Geometric Spin Hall Effect of Light

Andrea Aiello, Norbert Lindlein,

Christoph Marquardt, Gerd Leuchs

MPL Olomouc, June 24, 2009

Olomouc, 24/6/2009 2

Optical angular momentum and spin-orbit coupling

• A suitably prepared beam of light may have both a spin and an orbital angular momentum (SAM and OAM).

• SAM circular polarization

• OAM spiraling phase-front

• SAM and OAM may be coupled!L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185,

(1992)

http://www.physics.gla.ac.uk/Optics/play/photonOAM/

SAM OAM

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Spin Hall effect of light

Onur Hosten and Paul Kwiat, Science 319, 787-790 (2008)

This effect is also known as

Imbert-Fedorov shift

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Geometrodynamics of spinning light

K. Y. Bliokh et al. Nature Photon. 2, 748–753 (2008).

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Geometric spin Hall effect of light

x

yz

z’x’

y’

L

R

tan

2

x

A. Aiello, N. Lindlein, C. Marquardt, G. Leuchs, arXiv:0902.4639v1[quant-ph] (2009).

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1. What is the physical origin of such a shift?

2. Is this shift measurable?

Questions

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Reminder: Helicity of light

x

yz

E

ˆ ˆ

ˆ ˆ

x y

R R L L

E x y

e e

1ˆ ˆ ˆ

21

ˆ ˆ ˆ2

R

L

i

i

e x y

e x y

22**RLyxyxi helicity

122 RL

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Linear and angular momentum of light

* 30 Re ( ) ( )2

d rJ r E r B r

r3d)(rP p Total linear and angular momenta

)()(

)()(Re2

)( *0

rrr

rBrEr

pj

p

Time-averaged linear and

angular momentum densities(per unit of volume) = Poynting vector = energy density flux

)(2 rpc

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Transverse angular momentum

zyxr ˆˆˆdd)( zyx PPPyx pP

zyxr ˆˆˆdd)( zyx JJJyx jJ

yx ˆˆ yx JJ J

Linear and angular momentum of light per unit length

Transverse linear momentumyx ˆˆ yx PP P

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Centroid (barycenter) of the intensity distribution

z

z

P

yxpyx

yxI

yxI

dd)(ˆˆ

dd)(

dd)(

ryx

r

rrr

),,()(2 zyxIpc z r

intensity integrated 2 zPc

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Angular momentum-vs-transverse shift

zxy

yzx

PxPzJ

PzPyJ

yxP

yxpy

P

yxpx

z

z

z

zyx

ry

rxr ˆˆ

dd)(ˆ

dd)(ˆ

)()()(

)()()(

)()()(

rrr

rrr

rrr

xyz

zxy

yzx

pypxj

pxpzj

pzpyj

zyxr ˆˆˆdd)( zyx JJJyx jJ

zyxr ˆˆˆdd)( zyx PPPyx pP

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Geometric Spin Hall Effect of Light

xPJ

yPJ

zy

zx

at z = 0

x

z j

y

z

x̂xjj

y

z’L

1helicity

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1. What is the physical origin of such a shift?

2. Is this shift measurable?

Questions

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The answer is: YES, but….

• Many detectors are sensitive to the electric field energy density

rather than Poynting vector flux,

• Such energy density contains the contributions given by the

three components (x,y,z) of the electric field:

• The flux of the Poynting vector across the observation plane

contains the contributions given by the two transverse

components (x,y) of the electric field only:

2222)()()()( rrrrE zyx EEE

22)()(fluxintensity rr yx EE

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• In practice, it will be sufficient to use a polarizer (non tilted!) in

front of the detector to attenuate either or in order

to measure a non-zero shift.

• The difference between energy density and linear momentum

distributions is also relevant, e.g., in atomic beam deflection

experiments:

( )xE r ( )zE r

Observation plane

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1. When a circularly polarized beam of light is observed from a

reference frame tilted with respect to the direction of propagation

of the beam, the barycenter of the latter undergoes a shift

comparable with the wavelength of the light

2. Extensive numerical simulations performed with the program

POLFOCUS agree very well with analytical predictions for well

collimated beams not too close to grazing incidence

Conclusions