Fluorescence lifetime imagingIn rapid lifetime determination algorithms, a common approach involves the use of two LIF images. Each pixel in these images corresponds to the same point in the image plane. The acquisition of these images involves employing different gate characteristics that capture different segments of the fluorescence lifetime decay curve. Figure 1 displays the two gate functions utilized in this study: GLong, which captures the entire signal, and GShort, which captures the early portion of the signal. As a result, two images, ILong(x, y) and IShort(x, y), are obtained. By forming a ratio image from these two images, the ratio value can be used to determine the fluorescence decay time. When performing FLI with DIME, this ratio image, denoted D(x, y), is formed by dividing IShort(x, y) by the sum of IShort(x, y) and ILong(x, y):$$D= \frac{{I}_{{{{{\rm{Short}}}}}}}{{I}_{{{{{\rm{Short}}}}}}+{I}_{{{{{\rm{Short}}}}}}}$$
(1)
Fig. 1: Methodological description of Dual Imaging with Modeling Evaluation.In (a), gate functions are presented alongside an example of mono-exponential decay. Meanwhile, b displays the resultant lifetime model, with the shaded area indicating the estimated uncertainty associated with jitter.An analytical model is constructed to establish a correlation between the experimental ratio image and the fluorescence lifetimes. This model incorporates the known time gate functions Gj(t), which are measured experimentally, and the signal S(t). As a result, the detected intensity in each camera can be simulated using Eq. (2):$${I}_{j}= \int \, {G}_{j}(t)S(t)dt$$
(2)
In this investigation, the function S(t) is assumed to be a mono-exponential function, an assumption that is further discussed in the experimental section, where the gate functions (indexed by j) represent either the long or the short gate. By simulating the relative detected intensities for various fluorescence lifetimes using Eqs. (1) and (2), a function is generated that can correlate a ratio to a unique lifetime. This function, τ(D), is shown in Fig. 1b and can be utilized to convert the image ratio D(x, y) into a lifetime image, τ(x, y). A more detailed explanation of the DIME evaluation algorithm and a comprehensive review of the experimental considerations is found in our previous study27.Theoretical estimation of OH fluorescence lifetimeThe quenching behavior of OH fluorescence by major species in air, specifically N2, O2, and H2O, were simulated and the results were then compared with experimental data40. The quenching cross-sections were determined using empirical expressions presented by Heard and Henderson41. In this analysis, our focus was exclusively on collisional quenching, with the assumption that factors such as photolysis had a negligible influence. Subsequently, quenching rate constants were computed using the following equation:$${k}_{Q}= \langle v(T) \rangle {\sigma }_{Q}(T)$$
(3)
where 〈v〉 represents the average thermal velocity, and σQ denotes the quenching cross-section, which is temperature dependent. The average thermal velocity, 〈v〉, can be calculated as \(\sqrt{\frac{8{k}_{B}T}{\pi \mu }}\), where kB is the Boltzmann constant, T is the temperature, and μ is the reduced mass of the colliding molecules. To determine the overall quenching, the following expression was used:$$Q= {\sum}_{i}{k}_{Qi}{N}_{i}$$
(4)
Here, kQi represents the quenching rate constant for species i, while Ni is the number density of that specific species. The lifetime can then be calculated using 1/(A + Q), where A is the Einstein coefficient for OH, with a value of 1.467 ⋅ 106 s−1 42.3D tomographic reconstructionThe method used in this study to perform the 3D tomographic reconstructions have been previously described and thus only a concise overview is provided43. To allow for practical computations the continuous probed volume Ω is initially discretized into Nv voxels. Thereafter, 2D line-of-sight projections q of Ω are acquired by cameras from various viewpoints, each view having individual pixel projections p in the form of a matrix (m × n). Each pixel projection measurement p corresponds to an integral trough the probed volume. The continuous luminescence field \(f(\overrightarrow{s})\) within the probed volume Ω represents intensity at spatial positions \(\overrightarrow{s}=(x,y,z)\). The method revolves around mapping the plasma luminosity onto each view projection q p using a first-kind Fredholm integral equation derived from the radiative transfer equation44:$${b}_{q \, p}= {\int}_{q \, p}f (\overrightarrow{s})dA.$$
(5)
In this equation, bq p denotes a camera projection measurement from pixel p in view q, and \(f(\overrightarrow{s})\) represents the plasma luminescence field. The model is a simplification of the radiative transfer equation as it neglects self-absorption and scattering effects. The removal scattering effects is deemed acceptable due to the small size and optical thinness of the plasma arc, low electron density (1013 cm−3), and short camera-to-volume distances leading to low scattering conditions. Self-absorption could also be excluded as the luminosity mainly originates from primarily excited \({N}_{2}^{* }\) emitting in the visible wavelength range which is not absorbed in the surrounding environment. Discretizing the plasma luminescence field into voxels, allows each measured projection shown in Eq. (5) to be approximated as a finite sum:$${b}_{q \, p}={\sum}_{v=1}^{{N}_{v}}{w}_{q \, pv}{x}_{v}$$
(6)
where, xv represents a voxel within Ω, and wq pv denotes the contribution of that voxel to the complete projection q p. Smoothness was imposed on the solution, employing a sparse discrete Laplacian matrix with homogeneous Dirichlet boundary conditions for all boundaries. Implementing the Laplacian matrix promotes continuous solutions, aligning with the expected physical behavior of the investigated plasma arcs. Furthermore, the strategy mitigates any negative consequences arising from the potential ill-posed nature of the problem. The reconstruction problem can therefore be stated as a quadratic problem:$${\min}_{x}| | b-A{{{{\bf{x}}}}}| {| }_{2}^{2}+\lambda {{{{{\bf{x}}}}}}^{T}{{{{\mathscr{L}}}}}{{{{\bf{x}}}}}.$$
(7)
In this equation, x is a vector representing the discretized field of \(f(\overrightarrow{s})\), \({{{{\mathscr{L}}}}}\) is the Laplacian matrix, b is the measurement data vector, λ is a penalty term, and A is the projection matrix encompassing all linear voxel projections. This projection matrix A maps contributions from each 2D camera projections to each voxel in Ω based on Tsai’s pinhole camera model45. Iterative methods are commonly employed to solve this type of problem and by doing so, reconstructing the full volume Ω. In this work the preconditioned conjugate gradient (PCG) method was used to solve the inverse problem, with convergence criteria defined by a relative residual tolerance of ∣∣b − Ax∣∣/∣∣x∣∣ < 10−6.Experimental setupThe gliding arc discharge system functions in an open-air setting and comprises three key components: electrodes, airflow, and a power supply. The electrodes are constructed from hollow stainless steel tubes and are affixed to a teflon plate as seen in the center of Fig. 2. These electrodes, with a 6 mm outer diameter, incorporate internal water cooling. One electrode is connected to the high-voltage power supply as the powered electrode, while the other serves as the ground electrode. The airflow enters through a 3 mm diameter nozzle at the center of the Teflon plate, positioned between the two electrodes. The air flow rate was regulated using a mass flow controller (MFC), and the gliding arc was investigated for four different flow rates, see Table 1. The gliding arc discharge is powered by a Generator 6030 from SOFTAL Electronic GmbH. This AC power supply operates at a frequency of 35 kHz and was adjusted to provide a maximum input power of 400 W. This setting was chosen to accommodate the constraints of the tomographic reconstruction volume. The current was measured using a Pearson 6585 current monitor and the voltage using a Tektronix P6015A on the high-voltage supply cable. The power supply runs in burst mode with burst duration’s of 100 ms at a 5 Hz repetition rate. The power supply, in conjunction with the gliding arc, has been employed in numerous prior studies. These investigations have detailed studies of electrical characterization, electron density, optical emission spectroscopy, plasma-assisted combustion, as well as high-speed spatial and temporally resolved imaging7,35,46,47,48,49,50,51,52.Fig. 2: An overview of the experimental setup.Including sample raw images from the tomography (Ti) fluorescence cameras (Fj) together with a sample voltage and current curve from the 10 l/min case.Table 1 Summary of different flow conditions, provided with their corresponding Reynolds numbers and deposited powerThe power supply, the cameras for detecting laser-induced fluorescence and the cameras for tomography were all synchronized with the laser system to enable simultaneous measurements.The OH radicals are generated in the plasma discharge where hydrogen is supplied by the water vapor in the air, throughout the experiment the relative humidity was measured to be 40%. Laser pulses, having a 1.4 mJ energy, well within the linear regime for OH-induced fluorescence, and a 90 ps duration, were generated using a custom-built UltraFlux laser system from Ekspla, capable of producing tunable femtosecond and ps laser pulses. The OH radicals were excited using a wavelength of 283 nm (\({A}^{2}{\Sigma }^{+}({\nu }^{{\prime} }=1)\leftarrow {X}^{2}\Pi ({\nu }^{{\prime\prime} }=0)\)), more specifically this excitation is the Q1(6) which rotational population does not change much within the rotational temperatures 2000K–4500K. The chosen wavelength was also selected due to its limited susceptibility to temperature fluctuations with respect to its spectral position. Additionally, the utilization of ps excitation contributes to reducing this susceptibility, as ps excitation typically presents a broad linewidth, which in this case is 9 cm−1, thereby covering multiple rotational levels. Utilizing LIFBASE simulation software, it was estimated that the absorption changes by approximately 5% between 2000 and 3000K. The OH is excited to its first vibrationally excited state (A2Σ+), leading to subsequent fluorescence emission within the vibrational band 0–0 (306–314 nm)53,54. The predominant fluorescence observed is in the 0–0 band, primarily due to the high vibrational energy transfer (VET) rate, which surpasses the rate for 1–1 transitions under atmospheric pressure conditions, where major colliding partners being N2, O2, and H2O55,56,57,58. Two cylindrical plano-convex lenses were used to create a thin laser sheet (~100 μm), with a height of 30 mm, in the probe volume. The emitted light from the laser-excited OH radicals (OH*) was captured using a stereoscopic configuration of two Andor iStar IsCMOS cameras, each with UV-sensitive Gen II image intensifiers. The resulting images were 3 × 3 software binned, resulting in an image size of 853 × 720 pixels. Each camera was equipped with UV objectives (Bernhard Halle, f = 100 mm, f/2) and 32 mm extension rings. Spectral isolation of the laser-induced OH signal was achieved by attaching Semrock 320/40 nm (FF02-320/40) band-pass filters to the cameras, this setup, combined with short camera gating, effectively isolates laser-induced OH signals both spectrally and temporally47. To optimize the dynamics for FLI measurements and minimize interference from plasma emission, the camera gates were set to 4 ns for the Short gate (GShort) and 60 ns for the Long gate (GLong). The optical resolution of the imaging system were estimated to be 40 μm per pixel. The photon economy is optimized by using a stereoscopic setup with separate detection channels with a viewing angle of 15°, see Fig. 2, generating high fidelity images. Sub-pixel overlap in the image pairs was accomplished through a two-step calibration process. Initially, the detection system was calibrated using a checkerboard target. Subsequently, this calibrated data was aligned to the same coordinate system using MATLAB’s Computer Vision Toolbox. To temporally resolve the lifetimes the cameras was replaced with a Microchannel Plate Photomultiplier Tube (MCP-PMT) Hamamatsu R5916U-50 in order to validate the lifetimes determined by the FLI setup. The data was captured using a WavePro 604HD oscilloscope, with a bandwidth of 6 GHz and a sampling rate of 20 Gs.To capture the three-dimensional luminescence field of the gliding arc, a total of 10 Basler acA1920-40gm CMOS cameras were employed. These cameras were positioned in a semi-circular arrangement around the electrodes, as depicted in Fig. 2. Each camera had an average distance of 37 cm from the reconstruction volume. To enhance the signal-to-noise ratio (SNR), the camera resolution was software binned to 900 × 600 pixels. Each camera was equipped with a Nikon f = 28 mm f/4 objective lens, providing good combination of depth of field and light admission. The camera’s exposure time was set to 100 μs, striking a balance between signal-to-noise ratio and motion blur induced by the arc’s movement within a single exposure. Before measurements, dark background images without any plasma activity were captured and subtracted from each data image. The final reconstructed volume consisted of 221 × 221 × 221 voxels with a spatial voxel resolution of 0.5 mm/voxel. Camera calibration was performed by capturing images of a checkerboard surface from unique positions and angles relative to each camera. The calibration process utilized the Computer Vision System Toolbox in MATLAB 2022a to minimize image distortions and align the cameras to a common coordinate system. Although initial image distortions were negligible due to the use of high-quality optics, calibration further reduced any remaining distortions. The calibration quality was assessed by estimating the re-projection error for each camera, which was found to be approximately one pixel.
Holistic analysis of a gliding arc discharge using 3D tomography and single-shot fluorescence lifetime imaging
