Measuring magnetic fields in plasmas is vital for a proper understanding of plasma dynamics, but such measurements are often difficult to implement. Among the diagnostics commonly used in laser–plasma experiments^[1–3], only proton radiography, in which (quasi) mono-energetic protons are deflected in the plasma by magnetic fields, has been used with some success at the OMEGA laser facility^[1]. Proton radiography often allows a spatially resolved measurement of the magnetic field. While reconstruction of magnetic fields from proton radiography images is possible^{[4, 5]}, proton images are difficult to analyze, and the presence of caustic structures can make the reconstruction of the path-integrated magnetic field nonunique in the presence of strong fields. Moreover, this method cannot distinguish easily between magnetic or electric field deflections. This is particularly important in plasmas where strong electrostatic turbulence is present along with magnetic field fluctuations. Alternative diagnostics such as induction probes^{[6, 7]} or polarimetry^[8] have been attempted on OMEGA, but they are either too invasive or lack the needed sensitivity for accurate measurements.

Here we describe an implementation of a Faraday rotation measurement that has a much higher sensitivity and can be used together with proton radiography for accurate field measurements. The diagnostic makes use of the Thomson scattering probe beam and so causes little disruption to the currently available diagnostics on OMEGA. The requirement is that , where is the magnetic field, the electron density, and the path length of the probe beam through the sample.

The Faraday rotation measurement was implemented on a low-density turbulent plasma. The experimental setup is shown in Figure 1 (see also Ref. [9]). Two chlorine-doped plastic foils were each irradiated with 351 nm, 5 kJ drive lasers in either a or 10 ns pulse. This generated two counter-propagating plasma flows, each of which then passed through a plastic grid, and collided with one another after the start of the laser drive. The velocity of these flows prior to their collisions is . The turbulence that resulted from this collision is expected to produce dynamically significant magnetic fields through the turbulent dynamo mechanism^[9].

The Faraday rotation measurement is built alongside the Thomson scattering diagnostics^[10], and the setup allows for a coincident measurement of the magnetic fields with proton radiography. The Thomson scattering beam is a 30 J, 1 or 3 ns pulse duration and frequency doubled (526.5 nm wavelength) laser that probes the plasma within a region. To implement a Faraday rotation measurement, a Wollaston prism was inserted into the Thomson scattering collection optics, as shown in Figure 2. The prism splits the beam into two polarization components, labeled S and P. The two polarizations are further split such that half of each signal goes to the ion-acoustic wave (IAW) channel, which resolves ion-acoustic fluctuations, and half to the electron plasma wave (EPW) channel, which measures the total scattered power across all wavelengths. Both the IAW and EPW diagnostics are streaked in time with a 50 ps temporal resolution. Although the IAW diagnostic is spectrally resolved, since the electron features (contrary to ion-acoustic peaks in IAW) extend over a large frequency range, both polarizations cannot fit onto the EPW spectrometer and so the EPW diagnostic is not spectrally resolved.

Proton radiography is implemented at the OMEGA laser facility using a DHe capsule and CR-39 nuclear track detectors. Protons are generated by fusion reactions occurring by laser-driven implosion of a spherical capsule containing DHe gas^[17]. This releases mono-energetic protons with energies of and 15 MeV (accounting for Doppler shift). The protons are emitted isotropically, and thus illuminate the interaction region of the two plasma jets (see Ref. [9] for additional details). The capsule is positioned far enough away from the plasma such that the protons pass through the plasma as a thin, planar sheet. Magnetic fields generated within the plasma deflect the protons, which are then imaged onto the CR-39 plates. These proton radiographs can be used to infer the magnetic field structure^[4] within the plasma when the protons passed through. In this manner, quantities such as the mean magnetic field can be determined from the proton radiographs and then used as comparison/validation with the Faraday rotation diagnostic.

Both the IAW and EPW diagnostics can be analyzed in a similar manner. The difference in intensity of the two polarizations depends on three different factors. First of all, in the absence of any magnetic fields, the intensity of the S and P polarizations is determined by the relative angle between the polarization of the light and the axis of the Wollaston prism. We denote this angle by . In addition, the difference in intensity between the two polarizations also depends on the unequal response of the optics, detector, etc. Finally, when a magnetized plasma is present in the path, the induced rotation of the polarization angle changes the angle at which the light enters the prism and thus the ratio of the two polarizations when there is no magnetic field. The intensity of the two polarizations can be written as follows:

 (1)

 (2)

 (3)

Using Equation (3), the degree of Faraday rotation () can be determined, provided that the calibration angle , the ratio of to and the ratio of to are known. To determine the ratio of to , a measurement with no (or weak) magnetic field was used. A half wave plate in front of the Wollaston prism is set such that incoming polarization at the prism is . The axis of the Wollaston is set so that the two images from each polarization are separated along the streak camera input slit. The intensity of each polarization, and , is found by integrating over the total signal for each polarization.

As always, there is some stray light entering the detector. This must not be included in the Faraday rotation analysis, since it has not been scattered and so has not been influenced by the magnetic fields within the plasma. The IAW data is spectrally resolved, and since the stray light occurs at the same wavelength as the probe beam, it can be separated from the Thomson scattered signal, which instead is shifted in wavelength as the probe photons interact with the plasma. In practice, however, isolating the stray light from the scattered signal is not always possible, if for example, the frequency shifts are such that there is still a large overlap between the two. Conversely, as the EPW measurement is not spectrally resolved, there is no obvious way to disentangle the stray light from the scattering measurement. In this case, the approach is to minimize as much as possible the stray light. It is thus important to minimize stray light for both the Thomson scattering and Faraday rotation diagnostic. For Thomson scattering with IAW, the requirement is that stray light remains spectrally separated and distinguishable from the IAW peaks. Given the limited dynamic range of a streak camera, this means the stray light signal must always be of the same order (ideally smaller) than the IAW signal. These requirements become even more stringent for Faraday rotation using EPW since, as mentioned above, the signal is not spectrally resolved and so stray light directly overlaps with the scattering signal (and it is an additional source of uncertainty in the data). For EPW, we have estimated the stray light contribution by measuring the signal before and after the Thomson scattering probe laser is fired.

While the IAW and EPW data should give the same rotation angle, there are other effects that could make the two measurements differ from one another. We have already mentioned stray light. Additionally, the IAW data (being spectrally resolved) tends to be a lot noisier than the EPW data, and so the error in the measurement is larger.

The calibration shot used here was a single plasma flow (where only one foil is irradiated and so there is no collision). The only magnetic fields present for this case are those from the seed fields generated from misaligned temperature and density gradients. These seed fields are small, ^[9], and below the detection threshold. There are of course caveats to using this calibration shot. It can be seen from the measured IAW signal, shown on the top of Figure 3(b), that the stray light from the probe laser is significant. For the IAW signal, the wavelength of the stray light is close to the lower wavelength feature. As such, only the lower wavelength peak is considered in this case. Additionally, for this particular calibration shot, while the signal is not saturated on the charge coupled device (CCD) the signal has almost certainly saturated the more sensitive streak camera later in time. Nevertheless, during the initial 1 ns, the signal remains unsaturated, has low noise and has a constant ratio between the two polarizations. Turning to the EPW data, for well over ns of the shot, the signal is reliable and has an excellent signal-to-noise ratio but the contribution from stray light cannot be easily assessed, as shown in Figure 3(a).

Figure 4 plots the initial angle obtained from Equation (3) (when setting ). By requiring that , this allows us to find the ratio for both the IAW and EPW channels. This gives and , where the values have been averaged over 1 ns. The difference between the calibration from IAW and EPW is likely due to the amount of stray light and intrinsic noise associated with the two channels.

In this section we will apply the Faraday rotation angle measurements to a few different shots in order to measure the magnetic field during and after the collision of the two plasma flows. Data from shot 1, taken 29.5 ns after the drive lasers were fired with a 10 ns pulse, is shown in Figure 5. The Thomson scattering probe has a 1 ns pulse duration and the IAW signal is close to saturation on the streak camera. The EPW data on the other hand is very good, with minimal noise, and it can be used to determine the magnetic field for this shot. In Figure 6, we report the calculated rotation angles () obtained from the IAW and EPW signals. There is a significant difference between the two curves. As mentioned, this is likely due to saturation of the IAW signal on the streak camera.

Shot 2, taken ns after the drive lasers have been fired with a 5 ns pulse, also uses a 1 ns long probe beam and it is shown in Figure 7. The signal is much weaker than for shot 1. Figure 8 shows the resultant rotation angles extracted from IAW and EPW traces. We notice again that EPW gives a much better signal-to-noise ratio, and thus allows for a more precise determination of the rotation angle .

The Faraday rotation angle is given (in SI units) by^[18]

 (4)

 (5)

In this experiment, the magnetic field is expected to be turbulent. This implies that, on average, the Faraday rotation measurement would give a null result since the mean magnetic field along the path is zero. However, while the Faraday rotation is zero on average, every single measurement must be understood in terms of a random walk through a random magnetic field, and thus the Faraday rotation value corresponds to the standard deviation of the line integral. This can be estimated by assuming a field with a correlation length, , equal to the grid size, , and so the typical deviation is that acquired across one structure multiplied by the square root of the number, , of such structures encountered, . We thus obtain

 (6)

To determine the electron density, we have employed a full photometric calibration of the IAW channel, as shown in Figure 9. This allows for the integrated number of counts on the detector to be converted into the total scattered power. To complete the full photometric calibration, a fiducial laser was used. The amount of light coupled through the Thomson scattering telescope was measured using an energy meter and then cross referenced to a pick-off monitor at the start of the laser path. Laser pulses were then recorded on the Thomson scattering system and the energy measured through the cross calibrated energy meter. The transmission of the Thomson scattering probe beam through the Faraday rotation package was characterized separately as and then included in the final calibration. The counts registered on the IAW spectrometer could then be converted to a value for the total scattered power as ADU/pJ.

The total scattered power can be used to determine the electron density via

 (7)

 (8)

Having determined the electron density and rotation angle, the path-integrated magnetic field from the Faraday rotation measurement can be calculated using Equation (5). For this experiment, the effective path length,  cm, is equal to the scale length of the electron density, as inferred from the measured self-emission X-ray images^[9] and FLASH simulations (see Figure 1(c)). The EPW data then gives a magnetic field of 40 kG and 160 kG for shots 1 and 2, respectively (the noisier IAW data gives values of 50 kG and 190 kG for shots 1 and 2). These data shot values are well above the 4 kG seed values of the calibration shot. The strength of the Biermann battery generated seed fields for this experiment is described in Refs [9, 13]. Taking a field strength kG, path length cm and electron density then gives a rotation angle .

The accuracy of the Faraday rotation diagnostic can be characterized through comparison with proton radiography, a diagnostic already commissioned at the OMEGA laser facility^{[11, 12, 20]}. Proton radiography was performed on the same shots as those discussed previously. Faraday rotation and proton radiography probed the plasma at essentially the same time, to within ns, which is much shorter than the hydrodynamic eddy turnover times at the largest scale, and so they both probe the same magnetic field structures. The path-integrated magnetic field reconstruction^[4] from the proton radiographs shows the entire interaction region, as opposed to the area sampled by the Faraday rotation diagnostic, as shown in Figures 11 and 12. As such, the proton radiographs can provide both a mean path-integrated field estimate for the entire region as well as the maximum path-integrated field produced at the time of the radiograph. To make a fair comparison between the two diagnostics, the mean field from the proton radiography images is calculated within a square region. This region is the extent that the two plasma jets could travel within the ns difference in diagnostic timing. Additionally, because of the geometry of the experimental setup, the Thomson scattering beam experiences about twice the path length that the protons encounter. Accordingly, to fairly compare the two diagnostics, the path-integrated magnetic field measured by the Faraday rotation diagnostic is reduced by a factor of from Equation (6).

Both the mean path-integrated magnetic field within the Thomson scattering region and the maximum path-integrated magnetic field as calculated from proton radiography are plotted in Figure 13. The Faraday rotation path-integrated magnetic field for each shot is estimated from the average rotation angle within the ns of signal.

The error bars in the proton radiography inferred mean path-integrated field are found by sampling different 0.4 mm square regions throughout the reconstructed radiograph. There is an additional error of around inherent in the reconstruction algorithm which we include; this uncertainty is the result of approximations employed in the derivation of the algorithm^[4]. The error from the Faraday rotation measurement comes from both the variability in the measured rotation angles and due to inferring the electron density from the absolute calibration. The main source of uncertainty in the Faraday rotation measurement is due to the electron density which is known to within . While the small difference in timing between the two diagnostics allows for changes within the path-integrated magnetic field to occur, there is a close similarity between the mean path-integrated field inferred from the two diagnostics. Additionally, the mean path-integrated field calculated from Faraday rotation is consistently smaller than the maximum path-integrated field found from the proton radiography reconstructions, as expected. The similarity in the mean magnetic field as calculated from the two diagnostics gives confidence in the results of both diagnostics, suggesting that Faraday rotation can indeed be used for on-shot analysis throughout a shot day at the OMEGA laser facility.

We have fielded a new Faraday rotation diagnostic at the OMEGA laser facility. The analysis of the results has been described and a comparison made with the results from proton radiography. The Faraday rotation results are similar to those from proton radiography. One substantial advantage of this diagnostics is the fact that it does not rely on films or passive detectors (as CR-39). As such, analysis can be performed immediately after the shot, allowing for temporally resolved magnetic field measurements to be performed in real time during the experiment.