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Angular Momenta, Geometric Phases, and Spin-Orbit Interactions of Light and Electron Waves Konstantin Y. Bliokh Advanced Science Institute, RIKEN, Wako-shi, Saitama, Japan In collaboration with: Y. Gorodetski, A. Niv, E. Hasman (Israel); M. Alonso (USA), E. Ostrovskaya (Australia), A. Aiello (Germany), M. R. Dennis (UK), F. Nori (Japan)

Angular Momenta , Geometric Phases, and Spin-Orbit Interactions of Light and Electron Waves

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Angular Momenta , Geometric Phases, and Spin-Orbit Interactions of Light and Electron Waves. Konstantin Y. Bliokh Advanced Science Institute, RIKEN, Wako- shi , Saitama, Japan In collaboration with: - PowerPoint PPT Presentation

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Page 1: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Angular Momenta, Geometric Phases,

and Spin-Orbit Interactions of Light and Electron Waves

Konstantin Y. BliokhAdvanced Science Institute, RIKEN, Wako-shi, Saitama,

Japan

In collaboration with: Y. Gorodetski, A. Niv, E. Hasman (Israel);

M. Alonso (USA), E. Ostrovskaya (Australia), A. Aiello (Germany), M. R. Dennis (UK), F. Nori (Japan)

Page 2: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

• Basic concepts: Angular momenta, Berry phase, Spin-orbit interactions, Hall effects• Theory of photon AM• Application to Bessel beams• Electron vortex beams• Theory of electron AM

Page 3: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Angular momentum of light

2. Extrinsic orbital AM (trajectory)

ext c c L r kkc

rc

3. Intrinsic orbital AM (vortex)

int c

c

dV

L r r k

κ

0 1

1 2

cS κ1. Intrinsic spin AM (polarization)

1 1 c c c/ kκ k

exp i

Page 4: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Different mechanical action of the SAM and OAM on small particles:

O’Neil et al., PRL 2002; Garces-Chavez et al., PRL 2003

spinning motion

orbital motion

Angular momentum: Spin and orbital

Page 5: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Geometric phase

,i ie E e E e e

2D:

x x

y y

x yi e e e

1 1

d ΩJ

AM-rotation coupling:Coriolis / angular-Doppler effect

3D: J Ω

Page 6: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Φ

zk

xk

E κykS κ

C

Geometric phase

, , : ii e e e κ e κ e e

2 1 c

2

oszd S

dynamics, S:

d k kAC

1 cos ,sink

eA 3k

kkF A Berry connection, curvature

(parallel transport)

geometry, E:

Page 7: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Rytov, 1938; Vladimirskiy, 1941; Ross, 1984; Tomita, Chiao, Wu, PRL 1986

X

YZ

e

w

v

te

w

v

tew

v

t 2W20 0

Spin-orbit Lagrangian and phase

SOI Lagrangian from Maxwell equationsSOI k S ΩAL

Bliokh, 2008

SOI 2ˆ ˆ ˆ

4em

p σ pL A E2

ˆˆ 1 ,

2m

p E

S pA for Dirac equation

SOI d LBerry phase:

Page 8: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Liberman & Zeldovich, PRA 1992;Bliokh & Bliokh, PLA 2004, PRE 2004; Onoda, Murakami, Nagaosa, PRL 2004

c cln ,k n k r t k F ray equations (equations of motion)

X

YZ

D2D

+2D

D

Spin-Hall effect of light

ˆ ˆ,i j ijl lr r i F

c c c const J r k κ

Page 9: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Berry phase Spin-Hall effect Anisotropy

Bliokh et al., Nature Photon. 2008

Spin-Hall effect of light

Page 10: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Fedorov, 1955; Imbert, 1972; ……Onoda et al., PRL 2004;Bliokh & Bliokh, PRL 2006;Hosten & Kwiat, Science 2008

D r

ImbertFedorov transverse shift

Dr

Spin-Hall effect of light

Page 11: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

ˆ / sin ,z yR

coty

coszS

1 cot

ykY k

c c :y yk k k egeometry, phase:

ext : J L S

dynamics, AM:sin coszJ k Y 0zJ

k

Spin-Hall effect of light

Page 12: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Remarkably, the beam shift and accuracy of measurements can be enormously increased using the method of quantum weak measurements:

Angstrom accuracy!

Spin-Hall effect of light

Page 13: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Orbit-orbit interaction and phase

Φ

zk

xk

phase κyk

int L κ

C

OOI LagrangiannOOI i t k L ΩL A

X

YZ

w

v

te

w

v

tew

v

t 20 0

Bliokh, PRL 2006; Alexeyev, Yavorsky, JOA 2006;Segev et al., PRL 1992; Kataevskaya, Kundikova, 1995

SOI d L Berry phase

Page 14: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Orbital-Hall effect of light

l vortex-depndent shifts and orbit-orbit interaction

X

YZ

012–1–2

Bliokh, PRL 2006Fedoseyev, Opt. Commun. 2001;Dasgupta & Gupta, Opt. Commun. 2006

c c c J r k κ

Page 15: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

• Basic concepts: Angular momenta, Berry phase, Spin-orbit interactions, Hall effects• Theory of photon AM• Application to Bessel beams• Electron vortex beams• Theory of electron AM

Page 16: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Angular momentum of lightˆ ˆˆ ˆˆ ˆ J r p S L S

ˆ ,i kr ˆ ,p k ˆa aijijS i

total AM = OAM +SAM?

ˆˆ ,z z zijijL i S i

ix yi e

E e e

z-components and paraxial eigenmodes:

- polarization, l - vortex

1

10

1

1

Page 17: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

( )E k

“the separation of the total AM into orbital and spin parts has restricted physical meaning. ... States with definite values of OAM and SAM do not satisfy the condition of transversality in the general case.” A. I. Akhiezer, V. B. Berestetskii, “Quantum Electrodynamics” (1965)

0 E k E κ transversality constraint: 3D 2D

Nonparaxial problem 1

ˆˆ , LE κ SE κ though ˆ JE κ

Page 18: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

“in the general nonparaxial case there is no separation into ℓ-dependent orbital and -dependent spin part of AM” S. M. Barnett, L. Allen, Opt. Commun. (1994)

Nonparaxial problem 2

i iii ee e

E e ee κ

nonparaxial vortex

spin-to-orbital AM conversion?..

Y. Zhao et al., PRL (2007)

, 1z zL S

zJ though

Page 19: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Spin-dependent OAM and position signify the spin-orbit interaction (SOI) of light: (i) spin-to-orbital AM conversion, (ii) spin Hall effect.

2/ˆ ˆˆ k k Sr r spin-dependent position M.H.L. Pryce, PRSLA (1948), etc.

Modified AM and position operators

ˆˆ ˆ ˆ ΔL L r k projected SAM and OAM cf. S.J. van Enk, G. Nienhuis, JMO (1994)

ˆ ˆ ˆˆ ˆ S S κ κ S κΔ ˆ ˆ Δ κ κ S

,ˆˆ ˆ J L S compatible with transversality:

Page 20: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Modified AM and position operators

3ˆ /ˆ ˆ, li j ijlr kr i k

Modified operators exhibit non-canonical commutation relations:

ˆ ,0ˆ,i jS S ˆ ˆ ,ˆˆ,i j i ljl lL L i L S

ˆ ,ˆˆ,i j lijlSL S i

ˆ ˆˆ ,i j ijl lJ O i O

But they correspond to observable values and are transformed as vectors:

cf. S.J. van Enk, G. Nienhuis (1994); I. Bialynicki-Birula, Z. Bialynicka-Birula (1987), B.-S. Skagerstam (1992), A. Berard, H. Mohrbach (2006)

Page 21: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Helicity representation

ˆ : , , , ,x y zU κ e e e e e κ †ˆ ˆˆ ˆU UO O

00

0 0

3D 2D reductionand diagonalization:

ˆ ˆ ̂ L L kA all operators

become diagonal !

ˆ ˆ ̂ r r A

ˆ ̂S κ

ˆ diag ,11, 0

cf. I. Bialynicki-Birula, Z. Bialynicka-Birula (1987), B.-S. Skagerstam (1992), A. Berard, H. Mohrbach (2006)

kA - Berry connection

Page 22: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

• Basic concepts: Angular momenta, Berry phase, Spin-orbit interactions, Hall effects• Theory of photon AM• Application to Bessel beams• Electron vortex beams• Theory of electron AM

Page 23: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Bessel beams in free space

ˆE E O O mean values 0 iE A e

field

( )E k

20 02 1 cos ~

C

d k A Berry phase

The spin-to-orbit AM conversion in nonparaxial fields originates from the Berry phase associated with the azimuthal distribution of partial waves.

zL 1 ,zS SAM and OAMfor Bessel beam

Page 24: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Quantization of caustics

k R quantized GO caustic: fine SOI splitting !

1 1

4

1

22

2 2 2 22b bI r a J r J r a J r

spin-dependent intensity 1 / 2)/ 2( , ab

Page 25: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Plasmonic experiment

Y. Gorodetski et al., PRL (2008)

2B 0 / 2 ,

1

- circular plasmonic lens generates Bessel modes

B

1 1

Page 26: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Spin and orbital Hall effects

Bk Y orbital and spin Hall effects of light

1

1

4 0 4

,

azimuthally truncated field (symmetry breaking along x)

B. Zel’dovich et al. (1994), K.Y. Bliokh et al. (2008)

Page 27: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Plasmonic experiment

K. Y. Bliokh et al., PRL (2008)

0Plasmonic half-lens produces spin Hall effect:

~Y

1

1

Page 28: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

• Basic concepts: Angular momenta, Berry phase, Spin-orbit interactions, Hall effects• Theory of photon AM• Application to Bessel beams• Electron vortex beams• Theory of electron AM

Page 29: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Angular momentum of electron

2. Extrinsic orbital AM

ext c c L r k

kc

rc

zsS e1. Spin AM

k k

1/ 2s 1/ 2s ‘non-relativistic’

3. Intrinsic orbital AM ?

int L κ

0 1

1 2‘relativistic’

Page 30: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Scalar electron vortex beams

Page 31: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Scalar electron vortex beams

Page 32: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Scalar electron vortex beams

Page 33: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Murakami, Nagaosa, Zhang, Science 2003

, /e m k E r k k F equations of motion

Orbital-Hall effect for electrons

const J r k κ

Bliokh et al., PRL 2007

Page 34: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Scalar electron vortex beams

Page 35: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Scalar electron vortex beams

Page 36: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Scalar electron vortex beams

Page 37: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

• Basic concepts: Angular momenta, Berry phase, Spin-orbit interactions, Hall effects• Theory of photon AM• Application to Bessel beams• Electron vortex beams• Theory of electron AM

Page 38: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Bessel beams for Dirac electron

1i E tp W e

p rp plane wave

12 ˆ

E m wW

E E m w

σ κ

polarization bi-spinor

ˆˆ ˆti m α p Dirac equation: ... ... ... ...... ... ... ...

... ... ... ...

... ... ... ...

E > 0

E < 0cross-terms

cross-terms

w

spinor in the rest frame1 1

1 200 1,

2, zw S

Page 39: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Bessel beams for Dirac electron

2 221 / 2 / 2ssr J r J r D D

spin-dependent intensity

201 sinm

ED

SOI strength

1/ 2s 1/ 2s

4

1

0s s iW e p spectrum

( )E k

s

s

Bliokh, Dennis, Nori, arXiv:1105.0331

Page 40: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Angular momentum

ˆFWU p Foldy-Wouthhuysen

transformationˆP projector onto E>0 subspace

... ... ... ...

... ... ... ...

... ... ... ...

... ... ... ...

E > 0

E < 0cross-terms

cross-terms

†ˆ ˆ ˆˆ ˆ :FW FWU UO OP

ˆ ˆ ˆ L L Δˆˆ i kr A

ˆ ,ˆˆ S S Δ

2

ˆˆ 12

mp E

p σA

ˆ ˆ Δ pA

OAM and SAM for Bessel beams , 1z zL S ss D D

Bliokh, Dennis, Nori, arXiv:1105.0331

Page 41: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

Magnetic moment for Dirac electron

†ˆ ˆˆ ˆ ˆ ˆ ˆ 22FW FWeU UE

M M L SP

2e dV M r j † ˆ /E j α p

magnetic moment from currentˆ , M M ˆ ˆ ˆ

2e M r α

operator !

22e sE

DM

final result

slightly differs from Gosselin et al. (2008); Chuu, Chang, Niu, (2010)

Page 42: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

• Spin and orbital AM of light, geometric phase, spin- and orbital-Hall effects of light

• General theory for nonparaxial optical fields:

modified SAM, OAM, and position operators.• Bessel-beams example, spin-orbit splitting of

caustics and Hall effects• Electron vortex beams, AM of electron,

orbital-Hall effect.• Relativistic nonparaxial electron: Bessel

beams from Dirac equation• General theory of SAM, OAM, position, and

magnetic moment of Dirac electron.

Page 43: Angular  Momenta , Geometric Phases, and Spin-Orbit Interactions  of Light and Electron Waves

THANK YOU!