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Abstract
We developed a method for directly measuring displacement of a moving surface for use with dynamic or high explosive driven experiments. The technique, called “Modulation Based Ranging” (MBR), overcomes the errors associated with integrating a velocity history of an object undergoing non-radial flow, while also providing the exact displacement of the object with sub 100 µm resolution. A discussion of sources of phase sensitive errors is presented along with a demonstration of the applied corrections. Excellent agreement between MBR and integrated velocity from the Photonic Doppler Velocimetry (PDV) technique was observed when no non-radial flow was present. We then demonstrated the ability of MBR to accurately measure true displacement of a surface subjected to a strong non-radial component.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Photonic Doppler Velocimetry (PDV) has been used in the past several years for making measurements of the velocity history of a dynamic experiment [1–3]. Basically a fiber-coupled Michelson interferometer, the technique measures the beat frequency between a Doppler shifted signal that reflects from a moving target and a stationary reference signal. A fast Fourier transform (FFT) of the resultant signal creates a 2-dimensional spectrogram that gives a visual representation for the velocity history of the experiment. There are two types of PDV typically employed: homodyne PDV where the reference light is of the same wavelength of the signal, and heterodyne PDV where the reference light is frequency shifted from the signal [4]. Heterodyne PDV has one main advantage in that the frequency shift of the reference laser ensures zero velocity does not occur at zero frequency. This improves accuracy for low velocities by moving the beat frequency away from the DC component of the FFT while also removing sign ambiguities in the velocity that arise from signals born at zero frequency.
PDV has been compared with electrical shorting pins which were the standard diagnostic for making discrete, dynamic measurements of the position of a surface during an explosively driven test [5]. The results have shown that PDV exhibits excellent agreement with the legacy diagnostics while also giving a continuous measurement of the velocity. PDV measurements have now been used for measuring a wide range of dynamic phenomena including explosively driven hemispheres [6], laser driven flyer plates [7], shock compressions of matter [8], and fragment generation [9]. As experiments have gained complexity, additional channels of PDV were sought while minimizing costs associated with the diagnostic. Multiplexed Photonic Doppler Velocimetry (MPDV) techniques were more recently developed to increase the number of individual PDV signals recorded on a single scope channel. These techniques have included both frequency [6,10] and time [11] multiplexing.
A strong requirement for a velocity history measurement is that the velocity component that is being measured is either parallel to the measurement probe line-of-sight for all times, or that the experiment is sufficiently well-behaved that the time varying velocity vector can be accounted for. Simple experiments, such as explosively driven flyer plates [12,13], can be designed so that the object being measured always moves parallel or radially along the probes line-of-sight. However, for complex explosively driven hydrotests this can be either very complicated or even unknown before the experiment is conducted. As the velocity history is often integrated to yield an accurate measurement of position of a moving body, uncertainties in the true velocity can lead to significant errors in the measured position. For non-radial motion, the component of the velocity that lies along the probe line-of-sight can vary rapidly during an experiment. This can lead to a gross underestimate of the displacement of the body as a function of time. This data must be corrected after the fact which is not always an option, particularly for explosive surfaces that undergo very complex motion, such as shaped charges [14]. Newly developed techniques such as broadband laser ranging [15] are able to overcome non-radial effects using pulsed lasers, although at a reduction of temporal resolution.
In our work, standard MPDV was modified to make direct displacement measurements using an amplitude modulator. Microwave modulated PDV has been used previously to improve PDV recording with low bandwidth scopes [16], however our method utilizes high-bandwidth scopes. The technique, called Modulation Based Ranging (MBR), modulates an existing MPDV laser in the GHz range at a particular wavelength. Through simple fringe counting at the modulation frequency, the range to the device under test (DUT) can be made, similar to frequency modulated continuous wave (CW) radar [17]. The displacement measurement is also directly complementary to the velocity measurement made with MPDV since the two diagnostics share the same line-of-sight to the surface, allowing for simultaneous velocity and position measurements. Figure 1 shows a schematic of the MBR setup. The MBR components are housed in a separate rack from the MPDV hardware in order to make the system more modular. The PDV lasers are simply sent into the MBR rack, modulated, and returned to the MPDV system. Our MBR setup was built using off the shelf components with low loss. The components are relatively low cost as compared to the cost of a full MPDV system and therefore adds an almost negligible cost per MBR channel to the total cost of the setup. The 64 channel MPDV system was built for a cost of ∼43 K per channel while MBR can be added for less than 1.5 K per channel.
Fig. 1. Experimental setup of a simple rack-mounted MBR system. Such a system is required for each PDV laser in an experiment but can be made entirely modular so that it can be included very unobtrusively to existing PDV architecture. MW: Microwave, OSA: Optical spectrum analyzer.
Our proof-of-concept experiments consisted of a simple flyer-plate geometry under ideal conditions. This was to show the agreement with integrated MPDV data when no non-radial flow is expected. This was contrasted against a far more complex shot where significant non-radial flow was expected. Results from these experiments will be shown in this manuscript along with a discussion of corrections that must be made to ensure the data is representative of the actual displacement.
2. Principles and methods
2.1 Configuration
As seen in Fig. 1, the PDV lasers first pass through a polarizer which is used to optimize the modulation depth through the modulator. The modulator chosen was an Optilab 20 GHz, biased intensity modulator that covers the bandwidth of our MPDV system. The modulator is a Mach-Zehnder interferometer where an incoming signal is split into two equal signals and pass through two distinct paths. One path is electro-optically modulated at the frequency of an external microwave source and offset with a DC bias, while the other path remains unchanged. This introduces interference between the two paths that can be measured as a time-varying amplitude. Typically the modulator was operated between 8 and 15 GHZ, depending on the bandwidth of the recording scope. The modulator is driven by a Keysight E8257D analog microwave source, capable of frequencies up to 20 GHz and with an accuracy of ±4 × 10−8. Additional control for the modulator is provided by a DC-bias to give an overall voltage offset to one arm of the interferometer. The modulated signal is sent through a 99:1 splitter to pick off a small amount of light to be sent to an optical spectral analyzer (OSA).
The modulator has the effect on the PDV base laser of changing the signal from one distinct frequency into three separate frequencies. The original PDV frequency, ωpdv, remains unchanged but two new sidebands located at frequencies of ωpdv ±ωm are created. Figure 2 shows the results from the OSA for one particular PDV laser (a) without and (b) with the modulation activated. The relative amplitude of the sidebands Asb, with respect to the main PDV peak Apdv, can be controlled with the bias voltage Vbias, and the peak to peak modulation voltage from the microwave source Vm. Equations (1) and (2) show the amplitudes of each peak [18].
Fig. 2. Output from an OSA for a single PDV laser centered at 1555.75 nm (a) without and (b) with the amplitude modulating. In figure (b) the modulation frequency is set to 15 GHz resulting in two equal sidebands separated from the carrier by this offset.
In Eqs. (1) and (2), Vπ is the modulator-specific half wave driving voltage which results in a phase shift of π, E0 is the initial laser signal amplitude and J0(x) and J1(x) are the zeroth and first order Bessel functions, respectively.
In our configuration, the sidebands are usually equal amplitude although other schemes, such as carrier suppression, are also possible [19]. By choosing the appropriate values for Vbias and Vm, we generally try to achieve Asb = Apdv as this gives three strong and equal amplitude signals in our final data. The three signals are then sent to the DUT, bounce from the diffuse surface, and return to the MPDV setup. In heterodyne PDV, the return base laser light is mixed on a 20 GHz photodetector with a reference laser/local oscillator (LO). The LO is slightly detuned (typically 1-2 GHz) from the base laser to generate a beat frequency in the GHz region which is easily measured with off-the-shelf oscilloscopes. Figure 2(b) shows what an OSA might record from modulated PDV lasers [20].
2.2 MBR analysis
For MBR, three distinct signals return from the surface and are each mixed with the LO. The signals are all squared together on a photodetector. Equation (3) shows the field equation of MBR.
In Eq. (3), the θ terms represent the frequencies and relative phases of each laser. The subscripts B, D represent the baseline and Doppler shifted phases, respectively. The value of θB can be written as θB=2πfBt where fB is the frequency of the base laser and t is time. Since the signal bounces off a moving surface thereby imposing a Doppler shift, the value for θD is written as θD=2πfD(t-2x/c) where fD is the frequency of the Doppler shifted laser, c is the speed of light, and x is the displacement to the moving surface. The sign of x determines whether the frequency shift from the Doppler Effect increases or decreases the laser frequency. The baselines are signals that reflect from some location within the MPDV hardware and never travel to the surface, for example a bad connector. They contribute to a homodyne PDV signal since the portion of the light that does make it to the surface and undergoes a frequency shift can mix with the unshifted baseline signal. The Doppler shifted signals are the dynamic PDV traces arising from the laser bouncing off the DUT. The m and s subscripts in Eq. (3) represent the phases of the modulation and shifted modulation, respectively. Similar to the baselines and Doppler shifted signals, the modulation signal remains internal to the MPDV hardware while the shifted modulation signal bounces off the DUT. The LO subscript represents the local oscillator. Equation (3) gives new signals arising from mixing with the LO, signals from mixing between the shifted and unshifted signals, and mixing between the three distinct signals that were sent to the DUT.
Carrying out the multiplication of all of the electric fields shown in Eq. (3) results in the creation of 21 individual signals. The 21 signals can be significantly reduced by dividing them into 5 categories.
- 1. Signals at frequencies that are either harmonics or frequency sums that lie outside the bandwidth of the recording oscilloscope and can be ignored.
- 2. Signals associated with the unshifted baseline that are inconsequential to the dynamic measurements of the DUT.
- 3. Signals associated with homodyne PDV which can be eliminated by reducing back reflections in the MPDV hardware.
- 4. Signals associated with the GHz modulation.
- 5. Signals associated with heterodyne PDV.
Figure 3 shows a simulated spectrogram containing many of the calculated MBR signals. The simulation is limited to those signals that would exist in a spectrogram recorded by a scope with an upper limit on the bandwidth being only slightly above the microwave modulation frequency. The associated phase for each dynamic signal born via one of the operations listed in Eq. (3) is shown to the right of each line. The baseline signal phases are listed on the left sides on the figure. The simulated signal mimics a velocity that jumps off and accelerates rather slowly before drifting towards the detector at a constant velocity, similar to a detonator-driven flyer plate. The amplitudes of each signal were chosen so that they could all be seen in the simulation. Experimentally the signal amplitudes vary significantly depending on the type of PDV and condition of the experimental setup. For example, heterodyne PDV typically employs a very strong local oscillator to improve the signal to noise ratio in the experiment. This causes any signals that do not mix with the LO (all homodyne signals for instance) to be relatively suppressed to the level of not being detectable. After eliminating all signals that lie outside the bandwidth of the scope, all signals too weak to detect, and baseline signals that do not carry dynamic information, the remaining signals include:
Fig. 3. Simulated spectrogram with PDV signals and MBR modulation included. The phase of each baseline is included with in the spectrogram while the phases of the dynamic signals are included to the right of the spectrogram.
The value of θs = 2πfm(t-2x/c), where fm is the modulation frequency, must be extracted as this contains the instantaneous phase of the microwave signal and can therefore be used to determine the displacement. Since the value of the unshifted modulation phase, θm, is very accurately known, the displacement can be determined once θs is extracted.
As Eqs. (4–6) represent the dominant signals present in the time-domain experimental data, a three-cosine fit can be used to determine the relative phases and hence the displacement of the DUT. A fitting function using relative amplitudes (c0, c1, c2) of the three frequencies can be written as:
In all three cosine functions, ϕ can be written as ϕ = θD-θLO = p0+(ωh+p1)t + p2t2 with fitting terms (p0, p1, p2) representing the initial phase, velocity and possible acceleration of the DUT, respectively. The value of ωh is the heterodyne beat frequency. This can be further expanded to higher order although that has not been found to be necessary. All of the fitting parameters are varied for a least-squares fit of a small time-window of the digitizer record which is then shifted across the entire time-domain digitizer record. A spectrogram is not required for extraction of the MBR displacement. The displacements, xU and xL, are the outputs of the least-squares fit. The two values may differ as the result of frequency dependent phase shifts associated with experimental recording equipment and are therefore denoted with the U and L subscripts. These correspond to the upper and lower MBR traces, respectively. Additionally, Eqs. (4–7 show that the MBR signal is doubly-redundantly encoded in the time domain signal, as long as the microwave frequency is set low enough to be fully captured in the scope bandwidth. Additional details for the analysis and extraction of displacement from a raw time-domain signal are given in the Supplement 1.
3. Results and discussions
3.1 Photodetector phase
As a proof-of-concept experiment for MBR, a simplified PDV system was designed to act as a benchmark for all MBR tests. Figure 4 shows the experimental benchmark PDV/MBR setup.
Fig. 4. MBR included in a single channel PDV system. MW: microwave, OSA: optical spectrum analyzer, EDFA: erbium doped fiber amplifier, PD: photodetector.
A laser is first amplitude-modulated by a Mach-Zehnder interferometer. The output is sent to a Keopsys erbium doped fiber amplifier (EDFA) for pre-amplification of the launch light. The launch power is typically up to ∼200 mW. The laser then passes through a 3-port circulator where it first passes to a discrete optical probe and impinges on the surface of the DUT. A portion of the light is recollected upon reflection from the surface and returns to the circulator. The light is then combined with a reference laser that serves as the local oscillator via a 90:10 beam coupler. The mixed signal generates a beat frequency that is within the bandwidth of a MITEQ 20 GHz photodetector (PD) and the resulting signal is recorded on a Tektronix 23 GHz oscilloscope. This setup is a single channel of PDV and was kept as simple as possible with all fiber lengths minimized to eliminate the effects of chromatic dispersion.
The experimental geometry was also chosen to be as simple as possible in order to eliminate any non-radial components in the measured velocity. This would enable MBR to be compared directly to integrated PDV to show any discrepancies. The setup consisted of a polycarbonate housing that held a 1/2 inch (1 inch = 2.54 cm) diameter PBX-9501 high explosive (HE) pellet detonated by an RP-1 detonator. The HE pushed a metal flyer plate towards an optical probe also held in the same polycarbonate housing. The flyer plate was a 1/32 inch thick stainless steel plate that, after an initial acceleration, flew towards the optical probe at a uniform velocity. The flyer plate was set against a 1/8 inch polycarbonate spacer with a hole to act as an air gap for the flyer plate. This cushioned the detonation so that the acceleration was not too sharp. This ensured the MBR diagnostic had many points during the acceleration to extract the displacement. Although the initial flyer plate was 1 inch square, the detonation actually “cuts” a ∼1/2 inch diameter circular flyer. The optical probe was a 100 mm working distance focused probe from AC photonics. The standoff between the flyer plate and the probe was 117 mm before jump-off. Figure 5(a) shows the experimental setup. Figure 5(b) shows the circular flyer, mid-flight after detonation, as taken by an x-ray camera timed with the detonator. The damaged HE pellet housing can be seen on the right and the optical probe can be seen on the left.
Fig. 5. (a) experimental setup for an explosively driven flyer plate. (b) x-ray photograph of the flyer plate in motion (moving right to left) after detonation.
For the experiment, the modulator was run at 15 GHz and the DC bias of the modulator was set to the quad point (Vbias = Vπ/2). This balanced the sidebands with respect to the main carrier. The frequency of the modulator determines the resolution of the system. We expect to be able to discern a phase change of 1/100th of a fringe. For a 15 GHz signal, the half wavelength is ∼10 mm. This gives an expected displacement resolution of 100 µm. Figure 6(a) shows the recorded spectrogram. The launch power for this experiment was 200 mW, or 23 dBm, and the LO power was set so that the returned signal at the photodetector was ∼0 dBm. The main PDV signal is seen at the bottom of the spectrogram with an offset from the heterodyne of ∼1.55 km/s (2 GHz). Taking into consideration the heterodyne offset, the maximum velocity achieved was ∼2.3 km/s. Centered at 11.6 km/s (15 GHz), the upper and lower MBR traces can be seen, each 1.55 km/s away from the modulation frequency, again as set by the heterodyne offset. The spectrogram is very similar to that seen in the simulated spectrogram of Fig. 3. The heterodyne laser was sufficiently high however that the homodyne signals are basically nonexistent.
Fig. 6. (a) Resultant spectrogram for an explosively driven flyer plate, including PDV and MBR branches. (b) Extracted displacement from the upper (red) and lower (black) MBR traces. The inset shows the amount of discrepancy at early times.
Figure 6(b) shows the extracted MBR displacement for both the lower (black dots) and upper (red dots) MBR traces. The datasets have been arbitrarily shifted so that they agree during the static part of the record before jump-off. While the data shows very similar behavior, there is clearly an offset that occurs which causes the two traces to give different values for displacement.
As seen in the inset of Fig. 6(b), there is a significant difference in the displacement at early times during the accelerating portion of the velocity record. Since MBR is a phase extraction of a temporally varying signal, if the phase response of the photodetector is not perfectly flat, it can play a role in the error of the extracted phase. Once the experiment jumps off, the system accelerates for a short time. During this acceleration, the frequency changes rapidly, representing a fast phase change between the MBR branches. Once the flyer begins to move at a constant velocity, the frequency stops changing and is just a constant offset between the upper and lower MBR branches. To overcome this effect, the phase response of each photodetector was measured independently. This was accomplished by sending the output of our microwave source to a 1 × 2 splitter. Half of the signal was sent to a modulator with a PDV laser passing through it. The output of the modulator was then recorded by the photodetector under test and the signal was recorded by a digital oscilloscope. The other half of the microwave source signal was sent directly to another channel of the oscilloscope. The delays and types of cables were kept as similar as possible. Next, the microwave source was varied across all frequencies from 0 to 25 GHz in steps of 2.1 MHz. The phase difference between the two channels was then measured in software and a correction plot was made. Figure 7(a) shows three scans of a photodetector. Clearly the phase of the photodetector changes by several radians across the full spectral bandwidth of the detector. The three rectangles in Fig. 7(a) represent (from left to right) the PDV, lower MBR, and upper MBR traces with the full changes in frequency, respectively. Based on this result, it is not expected that a photodetector that is not corrected for its intrinsic phase shift will give the same answer from displacement for both the lower and upper MBR traces. Once the result in Fig. 7(a) was obtained, it was used as a phase correction on the raw time-domain data. The results of the photodetector phase correction can be seen in Figs. 7(b) and (c). The negative value of the upper trace from 0 to 1.5 µs is due to the photodetector phase shift being more negative than the calculated displacement.
Fig. 7. (a) Photodetector phase shift measurement for the detector used in the flyer plate experiment. The three rectangle represent the spectral bands associated with the PDV, lower MBR, and upper MBR traces. (b) The uncorrected upper (red line) and lower (black line) MBR traces. (c) The upper (red line) and lower (black line) MBR traces after applying the photodetector phase correction from Fig. 7(a).
Additional sources for phase error may include the response of the recording oscilloscope and the modulator itself. However, it was expected that these two should be significantly smaller than the phase effect of the photodetector itself and could be ignored for now. This assumption seems to be consistent with the results seen in Fig. 7(c). Besides the photodetector phase correction, there were no other phase corrections that were considered for the benchmark system.
The results for the flyer plate test can be seen in Fig. 8. The extracted velocity used is shown as the blue line in Fig. 8(a). This is the velocity extracted from the spectrogram in Fig. 6(a). The MBR displacement (black dots) is plotted on top of the integrated PDV data (red line). On this scale there is excellent agreement and no differences can be seen. Figure 8(b) shows the discrepancy between each as the difference between the MBR displacement and the integrated PDV value. A root mean square (RMS) value for the spread in the data is below 50 µm which is our predicted resolution. The error increases as the data gets weaker which is an expected artifact of the three cosine fitting algorithm. Over the course of the total data set, the RMS spread in the data varies from ∼15 um at early times to over 50 um at late times. This represents a phase change of as small as 1/660th of a fringe which exceeds our 1/100th of a fringe estimate. More accurate phase extractions could improve the expected resolutions but this has not yet been a driver in the predictive models used with our experiments. It should be noted that this error also exists in the direct velocity quantification as well. Error in MBR and in velocity are both related to the signal to noise ratio. Overall the two diagnostics give identical results for the flyer plate motion.
Fig. 8. (a) The extracted MBR displacement (black line) and Integrated PDV (red line) plotted against each other. Differences are not visible on this scale. The extracted PDV velocity (blue line) is also shown for reference. (b) The difference between the MBR and Integrated PDV displacements. RMS errors are less than 100 µm.
3.2 Chromatic dispersion
Up to this point, the results shown were taken with a simplified single-channel PDV system. Due to the nature of large scale dynamic hydrotests, however, over a hundred PDV points are typically fielded on a single experiment. This necessitates the use of multiplexed PDV systems which add a significant degree of complexity to the experiments. Figure 9 shows a simplified schematic of an MPDV system that has been time multiplexed with 100 µs between each consecutive channel [11].
Fig. 9. (a) Simplified multiplexed PDV system schematic with MBR. The system is very similar to the benchmark setup from Fig. 4 until the light returns to the circulator. The outputs of each circulator are combined onto a single fiber via a wavelength division multiplexer (WDM) and then filtered and delayed. The value of the delay, τ, is set by the length of the dispersion module and each channel is an integer multiple of τ.
The setup is similar to the benchmark system shown in Fig. 4, just scaled up for additional channels. A bank of 8 separate lasers from NKT photonics is used as the base PDV lasers. After modulation, the lasers art split 8 ways (not shown for simplicity) and are sent to a Keopsys EDFA. The power of each laser is amplified to ∼10 mW before being sent to the DUT. The return light from the DUT is muxed with a wavelength division multiplexer (WDM) and sent to a reconfigurable optical add-drop delay module (ROADDM). This allows each signal to be picked off and delayed by a certain amount based on the wavelength of the launch light [11]. The ROADDM allows for 16 distinct PDV signals to be multiplexed onto a single scope channel although only the first 8 are shown in the figure for simplicity. The output from the ROADDM is combined with a set of muxed reference lasers to generate the low frequency beat note. The time-multiplexed signals are then sent to a MITEQ 20 GHz bandwidth photodetector. The signal is recorded on a Tektronix 23 GHz oscilloscope.
The ROADDM represents a total additional delay length of ∼20 km per channel to give a total delay of 100 µs. This produces a substantial amount of chromatic dispersion added to each channel that only increases as the channel count increases. Indeed, the final channel of a 16 channel time multiplexed system goes through 300 km of additional fiber as compared with the first channel. This is done to increase the number of signals recorded on a single oscilloscope channel, significantly reducing the number of oscilloscopes required. The photodetectors are the same type used for the benchmark system so an identical photodetector phase correction is done for each detector and can be applied to all time multiplexed channels recorded by that particular detector. However, because the delay increases by 20 km for each sequential channel, a different amount of dispersion is experienced and must be individually corrected.
The chromatic dispersion acts on the three distinct signals seen in Fig. 2(b) that are sent to the device and propagate through the system. While the three signals are only separated by 15 GHz in a typical experiment and would have a nearly negligible amount of phase distortion, this can become significant by the later delayed channels. The phase offset between the upper or lower bands and the carrier can be estimated by:
In Eq. (8), n(λU,L) is the wavelength dependent refractive index of the fiber for the upper (U) and lower (L) MBR sidebands, respectively. The carrier wavelength of typically 1550 nm is denoted by λ0. λU,L are therefore (λ0+c/fm) and (λ0-c/fm) for the upper and lower sidebands, respectively. The length, L, represents the delay of each channel and will just be an integer multiplied by 20 km for our particular MPDV system. This is a similar effect to the photodetector phase correction in that the phase correction is frequency dependent and must include the effect across the total bandwidth of the recording scope. For constant velocities, the phase shift from dispersion is a constant while the phase offset changes rapidly during acceleration of the DUT. An estimate for the refractive index of the optical fiber used in our setups was found via a literature search for Sellmeier equations of corning SMF-28 optical fiber [21]. Estimates for the expected phase delay shows errors on the order of 1-2 mm. However, it should be noted that the Sellmeier equations are not accurate enough for determining the phase offsets of two wavelengths centered at 1550 nm but separated by only 15 GHz. Very small errors in the calculated index can change the phase offset substantially. Because of this, an experiment was devised to measure the phase offset for our system. This was done using a flyer plate geometry identical to that shown in Fig. 5(a) but with all 16 time multiplexed outputs from a single scope channel multiplexed and sent to a single PDV probe. By doing this, all 16 channels recorded the exact same velocity since they interrogated the same point on the flyer plate. The flyer plate achieved its maximum velocity of 2 km/s after ∼5 µs and then drifted at this velocity for an additional ∼7 µs. MBR displacement was then calculated for all channels and compared with the integrated PDV. Figure 10(a) shows the difference between MBR and integrated PDV. It should be noted that the launch power of 10 mW per channel was significantly less than the 200 mW of launch power used with the benchmark system of Fig. 4. This causes an increased level of noise between the two experiments.
Fig. 10. (a) The difference between the MBR and the integrated PDV for the odd channels of the 16 channel MPDV system. Only the odd channels are shown for clarity. Each consecutive channel shows a larger amount of error due to the increased dispersion. (b) The resulting difference of each channel after correcting for the dispersion. Note that the vertical scales are not the same.
Only the odd channels are shown for clarity in the figure. Since we can assume all velocity traces should be exactly the same, we can see that the effect of 20 km of dispersion adds ∼ 1 mm of phase delay that must be corrected. Each channel was also found to be simply an integer multiple of this correction, just as predicted. These individual phase corrections can be added spectrally or, since we can assume the effect is uniform across the full bandwidth of the scope up to the second order phase, in the time domain. For our correction, the displacement, x, needs to be shifted by xcorr = [x – cvn], where xcorr is the chromatic corrected displacement, v is the instantaneous velocity, n is the integer multiple of 20 km dispersion and c is a calibration constant. For our experiments, c was found to be ∼550 ns. After applying the correction, the results can be seen in Fig. 10(b). It should be noted that the vertical scales for Figs. 10(a) and (b) are not the same. The correction results in a spread that approaches our expected limit of error but is limited by the reduced signal strength of the multiplexed system as compared with the benchmark system.
3.3 Error estimates
The fitting parameters in the three cosine fit can yield some uncertainty in the extracted values of the phase for each of the three signals. Typically a threshold is set on the data and if the fitting algorithm fails, the extracted phase values are omitted. However, these are much smaller than the errors so far discussed. Table 1 shows the amount of error introduced before and after the phase corrections are included. By far the largest source of error arises from the chromatic dispersion in the fiber. This is a systematic source of error but, by limiting the fiber runs to much less than 1 km, the effect is negligible. However, this is not really feasible for the next generation multiplexed PDV systems, so the correction will have to be considered. We have also found that this can be reduced to a negligible level even when higher order phase distortions are not considered.
Table 1. Sources and estimated contributions of error in MBR
The photodetector phase correction is the second largest source of systematic error but this too can be reduced to nearly zero with the proper correction applied. Other sources of electronic equipment errors could be the recording scope, microwave source, cables, and possibly the modulator itself. However, these have thus far been found to be negligible compared to other sources of error and have not yet been considered. In the static portions of the records we are able to extract the phase stability of the microwave source so we can verify on each experiment. We have also conducted measurements where the static signal is recorded independently and we have found the source and modulator to be sufficiently stable.
The next source of error is the bias in the dataset itself. As this is a displacement measurement and not an absolute position measurement, the data can always be shifted in position to agree with the PDV. The source of error arises from the fact that an accurate measurement of the start of the data may not be known. For example, if the PDV signal is weak at early times, the three cosine fit might fail for several of the first data points. The starting position would therefore be arbitrarily shifted to an incorrect offset. We have found that this bias results in an error of up to 200 µm although by paying extra attention to where jump-off occurs this too can be reduced to negligible levels. This source of error is channel specific because it depends heavily on the experimental conditions that an individual PDV line-of-sight is subjected to.
The final source of error listed in Table 1 is the scatter in the experiment. This is a random source of error and is intrinsic to the recorded data. By using higher power lasers, we can reduce this level to 50 µm while lower power lasers can increase this error substantially.
All of the values together in Table 1 give a final accumulated error of 50 µm for a high power PDV/MBR setup after corrections are made. The extracted phase uncertainty is much less than this meaning the dominant source of error will always be the signal to noise level of the experimental setup. It should also be noted that, with the exception of the scatter error, all other sources of error are found in the analysis of the raw data. As the error corrections are further refined, the data can be revisited and reanalyzed for improved datasets in the future.
3.4 MBR sensitivity to non-radial flow
The final test of the MBR diagnostic was on an experiment where a significant amount of non-radial flow was expected. The point of this test was to demonstrate that MBR can give additional information on the behavior of a complex dynamic experiment that is unobtainable by PDV alone. Figure 11(a) shows the experimental geometry of the setup. A cylindrical charge of PBX 9501 was manufactured with a hemisphere of 2 inch radius removed from the center. A 2.54 mm thick shell of a titanium alloy (Ti-6Al-4V), was fit into the hemisphere.
Fig. 11. (a) Experimental setup of a shaped charge that results in significant non-radial flow. Points A and B are two probe locations that represent highly radial flow, and highly non-radial flow, respectively. (b) Expected velocity components at early (t0) and late (t1) times after detonation for probe position A and B. In the case of probe A, the velocity magnitude increases but the direction remains constant while probe B has both a magnitude and direction change.
The geometry of the HE compresses the shell and acts as a shaped charge, causing significant jetting of the shell at late times. An estimate of the shell behavior is shown shortly after jump off at t0 and at a later time t1. A series of PDV probes were mounted inside a hemisphere cup equal distance from the shell. Two points are shown for clarity. Point A viewed directly up towards the location of the detonator. A formed jet would then propagate directly towards the probe for all times. Point B was located at a large polar angle. Since the shell does not converge onto the center of the system, this point would show significant non-radial flow of the metal surface. Figure 11(b) shows the effect of the probe geometry. For point A, the velocity vector always points toward the PDV probe for fully radial flow and as such the total velocity will be recorded at all times. For point B, the velocity vector significantly changes direction as time progresses. Since the PDV diagnostic only measures the component of the velocity that lies along the probe’s line-of-sight, this eventually represents a significant underestimate of the total velocity. Integrating the velocity to determine the displacement would also result in a significant error. In both cases however, MBR is simply measuring the displacement of the surface along the probe’s line-of-sight and will not be subject to this error.
Figure 12 shows a summary of the results from these two probe’s points. Figures 12(a) and (b) show the spectrograms for PDV point A and B, respectively. The shell in the experiment was expected to fragment and as such it was unknown what the final velocity would be. It was decided to invert the heterodyne laser by shifting the LO center wavelength to a shorter value. This meant that at some point the beat frequency would be zero as the base laser blue-shifted to the same value as the LO. This could be easily handled by the MBR and PDV extraction algorithms and gave the benefit of increasing the useable bandwidth of the scope. Figure 12(a) shows that without inverting the heterodyne laser, the velocity trace for the upper MBR branch would have gone outside the bandwidth of the scope.
Fig. 12. (a) and (b) show spectrograms for a highly radial and highly non-radial flow point for the experiment shown in Fig. 11, respectively. (c) and (d) show the extracted MBR (black dots) and integrated PDV (red line) displacements for the corresponding spectrograms. The velocities (blue lines) are shown for references. Figure (d) shows significant disagreement at late times between the MBR and integrated PDV. (e) and (f) show the differences between the MBR and integrated PDV displacements from (c) and (d), respectively.
In the spectrograms, point A shows a very well defined velocity while point B shows significant discontinuities in the velocity. This is explained as the shell fragmenting and various surfaces moving into and out of the line of sight of the probe. Point A would be less sensitive to this effect due to its line-of-sight viewing the highest velocity fragments in the experiment. In the actual experiment, point A was found to be off the center by a 5.2° polar angle. This does impart a small amount of non-radial flow sensitivity to the PDV measurement but it was expected to be much less than point B. Figures 12(c) and (d) show the extracted MBR (black dots) plotted with respect to the integrated PDV data (red line) for point A and B, respectively. The extracted velocity (blue line) is also shown for both points. Point A had a relatively smooth and continuous velocity trace while point B, as expected from the spectrogram, showed a lot of discontinuities in the velocity trace. The MBR fitting algorithm was able to read across the discontinuities with very little difficulty giving many points of direct displacement measurements as seen by the black dots in Fig. 12(d). The gaps in the MBR data arise from low signal. These are present in the integrated PDV data as well but the data is interpolated through the missing points. The deviation of the MBR from the integrated PDV is very significant for point B, as expected. Figures 12(e) and (f) show the difference between the MBR and the integrated PDV result. For point A, the error of Fig. 12(e) is much less than 1 mm for almost all of the trace. It is not really visible on the scale of the figure, but the error did increase to ∼1 mm starting around 20 µs. This can be attributed to the small non-radial component of the velocity at late times. The effect in Fig. 12(f) shows a significant difference of over 20 mm by the end of the trace. Relying on integrated PDV alone would have resulted in a significant error.
4. Conclusions
In this work we developed a diagnostic capable of measuring directly the displacement of a dynamic surface that serves as a complement to PDV diagnostics. By measuring the displacement of the surface directly, PDV velocities can be corrected. However, since the point of PDV is often to determine the displacement via integration of the velocity, MBR can provide the data unambiguously. In the benchmark validation experiments using a simplified flyer plate geometry and experimental setup, the error of the MBR displacement measurement was found to be less than 100 µm. Using lower power PDV lasers with lower signal to noise ratios increases this error but this is not a limitation in the hardware. As MBR is a phase extraction diagnostic, the spectral phase response of the operating hardware and the chromatic dispersion of the optical fiber can play roles in introducing errors to the extracted MBR displacements. These are however straightforward to estimate and correct experimentally. Under these most ideal conditions, integrated PDV and MBR agreed to less than 100 µm RMS.
With the expected errors characterized and a comparison made between integrated PDV and MBR, a more complex experiment was conducted. The experiment was expected to exhibit significant non-radial flow for PDV probes interrogating at large polar angles. MBR demonstrated an ability to directly observe the displacement of the surface and indicated that integrated PDV alone would result in an error in the displacement of over 2 cm. For experiments where the flow is expected to be non-radial, MBR would be the preferred method for determining the displacement of the surface. Further refinement of the fitting algorithm and improvements in the signal to noise ratio of the data can drive down the error in direct displacement measurements in the future.
Acknowledgements
The authors wish to thank Phil Miller, Larry Hull, and Brian Esquibel for assistance in fielding the non-radial experiments.
Disclosures
The authors declare no conflicts of interest.
Supplemental document
See Supplement 1 for supporting content.
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