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
2
Airy wave-packets in quantum mechanics
  2  2
Free particle Schrödinger equation i

0
2
t 2m x
Airy wave-packet solution
Non-spreading
Airy wave-packet
solution
|Ψ|
2
t>0
acceleration
x
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
 1  2
i

0
2
 2 s
  2  2
i

0
2
t 2m x
Normalized paraxial
Helmholtz equation
Free particle
Schrödinger equation
|Φ|2
|Ψ|
Infinite energy
wave packet
2
Finite energy beam
Ai( s )e as
Berry and Balzas, 1979
• Non diffracting
• Freely accelerating
x
Siviloglou and Christodulides, 2007
• 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).
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s
Accelerating Airy beam




  , s   Ai s   2  exp i  s 2   i  3 12 


Siviloglou et al,,PRL 99, 213901 (2007)
  electric field envelope,
2
s  x x0  normalized transverse coordinate
z
kx02
 normalized propagation coordinate
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
Kaganovsky and Heyman, Opt. Exp. 18, 8440 (2010)
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1D and 2D Airy beams
1-D Airy beam
2-D Airy beam
-2
0
-2 -1
0
 x
Ai  
 x0 
1
2
2
-2
0
2
 x  y 
Ai   Ai  
 x0   y0 
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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 (k )  e
 ak 2 i k 3 3
e
Now we can create Airy beams easily:
Take a Gaussian beam
Impose a cubic spatial phase
Perform optical Fourier transform
lens
f
Optical F.T.
f
• 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
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
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Chong et al, Nature Photonics 4, 103 (2010)
Microchip laser (S. Longhi, Opt . Lett. 36,
711 (2011)
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
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)
 x
1
Ai    acceleration  2 3
4kB x0
 x0 
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:
1. Diffraction from the periodic mask:
2. 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 nanofabricated 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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