Let (X ) be a separable Banach space, (Y ) a Banach space and (f:X to Y ) a Lipschitz function. We show that the set of all G ^ ateaux non-differentiability points at which (f ) has all one-sided or two-sided directional derivatives can be covered by (special subsets of) Lipschitz surfaces of codimension 1 or codimension 2, respectively. This paper reviews the many recent advances at the Center for Ultrafast Optical Science (CUOS) at the University of Michigan in multi-MeV ion beam generation from the interaction of short laser pulses focused onto thin foil targets at intensities ranging from 10 17 to 10 19 W/cm 2.Ion beam characteristics were studied by changing the laser intensity, laser wavelength, target material.

• 117 Downloads • Abstract This paper reviews the many recent advances at the Center for Ultrafast Optical Science (CUOS) at the University of Michigan in multi-MeV ion beam generation from the interaction of short laser pulses focused onto thin foil targets at intensities ranging from 10 17 to 10 19 W/cm 2. Ion beam characteristics were studied by changing the laser intensity, laser wavelength, target material, and by depositing a well-absorbed coating. We manipulated the proton beam divergence using shaped targets and observed nuclear transformation induced by high-energy protons and deuterons. Qualitative theoretical approaches and fully relativistic two-dimensional particle-in-cell simulations modeled energetic ion generation. Comparison with experiments sheds light on ion energy spectra for multi-species plasma, the dependences of ion-energy on preplasma scale length and solid density plasma thickness, and laser-triggered isotope yield. Theoretical predictions are also made with the aim of studying ion generation for high-power lasers with the energies expected in the near future, and for the relativistic intensity table-top laser, a prototype of which is already in operation at CUOS in the limits of several-cycle pulse duration and a single-wavelength spot size.

• A. Maksimchuk • 1 • K. Flippo • 1 • H. Krause • 2 • G. Mourou • 1 • K. Nemoto • 3 • D. Shultz • 2 • D. Umstadter • 1 • R. Vane • 2 • V. Yu. Bychenkov • 4 • G. I. Dudnikova • 5 • V. F. Kovalev • 6 • K. Mima • 7 • V. N. Novikov • 4 • Y. Sentoku • 8 • S. V. Tolokonnikov • 9 • 1. Center for Ultrafast Optical Science University of Michigan Ann Arbor USA • 2. Oak Ridge National Laboratory Oak Ridge USA • 3. Central Research Institute of Electric Power Industry Tokyo Japan • 4.

Lebedev Physics Institute Russian Academy of Sciences Moscow Russia • 5. Institute of Computational Technologies Siberian Branch of Russian Academy of Sciences Novosibirsk Russia • 6. Institute for Mathematical Modeling Russian Academy of Sciences Moscow.

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