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By focusing petawatt peak power laser light to intensities up to 1021 Wcm−2, highly relativistic plasmas can now be studied. The force exerted by light pulses with this extreme intensity has been used to accelerate beams of electrons and protons to energies of a million volts in distances of only microns. This acceleration gradient is a thousand times greater than in radio-frequency-based accelerators. Such novel compact laser-based radiation sources have been demonstrated to have parameters that are useful for research in medicine, physics and engineering. They might also someday be used to ignite controlled thermonuclear fusion. Ultrashort pulse duration particles and x-rays that are produced can resolve chemical, biological or physical reactions on ultrafast (femtosecond) time scales and on atomic spatial scales. These energetic beams have produced an array of nuclear reactions, resulting in neutrons, positrons and radioactive isotopes. As laser intensities increase further and laser-accelerated protons become relativistic, exotic plasmas, such as dense electron–positron plasmas, which are of astrophysical interest, can be created in the laboratory. This paper reviews many of the recent advances in relativistic laser–plasma interactions.