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)
• Basic concepts: Angular
momenta, Berry phase, Spinorbit interactions, Hall effects
• Theory of photon AM
• Application to Bessel beams
• Electron vortex beams
• Theory of electron AM
Angular momentum of light
1. Intrinsic spin AM (polarization)
 1
S   κc
  1
2. Extrinsic orbital AM
(trajectory)
κ c  k c / kc
kc
rc
3. Intrinsic orbital AM (vortex)
0
1
 1
2
Lext  rc  k c
  exp i  
Lint    r  rc   k dV
 κc
Angular momentum: Spin and orbital
Different mechanical action of the SAM and OAM on small
particles:
orbital motion
O’Neil et al., PRL 2002; Garces-Chavez et al., PRL 2003
spinning motion
Geometric phase
2D:
e   e x  i e y 
x
x
e  ei e , E  ei  E
 1
y
  1
y
   
3D:
  J  Ω
    J  Ω d 
AM-rotation coupling:
Coriolis / angular-Doppler effect
Geometric phase
S κ
Eκ
e  , e  , κ  : e  κ    e  i e  ei
ky
dynamics, S:

Φ

kx
  2   d S z
 2 1  cos  
geometry, E:
C
kz
    A  k   dk
C
1  cos 
k
A 
e , F   k  A  3  Berry connection, curvature
k sin 
k
(parallel transport)
Spin-orbit Lagrangian and phase
L SOI   A  k  S  Ω
 SOI Lagrangian from
Maxwell equations
Bliokh, 2008
2
2W0
t
Berry phase:
   L SOI d
e
t
W0 e
X
t e
w
w
Z
v
Y
w
v
v
Rytov, 1938; Vladimirskiy, 1941; Ross, 1984; Tomita, Chiao, Wu, PRL 1986
ˆ p  m  ˆ
e
S
ˆ
ˆ
L

A

p

σˆ  E  p   for Dirac equation
A 
1   , SOI
2
2 
4m
2p  E 
Spin-Hall effect of light
k c  k ln n , rc  t   F k

2D
2D
+
Z
D
 ray equations
(equations of motion)
 rˆi , rˆj   i ijl Fl
X
Y
J  rc  k c   κc  const
Liberman & Zeldovich, PRA 1992;
Bliokh & Bliokh, PLA 2004, PRE 2004; Onoda, Murakami, Nagaosa, PRL 2004
Spin-Hall effect of light
Berry phase
Spin-Hall effect
Anisotropy
Bliokh et al., Nature Photon. 2008
Spin-Hall effect of light
Dr  
Dr
ImbertFedorov
transverse shift
Fedorov, 1955; Imbert, 1972;
……
Onoda et al., PRL 2004;
Bliokh & Bliokh, PRL 2006;
Hosten & Kwiat, Science 2008
Spin-Hall effect of light
geometry, phase:
k c  k c  k ye y :
k

Rˆ z  y / sin   , S z   cos 







   y cot 

Y   k y   k  cot 
1
dynamics, AM:
J  Lext  S :  J z  k Y sin    cos   J z  0
Spin-Hall effect of light
Remarkably, the beam shift and accuracy of measurements
can be enormously increased using the method of quantum
weak measurements:
Angstrom
accuracy!
Orbit-orbit interaction and phase
Lint  κ
phase  κ
L OOI  A  k  Lint Ω
ky
 OOI Lagrangian

Φ

   L SOI d
kx
20
t
C
kz
e
t
0
 Berry phase
X
e
t e
w
w
Z
v
w
v
v
Bliokh, PRL 2006; Alexeyev, Yavorsky, JOA 2006;
Segev et al., PRL 1992; Kataevskaya, Kundikova, 1995
Y
Orbital-Hall effect of light
  l
 vortex-depndent shifts
and orbit-orbit interaction
–2–10 1 2
X
Z
Y
J  rc  k c   
Bliokh, PRL 2006
 κc
Fedoseyev, Opt. Commun. 2001;
Dasgupta & Gupta, Opt. Commun. 2006
• Basic concepts: Angular
momenta, Berry phase, Spinorbit interactions, Hall effects
• Theory of photon AM
• Application to Bessel beams
• Electron vortex beams
• Theory of electron AM
Angular momentum of light
ˆ  Sˆ
Jˆ  rˆ  pˆ  Sˆ  L
rˆ  ik , pˆ  k,
 
Lˆz  i , Sˆz
ij
 1
 
Sˆa
 i zij
E    e x  i e y  e
  1
 total AM = OAM +SAM?
i 
0
ij
 i aij
 z-components and
paraxial eigenmodes:
 - polarization,
l - vortex
 1
1
Nonparaxial problem 1
Ek  0  E  κ
E(k )
 transversality constraint:
3D  2D
“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)
ˆ κ
ˆ  κ, SE
LE
though
Jˆ E  κ
Nonparaxial problem 2
E    e  i e  ei ei   e  κ  ei   nonparaxial
vortex

Lz    , Sz  1   
though
Jz  
“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)
spin-to-orbital
AM conversion?..
Y. Zhao et al., PRL (2007)
Modified AM and position operators
ˆ   Sˆ , compatible with transversality:
Jˆ  L
 
Sˆ   Sˆ  Δˆ  κ κ  Sˆ  κˆ
ˆ  rˆ   k
ˆ L
ˆ Δ
L



ˆr  rˆ  k  Sˆ / k 2

Δˆ  κ  κ  Sˆ

 projected SAM and OAM
cf. S.J. van Enk, G. Nienhuis, JMO (1994)
 spin-dependent position
M.H.L. Pryce, PRSLA (1948), etc.
Spin-dependent OAM and position
signify the spin-orbit interaction (SOI) of light:
(i) spin-to-orbital AM conversion, (ii) spin Hall effect.
Modified AM and position operators
Modified operators exhibit non-canonical commutation relations:
 Sˆi, Sˆ j   0,




 Lˆi, Lˆ j   i ijl Lˆl  Sˆl ,


 Lˆi, Sˆ j   i ijl Sˆl,


 rˆi, rˆj   i ijlˆ kl / k 3
cf. S.J. van Enk, G. Nienhuis (1994);
I. Bialynicki-Birula, Z. Bialynicka-Birula (1987),
B.-S. Skagerstam (1992), A. Berard, H. Mohrbach (2006)
But they correspond to observable values
and are transformed as vectors:  ˆ ˆ 
ˆ

J
,
O

i

O
ijl l
 i j
Helicity representation

Uˆ  κ  :  e x , e y , e z   e  , e  , κ
Sˆ   κˆ
ˆ L
ˆ  ˆ A  k
L
rˆ   rˆ  ˆ A

ˆ  Uˆ †O
ˆ Uˆ
 O
 all operators
become diagonal !
ˆ  diag 1, 1, 0
A  k   Berry connection
cf. I. Bialynicki-Birula, Z. Bialynicka-Birula (1987),
B.-S. Skagerstam (1992), A. Berard, H. Mohrbach (2006)
3D  2D reduction
and diagonalization:
 

 

 0 0

0

0



• Basic concepts: Angular
momenta, Berry phase, Spinorbit interactions, Hall effects
• Theory of photon AM
• Application to Bessel beams
• Electron vortex beams
• Theory of electron AM
Bessel beams in free space
E  A   0  ei   field
ˆ  E
O  E O
Sz   1    , Lz    

 SAM and OAM
for Bessel beam
2
A

d
k

2

1

cos

~



0
0

C

 mean
values
 Berry phase
E(k )
The spin-to-orbit AM conversion in
nonparaxial fields originates from the
Berry phase associated with the
azimuthal distribution of partial waves.
Quantization of caustics
I   r    a 2 J 2  r   b2 J 2 2  r   2abJ 2  r  
 spin-dependent intensity (b   / 2, a  1   / 2)

k R    
  1
4
1
 1
 quantized GO caustic:
fine SOI splitting !

Plasmonic experiment
  B   
0   / 2 , B  2
  1
1
 circular plasmonic lens
generates Bessel modes
Y. Gorodetski et al., PRL (2008)
 1
Spin and orbital Hall effects
 azimuthally truncated field
(symmetry breaking along x)
B. Zel’dovich et al. (1994),
   ,  
K.Y. Bliokh et al. (2008)

kY      B 

 4
 1
  1
0
 orbital and spin
Hall effects of light
4
Plasmonic experiment
Plasmonic half-lens produces spin Hall effect:
0
 1
  1
Y ~
K. Y. Bliokh et al., PRL (2008)
• Basic concepts: Angular
momenta, Berry phase, Spinorbit interactions, Hall effects
• Theory of photon AM
• Application to Bessel beams
• Electron vortex beams
• Theory of electron AM
Angular momentum of electron
1. Spin AM
k
k
s  1/ 2
s  1/ 2
2. Extrinsic orbital AM
S  s ez
‘non-relativistic’
kc
rc
Lext  rc  k c
3. Intrinsic orbital AM ?
0
 1
1
2
Lint  κ
‘relativistic’
Scalar electron vortex beams
Scalar electron vortex beams
Scalar electron vortex beams
Orbital-Hall effect for electrons
k  eE , r  k / m  F k
 equations
of motion
J  r  k  κ  const
Murakami, Nagaosa, Zhang, Science 2003
Bliokh et al., PRL 2007
Scalar electron vortex beams
Scalar electron vortex beams
Scalar electron vortex beams
• Basic concepts: Angular
momenta, Berry phase, Spinorbit interactions, Hall effects
• Theory of photon AM
• Application to Bessel beams
• Electron vortex beams
• Theory of electron AM
Bessel beams for Dirac electron


i  t  αˆ  pˆ  ˆ m 
p
 Dirac equation:  ... ...
E>0
...
...


1
i  pr  E t 
... ...
 cross plane wave
 W p  e
 terms
 ... ...
Emw 
1 
W


2 E  E  m σˆ  κ w 
... ... 
cross
terms
... ...


... ... 
E<0 
... ... 
 polarization bi-spinor
 
w     spinor in the rest frame
 
1  0
1
1

w    ,    Sz  , 
2 2
 0 1
Bessel beams for Dirac electron
 s p  W s  0  ei 
 spectrum
  r   1  D / 2 J  r   D J
s
2
2
2 s
r  / 2
s
s
E(k )
 spin-dependent intensity
s  1/ 2
s  1/ 2
4
 m 2
D  1   sin 0  
 E
1
 SOI strength
Bliokh, Dennis, Nori, arXiv:1105.0331
Angular momentum
Uˆ FW  p  Foldy-Wouthhuysen
Pˆ 
transformation
 projector onto E>0 subspace


ˆ   Pˆ  Uˆ † O
ˆ Uˆ
O
FW
FW :
ˆ, L
ˆ
ˆ L
ˆ Δ
Sˆ   Sˆ  Δ
rˆ   ik  Aˆ
Lz   sD, Sz  s 1  D 
 ... ...
 E>0
 ... ...

... ...
 cross terms
 ... ...
... ... 
cross
terms
... ...


... ... 
E<0 
... ... 
ˆ   Aˆ  p
Δ
p  σˆ  m 
ˆ
A 
1 
2 
2p  E 
 OAM and SAM for
Bessel beams
Bliokh, Dennis, Nori, arXiv:1105.0331
Magnetic moment for Dirac electron
e
M   r  j dV
2
j   †αˆ   p / E
 magnetic moment from current
ˆ  ,
M  M

e
ˆ
M  rˆ  αˆ
2  operator !


e ˆ

†
ˆ
ˆ
ˆ
ˆ
ˆ
M  P U FW MU FW 
L  2Sˆ 
2E

e
M
  2s  D 
2E

 final result
slightly differs from Gosselin et al. (2008); Chuu, Chang, Niu, (2010)
• 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.
THANK YOU!