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Self Accelerating Beams of Photons and ElectronsAdy 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

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xmti

M.V. Berry and N. L. Balazs, “Nonspreading wave packets,Am. J. Phys. 47, 264 (1979)

|Ψ|2

x

Non-spreading Airy wave-packet solution

t>0acceleration

Free particle Schrödinger equation

Airy wave-packet solution

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Airy wavepackets in Quantum Mechanics and Optics

02 2

22

xmti

|Ψ|2

x

Free particle Schrödinger equation

Berry and Balzas, 1979

Infinite energywave packet

• 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).

021

2

2

si

Normalized paraxial Helmholtz equation

s

|Φ|2

Siviloglou and Christodulides, 2007

Finite energy beam

( ) asAi s e

• Nearly non diffracting• Freely accelerating

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Accelerating Airy beam

2 3

020

, 2 exp 2 12

,

is A s i s i

electric field envelopes x x normalized transverse coordinate

z kx normalized propagationcoordinate

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

In a ray description, the rays are tangent to the parabolic line but do not cross it.

Kaganovsky and Heyman, Opt. Exp. 18, 8440 (2010)

Caustic – a curve of a surface to which light rays are tangent

Curved caustic in every day life

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1D and 2D Airy beams

-2 -1 0 1 2

-2 0 2

-2

0

2

2-D Airy beam1-D Airy beam

0

xAi

x 0 0

x yAi Ai

x y

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332

)( kiak eek

• 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).

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

phase mask f f

lens

Optical F.T.

Linear Generation of Airy beam

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Transporting micro-particles

Baumgartl, Nature Photonics 2, 675 (2008)

Curved plasma channel generation in air

Polynkin et al , Science 324, 229 (2009)

Chong et al, Nature Photonics 4, 103 (2010)

Airy–Bessel wave packets as versatile linear light bullets

Applications of Airy beam

Microchip laser (S. Longhi, Opt . Lett. 36, 711 (2011)

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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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3cif yNLS Ce

3 1 2 3

cif yNLS Ce3 1 2

The phase mismatch values for up-conversion and down-conversion processes that involve the same three waves have opposite signs

* I. Dolev, T. Ellenbogen, and A. Arie, Optics Letters, 35, (2010).

Gaussian Pump

f f

Optical F.T.

ω1

1-2

xy

ω2

SFG

DFG

Lens

1+2

Switching the propagation direction of Airy beams

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Measured DFGMeasured SHG

acceleration acceleration

Beam profile Beam profile

Switching the propagation direction of Airy beams

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Airy beam laser

G. Porat et al, Opt. Lett 36, 4119 (2011)Highlighted in Nature Photonics 5, 715, December (2011)

Output coupler pattern:

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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?

Airy wave-packet of massive particle?

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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 equationThe 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 goldPatterned 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 ordersCentral 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 beamBragg 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)

2 30 0

14 B

xAi acceleration

x k x

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 trajectoriesIt is possible to construct finite energy beam that will accelerate along arbitrary convex trajectoriesIn 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

80 microns

50 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

Gil Porat

Ido DolevTal Ellenbogen Noa Voloch-Bloch

Itai Epstein