Streaked optical pyrometer for proton-driven isochoric heating experiments of solid and foam targets Open Access

Warm dense matter (WDM) is broadly defined as the state of matter on the cold edge of plasma physics (spanning 1–100 eV) and a range of densities within about an order of magnitude of solid density. Experimental studies of warm dense matter (WDM) are important for determining equations of state (EOS), as theoretical treatments are challenging. Figure 1 shows a density/temperature diagram that outlines the region of WDM and highlights the parameters that are accessible by our experimental platform and have been achieved in the experiments presented here (temperatures 1–20 eV at solid density and lower, attainable by heating solid foils and foams).

A key challenge in warm dense matter experiments is the measurement of temperature, which is a key variable of the EOS, thermal conductivity, and other physical quantities of interest. Several mechanisms for temperature measurements have been successfully explored, including optical pyrometry, x-ray Thomson scattering (XRTS),² and neutron resonance spectroscopy (NRS).³ Both NRS and XRTS have the advantage of probing the bulk material, while pyrometry is a surface measurement. XRTS is impractical for experiments with the conditions reported here because a bright, short-pulse x-ray backlighter, such as an x-ray free electron laser (XFEL), is needed. NRS is impractical for a similar reason, since it would require a large neutron source, either spallation driven or from a nuclear reactor (the latter of which is not pulsed), and would therefore not allow time-resolved measurements. Neither of those facilities currently exists in conjunction with the high intensity short pulse lasers required for isochoric heating, although laser generated pulsed neutron sources have been proposed for temperature measurements⁴ and could, in principle, be developed on future multi-beam petawatt lasers. Streaked optical pyrometry (SOP) then becomes the preferred method for this type of experiment as it uses the light emitted by the plasma itself to determine a temperature, with no backlighter needed, and is also a time-resolved measurement.

Streaked optical pyrometry (SOP) is a common diagnostic for time-resolved heating in plasma first developed in 1985.⁵ Since then, various facilities have developed SOP systems for their own applications.^6–9 The concept is to accurately measure the thermally emitted light from a plasma, with appropriate spatial, temporal, and spectral resolution to discern the instantaneous spectral radiance at the points of interest, ideally in a wavelength range for which the plasma emits similarly to a blackbody. A streak camera is employed to achieve fast diagnostics of a dynamically evolving plasma. To successfully employ SOP measurements, precision is needed in imaging, timing, solid angle, and photon count, meaning that the system must be carefully calibrated. No appropriate reference for radiance exists at these timescales (∼ps) and radiation intensities [∼10⁸ (W/sr)/m²], so calibration of the optics must be done separately from calibration of the streak camera, and the system’s imaging resolution must be carefully measured.

The streaked optical pyrometer (SOP) measures the surface brightness temperature of the plasma, that is, the temperature is inferred from converting the intensity of light measured by using the SOP to an equivalent blackbody temperature. (Implicit in this method is the assumption that the plasma is emitting similar to a blackbody.) The spectral intensity of a given wavelength emitted by a blackbody source at any temperature is given by the following blackbody formula:

where ε_λ is the emissivity at the given wavelength, h is Planck’s constant, c is the speed of light, k is the Boltzmann constant, and λ is the radiation wavelength (all in SI units). The absorptivity is related to the emissivity for a blackbody in local thermal equilibrium by Kirchhoff’s law: α_λ = ε_λ (where α_λ is the absorptivity). In this way, reflecting light off the plasma at the same wavelength as the pyrometer measures can give the emissivity.

Although there are a variety of ways to achieve isochoric heating,^10–12 for this study, we used MeV energy protons accelerated via the target normal sheath acceleration (TNSA) mechanism^13–15 that arises from the short pulse laser interaction with a micrometer thick target. Understanding that low energy (<5 MeV) protons can heat a 10 µm thick foil with no need for the higher energy protons or carbon ions that are produced by a higher intensity laser/target interaction, our aim with this experiment was to produce many low energy protons over a relatively large focal area to heat a target.

These protons hit a secondary target and deposit their energy, heating it over ∼10 ps, faster than the hydrodynamic response time of the heated foil. The SOP imaged the blackbody emission from the rear surface of the secondary target over time, which was used to calculate the time-resolved brightness temperature. Over two experimental runs, we demonstrated the use of this SOP to measure temperatures of heated aluminum foils and carbon foams with peak temperatures from 1 eV to 10 eV. This experimental setup is a starting point from which additional diagnostics and targets can be developed to measure a variety of parameters under warm dense matter conditions.

This manuscript details the design, analysis, and calibration of the SOP and shows its application to isochoric heating experiments at the Texas Petawatt Laser facility. The work builds on previous short pulse laser ion acceleration and isochoric heating experiments at the University of Texas Center for High Energy Density Science.^16,17 We note that this pyrometer has also been fielded on WDM experiments at the Trident Laser Facility,¹⁸ where solid foils were heated by means of an aluminum ion beam to similar temperatures (1–3 eV).

We have designed the SOP specifically to measure temperatures with a lower limit of 0.4 eV. It is calibrated to measure temperatures up to 25 eV but could measure higher brightness temperatures with calibrated intensity reducing filters—we observed peak temperatures around 1 eV on most shots, and our hottest shot reached 10 eV, but the design allows for the measurement of a range of temperatures by using a variety of filters and streak camera settings. (The settings must be changed shot to shot to achieve the full range of the diagnostic—on a single shot, the measurable temperature range is more limited.)

The pyrometer images the 400 nm blackbody radiation from a heated plasma onto a Hamamatsu C7700-01 streak camera, which temporally sweeps with ∼5 ps time resolution on the fastest setting. The 400 nm wavelength was chosen because it lies in the visible range and avoids a harmonic of the ∼1 µm wavelength glass lasers, such as the Texas Petawatt and Trident Laser at LANL. (Additionally, we had a ready source of 400 nm light from a doubled Ti:sapphire laser, which we used to calibrate the pyrometer.)

The optical schematic for our pyrometer is shown in Fig. 2. An f/5 achromatic lens collects light from the heated matter, which is relay-imaged onto the streak camera slit by means of a second achromatic lens. A bandpass filter is placed before the streak camera to admit light of a narrow frequency band (±5 nm or ±20 nm depending on the filter) centered around 400 nm. The light that passes through the filter is sent through the slit to the streak camera, which produces a streaked image of the heating over time. See the supplementary material for the specific lens and camera details.

The total transmission of the optical system (see Fig. S1 of the supplementary material) allows one to calculate the radiance of the source from the number of counts measured by using the streak camera detector.

When converting the streak camera measurement to a temperature, we set the emissivity equal to 1 based on calculations from the work of Celliers and Ng,¹⁹ which shows that ε ≅ 1 for conditions similar to this experiment, particularly at later times (>20 ps). Knowing that the emissivity is close to 1 around the critical density layer, we can use Eq. (1) to calculate the radiance for a perfect blackbody at the desired temperature range. We note that the most common way to correct for gray body emissivity is by performing a time dependent reflectivity measurement since the reflectivity is equal to 1 − ε. This measurement can be accomplished by using Fourier domain interferometry (FDI)²⁰ or velocity interferometry (VISAR)²¹ and has been demonstrated in several recent papers.^9,22 We plan to field FDI concurrently with the SOP at the Texas Petawatt for this purpose in future experiments.

The wavelength-integrated radiance as a function of temperature [generated by Eq. (1)] scaled by the spectral transmission curve (Fig. S1) is shown in Fig. 3.

The expected number of streak camera counts as a function of temperature depends on the wavelength-integrated radiance [R(T)] as well as the following factors (the measurement of which is discussed in subsequent paragraphs, along with their estimated uncertainties): (1) light collection solid angle Ω (sr), which is controlled by the entrance pupil iris; (2) system resolution r (μm²/pixel), which can only be changed by swapping out the achromatic lenses; (3) the sweep speed (sweep) controlled on the streak camera; (4) the slit size multiplier controlled on the streak camera by manually opening and closing the slit; and (5) the streak camera sensitivity S (counts/J). The following formula is used to compute the counts registered by using the streak camera as a function of brightness temperature, measured in eV:

Figure 4 shows an example of calculated brightness temperature as a function of streak camera counts for shot 11441 (10 nm bandwidth filter, 60 µm slit, and 1 ns sweep speed). The data have 12-bit resolution, so the maximum number of counts before saturation is 4095.

We measured the streak camera sensitivity using the upgraded THOR laser at UT Austin (Ti:sapphire, 800 nm, 30 fs, and 10 Hz repetition rate).²³ We sent the frequency doubled laser pulse (filtered with the same 405 nm interference filter used in the SOP) to the streak camera and a calibrated energy meter using a calibrated beam splitter to give streak camera and energy data concurrently. A schematic of the measurement is shown in Fig. 5.

We integrated our measurements over multiple shots (with the slit fully open and the gain set to maximum) and thus were able to determine the number of counts registered by using the streak camera per joule of incident energy. We took calibration data at several sweep speeds relevant to our experiment: 1 ns, 2 ns, and 5 ns, as well as 100 ns. Our results are summarized in Table I; the counts/J are consistent to within 2σ (∼4%) for all sweep speeds, and there is no evidence for a trend. For all our measurements, the gain was at the maximum setting. We show a sample calibration image in Fig. 6.

We calculated the solid angle of light collection by measuring the distance from the light source (heated package) to the entrance pupil of the optical system (located at the first lens) and the diameter of the entrance pupil. An error of ±2 mm in positioning the target exactly at the prescribed position on every shot contributes to a 6% uncertainty in the solid angle calculation.

We measured a (1.5 ± 0.2) × 10⁻¹¹ m²/pixel system spatial resolution in situ using a resolution target and a blue laser pointer as a backlighter.

The Texas Petawatt^24,25 laser pulse (150 fs, 100 J) was focused by an f/40 spherical mirror to an ∼80 μm radius, corresponding to an on-target intensity of I ∼ 10¹⁸ W/cm², and irradiated a solid gold target, producing protons through TNSA. A secondary target (called the package) was mounted 300 μm behind the ion source and was heated by the proton pulse to WDM conditions. Figure 7 shows a schematic of the experimental setup.

The target was positioned nearly normal to the laser direction. A Thomson parabola spectrometer^26,27 was placed directly behind the package in order to measure the energy spectrum of the TNSA protons. The SOP imaged the rear surface of the package at a 45° angle.

The targets for proton production were 5 μm thick Au foils, while the heating packages were 10 μm thick Al foils, both of which are commercially available. The foils were mounted on either side of 300 μm thick Al stalks to provide the required separation for proton heating.

During the first experiment in the campaign, we took measurements with proton source targets ranging from 1 μm to 5 μm of Au [Fig. 8(a)]. This figure reports the number of counts obtained by the Thomson parabola as “arbitrary units”—while the Thomson parabola spectrometers are calibrated for energy, the individual imaging plates are not calibrated for conversion from “counts” to “number of protons.” There are established methods for calibrating imaging plates for a number of ions,^28,29 but this calibration was not performed for these imaging plates on this experiment.

From the data in Fig. 8(a), we determined that 5 µm Au was a good choice for the proton source target. For the subsequent experiments in this run, we used 5 µm Au proton source targets and measured several shots with proton spectra using the Thomson parabola shown in Fig. 8(b).

During our campaign on the f/40 beamline at the Texas Petawatt Laser, we observed peak brightness temperatures from 10 µm thick aluminum foils ranging from 1 eV to 10 eV. The variation in temperature arose from variations in on-target focusing and energy, with no deliberate changes in laser parameters. (The laser energy was systematically lower in the second set of experiments of the campaign than in the first experiments of the campaign, however.)

Using formula (1) with the conversion R(T) given in Fig. 4 (generated for the appropriate streak camera and filter settings), we calculated the brightness temperature in eV using the number of counts registered on the streak camera. From the processed SOP image, we took a lineout of the brightest region and included the uncertainty as calculated for the appropriate experiment settings, as shown in Fig. 4 by the red dashed lines. An example (background-subtracted counts and converted brightness temperature) is shown in Fig. 9. The lineout in Fig. 9(c) of the heating has t = 0 corresponding to the heating onset (i.e., any temperature higher than background).

The brightness temperatures measured in the first experiment of the campaign were hotter overall (we hypothesize that higher laser energy and intensity led to acceleration of more protons). Three heating images from the first experiment of the campaign are shown in Fig. 10.

The streaked optical pyrometer is able to spatially resolve the extent of the ion beam heating as well. This capability is useful for experiments on the Trident Laser Facility,^18,30 where this pyrometer was also fielded in which it is important to see the differences in heating of two foils placed side by side that are heated by the same ion beam. For the lens setup of the first experiment, the SOP has the spatial scale 0.48 ± 0.04 pixels/μm (measured with a resolution target). We took horizontal lineouts of the streak camera images shown in Fig. 10 at the heating onset to demonstrate the spatial extent of the heating (Fig. 11).

For the second experiment, the results are much the same. For this experiment’s lens setup, the spatial resolution was 0.26 ± 0.02 pixels/μm. We also tried two different aluminum thicknesses for the heated packages, 10 μm and 7 μm. The spatial extent of the heating, shown in Fig. 12, is consistently 300–500 μm, even though the overall brightness temperatures are lower. We note that the 7 μm Al shots are (as expected) slightly hotter than the 10 μm Al shots, with the exception of shot 11443 that was the coldest shot of the campaign (where heating was still observable by the SOP).

In the second set of experiments in the campaign, we also explored the possibility of heating lower density carbon foam targets as a possible candidate for a laboratory astrophysics target that would emulate some of the plasma conditions found in the atmosphere of white dwarfs. Heating results for two of the foam shots are shown in Fig. 13. Our foam packages were 100 µm thick and had a density of 60 mg/cm³. We found that the foam cools much more slowly than the aluminum. Lengthening the time window for shot 11485, we observed that the foam heated up again after ∼1 ns. Despite these differences, WDM conditions were still achieved when heating the foam targets with the laser-driven MeV proton beam.

In order to explore the relationship between proton energy deposition and the observed brightness temperatures, we have employed some preliminary 1d simulations in xRAGE,³¹ an Eulerian radiation hydrodynamics code developed at Los Alamos National Laboratory. For the simulations shown in this section, we included the physics models for hydrodynamics, 3T (electron, ion, and radiation temperatures), and partial ionization.

We model the cooling and expansion of the secondary package with an initial density and temperature for each region of the heated material. The starting density and temperature of the heated material are dependent on the energy source—in this case, the TNSA proton beam. There is no option in xRAGE to input a proton energy source directly, so we use SRIM and PSTAR^32,33 to calculate the proton stopping power, which, in turn, provides an internal energy source. This process is complicated by the fact that we have to estimate the number of low energy protons in the TNSA source because the Thomson parabola spectrometer has a low energy cutoff to its measurements. For this set of simulations, we made a lower bound estimate for the low energy proton trend, as shown in Fig. 14(a).

We then use the SRIM cold stopping power data (for solid density aluminum) to calculate the energy deposited by the proton spectrum into each zone of the material. The energy deposited per proton into each zone of the material is shown in Fig. 14(b).

The dE/dx values are converted into a specific internal energy (SIE) input to xRAGE. Previous calculations of SIE from ion energy deposition have been carried out by Bang et al.¹¹ We calculate the SIE in a given zone with the following:

The parameter N_ions is selected in order to give the starting temperature that matches best with the SOP observations. For the sample shot in Figs. 14 and 15 (shot 9626), the fraction of laser energy on shot that would have to go to the TNSA protons to produce the observed heating is ∼2%. This is not unreasonable for TNSA, which has seen laser → ion energy conversion efficiency up to 9%.³⁴ The “fraction not stopped in previous zones” is calculated for each zone with range data from SRIM [Fig. 14(c)].

We specify a starting density (solid density) and starting SIE for each zone of the material, which is converted to starting temperature with the equation of state SESAME 3720.³⁵ After an initial energy input, the material expands into a low-density background gas (deuterium at 1 × 10⁻⁴ g/cm³).

In order to compare simulation results to the experimental data as seen by the streaked optical pyrometer, we track the overall material temperature at the rear surface of the target. Since the target is expanding over the time of our simulation, we track the rear surface by determining the critical density for 400 nm at each time step. For an optically thick medium such as these aluminum foils, the thermal emissions at a particular wavelength originate in the material’s critical density surface for that wavelength. For underdense media such as the carbon foams, other methods for determining emission are needed, which is an area of ongoing work.

For each time step, we output data for the temperature and electron density from which we calculate the position of the critical density surface and then record the temperature at that location.

When we compare the xRAGE output of material temperature at the 400 nm critical surface with SOP lineouts for a sample shot (shot 9626, Fig. 15), the simulation compares well with the observed cooling trend. Our analysis of the other shots is ongoing and will be reported in a future paper.

We describe the design and calibration of a new ultrafast streaked optical pyrometer (SOP) with ∼5 ps resolution for measuring the time-resolved surface blackbody temperature of an isochorically heated plasma. With its variety of settings and calibrated filters, the pyrometer can perform time-resolved temperature measurements for WDM experiments at a variety of short-pulse laser facilities (ideal short-pulse wavelength of 1 μm) for a range of target densities.

Proton energy deposition-based modeling of heating is complicated by the presently available Thomson parabola data. We are able to measure the proton spectrum for energies > ∼2 MeV concurrently with the heating on each shot. With calibration of the Thomson parabola diagnostic to measure the number of ions and a better model for the low energy ion spectrum, we would be able to use our modeling techniques to more accurately relate the proton dE/dX to the observed heating on each shot. Currently, we demonstrate that we have the infrastructure for calculating proton dE/dX using our best estimations of the number of protons and spectrum for protons at energies <∼2 MeV.

We have demonstrated this pyrometry diagnostic’s utility with time-resolved brightness temperature measurements of solid aluminum and carbon foam heated to WDM conditions at the Texas Petawatt Laser facility f/40 beamline. Simulation temperatures from xRAGE agree well with the sample data, and analysis is ongoing for other experiments.

See the supplementary material for the measured transmission of all the SOP optics and the part numbers for the commercially available optics and hardware.

This work was supported by NNSA Cooperative Agreement No. DE-NA0002008, the DARPA PULSE program (No. 12-63-PULSE-FP014), and the Air Force Office of Scientific Research (No. FA9550-14-1-0045).

This work was performed under the auspices of the U.S. Department of Energy by the Triad National Security, LLC (Contract No. 89233218CNA000001), Los Alamos National Laboratory, and was supported by the LANL Office of Experimental Sciences programs. Simulations were run on the LANL Institutional Computing Clusters.

The authors would like to thank the staff of the Texas Petawatt Laser facility who were helpful and generous with operations and support during these experiments. The authors would like to thank Juan Fernandez, Brian Albright, Cort Gautier, Sasi Palaniyappan, Woosuk Bang, and everyone else involved with the Trident WDM experiments for which this pyrometer was first fielded. The authors would also like to thank the anonymous referees for their comments and careful reading of this manuscript, which improved the paper.