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Self Accelerating Beams of Photons and Electrons Ady Arie

Self Accelerating Beams of Photons and Electrons Ady Arie Dept. of Physical Electronics , Tel-Aviv University, Tel-Aviv, Israel. Heraklion , Crete, September 20 th 2013. Outline. The quantum-mechanical Airy wave-function and its properties

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Self Accelerating Beams of Photons and Electrons Ady Arie

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  1. Self Accelerating Beams of Photons and Electrons Ady Arie Dept. of Physical Electronics, Tel-Aviv University, Tel-Aviv, Israel Heraklion, Crete, September 20th 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 • Self accelerating plasmon beams with arbitrary trajectories • 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. 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).

  9. 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)

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

  11. Diffraction of fundamental and SH T. Ellenbogen et al, Nature Photonics 3, 395(2009)

  12. ω2 y x Switching the propagation direction of Airy beams • The phase mismatch values for up-conversion and down-conversion processes that involve the same three waves have opposite signs DFG w1-w2 Lens ω1 f f Optical F.T. Gaussian Pump SFG w1+w2 * I. Dolev, T. Ellenbogen, and A. Arie, Optics Letters, 35, (2010).

  13. Switching the propagation direction of Airy beams Measured SHG MeasuredDFG Beam profile Beam profile acceleration acceleration

  14. Airy beam laser Output coupler pattern: G. Porat et al, Opt. Lett36, 4119 (2011) Highlighted in Nature Photonics 5, 715, December (2011) 14

  15. 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?

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

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

  18. 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 18

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

  20. 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

  21. Acceleration measurements

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

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

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

  25. Self healing of electron Airy beam N. Voloch-Bloch et al, Nature494, 331 (2013) 25

  26. 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.

  27. Acceleration along arbitrary trajectories It is possible to construct finite energy beam that will accelerate along arbitrary convex trajectories In free-space, the caustic trajectory can be defined through the transverse phase of the beam at the input plane Greenfield et al, PRL106, 213902 (2011) Froehly et al, Opt. Exp. 19, 16455 (2011)

  28. Airy plasmon Salandrino and Christodoulides, Opt. Lett. 35, 2082 (2010) Minovich et al, PRL107, 116802 (2011)

  29. Can we make self-accelerating plasmons with arbitrary trajectories? New challenges: Phase mismatch between free-space beam and plasmon beam Excitation along an area (vs. line definition of transverse phase in free-space) Short plasmon propagation and measurement distance (<100 microns), thus requiring fast acceleration (=non-paraxial conditions) Flexible beam shapers (e.g. Spatial Light Modulators) do not exist for plasmon beams.

  30. Arbitrary bending plasmonic beams Excitation through special binary coupler Near field characterization with NSOM Key element: Plasmonic coupler – provides wave-vector matching and sets the transverse phase

  31. Bending plasmonic beams along polynomial and exponential trajectories Theory Experiment Theory Experiment 50 microns 80 microns

  32. Summary Three examples of self-accelerating beams: Generation and mixing of Airy beams in quadratic nonlinear medium Generation of Airy beam of a massive particle (an electron) Arbitrary bending plasmonic beams

  33. Acknowledgement Tal Ellenbogen Ido Dolev Noa Voloch-Bloch Itai Epstein Gil Porat

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