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Nonlinear interactions in periodic and quasi-periodic structures

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Title: Nonlinear interactions in periodic and quasi-periodic structures Author: ady arie* Last modified by: Omer Korech Created Date: 1/7/2000 4:06:20 PM – PowerPoint PPT presentation

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Title: Nonlinear interactions in periodic and quasi-periodic structures


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
Airy wave-packets in quantum mechanics
Free particle Schrödinger equation
Airy wave-packet solution
Non-spreading Airy wave-packet solution
tgt0
acceleration
M.V. Berry and N. L. Balazs, Nonspreading wave
packets, Am. J. Phys. 47, 264 (1979)
4
Airy wavepackets in Quantum Mechanics and Optics
Free particle Schrödinger equation
Infinite energy wave packet
Berry and Balzas, 1979
  • 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,,PRL 99, 213901 (2007) Berry
and Balazs, Am J Phys 47, 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
2-D Airy beam
1-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
Transporting micro-particles
Curved plasma channel generation in air
Polynkin et al , Science 324, 229 (2009)
Baumgartl, Nature Photonics 2, 675 (2008)
AiryBessel 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, Nature 494, 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
23
N. Voloch-Bloch et al, Nature 494, 331 (2013)
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
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