We describe measurements of the DC electrical conductivity of warm dense matter using ultrafast terahertz (THz) pulses. THz fields are sufficiently slowly varying that they behave like DC fields on the timescale of electron–electron and electron–ion interactions and hence probe DC-like responses. Using a novel single-shot electro-optic sampling technique, the electrical conductivity of the laser-generated warm dense matter was determined with <1 ps temporal resolution. We present the details of the single-shot THz detection methodology as well as considerations for warm dense matter experiments. We, then, provide proof-of-concept studies on aluminum driven to the warm dense matter regime through isochoric heating and shock compression. Our results indicate a decrease in the conductivity when driven to warm dense matter conditions and provide a platform for future warm dense matter studies.
I. INTRODUCTION
The zero-frequency (DC or electrical) conductivity is a critical parameter for modeling processes relevant for high energy density science, including predicting the growth of Rayleigh–Taylor instabilities in inertial confinement fusion (ICF)1,2 or for modeling magnetic fields produced by planetary dynamos.3,4 The challenge in predicting electrical conductivities for these situations is that the relevant conditions lie in the warm dense matter (WDM) regime, where the material temperature is on the order of ∼0.1–10 eV and the material density is ∼0.1–10 that of a solid.5–7 At these conditions, the thermal energy is comparable to the electron energy at the Fermi level, and this, combined with the strong Coulomb coupling, results in significant deviation from ideal plasma behavior. As such, conventional conductivity models, such as those of Spitzer,8 Kubo–Greenwood,9 Ziman,10 or Lee–More11 breakdown in the WDM regime. Recently, density functional theory calculations have been used to predict the conductivity of WDM based on the Kubo–Greenwood formalism;12–18 however, experimental data are needed to test and improve theory.
With the development of high-intensity and high-energy lasers, WDM can be generated for a short time (few picoseconds to nanoseconds) in a laboratory before irreversibly changing into a plasma.19–26 Given that laboratory WDM exists transiently, single shot diagnostics with temporal resolution high enough to capture short-lived states are vital for interrogating WDM. The high-frequency (AC or optical) conductivity at visible frequencies can be determined by measuring the optical reflectivity using nanosecond pulses and a time-resolving detector22,24,27,28 and using ultrafast laser pulses with a duration of a few to hundreds of femtoseconds.20,29–31 In principle, the optical conductivity can be used to determine the electrical conductivity. However, this requires knowing the frequency dependence of the dielectric function from high-to-low frequency which is a priori unknown. Another approach is to use inelastic x-ray scattering (x-ray Thomson scattering, XRTS),32 to extrapolate the conductivity from the plasmon response.33,34 However, these extrapolations are highly model dependent.
In conventional THz-TDS, the electric field of a coherent broadband THz frequency pulse is measured in the time-domain, E(t), and the complex-valued spectrum, , is obtained by a numerical Fourier transform.39–41 Coherent THz pulses can be produced by optical rectification in nonlinear crystals40,42,43 or by free-electron lasers (FELs).44–48 By measuring changes in the spectrum of a THz pulse that has interacted with a sample, the complex-valued dielectric function, , can be obtained, and this is related to the complex conductivity. Conveniently, this is a noninvasive method, meaning that no leads or wires need to physically touch the sample. THz-TDS has been used to characterize materials in condensed phases and integrated into pump-probe geometries to study the evolution of conductivity following photoexcitation.49–52 The most commonly used approaches for THz time-domain detection require hundreds to thousands of laser pulses, however, and are implemented using laser systems that operate at kHz to MHz repetition rates. This is impractical for WDM experiments where the sample is destroyed after a single laser exposure.
In order to implement THz-TDS for WDM studies, single-shot techniques for capturing the THz time-domain waveform are required. The development of single-shot schemes has been an active area of research. Recently, a single-shot approach has emerged using an echelon-based scheme that meets the requirements for THz measurements.53 The combination of bright THz sources and detection with the sufficiently high signal-to-noise of this method have opened the door for studies of irreversible processes, as has been demonstrated.54–57
In this paper, we discuss the implementation of THz spectroscopy for studies on WDM. Following this Introduction, Sec. II discusses electro-optic (EO) sampling, with a review of the convectional approach used for studies of reversible phenomena, including the details of the non-linear optics approach to conventional THz detection. Section III discusses single shot detection with echelons, with a focus on signal-to-noise reduction and considerations for WDM experiments. Section IV then presents applications of THz experiments on two cases of interest to WDM studies: isochoric heating of the thin film and measurements of the laser compressed matter. Finally, we conclude with an overall summary, suggesting areas for improvement and future directions.
II. PRINCIPLES OF ELECTRO-OPTIC SAMPLING
One of the most commonly used methods for free-space THz detection is electro-optic (EO) sampling.39,40,58 A schematic of an EO sampling setup is shown in Fig. 1. In conventional EO sampling, the THz pulse propagates co-linearly with a linearly polarized femtosecond laser pulse, the sampling pulse, through a nonlinear optical medium, hereafter referred to as the EO sampling crystal or EO crystal; in the presented case, it is a (1 1 0)-cut of either zinc telluride (ZnTe) or gallium phosphide (GaP). The THz electric field changes the optical properties of the EO crystal, and these changes are encoded as a polarization change in the sampling pulse. By measuring the polarization change, the THz electric field can be extracted.
A. Theoretical background
B. Methods for noise reduction
While measuring I alone is, in principle, sufficient to extract , only measuring I limits of the sensitivity of the EO sampling. For laser systems with slow stochastic drifts in laser energy, this can produce modulations that are indistinguishable from those induced by the THz signal. Similarly, large shot-to-shot fluctuations lead to significant noise. These effects can be mitigated and the sensitivity of EO sampling can be greatly improved by normalizing the intensity of the sampling pulse and using balanced detection to improve the signal-to-noise ratio.
III. ECHELON-BASED SINGLE-SHOT DETECTION
The main limitation of EO sampling for WDM studies is that multiple shots are required to measure the time-domain waveform. Often, a mechanical delay stage is moved to vary the arrival time of the sampling pulse relative to the THz pulses, and I and I0 are measured at each relative delay. Consequently, measurements where the sample is irreversible changed or destroyed require more sophisticated detection methods. This has been one motivation for the development of single-shot EO sampling techniques. Recent work by Teo et al.53 investigated multiple different approaches and determined that the use of echelons was the most promising method for single-shot THz detection. In this context, an echelon is an optical element with multiple steps cut into a bulk material.54,67
Figure 2(b) shows an image collected when the THz pulse is overlapped with the sampling pulses in a 2 mm thick ZnTe crystal. The two squares correspond to the different polarization states produced by the Wollaston prism. Figure 2(c) shows the result of dividing the image in Fig. 2(b) by an image collected with the THz pulse is absent, highlighting where the THz field has changed the polarization of the sampling pulses. Here, time is mapped to the horizontal position of the echelon image as the steps of the echelon run along the vertical axis. The time axis is pre-calibrated by adjusting the arrival time of the THz pulse and sampling pulses using a mechanical stage. This calibration can be re-used provided the imaging system remains unchanged.
A. Signal extraction and effects of noise reduction
In the data, the modulation is uniform along the vertical axis, and so the THz waveform can be extracted by averaging along the vertical axis. Doing so for the raw images yields the waveforms shown in Figs. 3(a) and 3(b). The THz signal can be seen on top of a large slowly vary background, which arises due spatial inhomogeneities in the illumination of the echelon square. This is analogous to the effect of slow laser energy drift in scanning EO sampling. The difference in intensity was due to intentional misalignment of the quarter wave plate in that dataset to compensate for parasitic static birefringence of the ZnTe crystal used in these measurements. By comparison, Figs. 3(c) and 3(d) show the waveforms extracted from normalized images. After normalization, the background is flat, and the signals have opposite sign as expected from the Jones calculus analysis. Figure 3(e) shows the waveform obtained by balancing, demonstrating the doubling of the signal intensity.
One important advantage of both conventional and single-shot EO sampling are that they record the THz field, rather than the intensity. From the field, the amplitude and phase of the frequency-domain spectrum, can be determined through a numerical Fourier transform. This permits direct access to the real and imaginary parts of optical and material properties, such as the refractive index or the conductivity,35 eliminating the need for full knowledge of the dielectric function which is a requirement for Kramers–Kronig analysis.
Figure 3(f) shows the THz amplitude spectrum. In this system, the maximum detected frequency slightly exceeds 2 THz. This is set by the details of the THz source, the type of detection crystal and its thickness, and the duration of the sampling pulses. For the data presented, the THz pulse was generated by optical rectification in a 2 mm thick ZnTe crystal. With this setup, we are able to record THz waveforms in a single shot with a SNR of >200:1. Additionally, the reproducibility of the system allows for averaging of multiple measurements to accurately investigate WDM.
B. Considerations for warm dense matter experiments
In the data in Fig. 3, the THz waveform begins with a pulse centered at 5 ps and has subsequent oscillations. These oscillations are due to absorption of the THz field by ambient water vapor. Ambient water vapor absorbs THz radiation at well-defined frequencies due to resonant rotational modes and vibrational bands as well as broadband absorption.68,69 This modifies E(t) directly, and the oscillations cannot be observed without the THz pulse. The resonant absorption is also visible as dips in the frequency domain. These can be seen in in the teal curves in Figs. 4(a) and 4(b), where the relative humidity measured by an independent sensor was ∼53%. These result in unwanted contributions and limit the sensitivity of the measurement, in particular, in the frequency domain where the regions of low signal have a lower dynamic range.
These effects can be removed by performing experiments in a vacuum chamber, as commonly done in WDM experiments, or in an enclosure flushed with an inert gas (e.g., nitrogen). The data in Figs. 4(a) and 4(b) show the time-domain waveform and spectrum collected using the same instrument; for the teal curves, the enclosure contained ambient air and for the black curves, the enclosure was filled nitrogen gas, thus reducing the relative humidity to ∼3%. Although at this humidity there are still some very weak oscillations present in the THz time-domain waveform, their contributions are small and do not significantly impact the dynamic range of the measurements.
The other key consideration is the effect of the THz spot size. Compared with optical lasers, the long THz wavelength (300 m at 1 THz) makes tight focusing a challenge because of diffraction effects. This is illustrated in Fig. 4(c), where a series of pinholes with various diameters are placed at the focus of a 3-in. diameter, 3-in. effective-focal-length, , 90° off-axis parabolic reflector. Here, even for a 1000 m diameter pinhole, there is a slight reduction in the measured THz field strength as some of the pulse is clipped by the pinhole. This is more apparent for a 500 m diameter pinhole, where the signal has been reduced by almost a factor of two and continues for smaller pinhole diameters. This effect can also be seen in the spectrum, where the lower frequency (longer wavelength) components are more dramatically clipped as they cannot be as tightly focused. This also indicates the importance of ensuring that sample is uniformly driven into the WDM regime and that a large laser spot is used to produce a uniform sample. When using a pump spot that is smaller than the THz probe beam, only the high frequency data will provide reliable information unless the THz pulse can be spatially filtered. This is particularly important when trying to reach extremely high energy density states.
IV. APPLICATIONS FOR WDM STUDIES
Performing single-shot THz measurements on WDM essentially requires combining two different pump-probe setups into one experiment and as such require three different kinds of laser pulses: (1) a drive pulse for exciting the sample, (2) a few mJ femtosecond pulse for making the THz pulse, and (3) a few J femtosecond pulse to readout the THz pulse. Different drive pulses can be used to achieve different WDM conditions. For example, pulses with Joule-level energy and tens of nanosecond duration can be used for laser driven compression while pulses with tens to hundreds of millijoule energy and femtosecond duration can be used for isochoric heating of thin films. For laser based THz generation, optical rectification is often used, and mJ energy pulses are chosen to avoid damaging the optics required. This process yields THz pulses with typically <1 J pulse energy. In the first pump-probe setup, the sample is driven to the WDM state with an intense or high energy laser pulse, and a time-delayed THz pulse delivered to the sample to probe. The other pump-probe setup is the echelon-based single-shot detection, where the THz pulse is readout as described above. Presently, measurements on WDM can be performed with the THz probing in a transmission geometry55,57 or in a reflection mode. These two experimental platforms allow for measurements of different parts of WDM phase space—either isochorically heated or under shock compression.
A. Laser-heated thin films
Laser heating films can produce non-equilibrium WDM and study WDM at up-to solid-density conditions.19,20,70,71 In these experiments, an intense femtosecond laser irradiates a nanometer-thin sample of material, thereby heating the electrons in the material to electronvolt temperatures. The electrons and ions thermalize between 1 and 100 ps, depending on the electron–phonon coupling strength. During this time, a THz pulse can be used to probe the sample. On sufficiently short timescales, the sample will not undergo hydrodynamic expansion, and thus, the THz pulse will probe an isochorically heated material.
1. Experimental setup
A schematic of the experiment is shown in Fig. 5(a). The data presented here were collected using an amplified femtosecond laser system which produced two separate 8 and 5 mJ = 800 nm, 50 fs pulses. The 8 mJ pulse served as the drive, while the 5 mJ pulse was split 90:10 for THz generation, performed by optical rectification in a 2 mm (1 1 0)-cut ZnTe crystal41–43 and THz detection by single-shot electro-optic sampling described above. The THz pulse was focused to the target plane using the same parabola. The drive pulse was spatially filtered and imaged onto the target plane. Overlap was accomplished by delivering the drive pulse through a 3 mm hole in the parabolic mirror focusing the THz pulse. A mechanical delay stage was used to set the arrival time of the drive pulse relative to the THz pulse. The samples were 40 nm aluminum films.
For data collection, computer-controlled mechanical shutters were used to block and unblock beams to collect echelon images with the THz pulse absent, only the THz pulse present, and both the drive and the THz pulses present. The measured echelon images were used to construct waveforms corresponding to the THz pulse transmitted through an empty hole (the reference), through the cold film, and then the heated material.
Compared with our initial demonstration,55 the films were deposited onto a thick aluminum plate with 1 mm diameter holes spaced 3 mm apart. This resulted in free-standing films over the 1 mm diameter holes, while the plate acted as a mask and ensured that the THz pulse only probed heated material. The experimental setup was also enclosed in a nitrogen atmosphere. These improvements permit accurate analysis of the THz spectrum, which was not possible previously. The use of nm pulses enables higher fluences, so targets can be driven to higher energy density WDM states.
Representative waveforms are shown in Fig. 5(b), with the black, blue, and red waveforms corresponding to the THz transmission through an empty hole, the cold film, and the heated film, respectively. The incident drive pulse fluence was ∼100 mJ/cm2, a factor of two higher than our previous measurements, corresponding to an intensity of 2 W/cm2, and the THz pulse was delayed by 4 ps. The data clearly show a significant decrease in the THz transmission when the THz pulse passes through the unheated thin film (blue) compared to the 1 mm hole (black). This is expected, as the high conductivity of the film makes it reflective to the THz field. When the film is heated with the drive, the THz transmission increases, indicating a change in the conductivity of the sample.
2. Conductivity determination
B. Dynamically compressed matter
Another common method for producing WDM is by laser-driven shock compression.21,26 In this technique, a thick (e.g., <1 m) sample is irradiated with high energy nanosecond long laser pulse. The laser sample interaction generates a plasma at the surface of a target which ablates away. Consistent with Newton's third law, this ablation consequently launches a shock wave which drives the material into a high pressure and high temperature state. Typical conditions achieved using laser driven shock compression are in the megabar pressure range. Furthermore, the resultant WDM state can be maintained for the duration of the drive laser pulse (e.g., tens of nanoseconds). During this time, the THz pulse probes the reflectivity opposite to the drive side of the sample.
1. Experimental setup
A schematic highlighting the key elements of a setup for THz probing of the compressed matter is shown in Fig. 6. Data presented here were measured at the Janus Target Area 1 of the Jupiter Laser Facility at Lawrence Livermore National Laboratory. Pulses for generating and measuring the THz pulse were produced by a Coherent Hidra Ti:sapphire amplified laser system that delivered nm pulses with a duration of 50 fs at 10 Hz repetition rate. The THz pulse was generated using a 2 mm thick ZnTe crystal and directed to the target using a pair of off-axis parabolic reflectors. The focusing THz pulse had an incidence angle of 60° relative to the target normal. The reflected THz pulse was collected and imaged using a matched set of off-axis parabolic reflectors onto a 2 mm ZnTe crystal, where single-shot EO sampling was accomplished using a reflective echelon. The target assembly consisted of a 100 nm layer of Al, a 25 μm CH ablator layer, a second 100 nm layer of Al, and a 500 μm lithium flouride (LiF) window. The 100 nm of Al was coated on the 25 μm CH, and the LiF was glued to the target stack. The THz pulse reflected from the vacuum–LiF and LiF–Al interfaces, while the drive arrived at the exposed 100 nm Al-coated side of the target.
The drive laser was configured to produce 10 ns square pulses with up to 300 J at 2 ( nm). Continuous phase plates (CPPs) were used to generate a 1 mm diameter spatially averaged uniform-intensity drive spot on the target. The maximum on-target intensity was W/cm2. A two channel velocity interferometry system for any reflector (VISAR) was used to determine the shock breakout time and hence shock velocity.76,77 The arrival time of the THz pulse first calibrated by sending 800 nm pulses that are co-timed with the THz pulse to the target position. The 800 nm pulses were subsequently scattered and collected by the streak cameras in the VISAR system. The drive beam arrival time was calibrated using a reference laser co-timed with the drive laser. This reference laser was similarly directed to target position and the reflected into the VISAR streak cameras. Fiducials were then set on the VISAR streak camera windows to mark the arrival of the THz pulses and the drive laser as shown in Fig. 8. The error in the arrival time, estimated from the width of the fiducial signals, is 30 ps.
2. Data example and data analysis
Finally, the reflectivity ratio can be calculated as .
A representative set of data collected during a shot is shown in Fig. 8. The drive energy for this shot was 282 J as measured using calibrated energy meters. Figure 8(a) shows a VISAR trace annotated to show the arrival times of the drive laser to the target as well as the THz pulses; these were determined by the positions of the fiducials discussed previously. The VISAR image shows a loss of reflectivity just before 5 ns on the streak camera window, which is attributed to the shock breakout through the 100 nm of Al into the LiF window. For the purpose of this demonstrative work, we only use this change to indicate that the shockwave has passed into the LiF.
Figure 8(b) shows data collected by single-shot EO sampling. The data show measurements taken both with and without the drive present. The reference data are an average of 50 undriven measurements. The “with drive” measurement is a single shot. For the driven measurement, a clear change is visible for the pulse reflected at the original LiF–Al interface, while the pulse at the vacuum–LiF interface overlaps well. This indicates the repeatability of the THz pulse produced for the measurements. Because the Al layer used was only 100 nm, we do not expect a sustained shock in the Al and thicker targets are more appropriate for extracting the conductivity. However, the data still indicate the ability to determine shock induced changes in the THz reflectivity.
Figure 8(c) shows the same data smoothed to remove noise. In the both the reference and driven measurements, THz pulses originating from reflections at the vacuum–LiF and the original LiF–Al interface are present at 1.8 and 11.5 ps, respectively. In the driven measurement, a weaker THz pulse at about 10 ps. This is likely due to a reflection from the shock front in the LiF window, indicating that the optical properties of the LiF have changed. The new pulse arrives earlier than the one reflected at the original LiF–Al interface because the shock has created a new interface within the LiF, effectively reducing in Eq. (11) and, thus, reducing .
V. CONCLUSIONS AND OUTLOOK
In conclusion, we have presented the details of single-shot THz measurements on WDM and provided details of demonstrative experiments illustrating the technique. A reflective echelon enables THz time-domain waveforms to be recorded in a single shot with a high signal-to-noise, thus enabling spectroscopic characterization of materials driven in to the WDM regime. We describe details of the design of our experiments to optimize the signal-to-noise ratio and show the results of experiments where single cycle THz pulses are used to characterize WDM produced by isochoric heating and shock compression. In both cases, the electrical conductivity decreases when driven to the WDM regime and can be calculated from the measurements.
Many areas for improvement exist for making THz studies on WDM. Laser based sources offer a variety of different methods for producing THz radiation.82 While not discussed, the work here exclusively used ZnTe as a THz source through optical rectification, a process that uses the interaction of an ultrafast pulse a non-linear crystal to produce THz radiation. Lithium niobate has been used as a non-linear crystal to produce intense THz pulses;83–86 however, the resulting THz pulses have a longer duration than those produced by ZnTe, reducing the temporal resolution of the measurement. An active area of research is the synthesis of new materials for use,87,88 which could improve the THz field strength or the bandwidth. In particular, N-benzyl-2-methyl-4-nitroaniline (BNA) has been shown to produce intense THz pulses with comparable duration to that of ZnTe. Furthermore, intense laser–matter interactions can produce intense THz radiation,89–91 facilitating WDM experiments with currently available facilities. While the measurements discussed focus on solid samples, the use of liquid jets92 offers possibilities for introducing different kinds of samples and also for high repetition rate experiments. Finally, this diagnostic can be combined with x-ray free electron lasers, enabling simultaneous determination of multiple parameters (e.g., conductivity via THz measurements, density via x-ray diffraction, electron temperature via XRTS) and, thus, providing an exciting avenue for characterizing WDM.
ACKNOWLEDGMENTS
This work was funded by the DOE Office of Science, Fusion Energy Science, under No. FWP 100182 as well as the Department of Energy, and Laboratory Directed Research and Development program at SLAC National Accelerator Laboratory, under Contract No. DE-AC02-76SF00515 and as part of the Panofsky Fellowship awarded to EEM and BOO. A.D. and E.E.M. were supported by the UK Research & Innovation Future Leaders Fellowship (No. MR/W008211/1). The use of the Jupiter Laser Facility was supported by the U.S. Department of Energy, Lawrence Livermore National Laboratory, under Contract No. DE-AC52-07NA27344.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Benjamin Ofori-Okai: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Visualization (lead); Writing—original draft (lead); Writing—review & editing (lead). Siegfried Glenzer: Conceptualization (equal); Funding acquisition (lead); Supervision (lead); Writing—review & editing (equal). Adrien Descamps: Investigation (equal); Writing—review & editing (equal). Emma Elizabeth McBride: Conceptualization (equal); Investigation (equal); Writing—review & editing (equal). Mianzhen Mo: Investigation (equal); Writing—review & editing (equal). Anthea Weinmann: Investigation (equal); Writing—review & editing (equal). Lars Seipp: Investigation (equal); Writing—review & editing (equal). Suzanne Jihad Ali: Investigation (equal); Writing—review & editing (equal). Zhijiang Chen: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing—review & editing (equal). Luke Bennett Fletcher: Conceptualization (equal); Investigation (equal); Writing—review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.