Frequency-division multiplexing (FDM) for...

17th International Symposium on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2014

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Frequency-division multiplexing (FDM) for interferometric referencing and multi-component measurements in Planar Doppler Velocimetry (PDV)

Thomas O.H. Charrett*, Ian A. Bledowski,

Stephen W. James, Ralph P. Tatam

1: Engineering Photonics, Cranfield University, Cranfield, UK

* correspondent author: [email protected] Abstract The use of Frequency-Division Multiplexing (FDM) in interferometric Planar Doppler Velocimetry (I-PDV) measurements is proposed and demonstrated for the simultaneous acquisition of the signal and reference data. Its application for interferometric referencing is demonstrated experimentally with measurements of a single velocity component of a seeded axisymmetric air jet. The technique is suitable for time-averaged flow measurements, requires only a single camera, and offers reduced camera noise and automatic background light suppression. Crosstalk between the channels is typically less than 10% and as this crosstalk is dependent upon the frequencies used, it can be corrected for theoretically in post-processing. The approach has the potential to be expanded to facilitate the multiplexing of measurement channels in multi-velocity-component I-PDV and molecular filter based Planar Doppler Velocimetry (PDV) systems. The expansion of the technique to measure multiple velocity components is investigated, theoretically and experimentally, and the bandwidth, cross-talk and dynamic range limitations are explored. 1. Introduction There is great interest in optical techniques that can provide velocity measurements quickly and over large areas without disturbing the flow. Planar Doppler velocimetry (PDV)[4; 16], also known as Doppler Global Velocimetry (DGV)[13], is one such technique that allows the measurement of the Doppler shift over a plane defined by a laser light sheet. This is most commonly undertaken using an absorption line of molecular iodine to transduce the Doppler frequency to intensity. However, the use of molecular gas has several disadvantages. The first is that the choice of lasers is limited by the requirement to tune the output wavelength to an appropriate absorption line. The second is that the filter transfer function is determined by the form of the gas absorption line and therefore the sensitivity cannot be varied greatly, although it is possible to change the shape of the spectral feature by varying the gas concentration, the cell temperature or by the addition of buffer gases [3]. In an attempt to overcome some of the limitations of molecular filter based PDV, several interferometric approaches have been demonstrated which together can be collectively termed Interferometric planar Doppler velocimetry (I-PDV). The reported I-PDV techniques include Near-resonant interferometry [9; 10], Mach-Zehnder-Interferometric PDV [11; 12], and Doppler picture velocimetry [17; 18]. In I-PDV, the molecular filter is replaced with a path-length imbalanced interferometer and the Doppler-shifted light causes a change in the phase of light intensity distribution in the recorded interference pattern. The magnitude of this change is proportional to the flow velocity and the path-length imbalance in the interferometer. The relationship between the Doppler frequency shift, ∆ν, the optical path length imbalance, ∆l, the change in the phase difference, ∆φ, and the free-space speed of light, c, is given by:

νπ

ϕ ΔΔ=Δ lc2

(1)

Although I-PDV systems still require a laser offering high-power and single-optical-frequency output, the laser does not have to be tuned to a specific absorption line. This greatly increases the choice of laser sources and operating wavelengths. Another significant advantage of I-PDV systems is that the filter transfer function can be adjusted by changing the path length imbalance of


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the interferometer, allowing the measurement range and resolution to be adjusted significantly. As the Doppler shift is measured as a change in interferometric phase, it is necessary to evaluate the phase under both flow (signal), and zero-flow (reference) conditions. This can lead to a significant issue if the images necessary to evaluate the reference and signal phases are captured sequentially, with errors introduced due to drifts in both the laser frequency and interferometric path lengths [12]. The majority of previously reported I-PDV systems take the sequential approach, with only the work by Seiler et al. [18] proposing a solution in the form of polarization multiplexing to capture a zero-shift reference image simultaneously with images of the flow. However, this approach suffers from crosstalk between the signal and reference channels, which distorts the phase measurements and requires a more complex setup using polarization sensitive optics and a second camera. Here we propose and demonstrate a new method for acquiring simultaneously the signal and reference channels in I-PDV using a single camera by employing frequency-division multiplexing (FDM) of the channels. The method is suitable for time-averaged flow measurements and has the added benefits of reduced camera noise and automatic background light suppression. Furthermore, it has the potential to be expanded to multiplex multiple measurement channels and thus facilitate multicomponent velocity measurement. 2. Frequency-division multiplexing for I-PDV Frequency-division multiplexing (FDM) [2] allows the simultaneous capture of multiple interferometric fringe images using a single imaging sensor at full-resolution by applying an amplitude modulation that is unique to each channel to be multiplexed. A schematic of its potential application in a generalised I-PDV experiment is shown in figure 1Error! Reference source not found.. Here, a 3-component I-PDV system is shown, using three illumination directions and a single I-PDV imaging head to measure three different velocity components. A Michelson interferometer is shown as the I-PDV imaging head, as this is the most commonly reported configuration[9; 10; 18]. The technique is equally applicable to single component velocity measurements, and to the Mach-Zehnder interferometer (MZI) configuration which we have previously reported [11; 12] and which is used in the experimental section of this paper. The Doppler-shifted scattered light from each measurement channel and the zero-shift reference light from the reference channel are imaged through the interferometer and the combined fringe patterns recorded by the camera. The fringe patterns from each channel can then be demultiplexed by acquiring a time-series of images over a number of modulation cycles via the process illustrated in figure 2, and using the relationship:

nN

nN

nRMS fHfXN

txN

I σ=⋅== ∑∑ 22

2, )()(1)(1

(2)

Where X(f) is the discrete Fourier transform (DFT) spectrum of the signal, x(t), N is the number of samples and Hn(f) is the window function applied in the frequency domain to isolate the nth channel. As the background component of the signal is removed when windowing, IRMS,n is equivalent to the standard deviation of a channel’s signal, σn. Hence, the peak-to-peak intensity, Ipp,n of the modulating signal can be found by dividing the extracted RMS intensity, IRMS,n, by the standard deviation of the normalized waveform, σcal, the theoretical or experimentally measured standard deviation of the modulation with a peak-to-peak amplitude of 1:

cal

nRMSnppI

Iσ

,, =

(3)

This step provides a useful quantity for the comparison of signal and noise levels between images recovered from the FDM approach and images recorded using continuous, non-modulated


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illumination. However the RMS intensity is generally sufficient for measurements. The demultiplexed images are then equivalent to fringe patterns captured with only a single illumination channel present, and processed as in conventional I-PDV, typically using the favoured single interferogram, FFT-based phase evaluation method [9; 12].

Fig. 1 Schematic showing the application of FDM to a general, 3-component I-PDV system consisting of three signal channels illuminating the flow from different directions together with an un-shifted common reference channel.

Fig. 2 The steps involved in demultiplexing images from a FDM image time-series. The example shown is for a single velocity component system, consisting of two multiplexed channels.

3. Experimental Arrangement To demonstrate the FDM technique, the 1-component I-PDV system shown in figure 3 was constructed. The laser source was a high-power, external-cavity, tapered diode laser (Sacher Lasertechnik Tiger) operating at 1.1 W output power at a wavelength of 780 nm. This was sampled using an anti-reflection coated wedge, splitting off 1% of the beam power to provide the reference channel illumination. The reference beam was then chopped using a mechanical chopper at a frequency, f0=17.6 Hz, before being coupled into a single mode (SM) optical fibre (Fibrecore SM750) and guided to the interferometer head. The output of the fibre illuminated a ground glass diffuser that rotated slowly (<5 Hz) to blur laser speckle, and entered the interferometer via reflection from a glass slide positioned in front of and at 45o to the imaging lens. The remaining output from the laser, the signal channel, was chopped using a second mechanical chopper at frequency, f1=25.2 Hz, before passing through a combination of cylindrical and spherical lenses that formed the beam into a light sheet that was used to illuminate the flow. The flow was an axisymmetric air jet angled at 45o to the light sheet. This geometry is illustrated in Error! Reference source not found., with the light sheet cutting across the jet flow at a 45o degree angle, and the interferometer collecting scattered light at an angle of 90o to the light sheet. This


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geometry gives a sensitivity vector that is aligned with, but opposite to, the main flow direction of the jet, giving a Doppler frequency shift sensitivity of approximately 1.4 MHz per ms-1. The jet used had a 20 mm diameter, smooth contraction nozzle, and the air flow was seeded using a Concept Engineering Vicount compact smoke generator, which generated smoke particles of diameter in the range 0.2–0.3 μm. By sliding the position of the jet nozzle forwards and backwards, the position of the measurement slice downstream of the nozzle exit could be varied, allowing the jet flow to be mapped. The jet was generated inside a chamber to contain the seeding particles. The scattered light was imaged through a Mach-Zehnder interferometer (MZI) [11; 12]. The MZI was constructed using an infinity-corrected optical system (Olympus PlanApo, 1.25x objective with matched tube lens). The arrangement used here was modified from our previous work [11; 12] by the addition of a second 80mm glass block in a single-pass configuration. This glass block not only added extra optical path difference, but, when used in conjunction with infinity corrected optics, allowed the carrier fringes to be generated easily by tilting the block off-axis, with both the spatial frequency and direction of the fringes controlled by varying the tilt direction and magnitude. This allow easier alignment and greater control over the linear carrier fringes required for the single interferogram, FFT-based phase evaluation method that was used [9; 12]. The total optical path difference was 360 mm, giving an FSR of 833 MHz. For the geometry used, this corresponds to a phase shift of approximately 0.01 radians per ms-1. A Baumer HXC-13 (pixel size 14 µm x14 µm) camera was used to record the resulting image time-series, using a region of interest of 328 pixels x328 pixels that was selected to match the image size, which was defined by the vignetting in the interferometer and the f=25 mm, F#=1.6 imaging lens used to image the flow. This gave a field of view of 35 mm x35 mm. The camera frame rate was set to 60 fps to achieve sufficient signal levels in the flow measurement channel. The development and implementation of the signal processing algorithms was conducted in the open-source Python programming language[14], together with the NumPy and SciPy[8] modules and a custom C shared library using the FFTW3 library[6], in order to increase memory efficiency and speed of the FFT based processing.

Fig. 3 Schematic showing the experimental FDM I-PDV system used and inset is a photograph of the imaging MZI and the chamber containing the jet.


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4. Results An image time-series was acquired at each slice through the jet flow, with centre positions at approximately 15, 24, 59 and 70 mm downstream from the nozzle exit plane. The image time series were then processed, and the results are shown in figure 4. Due to the measurement geometry used, the jet nozzle impinged in the field of view on the left side of the slices recorded at 15 mm and 24 mm downstream of the nozzle. In figure 5, a vertical profile through the centre of the each slice is shown as the red data points. As the seeded air was extracted from the smoke chamber, the air surrounding the jet was not stationary, hence a model of an axisymmetric compound jet[15] was applied and fitted to each of the measured profiles, shown as the solid black lines in figure 5. The experimental results showed that the expected top-hat velocity structure was present near the nozzle, becoming smoother as the jet developed. This was in good agreement with the model. The experiment demonstrated that the signal and reference fringe patterns can be successfully multiplexed using the FDM technique.

Fig. 4 Velocity measurements taken at 45° slices across the jet at different x positions of the center of the flow downstream from the nozzle. The field of view was 35x35 mm. On the 15 and 24 mm images, the jet nozzle can be seen impinging in the field of view on the left side.

Fig. 5 A comparison of velocity profiles taken vertically through the centre of the jet at different positions downstream from the nozzle (red points) and an empirical model (black lines) of an axisymmetric compound air jet [15].

4.1. Camera noise It was anticipated that the FDM technique may have some favourable characteristics with regard to the level of camera noise, due to the windowing / band-pass filtering applied as part of the demultiplexing process. To examine the camera’s noise characteristics when using the FDM technique, a lens cap was placed over the camera sensor and an image time-series was captured. From this, the two measurement channels were demultiplexed to give a measurement of the camera noise present in each channel. This was repeated to give 100 measurements from which the background level and standard deviation of a typical camera pixel, shown in table 1, can be found. As the camera used was a CMOS sensor, each pixel has individual noise characteristics; hence the typical values presented here are the mean values for all pixels. FDM intensities scaled to the peak-to-peak amplitude are presented here for comparison with a single frame measurement and with


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frame-averaging measurements. It can be seen that the FDM process suppresses the pixel background level and that the remaining noise is significantly reduced when compared to a single frame measurement, and is only slightly worse when compared to simply averaging 128 frames. However, it should be noted that, for a full 3-component system (consisting of 3 signal channels and 1 reference channel), the full 128 frames will be available for each channel using FDM as opposed to only 32 frames available when using frame averaging and sequential recording in the same overall time period. In this case the FDM technique will have favourable noise characteristics.

Table 1 Noise characteristics of a typical camera pixel using a) FDM demultiplexing at f = 25.2 Hz, with a 1.5 Hz window using 128 frames, b) single frame measurements, c) the average of 128 frames, and d) the

average of 32 frames. Method Mean background level (counts) Standard deviation (counts)

a) FDM, Ipp 0.21 0.063

b) Single frame 3.32 0.440

c) Frame-averaging 128 frames 3.32 0.054 d) Frame-averaging 32 frames 3.32 0.081

Expressions for the standard deviation of the noise in FDM measurements and for the standard frame averaging case can also be found via analysis of the propagation of errors through equation 3, using [1].

Ncal

xpp 2σ

σσ =

(4)

Nx

AVGσ

σ = (5)

Here σx, σpp and σAVG are the standard deviations in a single measurement, the computed FDM peak-to-peak intensity and the frame-averaged result respectively. N is the number of frames in the time-series and σcal is the standard deviation of the normalized waveform used to scale the signals to the peak-to-peak intensities. The theoretical noise levels and the experimental measurements, scaled to the peak-to-peak intensities, are shown in figure 6 for various values of N. While there is good agreement between the theory and experimentally obtained FDM points, the results for frame-averaging diverge for larger values of N, possibly due to the presence of drifts in the pixel amplifier gains that become apparent over longer timescales. It can be seen from equation 4 that the noise level in the FDM measurements is independent of the width of the window function used in the demultiplexing. However, in practice, the window width will influence the signal-to-noise ratio (SNR) of a measurement if the noise power-spectrum contains peaks at distinct frequencies that lie within the window function used. Similarly, if the channel’s power is spread beyond the window used, via peak broadening or higher harmonics, then the recovered signal intensity will be lower and the SNR will be reduced. 4.2. Channel crosstalk A further important potential source of error is the crosstalk between measurement channels. In the work by Seiler et al. [18], in which polarization-division multiplexing (PDM) was used to separate the signal and reference fringe patterns, incomplete separation of the two images was reported, but the level of this crosstalk was not quantified. However, in similar full-field imaging instrumentation using PDM, crosstalk of up to 20% has been reported [7]. To measure the level of crosstalk using the FDM technique, an image time-series was recorded for each of the two channels, with only one channel present in each. From each of these time-series, an image was demultiplexed at the modulation frequency of the illuminating channel, and was


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used as a measure of the signal for that channel. The level of crosstalk from the channel could then be evaluated by demultiplexing images at other frequencies. The FDM crosstalk from a channel is typically found to be below 10%, with a minima of around 1.5%, which compares favourably with reported levels of crosstalk when using PDM. In this work, the crosstalk was 3.6% and 8.1% for channels 1 and 2 respectively, although this could in principle be improved by adjusting the channel frequencies slightly such that they correspond to a minimum in the crosstalk spectrum.

Fig. 6 Noise standard deviation vs. number of frames in the time-series, N. The solid lines show the theoretical noise propagation for frame averaging (black) and FDM scaled to the peak-to-peak intensity (blue), and experimental results are shown using two FDM window widths, 3.75 Hz (red crosses) and 11.25 Hz (red dots), and frame averaging (black crosses).

Fig. 7 Experimentally measured crosstalk spectra between four FDM channels. The lines show the total crosstalk into a channel from the other three channels as a percentage of the channel’s RMS intensity, IRMS,n, at N=128 frames (red/dashed lines) and N=256 frames (blue/solid lines). Also shown, as the shaded areas, are the power spectrums of the channel itself calculated at N =128 (red/hatched shading) and N=256 (blue/solid shading).

4.3. Extension to multiple velocity components When considering the extension of the technique to the multiplexing of additional measurement channels, there are two important considerations: 1) The available camera dynamic range and 2) the available modulation bandwidth and related cross-talk. The available dynamic range of the camera (camera range minus the background signal) will be shared between all channels, potentially limiting the signal levels available. For example, in a system using an 8 bit camera with four channels, each channel can have a maximum peak-to-peak amplitude of 63 counts. Frankowski et al. [5] report that, for a digitization of 6 bits (I0=63 counts), a standard deviation in the phase measurement of better than 0.006 radians can be achieved when using the FFT-based phase evaluation method used in this work. This corresponds to a potential velocity uncertainty of ~0.6 ms-1 with the MZI described in section Error! Reference source not found.. The modulation peak-to-peak intensity, Ipp, required to give the same signal-to-noise ratio, and hence velocity uncertainty, as a single exposure measurement with an intensity, I0, can be determined by equating the signal-to-noise ratios in each case:


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NII

calpp 2

0

σ=

(6)

Applying equation (6) using I0=63 counts, N=128 and σcal~0.45 gives an equivalent SNR when Ipp~9 counts. Therefore it can be concluded that there is sufficient dynamic range available for the multiplexing of the four channels required for three-component velocity measurements, even in the presence of a strong background signal. The second consideration is whether there is sufficient bandwidth to multiplex the required number of channels and that the resulting crosstalk is not excessive. This is more complicated and will depend largely on the choice of modulation waveforms and windowing functions. A typical scheme was investigated experimentally using four FDM channels, at frequencies 5, 12, 20, and 29 Hz, with equal peak-to-peak modulation amplitudes. Image time-series, with only a single illumination channel present in each, were recorded for each of the four channels and the resulting power spectra were calculated. From these, the total crosstalk into a channel with a given demultiplexing frequency from the other three channels was estimated by summing the power spectra of the other three channels. The results are shown in Figure 7 for two values of N; N = 128 (red/dashed lines) and N = 256 (blue/solid lines). Here the total crosstalk is scaled to a percentage of the channel’s RMS signal. Also shown, as the shaded areas, are the power spectra of the channel calculated at N =128 (red/hatched shading) and N=256 (blue/solid shading), allowing the location of the channel relative to the crosstalk spectrum to be seen. For N = 128 frames, the total crosstalk for each channel is ~20% or less, which is comparable with the level of crosstalk achieved using PDM for two channels[7]. If N is increased, the power spectrum peaks become sharper and more localized, reducing the spread of frequencies, hence reducing crosstalk. Increasing N to 256 frames resulted in significantly lower crosstalk, with all channels now below 10% and comparable with the crosstalk levels measured in the two channel results presented above. Another important point to note is that unlike PDM, where the crosstalk is mainly due to depolarization of the scattered light and is thus unknown and potentially variable, crosstalk in FDM is due to the power spectra of the modulations used. As this should remain constant and is measurable, it may be possible to correct for the effects of crosstalk in post-processing. 5. Conclusions The frequency division multiplexing of the signal and reference fringe patterns required for I-PDV has been described and demonstrated by undertaking single-component velocity measurements on an axisymmetric air jet. The FDM technique was shown to suppress background light levels and reduce the effective camera noise as a consequence of the windowing/filtering process used as part of the process of demultiplexing the measurement channels. Expressions for the camera noise present in a FDM measurement were presented, with the noise level in a FDM measurement being slightly better than that achievable using frame averaging and sequential recording for a 3-component system. The dynamic range limitation of the camera was investigated, with the 63 counts available per channel when using an 8 bit camera and multiplexing four channels being shown to be more than sufficient to allow velocity uncertainties of <0.6ms-1. Finally it was shown that, by careful consideration of the channel frequencies, it is possible to multiplex four channels with crosstalk levels of 10% or less using 256 frames. Additionally, as the crosstalk is dependent upon the modulation used, it should be possible to compensate for via calibration and post-processing. Acknowledgements The authors acknowledge the support of the Engineering and Physical Sciences Research Council (EPSRC) UK, via grant EP/G033900/1 and a studentship for IAB.


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References

1. Betta G, Liguori C, Pietrosanto A (2000) Propagation of uncertainty in a discrete Fourier transform algorithm. Measurement 27:231–239. doi: 10.1016/S0263-2241(99)00068-8

2. Bledowski IA, Charrett TOH, Francis D, et al. (2013) Frequency-division multiplexing for multicomponent shearography. Appl Opt 52:350–8.

3. Chan V, Heyes A, Robinson D, Turner J (1995) Iodine absorption filters for Doppler global velocimetry. Meas Sci Technol 6:784–794.

4. Elliott GS, Beutner T (1999) Molecular filter based planar Doppler velocimetry. Prog Aerosp Sci 35:799–845. doi: 10.1016/S0376-0421(99)00008-1

5. Frankowski G, Stobbe I, Tischer W, Schillke F (1990) Investigation Of Surface Shapes Using A Carrier Frequency Based Analysing System. In: Jaroszewicz Z, Pluta M (eds) Proc. SPIE 1121, Interferom. ’89. pp 89–100

6. Frigo M, Johnson SG (2013) FFTW - Fastest Fourier transform in the west. http://www.fftw.org/.

7. Groves RM, James SW, Tatam RP (2000) Polarization-multiplexed and phase-stepped fibre optic shearography using laser wavelength modulation. Meas Sci Technol 11:1389–1395. doi: 10.1088/0957-0233/11/9/320

8. Jones E, Oliphant T, Peterson P, and others (2013) SciPy - Scientific Tools for Python. http://www.scipy.org/.

9. Landolt A, Roesgen T (2009) Global Doppler frequency shift detection with near-resonant interferometry. Exp Fluids 47:733–743.

10. Landolt A, Roesgen T (2009) Anomalous dispersion in atomic line filters applied for spatial frequency detection. Appl Opt 48:5948–55.

11. Lu Z, Charrett TOH, Ford HD, Tatam RP (2007) Mach–Zehnder interferometric filter based planar Doppler velocimetry (MZI-PDV). J Opt A Pure Appl Opt 9:1002–1013. doi: 10.1088/1464-4258/9/11/006

12. Lu Z, Charrett TOH, Tatam RP (2009) Three-component planar velocity measurements using Mach-Zehnder interferometric filter- based planar Doppler velocimetry (MZI-PDV). Meas Sci Technol. doi: 10.1088/0957-0233/20/3/034019

13. Meyers JF, Komine H (1991) Doppler global velocimetry: a new way to look at velocity. Laser Anemometry 1:289–296.


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14. Python Software Foundation (2013) Python Programming Language – Official Website. http://www.python.org/.

15. Rajaratnam N (1976) Chapter 4 Compound Jets. Turbul. Jets. Elsevier, pp 63–86

16. Samimy M, Wernet MP (2000) Review of planar multiple-component velocimetry in high-speed flows. AIAA J 38:553–574. doi: 10.2514/2.1004

17. Seiler F, George A, Leopold F, et al. (1999) Flow velocities visualization using Doppler picture interference velocimetry. ICIASF 99. 18th Int. Congr. Instrum. Aerosp. Simul. Facil. Rec. (Cat. No.99CH37025). IEEE, pp 11/1–11/8

18. Seiler F, George A, Srulijes J, Havermann M (2006) Progress in Doppler Picture Velocimetry (DPV) Technique. Proc. 12th Int. Symp. Flow Vis. Göttingen, Ger. pp 1–10

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