Micron-scale Fast Electron Filamentation and Recirculation determined from Rear Side Optical Emission in High Intensity Laser-Solid Interactions

The study of electron transport through dense plasmas is an active area of research of importance to many applications. Relativistic electrons generated in high intensity laser-solid interactions can pass through the critical surface, where the laser energy is mostly absorbed, and continue, free from the influence of the laser, through to the rear surface of the target. The propagation of this large current of electrons can be affected by self-generated electric and magnetic fields, as well as collisions. A better understanding of this transport is a key issue for the success of the fast ignitor approach to inertial confinement fusion [1, 2] and may lead to optimization of ion acceleration from laser-irradiated solid targets [3, 4].

Electron transport has been studied by $K_{\alpha}$ and XUV emission [5, 6], shadowgraphy [7, 8, 9] and optical emission [10, 11]. In particular, the divergence and temporal modulation of the electrons have been investigated by spatially and spectrally resolving rear surface optical radiation [10, 11, 12, 13, 14]. This radiation can be attributed to thermal, synchrotron, transition radiation or coherent wake emission (CWE) [15].

These different mechanisms of emission can be differentiated by their polarization characteristics. Thermal (blackbody) radiation is not expected to be polarized. Synchrotron radiation (SR) is mainly polarized in the plane of motion of the electrons [16]. In our case it would be produced by electrons being pulled back by the electrostatic field that builds up at the back of the target [10], so that the polarization would vary across the emission region, depending on the direction in which electrons travel before restriking the surface. CWE should be polarized in the plane of polarization of the laser field and is mainly emitted in the direction of propagation of the laser field [15]. Transition radiation (TR) is also polarized; its properties will be discussed in section 2.

In this paper we present the first spatially resolved measurements of the polarization of optical emission from the rear of solid targets irradiated by high intensity lasers. The radiation is found to be uniformly polarized over the emission region, with a degree of polarization dependent on the orientation of the target rear surface, demonstrating that it is predominantly TR. The strong variation of the signal with the orientation of the target rear surface (or, in other words, of the angle of observation) reveals that, at high laser intensities, fast electrons propagate in micron-size filaments. Moreover, imaging the TR far from the laser axis gives direct evidence for the presence of recirculating currents.

This paper is organized as follows. The polarization and general properties of transition radiation are briefly discussed in section 2. In sections 3 we describe the results of two different experimental campaigns, which have been performed on the Vulcan CPA laser system at the Central Laser Facility. The results of section 3 are further discussed in section 4 and a brief summary of the paper is given in section 5.

The total emitted energy of the transition radiation per unit angular frequency and unit solid angle can be written as [18]

	$\displaystyle\frac{\partial^{2}W}{\partial\omega\partial\Omega}=\frac{\textrm{e}^{2}N}{\pi^{2}c}\left[\int\textrm{d}^{3}\textbf{p}\left(\mathcal{E}_{\parallel}^{2}+\mathcal{E}_{\perp}^{2}\right)\right.+$				(1)
			$\displaystyle\ \ \ \ \ \ \ \ \ \ \left.\left(N-1\right)\!\left(\left|\int\!\!\textrm{d}^{3}\textbf{p}\,g(\textbf{p})\mathcal{E}_{\parallel}F\right|^{2}\!\!+\!\left|\int\!\!\textrm{d}^{3}\textbf{p}\,g(\textbf{p})\mathcal{E}_{\perp}F\right|^{2}\right)\right]\!\!,$

where the first integrals (first line) refer to the incoherent component of the transition radiation (ITR) and the second integrals (second line) to the coherent one (CTR). Here $g(\textbf{p})$ is the momentum distribution function, $\mathcal{E}_{\parallel}$ and $\mathcal{E}_{\perp}$ are the Fourier-transforms of the electric fields in the plane parallel and perpendicular to the radiation plane (defined by the directions of the target normal and the direction of observation) and $F$ is a coherence function that takes into account the exact time and position at which electrons reach the interface.

Expressions for $\mathcal{E}_{\parallel}$ and $\mathcal{E}_{\perp}$ can be found in [17] for the case of two media with generic dielectric constant. However, for our purposes an interface that separates a perfect conductor with the vacuum can be assumed, in which case the Fourier-fields simplify becoming [17, 18]

	$\displaystyle\hskip-56.9055pt\mathcal{E}_{\parallel}(\beta,\mu,u,\phi,\psi)=\frac{u\cos\psi[u\sin\psi\cos(\phi-\mu)-(1+u^{2})^{1/2}\sin\beta]}{[(1+u^{2})^{1/2}-u\sin\psi\cos(\phi-\mu)\sin\beta]^{2}-u^{2}\cos^{2}\psi\cos^{2}\beta}\ ,$		(2)
	$\displaystyle\hskip-56.9055pt\mathcal{E}_{\perp}(\beta,\mu,u,\phi,\psi)=\frac{u^{2}\cos\psi\sin\psi\sin(\phi-\mu)\cos\beta}{[(1+u^{2})^{1/2}-u\sin\psi\cos(\phi-\mu)\sin\beta]^{2}-u^{2}\cos^{2}\psi\cos^{2}\beta}\ ,$		(3)

where $u$ is the normalized momentum, $u=\sqrt{\gamma^{2}-1}$, and all the other variables are shown in Fig. 1a.

In the simplest case of a single particle traversing normal to a plasma-vacuum interface, transition radiation is radially polarised, since $\mathcal{E}_{\perp}=0$ for $\psi=0$. In other words, the electric field oscillates in the radiation plane (Figure 1b). For a particle that does not traverse normal to the interface the radiation is not radially polarized and a component normal to the radiation plane appears (Figure 1c). In this general case, the polarization state is dependent on both the particle’s momentum vector $\bf{p}$ and the direction of observation $\bf{k}$. Thus for a given set of observation angles the polarization state of the radiation carries information on the direction of the particles as they escaped the rear surface. This will be used for later considerations, to estimate the direction of the electrons from the polarization state of the radiation.

The experiments were performed with the Vulcan CPA laser operating with wavelength $\lambda_{\textrm{0}}=1.054\,\mu$m and pulse length $580\pm 114\>\textrm{fs}$.

Numerous numerical results have demonstrated the formation of $\mu\textrm{m}$-scale filaments directed over a wide range of angles [24, 25]. For our conditions, these filaments would have to propagate through $\sim 50\ \mu$m of solid Cu, where angular scattering would be expected to cause rapid transverse expansion. Neglecting energy loss and assuming small total angular deflection, this “beam blooming”, defined as the variance of the transverse beam size due to scattering [26], can be written as

$$B\approx\frac{e^{2}}{2\sqrt{3\pi}\varepsilon_{0}}Z\sqrt{n_{a}\ln\Lambda_{s}}\frac{s^{3/2}}{pv},$$

(5)

where $s$ is the distance travelled, $Z$ the atomic number, $n_{a}$ the atom number density, $\Lambda_{s}$ is a term that depends on the electron-atom scattering cross section and $p$, $v$ are the momentum and velocity of the electron, respectively. This result gives excellent agreement with Monte Carlo modelling [27]. Eq. (5) indicates that, in order to obtain features $\simeq 1.0\ \mu\textrm{m}$ FWHM ($\equiv 2\sqrt{2\ln 2}B$), electrons with energies $>50$ MeV are required for all realistic values of $\Lambda_{s}$. This is well above the ponderomotive potential of the laser ($\sim 4$ MeV) and contradicts the maximum of $30$–$40$ MeV measured with an electron spectrometer positioned on laser axis in this experiment.
Direct imaging of rear surface emission has recently suggested the presence of filaments as small as $2\;\mu$m and with a mean size of $4\;\mu$m for intensities of $\sim 10^{19}\ \textrm{Wcm}^{-2}$ onto a range of different targets [28, 29]. However, we show that at higher intensities, the filament size can be below that observable with traditional imaging techniques. This small scale structure can only be explained by the influence of self-generated electric or magnetic fields inside the target. This could be due to the resistive growth of the magnetic field inside the target [30]. Results from the code LSP for parameters close to those of this experiment show no sign of filaments at $0.5$ ps for intensities up to $10^{19}$ Wcm^-2, but exhibit filaments $1$–$2$ $\mu$m wide at $10^{20}$ W cm^-2 (see Fig. 12 in [25]). In addition, rear surface magnetic focusing [31] and instabilities associated with the electron sheath at the back surface may contribute to the observed small emission size.

In conclusion, we have shown that the Polarization of CTR can be used to directly characterise non-linear propagation phenomena of intense electron beams such as recirculation and filamentation with unprecedented spatial resolution. These results imply that the transport is strongly influenced by collective effects at high laser intensity. Understanding the formation and behavior of these small scale structure will be of vital importance for applications of high intensity laser physics.

We acknowledge discussions with S. Atzeni and R. J. Kingham and the assistance of the staff of the Central Laser Facility at the Rutherford Appleton Laboratory.