Inertial confinement fusion facilities generate implosions at speeds greater than 100 km/s, and measuring the material velocities is important and challenging. We have developed a new velocimetry technique that uses time-stretched spectral interferometry to increase the measurable velocity range normally limited by the detector bandwidth. In this approach, the signal is encoded on a chirped laser pulse that is stretched in time to reduce the beat frequency before detection. We demonstrate the technique on an imploding liner experiment at the Sandia National Laboratories’ Z machine, where beat frequencies in excess of 50 GHz were measured with 20 GHz bandwidth detection.
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Velocity measurements are central to the study of the shock wave compressions of matter. From the measurement of material velocity and shock wave velocity, the pressure, density and internal energy of the compressed state can be determined using the Hugoniot conservation equations . Additionally, estimates of phase boundary locations and material strength can be made based on the shapes of the velocity-time records (also called wave profiles) . For several decades VISAR (velocity interferometer system for any reflector) [3,4] and Fabry-Perot interferometers [5,6] have served the shock physics community well as velocimetry diagnostics. The emergence of high-frequency digitizers and three-port fiber-optic circulators made possible the development of heterodyne interferometry, which is often called photonic Doppler velocimetry (PDV) . PDV determines a target velocity by measuring the beat frequency between a Doppler-shifted laser beam reflected from the moving target and a reference beam, which is commonly an unshifted beam from the same laser. The beat frequency is recorded using a high-bandwidth photodetector and digital oscilloscope. Today PDV is often the preferred velocity diagnostic on shock experiments because of its high dynamic range, ability to record multiple velocities simultaneously, and the ability to trade between velocity and time resolution in the data analysis, all of which are limitations for VISAR and Fabry-Perot velocity techniques.
Modern (and expensive) digitizer and photodetector PDV systems can have bandwidths up to 70 GHz, giving a practical PDV velocity limit of around 50 km/s using 1550 nm lasers. With more affordable equipment, the typical recording bandwidth is <30 GHz and velocities are measured up to ∼20 km/s. PDV measurements at speeds higher than 50 km/s are needed for experiments on the Z pulsed-power machine  at Sandia National Laboratories (SNL) and on high-power lasers, such as at the Laboratory for Laser Energetics and the National Ignition Facility, where implosion velocities can exceed 100 km/s [9–11]. Because of the high bandwidths needed for PDV measurements, VISAR continues to be used for these experiments despite its limitations.
It is possible to increase the maximum recordable PDV velocity by upshifting the optical frequency of the reference laser, or local oscillator (LO), relative to the signal laser. This way, the effective velocity range of the system can be nearly doubled by starting with a beat frequency near the detection bandwidth limit at zero velocity and “bouncing” the Doppler shift off of the zero frequency floor as velocity increases. In a similar manner, by choosing an LO with an optical frequency offset by an amount larger than the detection bandwidth, one can tune the system to window a higher velocity range. Multiple PDV systems can be linked together in this manner to cover a larger velocity range. Dolan et al. have reported this technique, which they call Leapfrog PDV . They interleave several receivers, digitizers, and individual PDV units with different LO frequencies (all higher than that of the PDV laser, so the beat frequency is recordable only when the Doppler shift is large) to cover a large range of velocities >20 km/s. For a very large velocity range, the number of lasers, detectors, and digitizer channels required can become impractical. Additionally, while this approach extends the velocity capabilities of PDV, it does not extend the Nyquist limit and therefore cannot measure events that occur on a timescale comparable to the inverse bandwidth limit of the detectors.
Here we report a new approach capable of bypassing the velocity and Nyquist limits of a standard PDV system. This method increases the maximum recording velocity range by reducing the beat frequency of the PDV signal before detection, thereby trading some of the required bandwidth for a longer recording time. The technique, referred to here as time-stretch PDV (TSPDV), uses spectral time-stretch methods introduced by Bhushan , which have found uses in the study of laser dynamics [13,14], range finding [15–18], digitizer technology , various spectroscopy applications [20–23], and a variety of other areas [24–26]. The method involves transference of spectral information into the time domain where data can be recorded very quickly with fast photodetectors and digitizers. Time-stretch methods are finding widespread applications, due in part to the recent development of fiber-based laser technology brought on by growth of the telecommunications industry, and its broad potential was the subject of a recent review article .
2.1. Equipment design
Figure 1 shows a simplified schematic of the TSPDV system. A Menlo Systems T-light laser produces a 100 MHz train of 100 fs pulses of nominally 1550 nm light. A pulse picker selects a single pulse, properly timed to arrive at the experiment just before the motion starts. Initial stretching of the pulse from 100 fs to 150 ns is achieved by passing it twice through a highly dispersive spool of optical fiber (FS1) that is 21 km in length. To increase signal levels and make up for losses in the fiber, the spool of fiber is pumped with laser light to achieve Raman amplification while stretching. Four different pump wavelengths (1500, 1445, 1485, 1465 nm) are used to flatten and broaden the gain spectrum to increase the useable pulse width. Further amplification is achieved with an erbium-doped fiber amplifier (EDFA) after stretching. The result is a chirped ∼150 ns pulse with sufficient intensity for the experiment. The pulse enters a three-port circulator and is divided into a signal leg (90%), which is reflected from the target, and a reference leg (10%), which is reflected from a mirror (M2). The delay time of the reference leg is fine-tuned with an adjustable delay stage to ensure that the two legs have identical path lengths. Upon reflection from the target, the signal pulse is Doppler shifted to a frequency ω = ωs(t) (1 + 2v/c), where ωs(t) is the chirped laser frequency, v is the instantaneous velocity of the target, and c is the speed of light. The Doppler-shifted signal is then combined with the reference signal to achieve a beat frequency, which is zero Hz if there is no Doppler shift and no mismatch in the lengths of the two legs.
To reduce the beat frequency from a high-speed target enough for recording, the combined signal is then stretched again. To obtain a large amount of dispersion and make up for losses in the fibers, the pulse is passed several times through a fiber-based regenerative loop (labeled REGEN in Fig. 1) consisting of another dispersive fiber-optic spool (FS2), an optical switch, and Raman amplifiers. Here the pulse is trapped, stretched, and amplified. The optical switch can be adjusted for any desired number of passes through FS2. In the experiments described here, the pulse passes 11 times through FS2, which is 10.2 km long. After 11 passes the pulse is switched out of the loop and detected on a photodetector that is recorded on a 20 GHz digitizing oscilloscope with a sampling rate of 50 GS/s. Typical peak powers within a single pulse were on the order of several hundred mW going to the target and at the detector after the REGEN. Velocity is extracted by applying a scaling factor to the digitizer time base and performing a moving window fast Fourier transform (FFT) analysis to obtain the beat frequency versus time. The beat frequency Ω0 = (2v/λ) is proportional to velocity; here v is the target velocity and λ is the laser wavelength.
To determine the time-scaling factor and extract velocity from the data, the effects of dispersion must be known. A mathematical description of the dispersion effects is determined analytically in the following section. The result is surprisingly simple. The system reduces the beat frequency linearly by a scaling factor determined by the ratio of the lengths of dispersive paths before and after the Doppler shift ; Ω = (L1/L) Ω0, where L1 is the length of the first fiber path (2 × 21.4 km in this case), L2 the length of the second fiber path (11 × 10.2 km), and L = L1 + L2. Thus for this work the scaling factor L1/L = 0.276. Consequently, equipment capable of recording speeds up to 20 km/s can now record as fast as 70 km/s. Even smaller scaling factors are obtainable by adding more passes in the regenerative optical amplifier or reducing the dispersion before the target. As more passes are added, amplifier noise increases and spectral narrowing caused by a gain spectrum that is not flat can eventually result in data loss. We tested static scaling factors as low as 0.1, meaning 200 GHz beat frequencies could in principle be recorded with 20 GHz bandwidth detection.
2.2. Concept and derivations
In this section we derive the relationship between velocity and beat frequency, taking into account the effects of spectral dispersion. Dispersion causes a change in phase with frequency. Relative to some frequency, ω0, and accounting for dispersion up to the third order, the phase, φ, after propagating through a fiber of length L can be written as a Taylor series  expanded around ω0
Initial tests of the TSPDV system were done in the lab with targets comprising aluminum foils accelerated up to 2 km/s with a pulsed laser. The laser pulse (1064 nm, 10 ns, 100 mJ) was focused through a 13 mm thick fused silica window with the foil glued to the back surface. The glue was mixed with carbon powder to increase laser absorption. Laser-induced breakdown occurs at the glue interface and launches a small piece of the foil toward a probe aligned to send and receive reflected light from the projectile. A narrowband wavelength filter is used to combine a continuous wave laser with the chirped pulse in the fiber for simultaneous recording of regular PDV and TSPDV data. Figures 2(a) and 2(b) show the PDV and TSPDV spectrograms for one such experiment. The effect that the time-stretch has on the data is immediately apparent; compared to the PDV data the TSPDV beat frequencies have been reduced and the signal is stretched out in time. Next, the time axis of the TSPDV data was multiplied by the theoretically calculated scaling factor L1/L = 0.276, and a new spectrogram was created. Lineouts were then created for both datasets, and beat frequency was converted to velocity using standard PDV analysis . The velocities are overlaid in Fig. 3. The TSPDV data matches the PDV data to within 1%, indicating that the theory and assumptions used to calculate the scaling factor are accurate and validating the TSPDV technique.
We used the system on an imploding liner experiment (Z3138) at the Sandia Z pulsed power machine [8,30]. A ramping current from the Z capacitor bank produced an increasing magnetic field that drove the implosion of a cylindrical aluminum liner to a final velocity of about 40 km/s (Fig. 4). Inside the liner, centered on the axis, was a fused silica tube filled with liquid xenon. A PDV probe was placed at the center of the cylinder and reflected 90 degrees by a prism so that the light was directed along the implosion direction (perpendicular to the axis) through the xenon and the tube to the liner. Reflected light from the liner (and from the other surfaces between the liner and mirror) went back into the probe. PDV and TSPDV were fielded simultaneously on the same probe.
The PDV system used a local oscillator with a + 21 GHz frequency shift relative to the signal laser such that the initial beat frequency at zero velocity was –21 GHz. As the velocity increases, the apparent velocity decreases from 16 to 0 km/s, reflects from the baseline, and rises to + 17 km/s, a total velocity increase of 33 km/s. PDV recorded the liner motion until the beat frequency surpassed the 25 GHz bandwidth limit of the detection system. The TSPDV recorded up to 95 GHz. The bandwidth limits of PDV and TSPDV are evident by the disappearance of the noise floors (where the background turns from blue to purple) at the tops of the spectrograms, which are shown in Figs. 5(a) and 5(b). The arrival time of the 150 ns TSPDV pulse was timed to record events late in the implosion, where the higher velocities were expected.
The spectrograms in Figs. 5(a) and 5(b) have labels referencing the sources of various interference signals seen in the PDV and TSPDV data. Both spectrograms have baseline signals that reference zero velocity as a result of reflections from the quartz tube. For the PDV, the baseline at 21 GHz (A) is a mix between the quartz reflection and the LO. The moving liner (E) appears as a result of a mix between the LO and the Doppler-shifted reflection from the liner. The TSPDV baseline (D) is a mix between the quartz reflection and the reference pulse from the interferometer. This baseline signal (representing zero velocity) appears at 3.7 GHz, along with second and third harmonics that appear as a result of nonlinear distortions to the 3.7 GHz signal in the detector. The TSPDV sees the moving liner (C) as a mix between the reference signal and the Doppler-shifted reflection from the liner. The TSPDV also sees the moving liner at a slightly shifted frequency (B) caused by the mix between reflections from the quartz tube and the liner.
Figure 6 shows the extracted velocities from the PDV and TSPDV spectrograms overlaid. The velocity traces agree well during the time that they are both providing signal; however, the projectile continues to accelerate and the velocity eventually exceeds the measuring capability of the PDV system. This occurred just prior to the liner impact with the fused silica tube, at which point a frequency and velocity jump was observed. The jump occurs at beat frequencies ranging from 48 to 57 GHz, beyond the bandwidth of the PDV but well within the limit of the TSPDV system. This jump is consistent with the creation of a shock wave inside the fused silica tube. Accounting for the refractive index of material ahead of the shock front, the front begins around 25 km/s and increases to 30 km/s. A subsequent shock in the liquid xenon sample was expected but not observed in this experiment. The loss of signal in the xenon is likely due to radiation blanking, and efforts to mitigate this effect are in progress .
The use of time-stretched spectral interferometry applied to velocimetry can be used to help track fast-moving objects that are difficult to see using other methods. As velocities reach 100 km/s, important dynamics can occur on a sub-nanosecond timescale. Matter undergoes phase changes, plasmas begin to form, and energy densities become high enough to create fusion. Under these conditions a velocimetry technique with increased sensitivity and bandwidth is required. Interferometric-based approaches, such as PDV, benefit from heterodyne gain and subsequently higher sensitivity, but they can be bandwidth limited. The single-pulse TSPDV approach presented here bypasses the bandwidth limitations of PDV but retains the sensitivity, although TSPDV requires a longer record length. It is especially well-suited when measuring velocities >10 km/s over recording times of hundreds of nanoseconds (or less), conditions commonly seen in high-energy implosion experiments.
Office of Defense Programs (DE-NA0003525, DE-NA0003624).
This manuscript has been authored by Mission Support and Test Services, LLC, under Contract No. DE-NA0003624 with the U.S. Department of Energy and supported by the Site-Directed Research and Development Program, National Nuclear Security Administration, Office of Defense. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective 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.
The authors declare that there are no conflicts of interest related to this article.
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