Optical Angular Momentum
Laguerre-Gaussian Modes of Laser Light
Dimitri Dounas-Frazer
drdf@berkeley.edu
Department of Physics
University of California, Berkeley
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Overview
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Overview
Beth’s 1936 experiment
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Overview
Beth’s 1936 experiment
Laguerre-Gaussian (LG`p ) modes
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Overview
Beth’s 1936 experiment
Laguerre-Gaussian (LG`p ) modes
Production of LG`p modes
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Overview
Beth’s 1936 experiment
Laguerre-Gaussian (LG`p ) modes
Production of LG`p modes
Circularly polarized LG`p beams
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Beth’s 1936 experiment
Each photon in a σ± beam
carries ±~ spin angular
momentum
Birefringent plate transforms
σ− beam to σ+
Plate experiences torque
Both classical and quantum theories predict same value
R. A. Beth, Phys. Rev. 50, 115 (1936)
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Laguerre-Gaussian
u`p (r, φ)
∝r
`
`
(LGp
L`p (2r2 /w2 )
e
) modes
−r 2 /w2
e−i`φ
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Laguerre-Gaussian
u`p (r, φ)
∝r
`
`
(LGp
L`p (2r2 /w2 )
e
) modes
−r 2 /w2
e−i`φ
Laguerre polynomial with p + 1 radial nodes
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Laguerre-Gaussian
u`p (r, φ)
∝r
`
`
(LGp
L`p (2r2 /w2 )
e
) modes
−r 2 /w2
e−i`φ
Laguerre polynomial with p + 1 radial nodes
Gaussian with beam waist w
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Laguerre-Gaussian
u`p (r, φ)
∝r
`
`
(LGp
L`p (2r2 /w2 )
e
) modes
−r 2 /w2
e−i`φ
Laguerre polynomial with p + 1 radial nodes
Gaussian with beam waist w
Vortex with topological charge `
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Laguerre-Gaussian
u`p (r, φ)
∝r
`
`
(LGp
L`p (2r2 /w2 )
e
) modes
−r 2 /w2
e−i`φ
Laguerre polynomial with p + 1 radial nodes
Gaussian with beam waist w
Vortex with topological charge `
Orbital angular momentum `~ per photon
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Laguerre-Gaussian
u`p (r, φ)
∝r
`
`
(LGp
L`p (2r2 /w2 )
e
) modes
−r 2 /w2
e−i`φ
Laguerre polynomial with p + 1 radial nodes
Gaussian with beam waist w
Vortex with topological charge `
Orbital angular momentum `~ per photon
Cylindrically symmetric solutions to the wave
equation
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`
LGp
modes cont.
|u10 |2
|u50 |2
|u21 |2
|u32 |2
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Production of
`
LGp
modes
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Production of
`
LGp
modes
Conversion from Hermite-Gaussian (HGnm ) modes
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Production of
`
LGp
modes
Conversion from Hermite-Gaussian (HGnm ) modes
Spiral phase plates
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Production of
`
LGp
modes
Conversion from Hermite-Gaussian (HGnm ) modes
Spiral phase plates
Holographic gratings
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Production of
`
LGp
modes
Conversion from Hermite-Gaussian (HGnm ) modes
Spiral phase plates
Holographic gratings
Diffractive optics
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Conversion of HGnm to
`
LGp
modes
Cylindrical lens converter:
from HGnm to LG`p
` = m − n and p = min(m, n)
Introduces a Guoy phase shift
on incident beam
Combination of two lenses will
experience torque
For most lasers, high-order
HGnm modes are impractical
L. Allen et al., Phys. Rev. A 45, 8185 (1992)
J. Courtial and M.J. Padgett, Optics Comm. 159, 13 (1999)
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Spiral phase plates
Azimuthally dependent phase delay
Converts HG00 to LG`p mode with charge ` = hs (n − n0 )λ
Converts between any two LG`p modes
G.A. Turnbull et al., Optics Comm. 127, 183 (1996)
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Holographic gratings
Helical mode ∝ ei`φ
Topological charge ` = 40
Particle travels around
circumference of vortex
Radius of orbit R` ∝ `
Photons contribute `~ to angular momentum flux
T` = R`3 /(` × power) ∝ `2 /power
J.E. Curtis and D.G. Grier, Phys. Rev. Lett. 90, 133901 (2003)
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Diffractive optics
Etch small scale structures on
optical elements
Use two diffractive optics to
control intensity and phase
Converts HG00 to arbitrary LG`p
mode
High mode purity
S.A. Kennedy et al., Phys. Rev. A 66, 043801 (2002)
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Circularly polarized
`
LGp
modes
Shine a He-Ne laser on CuO particles in a viscous fluid
Phase hologram produces LG03 mode
Particles experience torque τ ∝ ` + σz
... and a drag torque proportional to angular velocity
Measure equilibrium angular velocity for σz = -1, 0, +1
M.E. Friese et al., Phys. Rev. A 54, 1593 (1996)
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Spin-to-orbital conversion
Anisotropic media transfer spin angular momentum to matter
(e.g., birefringent media and liquid crystals)
Inhomogeneous isotropic media transfer orbital angular
momentum
“q-plates” are both anisotropic and inhomogeneous
Optical axis orientation specified by α(r, φ) = qφ + α0
L. Marrucci et al., Phys. Rev. Lett. 96, 163905 (2006)
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Spin-to-orbital cont.
Jones matrix for the q-plate:




cos 2α sin 2α
1 0

 · R(α) = 
M = R(−α) · 
sin 2α − cos 2α
0 −1
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Spin-to-orbital cont.
Jones matrix for the q-plate:




cos 2α sin 2α
1 0

 · R(α) = 
M = R(−α) · 
sin 2α − cos 2α
0 −1
Incoming is left-circularly polarized plane wave.
Outgoing wave is right-circularly polarized with charge ` = 2q:

 

1
1
i2qφ 



Eout = M ·
∝e
i
−i
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Spin-to-orbital cont.
Jones matrix for the q-plate:




cos 2α sin 2α
1 0

 · R(α) = 
M = R(−α) · 
sin 2α − cos 2α
0 −1
Incoming is left-circularly polarized plane wave.
Outgoing wave is right-circularly polarized with charge ` = 2q:

 

1
1
i2qφ 



Eout = M ·
∝e
i
−i
Each photon exchanges ±2~(q − 1) angular momentum with plate.
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Summary
Circularly polarized light has σz ~ angular momentum per photon
Laguerre-Gaussian modes have `~ angular momentum per photon
Beams with angular momentum can make small things rotate
Fancy materials can convert spin to orbital angular momentum
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