Technology and times scales in Photonic Doppler Velocimetry (PDV)

Photonic Doppler Velocimetry (PDV) is a fiber-based measurement amenable to a wide range of experimental conditions. Interference between two optical signals—one Doppler shifted and the other not—is the essential principle in these measurements. A confluence of commercial technologies, largely driven by the telecommunication industry, makes PDV particularly convenient at near-infrared wavelengths. This discussion considers how measurement time scales of interest relate to the design, operation, and analysis of a PDV measurement, starting from the steady state through nanosecond resolution. Benefits and outstanding challenges of PDV are summarized, with comparisons to related diagnostics.


Introduction
Velocity is fundamental to most observations of a mechanical system.Natural questions like: • How quickly is an object moving?(steady motion) • Does the object's velocity change over time?(acceleration) • Do adjacent points on the object stay together or move apart?
(strain) • How do physical distortions travel through the object?(wave propagation) can all be expressed in terms of velocity.Measuring velocity may thus be important on its own or for determining another physical quantity.Characteristic time scales for these measurements range from real-time experience (∼1 s) to far below the limits of human perception (≪1 ms).
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Time-resolved velocity measurements are particularly useful in shock-wave physics.Knowledge of material velocity u p and wave velocity U s can be combined with conservation laws: [1] to determine material density ρ and longitudinal stress P (pressure in the hydrostatic limit) behind a one-dimensional steady shock.Such experiments reveal material conditions well outside human experience (thousands to millions of atmospheres) from velocity measurements.Photonic Doppler Velocity (PDV) [2,3] is an optical technique for tracking velocities from less than 1 m s −1 to more than 10 km s −1 .The diagnostic is conceptually simple but was entirely impractical for many applications until the beginning of the twenty-first century [4,5].This article considers how PDV measurements are defined by the experimental time scale of interest.Key principles of PDV-underlying technologies, basic analysis, and its benefits-are described in section 2. Section 3 journeys through experimental time scales where PDV is used.Section 4 summarizes the current state of PDV and outstanding challenges.Appendix A discusses related velocity diagnostics.Appendix B considers noise limits in PDV, demonstrating that multi-signal measurements are not intrinsically superior to the standard approach.

Diagnostic overview
Figure 1 shows a schematic diagram for a typical PDV system.Solid lines indicate single-mode fibers carrying optical signals, dotted lines illustrate the free-space path to/from the target, and the dashed line shows the electrical connection between the optical receiver and a real-time digitizer.One coherent source is connected to port 1 of a fiber circulator, while a second coherent source is connected to a 2 × 2 fiber coupler.The former emerges from circulator port 2 and travels to an optical probe, which illuminates a moving target and collects the reflected light.Return light enters circulator port 2 and exits from port 3, where it is mixed with light from the second coherent source.Interference between the two optical signals leads to beat frequencies at the receiver, which are recorded by a high-speed digitizer.There are a variety of alternate PDV configurations, [3] but this is the most common.Common support devices, such as inline power meters and attenuators, are omitted here for conceptual clarity.

Enabling technologies
PDV is built around technologies from the telecommunication industry [6].Minimum light absorption for silica fiber occurs near 1550 nm, and the vast majority of PDV systems operate in this range.Single-mode fiber, such as Corning SFM-28, transmits light at the 0.2 dB km −1 level with minimal dispersion.These fibers typically have a germanium-doped core and pure silica cladding, though puresilica core fiber (with fluorine-doped cladding) is also an option [7].
Erbium-based fiber amplifiers are particularly effective in the 1530-1565 nm (the 'conventional' C band) range.Turnkey systems readily boost ∼1 mW seed lasers to continuous power levels of several Watts.These seeds are typically centered around standard grid points, [8] such as ITU 34 (1550.12nm, 193.400THz), and may be tunable over 50-100 GHz (or more) of optical frequency.Remarkably, the absolute stability of these lasers can be a few parts per billion (MHz scale) [9,10].
A host of fiber components have been developed because of the above technologies.Particularly relevant to PDV are the optical circulator and 2 × 2 coupler shown in figure 1. Circulators transport light from ports 1 → 2 and 2 → 3 with very low loss (<1-2 dB), while at the same time blocking 1 → 3 and 2 → 1 transit with 40-60 dB isolation.Couplers mix two input signals at some power fraction, usually 90% from the target and 10% from the reference (source 2).Due to energy conservation, [11] light is always divided between two outputs, only one of which is typically used.
High-speed signal acquisition is the final component of modern PDV systems.Starting at the beginning of the 21st century, real-time GHz signal measurements have gone from state of the art to routine.Amplified InGaAs detectors, which are highly sensitive to 1550 nm light, are now available with >20 GHz bandwidths.Realtime digitizers supporting ⩾100 GHz bandwidth are also commercially available.With few exceptions (noted below), there is ample recording bandwidth for most PDV measurements.

PDV in practice
Figure 2 shows one implementation of the passive components for a PDV system.Five FC/APC fiber connectors bring light into and out of the left side of this printed-plastic box.Sources 1-2 enter through the top two connections.The bottom three connections are for the optical probe, an external power monitor, and the optical receiver (respectively).Power adjustments can be made with the manual attenuators between connector groups.The simple appearance of various components, especially the circulator, belies remarkable sophistication within these devices.
Figure 3 shows a complete 'PDV in a box' system within a PXIe chassis.This particular implementation supports four independent probes, with: • Eight tunable lasers in the left two slots.
• Four circulator/coupler (Doppler) modules in the center slots.• Four optical receivers in the right two slots.
Each Doppler module in this system performs similar functions as the module shown in figure 2, with computerized power adjustment and monitoring.This system is configured for an external digitizer, although that capability could be incorporated into adjacent chassis slots for some bandwidth ranges.
Although there is no fundamental distinction between PDV and Laser Doppler Velocimetry (LDV), [12,13] the latter typically operates at lower velocity and longer time scales than the former.For example, an LDV measurement may be scanned across a moving surface, [14] whereas beam rastering is typically too slow to be useful in PDV measurements.PDV systems are almost exclusively fiber-based and strongly favor the optical C-band, whereas LDV may use bulk components and often operate at visible wavelengths; having said that, fiberbased LDV is not uncommon [15].Distinguishing a 'slow PDV' from a 'fast LDV' largely comes down to application and community practice.PDV is the favored term in singleevent, destructive experiments: plate impact (gas/powder gun launchers), explosive detonation electrical pulsed power, and intense laser drive; many examples are documented in the ongoing series of PDV user workshops [16].LDV is more typically found in non-destructive settings, such as vibration and fluid flow [17] studies.

Measuring Doppler shift
PDV systems present the receiver with two optical signals: one that has travelled to/from the moving target and one that has not.Optical interference between these signals: depends on the optical power (P 1 and P 2 ) from each path and the optical phase difference Φ(t) between paths.The electrical output: depends on the receiver's responsivity K (V W −1 ) and the optical phase difference Φ(t) ≡ ϕ 1 (t) − ϕ 2 (t).Only the interference term is shown here because most PDV systems use AC coupled receivers [3].Electrical beat frequency is time derivative of optical phase difference: which depends on source wavelengths λ 1 and λ 2 as well as the target velocity v. Beat frequency changes with velocity scaled by 2/λ, which at 1550 nm is roughly 1.3 GHz per km s −1 .This scaling sets bandwidth requirements for measuring velocity with PDV: a 1 GHz digitizer/receiver combination is capable of measuring velocity changes as large as 775 m s −1 .Shorter wavelength systems are more sensitive to motion, e.g. a 532 nm PDV [18] is nearly 3× as sensitive as a 1550 nm PDV, yielding 3.8 MHz of Doppler shift per m s −1 instead of 1.3 MHz.However, the maximum velocity range of such a system would be reduced to 266 m s −1 for 1 GHz bandwidth, and many of the technology benefits mentioned in the previous section are lost.Alternate wavelength systems can also have experimentspecific advantages: mid-infrared PDV [19] can see through non-conductive materials opaque to visible and near-infrared light.Figure 4 illustrates how beat frequency varies with velocity.For systems having only one optical wavelength, i.e. source 2 is merely a fiber split from source 1, equation ( 5) reduces to 2|v|/λ.This means that the beat frequency is zero when the target is at rest, and beat frequency always increases with speed.PDV measurements using this configuration are thus directionally blind.When source 1 has a shorter wavelength than source 2, the system is said to be 'upshifted' because beat frequency increases with motion towards the probe; motion away from the probe has the opposite effect.When the target is not moving, the receiver continuously beats at initial frequency B u , providing an initial signal for digitizer optimization.The opposite configurationsource 1 having a longer wavelength than source 2-is less common but has important applications.Here velocity initially decreases with velocity, until at some point there is a 'bounce' after which beat frequency increases with velocity.Initial frequency B d can be set as high as the recording system's bandwidth, potentially doubling the range of measurable velocity with detectable beat frequency on either side of that bounce.Combining several frequency-shifted measurements allows 'leapfrog' PDV to span enormous velocity ranges [20].Figure 5 shows an example signal for a hypothetical PDV measurement.In this measurement, the target begins at rest and then constantly accelerates from some point in time.Using an upshift configuration, the PDV is always beating, even when the initial velocity is zero.Beat frequency increases with target velocity, so signal cycles occur more quickly as the target accelerates.Dense signal beats can be difficult to resolve by eye, but this is a limitation of human vision and/or graphic display, not PDV itself.To the uninitiated eye, PDV signals often appear to be noise, but careful zooming usually reveals a periodic signal.

Benefits of PDV
Operational simplicity is a key benefit of PDV, which is why other diagnostics (notably VISAR and Fabry-Perot velocimetry) have largely been replaced by PDV in many applications.Apart from the probe-target region, the entire optical path is confined to optical fiber, making internal misalignment essentially impossible.PDV systems can be modularized, miniaturized, and ruggedized to an extent unimaginable for many optical diagnostics.Furthermore, PDV uses one signal measurement per optical probe, whereas most optical measurements require 2-4 (sometimes as many as 8) signals per probe.
Many telecommunication practices can be applied to PDV.For example, time-domain and frequency-domain multiplexing can combine several probe measurements onto a single channel, vastly reducing the number of digitizers needed.A 32-channel multiplexed PDV system using one fourchannel digitizer received an R&D 100 award in 2012, [21] which would be largely impractical outside of the 1530-1625 nm range.Optical paths can also be modified on millisecond to nanosecond time scales with low-loss fiber switches in this spectral range.Furthermore, ultra-precise (≪1 ns) transit time measurements are possible via optical backscatter reflectometry; [22] unlike time-domain reflectometry, these swept-wavelength measurements have no minimum path length.
The combination of weak Doppler-shifted light with much stronger reference light allows PDV to handle a wide range of targets and probe designs.For example, interference between 1 µW and 1 mW optical signals has an amplitude of 63 µW (equation ( 4)).This leads to more than 100 mV peak-peak electrical signals for a 1000 V W −1 receiver, well above the noise floor of a high-speed digitizer.Interference of independent optical signals causes electrical amplitude to scale with the square root of target  return (equation ( 4)), dramatically enhancing PDV's dynamic range.
Another important benefit in PDV measurements is that time-domain linearity is not required.For example, real-time signals might clip unexpectedly during a measurement, introducing flat regions catastrophic to amplitude-and phase-based measurements.Frequency-based PDV merely see additional harmonics of the fundamental frequency, so measurements proceed without interruption.As such, PDV often functions in situations where other diagnostics would have failed; note this benefit does not apply to frequency-multiplexed PDVif one signal component clips at the digitizer limits, every component of that channel is clipped.Signal balancing and/or impulse response corrections are not required-as long as the detector/digitizer can keep up with the beat frequency, velocity measurements can be made.
A final benefit of PDV is that it does not require a single velocity for the measurement to be successful.This is a stark difference from VISAR (appendix A.1), where the presence of more than one velocity at a time destroys interference contrast, entirely compromising the measurement.Multiple velocities may arise from actual physical conditions, such as distribution particle motion in a debris cloud, [23] or because of optical reflections in the measurement [24].Human-guided region of interest selection may be needed to extract distinct velocities, but the presence of more than one velocity can be manageable for PDV.

Measurement time scales
The beat frequency of a PDV signal cannot be inferred from individual points-local analysis regions of time duration τ are required.Figure 6 illustrates this process for short-time Fourier transform (STFT) analysis, the most common technique used in PDV data processing.A digital window [25] is swept across the measured signal, applying a fast Fourier transform at each location.The red signal s(t) in the top-left plot of figure 6 shows a hypothetical PDV signal.Black curves indicate two positions for the window function, with the window-signal product shown in the top right plot; time shift is defined with respect to window center.The bottom plot shows power spectra calculated from each signal-window product.This process is repeated many times in PDV analysis, moving the window center to every location of interest and storing results (either the full power spectrum or merely peak locations) along the way.
There are three fundamental time scales in every PDV measurement.The duration τ controls the digital window size for every FFT, while the advance δ defines the time step between FFTs.The total measurement time ∆ depends on the system under study and has implications for PDV system design.In general:

Local analysis duration τ
Although τ can be selected after the measurement is complete, its value defines the limiting rise time.A general rule of thumb is that the 10%-90% response to instantaneous velocity change is roughly τ /3, with some variation due to the digital window shape [26].Maximum values for τ are therefore constrained by underlying dynamics of interest.For example, 1 ns velocity changes cannot be effectively resolved with analysis durations greater than about 3 ns.The well-known uncertainty principle links spectral width to τ −1 : [27] longer durations concentrate spectral power more tightly than shorter domains.However, the limiting beat frequency uncertainty σ B is actually proportional to τ −3/2 , not τ −1 [26] Here f s is the digitizer sample rate, σ s is the RMS signal noise, and A s is the beat signal amplitude.The number of samples per FFT (N ≡ f s τ ) strongly influences beat frequency uncertainty.Noise fraction (σ s /A s ) in PDV can be 1%-10% in ideal measurements (fibers imaged onto a specular surface), so part per million uncertainties are plausible even at modest sampling, e.g.N = 100 points per local analysis.For 1550 nm measurements, this works out to 0.2-2 m s −1 for a 10 GS s −1 digitizer.Every 100× increase in the analysis duration results in a 1000× decrease in velocity uncertainty.The noise fraction σ s /A s is related to the inverse square root of target power (Appendix B).
Analysis duration is also related to the minimum resolvable beat frequency.This is most obvious in conventional systems (λ 1 = λ 2 ), where beat frequency is directly proportional to target velocity.At low velocity, say 1 m s −1 , beat frequency is 1.29 MHz and the beat period is 775 ns.Beat frequency is poorly defined on time scales shorter than the period, so even steady velocities cannot be tracked until at least a full cycle has elapsed; any sense of velocity change is completely obscured on such time scales.Frequency-shifted measurements (λ 1 ̸ = λ 2 ) avoid this problem by providing an arbitrary number of fringes when the target is at rest, so minute velocity changes (subject to limiting uncertainty in equation ( 7)) can be detected on short time scales.However, frequency shifting does not completely solve this problem-it merely moves it to some other velocity range.Downshifted PDV approaches B = 0 at the bounce velocity, and measurements in this domain suffer from the incomplete cycle problem.The solid black line in figure 4 illustrates this 'no man's land', a region where frequency cannot be reliably estimated using time scale τ .The vertical span of this region is proportional to 1/τ and varies with digital window shape; for the Hann window, it happens to be 2/τ .Mathematically, full-cycle requirements can be expressed by the separation of spectral peaks at positive and negative frequency.
Target acceleration is a final consideration in selecting PDV analysis time scale.Spectral peak width only decreases with τ for fixed-frequency signals.Significant velocity changes during τ broaden peaks beyond the uncertainty principle limit.This effect can be seen in figure 6, where the later transform (dotted line) is wider than the earlier transform (solid line); peak amplitude is also reduced due to energy conservation.Reducing τ minimizes acceleration broadening, also known as chirp, but once again there are practical limits.Not only does limiting uncertainty rapidly increase with diminishing τ (equation ( 7)), it becomes increasingly difficult to meet fullcycle requirements for ultrashort analysis duration.If the minimum detectable beat frequency is 2/τ for a Hann window, analysis durations must be longer than 4/f s for any usable information to fall within the Nyquist limit.Minimum measurable frequency needs to be no more than perhaps 5%-10% of the Nyquist frequency to utilize most of the digitizer's bandwidth, which is to say that local analysis usually needs at least 40-80 sample points per FFT (before zero padding).

Local analysis advance δ
The time advance parameter δ controls the amount of time between local FFTs.For best results, δ should always be less than τ and possibly as small as the time between digitizer samples.Smaller values of δ more finely mesh velocity analysis but do not determine the limiting rise time or limit measurable velocity in any way.The two window functions shown in figure 6 are separated by several advances for visual clarity; adjacent calculations would be strongly overlapped.
Computational time scales inversely with δ.Analysis usually begins with δ = τ /2, incrementally stepping down to δ = 1/f s only after regions of interest have been selected from a coarse spectrogram [28].Small time advances are often practical when only peak positions, not entire spectra, are stored.When each FFT contains N sample points, there are at least N/2 frequency points to be stored.The minimum number of time-frequency points in a full-resolution spectrogram: can easily reach L = 10 9 -10 12 for modern digitizers.

Experiment duration ∆
There are intrinsic links between total measurement time ∆ and velocity magnitude-typically the latter decreases with the former-in most experiments.For example, 10 km s −1 velocities tend not to persist for more than a few microseconds, whereas subsonic motion may last for milliseconds to seconds (or longer).Furthermore, time resolution requirements tend to scale with measurement time, where τ = 0.1%-1% of ∆, even for sample rates that could support shorter analysis durations.
Although memory depth can be a limiting factor, most PDV digitizers can support almost any experiment of interest unless velocities are particularly fast or signals are sequentially multiplexed onto the same digitizer channel.Measurement uncertainties can be extremely small in millisecond to second experiments.Consider a 10 000 point FFT, e.g. a τ = 10 µs at 1 GS s −1 , characteristic of millisecond velocity measurements.The -3/2 power in equation ( 7) limits random uncertainty below 1 ppm, even when random noise equals signal amplitude.Absolute frequency uncertainties are therefore below 1 kHz or better, putting velocity uncertainty less than 0.001 m s −1 .Beat frequency drift is a bigger concern for such experiments, particularly for two-laser systems.Without active stabilization, beat frequencies can vary by 1-100 MHz over millisecond time scales; variations can be random (thermal drift of each lasers) or systematic (dithered laser tuning).An acousto-optic frequency shifter is generally a better solution than a second laser in this domain-even if the primary laser wavelength drifts over time, that variation is carried on both target and reference paths and automatically cancels out.Most LDV systems rely on fixed-frequency shifters, but this approach is less common with PDV.
Microsecond to millisecond experiments still benefit immensely from the −3/2 power uncertainty scaling.At 1000 points per FFT, e.g.τ = 100 ns at 10 GS s −1 , beat frequency uncertainties are tens of parts per million, or <1 MHz for a moderately fast (10 GS s −1 ) digitizer.Velocity uncertainties below 1 m s −1 are certainly plausible, even for poor light return.Single-event triggering becomes more important than in longer/slower experiments, and the observation of multiple velocities may be more likely.Other than increased bandwidth requirements, few technology changes are required in this domain.
Several PDV modifications become important for nanosecond to microsecond experiments, where 100 points or fewer are used per FFT.As N −3/2 moves into the 0.1% range, it is desirable for signal amplitudes to be larger (10-100×, if possible) than the noise floor.The ratio σ s /A s is fundamentally limited by photon noise, so having more target light P 1 is beneficial; although reference light P 2 is important for obtaining signals above the digitizer noise floor, additional power increases signal amplitude and shot noise in the same way (appendix B).Probe efficiency and laser power both contribute to the success of PDV measurements in this domain.Whereas eye-safe lasers may be sufficient in longer/slower experiments, 10-100 mW power delivery is often needed here.Consider an optical probe with −40 dB return (0.01%) and +20 dBm (100 mW) of input power.Doppler-shifted power levels would be -20 dBm (10 µW), which when mixed with 0 dBm (1 mW) of reference light yields as much as 200 µW of interference power (200 mV for a 1000 V W −1 receiver).For ∼2 mV noise floor digitizers, noise fractions of 10% are plausible, supporting 80 ppm beat frequency uncertainties.Sample rates tend to be >10 GS s −1 , however, so absolute uncertainties are 1 MHz or larger.Velocity uncertainties tend to be 1-10 m s −1 , often acceptable when motions of interest are 0.1-10 km s −1 .
PDV measurements in the nanosecond to microsecond range tend to rely on a second laser instead of an acoustooptic frequency shifter.Beat frequency drift is less of a problem in short experiments-even though this drift is obvious in sequential digitizer sweeps, it is too slow to be noticed over the interesting part of the actual measurement.While the absolute position of the first spectral (solid) peak in figure 6 depends on the lasers' exact optical frequency, the horizontal difference between that peak and a later (dotted) peak does not.Twolaser PDV systems are also very flexible: upshift frequency can be arbitrarily changed and/or swapped to a downshift.The extreme stability of an acousto-optic frequency shifter, which can span six or more significant digits, and limited range (<1 GHz) are less useful in nanosecond to microsecond experiments.
Ultrashort experiments, where τ < 1 ns, can be difficult for PDV.Although 100 GS s −1 and faster sampling is commercially available, the −3/2 power scaling becomes a problem for ∼10 points per FFT.Even at 10% noise fraction, relative frequency uncertainty reaches 0.25% of an already large number, pushing absolute uncertainties to the GHz range (km s −1 velocity uncertainty).At the same time, the range of measurable frequencies is limited by full-cycle requirements mentioned above.For example, a 0.10 ns FFT cannot detect frequencies below 20 GHz.Presumably the bandwidth of such a system would be larger than 20 GHz, but a significant portion of the frequency range would be useless.
Time-stretch PDV [29] converts large Doppler shifts on short time scales to measurable signals using femtosecond lasers; more recent variations include time-lens PDV [30,31].These approaches are substantially more complicated to build/operate than continuous PDV and the results have an intrinsic sparseness: one must either choose limited total coverage (say 10-100 ns) or tolerate intermediate gaps during the measurement.Synchronization of the laser pulses with an event one does not have complete control over is often a significant problem.
Quasi-continuous PDV might be a useful middle ground, but there are problems here as well.A pulsed target laser with high instantaneous power (1 W or more, depending on probe efficiency) could reduce noise fraction (equation (B.4)) well below 1% without burning fiber connections or the target.This approach has not yet been pursued, perhaps due to the challenge of finding a suitable pulse duration (100-1000 ns) laser.Every 10× decrease in the analysis duration requires 1000× more target power to maintain the same resolution limit; this might be viable for the 0.1 ns domain, but probably not beyond that.Furthermore, PDV measurements are most robust when reference power exceeds target power by 10-100×.Target returns of 0.1-1 mW would need to be mixed with 10-100 mW of reference power, requiring a 0.1-1 W laser to overcome the 10% of the reference path.Even if such a reference laser were itself pulsed, such power levels vastly exceed the damage threshold of standard InGaAs receivers.

Summary and ongoing challenges
PDV is a specific implementation of a simple concept: Doppler-shifted light mixed with an optical reference yields measurable beat frequencies.Telecommunication technology heavily influences PDV design/operation: near-infrared light (usually near 1550 nm) illuminates a moving target, and the Doppler-shifted reflection is mixed with a reference (typically at a slightly different wavelength).Single-mode optical fiber, tunable fiber lasers, optical amplifiers, sophisticated fiber components, and high-speed receivers digitizers allow velocity measurements over a vast range of experiment conditions.Encoding velocity as signal beat frequency has numerous advantages when adequate recording bandwidth is available.Continuous PDV measurements are well suited to nanosecond through second time scales, although systems designed for one domain may not be well suited to another.
Recording bandwidth was an ongoing challenge for PDV, but for velocities below 10 km s −1 (13 GHz Doppler shift) is largely a solved problem.Ongoing challenges for PDV are summarized below.
• The limited power collection of a single mode fiber, with core diameter ∼0.01 mm, can be a problem.The combination of bare optical fiber and rough surfaces is particularly acute-geometrically, the fiber may be unable to gather enough light for suitable measurement.Bare fibers can be used with specular surfaces, collimated probes with rough surfaces, and imaged probes with virtually any surface condition.• Dynamic speckle effects can also be a problem for rough surfaces [32].Random intensity variations from highlycoherent light bouncing off a rough surface can obscure beat frequencies of interest.Redundant measurement using 2+ independent probes (receiver diversity) is currently the best solution.
• Fiber birefringence is known to occur along long paths, especially when the fiber is mechanically agitated.Statistically, the chance that a PDV measurement will have zero interference amplitude is small, though significant variations are routinely observed between digitizer acquisitions [3].Active polarization control can help, but this may not easily scale to many-channel systems.• Analysis of overlapping velocities in a PDV measurement is an ongoing effort [3].
Some of these challenges are less problematic for the related diagnostics described in Appendix A. Each alternative has its own difficulties; however, and none of them are as widely applicable as PDV.
technical results and analysis.Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

Appendix A. Related diagnostics
This section briefly compares optical diagnostics for measuring motion on similar time scales as PDV.Emphasis is placed on the fundamental differences with PDV and relative advantages/disadvantage; for additional details, refer to the cited references.Different PDV implementations [3] are not covered here because they have the same limiting uncertainty (equation ( 7)).
A.1.VISAR VISAR (Velocity Interferometer System for Any Reflector [33,34]) is a definitive optical diagnostic in shock-wave physics.Its creation was preceded by a free-space (non-fiber) version of figure 1 and the realization that velocities above 200 m s −1 were impractical with existing technology [35,36].Where PDV mixes Doppler-shifted light with a distinct reference, VISAR combines Doppler-shifted light with a time-delayed version of itself.This critical difference causes VISAR signals to vary only when velocity changes, severing the link between velocity and recording bandwidth; strictly speaking, that is only true in the steady-state limit [37].Target velocity is encoded as signal phase, which means at least two quadrature signals must be recorded to ensure detectable motion at all times.Each signal is insensitive to motion at its peak/trough, but its complement is steepest precisely when that happens.
A second benefit of VISAR is its tolerance of 'any reflector'.This under-appreciated feature means that interferometer behavior is not diminished by a rough target, if enough light can be collected from that surface.The same cannot be said for PDV, where rough surfaces and/or multimode fiber delivery degrades optical coherence of the return signal [38].Some of that loss is offset by the scaling of signal amplitude with optical power ( √ P 1 ) in PDV, whereas VISAR signal amplitudes scale with P 1 .For example, a 100× reduction in target power cuts VISAR by 100× while PDV only sees a 10× reduction.VISAR also requires all potions of the reflector to move at precisely the same velocity; multiple velocities destroy interferometer contrast [39].
PDV has largely replaced VISAR outside of facilities where >10 km s −1 velocities are common, such as the Sandia Z machine [40].Operational simplicity is a large driver for this change.Once the 1.3 GHz of bandwidth per km s −1 of velocity requirement is met, the advantages of PDV become too compelling to support VISAR.Furthermore, the optical delay used in VISAR is a physical construct, whereas PDV analysis time scale is an easily modified software parameter.There is a clear path for PDV operations with nanosecond to millisecond time scales (section 3), whereas building a VISAR system with ≫ 10 ns delay is not physically practical.
Line [41] and area [42] imaging are a domain where VISAR retains some advantage over PDV.Rather than using discrete detectors, these diagnostics image a sample line onto a streak camera or a sample area onto a framing camera, providing spatially-resolved velocity measurements.This type of continuous imaging is stymied in PDV by the lack of suitable photocathodes at telecommunication wavelengths.Discrete PDV measurements can be implemented along line [43] or any fiber packing geometry.Multiplexing [21] is typically needed to support more than ∼16 PDV channels for spatially-resolved measurements.

A.2. Fabry-Perot velocimetry
Fabry-Perot velocimetry [44] is an optical diagnostic capable of detecting more than one simultaneous velocity.Dopplershifted light is passed through an etalon to produce interference fringes, which are recorded on a streak camera.Considerably more complex than VISAR, this diagnostic was not widely adopted and has been largely abandoned because of PDV's ability to handle multiple velocities [5].

A.3. WEIRD
WEIRD (Wavelength Encoded Intensity Ratio Diagnostic) determines velocity from indirect measurements of optical wavelength [45].The ratio target power passing through spectral filter to target power bypassing the filter provides a measure of the Doppler shift.This ratio can increase or decrease with motion, depending on the filter design.
One implementation called RALF (Rubidium Atomic Line Filtered) operates near 780 nm [46].Hydrogen-cyanide gas cells could support similar measurements in the same spectral range as PDV [47].Unlike PDV, WEIRD measurements become easier at high velocity because absorption edges (dB/nm) do not need to be as steep.Velocity resolution is intrinsically tied to the maximum measurable velocity and probably limited to 0.1%-1% level.Dynamic range, not recording bandwidth, is the resolution-limiting factor for this diagnostic.
Although WEIRD is not as thoroughly tested as other diagnostics, there are some compelling advantages.Unlike PDV, it is insensitive to fiber birefringence and does not require single-mode fiber.Furthermore, its uncertainty scales as τ −1 rather than τ −3/2 , which could be beneficial at short time scales.However, WEIRD cannot handle multiple velocities, extreme power variation, or signal non-linearity.It remains to be seen if this diagnostic will be widely adopted.

A.4. Optical ranging
Optical ranging is a complementary diagnostic that can operate simultaneously with PDV [48].Unlike PDV, which measures velocity projected onto the optical beam path, [49] ranging directly measures the time of flight between the optical probe and moving target.Integration of PDV measurement is consistent with optical ranging when all motion occurs along the probe axes, but the two results may differ when transverse motion occurs.
Femtosecond lasers play a critical role in optical ranging.Interference between light reflected from the sample and a reference path yields beat signals too fast for direct detection.After a time stretch is performed, distance spectrograms can be generated in a very similar manner as velocity spectrograms in PDV.Time-stretch PDV [29] is actually a variation on optical ranging, where femtosecond pulses are partially stretched prior to the target; a second stretching is performed on the mixed signal.
Sparsity and repetition rate are key limitations in optical time stretching.For best results, femtosecond pulses are stretched as much as possible to increase distance coverage and improve measurement performance.However, adjacent laser pulses cannot be allowed to overlap, so there is a firm upper limit to the amount of permissible stretching.The repetition rate also sets an intrinsic measurement clock, often an integer fraction of 80 MHz.The desire for high repetition rate (time resolution) works in opposition to the need for low repetition rate (spatial range/resolution).While not as complicated to as Fabry-Perot velocimetry, optical ranging is more difficult (and less frequently fielded) than PDV.Transverse motion can be detected with multiple PDV probes at different angles, [50] and this is often easier than optical ranging.

Appendix B. Noise limits
Photon noise is a limiting factor in PDV uncertainty (equation ( 7)).Although worse performance is always possible, this limit is instructive for the design and operation of PDV systems.Measurements rely on (Doppler-shifted) target power P 1 and reference power P 2 , leading to a noise fraction: The subscript 's' indicate signal quantities (electrical voltages), which are proportional to electrical currents denoted by the subscript 'i'.Transimpedance scaling between these quantities is not specified because it cancels out in the noise fraction, as does any additional electrical gain that might be present.Interference current amplitude: depends not only on measurement power, but how that power is mixed for the receiver.Coupling factors ρ 1 and ρ 2 are often 90% (target to detector) and 10% (reference to detector), although this not the only possibility; neglecting losses, ρ 1 + ρ 2 = 1.The quantum efficiency η can be extremely close to 100% for InGaAs, [51] and values ⩾80% (1 A/W sensitivity) are common.Current noise is related to average photocurrent: [52] σ 2 i = 2e (η P) B max ≈ 2e (ηρ 2 P 2 ) B max (B.3)where e is electron charge and B max is the measurement bandwidth.The approximation is based on the fact that most PDV measurements use far more reference power than target power [3].
Combining equations (B.2) and (B.3): reveals that noise fraction is independent from reference power and its coupling.This statement seems to contradict user experience, where more reference light makes PDV signals appear stronger.However, increasing signal amplitude is entirely cancelled by increasing photon noise, at least once the former becomes significantly larger than the digitizer's noise floor.For 25 GHz bandwidth, noise fraction cannot be less than about 1.4% for 10 µW (−20 dBm) of target power, increasing to about 4.5% at 1 µW (−30 dBm).Lower fractions can obviously be obtained at lower bandwidth, e.g. the preceding values are 5× smaller for a 1 GHz system.Consider the ratio of noise fractions for different coupling factors, keeping all other parameters fixed For example, the noise fraction for a 3 × 3 coupled PDV (η A = 1/3) is roughly 1.64× higher than a 90-10 coupler.This difference nearly offsets the √ 3 benefit for recording three signals where the former is typically used [53].Furthermore, target and reference power levels are usually balanced in three-signal measurements, so noise fraction is a factor of √ 2 larger than the above estimate (roughly 2.32 per signal).Although there may be reasons to use three-signal PDV, noise performance is not necessarily better than one-signal systems.

Figure 1 .
Figure 1.PDV schematic using a fiber circulator (C) and two optical sources.

Figure 2 .
Figure 2. Interior photograph of a Doppler module (courtesy of Christopher Johnson, Sandia National Laboratories).

Figure 3 .
Figure 3. Exterior photograph of a 'PDV in a box'.

Figure 4 .
Figure 4. Beat frequency trends for different PDV configurations.Horizontal lines indicate the maximum measurable frequency (bandwidth limit, dashed) and minimum measurable frequency (full cycle requirement, solid).

Figure 5 .
Figure 5. Frequency-shifted PDV signal for a linear velocity ramp.

Figure 6 .
Figure 6.Local FFT analysis in the time domain (above) leads to a power spectrum in the frequency domain (below).