Multi-angle multi-pulse time-resolved Thomson scattering on laboratory plasma jets

Thomson scattering is a powerful plasma diagnostic that can be used to measure multiple plasma properties simultaneously. In the collective scattering regime, α ≡ 1/kλ_De ≳ 1, plasma properties that can be measured include electron temperature, ion temperature, electron density, and flow velocity of the plasma. Here λ_De is the electron Debye length and the wave number k is defined by k = k_s − k_l, where k_s and k_l are the scattering and laser wave vectors, respectively. These types of measurements have often been performed with time resolution in order to study the evolving plasma properties of laser produced plasmas.^1–3 However, for pulsed power plasmas, likely due to the slower time evolution and larger spatial scales, Thomson scattering is traditionally not carried out with time resolution.^4,5 These plasmas can also change rapidly over a few nanoseconds, such as during pinch time of a Z-pinch.⁶ Previously we reported the first sub-ns time-resolved Thomson scattering diagnostic system fielded in a pulsed power-driven high energy density (HED) plasma using a single laser pulse from a single scattering angle.^7,8 Since total time scales of interest in such experiments can be >10 ns, we have now split the laser beam into two pulses and time delayed one relative to the other. Time resolution is achieved within these two pulses by using a streak camera to record the Thomson scattering spectra. Furthermore, spectra from two scattering angles are now multiplexed onto a single streak camera using optical fibers.⁹ The collection of scattered light from two different angles was undertaken to have the possibility of measuring the electron density in addition to the electron temperature.¹⁰ Scattering from two different plasma volumes could also be recorded with this technique. Thus, we now have the capability to follow plasma conditions with sub-ns time resolution at two times and simultaneously at two angles or two spatial locations in plasma, the first such capability in pulsed power-driven HED plasma.

As a test bed for these Thomson scattering diagnostic system improvements, plasma jets produced from radial foils^5,7,11 were chosen as the load as they produce reproducible target plasmas that last for ∼100 ns and are located on the experimental chamber axis. Prior Thomson scattering measurements on these jets determined the flow velocity⁵ and time-resolved heating of the jet by the laser through inverse bremsstrahlung.⁷ The series of experiments discussed in this paper involved measuring the effect of the direction of current flow in the radial foil on the plasma jet. Both experiments and theory, in jets with and without an external magnetic field, have shown that current polarity affects the formation of the jets, a result of the Hall term in generalized Ohm’s law.^5,12–15 The experiments reported here showed differences in the measured electron temperature, and in the widths of the ion-acoustic peaks, which could indicate differences in the ion temperature.

Experiments were performed on the COBRA pulsed power generator¹⁶ at Cornell University operating in the short pulse mode (100 ns rise time, 1.2 MA peak current). Data were gathered about 190 ns into the current pulse (Fig. 1). The load was a 15 μm thick aluminum (Al) radial foil, with a 5 mm diameter brass center pin.^7,15 The center pin was used as either the anode [Fig. 2(b)] or the cathode (i.e., “reverse polarity” and “standard polarity,” respectively).

The Thomson scattering probe is a Nd-YLF laser that is frequency doubled to produce a 10 J, 526.5 nm laser pulse with a full width half maximum (FWHM) of 2.3 ns.^5,7 This beam is divided into two temporally separated, coaligned 2.5 J pulses using the optical arrangement shown and described in Fig. 3. At present, the delay between the start of the two beams can be changed from 3 ns, effectively making a continuous 6 ns laser pulse, to about 14 ns. The 35 mm diameter beams are focused down to 350 μm at 5 mm above the center of the pin using a 2.5 m focal length lens.

Scattered light was collected from the plasma with 100 μm diameter multimode fibers. These fibers were set up at 60° and 150° and imaged the plasma through a 125 mm focal length lens with a 50 mm diameter. The lens was 570 mm from the load, resulting in collection at f/11.7 and a magnification of 0.28. The scattering length along the laser captured by a fiber was therefore 380 μm. The fibers from the two scattering angles were set up to enter the spectrometer at the same height to image both of them onto the streak camera⁹ and had enough separation, so the spectra were clearly resolvable. Sample Thomson scattering spectra with this fiber setup and a 12 ns delay between laser pulses are shown in Fig. 4.

The spectrometer is a 750 mm Czerny-Turner instrument with a 2400 l/mm grating. The 100 μm fiber diameter acts as the slit width, resulting in a spectral FWHM of 0.6 Å. Data were recorded using a streak camera with a sweep speed of either 10 ns or 20 ns, determined by the delay time between the two laser pulses. A 125 μm entrance slit width resulted in a possible temporal resolution of 150 or 250 ps, respectively, for the two sweep speeds. However, a temporal bin size of 25 pixels, used to increase the signal to noise ratio (SNR), resulted in temporal resolution of 270 ps and 520 ps, respectively.

The experimental ion-acoustic wave spectra are compared to a collisionless theoretical spectral density function, S(k, ω), with ω being the difference between the scattered and laser frequency.¹⁷ Of specific interest here is the wavelength separation between the two ion-acoustic peaks, which can be approximated by¹⁷ 

where c is the speed of light, θ is the angle between k_s and k_l, Z is the average ionization state, m_i is the ion mass, and T_e and T_i are the electron and ion temperatures, respectively. If ZT_e > 3T_i, then the peak separation will depend primarily on the electron temperature in the $k2λDe2<1$ regime. Therefore, by finding Z, a function of T_e and n_e, from a FLYCHK table for Al,¹⁸ we can have a good measure of both T_e and Z, which was found to be between 6 and 9 for the range of T_e obtained below. Each Thomson scattering profile is fit twice, under different key assumptions. The first fit assumes that T_e = T_i and that most of the width of the profile is due to additional broadening mechanisms to be discussed shortly. The second fit has no additional broadening but allows T_i to vary. As T_i is found to be higher than T_e when it is allowed to vary, the first assumption creates an upper limit for T_e and the second creates a lower limit, as higher T_i will cause the peaks to be further separated for the same electron temperature. For most of the fits, we also assume that n_e = 5 × 10¹⁸ cm⁻³. For both fitting methods, we use a Monte Carlo technique to calculate error bars.^1,7

Factors other than the ion temperature that broaden the ion-acoustic peaks include collisions,^19,20 velocity gradients, and turbulence.^1,21 To first order, these can be included by convolving S(k, ω) with a Gaussian function.^1,21 Accounting for this additional broadening often improved the quality of the fit to the central dip in the spectrum, compared to just varying T_i.⁷ At the very least, we expect ion-ion collisions to affect our Thomson scattering profile as kλ_ii, with λ_ii being the collisional mean free path, is 0.44 when T_e = T_i = 50 eV, n_e = 5 × 10¹⁸ cm⁻³, and Z = 8, which puts it in the intermediate range for collisionality.²⁰ 

The electron density can be measured from the ion-acoustic features by collecting from two significantly different wave numbers due to the $k2λDe2$ dependence in the peak separation, see Eq. (1).¹⁰ Assuming that the density and temperature are the same for both wave numbers, the smaller wave number is used to measure the electron temperature and that temperature is used to find the electron density from the large wave number. For plasma jets, T_e = 50 eV and n_e = 5 × 10¹⁸ cm⁻³, scattering at 60° and 150° results in $k602λDe2=0.0786$ and $k1502λDe2=0.2934$⁠, which is a 10% difference in peak separation.

To compare the degree of laser heating with only 2.5 J of laser energy to earlier experiments,⁷ which used 10 J of laser energy, the first experiments were performed in reverse polarity. In the earlier experiments,⁷ the Al plasma jet heated from 25 eV up to about 85 eV before cooling back down at the end of the laser pulse. This heating is likely due to inverse bremsstrahlung, as discussed in that work.⁷ For this reverse polarity setup, the laser pulse separation was 12 ns. The streaked scattering spectra from a typical reverse polarity shot is shown in Fig. 4, and the electron temperatures determined from the spectral profiles as a function of time are shown in Fig. 5. Figure 4 shows that early in time the two peaks of each Thomson scattering profile separate due to the heating of the plasma. Figure 5 shows the electron temperature starting at 20 eV and increasing to about 35-45 eV. As expected, this is less than earlier experiments with 10 J of laser energy. Figure 5 also shows that the electron temperature is again at the initial temperature when the second pulse arrives. This is due to the jet plasma propagating up through the scattering volume. Based on a measured axial velocity for magnetized plasma jets of about 60 km/s,⁵ the plasma moves a distance of 600 μm during the 10 ns between pulses. Since the laser spot size is only 350 μm, an entirely different volume of plasma is probed by the second pulse.

Another series of experiments were performed to compare the Thomson scattering spectra based on the current polarity through the foil, as earlier work, discussed in the introduction, has shown the current to have a significant impact on the plasma jet.^5,12–15 For these experiments, the time delay between the laser pulses was shortened to 4 ns, in order to probe and heat the same plasma volume with a nearly continuous laser pulse. The electron temperature as a function of time for a typical standard polarity shot (Fig. 6) exhibits an increase from 15 eV up to 30–35 eV, compared with the temperature maximum of 40–45 eV measured in reverse polarity shots under the same conditions (Fig. 5).

Another significant difference between the two polarities is the widths of the ion-acoustic peaks. Figure 7 shows that when assuming additional brocading and T_e = T_i, the Gaussian FWHM is substantially smaller for the standard polarity experiments. This means that either the additional factors affecting the ion-acoustic peak width are more important or T_i is significantly higher than T_e in reverse polarity shots. If the profile width was solely due to the ion temperature, it would imply a maximum ion temperature of 200 eV for reverse polarity shots and only 100 eV for standard polarity. The broader peaks caused the reverse polarity shots to have a larger difference in T_e between the two fitting techniques than standard polarity as the higher value of T_i, when T_i ≠ T_e, reduces the electron temperature. Therefore, though there is always a difference between the measured heating of the two jet polarities, it is more extreme if we assume that the widths of the profiles are due to the additional broadening instead of the ion temperature. Comparing the time scale of the experiment to the electron-ion energy collision time, around 10 ns, suggests that the two temperatures could be slightly different. Therefore, it is likely that T_e lies somewhere between the two extremes.

Although it is, in principle, possible to estimate an electron density by comparing the measured peak separation from two different angles,¹⁰ the accuracy of this measurement technique depends critically on having the larger kλ_De close to 1. For plasma jets, we fit the scattering profiles for both angles assuming a variety of different plasma densities and then compared the ratio of the measured temperatures between the two angles assuming a 10% error on each temperature measurement. The ratio should be unity as the measured electron temperature should be the same for the two different angles. Figure 8 shows the results of this comparison for a typical reverse polarity shot. It is clear that for our laser and plasma parameters the density does not significantly affect the measured temperature ratio if it is higher than 1 × 10¹⁹ cm⁻³. This is because changes in density do not affect the peak separation when $k2λDe2<<1$ for both scattering angles. While the data from several shots show that accurate density measurements cannot be made since, for most densities, the error bars for T₆₀/T₁₅₀ overlap with one, we do see that the electron density is at least above 2 × 10¹⁸ cm⁻³, which is in agreement with previous interferometry data.²² 

In order to increase the viability of obtaining a density estimate, the effect of α on peak separation was considered. For a given electron density, the region of viable electron temperatures can be upper bounded when α is large enough to be in the collective scattering regime and lower bounded when $1+k2λDe2≈1$⁠. Figure 9(a) shows these bounds for our current scattering setup, as well as the parameter range where we want to make this measurement. The upper bound is set by having α > 0.5 for both scattering angles. The lower bound is set by ensuring that the ratio between the density-dependent term in the peak separation, $1+k2λDe2−1/2$⁠, is greater than 20%. Figure 9(b) shows how a possible redesign of the system to collect from two different scattering wavelengths (526.5 and 263.25 nm) at 150° would enable measuring the density from the ion-acoustic waves in the plasma jet.

We have demonstrated the use of a multi-pulse, multi-angle, streaked Thomson scattering system to measure the heating of the electrons by inverse bremsstrahlung in Al plasma jets. The multi-pulse laser scattering system was created by splitting the laser beam and time delaying one pulse relative to the other, while the multi-angle capability in a single streak camera was achieved by a nontraditional coupling scheme between the fibers and the spectrometer. Reverse polarity jets showed more heating and wider peaks than similar standard polarity jets, which implies that the jet plasma properties depend on the direction of current flow. Electron densities of the jet were estimated to be at least 2 × 10¹⁸ cm⁻³.

The Thomson scattering system described here can effectively measure plasma properties as a function of time and space in several high energy density experiments in our lab, including gas puff z-pinches,^8,23 shock structures, and cylindrical foil lines. Further improvements of the system, such as splitting the beam into four parts, are currently being evaluated.

The authors thank T. Blanchard, D. Hawkes, and H. Wilhelm for hardware fabrication and COBRA operation. This research was supported by NNSA Stewardship Sciences Academic Programs under DOE Cooperative Agreement Nos. DE-NA0001836 and DE-NA0003764.