1. 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

2. Caustics Menaechmus ~ 400 BC

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

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

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

6. 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

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

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

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

10. 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

11. Pump Diffracting grating

12. Reflecting stairs

13. Accelerating beam in time

14. Multistage accelerating beam FWM Stage 1 signal idler

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

16. Experimental setup

17. Accelerating beam in time

18. Multistage accelerating beam in time

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