Magnetic reconnection is a fundamental process where magnetic field lines break and reconfigure in a lower energy state, thereby releasing energy to heat the plasma. It is an important mechanism in many astrophysical situations, such as powering coronal mass ejections and solar flares, the solar wind interacting with the Earth's magnetic fields [1], as well as in the universe's most violent and energetic objects like pulsars [2], active galactic nuclei [3] or gamma ray bursts [4]. Direct measurements of the fields and particles are either difficult in the case of the near-Earth environment [5], or impossible at greater distances. Furthermore, these phenomena cover a wide range of plasma parameters and field conditions making the topic diverse. Studying reconnection processes in the laboratory is therefore a valuable method for enhancing our theoretical knowledge.
Terrestrially, magnetic reconnection can occur within tokamak plasma [6], or dedicated magnetic reconnection experiments such as the MRX machine [7]. Over the last decade, laser-driven magnetic reconnection experiments have been developed using high-energy nanosecond laser pulses where self-generated magnetic fields are driven together by the plasma flow [8][9][10][11][12][13][14][15]. At intensities of ∼ 10 14 Wcm -2 , a laser pulse can heat a target to form a plasma containing non-parallel tempera-ture and density gradients, thus generating azimuthal magnetic fields through the Biermann battery [16][17][18]. The megagauss-strength magnetic fields are transported by the bulk plasma motion at the plasma sound speed, c s = (Zk B T e /m i ) 1/2 where Z and m i are the ion charge and mass respectively and k B T e is the electron temperature; this is described as "frozen-in-flow". Focusing two laser pulses onto a target in close proximity produces a geometry where two opposing direction magnetic fields are driven into one another in the midplane. Fox et al. found in this strongly driven reconnection regime the compression of the magnetic flux means the Alfvén speed is time dependent [19], an important consideration for understanding the reconnection rate.
Increasing the laser intensity generates hotter electrons. The inverse-velocity dependence of the collision operator means the mean-free-path for the hottest electrons is large compared with the system size. Therefore, Braginskii's collisional transport theory breaks down and kinetic effects become important so that heat flows are "non-local". The magnetic field can then travel faster than the ion fluid velocity [20][21][22]. Driving a magnetic reconnection under these conditions means the reconnection rates are dictated by heat flows rather than the Alfvènic flows [23].
At intensities of Iλ radians for 15 µm thick aluminum and 17 milliradians for 30 µm thick polypropylene. In the detector plane, this would blur the images to give spatial resolutions of about 170 µm, 110 µm and 50 µm respectively. The azimuthal magnetic fields generated by the two spatially separated, high intensity laser pulses interacted produces distortion of the fields from the purely circular fields observed around a single laser pulse.
In the metal targets the significant scattering of the proton beam makes quantitative retrieval of the fields challenging. Furthermore, the proton flux had considerable shot-to-shot variation meaning the proton flux and spatial distribution are not sufficiently stable to use as an unperturbed reference for the radiography calculation. However, two local reductions in the measured proton flux with flux enhancement at the edges are observed. These structures are noticeably smaller compared with those of the plastic targets taken at the same time and laser spot separation, with the smallest structures ob-served for the copper target. This implies that for this material the fields were either weaker, thinner (occupied a shorter path length along the proton trajectory) or their transverse extent smaller.
The scattering of the proton probe is minimize for the polypropylene target. Figure 3 shows the raw proton images for a polypropylene target where the focal spots are separated by 820 µm (shot A presented in figure 4). The times indicated are the time after the arrival of the leading edge of the main interaction pulses at the target.
To generate quantitative field measurements, these images are first processed, using the method described in Appendix A, to convert the scanned RCF images into number of protons. Then the proton data is processed using the field-reconstruction technique described in Appendix B to extract path integrated magnetic field maps using the Kugland image-flux relation and Ampére-Monge equation [41]. Appendix B describes the methods used to determine the undisturbed proton beam profiles, a crucial step in the retrieval process. A masked 2D polynomial fit was used on for shot A to retrieve the field structures presented in figure 4 and a masked Gaussian fit was used for shot B. It is important to note that small discrepancies between the assumed undisturbed beam profile and the measured proton beam at the edges of the beam can lead to the retrieval of nonphysical magnetic fields. Also, although strong time-varying electric fields will be present during the interaction these are primarily directed normal to the target surface (along the direction of proton propagation) they should not contribute significantly to deflection of the protons. In this experiment, the proton beam dimension at the main interaction plane was a similar size to the features of interest, meaning the edge effects are particularly detrimental. We estimate the accuracy of the fields within the region of interest to have an error of ∼ 20%. Also the accuracy of the retrieved fields are strongly dependent on the overall flux and beam uniformity.
Figure 4 shows the evolution of the fields using data from two different shots. The absolute timings are given in reference to the first appearance of deformation within the proton beam in shot A, which was assumed to correspond to the arrival of the laser pulse. The first snapshot (t = 6 ps) likely illustrates fields driven during the 10 ps laser pulse duration, the later images follow the evolution of the fields. Shot B has a later timing for the proton beam and extends the temporal window to up to 69 ps with an estimated error of ±1 ps. Note that shot B had 10% more energy in the main interaction laser pulses than shot A. The retrievals still produce much stronger fields for shot B compared to shot A. This could either indicate strong shot-to-shot variation, or the limited accuracy of the retrieval method based on the assumptions made to determine the magnetic field maps. The overall trend suggests an increase in field strength to a maximum shortly after the laser pulse with the maximum field strength decaying at later times.
Although the absolute magnitude of the fields retrieved is dependent on the choice of undisturbed beam and therefore has a large error we estimate to be ∼ 20%, and is perhaps affected by the curved 'beam'-front of the protons reaching the target, the qualitative shape of the fields is as expected, with the azimuthal fields around the focal spot disrupted in the region close to the second laser spot by the opposing azimuthal fields associated with that laser focus. The time series in figure 4 indicates strong fields generated during the first 6 ps of the interaction. The field-vectors also indicate the opposing direction magnetic fields in the midplane region required for magnetic reconnection.
The peak path integrated azimuthal magnetic fields retrieved here from the proton radiographs of the dual laser foci are in the region of 50-100 µmMG, and are of similar strength to those reported in measurements of single, relativistically-intense laser spots (∼ 1 µm × 100 MG [30] and ∼ 10 µm × 40 MG [29]). Note these fields are much stronger than those typically measured from nanosecond duration laser interactions, where the magnetic fields are of the order 1 MG [16,18].
In the absence of a measurement of the scale length of the magnetic field within the preplasma at the target surface, we have left the retrieved field in its path integrated form. It is tricky to estimate the likely front side scale length, l, since the target temperature and expansion will be highly dynamic over the temporal evolution of the laser pulse (relatively long at a FWHM of τ L = 9.6 ps). Using an isothermal expansion estimate, l = c s τ L [42],
with T e = 170 keV, and for a pure proton plasma gives l ≈ 38 µm. However, this likely significantly overestimates the scalelength because the T e is estimated from the peak intensity and the carbon component of the plasma will reduce the sound speed. Sarri et al., using the same laser system, with shorter pulse (1 ps), higher intensity (10 19 Wcm -2 ) found field thicknesses of 10 µm best matched their results [29]. Therefore an estimate of the path averaged field strengths in this experiment is made to be between 2 -10 MG.
Further information can be obtained from the reconstructed field images by considering the dimensions of the reconnection layer. The ratio of the width, δ, to the length, L, of the region can be used to determine the reconnection rate, the time it takes a magnetic field line to enter the diffusion region, reconnect and then exit the layer in the outflow plasma. For our retrieved magnetic fields, this ratio was estimated to be δ/L ≈ 0.14, using the FWHM of the best-fit Gaussian of the width, and defining the length of the region by the intersection point of the two bubbles. This is possibly an underestimate since the L is not FWHM as with δ. Raymond et al. observed δ/L ≈ 0.3 using copper K α emission and numerical modeling in a similar regime [31]. It is, however, consistent with fast, collisionless reconnection which predicts rates of 0.1-0.2v A [19].
Figure 5 shows lineouts of path integrated magnetic fields along the axis of symmetry perpendicular to the midplane for shot B. There are a couple of features to note. Firstly, the unsurprising observation that the magnetic field strengths decrease with increasing time. The fields persist for many pulse duration's (τ L = 9.6 ps), an observation that is consistent with Sarri et al. who made similar single spot measurements using τ l = 1 ps pulses [29]. Secondly, the relative strength of the magnetic fields decays quicker for the "internal" fields on both sides of the midplane region compared to the external fields on either side. This is likely because magnetic reconnection is taking place in the midplane region.
In conclusion, we have utilized proton radiography to probe the evolving azimuthal magnetic fields at the surface of thin targets of metal and plastic by two, co-timed, high intensity laser pulses. The data suggests reduced size fields on the metal targets in comparison with the plastic foils. An algorithm exploiting the Kugland imageflux relation together with the Monge-Ampére equation to retrieve the path-integrated magnetic fields assuming an initial proton flux distribution. As expected, two azimuthal fields were retrieved with field strengths up to B • L ∼ 100 µmMG. The field maps indicated the magnetic fields in the midplane were compressed and the field strengths reduce at a faster rate compared to the external fields. The width to length ratio of, δ/L ≈ 0.14, suggests a fast collisionless reconnection mechanism would be appropriate in this regime.
Future experiments could explore the differing field formation on metals and plastics in more depth utilizing higher energy proton probing. Finer temporal resolution in the design of the proton probing diagnostic would permit measurement of the rapid (6 ps) growth of the fields. With respect to using a field retrieval algorithm on TNSA proton radiography, the calculation of the assumed unperturbed proton beam is most successfully recreated using a custom 2D 3rd order polynomial or Gaussian filter, that is masked to recreate the beam edge. We note that using a larger distance between the source foil to the interaction, so that the proton beam overfills the region of interest would also improve the field reconstruction.
Radiochromic film (RCF) is a dose dependent radiation detector that darkens on exposure to radiation. Used in a stack configuration, as was for this experiment, the film can be used to record the proton beam transverse profile for discrete proton energies. Following exposure, the films were scanned, after a wait-period of 24 hours, with a three color transmission scanner (Nikon CoolScan9000). The same device, and settings, were used to scan a set of calibration HD-V2 type films. These had previously been exposed, at the Birmingham synchrotron, to known doses of a 29 MeV proton beam between 0.1 and 200,000 Gy. The contributions of the three colour channels were combined and a custom fit used to obtain a pixel value to dose conversion.
Before conversion of the pixel values from raw data to dose, dark points due to the presence of dust on the films was removed. The dust removal method used was based on the technique developed by G. Hicks [43]. Here, a 2D-histogram of the pixel values of the green and blue channels is generated. Points that fall outside of ±2.5σ, where σ is the local rms of the distribution, are labelled as dust and the values are in-painted from the surrounding film. Since darker regions of film correspond to higher proton signal, and exclusion of regions of the film would affect the field retrieval algorithm, this step is very important.
Dose per pixel in Grays (1 G = 1 J/kg), is converted to energy per pixel by considering the density and volume of the active layer in which the proton energy is deposited. Here, density was assumed to be 1.2 gcm -3 [44] with the pixel volume of (84 µm 2 × 12µm).
Conversion of energy deposited in the film in Joules to number of protons must consider that all protons with energies sufficiently high to reach a particular layer, will contribute to the deposited energy of that layer. This is illustrated in Figure 6, which shows response curves for the RCF stack, that is the energy deposited in each RCF layer as a function of initial proton energy (before entering the stack). These curves were calculated using proton stopping powers from SRIM [40] and a GUI developed by D. C. Carroll [45]. For the retrieval of the proton number, the signal on each layer is assumed to be due only to protons with energies falling within a bandwidth defined by deposition above 1/e of the maximum. The mean energy deposited by protons within this binwidth is used to estimate the number of proton in this energy bin from the deposited dose.
Typically, protons generated by TNSA exhibit a spectrum that decays exponentially with increasing proton energy up to the cut-off energy. By starting at the rear of the RCF stack, it is possible to remove the contribution of higher energy protons to pieces of film earlier in the stack. However, the absolute protons/pixel is important in determining how the proton flux at the stack has been locally affected by fields in the target. In order to correct the proton flux it is necessary to track the protons through the film so that the extra dose of higher energy protons is removed from the correct location in the earlier films. Due to the hundreds of micron (multipixel) positioning accuracy of the films relative to one another and the strong but spatially small flux modulation introduced by the mesh it was not possible to perform relatively small (∆E/E < 10%). The contribution of a small range of proton energies to each layer will lead to low levels of 'blurring' in the flux distribution.
The duration of passage of monoenergetic protons through a 100 µm plasma is < 0.5 ps for all energies utilized here. While the evolution of the fields in a relativistic laser plasma interaction can be extremely rapid, the similarity of the images obtained for different probing times implies that our field structures are not evolving significantly over this timescale.
5. paraxial approximation: assuming a point source of protons at the proton foil, the half-angle divergence of the proton beam was calculated. This varies with proton energy from 9 • to the maximum acceptance of the RCF stack ( 27• ) implying that the proton beam cannot be approximated as planar at the interaction. This does not qualitatively change the contrast regime but can influence quantitative analysis.
6. initial proton profile: the proton beam profile is clearly non-uniform and estimation of this 'undisturbed' profile represents the largest source of error. This is discussed in greater depth below.
The basis of the field retrieval algorithm is that a change in flux distribution of the proton image results from local flux being redirected by the magnetic fields. Therefore a lack or excess of protons in a particular region can be used to infer the fields. This is only true if the undisturbed proton flux distribution is known. In many cases, for example capsule implosions, the undisturbed beam profile is isotropic and smooth and can be approximated by using the mean flux with any large-scale modulations estimated using a low-pass Fourier filter. In the case of TNSA-produced proton beams, this is not the case. The flux cut-off marking the 'edge' of the beam can have a strong gradient relative to flux variations within the beam. Shot-to-shot variability in beam profile and varying beam profile with proton energy, mean that it is difficult to infer the shape of the beam from 'reference' shots or RCF pieces corresponding to high energy/early time protons. In addition, in this case, the presence of the modulation imposed by the mesh adds an extra challenge since the frequency of this modulation is on a similar scale to the size of the signal and therefore Fourier filtering is ineffective.
In attempting to estimate the undisturbed proton flux profile, we have utilized a) flat mean-field, b) large sigma Gaussian filtering and c) 2D 3rd order polynomial fit. We present here the magnetic field retrievals for these methods. In figure 8, the different backgrounds are shown for the same film (figure 3c), together with the associated field-retrieval. It can be seen that in the case of the flat undisturbed beam (a), the beam edge, which is y RCF (cm) y target (mm)
x target (mm)
x RCF (cm) not accounted for in the background, results in strong, non-physical fields. The algorithm deduces that the lack of flux in these regions compared with the flat distribution results from strong magnetic fields which dominate over the structure within the beam. The Gaussian filter (b) fairs better, with the gradual fall-off at the edge of the undisturbed beam reducing these spurious fields. However, in this case, the fall-off in flux is too gradual, such that in some cases there is a mismatch between the initial and final beam profile resulting in retrieval of nonphysical fields.
In contrast, the masked 2D polynomial fit exhibits a much sharper drop-off in flux at the edge of the beam and more accurately follows the initial profile of the beam. This can be seen clearly in the central row of figure 4 which illustrates the undisturbed beam profile calculated using this custom polynomial for each of the different films within the same stack (the same shot). It is clear from the measured data that the beam edge is relatively sharp with changing beam size with proton energy. This custom fit, utilized a 2d 3rd-order polynomial fit to the measured data after the data had been smoothed with a large kernel Gaussian filter. This also leads to an overestimation of signal at the beam edge and so the fit was then masked to exclude values 'outside' the beam edge, where the beam edge was defined as pixels with values falling below 25% of the measured maximum. The pixels outside of the edge of the beam were replaced with their values from the original data. This masked fit was then smoothed with a small (20-pixel) Gaussian filter to prevent sharp edges. As in all other cases, the total flux in the undisturbed image was adjusted to match the total flux within the measured image.