Plasma wakefield acceleration for Protons Laser plasma acceleration is a promising technology that could replace the current method of accelerating particles (which is via electric fields). It is somewhat cheaper also as it makes the accelerator compact.
But the problem is, that so far I have only seen it's application to accelerating electrons. Is there a method by which laser plasma acceleration could be used to accelerate protons?
 A: Plasma wakefield acceleration works by introducing a driver into plasma and accelerating particles in its wake. The driver can either be a laser pulse or an electron beam. 
In both cases, the most common mechanism of acceleration is the so-called bubble (or blow-out/cavity) regime. This works by the driver pushing electrons in front of it while due to their higher mass, the ions are stationary, effectively creating a bubble-like cavity behind the driver. More than one of such ion channels can be created, but the first one is the most prominent for acceleration.

Because of the relatively higher abundance of ions in this cavity, the electric field pointing to its centre accelerates electrons that get trapped into it. 
The figure below neatly shows the driver pushing the electrons away (the outer orange points) while also permitting you to see the electron bunch being accelerated in its wake (column-like blue structure in the central region), especially obvious in the 2D projections.

After understanding the mechanism described above, you can see that the whole concept is based on the discrepancy of ions' and electrons' mass. It is crucial for the former to remain more or less stationary. I am not aware of a way to utilize this for accelerating protons.
Note, however, that proton-driven wakefield acceleration has been proposed [1,2]. The idea is simply using a proton bunch as a driver, where the bunch is first accelerated by different means (e.g. in a traditional accelerator). Secondly, laser-plasma interaction can be used for accelerating protons [3], but this is distinct from wakefield acceleration.


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*Caldwell, A., Lotov, K., Pukhov, A., & Simon, F. (2009). Proton-driven plasma-wakefield acceleration. Nature Physics, 5(5), 363-367. doi:10.1038/nphys1248

*Assmann, R., Bingham, R., Bohl, T., Bracco, C., Buttenschön, B., Butterworth, A., ... & AWAKE Collaboration. (2014). Proton-driven plasma wakefield acceleration: a path to the future of high-energy particle physics. Plasma Physics and Controlled Fusion, 56(8), 084013. doi:10.1088/0741-3335/56/8/084013

*Schwoerer, H., Pfotenhauer, S., Jäckel, O., Amthor, K. U., Liesfeld, B., Ziegler, W., ... & Esirkepov, T. (2006). Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets. Nature, 439(7075), 445-448. doi:10.1038/nature04492
A: Laser proton acceleration is always a secondary effect, since the laser itself predominantly interacts with the electrons of the plasma due to their high charge to mass ratio.
Nethertheless there are different possibilities to accelerate protons or ions with laser pulses.
The oldest mechanism is the so called target normal sheath acceleration or TNSA. Here you basically take a metal foil of a few micron thickness, where your laser pulse is focussed onto. The laser generates hot electrons, which travel through the target. They are exiting on the back surface, setting up a charge seperation, which generates fields of the order of 1TV/m. Within this field ions can be accelerated to several ten MeV energy.
See for example:
Pfotenhauer et al., Spectral shaping of laser generated proton beams, New Journal of Physics 10 (2008) 033034 doi:10.1088/1367-2630/10/3/033034 and references herein.
With higher intensities the light pressure of the focussed laser pulse becomes significant. Then you're able to accelerate you target as a whole by radiation pressure acceleration (RPA). To work efficient with up to date lasers your target foils are just several nanometers thick. This mechanism has been showed to work in principle experimentaly. Theoreticians predict the possibility to accelerate protons up to GeV energies with future laser systems. E.g.:
Aurand et al., Radiation pressure-assisted acceleration of ions using multi-component foils in high-intensity laser–matter interactions,  New Journal of Physics, 15, (2013), 033031 doi:10.1088/1367-2630/15/3/033031  and references herein
A third possibility to accelerate ions is the so called shock acceleration. If an obstacle moves through a medium with a velocity higher than the specific velocity of sound, then a shock wave forms. This is a long known phenomenon from supersonic jets. In a plasma such a perturbation generates collisionless shock waves, which accelerate particles, if they move faster than a critical Mach number. To use this acceleration mechanism you prefer a target which is close to the critical density. Unfortunately these type of targets are hard to generate for near infrared lasers, since gases are much to thin and solids are to dense. But that this mechanism works, has been shown with ${\rm CO_2}$ lasers, where near-critical targets can be realised with gas jets. E.g.
Haberberger et al., Collisionless shocks in laser-produced plasma generate monoenergetic high-energy proton beams, Nature Physics 8, 95–99 (2012) doi:10.1038/nphys2130
