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Abstract
Temporal and spatial evolution of temperature in femtosecond laser filamentation is investigated using planar Rayleigh scattering combined with optical flow algorithm, the corresponding mechanism is analyzed. The temperature increases sharply with a characteristic time of 4.53μs and reach a maximum value of 418 K within 1∼10μs, then decreases slowly to around 300 K with a characteristic time of 136μs. While the temperature first diffuses rapidly in the radial direction and then diffuses very slowly, an obvious step is observed around 2μs. The mechanism of heat transfer is the result of energy exchange between electron and heavy particles and heat conduction. Within 1 ns to 10μs, molecules obtain energy continuously due to collision with electrons, which is much larger than the energy loss due to thermal conduction, leading to rise of gas temperature and the high-speed movement of the filament edges. After 10μs, thermal conduction becomes the dominant factor, resulting gas temperature decreasing and slower movement of the filament edges.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Femtosecond laser filamentation is a highly nonlinear process that occurs as the self-focusing from the optical Kerr effect is balanced by diffraction, plasma defocusing, and other nonlinear mechanisms [1]. Filamentation in air has especially attracted considerable attention for applications in remote sensing [1], active flow control [2,3], velocimetry of gas flow [4–6] and combustion diagnosis [7]. In particular, femtosecond laser tagging based seedless velocimetry and corresponding variants have been applied in supersonic flow [8,9] and boundary-layer [10] with kilohertz even megahertz repetition rates. Usually, it is unavoidable that a femtosecond laser has sufficient high intensity (typical with ∼1014 W/cm2) will ionize the gas due to multi-photon ionization (MPI) and produce local gas heating in focused region [11]. Furthermore, with fast gas heating, the acoustic wave, which is generated by plasma, may interact with flow structures and produce an effect on flows, especially in turbulent. In the application of active flow control, interaction process between filament and gas flow should be strengthened for high efficiency. In the application of noncontact measurement, interference from filament to gas flow should be eliminated to ensure measurement accuracy [12–14]. A practicable method is to increase the energy deposition efficiency by using the multiphoton-resonance enhancement of rare gas, thereby reducing the required input excitation-pulse energies [15]. When a femtosecond filament pulse train is suddenly initiated in a gas, the local density will be reduced, leading to a beam deflection [16]. Besides, the heating widens the width of femtosecond laser filament, and decrease the measurement accuracy [17,18]. Therefore, it is important to characterize the gas heating and heat transfer processes and to investigate the corresponding mechanism in femtosecond laser filamentation.
Up to now, there are only a few related studies about heating process in femtosecond laser filamentation and corresponding mechanism is lacking. C. M. Limbach, et al. [19,20] measured molecular dissociation and temperature in the afterglow (from 1μs to 50 μs) of a femtosecond laser filament in nitrogen. The generation of shock waves and a high-temperature region, with a corresponding temperature increase of 280 K at least, were observed. The results show the temperature decrease with delay time within 1 μs to 50 μs. M. R. New-Tolley, et al. [18] explored the effect of localized thermal perturbations on laminar flow fields by using a hot channel model, and there was an artificial thickening in the shear layer. A. Ryabtsev, et al. [21] investigated the fluid flow caused by femtosecond filaments in dry air, a similar result was obtained by a hot resistance wire. The results show that the heat deposition play an important role in flow disturbance and should be considered in femtosecond laser filamentation. N. Jiang, et al. [22] explored the particularly valuable of 100 kHz PLEET (Picosecond laser electronic-excitation tagging, which is a variant of femtosecond laser electronic-excitation tagging) for high-speed velocimetry of turbulence, the results show a local flow temperature of 550 K in focal region, and a 0.08 mm/s of the induced buoyancy velocity bias was estimated.
In this work, the temporal character of heating in femtosecond laser filamentation is investigated using planar Rayleigh scattering (PRS), the plasma emission spectra are used to obtain the plasma temperatures. The effect of plasma on scattering cross-section in focus region due to MPI is estimated by using a two-temperature model. Heat transfer process is investigated using time-resolved 2D (two dimension) temperature distribution which is derived from PRS, and corresponding mechanism is explained by using a two-temperature heat conduction model. Finally, Lucas-Kanade optical flow algorithm is employed to investigate the motion of the local high temperature region at the focused region due to the heat transfer, and the heat transfer mechanisms are analyzed.
2. Experiment
The experimental setup is schematically shown in Fig. 1. An amplified Ti:Sapphire laser system (with 800 nm center wavelength, 45 fs pulse duration) is used to generate filament in air. The initial beam diameter is about 10 mm. The pulse energy is tuned by polarizers pair and measured to be 2.8 mJ, which is focused with a f=500 mm focal length convex lens. The width (i.e. full width at half height) of filament is 400 μm, which is obtained by fitting the spatial profiles of emission with a Gaussian model. Rayleigh scattering probe laser is a frequency doubled Nd:YAG laser (with 532 nm center wavelength, 10 ns pulse duration, less than 1 cm-1 of FWHM). The pulse energy is tuned by polarizers pair and measured to be 180 mJ, which is reshaped to a sheet light by combined with planoconvex lens and cylindrical lens. The thickness of the sheet light is about 1 mm to ensure the sheet light can cover the filament. The Rayleigh scattering signal is acquired by a gated and intensified charge-coupled device (ICCD) camera with a 50 mm zoom lens, leading to an image scale of 123.5 μm/pixel. The depth of field of the imager is greater than the thickness of Rayleigh scattering sheet to ensure the Rayleigh scattering signal from filament can be collected completely. All Rayleigh scattering measurements are acquired with a 532 nm bandpass filter and a 100 ns gate, producing complete suppression of plasma emission during Rayleigh measurements. The plasma emission is detected and recorded by a spectrometer system (equipped with a 300 grooves/mm grating and another ICCD). For plasma spectrum measurements, the slit width of the spectrometer is set to 100 μm. The wavelength of the spectrometer is calibrated by a mercury-argon lamp, and its spectral response is calibrated by a deuterium-halogen lamp. Instrumental line-broadening is measured by the frequency doubled Nd:YAG laser. With 100 μm slit width, the instrumental broadening shows a Gaussian profile with a FWHM of 1.05 nm, and its effect on the measured spectrum is considered in the data processing. The delay between the femtosecond pulse and diagnostics (probe laser, ICCD camera and spectrometer system) is controlled by a pulse delay generator, which is varied from 0 ns to 800 μs. The femtosecond pulse and diagnostics (probe laser, ICCD camera and spectrometer system) are operated at 10 Hz repetition rate.
Fig. 1. Schematic illustration of the experimental setup. P: polarizer, L: lens, CL: cylindrical lens, BP: bandpass filter.
3. Results and analysis
3.1 Rayleigh scattering and emission spectra
Figure 2 shows the Rayleigh scattering with femtosecond laser only, nanosecond laser only, both femtosecond and nanosecond laser respectively. As shown in Fig. 2(a), there is a uniform signal when only the femtosecond laser is focused into the air, and the Rayleigh scattering from femtosecond laser, which is usually used as a multiparameter measurement method and it will interfere the Planar Rayleigh scattering measurement results of filament, is effectively suppressed [23,24]. As shown in Fig. 2(b) and Fig. 2(c), the Rayleigh scattering obviously changes in filament when femtosecond laser focus to air. The results show that the plasma emission is effectively suppressed.
Fig. 2. Rayleigh scattering intensity with (a) femtosecond laser only, (b) nanosecond laser only, (c) both femtosecond and nanosecond laser.
The measured Rayleigh scattering signal S, which is related to gas, laser and optical parameters, is expressed as follows [25]:
The composition of the focusing region is more complex due to the dissociation and ionization, and the differential cross section $\bar{\sigma }(T)$ depends on the species concentrations. In order to evaluate the influence of the composition of the focusing region on the differential cross section $\bar{\sigma }(T)$, the emission spectra of focusing region are used to obtain the plasma temperature by fitting the second positive system and first negative system of nitrogen [27–29]. Figure 3 shows a spectrum attained from 200 nm to 850 nm for 1 ns delay, the bands from 250 nm to 500 nm are due to transitions of the N2 2nd positive system ($C{}^3{\Pi _u} - B{}^3{\Pi _g}$) and N2+ 1st negative system ($B{}^\textrm{2}\Sigma _u^ +{-} X{}^\textrm{2}\Sigma _g^ +$), the bands from 600 nm to 800 nm are due to corresponding second order diffraction, and the maximum peak at 800 nm comes from the femtosecond laser. The inset shows the spectrum from 365 nm to 397 nm and corresponding fitting result, the maximum peak at 391.2 nm is due to the 0-0 vibrational transition of the N2+ 1st negative system, the smaller peaks at 365 to 382 nm are due to the 0-2 vibrational transition of the N2 2nd positive system. The fitting result shows a vibrational temperature of 5010 K, a rotational temperature of 712 K (which is close to the local flow temperature excited by PLEET in Ref. [22]), N2+/N2 = 7.15×10−8, and indicate the plasma is in thermal non-equilibrium. Moreover, as shown in Fig. 3, there are no atomic lines (usually are OI 777.3 nm, OI 777.417 nm, OI 777.539 nm, NI 742.364 nm, NI 744.229 nm and NI 746.831 nm) from O or/and N, while atomic lines in LTE (local thermal equilibrium) air plasma are usually significant.
The species concentrations are calculated from two-temperature model [30] based on the fitted temperature. In the calculation, it is assumed that air is composed of a total of 11 different species (i.e. N2, N2+, N, N+, O2, O2+, O2−, O, O+, NO, e−) within the temperature range. As listed in Table 1, eight different reactions, which these species are generated, are considered in the calculation, and associated non-equilibrium Saha/ Guldberg-Waage equations are expressed as follows:
Table 1. Reactions considered in the calculation and corresponding reaction energies(eV).
In this work, the molecular/atom constants involved in the calculation are derived from Ref. [30–33]. The calculated concentration, corresponding differential cross section and temperature dependence are tabulated in Table 2.
The column 4 represents the relative deviation of the differential cross section caused by every 1000 K increase in temperature [11]. The results show that O2 and N2 are still the dominant components, the sum of mole fraction of other components is about 4.0×10−5 in the non-equilibrium plasma. Compared with experimental value of N2+/N2 = 7.15×10−8, there is a lower calculated value of N2+/N2 = 1×10−8. There are many factors may cause the deviation between the experimental and calculated N2+/N2, such as the spatial average effect and the limitation of spectral resolution will cause measurement error, and the incompleteness of the two-temperature model will cause calculation error. According to the scattering cross-section and corresponding temperature dependence data for molecular, atomic and ionic species (Table 2.1 and Table 2.4 of Ref. [11]), the difference of scattering cross-section caused by the non-equilibrium plasma (with Tvib = 5010 K and Trot = 712 K) is less than 2%. Therefore, the influence of the non-equilibrium plasma on the scattering cross section $\bar{\sigma }(T)$ can be neglected (i.e. $\bar{\sigma }(T)\textrm{/}\bar{\sigma }({T_\textrm{0}}) \approx \textrm{1}$), which there is a similar conclusion can also be inferred from Ref. [28], while is a $\bar{\sigma }(T)\textrm{/}\bar{\sigma }({T_\textrm{0}}) \approx \textrm{0}\textrm{.78}$ in LTE plasma with T = 5010 K [11].
Table 2. The calculated concentration, corresponding differential cross section and temperature dependence
3.2 Temporal characterization of heating
Figure 4 shows the temperatures derived from Rayleigh scattering of femtosecond laser filament with different delay time. The temperature increases firstly and then decreases. The width of filament increases with delay time and finally splits into two. According to the temperature distributions after 100 μs, there is a lower temperature region appear in the center of tagging line, which is not caused by the measurement or calculation error, it is real exist. Actually, if the Coulomb field modification is considered, the pressure will be reduced by ΔP, which is resulted from the charged particles interactions and can be expressed as follow [34]:
Where, ε0 is vacuum permittivity, λd is the Debye length, Zi and ni are the charge number and number density of species i respectively. The pressure correction is proportional to the number density of charged particles. According to the Dalton’s law taking into account the Coulomb field modification (i.e. $P + \Delta P = k{T_e}{n_e} + k{T_h}\sum\limits_{i \ne e} {{n_i}}$), the pressure correction ΔP will cause an increase of observed total number density, which will cause a lower temperature when using the ideal equation of state and Eq. (2) to calculate gas temperature of filament, i.e. ${T_{cal}} = \frac{P}{{P + \Delta P}}{T_{true}}$. However, with the plasma decaying, the charged particles number density decrease, leading to the pressure correction ΔP decreases and the lower temperature region disappear. It is contrary to the experimental results. Moreover, correction ΔP≈2.0×10−4 Pa in the non-equilibrium plasma with Tvib = 5010 K and Trot = 712 K. The effect of pressure correction ΔP on the temperature derived from Rayleigh scattering can be neglected. It is shows that the lower temperature region after 100 μs is not due to the calculation error caused by the pressure correction ΔP due to the charged particles interactions.
Fig. 4. Temperatures derived from Rayleigh scattering of femtosecond laser filament with different delay time.
Figure 5 shows the maximum temperatures derived from Rayleigh scattering of femtosecond laser filament. The result shows that the temperature increases sharply and reach a maximum value of 418 K within 1μs ∼ 10 μs, and then decreases slowly to around 300 K.
In the non-equilibrium plasma, due to the axial length is much larger than the radial, a one-dimensional transient model is used to describe the evolution of electron temperature Te and heavy-particle temperature Th (i.e. gas temperature):
Where, ρe, ρ, Cpe, Cp, κe, κeff, me, mj, ne, nj are electron mass density, heavy-particle mass density, electron specific heat at constant pressure, heavy-particle specific heat at constant pressure, electron translation thermal conductivity, heavy-particle effective thermal conductivity, electron mass, species j mass, electron number density, species j number density, respectively. ${\bar{v}_e}$ is electron mean velocity and ${\bar{v}_e} = \sqrt {8{k_B}{T_e}/\pi {m_e}}$. Qej is effective cross section between electrons and particles j, and those values can be calculated from Ref. [35]. The last term Eeh in Eq. (8) is used to describe the coupling between the electron temperature Te and the heavy-particle temperature Th due to energy exchange by elastic collisions. Inelastic collisions are involved through the radiation loss term U and implicitly through the reaction thermal conductivity.
According to the experimental results, the $\rho {C_p}\partial {T_h}\textrm{/}\partial t$ is about 109 ∼ 108 W/m3 within 1 ns to 300 ns and is about 108 ∼ 106 W/m3 after 10 μs. The electron mean velocity ${\bar{v}_e}$ is about 105 m/s and the energy exchange term Eeh is about 108 W/m3 within 1 ns to 300 ns. The first term in the right of Eq. (7) (i.e. thermal conduction term) is about 106 ∼ 105 W/m3 within 1 ns to 800μs. The radiation loss is estimated from Ref. [36] and has a value of U∼105 W/m3 with Te = 5010 K. From Ref. [30,34,37], the electron translation thermal conductivity κe has a value of 10−5 W/m·K with Te = 5010 K in LTE and which value of κe decreases with the increase of non-equilibrium θ (defined as θ = Te/Th), so it can be inferred that thermal conduction is much less than radiation loss and energy exchange through elastic collisions. It is concluded that the energy exchange through elastic collisions between electrons and heavy-particles is the main reason for causing the loss of electron energy and the rise of gas temperature. Within 1 ns to 10μs, the electrons lose energy continuously due to collision with molecules, the electron temperature decreases, while the molecules obtain energy through collision with electrons, which is much larger than the energy loss due to thermal conduction, and the gas temperature rises. After 10 μs, thermal conduction becomes the dominant factor, which resulting the gas temperature decreases. In order to quantify the gas temperature evolution process, a double-exponential model, $T = {T_0} + {a_1}{e^{ - t/{\tau _1}}} - {a_2}{e^{ - t/{\tau _2}}}$, is used to fit the gas temperature. The result shows that the characteristic time of gas temperature rising is 4.53 μs, while decaying is 136 μs.
In order to investigate the heat transfer process of femtosecond laser filament, Lucas-Kanade optical flow algorithm [38] (optical flows from adjacent pixels are approximately identical, usually be used to calculate motion between two image frames which are taken at different delay times) is employed to investigate the movement of the local high temperature region at the focused region due to the heat transfer. Figure 6 shows the velocity and vector distribution of the motion of the local high temperature region. The heat mainly transfers along the radial direction, and the corresponding velocity decrease sharply with delay time.
Figure 7 shows the radial directional velocity distribution of the local high temperature region with different delay time at center position (i.e. x = 30 mm) of filament. The results show that the highest velocity is observed at radial center of filament (i.e. y = 18 mm), and have a velocity of 5.8 km/s at 0.35 μs. Furthermore, there is one maximum on each side of the highest velocity, which represent the movement of the edges of filament.
Fig. 7. Radial directional velocity distribution of the local high temperature region with different delay time at center position.
Figure 8 shows the evolution of the velocity with different delay time at three peak positions. The velocity decreases sharply within 200 ns to 2 μs, then decreases very slowly after 2 μs. The obvious step around 2 μs indicate that there are two heat transfer mechanisms, which leading to the motion of local high temperature region. Within 0 μs to 2 μs, the high-speed movement of the edge with have a velocity of 103 m/s ∼ 104 m/s, which is close to the electron mean velocity, may be caused by the movement of electron. When the electron moves to the edges of filament, the energy exchange through elastic collisions between electrons and heavy-particles leading to the rise of gas temperature, which causes the fast radial motion of the filament edges. With the electrons lose energy continuously due to collision with molecules, the electron temperature decreases, leading to the decrease of electron velocity and energy exchange between electrons and heavy-particles. As a result, the rising rate of gas temperature and the radial movement velocity of filament decrease. After 2 μs, thermal conduction becomes the dominant factor, which resulting the gas temperature rising and radial movement of the edge, but it is very slowly.
4. Summary and conclusion
In sum, temperature distribution and corresponding evolution of femtosecond laser filamentation is preliminary investigated by time-resolved planar Rayleigh scattering (PRS). Firstly, emission spectra from the femtosecond laser filament are analyzed to estimate the influence of the plasma on scattering cross-section, which is a very important parameter for accurate temperature calculation of FRS. By fitting the N2 2nd positive system and N2+ 1st negative system, a vibrational temperature of 5010 K, a rotational temperature of 712 K, N2+/N2 = 7.15×10−8 are obtained, and indicate there is a thermal non-equilibrium plasma. Then species concentrations of the plasma are calculated using a two-temperature model. The result shows the difference of scattering cross-section caused by the non-equilibrium plasma is less than 2%, and it can be neglected(i.e. $\bar{\sigma }(T)\textrm{/}\bar{\sigma }({T_\textrm{0}})$≈ 1), while is a $\bar{\sigma }(T)\textrm{/}\bar{\sigma }({T_\textrm{0}})$≈ 0.78 in LTE plasma with T = 5010 K. Based on these results, temperature of filament can be derived from time-resolved PRS accurately, which increases sharply and reach a maximum value of 418 K within 1∼10μs, then decreases slowly to around 300 K. A dual-exponential model is used to fit the gas temperature with 4.53 μs characteristic time of gas temperature rising and 136 μs decaying time. By analyzing various factors of temperature evolution, it is finally concluded that the mechanism of temperature evolution is the result of energy exchange between electron and heavy particles and heat conduction. Within 1 ns to 10 μs, the electrons lose energy continuously due to collision with molecules, the electron temperature decreases, while the molecules obtain energy through collision with electrons, which is much larger than the energy loss due to thermal conduction, and the gas temperature rises. After 10 μs, thermal conduction becomes the dominant factor, which resulting the gas temperature decreases. Finally, Lucas-Kanade optical flow algorithm is employed to investigate the motion of the local high temperature region at focused region due to the heat transfer. The results show that the axial and edges of filament have the maximum radial velocity, which decrease sharply within 200 ns to 2 μs, then decreases very slowly after 2 μs. The obvious step around 2 μs indicate that there are two heat transfer mechanisms, the electron motion causes the high-speed movement of the edges and the thermal conduction causes the slower movement.
Funding
National Natural Science Foundation of China (11904396).
Acknowledgments
We gratefully thank Prof. Bincheng Li, University of Electronic Science and Technology of China, for helpful discussions and suggestions concerning the heat transfer process.
Disclosures
The authors declare no conflicts of interest.
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