Moti Fridman 1, Alessandro Farsi 2 and Alexander L. Gaeta 2 1 School of Engineering, Bar Ilan University, Ramat Gan, 52900, Israel 2 School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA Multistage Accelerating Beams

Caustics Menaechmus ~ 400 BC

Accelerating beam of light G. A. Siviloglou et. al. (2007)

Accelerating beam in space Input beam Mask imposing cubic phase Accelerating beam Lens

Accelerating beam of electrons N. V. Bloch et. al. (2012)

Akhmanov, Sukhorukov, and Chirkin, Sov. Phys. - JETP, 28, 748, (1969). Kolner, IEEE J. Quantum Electron. 30, 1951 (1994). Optical processing based on time-space duality Spatial diffraction  Dispersive propagation Spatial domain Temporal domain x z z diffraction dispersive broadening

Imaging in space Imaging achieved by adding a quadratic phase to wavefront. lens quadratic phase shift in space

Temporal imaging Imaging achieved by adding a quadratic phase to the signal. quadratic phase shift in time dispersive elements

Accelerating beam in time cubic phase shift in time and a lens dispersion

Parametric mixing with a quadratic chirped pump D quadratic frequency shift Quadratic dispersion pump pulse Look for interaction that imparts quadratic phase shift. Imposing cubic phase in time

Pump Diffracting grating

Reflecting stairs

Accelerating beam in time

Multistage accelerating beam FWM Stage 1 signal idler

Multistage accelerating beam FWM Stage 1 FWM signal idler Idler #2 Stage 2

Experimental setup

Accelerating beam in time

Multistage accelerating beam in time

Conclusions Accelerating beam in time. Multistage acceleration. More complex dispersion relations. Space-time equivalent can open a rout for many more novel demonstrations.