A mechanism that has gained great attention recently is rear surface acceleration of ions with lasers (Cowan et al., 2000; Pegoraro et al., 2000; Murakami et al., 2001; Ruhl et al., 2001) due to the remarkable properties of the accelerated beam. An interesting prediction is that protons accelerated from the rear surface of a laser-irradiated target might be capable of generating images of micro-structures present on this surface. The protons at the rear surface could stem from water contamination. In this paper, we address the mechanism of the generation of images of micro-grooves machined into the rear surface of the target with laser-accelerated protons.

Essentially images are generated in two steps: The micro-structures trigger perturbations in momentum space of the accelerated protons at early times which are embedded into the expanding proton beam. Due to persistent acceleration of the protons by hot electrons they eventually expand to a point where the micro-perturbations are well separated from each other in proton momentum space. This is when the image is supposed to form. The understanding of image formation might represent a new approach to the measurement of the emittance and source size of the laser-accelerated flow of protons.

Intensity distributions of protons are recorded in radio-chromic stack detectors (RCF-detectors). Let us briefly explain which part of the proton momentum space RCF-detectors measure. Readers interested in physics details of the material used in RCF is referred to the literature (Klassen et al., 1997) and references therein. As soon as the expansion of the protons comes to a rest individual protons propagate force free. We assume that this is the case after a few 100 μm based on one-dimensional (1D) PIC simulations of proton acceleration for sub-picosecond laser pulses. RCF-stacks, however, are typically located several cm away from the laser-irradiated target. Hence, the information recorded on the RCF-detector represents the divergence angles α_x and α_y of the flow. We find α_x = p_x /p_z = x_RCF/z_RCF and α_y = p_y /p_z = y_RCF/z_RCF, where x_RCF, y_RCF, and z_RCF are the positions and p_x, p_y, and p_z the momenta of a proton in the detector. The laser is assumed to propagate along the z-direction.

An example of how an image with laser-accelerated protons might look like in an experiment is shown in Figure 1. Plot (a) shows a RCF-image calculated for protons with energy of approximately 8 MeV. Plot (b) shows a calculation for 10 MeV. Both plots have been obtained with the effective particle code CPT which we will not explain here. We only note that CPT allows us to perform three-dimensional (3D) simulations of image generation from rear surface acceleration. For the calculation, we assume that the rear surface has sinusoidal micro-grooves with a wavelength of 2.5 μm and an amplitude of about 0.2 μm. Each line in plots (a) and (b) corresponds to a micro-groove on the rear target surface.

Theoretical prediction of image formation in RCF with CPT. A laser with a Gaussian intensity distribution is incident on a thin solid density foil of 60 μm thickness. The focal spot diameter is 10 μm at FWHM. The rear surface of the foil has sinusoidal micro-grooves in one direction. Plot (a) shows the angular proton distribution obtained from 8 MeV protons. Plot (b) shows the divergence angles of the flow predicted for 10 MeV. The units are in radians. The size of the proton film on the back of the target is 280 × 280 × 0.01 microns. The FWHM of the momentum envelope for Ep > 3.0 MeV is ≈140 μm. CPT makes a reasonable assumption for the beam expansion. However, free parameters would have to be determined with the help of measured RCF-images at various energies. Further details are beyond the scope of the paper and will be published elsewhere.

To obtain an idea of how such images form and how the CPT code works we proceed by performing a Particle-In-Cell (PIC) simulation with the PSC code (Bonitz et al., 2004) PSC is fully self-contained. The simulation box for the PSC-simulation is 1 μm × 40 μm × 15 μm large to save computational expense. The numerical resolution is chosen adequately. The target in the simulation consists of a substrate of heavy material with a thickness d_s = 1 μm and m_s = 100 m_p, where m_p is the proton mass. The substrate is coated with a proton film of thickness d_f = 0.1 μm to model the real world situation with water vapor attached to a foil. The whole target in the simulation is singly ionized. The ion densities of the substrate and the film are both n_i = 5·10²² cm⁻³, about 1/10 of solid density. The initial electron and ion temperatures are T_e = 1 keV and T_i = 0 keV, respectively. The laser intensity is Iλ² = 5·10¹⁸ Wcm⁻² μm². The laser beam has a line focus along x. The focal spot diameter along y at Full-Width-Half-Maximum (FWHM) is 10 μm. The temporal envelope of the laser pulse is Gaussian with 200 fs at FWHM. The peak of the laser pulse is initially located 60 μm in front of the foil. The rear surface of the foil has sinusoidal micro-grooves with a wavelength of 2.5 μm and an amplitude of 0.1 μm.

Figure 2 shows the proton phase space obtained with the PSC simulation. Plots (a,b) give the zp_y- and zp_z-planes and plots (c,d) the contrast pattern generated by the divergence angle α_y in color scales at 300 fs and 613 fs. While only a single bright dot is visible in plot (c) four individually resolved dots can be seen in plot (d). These four dots stem from four micro-grooves on the rear surface of the target. To understand the formation of the contrast patterns in plots (b) and (c) we look at the zp_y (a)- and zp_z (b)-projections of the proton phase space. They show that the space-momentum relation for laser-accelerated protons is approximately a monotonic function. We call this property the luminosity of the flow. They also show that the part of the proton flow with large p_z expands in p_x/y and flow perturbations become visible in the flow. The expansion of the momentum flow in the simulation is due to persistent acceleration of the expanding protons by fast electrons. At 613 fs the expansion of the flow is effectively a free drift.

PSC simulation of flow envelope expansion and image formation. Plots (a,b) show the zpy- and zpz-planes of the proton phase space at t = 300 fs (red), t = 454 fs (green), and t = 613 fs (black). Plot (a) shows micro-perturbations embedded into an expanding beam envelope due to the grooves at the rear target surface. The blue lines in plot (b) represent pz /mp c = (z − z0)/c(t − t0) for the above times, where z0 = 2.5 μm and t0 = 170 fs. The parameter z0 is the location of the rear target surface and t0 the time when the first protons accelerate. Plots (c,d) show the divergence angles αx and αy n color scales at 300 fs, where 0.06 ≤ pz /mp c ≤ 0.07 and 613 fs, where 0.08 ≤ pz /mp c ≤ 0.09. Both plots are generated by the same particles. Careful analysis shows that they follow a logarithmic flow envelope. For illustration the dashed lines in plots (a,b) indicate the proton populations centered at pz = .065 mp c and pz = 0.085 mp c.

To illustrate the mechanism of image generation in the PSC-simulation we derive an effective model. Details are summarized in Figure 3. Further details are discussed in the literature (Ruhl et al., 2004; Cowan et al., 2004). CPT uses a sophisticated 3D version of the effective model explained here. As the PIC simulation shows (see plots (a) and (b) of Fig. 2) significant expansion of the proton momentum space takes place between t = 300 fs and t = 613 fs. Based on these results we assume that the expansion of the flow can be described with the help of envelope functions p_y/z. On top of the expanding flow momentum perturbations propagate. The following equations give details of this model

where the embedded flow perturbations are given by δp_yi = A_y sin(k_y y_i(0)). For the longitudinal flow expansion we write

Effective model of image generation. A laser with a line focus and Gaussian intensity distribution along y is incident on a thin near solid density foil. The rear surface of the foil has sinusoidal micro-grooves in y-direction. Plot (a) shows the normalized lateral momentum distribution at t = 300 fs. Plot (b) shows the same at t = 613 fs. The white lines in both plots indicate the lateral flow envelope. The sinusoidal micro-perturbations in both plots originate from the micro-grooves during sheath acceleration of the protons. Plot (c) shows the divergence angles at t = 300 fs and plot (d) at t = 613 fs. The plots basically represent the projections of the lateral distributions along y.

The quantities p_yi(0), p_zi(0), y_i(0), and z_i(0) describe the proton distribution at t = 0. Equation (1) means that the lateral momentum spread of the protons increases faster the further away a proton is from the center of the target as is observed in the simulation. The quantities y_i and y₀ represent the lateral position of a proton and the target center. The quantity C_y determines how fast the lateral flow expands and S_y represents the initial inclination of the momentum envelope, which is determined by the early rapid expansion of the flow in the sheath. The quantities C_z, and S_z mean the same for the longitudinal expansion of the flow. The parameters z_i and z₀ represents the longitudinal position of a proton in the flow and the position of the rear target surface. Equations (1) and (2) can be solved analytically. We find for the divergence angle α_y of a single proton as a function of time

where t < t_f has to hold. The time t_f is the time when the flow begins to expand ballistic ally. For simplicity we did not sum over all individual protons in the detector to determine the contrast patterns. The free parameters are determined by the details of the sheath physics and flow expansion. We have assumed that the acceleration of the protons along z is constant in time which implies that the flow velocity scales as

. This scaling is approximately correct in the tip of the flow where the images reside. For the parameters in Eqs. (1) and (2) we find from the PSC simulation S_y ≈ 6.5·10⁻⁴m_p c/μm, C_y ≈ 5·10⁻⁶ m_p c/μm fs, S_z ≈ 2.5·10⁻² m_p c/μm, C_z ≈ 4.2·10⁻⁵ m_p c/fs, A_y ≈ 0.003 m_p c, y₀ = 20 μm, z₀ = 2.5 μm, and k_y = 2.51/μm. We look into the momentum range 0.06 ≤ p_z /m_p c ≤ 0.07 at t = 300 fs. With the above provisions the same protons are within the interval 0.08 ≤ p_z /m_p c ≤ 0.09 at 613 fs. We note that at this time the PSC simulation predicts the formation of images.

Figure 3 shows the image formation process calculated with the effective model outlined above. At t = 300 fs no image has formed. However, at t = 613 fs an image of the rear surface micro-grooves is visible. At this time the expansion of the beam envelope indicated by the white line in plot (b) of the figure has progressed to a point where the projection of the divergence angle α_y along y shows separately resolved contrast patterns along the α_y-axis. This is when the image forms. The same mechanism applies for the PSC simulation as can be inferred from plots (a-d) of Figure 2. Image generation is based on the expansion of the flow envelope due to persistent proton heating by fast electrons. Depending on the target and groove geometry and the initial setup of the sheath proton phase space as identified in the model it can happen that no unique image of the grooves forms. This would be the case if the amplitude of the groove related micro-perturbations was larger than the steepness of the momentum envelope at the time when the flow becomes free streaming. Since

holds we can expand Eq. (3) to obtain

The divergence angles α_y(y_i(0),t) of the perturbations have to be separated far enough from each other on the scale length 2 π/k_y for an image to exist. Hence, we find the relation

Without going into too much detail the parameters S_y, C_y, and A_y depend on the laser and target configuration and the degree of target heating. The laser focal spot diameter and intensity have to be adapted adequately for a given target geometry for an image to occurs. The same is recovered from PSC simulations.

We have proposed the possibility of image formation in RCF-detectors with the help of micro-grooves machined into the rear surface of laser-irradiated targets. The process is based on three distinct mechanisms which are flow luminosity, generation and propagation of micro-perturbations in an expanding flow, and flow expansion. The physics described here is robust and could work over several centimeters of beam propagation between source and target.