Transcript Nonlinear interactions in periodic and quasi

Self Accelerating Beams of Photons and Electrons
Ady Arie
Dept. of Physical Electronics, Tel-Aviv University, Tel-Aviv, Israel
Heraklion, Crete, September 20th 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
•Self accelerating plasmon beams with arbitrary trajectories
•Summary
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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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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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Diffraction of fundamental and SH
T. Ellenbogen et al, Nature Photonics 3, 395 (2009)
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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
3  1  2
S
NL
3  1  2
ifc y3
 Ce
S
NL
ifc y3
 Ce
DFG
1-2
Lens
ω1
f
ω2
Gaussian Pump
y
x
f
Optical F.T.
* I. Dolev, T. Ellenbogen, and A. Arie, Optics Letters, 35, (2010).
SFG
1+2
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Switching the propagation direction of Airy beams
Measured SHG
acceleration
Beam profile
Measured DFG Beam profile
acceleration
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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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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, PRL 106, 213902 (2011)
Froehly et al, Opt. Exp. 19, 16455 (2011)
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Airy plasmon
Salandrino and Christodoulides, Opt. Lett. 35, 2082 (2010)
Minovich et al, PRL 107, 116802 (2011)
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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.
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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
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Bending plasmonic beams along polynomial and
exponential trajectories
Theory
Experiment
Theory
Experiment
50 microns
80 microns
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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
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Acknowledgement
Tal Ellenbogen
Gil Porat
Ido Dolev
Noa Voloch-Bloch
Itai Epstein
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