The Twist in Light’s Tail FRS FRSE Kelvin Chair of Natural Philosophy

follow me on Twitter @MilesPadgett Light beams carry a Linear Momentum

• Light carries a linear momentum – Force exerted = Power/c

• Laser pointer exerts 10-11 N

• Strong enough to move cells and bacterial

• Or (with a very large sail!) a spacecraft…. Light’s Angular Momentum in History

We now refer to this as the SPIN angular momentum of the photon

But light has an extra angular momentum in addition to photon spin (this extra momentum is needed for some rare atomic transitions) 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

E E

B B S S

Sθ = 0 Sθ ≠ 0 Helical phasefronts and orbtial angular momentum

• Phase cross-section ℓ= 0 Φ = exp (iℓφ) ℓ = 0, plane wave ℓ = 1, helical wave ℓ = 2, double helix ℓ= 1 ℓ = 3, pasta fusilli etc. 2π

ℓ= 2

Phase 0 x-section of helical wave ℓ = 3

Intensity Phase Angular momentum in terms of photons

• Spin angular momentum Polarisation

σ = +1 Circular – Circular polarisation σ" per photon • Orbital angular momentum

– Helical phasefronts σ = -1 ℓ" per photon

Helical Phasefronts

ℓ = -1 ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 A double-start helix (ℓ=2): Home improvments

Chambord castle (chateaux de la Loire) Making helical phasefronts

• Pass plane-wave through a spiral- phase plate (thickness α φ) – step height= ℓλ/(n-1) Designing helical phase hologram

! Spiral Phase-plate s= ℓλ/(n-1)

s 2π • Holographically + = e.g. ℓ = 1 ▹ 0 Making helical phasefronts with diffraction gratings Making OR measuring phasefronts with holograms

Make interactive by using SLM

Switching time Generate ≈5-20mSec Efficiency ≈50%

Light source OR detector Measure Richard Bowman

A gift for all the family….. Orbital angular momentum from Skew rays

Poynting vector

E

B S Orbital angular momentum from skew rays Angular Momentum Transfer

Spin Angular Momentum Orbital Angular Momentum (circular angular momentum) (helical phasefronts) The OAM in……

OAM in Quantum e.g. Zeilinger OAM in Communication OAM in tweezers OAM in Quatum Optics e.g. Willner e.g. Rubinsztein-Dunlop e.g. Zeilinger

OAM in Imaging e.g. Ritsch-Marte OAM in Lasers OAM in NOT optical e.g. Forbes e.g. Verbeeck OAM and SLM control OAM in Phenomenology in micromanipulation

OAM in Quantum OAM in Imaging OAM in Communications Optics Fractality and Topology of Light’s darkness

Kevin O’Holleran Florian Flossmann

Mark Dennis (Bristol)

Vortices are ubiquitous in

• Whenever three (or more) 2π plane waves interfere optical vortices are formed – Charge one vortices occur 2π wherever there is diffraction or scattering

2π

0 Map out the vortex position in different planes

• Either numerically or experimentally one can map the vortex positions in different planes The tangled web of speckle

• ≈1600 plane waves c.f. Gaussian speckle 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 = ℓ/kr photon ℓ =2ℓ • Does this upset a co-linear “cork screwing” 0 phase match? -No Poynting vector • Frequency & ℓ-index both double • “Path” of Poynting vector stays the same – phase matching maintained

2 infra red photons

ℓ = ℓ0 Correlations in angular momentum

+10

ℓs

-10 -10 +10 π ℓi

Near perfect (anti) Correlations in angular Δℓ momentum Prob.

-20 ℓs -ℓi +20 Correlations in angle

+π

φs

π π φi +π

Near perfect Correlations in angle Δφ Prob.

π φs -φi +π Angular EPR

Correlations in complimentary basis sets

2 2 Δ    Δ φ φ = 0.004752 << 0.252 [ ( s i) ] [ ( s i)]

Assuming we believe QM to be complete we have to accept that it is non-local

€- > EPR for angular momentum and angles! The OAM communicator

• Every OAM state forms an independent channel

Rotational Doppler Shifts

Martin Lavery

Steve Barnett and Fiona Speirits Linear vs. Rotational Doppler shifts

Light source Light source - exerts force = P/c - exerts torque = ℓP/ω

Ω v

Doing work on a light beam changes its energy and hence shifts it’s frequency Doppler shift from a SPINNING surface

N photons per second

Torque= ℓN"ω/ω

Work Done= ΩNℓ"

Work per photon Ω≠0 "Δω = Ωℓ"

Frequency shift Δω = Ωℓ

v=0 Illuminate with OAM at +/- ℓ and measure Δω

A B 0

L -5 dB

H -10 -15

Power -20 . -25 Rel -30 2200 2250 2300 2350 2400 2450 2500

fmod Hz C 500 H L 450 L

1 400 - s . 350 rad H 300 W 250 200 1200 1400 1600 1800 2000 2200 2400 2600

fmod Hz

H L Conclusions

• Beyond the field of Orbital Angular Momentum itself, the realisation that the Poynting vector can be engineered over the beam cross- section has triggered research into many beam types, and light-matter interactions in both the classical and quantum regimes.

• Better communication, better sensing, better imaging. Richard Bowman Students/RAs, Graham Gibson especially Martin Lavery Jonathan Leach Kevin O’Holleran Me Sonja Franke-Arnold Johannes Courtial Jackie Romero Neil Simpson

Collaborators Collaborators Les Allen Steve Barnett External Bob Boyd Mark Dennis Mervyn Miles Monika Ritsch-Marte Wilson Sibbett

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