Nonlinear interactions in periodic and quasi-periodic structures

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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
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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

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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)
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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).

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Accelerating Airy beam
Siviloglou et al,,PRL 99, 213901 (2007) Berry
and Balazs, Am J Phys 47, 264 (1979)
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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
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Kaganovsky and Heyman, Opt. Exp. 18, 8440 (2010)
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1D and 2D Airy beams
2-D Airy beam
1-D Airy beam
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Sir George Biddel Airy, 1801-1892
The Airy function is named after the British
astronomer Airy, who introduced it during his
studies of rainbows.
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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).

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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)
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Nonlinear generation of accelerating Airy beam
T. Ellenbogen et al, Nature Photonics 3, 395
(2009)
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Airy beam laser
Output coupler pattern
G. Porat et al, Opt. Lett 36, 4119
(2011) Highlighted in Nature Photonics 5, 715,
December (2011)
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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?
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Generation of electron vortex beams
J. Verbeeck et al , Nature 467, 301 (2010) B. J.
McMorran et al, Science 14, 192 (2011)
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Generation of Airy beams with electrons
N. Voloch-Bloch et al, Nature 494, 331 (2013)
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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
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The transmission electron microscope
Operating voltage 100-200 kV Electron
wavelength 3.7-2.5 pm Variable magnification
and imaging distance with magnetic lenses.
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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
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Acceleration measurements
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Comparison of Airy lattice with Bragg and vortex
lattices
The acceleration causes the lattice to lose its
shape
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Acceleration of different orders
Central lobe position in X (with carrier) and
Y. In Y, the position scales simply as (1/m)
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Non-spreading electron Airy beam
Bragg reference
Airy beam
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Self healing of electron Airy beam
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N. Voloch-Bloch et al, Nature 494, 331 (2013)
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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.

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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.
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Calibrating the distance in the TEM
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Is it a parabolic trajectory?
Yes, it is!
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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.

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