Light in a Twist: Orbital Angular Momentum - Miles Padgett Kelvin Chair of Natural Philosophy
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Light in a Twist: Orbital Angular Momentum Miles Padgett Kelvin Chair of Natural Philosophy
The talk today • Orbital Angular Momentum, what is it? • What has been done with OAM • A couple of example of what we have done and doing!
A question • A photon carries a spin angular momentum of • So how does a multi-pole transition (ΔJ > ) conserve angular momentum?
Linear momentum at a radius exerts a torque
k
r
k x r-> multipole transition
Providing the lever is long
enough, a fixed linear
momentum can exert an
arbitrary high torque
Getting started on Orbital Angular Momentum of Light
• 1992, Allen, Beijersbergen, Spreeuw and Woerdman
Miles Padgett
• 1994, Les meets Miles at dinner…...
Orbital Angular Momentum from helical phase fronts
r"
E"
z"
p!
B" p
E"
z"
B"
pθ = 0 pθ ≠ 0
Angular momentum in terms of photons
• Spin angular momentum
– Circular polarisation σ = +1!
σ per photon
• Orbital angular momentum
– Helical phasefronts σ = -1!
per photon
= 0! = 1! = 2! = 3! etc!
Optical vortices, Helical phasefronts , Angular momentum
• Intensity, I ≥0
= 0!
• Phase, 2π ≥ φ ≥0
= 0, plane wave
= 1, helical wave = 1!
= 2, double helix
Interference +/- #
= 3, pasta fusilli
etc.
= 2!
= 3!
= vortex charge!
I! φ& I!
Orbital angular momentum from Skew rays
Poynting vector
Making helical phasefronts with holograms
Making OR measuring phasefronts with holograms
Make interactive by using SLM
Switching time Generate
≈5-20mSec
Efficiency ≈50%
Light source
OR detector MeasureRichard Bowman A gift for all the family…..
And the point of shaping the spot is……
A double-start helix (=2) Chambord castle (chateaux de la Loire)
OAM in optical manipulation
He et al. PRL 1995
Curtis et al. Opt Commun. 2002OAM in quantum optics Mair et al. Nature 2001
OAM in imaging
Fürhapter et al. Opt. Lett. 2005
Swartzlander et al. Opt. Express 2008OAM in communication
Tamburini et al. New J Phys. 2012
Wang et al. Nature Photon 2012OAM in not just light
Volke-Sepulveda et al. PRL 2008
Verbeeck et al. Nature 2010The OAM communicator
Optical Vortices before Angular Momentum
Fractality and Topology of Light’s darkness Kevin O’Holleran Florian Flossmann Mark Dennis (Bristol)
Vortices are ubiquitous in nature
• Whenever three (or more) 2π
plane waves interfere optical
vortices are formed
– Charge one vortices occur 2π
wherever there is diffraction
or scattering
2π
0Map out the vortex position in different planes
• Either numerically or
experimentally one
can map the vortex
positions in different
planesThe tangled web of speckle • ≈1600 plane waves c.f. Gaussian speckle
Entanglement of OAM states
Quantum entanglement with spatial light modulators Jonathan Leach Barry Jack Sonja Franke-Arnold (Glasgow) Steve Barnett and Alison Yao (Strathclyde) Bob Boyd Anand Jha (Rochester)
OAM in second harmonic generation
• Poynting vector “cork screws”,
azimuthal skew angle is 1 green
θ = /kr photon!
=20!
• Does this upset a co-linear “cork screwing”
phase match? -No Poynting vector!
• Frequency & -index both
double
• “Path” of Poynting vector stays
the same
– phase matching maintained
2 infra red
photons!
= 0!Correlations in angular momentum
+10
s
-10
-10 +10
π i
Near perfect (anti)
Prob.
Correlations in angular Δ
momentum
-20 s -i +20Correlations in angle
+π
φs
π
π φi +π
Near perfect
Prob.
Correlations in angle Δφ
π φs -φi +πAngular EPR
Correlations in complimentary basis sets
-> demonstrates EPR for Angle and Angular momentum
2 2
[ ][
Δ( s i ) Δ(φ s φ i ) ] = 0.00475 2Entanglement of OAM states
Optical Activity /Faraday effect for OAM Sonja Franke-Arnold Graham Gibson Emma Wisniewski-Barker Bob Boyd
Poincaré Sphere! Poincaré Sphere for OAM!
The (Magnetic) Faraday Effect
• Rotation of plane
polarised light L"
Δθ pol = BLV
• V Verdet constant
• OR treat as phase
delay of circularly B"
€ polarised light
Δφ = σBLV Δθ = Δφ +σ ,−σ Δσ
But the magnetic Faraday effect does NOT rotate an Image• Photon drag, gives
Polarisation rotation
& ΩL
Δθ =
c
( )
n g − 1 nφ
σΩL
Δφ =
c
(
n g − 1 nφ )
€
• Mechanical Faraday
Effect
ۥ SAM -> Polarisation
rotation
• OAM-> Image rotation
• Look through a Faraday
isolator (Δθ≈45°), is the ?
“world” rotated - NO
– SAM and OAM are not
equivalent in the
Magnetic Faraday effect
– SAM and OAM are not
?
equivalent in the optical
activitiyEnhancing the effect…..
• Plug in “sensible
numbers” and get a
micro-radian rotation… ΩL
• Increase the group
Δθ image =
c
(
n g − 1 nφ )
index to enhance the
effect
ۥ Shine an elliptical laser
beam (≈LG, Δ=2) @
532nm through a
spinning Ruby bar.
Camera Laser
Spinning rod of ruby≈25Hz clockwise anticlockwise
A beam splitter for OAM
Martin Lavery
Gregorius Berkhout (Leiden)
Johannes CourtialAngular momentum
• Spin angular momentum
– Circular polarisation σ = +1!
σ per photon
• Orbital angular momentum
– Helical phasefronts σ = -1!
per photon
= 0! = 1! = 2! = 3! etc!Measuring spin AM
• Polarising beam
splitter give the
“perfect” separation of
orthogonal (linear)
states
– Use quarter waveplate
to separate circular
states
– Works for classical
beams AND single
photonsMeasuring Orbital AM
1
• OAM beam splitter
give the “perfect” 2
separation of
3
orthogonal states
– But how?
4
5It works for plane waves • A “plane-wave” is focused by a lens • A phase ramp of 2π displaces the spot
It works for plane waves
• Image transformation
φ -> x and r -> y
– i.e. Lz -> px
xReplacing the SLMs
• The principle works
• But the SLMs are
inefficient (≈50% x 2)
• Use bespoke optical
elements (plastic)
– Prof. David J
Robertson reformater phase corrector
– Prof. Gordon LoveDoughnut to hot-dog
• The principle works
• But the SLMs are
inefficient (≈50% x 2) Annulus
• Use bespoke optical Line
Reforma(er)
elements (glass/
plastic)
– Prof. David J
Robertson Corrector)
– Prof. Gordon LoveThe output
Evolution in time c.f. translation and rotation Φ = f(kz+ωt) Φ = f(kz+ωt+θ) = 0 time c.f. translation > 0 time c.f. rotation
Linear vs. Rotational Doppler shifts
Light source! Light source!
Beam of light!
Ω!
v!Doppler shift from a translating surface
θ& Δω= ω0cosθ v/c
v
But for rotation and OAM
Δω= Ω #
v = Ωr and θ = /krMartin Lavery
Rotational Frequency shifts
Scattered Light
Observe frequency shift
between +/- #
components
Illuminate#
DetectorRotational Doppler shift in scattered light
Δω = 2 Ω&
c.f. Speckle velocimetry? Albeit, in this case, angularMore talks in 238
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