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Abstract
Nanosecond laser-induced grating scattering/spectroscopy (LIGS) technique has been widely applied for measuring thermodynamic parameters such as temperature and pressure in gaseous and liquid media. Recently, femtosecond (fs) laser was demonstrated to induce the grating and develop the fs-LIGS technique for gas thermometry. In this work, we systematically investigated the fs-LIGS signal generation using 35 fs, 800 nm laser pulses at 1 kHz repetition rate in ambient air by varying the pump laser energies, the probe laser powers and the temporal delays between two pump laser pulses. The stability of single-shot fs-LIGS signal was studied, from which we observed that the signal intensity exhibits a significant fluctuation while the oscillation frequency shows a much better stability. A 4.5% precision of the oscillation frequency was achieved over 100 single-shot signals. By using a previously-developed empirical model, the fs-LIGS signals were fitted using nonlinear least-squares fitting method, by which crucial time constants characterizing the signal decay process were extracted and their dependences on the pump laser energy were studied. From the measured results and theoretical analysis, we found that the appropriate range of the overall pump laser energy for reliable fs-LIGS measurements is approximately located within 80 ∼ 300 μJ. The limitations on the accuracy and precision of the fs-LIGS measurements, the origin of destructive influence of plasma generation on the signal generation as well as the electrostriction contribution were also discussed. Our investigations could contribute to a better understanding of the fs-LIGS process and further applications of the technique in single-shot gas thermometry and pressure measurements in various harsh conditions.
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1. Introduction
Coherent light scattering from a grating induced by two crossing laser beams has been intensively studied, from which a nonlinear optical diagnostic technique termed as Laser-Induced Grating Scattering/Spectroscopy (LIGS) has been developed [1–3]. Fundamentally, LIGS is a four-wave mixing process, as briefly shown in Fig. 1. Two laser beams with the same polarization and wavelength intersect at a small angle (${\rm \theta}$), and interfere with each other when they overlap spatiotemporally, leading to the formation of an electric field interference pattern in the crossing region. Resonant absorption occurs when atoms or molecules absorbing the pump laser are present in the grating volume, and subsequent collisional quenching releases absorbed energy into the immediate vicinity and thus the local temperature increases at the regions of high laser intensity. The change in temperature leads to density oscillation and therefore an oscillation of the local refractive index, in which case the grating is normally termed as “thermal grating”. The thermal grating consists of two superimposed components, including a stationary pattern of density oscillation arising from the change in temperature (termed as “temperature grating”) and an acoustic wave generation due to sudden density oscillation (termed as “acoustic grating”). The induced grating pattern leads to two planar acoustic waves starting from its antinodes and propagating in opposite directions normal to the grating plane. The stationary temperature grating decays exponentially due to thermal diffusion whereas the acoustic grating also decays exponentially but as a result of viscous damping effect. Except for the temperature change that modulates the refractive index, non-resonant pump laser with sufficient intensity can also create modulations of the refractive index by the electrostrictive effect [4–7]. In an electrostriction grating, the electric field with an interference structure polarizes the dielectric medium, and the spatial inhomogeneity of the electric field causes a motion of molecules or atoms towards the regions of high laser intensity, resulting in an acoustic wave in the transverse direction. The time evolution of the density or refractive index grating can be probed by scattering a continous-wave (CW) laser beam incident to the grating plane at the first-order Bragg angle (${\rm \theta} _B$). In the direction that satisfies the phase-matching condition of four-wave mixing process, the scattering results in a coherent light, i.e. the LIGS signal, which reveals the temporal dynamics of the laser-induced refractive index grating.
Fig. 1. Schematic illustration of the principle of the LIGS process: (a) Two pump lasers intersect with a small angle $\theta$ and form a time-varying refractive index grating with a spacing of $\Lambda$; (b) A continuous-wave laser is sent to probe the grating region and interact with the refractive index grating, leading to a coherent signal scattered in the phase-matched direction of four-wave mixing process, i.e., LIGS signal.
Except for applications in molecular absorption spectroscopy [8–10] and molecular relaxation dynamics in gaseous media [11–14], LIGS has been widely applied for optical diagnostics of combustion chemical reactions and gaseous flow by measuring thermodynamic parameters such as sound speed, thermal diffusion coefficient, temperature, pressure, flow field velocity and element concentration [15–29]. Buntine et al. observed LIGS signals in four-wave mixing experiments in gas in 1994 and studied the rotational absorption spectra of OH molecule by measuring the dependence of its intensity on the wavelength of pump laser [8]. In 1998, Latzel et al. realized the potential of LIGS for temperature and pressure measurements and obtained the results with an accuracy of about 4% and 10% in high-pressure methane/air flames, respectively [19]. In 2006, Stevens et al. achieved one-dimensional temperature and pressure measurement for the first time in ${\rm NO_2}/N_2$ gas mixture by using a laser sheet [23]. In terms of practical applications, Williams et al. firstly used the LIGS technique to measure temperature with an accuracy of 0.4% in a gasoline direct-injection engine in 2014 [27]. In 2016, Sahlberg et al. accurately measured the absorption spectrum of water molecules and flame temperature using a 3-$\mu$m mid-infrared pulse laser as the pump laser [28], which triggerred the applications of mid-infrared LIGS technique for species concentration measurements and flame thermometry [30–33]. In 2017, Froster et al. demonstrated gas thermometry using the LIGS technique at rates up to 10 kHz in a static cell and 1 kHz in compressed gas flow, extending the technique to allow time-resolved measurements of gas dynamics [34]. Domenico et al. further increased the rate to 100 kHz using a burst laser in reacting flows and premixed laminar methane/air flame, showing the potential of high-repetition-rate LIGS technique for instantaneous detection of sound speed, temperature, pressure, etc [35]. In 2019, Sahlberg et al. achieved 1-3% accuracy and 4-7% single-shot precision of pressure measurements using the LIGS technique in the range of 0.5-5.0 bar in a gas cell filled with NO-${\rm N_2}$ mixture by applying an improved theoretical fitting method [36]. A more robust way to derive pressure from the LIGS signal was recently proposed by Willman et al. where the signal decay time was converted to pressure using a calibration database, based on which a measurement accuracy of about 10% in an internal combustion engine was achieved [37].
Recently, Ruchkina et al. employed 800 nm femtosecond laser as the pumping source to induce the grating and develop the so-called fs-LIGS technique [38]. Strong single-shot signal beam was generated that can be directly observed with naked eyes, showing striking contrast to nanosecond laser-based LIGS. By using an empirical model to extract temperature from single-shot data, the fs-LIGS technique was successfully applied for gas thermometry in heated nitrogen gas flows for up to 750 K with a deviation of approximately 10%. The generation of plasma with high laser pulse energy was found detrimental to the fs-LIGS signal by smearing out the oscillation and causing strong fluctuations. Very recently, we applied this technique to measure elevated gas pressures in the range from 0.2 to 3.0 bar in a static gas cell, and a quasi-linear relationship with a slope of 0.96 between the derived pressure and the value measured with a transducer was achieved, showing the feasibility of the fs-LIGS technique for gas-phase pressure measurements [39].
In this work, we report on thorough characterization of the fs-LIGS signal generated in ambient air by measuring the influence of laser pulse energy, from which the optimal energies level of both the pump and probe laser were determined. It was found that the fs-LIGS signal strength depends on the pump-laser-pulse energy in a highly-nonlinear manner whereas it generally increases with the probe laser power in a linear manner. We also investigated the dependence of the fs-LIGS signal strength on relative time delay between two pump laser pulses. The signals were fitted by using a previously-developed empirical expression and few crucial parameters, including the oscillation damping time constant $\tau _{th}$, the acoustic transit time $\tau _{tr}$ and the characteristic time constant of the fast relaxation process $\tau _{f}$, were extracted. Their dependence on the pump-laser-pulse energies were analyzed, from which the suitable energy range for reliable application of the fs-LIGS technique was eventually determined.
2. Experimental methods
The schematical illustration of the experimental setup is shown in Fig. 2(a). A commercial Ti: Sapphire femtosecond system (Astrella, Coherent Inc.) operating at a repetition rate of 1 kHz was used to deliver laser pulses with central wavelength at 800 nm, beam radius of 5 mm (1/$e^{2}$), pulse duration of 35 fs, and pulse energy up to 5.8 mJ. The laser beam was split into two arms (50:50) by a beam splitter (BS1). An optical delay line, composed of two high-reflective mirrors (M4 and M5) and a mechanical translation stage with a minimal time step of 2 fs, was installed to adjust the relative time delay between two pump laser pulses. These two parallel pump beams, separated by 18 mm, were focused by a spherical lens (L1, focal length of 500 mm, diameter of 50.8 mm) and intersect at a small angle of $\theta = 2.06^{\circ }$. By carefully adjusting the optical delay line and the mirror M3, two pump beams was managed to overlap in time and space at the intersection region to produce a grating. To acknowledge its structure and fringe spacing, a CMOS camera (MV-3000UC, spatial resolution: 2048 $\times$ 1536, pixel size: 3.2 $\mu$m $\times$ 3.2 $\mu$m) and a concave silver mirror were used to perform two-dimensional imaging of the plasma grating with relatively high pump laser energy. To calibrate the grating spacing, we placed a thin copper wire with a known diameter (1 mm) in the position of grating and recorded its image by the CMOS camera, by which the calibration factor of the imaging resolution can be obtained.
Fig. 2. (a) Schematic illustration of the experiment setup. M1-M10: high-reflection mirrors, BS: beam splitter, DM: dichroic mirror, L1-L3: spherical lenses, PMT: photomultiplier tube. (b)-(d) Two-dimensional images of the focal volumes in full scale with two pump laser beams with total energy of 670 $\mu$J, only the probe beam with power of 250 mW and all three beams, respectively.
A CW laser system (Verdi G8, Coherent Inc.) provides 532 nm laser as the probe, which was combined with two pump laser beams through a dichroic mirror (DM) and then focused by L1 into the grating at first-order Bragg angle ($\phi$). Full-scale imagings of the focal volumes produced respectively by two pump laser beams, only the probe laser beam and all three beams were conducted, as shown in Fig. 2(b)-(d), which helps to monitor the intersecting situation of the probe into the grating. Part of the probe laser was scattered by the refractive index grating, i.e. fs-LIGS signal light, whose scattering angle was determined by the phase-matching condition. We placed another lens (L2, focal length of 600 mm, diameter of 50 mm) at the position having an identical distance to the focal point to guarantee the four beams are parallel. The light trap blocked both the pump light and probe light so that only the fs-LIGS signal light passed through, which was then detected by a monochromator (WDG30-z) coupled with a photomultiplier tube (PMT, R943-02). The usage of the light traps could not remove all the stray light, so a relatively long (about 1 m) collection path was arranged, and the stray light was largely removed after several reflections. The central wavelength of the monochromator was set to 532 nm wavelength for all experiments to filter out light with any other wavelengths. A narrow-band filter (Semrock, $\sim$15 nm bandwidth) with optimal transmission at 532 nm wavelength was placed before the monochromator to further purify the incoming signal. The voltage applied for the photomultiplier tube is adjustable from 0 to 2000 Volts. Since the PMT is very sensitive, it was completely covered with black cloth. The fs-LIGS signal was finally displayed on an oscilloscope (Tektronix DPO 4054) with a bandwidth of 500 MHz and maximum sample rate of 2.5 GS/s.
In the laboratory, the room temperature could vary from $17^{\circ }$C to $22^{\circ }$C. Because of the high altitude of $\sim$1600 metres in Lanzhou city, air pressure keeps at 85.12 kPa. The air humidity is normally 30%.
3. Experimental results and discussions
3.1 Characterization of the fs-LIGS signal
The picture displayed in Fig. 3(a) exhibits the detailed luminescence image of plasma grating created with pump laser pulse energy of 670 $\mu$J in the air, in which about 4 fringes can be observed. The grating spacing was determined to be 22.40 $\mu$m. It has to be mentioned that the luminescence imaging of the grating can no longer be acquired if the pump laser pulse energy is below approximately 500 $\mu$J. Since the grating spacing is only determined by the crossing angle of two pump laser beams and the pump laser wavelength, it is reasonable to assume that the grating spacing keeps the same value for low pump laser energies. Based on this assumption, we can further calculate the grating spacing for varying laser energies with $\Lambda = \lambda _{pump} / \big (2 sin(\theta / 2 )\big )$ where $\lambda _{pump} = 800$ nm is the pump laser wavelength and $\theta = 2.06^{\circ }$. The calculated result is 22.23 $\mu$m, which is in good agreement with the measured value of 22.40 $\mu$m.
Fig. 3. (a) Close-up two-dimensional luminescence imaging of the plasma grating, where the measured fringe spacing $\Lambda _m$ was denoted. (b) Photo of the observed fs-LIGS signal beam (circled with the red dashed line) taken with a mobile phone.
The incidence of the CW probe laser into the grating at first-order Bragg angle ${\rm \theta} _B = \lambda _{probe} / (2 \Lambda ) = 0.61^{\circ }$ results in its own scattering in the phase-matched direction, and the scattered light is namely the so-called fs-LIGS signal. Compared to nanosecond laser pumping, as an obvious light beam, the fs-LIGS signal can be directly seen with the naked eye. Figure 3(b) shows a selected photo of the signal beam, which was taken by a mobile phone on a white board. From the photo, one can also see sideward the strong probe laser beam (partially blocked) and downward the 800 nm pump laser beam. The spatial positions of each beam could be better understood by referring to the setup in Fig. 2 and Fig. 1(a) in [38]. The signal beam appears elliptical and vertically stretched.
Fig. 4. (a)-(b) Selected single-shot fs-LIGS signal and corresponding FFT frequency result obtained with the pump laser energy of 482 $\mu$J and the probe laser power of 226 mW, respectively. (c)-(d) Instabilities of single-shot signal intensity and its oscillation frequency over 100 shots. (e) Instability of the pump laser pulse energy and probe laser power over 140 seconds.
Single-shot fs-LIGS signals were routinely generated, and a typical signal is shown in Fig. 4(a). The experiments were conducted with the pump laser pulse energy of $\sim$482 $\mu$J and the probe laser power of 226 mW. One can see that the signal exhibits a fast buildup and much slow decay superimposed with only few periodic oscillations. The single-shot signal shows a quite good signal-to-noise ratio (SNR), which is approximately 83. With a view to practical applications of the fs-LIGS technique for analyzing and time-resolving unsteady phenomena, we recorded single-shot signals over 100 shots and studied their instability in terms of variation percentage relative to the mean value. As shown in Fig. 4(c), the single-shot signal intensity exhibits a maximal instability of approximately 75%, and its standard deviations was determined as 30.7%. The corresponding Fast Fourier Transform (FFT) result of the signal in (a), which was processed within the time range from 0 to 500 ns (10% of the signal maximum), is shown in Fig. 4(b). Unexpectedly, the oscillation frequency exhibits a much better stability, where the maximal instability is less than 10%. The averaged FFT frequency was determined as 15.09 MHz while the standard deviation was calculated as 0.63 MHz, 4.5% of the average value. We also recorded the laser output instabilities of the pump and probe lasers within about 2 minutes, both of which shows less than 1% output variation as displayed in Fig. 4(e). Therefore, the laser instability should be a minor factor for causing the drastic variation of the signal intensity. Except for negligible air flow due to the operation of ventilators in the laboratory, the nonlinear nature of the fs-LIGS process could be the major contributor to the intensity variation. Nevertheless, the relatively small change of the oscillation frequency is critical, which would be beneficial for reliable measurements such as single-shot gas thermometry.
Figure 5 shows the averaged fs-LIGS signal and corresponding FFT result. Compared to single-shot signal, the averaged signal has a much higher SNR of $\sim$500. In addition, the peaks around 200 ns becomes more obvious, which contributes to a better determination of the FFT frequency peak with higher amplitude, as shown in Fig. 5(b). The optimal oscillation frequency is 15.27 MHz, which is in consistent with the value obtained from single-shot data. This consistence approximately meets the requirement that the average of the FFT peaks of the single-shot signals equals to the FFT peak of the averaged signal.
Further on, we could estimate the air temperature based on the experimental results. Given the measured grating spacing $\Lambda _{m} = 22.40 \ \mu$m and the oscillation frequency of the averaged fs-LIGS signal $f_{osc} = 15.27$ MHz, the sound speed can be then calculated as $\upsilon _{air} = \Lambda _{m} \times f_{osc} = 342.05$ m/s. The approximate equation for calculating the sound speed in air can be expressed as $\upsilon _{air} = 331.5024 + 0.603055 \times T_{lab} - 0.000528 \times T_{lab}^{2}$ by neglecting minor factors such as humidity and pressure, where $T_{lab}$ denotes the air temperature in the laboratory [40]. From this equation, we can calculate the air temperature as $17.7^{\circ }$C. It is slightly lower than the temperature of $20^{\circ }$C displayed on a thermometer, leading to an accuracy of approximately 13% for the temperature measurement in ambient air.
Essentially, the accuracy and precision is limited by the finite number of periodic oscillations so that the transformed frequency has a low amplitude. The diameter of the focused beam at the focus of the lens is $d = 4 \lambda f/(\pi D) \approx 51 \ \mu$m, where the focal length $f=500$ mm and the unfocused beam diameter $D \approx$ 10 mm. Given the grating spacing of 22.40 $\mu$m, only two or three fringes as well as the number of fs-LIGS signal oscillation can be expected. In order to perform more accurate fs-LIGS measurements, the fringe number needs to be significantly increased. The effective approach to do it is then two folds: (1) to increase $d$ by reducing the diameter of unfocused beam while keeping the grating spacing; (2) to reduce the grating spacing with larger crossing angle between two pump laser beams. For the second approach, however, the reducing contrast of the grating fringe visibility could be a problem.
The results shown above were obtained when two pump laser pulses were adjusted to reach spatiotemporal overlap. Optimal overlap was determined by the strongest dynamic interference of the supercontinuums generated by the pump lasers when they experience filamentation. The interference pattern is shown in Fig. 6(a), where two brightest spots are filament-forming pump lasers surrounded by their self-generated supercontinuum light. Optimal overlap is then regarded as the time zero of relative delays. Figure 6(b) shows the dependence of the first-peak intensity of the fs-LIGS signal on the relative time delay of two pump lasers. In this measurement, in order to avoid the influence of plasma generation, the total pump laser energy and the probe laser power were kept at 150 $\mu$J and 250 mW, respectively. Obviously, the signal intensity quickly decreases as the delay is adjusted away from the time zero, and the signal completely disappears when the delay reaches to approximately 75 fs. The full width at half maximum (FWHM) of the dependence curve with Gaussian fit is then determined as 50.1$\pm$1.8 fs, which is in agreement with the convolution time of two 35 fs laser pulses, i.e. 49.5 fs.
Fig. 6. (a) Photo of the interference pattern of the supercontinuum generated by the pump lasers when they experience filamentation. (b) The fs-LIGS signal intensity as a function of the relative delay between two pump laser pulses. Each point was averaged for five measurements.
In the experiments, there is no apparent electrostriction-induced fs-LIGS signal observed, which is expected to appear at the frequency twice that of thermal-grating-induced one [4–7]. Normally, electrostrictive LIGS signals require laser energies one order of magnitude higher than that needed for thermal ones [41]. At high pump laser energies, one negative factor for studying the electrostriction-induced LIGS is the plasma formation, but this laser-induced plasma is weakly-ionized in which only about 1% molecules are ionized by the fs laser pulses [42]. Therefore, the electrostrictive effect could still play an important role with high energy pump lasers, and further investigations on its contribution is needed.
3.2 Influence of laser pulse energy on fs-LIGS measurements
In order to study the influence of pump laser energy on the fs-LIGS signal generation and determine optimal experimental conditions for further applications, we recorded a set of fs-LIGS signals by varying the pump laser pulse energy from 0.029 to 2.225 mJ, as displayed in Fig. 7. The experiments were conducted in an environment with the recorded room temperature of 22 degrees. The graphs in left column show the results obtained with low energies from 29 to 136 ${\rm \mu} J$ whereas the graph in middle column shows that obtained with energies from 256 to 1068 ${\rm \mu} J$. From the results of low energies, we can see that the signal intensity drastically increases with the pump laser energy. The minimum laser energy that can generate detectable fs-LIGS signal was about 29 ${\rm \mu} J$, as shown by the log-scale plot in Fig. 7(b). Figure 7(c) is the corresponding FFT results.
Fig. 7. The fs-LIGS signals, signals in log scale and corresponding FFT results recorded for low pump laser pulse energy from 29 to 136 ${\rm \mu} J$ (a)-(c), for medium energy from 256 to 1068 ${\rm \mu} J$ (d)-(f), and for high energy from 1.229 to 2.225 mJ (g)-(i), respectively.
With the increase of the pump laser energy from 256 to 848 ${\rm \mu} J$, the fs-LIGS signal intensity continues to grow, as shown in Fig. 7(d). However, for the result obtained with 1068 ${\rm \mu} J$ shows a weaker signal strength accompanied with distortions of the oscillation structure if compared to that of 848 ${\rm \mu} J$. Note that the corresponding FFT frequency of 1068 ${\rm \mu} J$ result shows a significant shift compared to other frequencies in Fig. 7(f). This oscillation structure distortion is caused by the plasma generation in the probe volume when the laser intensity is strong enough to ionize the air molecules. With higher energies from 1.229 to 2.225 mJ, as shown in Fig. 7(g), the plasma generation completely smears out the oscillation structures and also brings drastic fluctuation to the fs-LIGS signal, as we reported previously [38].
To have a deeper understanding on how the plasma formation is affecting the fs-LIGS signal, we have looked into the literatures [43–46] and acknowledged that the lifetime of femtosecond laser-induced plasma grating is typically on the order of 100 ps $\sim$ few ns. The decay of plasma is a process in which ions and electrons recombine. As a result, heat will be released so that the thermal grating is still in the process of establishing when the plasma has completely decayed. Naturally, the plasma grating lifetime is much shorter than the lifetime of the LIGS signal. Due to the unstable nature of the laser-induced plasma, in which initial tiny fluctuation will be amplified during the plasma generation, the heat release following recombination inherits this instability and thus leads to unrepeatable single-shot fs-LIGS signal. The rapid heat release also might smear out the grating structure, resulting in a LIGS signal without oscillation patterns as we see in our results. The light scattering by laser-induced plasma grating is no longer resolvable within the traditional thermal-grating-based LIGS theory. The plasma formation, non-equilibrium thermodynamic equations, electron-ion dissociative recombination, and three-body recombination, etc. have to be taken into account to better understand the plasma-grating-mediated fs-LIGS results. Nevertheless, it would be an interesting topic to pursue in the near future since it could be useful for plasma diagnostics.
To obtain the variation of the fs-LIGS signal strength with the pump laser energy, we selected the intensities of three peaks to analyze, and the results are shown in Fig. 8(a). P1, P2, P3 successively represent the three peaks and the results are plotted in logarithmic coordinate. Firstly, all three peak intensities show similar trend that they rapidly increase with the pump laser pulse energy when it is below $\sim$500 ${\rm \mu} J$. Power function fitting to P1 suggests an exponent of $6.38 \pm 0.39$, implying a high nonlinearity of the scattering process. Secondly, the peak intensities start to be off the power dependence curve as the pump laser pulse energy increases beyond 500 ${\rm \mu} J$ and eventually saturate. Thirdly, when the pump laser energy is higher than 1.229 mJ, the fs-LIGS signal becomes so unstable and unmodulated that the peak positions can be no longer determined, because of which the energy range was limited up to 1.229 mJ.
Fig. 8. (a) The peak intensities of P1, P2 and P3 as a function of total pump laser energy. The dashed line represents power function fitting to the data. (b) The first peak intensity and integrated area of the fs-LIGS signal as a function of the probe laser power. The dashed line represents linear fitting to the data.
We also investigated the dependence of fs-LIGS signal on the CW probe laser power. Figure 8(b) shows the first-peak intensity and energy in terms of integrated area of the fs-LIGS signal as a function of the probe laser power. The pump laser energy was fixed at $\sim$100 ${\rm \mu} J$. Both the intensity and integrated area firstly increase linearly with the probe laser power and saturate when the power reaches to 1.5 W, and then decreases. It was noticed that there is no shift of the peak position or change of the temporal profile of the signal when the probe laser power increases. Hence, the experiment is best carried out under the condition when the probe laser power is lower than 1.5 W. However, higher probe laser powers cause stronger stray light spreading around the signal beam that brings difficulties for detection. In addition, one can note that there appears two dips in the curve that position around 500 mW and 800 mW, which is not understandable at this moment and requires further investigations. In the experiments, the probe laser power of 250 mW was applied to avoid the dips and mitigate the stray light.
To determine the pump laser energy for feasible applications, the FFT frequency and amplitude of the fs-LIGS signals for different pump laser energies were plotted in Fig. 9. It can be noticed that the frequency appears stable when the total pump laser pulse energy is below 600 ${\rm \mu} J$. The optimal pump laser pulse energy with highest FFT amplitude is approximately 150 ${\rm \mu} J$ for the current configuration of experiments. For pump laser energies exceeding 600 ${\rm \mu} J$, the FFT frequency drops with higher laser energies while its amplitude gets close to zero, suggesting disappearance of the periodic oscillations. The threshold of 600 ${\rm \mu} J$ is approximately in consistent with the occurrence of weakly-ionized filament plasma in which case shrill sounds can be heard from the focal volume. From the robustness perspective of the fs-LIGS technique for practical diagnostics, the small interval of the pump laser energy, within which the signal oscillation frequency stays approximately constant, is an obvious disadvantage. However, it also suggests a potential advantage. Since fs laser pulses with only few hundreds of microjoule energy are capable to generate strong LIGS signals with stable oscillation frequency, high-repetition-rate fs lasers could be applied for time-resolved combustion and reacting flow diagnostics.
Fig. 9. FFT frequency and amplitude of the fs-LIGS signal as a function of total pump laser pulse energy.
3.3 Theoretical analysis of the fs-LIGS signal
In order to extract more information from the measured fs-LIGS signal and focus on the impact of pump laser energy on the signal features, we fitted the temporal evolution of the fs-LIGS signal by using the empirical expression developed by Kozlov et al. [47,48], and the procedure is similar to our previous work [38]. From two-dimensional image of the grating in Fig. 3(a), we could estimate the value of $\tau _{tr}$ in our experiments. Given that there appears 4 fringes and the fringe spacing was measured as 22.4 $\mu$m, the whole length of the transverse dimension of the grating is around 67.2 $\mu$m. Considering that the sound propagates over half of the transverse length, i.e., 33.6 $\mu$m, one can obtain that $\tau _{tr} \sim 98$ ns. It is considerably smaller than $\tau _a$ which is on the order of $\mu$s [47]. Therefore, the contribution of acoustic damping was neglected in the expression by simplifying the decay factor from $\text {exp}(-(t/\tau _{tr})^{2}-t/\tau _a)$ to $\text {exp}(-(t/\tau _{tr})^{2})$.
Figure 10(a) shows a typical fs-LIGS signal generated with pump-laser-pulse energy of 136 ${\rm \mu} J$ and corresponding fit curve with the empirical expression using nonlinear least-squares fitting method and the Levenberg-Marquardt algorithm. In the fitting process, all three contributions to the generation of fs-LIGS signal, i.e., the instantaneous and fast energy distribution as well as the electrostriction effect, were considered. The initial values for the fitting parameters were estimated from the raw data. The initial value for $f_{osc}$ was set as the FFT result of the signal, i.e., $\sim$16.10 MHz. From the buildup time of the first peak, the initial value for $\tau _f$ can be estimated as 15 ns. From the decay of overall signal without considering the oscillation, we estimate $\tau _{th} \approx 250$ ns as the signal has decreased to only 1% of its maxima. The initial values for $S_0$, $M_i$, $M_f$ and $M_e$ were set to unity. From Fig. 10(a), one can see a fairly good fit except for slight mismatch of a weak peak appearing in the range from 200 to 250 ns. Nevertheless, the fitting process results in a residual less than 0.02 and gives a goodness of fit in terms of adjusted R-square as 0.998.
Fig. 10. Fs-LIGS signal obtained with laser pulse energy of (a) 136 $\mu$J and (b) 626 $\mu$J, and corresponding fitted curves. Low panels show the difference between measured fs-LIGS signal and the fitted one.
We have analyzed the fs-LIGS signals obtained with different pump laser energies by above fitting process, and extracted few temporal parameters including $\tau _{th}$, $\tau _{tr}$ and $\tau _f$. Figure 11 shows their variation as the pump laser energy increases. The fs-LIGS signal generated with the energy above 700 $\mu$J cannot be fitted since the temporal profile of the signal has been irregularly distorted due to the plasma generation. In fact, the goodness of fit of the signal has already become worse once laser filamentation phenomena start to happen, as shown by the result of 626 $\mu$J in Fig. 10(b). On the other hand, in the lower limit of energy less than 46 $\mu$J, the signal appears weak and noisy, and it also decays within the oscillation period, which inhibits a reasonable fit. From Fig. 11(a)-(c), a general trend can be observed that all the extracted parameters becomes smaller with the increase of pump laser energy. However, in the energy region of approximately 80$\sim$300 $\mu$J, $\tau _{th}$ and $\tau _{tr}$ almost keep constant at values of 280 ns and 102 ns, respectively. These two values are quite close to the initial values we set for $\tau _{th}$ and $\tau _{tr}$ before fitting, i.e., 250 ns and 98 ns. This region, which we hereby name it as the trusted region, is consistent with that in Fig. 9 where the FFT frequencies of the fs-LIGS signals have an amplitude larger than $\sim$0.02, half the maxima. Since $\tau _{th}$ is related to the thermal diffusivity $\chi$ by [47]
where $\chi$ depends on local temperature $T_{local}$, we can conclude that the pump laser does not introduce additional thermalization to the probed gas and the grating structure remains unchanged in the trusted region. An important implication of this result is that it provides us a selection region for choosing the pump laser energy for further fs-LIGS applications for either gas thermometry or pressure measurements.Fig. 11. Extracted fitting parameters including (a) $\tau _{th}$, (b) $\tau _{tr}$ and (c) $\tau _f$ as a function of the pump laser energy up to 700 $\mu$J. (d) The rising edge details of the fs-LIGS signals obtained with pump laser energies of 136, 256 and 626 $\mu$J, respectively.
The decrease of $\tau _{tr}$ beyond the trusted region could be attributed to femtosecond laser filamentation, which originates from the interplay between optical Kerr effect and plasma formation when the peak power of laser pulse exceeds a critical value $P_{cr} \approx 5.5 \times 10^{13} \ W/cm^{2}$ [42,49]. As a result, the laser focus appears slightly ahead of the geometrical focus and the laser intensity inside the formed filamentary column is clamped at $P_{cr}$ while its diameter is approximately fixed at 130 $\mu$m in air [50]. Given the pulse duration of $\tau _{l}=35$ fs and the filament diameter of $d=130 \ \mu$m, we could estimate the laser pulse energy required for forming a filament in our experiments by $E=\pi (d/2)^{2} \tau _{l} P_{cr} \approx 255 \ \mu$J. Thus, the total energy required for filamentation is approximately 510 $\mu$J by considering two pump lasers, which is consistent with the observations that conical emission, a typical signature of filamentation, takes place when the laser energy exceeds about 500 $\mu$J. The occurrence of laser filamentation leads to slight displacement of the focal point towards the focusing lens, which is approximately proportional to $1/ \sqrt {P_{laser}/P_{cr}}$ where $P_{laser}$ is the laser power [51]. Correspondingly, the crossing angle $\theta$ of two pump lasers slightly increases. According to the formula for calculating the grating spacing, i.e., $\Lambda = \lambda _{pump} / \big (2 sin(\theta / 2 )\big )$, $\Lambda$ would slightly decreases, thus leading to a decrease of ${\rm \tau} _{tr}$ in Fig. 11(b). Furthermore, this slight decrease of $\Lambda$ would directly explain the deduction of the oscillation damping constant $\tau _{th}$ since $\tau _{th} \propto \Lambda ^{2}$. However, the possible slight variation of the grating spacing using high pump laser energy could bring about additional uncertainties to the fs-LIGS measurements and thus undermine its feasibility.
Figure 11(c) shows that $\tau _{f}$ monotonically decreases with the increasing pump laser energy. Since $\tau _{f}$ defines the characteristic time constant of the fast relaxation process, a more rapid relaxation process at higher energies would in principle lead to a faster buildup of the thermal grating and therefore a quicker rising edge of the fs-LIGS signal. This was indeed observed in the experiments. Figure 11(d) shows the rising edges of the fs-LIGS signals obtained with the pump laser energies of 136, 256 and 626 $\mu$J, respectively. One can see the half-width at half maximum (HWHM) of the rising edges becomes longer with higher energies, which is in agreement with the decrease of $\tau _{f}$.
4. Conclusion
To conclude, we have systematically investigated the fs-LIGS technique using 35 fs infrared laser pulse to form the grating in ambient air. Single-shot fs-LIGS signals were presented and its stability in terms of the signal intensity and the oscillation frequency over 100 shots was studied, which shows that the intensity exhibits significant fluctuation while the oscillation frequency has a much stability with only 4.5% variation with repsect to the mean value. The signal generation was also characterized by varying the pump laser energies, the probe laser powers and the temporal delays between two pump laser pulses. The minimum energy required to generate detectable fs-LIGS signal was 29 $\mu$J whereas the optimal energy was determined to be around 150 $\mu$J under current experimental conditions. It was found that the signal strength rapidly increases with the pump laser energy when it is below $\sim 600 \ \mu$J, showing a very strong nonlinearity.
The fs-LIGS signals were fitted using a previously-developed empirical model. From the fitting, crucial time constants including $\tau _{th}$, $\tau _{tr}$ and $\tau _f$ that respectively characterizes the density oscillation damping due to thermal diffusion, the acoustic transmission over the grating and the fast relaxation process, were extracted. Their dependences on the pump laser energy were obtained and discussed. From the measured results and theoretical analysis, the appropriate region of the pump laser energy for reliable fs-LIGS measurements was approximately located within 80 $\sim$ 300 $\mu$J.
Due to its strong single-shot signal strength with rather low pump laser energy required, the fs-LIGS technique holds very promising potential for gas thermometry and pressure measurement in harsh environments such as gas turbine and aero-engine combustor. We believe that our investigations could contribute to a better understanding of the fs-LIGS process and further applications of the technique in thermometry and pressure measurements in various harsh conditions.
Funding
National Natural Science Foundation of China (12004147, 12027809, U1932133).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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