Self Accelerating Electron Airy Beams N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover and Ady Arie

1. Self Accelerating Electron Airy Beams N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover and Ady Arie Dept. of Physical Electronics, Tel-Aviv University, Tel-Aviv, Israel FRISNO-12, February 24, 2013

2. Outline • The quantum-mechanical Airy wave-function and its properties • Realization and applications of Airy beams in optics • Generation and characterization of electron Airy beams • Summary

3. |Ψ|2 x Airy wave-packets in quantum mechanics Free particle Schrödinger equation Airy wave-packet solution Non-spreading Airy wave-packet solution t>0 acceleration M.V. Berry and N. L. Balazs, “Nonspreading wave packets, Am. J. Phys. 47, 264 (1979)

4. |Ψ|2 |Φ|2 x s Airy wavepackets in Quantum Mechanics and Optics Free particle Schrödinger equation Normalized paraxial Helmholtz equation Infinite energy wave packet Finite energy beam Berry and Balzas, 1979 Siviloglou and Christodulides, 2007 • Non diffracting • Freely accelerating • Nearly non diffracting • Freely accelerating • Berry and Balzas, Am. J. Phys, 47, 264 (1979) • Siviloglou & Christodoulides, Opt. Lett. 32, 979-981 (2007). • Siviloglou, Broky, Dogariu, & Christodoulides, Phys. Rev. Lett.99, 213901 (2007).

5. Accelerating Airy beam Siviloglou et al,,PRL99, 213901 (2007) Berry and Balazs, Am J Phys47, 264 (1979)

6. Airy beam – manifestation of caustic Caustic – a curve of a surface to which light rays are tangent In a ray description, the rays are tangent to the parabolic line but do not cross it. Curved caustic in every day life 6 Kaganovsky and Heyman, Opt. Exp. 18, 8440 (2010)

7. 1D and 2D Airy beams 1-D Airy beam 2-D Airy beam

8. Sir George Biddel Airy, 1801-1892 The Airy function is named after the British astronomer Airy, who introduced it during his studies of rainbows.

9. Linear Generation of Airy beam Fourier transform of truncated Airy beam Now we can create Airy beams easily: Take a Gaussian beam Impose a cubic spatial phase Perform optical Fourier transform lens phase mask f f Optical F.T. • Siviloglou, G. A. & Christodoulides, D. N. Opt. Lett. 32, 979-981 (2007). • Siviloglou, G. A., Broky, J., Dogariu, A. & Christodoulides, D. N. Phys. Rev. Lett.99, 213901 (2007).

10. Applications of Airy beam Curved plasma channel generation in air Transporting micro-particles Polynkin et al , Science 324, 229(2009) Baumgartl, Nature Photonics 2, 675(2008) Airy–Bessel wave packets as versatile linear light bullets Microchip laser (S. Longhi, Opt . Lett. 36, 711 (2011) Chong et al, Nature Photonics 4, 103(2010)

11. Nonlinear generation of accelerating Airy beam T. Ellenbogen et al, Nature Photonics 3, 395(2009)

12. Airy beam laser Output coupler pattern: G. Porat et al, Opt. Lett 36, 4119 (2011) Highlighted in Nature Photonics 5, 715, December (2011) 12

13. Airy wave-packet of massive particle? So far, all the demonstrations of Airy beams were in optics. Can we generate an Airy wave-packet of massive particle (e.g. an electron), as originally suggested by Berry and Balzas? Will this wave-packet exhibit free-acceleration, shape preservation and self healing?

14. Generation of electron vortex beams J. Verbeeck et al , Nature 467, 301 (2010) B. J. McMorran et al, Science 14, 192 (2011)

15. Generation of Airy beams with electrons N. Voloch-Bloch et al, Nature494, 331 (2013)

16. Quasi relativistic Schrodinger equation The Klein-Gordon equation (spin effects ignored) Assume a wave solution of the form For a slowly varying envelope, the envelope equation is: Which is identical to the paraxial Hemholtz equation and has the same form of the non-relativistic Schrodinger equation 16

17. The transmission electron microscope Operating voltage: 100-200 kV Electron wavelength: 3.7-2.5 pm Variable magnification and imaging distance with magnetic lenses.

18. Modulation masks (nano-holograms) 50 nm SiN membrane coated with 10 nm of gold Patterned by FIB milling with the following patterns: Carrier period for Airy: 400 nm Carrier period for Bragg: 100 nm

19. Acceleration measurements

20. Comparison of Airy lattice with Bragg and vortex lattices The acceleration causes the lattice to “lose” its shape

21. Acceleration of different orders Central lobe position in X (with carrier) and Y. In Y, the position scales simply as (1/m)

22. Non-spreading electron Airy beam Bragg reference Airy beam

23. Self healing of electron Airy beam N. Voloch-Bloch et al, Nature494, 331 (2013) 23

24. Experimental challenges 1. Very small acceleration (~mm shift over 100 meters), owing to the extremely large de-Broglie wave-number kB (~1012 m-1) • 2. Location of the mask and slow-scan camera are fixed. • Solution: • Vary (by magnetic field) focal length of the projection lens in the TEM • And, calibrate the distances with a reference grating.

25. Calibrating the distance in the TEM • Two possibilities: • Diffraction from the periodic mask: • Difference between the Airy patterns in X (with carrier) • and Y (without a carrier) Periodic mask period: 100 nm Airy mask period: 400 nm.

26. Calibrating the distance in the TEM

27. Is it a parabolic trajectory? Yes, it is!

28. Summary • We have generated for the first time Airy wave-packet of a massive particle (an electron) • Generation enabled by diffraction of electrons from a nano-fabricated hologram • Airy wave-packet is freely accelerating and shape preserving. It can recover from blocking obstacles. • Possible applications: • New type of electron interferometers • Study interactions with magnetic and electric potentials and with different materials • Microscopy – large depth of focus • Nanofabrication – e.g. drill straight holes. 28