Self Accelerating Electron Airy Beams

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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
1
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)
3
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).
4
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)
5
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)
6
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.
8
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).
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
10
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)
11
Airy beam laser
Output coupler pattern:
G. Porat et al, Opt. Lett 36, 4119 (2011)
Highlighted in Nature Photonics 5, 715, December (2011)
12
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?
13
Generation of electron vortex beams
J. Verbeeck et al , Nature 467, 301 (2010)
B. J. McMorran et al, Science 14, 192 (2011)
14
Generation of Airy beams with electrons
N. Voloch-Bloch et al, Nature 494, 331 (2013)
15
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
The transmission electron microscope
Operating voltage: 100-200 kV
Electron wavelength: 3.7-2.5 pm
Variable magnification and
imaging distance with magnetic
lenses.
17
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
18
Acceleration measurements
19
Comparison of Airy lattice with Bragg and vortex
lattices
The acceleration
causes the lattice to
“lose” its shape
20
Acceleration of different orders
Central lobe position in X (with carrier) and Y.
In Y, the position scales simply as (1/m)
21
Non-spreading electron Airy beam
Bragg reference
Airy beam
22
Self healing of electron Airy beam
N. Voloch-Bloch et al, Nature 494, 331 (2013)
23
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.
24
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.
25
Calibrating the distance in the TEM
26
Is it a parabolic trajectory?
Yes, it is!
27
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.
28
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