viscosity and thermal conductivity), being of the same order in both cases. The signal temporal shape can be reasonably fitted by the analytical function similar to that employed in [22,26] and taken in the form corresponding to electrostrictive (Ar) or both electrostrictive and “instantaneous” energy exchange thermal (N₂) contributions to LIGs. To the right in Fig. 2 the twentyfold magnified weak tail of the signal in N₂ (to the left) and the result of its fitting in frame of the one-stage “instantaneous” energy exchange model are plotted. The level of the stationary contribution at delays larger than 1.2 µs after the pump laser pulse is well reproduced within this model.

In Ar, with the electrostrictive contribution only, the measured LIG signal strength is proportional to M²_e, i.e. it increases with gas pressure as P ². The signal strengths dependence on pressure, as obtained in the investigated molecular gases in the range of P=0.1-5 bar, is weaker than quadratic, being rather close to a linear one. This may possibly result from the increase with pressure of the difference Δy of the optical path lengths of the two pump beams, which was not being optimized in the course of measurements at different pressures. However, the validity of this assumption has to be further investigated.

As expected, the signals in the molecular gases are substantially stronger than in Ar, especially in CO₂ and C₃H₈ (compare the vertical scale in Figs. 2, 3 and 5). There the observed temporal profile of the broadband laser excited LIG signal is to a large extent defined by the multi-component thermal contribution. Compared to N₂, CO₂ and C₃H₈ have a more complicated and dense structure of rovibrational levels, which can be populated by collisions with Raman-excited molecules. The composite temporal form of the thermal contribution is, feasibly, a consequence of the larger number of paths of deactivation, migration and relaxation of the internal energy.

Thus, in CO₂ at pressures 0.1–3 bar the LIG signal shape is mainly determined by electrostrictive and “instantaneous” thermal contributions and can be described within the one-stage energy exchange model by using the respective expression. Note that the ratio of the amplitudes (M_i/M_e)_CO2/(M_i/M_e)_N2≈2.1, derived as a result of the fitting of the signals in CO₂ and N₂ at pressures of 0.1–0.5 bar, is in a good agreement with the scaling discussed above, which gives [BJ_maxβ²v_s/c_p(n_STP - 1)]_CO2/[(BJ_maxβ²v_sm/c_p(n_STP - 1)]_N2=2.3. Here, the ratio β²_CO2/β²_N2≈8.2 can be calculated from the data on rotational Raman line strengths in N₂ and CO₂ presented in [33].

 

Fig. 3. The temporal shape of a LIG signal in CO₂ at 4 bar (left), with the twentyfold magnified amplitude at large delays (right). The offset of 0.28 µs is introduced to the time scale. The short-dash line (- - -) shows the result of the calculation taking into account only the electrostrictive and “instantaneous” energy exchange contributions; the long-dash line (— — —) represents the result of the signal fitting when the “fast” thermal contribution is included into the model.

Download Full Size | PDF

At pressures of CO₂ above 3 bar (an example of a signal temporal shape at 4 bar is displayed in Fig. 3 as the leftmost curve) the slightly enlarged stationary thermal contribution to LIGs can be observed in the signal at time delays exceeding 2 µs after the laser excitation pulse. The experimental curve to the right shows the twentyfold magnified tail of the same signal. The above mentioned stationary contribution manifests itself as an excess of the signal level above the result of the calculation in frame of the one-stage energy exchange model, which is shown by the short-dash line in Fig. 3. The discrepancy can be eliminated by inserting a “fast” energy exchange contribution with the corresponding non-zero amplitude, M_f [22],

into the fitting expression. Here, Δε_f is the amount of energy exchanged with the environment in one “fast” energy exchange event, τ_R is the characteristic relaxation time, and τ_f is the LIG temporal evolution parameter, τ_f ^-1=τ_R ^-1+τ_D ^-1, where τ_D=Λ ²/(4π ² D) is the time of mass diffusion, defined by the diffusion coefficient D at a given pressure. A possible mechanism responsible for this contribution may be collisional V-T relaxation of the spatially-periodic excitation of CO₂ molecules to the lowest vibrational state (01¹0) at ν ₂=667.0 cm^-1. Since this state, in accordance with the selection rules, can not be excited via the stimulated Raman process, one can assume that the V-T relaxation follows a rapid collisional R-V energy transfer, which may be quasi-resonant for a collision of a Raman-excited CO₂ molecule with J=40 (E≈640 cm^-1) or J=42 (E≈705 cm^-1) and a non-excited one. This R-V energy transfer may also be quasi-resonant for a collision of two Raman-excited CO₂ molecules. The results of the LIG signal profile fitting using the one-stage energy exchange show that the ratio (M_i/M_e)_CO2, being approximately constant in the pressure range of 0.1–0.5 bar, decreases almost by a factor of 2 at 3–4 bar. This may also be regarded as an indication that a part of the rotational energy of Raman-excited molecules is collisionally transferred to the vibrational degree of freedom, rather than “instantaneously” released to the environment. Using the values of the V-T deactivation time for CO₂ (01¹0) molecules at room temperature τ_R·P=7.1 µs bar [34] and D=0.111 cm²/s at 1 bar one can calculate the parameter τ_f. The pressure dependence of τ_f obtained in this way is shown in Fig. 4. The values of τ_f, which have been used for the LIG signal fitting at 4 and 5 bar, are equal to 1.42 µs and 1.23 µs, respectively, and pointed in Fig. 4. Though the relative amplitude M_f/M_e. of the “fast” stationary contribution is increasing with gas pressure, the strength of this contribution is, however, relatively still too small in the range of 3–5 bar to derive τ_f values with reasonable accuracy by fitting the temporal profile of the LIG signals.

 

Fig. 4. “Fast” energy exchange in CO₂ and C₃H₈: pressure dependence of LIG temporal evolution parameter τ_f in the two-stage energy exchange model. For CO₂, the solid line (???) shows the result of calculation using the known rate of collisional V-T relaxation and the time constant of mass diffusion; the values of τ_f used for LIG signal fitting at 3 and 4 bar are indicated by dots (???). For C₃H₈, the dashed line (???) shows the result of fitting of the experimentally derived τ_f values (???).

Download Full Size | PDF

In case of C₃H₈ (see Fig. 5) a significant slowly decaying stationary contribution to the LIG signal due to “fast” energy exchange can be clearly observed above the electrostrictive and “instantaneous” energy exchange contributions at delays larger than 2 µs. This is already the case at pressures as small as 0.6 bar. In C₃H₈, the discrepancy between the result of calculation and the experiment is even more dramatic than in CO₂ (see Fig. 3), and it rapidly increases with gas pressure. The short-dash line in Fig. 5 shows the result of the calculation with the “fast” energy exchange contribution equal to zero. Note that at the same time the fitting of the signal profile in the frame of the one-stage energy exchange model satisfactorily reproduces the signal evolution at smaller delays.

Within the model of the two-stage energy exchange the LIG signal temporal shape is reasonably well fitted at various pressures (the long-dash line in Fig. 5 shows an example at 4 bar), and the values of the relative amplitude of the “fast” contribution M_f/M_e and of the parameter τ_f can be defined as a result. In C₃H₈ the relative amplitude M_f/M_e grows with gas pressure in the range of 0.6-4 bar by more than an order of magnitude steeper than in CO₂. In the same pressure range the derived values of the parameter τ_f in C₃H₈ appear to be small (about 0.30–0.36 µs) -at a rather large value of the mass diffusion time τ_D≈2.17 µs- and practically independent on pressure (see Fig. 4). The dashed line shows the fitting result of the obtained τ_f values as a function of pressure by the expression which includes the pressure-independent time τ_R≈0.40 µs of the rapid (intra-molecular) energy exchange and the mass diffusion time that is linearly increasing with pressure.

 

Fig. 5. The temporal shape of the LIG signal in C₃H₈ at 4 bar. The offset of 0.28 µs is introduced to the time scale. The short-dash line (- - -) shows the tenfold enhanced result of the calculation taking into account only the electrostrictive and “instantaneous” energy exchange contributions; the long-dash line (— — —) represents the result of the signal fitting which includes the “fast” thermal contribution.

Download Full Size | PDF

In case of C₃H₈ one can assume the following major mechanism of the “fast” energy exchange. Spatially-periodic stimulated Raman excitation of molecules occurs from the ground vibrational state to rovibrational states at ν=375 cm^-1, and, additionally, from the populated at ambient temperature low-lying vibrational states with the energies ν=216 cm^-1 and ν=268 cm^-1 [32]. Furthermore, as in CO₂ molecules, the “instantaneous” quasi-resonant collisional R-V energy transfer to the lowest vibrational states located at 216 cm^-1, 268 cm^-1, and 369 cm^-1 [35] may take place. This is also in agreement with the decrease of the ratio M_i/M_e by a factor of about 5 when pressure increases from 1 to 4 bar. This excitation of rovibrational states is followed by rapid collisionless intra-molecular V-R and/or V-V transformation of the vibrational states.

4. Conclusion

Thermal laser-induced gratings were generated via two-photon stimulated Raman excitation of a number of pure rotational (and low-lying vibrational) molecular states employing broadband radiation of a single pump laser with a few ns pulse duration. The moderate enhancement of the LIG signals due to Raman-resonant excitation, as related to the non-resonant one, has been observed in the neat molecular gases N₂, CO₂ and C₃H₈. At the initial stage of the signals’ temporal evolution the resonant (thermal) and the non-resonant (electrostrictive) contributions are comparable. The reason for this is that only a relatively small amount of non-equilibrium rotational energy is absorbed by a molecule as a result of spatially-periodic rotational Raman excitation and subsequently released as heat to the environment generating a thermal LIG. However, the stationary part of the thermal contribution may exist essentially longer than the electrostrictive LIG, having large integrated intensity. This is most probably due to a vibrational excitation of molecules which is characterized by lower energy exchange rates. Substantial increase of the thermal LIG signal strength with pressure in the range of 0.2–5 bar has been observed.

Note that the observed additional optical and/or collisional spatially-periodic excitation of vibrational degrees of freedom in gases of polyatomic molecules, which results in stronger and longer LIG signals, may provide measurements with higher sensitivity and precision, if the technique is used for gas diagnostics. To specify the presented description of the obtained LIG signals’ temporal profiles and their pressure dependence, a more detailed investigation of spectroscopic properties and excitation peculiarities of the molecules under study is planned to be accomplished.