INSTITUTE OF PHYSICS PUBLISHING
MEASUREMENT SCIENCE AND TECHNOLOGY
Meas. Sci. Technol. 12 (2001) 442–451
www.iop.org/Journals/mt
PII: S0957-0233(01)19589-5
Flow field imaging through sharp-edged atomic and molecular ‘notch’ filters Richard B Miles1 , Azer P Yalin2 , Zhen Tang1 , Sohail H Zaidi1 and Joseph N Forkey3 1 Department Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA 2 Stanford University, HTGL Building 520, Stanford, CA 94305-3032, USA 3 University of Pennsylvania, Pennsylvania Muscle Institute, The School of Medicine, D700 Richards Building, 3700 Hamilton Walk, Philadelphia, PA 19104-6083, USA
E-mail:
[email protected],
[email protected] and
[email protected]
Received 30 November 2000, accepted for publication 5 January 2001 Abstract Sharp cut-off atomic and molecular notch filters simultaneously provide high spectral resolution and allow imaging by collecting light over a wide field of view. Many important properties of flow fields can be observed by imaging light elastically scattered from small particles, molecules or electrons. In order to extract information about the flow field from elastic scattering, the spectrum of the scattering must be resolved and the background scattering must be suppressed. Very high resolution, on the order of a few tens of megahertz, is usually required. The spectrum of the scattered light is broadened and shifted by the motion of the scatterers. For particles, which have relatively little thermal or acoustic motion, the spectral shift is only a function of the velocity. For molecules, the scattering spectrum is a function of the temperature, velocity and pressure of the gas as well as its composition. For electrons, the spectrum is a function of the electron temperature and electron number density in a plasma. In this paper, sharp edged notch filters made of rubidium, iodine or mercury vapour are used to image shock wave and boundary layer structure by Rayleigh scattering from particles, to image gas pressure, velocity and temperature by molecular Rayleigh scattering, and to measure electron temperature and electron number density by Thomson scattering. For molecular scattering, filter transmission is generally a function of velocity, temperature and pressure, but, under some circumstances, it is a function of only one or two variables, so a notch filter can provide single-pulse images of a specific flow field parameter. Keywords: flow field imaging, atomic filters, molecular filters, velocity, temperature, pressure, background suppression, wide field of view, Rayleigh scattering, ‘notch’ filters, blocking filters, flow field diagnostics, Thomson scattering, spectroscopy, iodine, mercury, rubidium
1. Introduction In recent years there has been much attention devoted to the use of atomic and molecular filters for imaging flow fields [1]. A major focus of this work has been on the use of the iodine molecular filter for the acquisition of multicomponent velocity vectors across a two-dimensional image plane. For this application, researchers have relied on an 0957-0233/01/040442+10$30.00
© 2001 IOP Publishing Ltd
approximately linear variation of the transmission through the iodine cell as a function of frequency. Through normalization with a non-filtered image, the brightness of the scattering from particles seeded into the flow can be associated with their velocity. For these applications, a gradual cut-off with frequency is required, and researchers have used optically thin iodine cells [2] or iodine cells pressure broadened with foreign gases [3].
Printed in the UK
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Sharp-edged atomic and molecular ‘notch’ filters
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There are, however, a variety of applications that are derived from an extremely sharp cut-off, ‘notch’ type atomic or molecular filter used in conjunction with a tunable laser. An ideal ‘notch’ filter is one that has zero transmission over a small range of frequencies, and 100% transmission elsewhere. This filter profile is approximated by optically thick, thermally broadened atomic and molecular absorption where light is passed through a cell filled with the appropriate vapour and the ‘notch’ is created by strong resonance absorption at a specific frequency. Two primary types of application for sharp-edged filters can be identified. The first type of application is based on their ability to block a particular wavelength of light while transmitting others. This blocking function can be used to enhance contrast by blocking background light, or to give a quantitative measurement of temperature, pressure or velocity, if the other two variables are fixed. In some cases, quantitative measurements of one variable can be made with only one of the other two variables fixed, i.e., velocity or temperature at constant pressure, or pressure at constant velocity or temperature. The second type of application is based on the recognition that the sharp cut-off portion of the filter represents a step function, so the turn-on or the turn-off slope of the intensity recorded as a signal is swept in frequency across this filter edge is the spectrum of that signal. That is to say, the edge of the filter acts as a spectral integrator. The second edge is a negative integrator, so the full spectrum of the signal can be recovered. A real filter does not have infinitely sharp edges, so the spectrum is actually recovered by deconvolution. Through an absolute calibration of the transmission wavelength, this feature also leads to the measurement of spectral frequency shifts. The great advantage of these filters, as contrasted with similar filters which use narrow linewidth interferometers or spectrometers, is that the atomic and molecular filters have very wide acceptance angles. They are ideal tools for enhancing the signal strength at high resolution by collecting large acceptance angle scattering and for resolving the spectrum of light scattered from an image plane. Generally, the extinction of the atomic and molecular filters far exceeds that of interferometric filters and spectrometers, so they are particularly good at blocking background scattering. In order to utilize atomic or molecular filters for flow field diagnostics, a frequency-tunable, narrow linewidth laser is required. This laser source must be tunable in the vicinity of the filter, so that light that is scattered from the flow field can be modified as it passes through the vapour. Light that is scattered from the region of the experiment has frequency components associated with the characteristics of the flow field. Normally, elastic scattering from windows, walls and models dominates the scattering and occurs at the frequency of the illuminating laser. Light that is scattered from particles moving in the flow field is frequency shifted by the Doppler effect. Light scattered by the molecules is also frequency shifted by the Doppler effect, but the frequency shift includes thermal motion and acoustic motion as well as that of the flow itself. An example of the scattering lineshape from air at 1 atm, 300 K is shown in figure 1 [4]. This profile becomes Gaussian at high temperatures or low pressures where acoustic effects become negligible. In plasmas, the electrons act as strong scatterers
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Figure 1. Spectral shape of Rayleigh light scattering from 300 K, 1 atmosphere pressure air illuminated by a frequency doubled Nd:YAG laser at 532 nm and observed at 90◦ . The curve is not Gaussian because of acoustic contributions to the scattering.
(Thomson scattering) and the spectrum can be used to measure electron number density and electron temperature. Experiments are generally conducted either with the narrow linewidth laser held constant in frequency during the experiment, or with the laser tuned in frequency during the experiment. For instantaneous measurements of flow field properties, a single high-power laser pulse is generally utilized and frequency tuning is not practical. Unless tuning can be accomplished at a very high rate, flow field properties captured by frequency tuning are time averaged. Thus, the applications that do not require laser tuning are candidates for instantaneous field measurements.
2. Selection of the filter The filter should be chosen with the following characteristics. 1. Very steep cut-off and cut-on edges for high spectral resolution. 2. Strong extinction (many orders of magnitude) in the blocking region and transmission close to 100% in the transmission region. 3. Overlap with currently available, narrow linewidth, tunable laser sources. Optically thick resonant transitions in atomic or molecular vapours form a close to ideal medium for sharp cut-off filters. In the optically thick regime, the absorption at the centre of the resonance transition can be many orders of magnitude. Far away from the resonance there is very little attenuation, so filter transmission is close to 100%. The expression for the intensity of light transmitted through such a filter is given by I = I0 e−αlV (ω)
(1)
where α is the absorption constant on the line centre, l is the length of the filter and V (ω) is the linewidth function, which normally includes terms associated with thermal motion and collisional processes. In the Voigt approximation, the linewidth, V (ω), is a convolution of the Gaussian broadening associated with thermal motion, G(ω), and the Lorentzian broadening associated with collisional processes, L(ω). The 443
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Lorentzian profile drops quadratically with frequency offset from resonance: L(ω) =
( ωL /2)2 . (ω − ω0 )2 + ( ωL /2)2
(2)
ω0 is the resonance frequency, and ωL (= 2× collision frequency) is the Lorentzian full width at half maximum. The Gaussian profile drops exponentially with the frequency offset: −(ω − ω0 )2 ln 2 G(ω) = exp . (3) ( ω0 /2)2 √
ω0 = (2ω0 /c) 2kTv ln 2/Mv is the Gaussian full width at half maximum, c is the speed of light and Tv and Mv are the temperature and mass of the atoms or molecules of the vapour. Because the Gaussian drops much faster than the Lorentzian, a sharp cut-off is achieved by using an atomic or molecular vapour in which the collisional broadening processes are negligible compared to the thermal broadening processes. In this case, V (ω) = G(ω). Most allowed atomic or molecular transitions can be made optically thick by just increasing the length of the filter. The linewidth of the atomic or molecular filter is usually determined by the presence of hyperfine splitting and more than one isotopic form of the atom or molecule. In the models, these multiple states are accounted for by a simple summation of the spectral features. The sharp cut-off and cut-on features of the filter are then determined by the outermost transitions, and the location of the cut-off from the centre frequency of those outermost transitions is given by the expression ω0 2kTv
ωc = ln(α) (4) c Mv assuming that the optical density (α) is much greater than one. Note that the cut-off frequency of the filter can be changed somewhat by changing the optical density, which is achieved by changing either the vapour pressure or the filter length. In the optically thick case, the steepness of the filter in fractional transmission per unit frequency is √ 1 dI 2e−1 c ln(α) = . (5) √ I0 dω ω0 2kTv /Mv It becomes clear that low vapour temperature and high atomic or molecular mass leads to sharp cut-off and cut-on. Equation (5) gives guidance on the selection of the materials for an atomic or molecular vapour ‘notch’ filter. The absorption constant is not a critical factor, but should be large enough to avoid cells whose lengths are greater than a few centimetres. Assuming strong transitions, a figure of merit for the steepness of the cut-off can be approximated as the ratio of the molecular or atomic mass to the boiling point temperature [5]. Table 1 shows a list of various atomic species, plus iodine, their figures of merit based on the ratio of the atomic or molecular mass to the boiling point temperature and primary transition wavelengths where appropriate. It should be noted that iodine, as well as iron, lead, barium and calcium, have many transitions throughout the visible and ultraviolet portion of the spectrum. Of particular interest for iodine are transitions 444
Table 1. Figure of merit for FRS temperature measurement for various filter materials. Filter
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I2 Hg Cs Pb Rb Ba K Fe Ca
457 629 940 1781 958 2170 1029 3132 1755
254 200.6 132.9 207.2 85.5 137.3 39.1 65.8 40.1
0.56 0.32 0.14 0.12 0.09 0.06 0.04 0.02 0.02
many 254 852, 894 many 780, 795 many 764, 770 many many
which overlap the frequency-doubled Nd:YAG laser at 532 nm, and the argon-ion laser at 514.5 nm. This table suggests that iodine is the best choice for a filter; however, the maximum optical depth for iodine is approximately 10, corresponding to an extinction of 10−5 . At higher optical depths, absorption from a weak continuum transition becomes important and filter transmission away from the blocking region is seriously diminished. The optical depth of atomic vapour cells can be much higher due to the stronger oscillator strength of atomic transitions and the absence of weak overlapping continuum bands. Three different ‘notch’ filters have been used in our laboratory for experiments in the visible, ultraviolet, and near-infrared. The visible experiments have been conducted using an iodine molecular filter in conjunction with a narrow linewidth, frequency-doubled, Nd:YAG laser at 532 nm. Molecular iodine has many transitions throughout the visible and several over the tuning range of the frequency-doubled Nd:YAG laser. The modelled and measured absorption profile that was used for these experiments is shown in figure 2 [6]. The ultraviolet filter is atomic mercury vapour, which is used in conjunction with a frequency-tripled, narrow linewidth, Ti:sapphire laser. Mercury has seven naturally occurring isotopes and several hyperfine transitions, which lead to five or six optically thick absorption features in the vicinity of 253.7 nm. The particular features used for the experiments are either a single line from mercury 202 or a combined line from mercury 201, 204 and 199. The measured and modelled profiles of these lines are shown in figure 3 [5, 7]. In the infrared, atomic rubidium vapour was chosen because of its overlap with the high gain region of Ti:Sapphire. Here, again, several transitions occur from rubidium isotopes and from hyperfine splitting. The D2 transitions were chosen at 780 nm, and the measured and modelled filter profile is shown in figure 4 [8]. Each of these filter absorption profiles comes close to the ideal ‘notch’ filter contour, particularly when the filter is operated at low enough vapour pressures so collisional effects are minimized. In all cases, the true filter profile is only recovered if collisional effects are included in the model.
3. Background suppression The very sharp cut-off feature of these filters makes them particularly useful for suppressing high intensity, narrow band light that falls spectrally close to lower intensity light that one is interested in collecting. Atomic filters have been used in this context to block out laser Rayleigh scattering for Raman
Sharp-edged atomic and molecular ‘notch’ filters
Figure 2. Measured (solid curve) and predicted (dashed curve) transmission profile for the 9.88 cm long iodine absorption cell, with cell temperature of 353 K and cell pressure of 3.05 Torr (side arm temperature = 55 ◦ C). Rb Transmission Data Voigt Profile Prediction Gaussian Profile Prediction
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Figure 3. Measured (dotted curve) and predicted (solid curve) transmission profile near 253.7 nm for a 5 cm long mercury vapour absorption cell, with cell pressure of 0.0030 Torr. Isotopic species are indicated. These are two of the six separately observed features in this spectral region.
Figure 4. Measured (dots) and predicted transmission profile of the rubidium D2 curves at 780 nm for a 366 mm long cell, with cell pressure of 0.0030 torr. The theoretical Voigt and Gaussian profile predicted transmission curves are indicated by a solid curve and a dot-dashed curve, respectively.
measurements [9, 10]. With a sharp cut-off filter the frequency shift can be in the range of 1 GHz or less and still allow for significant suppression of the high intensity background light. An important example of this is the imaging of flow field structure and flow velocity by Rayleigh or particle scattering. In many cases, background scattering from windows and walls, or, in a wind tunnel situation, from the model itself, obscures light which is scattered from the flow field. If a narrow linewidth laser is used, however, it can be frequency tuned to fall within the blocking region of the atomic or molecular filter, thus eliminating, or suppressing by many orders of magnitude, scattering from the windows and walls. At the same time, light scattered from the flow field can pass through the filter due to the Doppler frequency shift associated with the fluid motion. A good example of this application is the set of images taken using the narrow linewidth, pulse-burst frequency doubled Nd:YAG laser in a Mach 2.5 flow field, shown in
figure 5 [11]. The pulse-burst laser system consists of a pulseburst master oscillator and power amplifier chain. A pulse sequence of 30 pulses at up to a 1 MHz rate is generated by high speed slicing of a 1.06 µm CW laser with a Pockels cells. The pulse-burst train is amplified up to approximately 100 mJ per pulse by a chain of flashlamp pumped Nd: YAG amplifiers and then frequency doubled to 0.532 µm. The CCD camera for visualization is developed by Princeton Scientific Instrument, Inc. and includes 30 image storage buffers built into each of the 180 × 180 pixels of the image sensor chip to achieve up to 1 MHz framing rate. The pulse-burst laser and CCD camera allow the high speed movies to be captured. The 500 000 frame per second images show turbulent shock wave/boundary layer interactions in a Mach 2.5 flow over a 24◦ angle wedge. In this case, the air is seeded with approximately 1% of CO2 gas upstream of the plenum, and the CO2 condenses in the cold core of the flow, generating a fog of 10 nm scale particles which 445
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Row 1
Row 2
Row 3 Figure 5. Mach 2.5 air flow over a 24◦ wedge taken at 500 000 frames per second with the pulse-burst laser and an iodine ‘notch’ filter. Flow is from left to right, and light is scattered from CO2 particles. Row 1 is with the filter set to block wall scattering, Row 2 with the filter set to suppress low velocity components and Row 3 with the filter set to enhance low velocity components.
Particle, Window and Wall Scattering
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Figure 6. Schematic diagram of the ‘notch’ filter used to block background particle scattering while transmitting molecular scattering. The transmitted light is sensitive to changes in gas temperature, pressure and velocity.
form the predominate light scatterers [12]. These particles are not condensed in the warmer boundary layer, and so there is strong contrast highlighting the outer portion of the boundary layer. Weak shocks are seen by an abrupt increase in particulate density, and strong shocks are seen by the absence of particles behind the shock due to rapid heating and sublimation. The laser was tuned to coincide with the absorption line of a molecular iodine vapour filter so that scattering from the model was eliminated from the images. The particle scattering spectrum has a narrow linewidth since the particles have little thermal or acoustic motion, but is shifted in frequency by the particle velocities associated with fluid motion. By tuning the laser, the velocity-shifted scattered light frequency can be changed relative to the filter cut-off frequency and various features of the flow highlighted. Row 2 of figure 5, for example, shows what happens when the laser is tuned to suppress scattering from the low velocity portions of the flow. Note the strong contrast behind the shock wave associated with the large change in velocity. The laser can also be tuned to 446
Figure 7. Conversion graph: signal ratio–temperature. Hg filter: 5 cm, P = 0.0030 Torr, T = 315 K, scatterer: argon 50 Torr, normalized to Toff = 308 K.
highlight the low speed portions, as shown in Row 3, although, in this case, some scattering from the walls is observed. With these rapid sequential images the velocity in the plane of the laser illumination can also be determined by image-to-image correlation in those regions of the flow where sufficient contrast is present [13].
4. Measurement of temperature, pressure or velocity with fixed laser frequency 4.1. Two variables fixed A very appealing feature of the sharp cut-off ‘notch’ type filters is their ability to simultaneously block background scattering and capture a quantitative measure of flow field variables. This is made possible by the fact that the scattering from molecules in the flow is frequency broadened by temperature and pressure effects, and frequency shifted by the velocity. This means that the amount of light transmitted through the filter is a function of the flow velocity, temperature and pressure. In general, measurements at a variety of frequencies must be made in order to extract these flow field variables from the transmission
Sharp-edged atomic and molecular ‘notch’ filters UV FRS Temperature Pro file o f Ar P la s ma, p=50 torr, i=20 mA
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Figure 8. Radial temperature profile in Ar glow discharge tube measured by UV filtered Rayleigh scattering.
intensity. It is important to realize, however, that if two out of the three variables are held constant, or their values are known, then the transmission intensity is uniquely a function of the third variable. In most cases, this third variable can be measured while the frequency of the laser is within the absorption region of the ‘notch’ filter, so background scattering can be simultaneously blocked. An example of this is the measurement of temperature by Rayleigh scattering. Often in Rayleigh scattering experiments, the Rayleigh scattering signal is obscured or contaminated with background scattering from windows and walls, and, in some cases, from particles in the sample volume. In the absence of the net flow velocity, the thermal motion of the molecules themselves causes a frequency broadening which permits the sharp cut-off atomic or molecular filters to be used to suppress this background scattering. This approach has been used to suppress background scattering from soot in flames [14, 15] and from atmospheric particles for LIDAR applications [16, 17]. Figure 6 shows a diagram of a filter used for this application. The strong scattering from the walls, windows and, in the absence of net flow, particles, falls at the same frequency as the illuminating laser. The scattering from the molecules, however, is broadened by the molecular motion, that is a combination of the thermal motion and coherent motion associated with propagating acoustic waves. At atmospheric temperatures, this broadening extends hundreds of megahertz away from the illuminating laser line position. By placing the filter in front of the detection system, the centre portion of the line is blocked, thereby eliminating scattering from windows, walls and particles, and the skirts of the broadened line pass through, giving a signal that is produced only by the molecules. If the composition of the gas is known (such as air), then the signal intensity is a function of both pressure and temperature for a constant laser frequency, assuming zero velocity. A useful application of this filtered Rayleigh scattering is for the measurement of neutral gas temperature in a weakly ionized plasma. Since the pressure is of the order of 50 Torr, and the plasma is contained within a glass vessel, background scattering becomes a major limitation on the detection of Rayleigh scattered light from the molecules. At these low
Figure 9. Schematic drawing of the Thomson scattering experimental set-up for the measurement of electron temperature and number density. A is a Glan–Taylor polarizer, and B is a 40 nm passband interference filter.
pressures, coherent scattering processes can be neglected and the lineshape can be accurately approximated as Gaussian. Since the pressure is uniform throughout the cell, it can be measured using a conventional high-sensitivity pressure gauge, so the scattering seen through the blocking ‘notch’ filter is a direct measure of the temperature. Experiments to measure plasma temperature were conducted using a frequency-tripled Ti:sapphire laser in conjunction with an atomic mercury vapour cell [18]. A model was used to predict the scaling of signal intensity with temperature, and a calibration was done at room temperature so the ratio of signal intensity to the calibration signal directly gave the ratio of the measured temperature to the calibration temperature. The calibration curve used for these experiments is shown in figure 7 [18]. The experiments were conducted in the ultraviolet in order to take advantage of the frequency to the fourth dependence of the Rayleigh scattering cross section, and to benefit from mercury vapour, which has extremely strong extinction. Figure 8 shows the temperatures measured across a 38 mm diameter, quartz tube containing a weakly ionized argon plasma operating at 50 Torr in the diffuse regime with a current of 20 milliamps. The curve is a truncated Gaussian fit to this temperature profile. Rayleigh scattering from the molecules dominates Thomson scattering from electrons since the ionization fraction of the plasma is less than 10−5 . The error bars in the temperature measurements indicate the standard deviation of the data. The measurement of this temperature profile proved critical to developing an 447
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600
Emission Thomson Scattering ASE Signal Thomson Model
Signal (arb. u.)
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ne=1.61*10 cm Te=0.82 eV 200
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Wavelength From Center (nm) Figure 10. Plot of the Thomson scattering spectral lineshape measurement, along with the spectrum of the plasma emission with the laser ‘off’ and the scattered laser light with the plasma ‘off’. A theoretical lineshape is fitted to the data to obtain the electron number density and electron temperature.
understanding of shock propagation through weakly ionized plasmas and the effect that temperature gradients have on the shock structure [19]. This same approach may be used for the measurement of Thomson scattering [20]. Thomson scattering is the scattering of light by free electrons in a plasma. The cross section is independent of the driving frequency so it is advantageous to operate in the near infrared where narrow linewidth pulses energies can be very high and competing Rayleigh scattering from neutral species is low. The spectral lineshape of Thomson scattering has contributions from two components. One component is the electron feature, which is due to the electron density fluctuations. The other component is the ion feature, which comes from the electron density fluctuations due to the electrostatic force influenced by the ions. The spectral width of Thomson scattering is much broader than that of Rayleigh scattering because free electrons have a much smaller mass and a much higher temperature. The theoretical line profile of Thomson scattering can be modeled by the Salpeter’s approximation [21]. Electron number density and electron temperature can be measured by fitting the experimental spectral profile with the theoretical model. For these experiments an injection-locked Ti: sapphire laser was used in conjunction with a rubidium ‘notch’ filter. The plasma source is an atmospheric pressure argon arc lamp from the National Institute of Standards and Technology. The laser was tuned to fall at the centre of the ‘notch’ filter, so the background scattering from windows and walls was eliminated. Since Thomson scattering is polarized, a Glan–Taylor polarizer is used to reduce the background plasma emission, which is unpolarized. The scattered light was then passed into a spectrometer so the Thomson scattering spectrum could be determined. Figure 9 shows a diagram of the experimental configuration, and figure 10 shows the scattering data together with fits giving the electron number density and temperature [22]. 448
4.2. One variable fixed Under certain conditions, the sharp cut-off ‘notch’ filter can be made insensitive to either temperature or velocity, and, in this manner, it becomes a device that is capable of measuring a single variable if the other one is fixed. For example, if the sensitivity to temperature can be removed, then the filter can be used to measure pressure in gases with zero velocity, or, to measure velocity in gases where the pressure is known. The fact that this mode of operation can be achieved becomes clear when one looks at the derivative of the transmission intensity as a function of the flow field parameters. This is seen by expanding the functional form of the intensity variation into its individual components, each of which includes a partial derivative of the intensity with respect to a specific flow variable. ∂I ∂T ∂I dI (P , T , v) = dT + dv + dP . (6) ∂T P ,v ∂v T ,P ∂P T ,v These derivatives are functions of the flow parameters as well as functions of the cut-off and cut-on frequencies of the ‘notch’ filter and of the frequency of the laser itself. The partial derivatives represent sensitivities of the intensity variation to changes of a particular parameter. In some circumstances, the sensitivity with respect to one variable can be made to go to zero. An example of this can be demonstrated for the simple case where the scattering is Gaussian in nature and the sharp cut-off filter is an ideal ‘notch’ filter. The Gaussian Rayleigh profile corresponds to scattering from low pressure or high temperature gas, where acoustic modes are not important. Under those circumstances, the scattering intensity, I (ωL ), observed through an ideal ‘notch’ filter with width W , and a centre frequency ω0 , is just the convolution of the Gaussian Rayleigh profile with the ideal filter transmission profile. Since the ideal filter has either zero or 100% transmission, this convolution can be written as just one minus the integral over
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Figure 11. Temperature sensitivity versus laser frequency for Gaussian scattering at 300 K seen through an ideal ‘notch’ filter. WD is the ratio of the Rayleigh scattering linewidth to the filter width.
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Figure 13. Pressure sensitivity versus laser frequency for Gaussian scattering at 300 K seen through an ideal ‘notch’ filter. WD is the ratio of the Rayleigh scattering linewidth to the filter width.
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Figure 12. Velocity sensitivity versus laser frequency for Gaussian scattering at 300 K seen through an ideal ‘notch’ filter. WD is the ratio of the Rayleigh scattering linewidth to the filter width.
the frequency-offset filter blocking region: +ω0 +(W/2) √ M c CP = 1− I (ωL ) = √ T 2πkT ω0 +ω0 −(W/2) (ω − ωL − 2(ω0 v/c) sin(θ/2))2 × exp dω (ω0 /c)2 (2kT /M)
Figure 14. UV FRS data from air at T = 295 ± 2 K, P = 1 atm (◦), as well as data from air at T = 330 ± 2 K, P = 1 atm (). The modelled filter transmission profile (thick curve) is shown, as well as the model fit for atmospheric pressure air at T = 295, 330, 500 and 700 K.
(7)
where ωL is the illumination frequency, v is the velocity, θ is the scattering angle, c is the speed of light, M, P and T are the mass, pressure and temperature of the gas in the sample volume, and C is a constant. The term 2ω0 v sin(θ/2)/c represents the Doppler frequency shift due to the flow velocity, and P /T is proportional to the number density of the scattering molecules. Figures 11–13 show plots of the normalized temperature, velocity and pressure sensitivities as a function of laser frequency for a variety of filter widths, assuming air with a Gaussian scattering profile at 300 K. WD is the Gaussian full width at half maximum normalized by the filter width. Note here that the sensitivity to temperature variation and velocity variation pass through zero under certain conditions. If, for a real atomic or molecular ‘notch’ filter those conditions can be met, then a one parameter measurement can be made, holding
Injection Seeded Nd: YAG laser
CW seed laser
Frequency Measurement
Camera Test Section
Iodine Cell
Figure 15. Schematic diagram of the experimental apparatus used to measure velocity, temperature and pressure with filtered Rayleigh scattering.
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Figure 16. Filtered Rayleigh scattering measurements of temperature (left), pressure (middle) and velocity (right) in the core of the Mach 2 free jet showing crossing shock structure characteristic of an over- or under-expanded jet. Flow is from bottom to top.
only one of the other parameters constant. Of particular interest are conditions where those parameters that need to be held constant correspond to practical situations. Two such situations often arise. In almost all flows in the absence of shocks, the static pressure is approximately constant. This means that, by operating at a point where the sensitivity to temperature is zero, the transmission intensity through the filter can be made a function of velocity alone. To achieve this, we pick a point with high velocity sensitivity and zero temperature sensitivity, such as WD = 0.5, and normalized frequency = ±0.25. This point is selected by choosing the geometry so the Rayleigh scattering linewidth is half the filter width and the laser frequency so that velocity shift associated with the average flow velocity is at 0.25 of the filter width. Changes in velocity shift the frequency away from this ideal point, so there will be an increasing contribution from velocity as the fluctuations become large. The velocity fluctuations need to be small compared to Mach 1, i.e.
