High field plasmonics and laser-plasma acceleration in solid targets

Plasmonics is a flourishing field of research [1] which exploits the unique optical properties of metal nano-structures to manipulate and confine light. These peculiar properties come from the collective motions of the electrons of a metal at the interface with vacuum or a dielectric medium. The excitation and propagation of surface waves or Surface Plasmons (SP) is obtained and controlled with an appropriate shaping of the nano-structured targets. Many configurations have been tested and allow for several present and foreseen applications, ranging from extreme light concentration beyond diffraction limit [2] to biosensors [3] and plasmonic chips [4]. Most of the tested experimental configurations consider laser light of sub-micron wavelength at moderate intensity (I  <  10¹² W cm⁻²) in conditions where the dynamics remains essentially linear.

The relativistic regime of laser-matter interaction (i.e. when the quivering motion of the electrons in the EM fields becomes relativistic at irradiances ${{\lambda}^{2}}\,I>{{10}^{18}}$ μm² W cm⁻²) with structured targets is still largely unexplored. Extending the study of plasmonic effects to the high field, relativistic domain could open new routes to the manipulation of intense light and the development of laser-driven sources of high energy electrons, ions and photons. To this aim, very short laser pulses (tens of fs) are required in order to both preserve the target structuring and maintain a steep plasma-vacuum interface during the whole interaction, allowing the excitation of SPs. In addition, ultra-high contrast systems [5–7] are necessary to avoid prepulses, typical of high power lasers, to cause early destruction of the structures before the short pulse interaction.

The simplest type of target allowing the coupling of a laser pulse with a SP is a grating, i.e. a target with a shallow periodic surface modulation. The survival of such gratings to prepulse effects has been demonstrated, e.g. by measurements of high harmonic emission [8]. An alternative, all-optical approach may be provided by laser-produced plasma gratings [9]. However, laser-grating interaction is not the only possible approach to SP excitation. For example, experiments and simulations have shown that the highly transient charge imbalance created by fast electron generated in a solid target creates strong EM pulses propagating along the target surface over long distances [10, 11]. This effect may be exploited for the generation of powerful pulses of THz radiation [12, 13].

It may also be noticed that apparently a theory of SP adequate for the high field regime, accounting for relativistic nonlinearities, is still lacking. However, evidence for SP excitation by ultraintense pulses was found in simulations [14] and their role in, e.g. enhanced absorption and electron heating [15] and high harmonic generation [16] was also shown numerically.

In this paper we focus on two cases of intense laser-plasma interaction where plasmonic effects arise because of the presence of either a pre-imposed or a self-generated grating. In the first case, we report on experimental observation of electron acceleration by SPs excited in high contrast laser-grating interaction. In the second case, we show by theory and numerical simulations how plasmonic field enhancement at a modulated interface strongly affects the acceleration of thin targets by radiation pressure.

1.1. Surface plasmon excitation in gratings

SPs are electron oscillation modes excited at a steep metal-dielectric interface. They can propagate along the surface remaining confined in a small region across the boundary. It is possible to show [17] that only transverse-magnetic modes can exist at the surface. Figure 1 shows a sketch of the EM components of the field associated with a SP. The dispersion relation of SPs reads

$$\begin{eqnarray}&&{{k}_{\text{SP}}}\left(\omega \right)=\frac{\omega}{c}\sqrt{\frac{{{\varepsilon}_{1}}{{\varepsilon}_{2}}}{{{\varepsilon}_{1}}+{{\varepsilon}_{2}}}}\end{eqnarray} \tag{ 1 }$$

where ${{\varepsilon}_{1}}$ and ${{\varepsilon}_{2}}$ are respectively the dielectric constants of the dielectric and the metal (${{\varepsilon}_{2}}<0$), c is the speed of light, ω and ${{k}_{\text{SP}}}$ are the frequency and the wavenumber of the SP. The condition for the SP to remain confined close to the interface (i.e. to have evanescent k_x modes, with the axes convention of figure 1) requires ${{\varepsilon}_{1}}{{\varepsilon}_{2}}<0$ (or more precisely the real parts of the dielectric constants need to be of opposite sign) [17].

Zoom In Zoom Out Reset image size

To simplify the configuration, instead of a generic dielectric, we can restrict to consider vacuum so that ${{\varepsilon}_{1}}=1$. The optical properties of the metal, over a large range of frequencies, can be described using the 'plasma model', where an electron gas freely moves in a background of positively charged ions. Considering a linear theory, where the plasma electrons are non-relativistic, the dielectric constant of the metal becomes

$$\begin{eqnarray}&&{{\varepsilon}_{2}}\left(\omega \right)=1-\frac{\omega _{p}^{2}}{{{\omega}^{2}}},\end{eqnarray} \tag{ 2 }$$

where ${{\omega}_{p}}$ is the plasma frequency ($\omega _{p}^{2}=4\pi {{n}_{e}}{{e}^{2}}/{{m}_{e}}$, with n_e, e and m_e being respectively the electron plasma density, the elementary charge and the mass of an electron). The dispersion relation of the SP depends solely on the plasma frequency ${{\omega}_{p}}$ and reads

$$\begin{eqnarray}&&{{k}_{\text{SP}}}\left(\omega \right)=\frac{\omega}{c}\sqrt{\frac{\omega _{p}^{2}-{{\omega}^{2}}}{\omega _{p}^{2}-2{{\omega}^{2}}}}.\end{eqnarray} \tag{ 3 }$$

An electromagnetic wave with frequency ω (i.e. a laser pulse) can be exploited to resonantly excite a SP (${{\omega}_{\text{SP}}}=\omega$) provided that a phase-matching condition is met. Considering a generic angle of incidence ${{\phi}_{i}}$, ${{k}_{\text{L}}}$ must be projected on the surface and the matching condition reads

$$\begin{eqnarray}&&{{\omega}_{\text{SP}}}=\omega ,\quad {{k}_{\text{SP}}}\left(\omega \right)=k_{\parallel}^{\text{L}}=\frac{\omega}{c}\sin \left({{\phi}_{i}}\right).\end{eqnarray} \tag{ 4 }$$

As shown in figure 2 it is not possible to fulfil equation (4) by direct illumination of a plane surface with EM light, as the dispersion relation of the SP resides below the light cone and no matching can be found.

Zoom In Zoom Out Reset image size

Among the techniques employed in classical plasmonics to resonantly excite SPs using lasers, a simple one consists in using metal surfaces with a shallow ($\delta \ll \lambda$ in figure 3) modulation of period d. The SP dispersion relation in this case is folded with period $q=2\pi /d$ in the $k-\omega$ plane (Floquet-Bloch theorem) and the matching condition hence reads

$$\begin{eqnarray}&&k_{\parallel}^{\text{laser}}={{k}_{\text{SP}}}\left(\omega \right)\pm nq.\end{eqnarray} \tag{ 5 }$$

Zoom In Zoom Out Reset image size

A solution for the matching conditions can be found (see figure 3). For a given value of ω, the condition of equation (5) only depends on the period of the grating d and the angle of incidence of the EM wave ${{\phi}_{i}}$:

$$\begin{eqnarray}&&\pm \,n{{\lambda}_{\text{L}}}/d=\sqrt{\left(1-\omega _{p}^{2}\,/{{\omega}^{2}}\right)/\left(2-\omega _{p}^{2}\,/{{\omega}^{2}}\right)}-\sin \left({{\phi}_{i}}\right).\end{eqnarray} \tag{ 6 }$$

If the plasma is strongly over-dense (${{\omega}_{p}}\gg \omega$) and we restrict to the solution with n  =  +1 (for the choice of the signs of angle ${{\phi}_{i}}$ and integer $\pm nq$, this corresponds to the situation represented in figure 3), the condition for d and ${{\phi}_{i}}$ becomes:

$$\begin{eqnarray}&&\frac{{{\lambda}_{\text{L}}}}{d}=1-\sin \left({{\phi}_{i}}\right).\end{eqnarray} \tag{ 7 }$$

As an example, for a grating with $d=2{{\lambda}_{\text{L}}}$ the expected angle of resonance is ${{\Phi}_{\text{res}}}\simeq {{30}^{{}^\circ}}$.

The development of laser systems able to deliver ultrashort high power pulses (>100 TW) with ultra-high contrast [5, 18] allows to explore the interaction of ultraintense (>10¹⁹ W cm⁻²) laser light with solid targets having sub-micrometric structures. Evidence of SP excitation were found in two recent experiments involving laser interaction with grating targets. In a first experimental campaign in 2012 at CEA-Saclay, grating targets were irradiated with intense laser pulses at the resonant angle for SP excitation. This configuration was proven to enhance the cut-off energy of protons accelerated from the rear surface of the target [19, 20], as previously predicted by few numerical studies [21]. A second experimental campaign was performed at the same facility in 2014 [22]; the activity was focused on the effects of SP excitation on electron emission from grating targets. A strong emission of multi-MeV electron bunches was observed along the target surface only when a grating target was irradiated with an angle close to the one expected for the resonant SP excitation.

2.1. Electron acceleration with SP

2.2. Experimental set-up

The experiment was carried out at the Saclay Laser Interaction Center (SLIC) facility with the UHI100 laser system. UHI100 provides a 25 fs laser pulse with a peak power of 100 TW and $\lambda \sim 0,8$ μm. The use of a double plasma mirror [6] allows to achieve a very high pulse contrast (∼10¹²).

Two devices have been used to detect and characterise the electron emission (figure 4). A scintillating Lanex screen was placed facing the irradiated side of the target and covering the full angular range $\phi ={{0}^{{}^\circ}}$–${{90}^{{}^\circ}}$. A 3 mm thick aluminium foil was placed in front of the Lanex to block the electrons with energy ${{E}_{\text{kin}}}<1$ MeV. The green light emitted by the Lanex was imaged by a 12bit CCD camera. A custom made compact electron spectrometer was mounted on a motorised arm able to change the angle of observation ${{\phi}_{\text{spec}}}$ in the range ${{0}^{{}^\circ}}$–${{60}^{{}^\circ}}$. The electrons entering the pinhole (500 μm diameter) were dispersed by two permanent magnets and imaged by a large ($49.2\times 76.8$ mm²) triggered 12bit CMOS. The strength of the magnets was chosen to provide an energy detection range of  ∼2–20 MeV. The ions accelerated toward the rear side were detected with a Thomson parabola was aligned with the target normal.

Zoom In Zoom Out Reset image size

The grating targets were produced at Czech Technical University, Prague by heat embossing of Mylar™ foils. The data shown below have been obtained using gratings with ${{\Phi}_{\text{res}}}\simeq {{30}^{{}^\circ}}$, i.e. $d=2{{\lambda}_{\text{L}}}$ (having assumed ${{\omega}_{p}}\gg \omega$). The average thickness was 10 μm and the peak-to-valley depth of the grooves 0.25 μm. Flat foils with the same average thickness were used for comparison. In a limited number of shots also gratings with a resonance angle of ${{\Phi}_{\text{res}}}\simeq {{15}^{{}^\circ}}$ (${{\lambda}_{g}}=1.35\lambda$) and ${{\Phi}_{\text{res}}}\simeq {{45}^{{}^\circ}}$ (${{\lambda}_{g}}=3.41\lambda$) were used, obtaining similar results.

2.3. Experimental results

When a grating target was irradiated at an angle close to that expected for resonant SP excitation, a strong emission of multi-MeV electron bunches was observed along the target surface. When compared to the case of simple flat foils, the electrons accelerated with grating targets not only were higher in number but also exhibited a much higher kinetic energy up to ≃20 MeV. These energetic electrons were emitted within a cone of few degrees around the target tangent in the direction expected for the SP propagation.

Figure 5 shows the distribution of the electron signal obtained on the Lanex screen for the case of grating targets and flat foils. In presence of a flat target the electron emission was diffused and peaked in an annular region around the position of the specular reflection of the laser pulse (m  =  0 in the figure). This is attributed to fact that electrons were swept away by the ponderomotive force of the EM pulse, leaving a 'hole' in the signal of the Lanex screen.

Zoom In Zoom Out Reset image size

On the other hand, for a grating target at resonance, the electron emission was localised on the plane of incidence close to the target tangent. The intensity of the electron emission was  ∼10 times larger than what obtained with a flat foil at the same angle. In figure 5(b) two minima are visible and they are located respectively in the position corresponding to the laser-pulse specular reflection and in the position of the first diffraction order (m  =  1 which is normal to the target surface).

Figure 6 reports the intensity of the signal of the electrons emitted close to the target tangent (collected with the Lanex screen). When increasing the angle of incidence (from ${{\phi}_{i}}=20$) a sharp increment of the emission was observed when ${{\phi}_{i}}$ approaches ${{30}^{{}^\circ}}$–${{35}^{{}^\circ}}$, where the maximum signal was observed. The signal decreases at a slower rate for ${{\phi}_{i}}>{{35}^{{}^\circ}}$.

Zoom In Zoom Out Reset image size

Figure 7 shows a typical spectrum of electrons emitted close to the target tangent (selected with a pinhole of 0.5 mm placed 8 cm from the interaction point) for a grating irradiated at the resonant angle for SP excitation. The energy spectrum is peaked at 5 MeV and it extends up to  ∼15 MeV, while a dip is observed at lower energies (∼3 MeV). This spectral shape is representative also for spectra collected with ${{\phi}_{i}}={{35}^{{}^\circ}}$, whereas for incidence angles greater than ${{\phi}_{i}}={{40}^{{}^\circ}}$ the observed spectra were less intense and lacked a peaked shape.

Zoom In Zoom Out Reset image size

A large three-dimensional (3D) Particle-In-Cell (PIC) simulation (see [22] for details on the parameters used) was in remarkable agreement (see figure 7) with the experimental results, closely reproducing the shape of the electron energy spectra. The simulation confirms the excitation of a SP and support its strong role in the electron acceleration process. Notice that 2D simulations failed to reproduce the details of the experimental spectrum, showing a smooth distribution instead of the broad peak. The 3D simulations also show a correlation between angle and energy, such that integrating over a solid angle much larger than that of the detector the peak is smoothed and eventually the spectrum is similar to the 2D case. This suggests that the energy gain of electrons depends on their direction with respect to the SP wavevector in a 3D geometry.

Electron acceleration in laser-grating interaction is a compelling evidence of the presence of a SP. It has been studied so far at moderate intensities [25, 26] below 10¹⁶ W cm⁻² (not enough to reach the relativistic regime), while the present study is to our knowledge the first one performed at relativistic laser intensities.

The Radiation Pressure Acceleration (RPA), in particular the Light-Sail regime (LS), is a laser-driven ion acceleration scheme which is expected to be particularly efficient at extreme intensities (I  >  10²² W cm⁻²) and to provide ion bunches with low energy spread [27, 28]. In the basic one-dimensional (1D) picture of LS an intense EM wave (ω) irradiates a thin plasma slab target with density higher than the cut-off value (${{n}_{e}}>{{n}_{c}}$). Provided that the target integrity and reflectivity are kept on a sufficiently long time scale, the thin plasma slab behaves almost as a perfect mirror and can be efficiently accelerated to relativistic velocities $V=\beta c$ thanks to the light pressure. The progress in target manufacturing and pulse cleaning techniques for the delivery of high contrast pulses allowed the first recent experimental evidence of LS using ultra-thin solid foils [29–35]. Next-generation laser systems may allow reaching the relativistic ion regime in LS acceleration, which is particularly appealing because of the foreseen high efficiency. Theory and PIC simulations have shown that the ion-energy evolution with respect to time is strongly dependent on the dimensionality of the system, with the energy gain being particularly fast in 3D with respect to 1D [28, 36]. The gain is ultimately limited by the onset of target transparency due to rarefaction, but energies above the GeV/nucleon barrier appear to be reachable with near-term laser technology. However, the stability of the acceleration process is an important issue. Using theory and simulations we show that LS acceleration may be further limited by the rise of a transverse Rayleigh–Taylor instability (RTI)at the accelerated plasma front [37] which can be detrimental for the ion beam quality and favour an early onset of transparency.

RTI occurs when a light fluid accelerates a heavier one, leading to rippling of the interface between the two fluids. RTI-like phenomena have been already observed in simulations of the laser-plasma interaction in the ultraintense regime [38] and in other studies devoted to the LS regime [39–44]. The image of the proton bunch obtained in an experiment of laser-driven ion acceleration also brought some experimental evidence of RTI in thin targets [34].

In the simulations, the plasma surface pushed by the light pressure exhibited strong transverse modulations with a size close to the laser wavelength ${{\lambda}_{\text{L}}}$ (see figure 8). The standard analytical model for the RTI of a thin foil accelerated by a generic pressure [45] predicts a growth rate ${{\gamma}_{\text{RT}}}(q)$ which increases monotonically with the wave-vector of the transverse mode, in contrast with the observations.

Zoom In Zoom Out Reset image size

We have proposed a model which shows how the rippling of the foil leads to a self-consistent transverse modulation of the radiation pressure allowing to explain the dominant spatial scales observed in the simulations [37] (a similar model, leading to essentially the same results, has been independently proposed [46]). The model considers a plane wave at normal incidence which is reflected by a perfect mirror with shallow modulation of the surface (grating), in a 2D geometry. For definiteness, we assume that the laser propagates along x and the target has modulations along y. In these conditions it can be demonstrated that the two polarisations S (electric field orthogonal to the plane of incidence, E_z) or P (parallel, E_y) behave differently: the fields are locally enhanced in the peaks for S-polarisation and in the valleys for P-polarisation. Because the radiation pressure depends on the intensity of the fields, with P-polarisation it will be stronger in the valleys and the opposite will happen with S-polarisation. Figure 9(a) shows the enhancement of the P-polarized field in the valleys of a preformed surface modulation which can be directly compared to the case of S-polarisation (figure 9(b)) where such enhancement does not occur. In particular, the field enhancement diverges for P-polarisation and a grating periodicity equal to the laser wavelength ${{\lambda}_{\text{L}}}$. This effect can be described as the resonant excitation of a standing SP. In fact, analogously to what obtained previously for a generic angle (see figure 3), the matching condition for normal incidence of the laser light (${{\phi}_{i}}={{0}^{{}^\circ}}$) can be found for a modulation with period $d={{\lambda}_{\text{L}}}$ (see figures 9(c) and (d)). Due to the inversion symmetry, a pair of oppositely directed SPs with the same frequency has to be excited, leading to a standing SP of wavelength equal to ${{\lambda}_{\text{L}}}$. The presence of the grating is essential for SP excitation also because the electric field is not strictly parallel to the target surface locally and may excite an electron oscillation. For circular polarisation (the preferred mode for LS acceleration), the P-component dominates the S-one near the resonant wavevector, seeding the RTI at the wavevector $q=2\pi /{{\lambda}_{\text{L}}}$.

Zoom In Zoom Out Reset image size

As the energy density is higher in the valleys of the modulation rather than on the tips, the radiation pressure tends to increase the modulation depth, which is the origin of the instability seeding. Figure 8 shows the structures in the electron and ion density observed in a 3D simulation where a plane-wave accelerates a thin e-H⁺ plasma. Such simulation is initially symmetric for transverse translation but the development of the RTI soon leads to symmetry breaking (further details on the simulation parameters can be found in [37]). In 3D geometry, the RTI leads to development of hexagonal-like structures, corresponding to the, highly recurrent in nature, p6 mm or 'wallpaper' symmetry [47].

Two examples of the impact of surface plasmon (SP) excitation in ultra-intense laser-plasma interaction have been shown. In the interaction with grating targets, electron acceleration in SPs has been demonstrated experimentally, with possible potential as an intense source of MeV electrons. In 3D simulations of light sail acceleration, the resonant excitation of a standing SP leads to modulation of the radiation pressure at the target surface, seeding a Rayleigh–Taylor instability at a spatial scale equal to the laser wavelength. This effect is a possible issue for high-gain, relativistic regimes of laser-driven radiation pressure acceleration. Besides their specific importance and applications, the above discussed results demonstrate that SPs can be excited in the high field, relativistic regime and open the way to develop novel schemes for super-intense laser-matter interaction and high field manipulation which may take inspiration from ordinary low-field plasmonics.

Exploiting target manufacturing to achieve SP excitation at relativistic intensities requires ultra-high contrast pulses. These examples support high field plasmonics as an emerging field in ultra-intense laser-matter interactions and related applications.

The experiment at the SLIC facility of CEA Saclay, France, has received funding from LASERLAB-EUROPE (grant agreement no. 284464, EU's Seventh Framework Programme. Additional support from 'Investissement d'Avenir' LabEx PALM (Grant ANR-10-LABX-0039), Triangle de la physique (contract nbr. 2014-0601T ENTIER) and 'Institut Lasers et Plasmas' is also acknowledged. The contribution of D Garzella, F Réau, I Prencipe, M Passoni, M Raynaud, M Květoň and J Proska to the design and realization of the experiment is gratefully acknowledged. We also acknowledge F Pegoraro and S Sinigardi for their contribution to the theory and simulations of the laser-driven Rayleigh–Taylor instability. The numerical simulations were performed on the BlueGene/Q machine FERMI at CINECA, Italy, with access sponsored by PRACE awarded projects 'LSAIL' and 'PICCANTE', ISCRA project 'FOAM2' and LISA project 'LAPLAST'.