Coherent Rayleigh-Brillouin scattering measurement of atmospheric atomic and molecular gas temperature

Jacob Graul; Taylor Lilly

doi:10.1364/OE.22.020117

1. Introduction

Beyond applications centering upon the kinetic modification and manipulation of gases [1–8], optical lattices have proven essential to the development of two valuable gas diagnostic techniques; coherent Rayleigh (CRS) [9–15] and coherent Rayleigh-Brillouin scattering (CRBS) [16–23]. As a result of the axially-periodic dipole forcing effect of an optical lattice on a polarizable gas, optical lattices are used to electrostrictively induce density gratings in a polarizable gas media with a periodicity on the order of the laser wavelength. This modulation of gas density produces fluctuations in the dielectric constant of the gas which can conveniently be used to coherently scatter a third laser beam. As these are induced fluctuations, with gas density perturbations orders of magnitude greater than those present naturally, these techniques offer greatly improved scattering efficiency, and thus characteristic signal intensities, over similar spontaneous scattering processes [24]. In gases approaching the free-molecular limit, where collision distances between molecules are long compared to the interference pattern characteristic length, this stimulated, elastic four-wave scattering process is called “coherent Rayleigh scattering,” with the characteristic scatter signal governed largely by the thermal motion of the individual gas molecules. The same stimulated process in the kinetic regime is called “coherent Rayleigh-Brillouin scattering,” with the line shape further exhibiting the contribution of a Brillouin component, arising from the laser-induced, bulk-motion of the gas medium. These coherent scattering processes are typically performed in the low-intensity regime for diagnostics, where the weakly perturbative effect of the lattice on the gas is considered minimal and therefore has a negligible impact on the thermodynamic state of the gas. Consequently, the resultant line shape, a frequency-dependent intensity profile of the scatter signal, can be used to gain valuable thermodynamic information about the gas at the instant of scatter.

A large contributor to the utility of coherent Rayleigh and Rayleigh-Brillouin scattering as gas diagnostic instruments lies in the potential breadth of information that can be gained from the technique, as well as its nonresonant, non-intrusive, and highly localized nature. Because the pump lasers non-resonantly produce density gratings in the gas medium, the only restriction on the laser used is that it is sufficiency far from any electronic or roto-vibrational resonances with the gas species as not to cause further perturbation in the gas. The technique further avoids restrictions imposed by absorptivity, path length, and other line shape considerations normally associated with resonant absorption approaches [25–28]. Much like their spontaneous scattering analogs, CRS and CRBS have been used or are suitable for a number of diagnostic applications. For example, CRS has previously been used in the measurement of flame [12], low density gas [13], and weakly-ionized plasma temperatures [29], with the potential to reveal information on gas pressure, composition and density. Similarly, CRBS has been shown to be a viable method in the measurement of pressure [18, 30], bulk viscosity [31], and relative mixture composition [18], for atomic and molecular gases as well as for the detection of nanoparticles [32]. To date, however, broadband CRBS has not been employed explicitly as a temperature measurement instrument. This work reports on the experimental implementation of broadband CRBS as applied to the measurement of translational temperatures of nitrogen, argon and methane at an ambient, atmospheric pressure of 0.8 atm, from 300 K to approximately 400 K. In addition to the CRBS temperature measurement data presented, this report also details the Fabry-Perot etalon-CCD spectrometer and analysis techniques used within this study, which offers significant improvements over previous broadband CR(B)S line shape acquisition techniques by reducing the number of laser shots and time required, while eliminating the need to modulate pump frequency. For collisional gas applications that require a non-intrusive, temperature diagnostic, or for those where temporal, spatial and experimental considerations prevent more conventional gas diagnostic techniques, this broadband CRBS temperature measurement approach is expected to be advantageous. In contrast to other temperature measurement techniques, CRBS nonresonantly circumvents the need for a laser source frequency-tunable to the species of interest as required by coherent anti-Stokes Raman spectroscopy (CARS) [33], while avoiding the relatively long interaction timeframes (100’s ns) required for measuring acoustic perturbations associated with transient grating spectroscopy [34]. Similarly, despite filtered Rayleigh scattering (FRS) being able to measure both temperature or pressure and flow velocity, the application of this study’s Fabry-Perot spectrometer to CRBS sidesteps the need for well-characterized atomic or molecular absorption filter cells [35]. As such, the following CRBS-based temperature diagnostic study was developed for potential implementation in ongoing optical lattice experiments, with a particular focus on the quantification of optical lattice gas heating [36], where small optical lattice interaction regions, nanosecond time-scales, physical constraints unique to optical lattice experimentation, and the non-intrusive requirement make the measurement of changing gas properties (e.g. temperature) difficult.

2. Theoretical framework

A conceptual diagram of the coherent Rayleigh-Brillouin scattering process within the larger context of this study’s gas heating apparatus is notionally shown in Fig. 1. Coherent Rayleigh-Brillouin scattering utilizes the interference pattern created by the spatial and temporal superposition of two counterpropagating and linearly polarized optical fields (pump 1 & pump 2), an optical lattice, to generate wavelike density perturbations in the gas medium [17]. This periodic density perturbation results from the electrostrictive interaction of the field upon the gas, in which optical lattices induce dipole moments in particles with polarizability (α). As a consequence of this induced dipole, the particles experience an axially-periodic, optical dipole force that pushes the particles to regions of high intensity, with a magnitude proportional to the gradient of the squared field strength Eq. (1) [37].

Fig. 1 CRBS conceptual diagram.

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F = - \nabla U = (\frac{α}{2}) \nabla {| E |}^{2}

CRBS relies upon the resulting laser-induced gas density grating and its effect on local changes in the indices of refraction to coherently scatter a third (probe) beam. The scattered signal obtained in CRBS studies has been found to be proportional to probe intensity and the square of the density perturbation, as given by Eq. (2), with the favored direction of the scatter governed by the four wave phase matching condition, Eq. (3) [16].

I_{signal} \propto I_{probe} {δρ}^{2}

k_{sig} - k_{probe} = k_{1} - k_{2}

Scattered light from the narrowband probe is Doppler shifted by the frequency difference between the pump beams and the induced density gratings. Since the strength of the density perturbation is dependent on the lattice (grating) velocity, the scattering signal power spectrum can be spectroscopically connected to the density perturbation of the gas. Consequently, the CRBS line shape can be used to glean thermodynamic, kinetic, and collisional information about the scattering medium through the application of models derived from numerical approximations to the Boltzmann equation. In practice, the two pump beams forming the lattice interaction can either be broadband or narrowband. In a narrowband approach, only lattices of a fixed velocity are created per set of laser pulses. Therefore, the acquisition of a full CRBS line shape requires scanning over discrete frequency differences between the narrowband pumps. In the broadband approach, however, a continuum of lattice velocities are created by a single set of laser pulses. The spread in density fluctuations results in a broadband scatter of the narrowband probe which then must be spectrally de-convolved to get a line shape. As the line shape evolved from the density perturbations in the gas, the shape is sensitive to the thermodynamic state, e.g. temperature, of the gas, as seen in Fig. 2.

Fig. 2 CRBS line shape “rainbow” of N2 at 1 atm illustrating a clear temperature dependence in the line shape.

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Traditionally, the experimental acquisition of CRBS line shapes has proven to be labor intensive. The narrowband CRBS studies of Cornella et al. [19] required scanning over a total pump frequency difference of ~5 GHz. To achieve the necessary resolution, this scanning was performed in increments of 15 MHz, with 100 shots taken per increment. So, for a given CRBS line shape presented, approximately 30,000 laser shots spanning a period of approximately an hour were required. Using broadband pumps, a narrowband probe, and an etalon to resolve the frequency structure of the CRBS signal, the experiments detailed in Pan et al. [17] still required thousands of laser shots to scan a single line shape, with 30 shots averaged for each discrete pump frequency difference. Similarly, the narrowband CRS measurement of flame temperature by Bookey et al. [12] required the discrete modulation of pump frequency, through the adjustment of seeder voltage, to create the desired pump frequency offset. Although this study only needed 25 data points to construct the CRS line shape, each of these data points were derived from the averaged intensity of 1000 laser shots. These cumbersome experimental approaches have previously limited the potential of CRBS as a laser gas diagnostic tool for dynamic processes. The following broadband CRBS analysis technique, used specifically by this study for gas temperature determination, offers a significant improvement to the CRBS line shape acquisition process by allowing the entire scattering power spectrum to potentially be sampled for every CRBS scattering event.

3. Experimental setup

The CRBS gas temperature detection setup is shown in Fig. 3, and is consistent with other CR(B)S experimental approaches [16, 17, 21, 30, 38]. Broadband optical lattice pumps were created by splitting a 5 ns FWHM, frequency-doubled, Q-switched Nd:YAG laser pulse (10 Hz repetition) into two 15 mJ pump beams using a polarizing beamsplitting cube (PBC). The spectral linewidth of the broadband emission was found to be approximately 30 GHz FWHM at 532 nm as imaged through a 100 GHz Fabry-Perot etalon. Pump polarization was controlled such that both beams were plane-polarized parallel to the optical bench surface. After passing through Galilean telescopes to control interaction region focal spot size and location, each pump was passed off-axis through a Keplerian telescope comprised of 5.1 cm (2”) diameter, 500-mm biconvex lenses, at an included crossing angle of approximately 178°. Similarly, the 2 mJ narrowband probe, plane-polarized normal to the optical bench surface and therefore orthogonal to the pumps, was produced by an injection seeded, frequency-doubled Nd:YAG, with a linewidth found to be approximately 250 MHz at 532 nm using a 12 GHz etalon. To satisfy the phase matching condition required by the four wave nature of the technique, Eq. (3), the probe was set to propagate counter to pump 1, which would in turn, allow the CRBS signal to be extracted at the second polarizing beamsplitting cube on the path of pump 2. The optical arrangement was designed such that the broadband CRBS signal and a portion of the narrowband probe, obtained as mirror pass-through, would utilize the same extended path around the table. This arrangement not only reduces the requisite number of optics, but the extended path reduces the possibility of stray light compromising the spectral analysis, while allowing each beam to be individually imaged by the etalon-CCD apparatus, as required by the spectrometry technique that follows.

Fig. 3 CRBS line shape “rainbow” of N2 at 1 atm illustrating a clear temperature dependence in the line shape.

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Broadband CRBS temperature measurements of nitrogen, methane and argon were all performed at an ambient, atmospheric pressure of 0.8 atm. As notionally illustrated in Fig. 1, the gases were heated using a resistive, rope heater controlled by a variable transformer, and flowed through a vertical free jet, $ϕ_{i n n e r}$ = 0.9525 cm (3/8”), with an approximate flow rate of 1000 sccm and velocity of 0.25 m/s. Experimental temperature measurements were made with a J-type thermocouple inside the jet just above the CRBS interaction region. The uncertainty of the thermocouple is advertised by the manufacturer as ~ ± 1% [39]. A knife edge apparatus was used to measure spot size dimensions within the interaction region, and each pump was found to be approximately 30 um compared to the 20 um diameter of the probe. External timing of both lasers was controlled using a Stanford Research DG645 delay generator. It should be noted that initially, the pumps path lengths were made to be as close to the same length as possible. However, upon imaging the CRBS scatter signal through a 12 GHz etalon, it was apparent that the lattices were narrowband, indicating that the pump path lengths were matching to within the coherence length of the pumps ( $L_{C o h e r e n c e}$ ≈3 mm) [21]. After inducing a 10.16 cm (4”) offset in path length between the pumps, the CRBS signal was again imaged and found to exhibit the expected broadband interference pattern.

CRBS spectral data were obtained by alternatively passing the narrowband probe and the broadband CRBS scatter signal through a 12 GHz FSR etalon and recording the resulting fringe structure as captured by a CCD. Prior to passing through the etalon, the two requisite signals were passed separately through a −150 mm plano-concave, 100 mm plano-convex Galilean telescope, followed shortly thereafter by a −50 mm plano-concave lens. All optics were contained within an optical cage system to reduce alignment-induced errors. The fused silica, one inch diameter, 532 nm AR-coated, 12 GHz FSR solid etalon was obtained from LightMachinery Inc., with a manufacturer-specified thickness $(t)$ , reflectivity $(R)$ , and finesse $(F)$ of 8.403 mm, 97.3% and 40, respectively. The resulting interference fringe patterns were focused by a + 250 mm plano-convex lens onto, and captured by, a digital Hammamatsu C4742-95 ORCA-ER CCD covered by an Edmund Optics 1 nm FWHM bandpass filter centered at 532 nm. All fringe patterns were taken with a 101 ms exposure (to capture one shot at 10 Hz); the narrowband probe averaged a single shot while the broadband CRBS signal averaged 1000 shots (100 seconds) to maintain fringe fidelity. As a consequence of the time required for data collection, averaging and file writing, each CRBS signal-probe image pair took four and a half minutes to acquire. Compared to the studies mentioned previously, Cornella et al. [19], Pan et al. [17], and Bookey et al. [12], this acquisition technique reduces the time and laser shots required to obtain an accurate CRBS line shape by factors of 10, 30, and 25, respectively.

4. CRBS signal Fabry-Perot spectral analysis

A Fabry-Perot etalon was used by this study in the de-convolution of a CRBS signal into an accurate CRBS line shape for temperature determination. The Fabry-Perot etalon consists of two internal, highly reflective surfaces separated by a transparent resonant cavity. When a light source is incident upon this reflective cavity, the etalon facilitates multiple reflections of the incident light at each reflective interface and produces a transmission consisting of a series of concentric illuminated ‘interference fringes’; the properties of which are explicitly defined by the properties of the etalon, and the optical frequency and divergence of the incident light. The transmission function of light through an ideal etalon $(A)$ can be modeled as an Airy function, shown below in Eq. (4), and depends upon etalon thickness (t), reflectivity (R), transmissivity $(T)$ , refractive index ( $μ$ ), vacuum wavelength ( $λ_{0}$ ), angle of incidence ( $θ$ ) of the ray relative to the etalon normal, and lastly, the phase shift incurred at the reflective surfaces ( $ϕ$ ). This transmission function is continuous and periodic, and is a function of the order of interference (m), with transmission maxima corresponding to integer values of m. When calibrated by a distant, quasi-monochromatic light source of known optical frequency, the etalon-CCD spectrometer allows for the determination of a 1-D wavelength spectra of an unknown source from the radial intensity distribution of the interference fringes, as long as the spectral bandwidth of the source is smaller than the free spectral range of the etalon [40].

A (m) = \frac{T^{2} {(1 - R)}^{2}}{1 + [\frac{4 R}{{(1 - R)}^{2}}] \sin^{2} (πm)} where m = \frac{2 μt \cos θ}{λ_{0}} + \frac{ϕ (λ)}{π}

The difficulties encountered in using an etalon to obtain high fidelity spectral information from experimental signals lie in a number of experimental, theoretical and analytical considerations. For example, the above transmission function assumes that the interference fringes are uniformly illuminated and that the transmission function is that of an ideal etalon. In reality, however, the illumination of interference fringes is never perfectly uniform. Other sources of deviation from the ideal etalon are physical imperfections and misalignments of the etalon and associated optics, as well as the finite size of the CCD detector pixels. Despite these factors and the tendency to broaden individual fringes, these imperfections in physical implementation and execution do not alter the relative separation of the fringes, and can be accommodated by using a slightly reduced value for $R$ . The analysis that follows not only allows for and incorporates various types of fringe distortions, but is able to more accurately model fringe center, magnification and distortions to take full advantage of the spectral resolution provided by the etalon.

To obtain CRBS temperature information from the interaction region within the jet, the post-etalon CRBS signal fringe set was captured on a 2D CCD array, and spectrally decomposed by mapping the CRBS signal intensity in frequency-space using the two-dimensional pixel-frequency information gained by imaging the narrowband probe, a known optical frequency reference, through the same spectrometric (etalon/CCD) apparatus. The experimental line shape was then fit using a parametric library of gas and pressure-specific, numerically-obtained CRBS line shapes to make a temperature assessment. For a more detailed explanation at the fringe analysis that follows, please reference Conde et al. [41] and the sources therein.

4.1 CRBS probe phase mapping

After capturing CRBS probe and signal fringe structures, seen in Fig. 4, and with the goal of building a lens function that connects pixel location with frequency, the probe image was analyzed to determine the fringe center, phase, magnification and distortions inherent to the optical system. By capturing the bright, concentric interference fringes on a rectangular CCD (1344 x 1024 pixels), it becomes necessary to effect a coordinate conversion that relates the $(x, y)$ position of an illuminated pixel with axial, $θ = L_{θ} (x, y)$ , and azimuthal, $ϕ = L_{ϕ} (x, y)$ , angles of a ray through the etalon. This allows the continuous order of interference function, $m$ , to be written

m (x, y, λ_{0}) ≃ m_{0} - \frac{L_{θ}^{2} (x, y) μt}{λ_{0}}

where

m_{0} = \frac{2 μ t}{λ_{0}}

is the order of interference at the center of the fringe structure with a perfectly collimated incident beam. Similarly, for a distortion-free ring structure, the axial incidence angle (

θ

) can be expressed in Cartesian pixel coordinates as

θ = L_{θ} (x, y) = α \sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2}}

The variable

α

is a magnification coefficient, and the ordered pair

(x_{o}, y_{0})

represents the center of the fringe pattern. Merging Eqs. (5) and (6), the order of interference function can then be expressed as

m (x, y, λ_{0}) ≃ m_{0} - β [{(x - x_{0})}^{2} + {(y - y_{0})}^{2}] where β = α^{2} μt / λ_{0}

With the physical properties of the etalon, the vacuum wavelength of the CRBS probe and the expression for interference order in Eq. (4), Eq. (7) allows for the plotting of a series of perfectly concentric, simulated interference fringes over the experimental image of the CRBS probe; the result of which is used to yield judicious initial guesses for the center, magnification and fractional order at the center

(m_{0})

of the fringe system.

Fig. 4 Raw CRBS probe (left) and signal (right) for N2 at 300 K as imaged through a 12 GHz FSR Fabry-Perot. Color denotes non-dimensional intensity value.

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To accurately capture the information contained within a fringe structure, however, the analysis technique must also accommodate optical distortions inherent to any realistic optical instrument and experimental fringe image. To account for azimuth angle-dependent distortions producing variations in fringe radius, the horizontal and vertical positions of a pixel, with respect the fringe center, are redefined as

\begin{array}{l} u = (x - x_{0}) + γ_{x} | x - x_{0} | \\ v = (y - y_{0}) + γ_{y} | y - y_{0} | \end{array}

To incorporate elliptical distortions, as well as barrel and pincushion aberrations in the fringe structure, the interference order function m can be further expressed as

m (x, y) = m_{0} - ψ (x, y) + {εψ}^{2} (x, y)

where ψ (x, y, λ_{0}) = β_{x} u^{2} + β_{y} v^{2} + β_{xy} uv

The nine free model parameters $(m_{0}, x_{0}, y_{0}, β_{x}, β_{y}, β_{x y}, γ_{x}, γ_{y}, ε)$ are all used in the development of the spectrometer lens function, which phase maps the CCD and ultimately allows for the extraction of a 1-D line shape from a CRBS signal fringe image. Before the lens function can be determined, the aforementioned model parameters must be optimized and fit to the experimentally observed CRBS probe fringes, which is a monochromatic collimated source of known frequency. After filtering the probe and signal images using a Minkowski function-processing filter to reduce the effects of uneven illumination and making judicious initial guesses for the free parameters, an iterative parametric grid search was used to fit simulated fringes, denoted $A_{M} (x, y)$ to the experimental CRBS probe fringes, $P_{r} (x, y)$ [42, 43]. At every iteration, each free parameter was optimized independently and sequentially, using a “golden section” search and parabolic interpolation [44]. Goodness of fit of the model fringes to the probe was used to assess each parameter at every iteration, using Eq. (11), and for a given iteration, the parameter value corresponding to a maximum κ was assumed optimal. It should be noted that while the probe fringes serve as the basis for the spectrometer lens function, the goodness of fit parameter (κ) is not intended to be used to quantify uncertainties relating to the free parameter estimates [41].

κ = \sum_{x = 1}^{N_{x}} \sum_{y = 1}^{N_{Y}} {\bar{A_{m}} [m (x, y, λ_{0})] \cdot \bar{P_{r}} (x, y)}

\begin{array}{l} \bar{A_{m}} [m (x, y, λ_{0})] = A_{m} [m (x, y, λ_{0})] - min (min (A_{m} [m (x, y, λ_{0})])) \\ a n d \\ \bar{P_{r}} (x, y) = P_{r} (x, y) - \min (\min (P_{r} (x, y))) \end{array}

After performing this iterative optimization scheme to obtain the optimal values for the nine free model parameters, a phase map of the CRBS probe, which serves as a lens function of the spectrometer, was constructed from $A_{M} [m (x, y, λ_{0})]$ . As shown in Fig. 5, the phase map consists of a series of concentric, polychromatic fringes, where the entire range of values (colors) spans a single order of interference and corresponds to an optical frequency span equal to the free spectral range of the etalon.

Fig. 5 Phase map (left) of CRBS probe derived from Fig. 4 (left) with filtered CRBS signal (center) and averaged intensity histogram over an order of interference range (right) corresponding to the shaded region of the center image.

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4.2 CRBS signal spectral construction

With the lens function of the spectrometer derived using the CRBS probe, the phase map facilitates the analysis and construction of a frequency-dependent intensity profile (CRBS line shape) from a CRBS signal fringe structure, $B (x, y)$ . The phase map correlates pixel position with phase, and as given by Eq. (13), allows the intensity at every pixel corresponding to a given phase to be summed, weighted by the number of pixels corresponding to, and plotted as a function of, that phase. The resulting intensity histogram (spectra) represents relative signal intensity as a function of optical frequency offset from the probe frequency center, and as such, characterizes the CRBS line shape. For illustration purposes in Fig. 5, an arbitrary range of $m$ from 0.90 to 0.95 is partially highlighted on the CRBS signal image, as well as the corresponding portion of the intensity histogram to show how, using pixel position as a frequency reference, signal intensity is averaged to produce an experimental CRBS line shape.

S (λ_{n}) = \frac{\sum B [m (x, y, λ_{n})]}{\sum_{x = 1}^{N_{x}} \sum_{y = 1}^{N_{y}} A_{m} [m (x, y, λ_{n})]}

λ_{n} = λ_{0} \pm z F S R a n d 0 \leq z \leq \frac{1}{2}

4.3 CRBS line shape temperature determination

To obtain analytical temperature predictions from the experimentally-obtained CRBS spectra, the data were fit using a parametric, gas and pressure-specific CRBS line shape library using a Levenberg-Marquardt nonlinear least squares algorithm . The CRBS line shape library was constructed from a series of experimentally-validated, atomic and six-moment kinetic line shape models stemming from numerical solutions to the Boltzmann equation [16, 17, 45]. For atomic gases, the CRBS line shape is obtained by taking moments of the linearized Bhatnagar-Gross-Krook (BGK) kinetic equation [9]. Similarly, for molecular gases in which internal degrees of freedom must be considered, the CRBS line shape is elucidated by taking moments of the Wang-Chang-Uhlenbeck (WCU) equation [16]. Importantly, the bulk and shear viscosity values used in the production of the six-moment line shape models for N2 are $η_{b} / η = .73$ and $η = 17.89 {[10}^{- 6} Pa \cdot s]$ [46]. Similarly, the values used for CH4 are $η_{b} / η = 2.16$ and $η = 11.13 {[10}^{- 6} Pa \cdot s]$ [47]. The shear viscosity value used in the atomic model for argon is $η = 22.74 {[10}^{- 6} Pa \cdot s]$ [48].

Figure 6 shows a raw CRBS line shape of nitrogen at a thermocouple-measured temperature of 300 K, alongside the same data after being filtered in the frequency domain using a Fourier transform (left), and regressively fitted with a model N2 CRBS line shape (right). The strong modulation observed in the raw CRBS signal spectra is due to the approximate 250 MHz longitudinal mode structure of the pump lasers [45]. In all cases, the fitting of CRBS model line shapes to the raw and filtered data produced spectrally-derived temperatures with variations less than 1 K.

Fig. 6 (Left) N2 300 K CRBS experimental signal spectra; raw and filtered. (Right) Same experimental spectra, with the fit CRBS line shape corresponding to a temperature of 296.8 K.

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5. Results and discussion

CRBS temperature data for nitrogen, argon and methane is collectively shown in Fig. 7, with CRBS spectral temperatures plotted as a function of measured thermocouple temperature (top), with percentage discrepancy between the two temperatures shown as a function of thermocouple temperature (bottom). For nitrogen, the linear CRBS spectral temperature trend line matches closely with the expected value, where the spectral temperature is equal to the thermocouple temperature; with the fit line exhibiting a fixed, approximately 2 K temperature shift over the expected trend. The percent error for the nitrogen tests range from −1.6 to 3.5%, with an average error over the entire temperature range tested of + 0.8%. Similarly, the argon CRBS spectral temperature fit exhibits a constant 4 K temperature shift over the expected trend, a range of percentage error of −0.9 to 5.2%, and an average error of approximately + 1.4%. In contrast to nitrogen and argon, the methane CRBS spectral temperature trend line exhibits a temperature shift down relative to the expected trend line, with a maximum deviation of −3 K occurring at a measured temperature of 298 K. The percent error for the methane spectral temperatures ranges from −2.4% to 1.8%, with an average error of approximately −0.5%.

Fig. 7 Nitrogen (Left), argon (Right) and methane (Bottom) CRBS spectral temperatures plotted as a function of measured thermocouple temperature, with corresponding percentage error (below).

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The temperature deviations observed in these tests are on the order of the manufacturer specified standard limits of error for the J-type thermocouple used in these tests (2.3-3K) [39]. Other sources that would contribute to differences between the CRBS spectral and thermocouple include uneven fringe illumination, optical plate defects in the etalon, optical imperfections in the CRBS and detection apparatus, as well as the finite pixel size comprising the CCD sensor. As described previously, these factors are contributors to fringe broadening and would lead to artificially inflated spectral temperatures, as was observed in the N2 and Ar tests. In addition to these experimental considerations, the CRBS process is susceptible to signal to noise variations relating to the width of the velocity distribution within the gas being measured. During testing it was noticed that imaging complete methane CRBS signal fringe structures proved to be much more difficult than either nitrogen or argon.

A similar observation was encountered for all three gases at temperatures approaching and exceeding 400 K. This signal attenuation is attributed to the reduction in strength of the density perturbation at each lattice velocity, relative to that at lower temperatures (or in methane’s case, for heavier molecules). Recalling that CRBS signal strength is proportional to the square of the density perturbation, shown in Eq. (2), the temperature-dependent (or mass dependent) broadening of the CRBS signal line shape dictates that the total perturbation to the gas, for fixed pump intensities, is limited to a proportionally smaller range of lattice velocities. This reduction in scattering intensity at discrete lattice velocities leads to a net reduction in scattering intensity over the entire line shape. Importantly, since the square of the density perturbation has also been found to be proportional to the product of the probe intensities, as illustrated in Eq. (15) [45],

{δρ}^{2} \propto I_{1} I_{2}

the application of higher pump intensities at elevated temperatures is expected to not only increase the gas density perturbation across the scattering spectrum, but more importantly, result in a concomitant increase in CRBS signal strength and the potential for broadband CRBS diagnostics at elevated temperatures.

Lastly, an additional source of error in the spectrally-derived temperature measurements of this study rests in the calculation of the scattering spectrum for methane. Although the six-moment model has been shown to be adequate for the prediction of CRBS line shapes for diatomic gases such as N2, the six-moment model has also exhibited a departure from experimental line shape data for some polyatomic gases [17, 47, 49]. These studies have highlighted the need for the accurate specification of a bulk viscosity term in the model, as is required by the equilibration of energy between translational and internal modes in diatomic and polyatomic gases. Notably, the lack of empirical data on bulk viscosity values for many polyatomic gases has driven the use of CRBS as an experimental technique with which to gather this information [31, 47]. Although the most recent estimates of methane shear and bulk viscosity were used in the calculation of this study’s line shapes [47], this may explain the non-constant shift in methane CRBS spectral temperatures as compared to the experimentally-measured value.

6. Conclusions

Translational temperatures were measured for nitrogen, argon and methane at 0.8 atm using broadband coherent Rayleigh-Brillouin scattering. CRBS scatter signals were passed through a 12 GHz Fabry-Perot etalon and the resulting fringe structures were captured on a 2D CCD. Signal fringe structures were spectrally de-convolved by mapping CCD pixel locations in relative frequency space, as obtained by analysis of narrowband CRBS probes, and averaging signal intensity over all pixels corresponding to a given phase. Importantly, this spectral analysis technique significantly improves upon previous CRBS line shape acquisition methods by reducing the number of laser shots and time required to construct complete line shapes, while removing the need to modulate laser frequency of narrowband pumps to produce a desired pump offset. To make temperature predictions, CRBS line shapes resulting from the above spectral analysis were regressively fit using a CRBS line shape catalog constructed from experimentally-validated atomic and six-moment kinetic line shape models. From 300 to approximately 400 K, CRBS spectral temperatures, when compared to experimentally-measured temperatures, were found to be within 5.2% of the expected temperature in all gases at all temperatures, with average errors over the entire range measured of 0.8%, 1.4% and −0.5%, for nitrogen, argon and methane, respectively.

Acknowledgments

The authors wish to recognize and thank Dr. Andrew Ketsdever, Dr. Sergey Gimelshein, and Dr. Mark Conde for their conversations and support. This work used, in part, the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575. This work was also supported, in part, by a grant of computer time from the DoD HPC Modernization Program (HPCMP) at the ERDC DoD Supercomputing Resource Center (DSRC).

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Coherent Rayleigh-Brillouin scattering measurement of atmospheric atomic and molecular gas temperature

Abstract

1. Introduction

2. Theoretical framework

3. Experimental setup

4. CRBS signal Fabry-Perot spectral analysis

4.1 CRBS probe phase mapping

4.2 CRBS signal spectral construction

4.3 CRBS line shape temperature determination

5. Results and discussion

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (15)

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