DC electrical conductivity measurements of warm dense matter using ultrafast THz radiation

Topic


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. 3,4The 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 $0.1-106][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-More 11 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][13][14][15][16][17][18] however, experimental data are needed to test and improve theory.
][21][22][23][24][25][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.0][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. 33,34However, these extrapolations are highly model dependent.
6][37] Unlike visible and near-infrared frequency pulses, the THz electric field oscillates slowly when compared with electron scattering time, s.In the context of the Drude model, where the frequency dependent conductivity, rðxÞ, is given by rðxÞ ¼ r 0 1 À ixs ¼ n e q 2 s=m e 1 À ixs ; where r 0 is the DC conductivity, n e is the free electron density, q is the charge on an electron, and m e is the free electron mass, s $ 10 fs in typical metals, 38 possibly shorter in WDM. 30 As such, in the range x=2p $ 1 THz, xs ( 1 and consequently rðxÞ !r 0 .This indicates that THz frequency radiation more appropriately probes the DC conductivity.
5][46][47][48] By measuring changes in the spectrum of a THz pulse that has interacted with a sample, the complex-valued dielectric function, ẽðxÞ, 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.0][51][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][56][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. 39,40,58A 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
0][61] In this section, we only describe the key results.In the Appendix, we provide a more thorough derivation of the starting equations from nonlinear optics.ZnTe or GaP are initially optically isotropic, with refractive indices n x ¼ n y ¼ n z ¼ n 0 .When the THz field is present, it changes the refractive indices of the EO crystal via the Pockels effect, a linear change of a material's refractive index in the presence of a static electric field, [62][63][64] and makes the EO crystal transiently anisotropic.For ZnTe-(1 1 0) or GaP-(1 1 0), the electric field of the THz pulse changes the refractive indices according to Here, E THz is the THz electric field, a is the polarization angle of the THz field with respect to the [0 0 1] axis, r 41 is the relevant nonlinear tensor element, and w is the angle between the [0 0 1] axis and the fast optical axis of the crystal, n z .The angle between the fast and slow axes is always 90 .w is related to a by In EO sampling, the THz and sampling pulses propagate along the [1 1 0] axis, parallel to n x .Therefore, the relevant induced anisotropy, or birefringence, is between the n y and n z axes.For the sampling pulse, this birefringence causes a change in the optical phase of the components of the sampling pulse projected along n y and n z .The optical phase difference accumulated when propagating through a thickness ' of the EO crystal is where k is the center wavelength of the sampling pulse.Using Eqs.(2b), (2c), and (3), one can determine that D/ is maximized for a ¼ 6p=2.This means that if the THz field is horizontally polarized in the lab frame, the [0 0 1] axis of the EO crystal should be oriented vertically.The fast and slow axes of the EO crystal are then at 645 relative to vertical (and horizontal).The maximum phase change will then occur if the polarization of the sampling pulse is either horizontal or vertical. 60he phase change in the sampling pulse manifests as a modulation of the polarization, making the initially linear polarization of the pulse elliptical.The ellipticity is measured using a quarter wave plate, a polarizer oriented perpendicular to the initial polarization, and a photodiode.The action of the waveplate and polarizer converts the polarization change into an intensity change, which is measured on the photodiode.Using Jones calculus 63 for the configuration described above, one finds that when the fast axis of the quarter wave plate is set 45 relative to the axes of the polarizer, the measured intensity on the photodiode, I, is where I 0 is the intensity of the sampling pulse when no THz field is present.The full THz time-domain waveform, E THz ðtÞ, is determined by measuring I for different timing between the sampling pulse and the THz pulse, for example, by using a mechanical stage, shown in Fig. 1, to change the path length traveled by the sampling pulse.

B. Methods for noise reduction
From Eq. ( 5), optimizing the signal can be accomplished through ', r 41 , n 0 .The choice of material for the EO crystal will influence all of these parameters; R 41 and n 0 are material specific.Typical values for n 0 are $2-3 (see Ref. 65), while r 41 can vary by up to an order of magnitude, making this term dominate.The maximum thickness, ' M , is dependent on the velocity mismatch between the pulses.We can determine ' M as Here, x THz is the THz angular frequency, n g is the group refractive index of the sampling pulse, n THz is the THz refractive index at x THz , and ' M , thus, corresponds to the distance over which the THz and sampling pulses will slip by a half-period of the THz cycle.Finally, the material will also limit the detectable bandwidth due to absorption of THz radiation.ZnTe is often chosen for THz pulses with spectral bandwidth less than 2.5 THz, as ' M $ 2:5 mm and r 41 is large.While GaP has a much lower r 41 and ' M $ 0:2 mm, it is weakly absorbing below 7 THz, and so it is also used.While measuring I alone is, in principle, sufficient to extract E THz , 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.
To normalize the intensity, a measurement of the sampling pulse without the THz field present is required.This yields I 0 , which can be used to determine the corresponding relative intensity change in the sampling pulse, DI=I 0 , Compared to measuring I alone, DI=I 0 has the advantage that it can be used to determine D/, which is a function of E THz .One strategy to determine I 0 is by averaging measurements of the sampling pulse either long before or after the measurement with the THz pulse.This, however, only re-scales the data and does not compensate for slow drift.Instead, measuring I 0 as close in time as possible as I is measured reduces effects from the long-term drift in the laser energy.
In addition to intensity normalization, balanced detection can be implemented to correct for shot-to-shot fluctuations in the laser energy.In traditional balanced detection, the sampling pulse is split and bypasses the EO crystal, and I and I 0 are measured on separate photodetectors.However, for EO sampling, an advantage can be leveraged by using a polarizing beam splitter or a Wollaston prism to measure pulses with orthogonal polarizations.Compared to the signal described by Eq. ( 5), referred to as I þ , the intensity in the other polarization is I À ¼ I À 0 ½1 À sin ðD/Þ.The signal S(t) can then be constructed as This version of balanced detection has the advantage that the signal is doubled, while at the same time fluctuations in the sampling pulse between the measurements of I 6 and I 6 0 are subtracted out.This leads to a greater than 2Â increase in the signal-to-noise ratio (SNR).In some cases, a 10Â improvement can be obtained. 66

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 I 0 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 schematic of a reflective echelon is shown in Fig. 2(a).Unlike a grating, the steps of an echelon are larger than the wavelength of the sampling pulse.As such, each step of the echelon acts as a mirror and specularly reflects a different part of the beam.Additionally, each reflected part of the beam travels a slightly different distance, which results in a time delay, Dt, given by where H is the step height, c 0 is the speed of light, and h i is the angle of incidence of the beam with respect to the echelon step normal.For the data in this work, H ¼ 5:00 6 0:02 lm and Dt % 33 fs.The overall effect turns a single input pulse into a series of time-delayed pulses, thereby replacing the need to scan the mechanical stage.The total time window was NDt ¼ 15 ps, where N ¼ 500 is the number of steps on the echelon.Using a lens, these pulses are focused into the EO sampling crystal and different segments of the THz pulse are encoded on different spatial locations of the sampling beam.The THz waveform is determined using the same polarization optics as described in Sec.II; however, a charge coupled device (CCD) or complementary metal oxide semiconductor (CMOS) camera replaces the photodiodes and the echelon must be imaged onto the camera.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, ẼðxÞ ¼ AðxÞ exp ½i/ðxÞ; 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. 68,69This 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 lm 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.effectivefocal-length, f =1, 90 off-axis parabolic reflector.Here, even for a 1000 lm 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 lm 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 lJ 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 lJ 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 geometry 55,57 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. 19,20,70,71In 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.

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 k ¼ 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 crystal [41][42][43] and THz detection by single-shot electro-optic sampling described above.The THz pulse was focused to the target plane using the same f =1 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 k ¼ 800 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/cm 2 , a factor of two higher than our previous measurements, corresponding to an intensity of 2 Â10 12 W/cm 2 , 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.

Conductivity determination
To determine the conductivity, the spectrum of the THz pulse transmitted through the film, Ẽs ðxÞ was normalized to the spectrum obtained from the field transmitted through the air hole Ẽr ðxÞ.This gives the complex-valued transmission function, tðxÞ ¼ Ẽs ðxÞ= Ẽr ðxÞ.For a free-standing thin film, the complex-value conductivity can be calculated using the Tinkham formula, 72 which accounts for multiple reflections of the field within the film, where Z 0 ¼ 377 X is the impedance of free space, and d is the film thickness and was determined independently by frequency-domain interferometry [73][74][75] to have increased by 20%.The determined rðxÞ are shown in Fig. 5(c) for the unperturbed and heated film.We plot only the real part of the conductivity as the imaginary part is several orders of magnitude smaller.Initially, the film has an average real conductivity of 13 6 1 Â 10 6 S/m, and upon heating into the WDM regime, the conductivity drops to 2:1 6 0:3 Â 10 6 S/m.The details of this change go beyond the scope of this manuscript and a more systematic study will be the subject of future investigations.

B. Dynamically compressed matter
Another common method for producing WDM is by laserdriven shock compression. 21,26In this technique, a thick (e.g., <1 lm) 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.

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 k ¼ 800 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 f =2 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 singleshot EO sampling was accomplished using a reflective echelon.The target assembly consisted of a 100 nm layer of Al, a 25 lm CH ablator layer, a second 100 nm layer of Al, and a 500 lm lithium flouride (LiF) window.The 100 nm of Al was coated on the 25 lm 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 x (k ¼ 527 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 3:82 Â 10 12 W/cm 2 .A two channel velocity interferometry system for any reflector (VISAR) was used to determine the shock breakout time and hence shock velocity. 76,77The 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 6 30 ps.

Data example and data analysis
Unlike the transmission measurements, the time-domain waveforms in Fig. 6(b) show two pulses.These arise from the reflections of the THz pulse at the vacuum-LiF interface and the LiF-aluminum interface.In general, anywhere there is a significant change in the THz refractive index, and this will lead to a pulse appearing in the time domain.In the undriven measurement, the temporal separation between the pulses is set by the optical path length of the THz pulse through the LiF, where i LiF is the THz refractive index of LiF at ambient conditions, ' is the LiF thickness, c 0 is the speed of light, and h is the propagation angle of the THz pulse within the LiF relative to the surface normal.Given our measured delay of Dt ¼ 9.8 ps and ' ¼ 500 lm, we determine a THz frequency refractive index of 2.8 for LiF, in good agreement with previous measurements of ambient LiF. 78hough the use of a window material is not mandatory, it has the added benefit that the reflection at the vacuum-LiF interface should not be affected by the shockwave produced at the other side of the sample.As such, the THz pulse reflected at that interface can act as an on-shot reference.This is particularly valuable when there are large shot-to-shot fluctuations in the THz pulse.On-shot referencing may be implemented by correlating the spectrum of the THz pulse reflected at the first interface to that reflected at the second interface.A visual representation of this procedure is shown in Fig. 7.This is based on a similar approach where a THz pulse was split into two, and the separate THz pulses measured at different EO sampling crystals. 79Here, the time-domain waveform from an undriven measurement was divided so that each THz pulse was isolated, and each pulse was separately Fourier transformed to obtain two THz spectra, Ẽw;0 ðxÞ and Ẽs;0 ðxÞ, for the vacuum-window and window-sample interfaces, respectively.Next, a correlation function was calculated by using these spectra, ĈðxÞ ¼ Ẽs;0 ðxÞ= Ẽw;0 ðxÞ.For measurements taken with the drive, the data were split in the same way, and the reference spectrum was calculated using the correlation function Ẽs;r ðxÞ ¼ Ẽw ðxÞ ĈðxÞ ¼ Ẽw ðxÞ Ẽs;0 ðxÞ Ẽw;0 ðxÞ : Finally, the reflectivity ratio can be calculated as r ¼ Ẽs ðxÞ= Ẽs;r ðxÞ.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  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 Dt.
For a sufficiently thick sample, the conductivity of the compressed material can be determined starting from the Fresnel relation for reflectivity: 80 where rp is the measured reflectivity assuming a p-polarized THz field, ñw is the refractive index of the window, ñs is the refractive index of the sample, and h is the angle of incidence of the pulse at the windowsample interface.In the case that ñw ( ñs , this expression can be approximated by rp % ñw À ñs cos ðhÞ ñw þ ñs cos ðhÞ : In this form, the expression can be inverted to solve for the refractive index of the sample, For WDM where there are a large number of free electrons, we can relate the refractive index to the conductivity by From this, we see that knowledge of ñw is required to determine the conductivity accurately, which is not yet determined in the THz regime at shocked conditions.Changes in the refractive index of LiF have been observed using visible and infrared frequency light. 81However, there is no evidence of a correlation between those changes and what would be expected at THz frequencies.Separate measurements are required to determine the relevant refractive indices at THz frequencies and those along with experiments with thicker targets will be the focus of future studies.

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. 82While 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][84][85][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-2methyl-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][90][91] facilitating WDM experiments with currently available facilities.While the measurements discussed focus on solid samples, the use of liquid jets 92 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.

APPENDIX: THE POCKELS EFFECT FOR EO SAMPLING
Here, we derive the expressions for Eqs.(2a)-(2c).The change in the refractive index via the Pockels effect is described by considering the optical impermeability tensor, g , a 3 Â 3 rank 2 tensor whose elements are the inverse of the elements of the dielectric tensor e (i.e., g ij ¼ 1=e ij ).Combined with the relationship between the dielectric constant and the refractive index, e ij ¼ n 2 ij , g ¼ g xx g xy g xz g yx g yy g yz g zx g zy g zz (A1) For a lossless medium, g is a symmetric matrix (i.e., g ji ¼ g ij ).When any of the elements is modified, one can then write the modified elements of the impermeability tensor as where g ij;0 is the original impermeability tensor element, Dg ij is the change in that element, and n ij;0 and Dð1=n ij Þ 2 are the associated refractive index and its change.If the impermeability tensor is diagonal, the elements can be used to derive the following expressions for the change in the refractive index: The applied electric field from the THz pulse, where the 3 Â 6 matrix is the electro-optic tensor with r ij as its elements.
The details of the EO tensor and the impermeability are determined from the symmetry and cut of the EO crystal.For ZnTe or GaP crystals, the dielectric and impermeability tensors are diagonal, with e ii ¼ e 0 .The only nonzero elements of the EO tensor are r 41 ¼ r 52 ¼ r 63 .In the case presented, the crystal is cut along the (1 1 0) planes, with the z-axis set parallel to the [0 0 1] crystallographic axis.In this geometry, the THz field can be written as In order to determine how the refractive indices of the crystal are modified, this matrix needs to be diagonalized.For this configuration, this is accomplished by first applying a 45 clockwise rotation matrix around the z-axis, yielding the following impermeability matrix: This places the [1 1 0] axis along the x-axis and reflects the use of a (1 1 0)-cut for the EO crystal.The form of this matrix indicates that the fast and slow axes are rotated around the current x-axis by an angle w, which can be determined diagonalizing this matrix.To do this, a second rotation about the x-axis is applied, yielding an impermeability matrix Setting this term to zero yields Eq. ( 3), thereby defining the relationship between a and w.

FIG. 1 .
FIG. 1.(a) Schematic of a setup for EO sampling.(b) Schematic of the EO crystal with the relevant axes and angles labeled.The gray arrow represents the THz polarization, which makes an angle a with the [0 0 1] axis of the EO crystal.Plots of (c) w and (d) change in the refractive indices Dn y and Dn z for different values of a.

FIG. 2 .
FIG. 2. (a) Schematic illustration of how the echelon used for single-shot EO sampling.(b) Raw images of echelon measured after passing through Wollaston prism.The two squares correspond to I þ and I À , as the THz pulse is present, leading to modulation of the intensity from left to right.(c) The result of normalizing the image in panel (b) with ones where the THz pulse is absent, yielding DI þ =I þ 0 and DI À =I À 0 .

FIG. 3 .
FIG. 3. (a) and (b) THz time-domain waveforms extracted from unnormalized images of different polarization states produced by the Wollaston prism.(c) and (d) Waveforms extracted from normalized images.(e) Balanced signal obtained by taking the difference between the waveforms.(f) Spectrum of THz pulse obtained by Fourier transformation of the time-domain waveform.

FIG. 5 .
FIG. 5. (a) Schematic of the experimental setup around the sample plane.(b) Timedomain waveforms for the THz pulse, offset for clarity.The data on the cold and heated films are scaled 40Â.(c) Frequency-dependent conductivity determined from THz measurements.

FIG. 6 .
FIG. 6.(a) Schematic of the experimental setup around the sample plane.(b) Timedomain waveform showing two THz pulses.The first is due to the reflection at the vacuum-LiF interface and the second from the LiF-sample interface.
FIG. 7. (a) Time domain waveform of THz pulses measured without drive.Data in blue (left half of the waveform) are used to determine Ẽ w;0 and data in orange (right half of waveform) provide Ẽ s;0 .Plots of (b) Ẽ w;0 and (c) Ẽ s;0 .(d) Frequency domain amplitude of ĈðxÞ.

FIG. 8 .
FIG. 8. (a) VISAR image showing fiducials and arrival times of the drive and THz pulses as well as the shock breakout.(b) Raw and (c) filtered time-domain waveforms show THz pulses reflected from vacuum-LiF and LiF-Al interfaces with no drive and with drive present.Data are overlaid to highlight the change in the THz pulse reflected at the original LiF-Al interface.Additionally, a weak THz pulse is seen likely due to changes in optical properties of the LiF.
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.

Dg xy 0 0 0 g 0 À
Dg xy Dg yz ffiffi E y ŷ þE z ẑ, changes the impermeability according to ¼ r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 r 41 r 42 r 43 r 51 r 52 r 53 r 61 r 62 r 63 ¼ E THz ðsin ðaÞ= Dg xx ¼ Dg yy ¼ Dg zz ¼ 0; (A6a)Dg xy ¼ r 41 E THz cos ðaÞ: Dg yz sin ð2wÞ À Dg xy sin 2 ðwÞIn order for this matrix to be diagonalized, w should be chosen such that the off-diagonal elements are zero.Substituting for Dg xy and Dg xz , the off-diagonal terms become r 41 E THz sin ðaÞ cos ð2wÞ þ