Experimental details

The experiment was carried out at HZDR, Dresden, using the 150 TW Ti:sapphire laser system delivering 30 fs (FWHM) at 10 Hz repetition rate, at a central wavelength of 800 nm. The p-polarized laser beam was focused to a focal spot of 8 µm diameter (FWHM) with an off-axis parabolic mirror with a focal length of 250 mm, reaching a peak intensity of the order of ~3 × 10²⁰ W/cm² (Schramm et al., Reference Schramm, Bussmann, Irman, Siebold, Zeil, Albach, Bernert, Bock, Brack and Branco2017). The dispersed ion beam was detected by a micro-channel plate (MCP) detector coupled to a 16-bit camera. A stack of radiochromic films (RCFs) was placed at 5 cm behind the target to detect the ion beam profile. RCF was used because it is an absolutely calibrated dosimetry medium, which changes its color on exposure to ionizing radiation. Protons of a given energy will traverse only through a well-defined thickness of the film before depositing the energy at the Bragg peak (Xu et al., Reference Xu, Liao, Wu, Geng, Li, Zhu, Li, Hu, Shou, Chen, Lu, Ma, Zhao, Zhu, Lin and Yan2019). Therefore, each layer of RCF stack could be associated with a definite minimum proton energy (Fig. 1).

Fig. 1. (Color online): A schematic diagram of the experimental setup showing the RCF position behind the target.

Results and discussion

In this paper, we report the observation of proton beam filamentation at relativistic laser intensities. The filamentary structures observed in the proton beam are observed to uniformly evolve as the ultrafast (ps) plasma channel expands, which is clearly evidenced by the proton beam distribution recorded on each layer (6–10 MeV) of RCF stack as shown in Figure 2. The fast electron beam generated at the critical density surface of the plasma is observed to be divergent in the laser-plasma interaction (Santala et al., Reference Santala, Zepf, Watts, Beg, Clark, Tatarakis, Krushelnick, Dangor, McCanny, Spencer, Singhal, Ledingham, Wilks, Machacek, Wark, Allott, Clarke and Norreys2000). Very large instantaneous currents (10⁶–10⁹ A) produced will lead to strong collective effects (Bell et al., Reference Bell, Davies, Guerin and Ruhl1997; Davies et al., Reference Davies, Bell, Haines and Guerin1997), which results in fine structure (i.e., filaments) (Bell and Kingham, Reference Bell and Kingham2003) and hence divergence of the beam. The target chosen for the present experiment is CH–Al–CH and Al–CH–Al multilayered targets. First, given the large resistivity expected in dielectric materials, plastic targets are ideal candidates to investigate the generation of resistive fields (Jung et al., Reference Jung, Osterholz, Wenbrück, Kiselev, Pretzler, Pukhov, Willi, Kar, Borghesi, Nazarov, Karsch, Clarke and Neely2005). Second, using multilayers, it is possible to change the resistivity gradient, which can have a direct impact on the magnetic fields generated.

Fig. 2. (Color online): Proton beam profiles (a) 10 MeV, (b) 8 MeV, and (c) 6 MeV recorded on RCF from the target rear surface CH–Al–CH (2.5 µm); (d) shows a smooth beam recorded on RCF from the target rear surface of Al–CH–Al (2.5 µm).

The theory behind collimation of fast electrons in a structured target could be understood by combining ohm's law with Faraday's law for the growth rate of magnetic field resulting in the following equation:

(1)$$\displaystyle{{{\rm \partial} B} \over {{\rm \partial} t}} = {\rm \eta} \nabla \times j + \lpar {\nabla {\rm\eta}} \rpar \times j, $$

where η is the resistivity and ј is the fast electron current density. The first term corresponds to the magnetic fields that act to push fast electrons toward regions of higher fast electron density, while the second term corresponds to the magnetic fields that push the fast electrons toward regions of higher resistivity. The first term of Eq. (1) is the origin of electromagnetic instability. A fast electron current density perturbation normal to the streaming direction (i.e., normal to the direction of j) will generate the magnetic field that would push more fast electrons into the high density region from the low density region (the higher the density of fast electrons is, the higher the current density is: j). More and more fast electrons are constricted together, which further strengthens the density perturbation. Consequently, the generated field gets stronger further according to Eq. (1), and the density ripple magnitude gets larger and larger. As a result, the Weibel instability is created and filaments (a sequence of high and low density structures) are formed.

Figure 2a–2c shows filamentation in the proton beam imprint from the rear side of the CH–Al–CH target, as recorded on an RCF arising due to self-generated resistive instabilities (Sentoku et al., Reference Sentoku, Mima, Kojima and Ruhi2000, Reference Sentoku, d'Humieres, Romagnani, Audebert and Fuchs2011). Figure 2d represents the proton beam profile of a 6 MeV RCF observed from the Al–CH–Al target, which represents a smooth collimated beam with no net-like pattern. Robinson and Sherlock (Reference Robinson and Sherlock2007) have highlighted the potential for harnessing these self-generated fields for guided electron beam transport in targets specifically designed to give rapid magnetic field growth at a well-defined boundary.

The divergence of the beam calculated for the CH–Al–CH foil is around 30° calculated from the recorded RCF data. The common inference from these figures is that whenever a low-Z thin transport layer is present on the rear side, we can observe a filamented proton beam imprint on the RCF. Similar experiments were conducted on the HZDR DRACO Laser facility using thin foil titanium (Ti) targets (2 and 3 µm), wherein they have reported transverse modulations in the proton beam profile (Metzkes et al., Reference Metzkes, Kluge, Zeil, Bussmann, Kraft, Cowan and Schramm2014). They have used a very thick hydrocarbon transport layer (3.5 µm) at the rear side of 2-µm Ti foil to control the spatial modulations in the beam. In this experiment, we have used the advantage of inherent resistivity gradient of the targets to control the filamentation without increasing the thickness of the rear side transport layer.

Figure 3a represents the magnified proton beam profile, which depicts an increasingly strong modulation for thinner targets. Particularly, striking is the profile which exhibits a high-contrast modulation with an elaborate net-like periodic uniform structure. Inspection of consecutive films illustrates that the structure persists deep into the stack with almost constant formation, indicating that it is formed well in advance of the detector surface. Also, the filamentary structures are constantly present in all ranges of energy as shown in Figure 2a–2c, indicating that the filamentation has occurred well inside the target with the fast electrons and the corresponding structure is mapped onto the protons through the sheath electric field.

Fig. 3. (a) A zoom in the profile depicting a net-like pattern in the proton beam imprint. (b) A line out of the pattern shows periodicity in the net pattern.

The typical structure size at the stack is 0.2 cm over a beam that extends 5 cm on the stack. Assuming a beam originating from a radial size of 100–200 µm (Borghesi et al., Reference Borghesi, Mackinnon, Campbell, Hicks, Kar, Patel, Price, Romagnani, Schiavi and Willi2004) and that the expansion of the structure is ballistic from the target surface where it was generated to the detector, implies an instability wavelength, of 6–8 µm or less. This is greater than the typical wavelength of surface modulations on the foils, which is about 0.1 µm. We have further investigated the mechanism for the filamentation of the beam observed in the experiment. Resistivity (Gremillet et al., Reference Gremillet, Bonnaud and Amiranoff2002; Debayle and Tikhonchuk, Reference Debayle and Tikhonchuk2008; Robinson et al., Reference Robinson, Kingham, Ridgers and Sherlock2008), ionization (Krasheninnikov et al., Reference Krasheninnikov, Kim, Frolov and Stephens2005), and Weibel (Fried, Reference Fried1959; Weibel, Reference Weibel1959; Grassi et al., Reference Grassi, Grech, Amiranoff, Pegoraro, Macchi and Riconda2017) instabilities are typically invoked to explain the filamentation of fast currents observed experimentally (Borghesi et al., Reference Borghesi, Mackinnon, Bell, Malka, Vickers, Willi, Davies, Pukhov and Meyerter-Vehn1999; Gremillet et al., Reference Gremillet, Amiranoff, Baton, Gauthier, Koenig, Martinolli, Pisani, Bonnaud, Lebourg, Rousseaux, Toupin, Antonicci, Batani, Bernardinello, Hall, Scott, Norreys, Bandulet and Pepin1999; Tatarakis et al., Reference Tatarakis, Beg, Clark, Dangor, Edwards, Evans, Goldsack, Ledingham, Norreys, Sinclair, Wei, Zepf and Krushelnick2003; Wei et al., Reference Wei, Beg, Clark, Dangor, Evans, Gopal, Ledingham, McKenna, Norreys, Tatarakis, Zepf and Krushelnick2004). In physical terms, resistive filamentation occurs because any transverse perturbation in the current density will resistively grow a magnetic field that drives fast electrons into the perturbation. As this increases the local current density, the magnetic field gets further enhanced and a positive feedback is created. Our experimental observation that the filamentation wavelength λ = 2π/k = 6–8 µm agrees well with the reported wavelength for Weibel instability to be about 5–15 µm (Romagnani et al., Reference Romagnani, Robinson, Clarke, Doria, Lancia, Nazarov, Notley, Pipahl, Quinn, Ramakrishna, Wilson, Fuchs, Willi and Borghesi2019) using the proton probe technique. From our observations, we conclude that Weibel instabilities at faster time scales may be the root cause to seed resistive instabilities in the target leading to the filamentation.

PIC simulations

To investigate the filamentation and corresponding correlation to the divergence of the proton beam from sandwich targets, 2D particle-in-cell (PIC) simulation was carried out using the EPOCH (Arber et al., Reference Arber, Bennett, Brady, Lawrence-Douglas, Ramsay, Sircombe, Gillies, Evans, Schmitz, Bell and Ridgers2015). The simulation domain was chosen to be 40 µm × 40 µm along x- and y-directions with a grid resolution of 20 nm along each direction accommodating 32 macroparticles per cell. The laser pulse of duration 30 fs, wavelength 0.8 µm propagates along the positive x-direction with a transverse Gaussian profile. The laser focal spot and intensity were taken to be 8 µm and 3 × 10²⁰ W/cm² in accordance with the parameters used during the experiment. The simulations were carried out for sandwich targets Al–CH–Al (1–0.5–1 µm) and CH–Al–CH (1–0.5–1 µm) with n _Al = 10n _c, and for the plastic CH layer, we assumed n _carbon = 10n _c, n _H = 20n _c, where n _c = 1.74 × 10²¹ cm⁻³. Both the targets were coated with a 50 nm proton layer on the rear side in order to account for the effect of contaminant layer in the experiment, structures of fast electrons are mapped on to the sheath electric field, which is directly correlated to the acceleration of protons. Therefore, protons accelerated from the rear side of the target would also exhibit filamentation as observed from the experiment.

Figure 4a represents that the magnetic field inside Al–CH–Al cease to grow once the modulations reach the boundary of the interface between CH–Al layers on the rear side, and therefore, the modulations are not propagated through to the rear surface of the target. Whereas in case of CH–Al–CH (Fig. 4b), it can be clearly seen that the filaments do not get suppressed near the boundary of the rear side layers and propagate through to the rear surface of the target. Similarly, the number density of electrons in Al–CH–Al (Fig. 4c) do not show filamentary structures in the layer corresponding to the rear side of the target, but CH–Al–CH (Fig. 4d) clearly has the filamentary structure in the layer at the rear side of the target. These filamentary structures of fast electrons are mapped on to the sheath electric field, which is directly correlated to the acceleration of protons. Therefore, protons accelerated from the rear side of the target would also exhibit filamentation as observed from the experiment.

Fig. 4. Magnetic field within the target for (a) Al–CH–Al and (b) CH–Al–CH. The electron number density within the target for (c) Al–CH–Al and (d) CH–Al–CH.

The parameter, which indicates the divergence of the proton beam, is the spread in the transverse momentum P_y of the protons. The spread in the transverse momentum P_y is measured at P_x = 20 × 10⁻²¹ kg m/s for both the targets. Al–CH–Al (Fig. 5a) has a spread of 2.98 × 10⁻²¹ kg m/s, whereas CH–Al–CH (Fig. 5b) has a spread of 4.11 × 10⁻²¹ kg m/s, indicating that the protons from CH–Al–CH are more divergent than that from Al–CH–Al as observed from the experiment. The histogram of the electric field over the entire simulation domain is shown in Figure 6. The maximum value of the sheath electric field reached for Al–CH–Al is about 10 TV/m, whereas for CH–Al–CH, the maximum is about 6 TV/m. This implies that the formation of filamentary structures inside the target rear surface may have caused perturbation in the sheath field and, therefore, may have direct correlation in the observed reduction in the sheath electric field for the target CH–Al–CH.

Fig. 5. The phase space distribution of protons at 1ps for (a) Al–CH–Al and (b) CH–Al–CH.

Fig. 6. A histogram of the electric field for both the targets.

In conclusion, we have shown that self-resistive fields are very important to understand the transport of fast electron/ion beams, which can control the flow of mega-ampere currents in the intense laser-plasma interaction. This provides a direction to understand the instabilities arising at the interface of resistivity gradient targets that will otherwise lead to filamentation/pinching of the electron beams. Our experimental findings are in good agreement with the qualitative analysis performed using 2D PIC simulations. This offers a route to control and investigate these instabilities, which may otherwise prove detrimental for fast ignition experiments. Also, the magnetic fields of about 10⁴ T are observed in the simulation, which may have implications in the astrophysical scenario (Cassam-Chenai et al., Reference Cassam-Chenai, Hughes, Reynoso, Badenes and Moffett2008; Suzuki-Vidal, Reference Suzuki-Vidal2015), that is, when a supernova collapses to form a neutron star a similar counter streaming charged particles may give rise to instability leading to generation of strong magnetic field. Charged particles streaming in such high magnetic fields may also contribute to gamma ray bursts.