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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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OutlineOutline

•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

02 2

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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 beamAccelerating 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 causticAiry 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 beams1D 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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Sir George Biddel Airy, 1801-1892Sir 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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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 beamLinear 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 beamApplications of Airy beam

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

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Nonlinear generation of accelerating Airy beamNonlinear generation of accelerating Airy beam

T. Ellenbogen et al, Nature Photonics 3, 395 (2009)

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Airy beam laserAiry 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?Airy wave-packet of massive particle?

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Generation of electron vortex beamsGeneration 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 electronsGeneration of Airy beams with electrons

N. Voloch-Bloch et al, Nature 494, 331 (2013)

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Quasi relativistic Schrodinger equationQuasi 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 microscopeThe 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)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 measurementsAcceleration measurements

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Comparison of Airy lattice with Bragg and vortex Comparison of Airy lattice with Bragg and vortex latticeslattices

The acceleration causes the lattice to “lose” its shape

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Acceleration of different ordersAcceleration 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 beamNon-spreading electron Airy beamBragg reference Airy beam

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Self healing of electron Airy beamSelf healing of electron Airy beam

N. Voloch-Bloch et al, Nature 494, 331 (2013)

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Experimental challengesExperimental 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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Calibrating the distance in the TEMCalibrating 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 nmAiry mask period: 400 nm.

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Calibrating the distance in the TEMCalibrating the distance in the TEM

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Is it a parabolic trajectory?Is it a parabolic trajectory?

Yes, it is!

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SummarySummary

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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.