The Angular Momentum of Gauge Fields:
The Case of Twisted Photons
Andrei Afanasev
The George Washington University
Washington, DC
QCD Evolution Workshop
Jefferson Lab, Newport News, VA
May 9, 2012
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
Objectives
.
.
Consider Abelian gauge fields (QED)
Discuss the properties of of photon beams with a large angular
momentum (>ħ) projection on the direction of propagation
. Generation
. Formalism
. Absorption by atoms
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
Introduction
. Photons carry linear momentum p=ħk (k=wave vector)
. Photons carry both spin angular momentum (SAM) and orbital angular
angular momentum (OAM) – may be separated in paraxial approximation
. Circularly polarized plane-wave photons carry Jz=±ħ along the
propagation direction z (Beth’s experiment, 1936)
. Heitler, Quantum Theory of Radiation (1954): larger Jz possible if the
EM wave is constrained in the transverse plane (cylindrical waves)
. Spherical waves: expansion in terms of angular momentum
eigenfunctions, position dependence of vector potential Aμ(x) contains
OAM information
. Beams of light with azimuthal beam dependence exp(ilϕ) (e.g,
Laguerre-Gaussian modes) can carry large values of OAM (Allen et
al, 1992).
Review: Yao, Padgett, Advances in Optics and Photonics 3, 161–204 (2011)
and references therein
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
Orbital vs Spin Angular Momentum (from Yao’11 review)
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Twisted Photons
. Quantization of light beams having azimuthal phase dependence
exp(ilϕ) lead to a concept of twisted photons
G. Molina-Terriza, J.Torres, L. Torner, “Twisted Photons”, Nature
Physics, May 2007.
The typical transverse intensity pattern of a light beam with orbital angular momentum, (a)
theory (b) experiment. The light beam exhibits a dark spot in the center, and a ring-like
intensity profile. (c) Azimuthal dependence of beam phase results in a helical wavefront.
(d) Orientation of the local momentum of the beam has a vortex pattern (hence another
name, an optical vortex).
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
Generation of Light Beams with
Orbital Angular Momentum
.
A diffraction grating with fork dislocation centered
on the beam axis, could convert the fundamental
Gaussian mode from any laser into a helically
phased mode [V. Bazhenov, M. V. Vasnetsov, and
M. S. Soskin, “Laser-beams with screw dislocations
in their wave-fronts,” JETP. Lett. 52, 429–431
(1990).] Commonly accepted method for producing
helically phased beams.
.
Spiral Phase Plates: Gaussian beam is
passed through optical media, with azimuthal
dependence in thickness
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
Generation of Twisted Photons with Helical Undulators
.
.
E. Hemsing, A. Marinelli, and J. B. Rosenzweig, “Generating Optical
Orbital Angular Momentum in a High-Gain Free-Electron Laser at the
First Harmonic,” Phys. Rev. Lett. 106, 164803 (2011).
AA, Mikhailichenko, On Generation of Photons Carrying Orbital
Angular Momentum in the Helical Undulator, E-print: arXiv
1109.1603
. Considered properties of synchrotron radiation by charged
particles passing through a helical undulator. Shown that all
harmonics higher than the first one radiated in a helical undulator
carry OAM. Large K-factors favor large values of OAM for
generated radiation.
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
Helical Undulators (cont)
. AA, Mikhailichenko, On Generation of Photons Carrying Orbital
Angular Momentum in the Helical Undulator, E-print: arXiv
1109.1603
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
Transfer of Angular Momentum
. For optical wavelengths, transfer of Orbital AM differs from Spin AM.
.
Important: wavelength vs the target size
. Wavelength >> target size: OAM transfer results in linear
momentum of the target as a whole
. Wavelength < target size: OAM results in target rotation
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
Absorption of Twisted Photons
.
Translation or rotation?
. Mechanism depends on the dimensions of target vs wavefront
cross section S
. Depending on a topological number and pitch angle, wavefront
cross section may be from a few to a few hundred wavelengths
. If the target size is of the order S or larger, an essential fraction of
photon energy is transferred to rotation of the target.
. For smaller targets, rotation is more likely is the target is next
the center of the optical vortex
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
Atomic Excitations with High-L Photons
.
Twisted photons may enhance (relatively) atomic transitions with large
transfer of angular momentum (Picon et al, 2010; AA, Carlson,
Mukherjee, arXiv:1304.0115).
. An atom must be close (relative to wavelength) to the beam axis,
but there is a dip in intensity there. Result: the probability to excite
high-L atomic levels is suppressed
. Reason: optical wavelength >> atomic size, atomic transitions are
caused by long-wavelength photons because the bound electrons
are non-relativistic
. Situation will change in hadronic physics: need photons with
wavelength < fm to excite a nucleon.
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Twisted Photon State
.
Use plane-wave expansion
.
Plane wave:
.
Twisted wave:
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Fields of the Twisted Wave
. Vector potential
.
Poynting vector
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Magnetic field
Transverse Beam Profile
. OAM light beam is characterized with a special transverse profile
(example from AA, Carlson, Mukherjee,)
. Intensity dip on the beam axis, with transverse size > wavelength
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
Atomic Photoexcitation
. Interaction Hamiltonian:
.
Matrix element of the transition:
.
b- an impact parameter w.r.t. the atomic center
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Matrix Element of Photoexcitation
. Photo-excite a hydrogen atom from the ground to the state with a
.
principal quantum number nf, OAM lf, and OAM projection mf.
Incoming twisted photon is defined by AM projection mγ, energy ω
and a pitch angle θk (with κ=kperp)
Matrix element:
.
Atomic factors
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Calculation Results
. Matrix elements as a function of an impact parameter b
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Helicity Asymmetry
.
.
.
.
.
Flip photon helicity Λ, keep the OAM projection the same =>
. Results a different twisted photon state with mγmγ-2Λ
Photoabsorption cross sections are different for a given impact parameter =>
(parity-conserving) helicity asymmetry
The largest asymmetry is near the center of optical vortex
Asymmetry is zero after averaging over the impact parameter b
May be observed for small-size targets or near-field geometry
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
Twisted Photons for Hadrons vs Atoms
.
For atomic transitions photon OAM is preferably transferred to internal
degrees of freedom if the target is near the center of an optical vortex
. Otherwise OAM results in linear momentum of the entire atom
.
For excitation of a baryon with a twisted photon, γTN->N* OAM will
be passed to internal degrees of freedom and will help to get insight
into nucleon structure
. Will need more helicity amplitudes to describe a baryon resonance
excitation
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
How to Generate Twisted Photons in MeV-GeV?
.
Serbo et al proposal (2010): Compton backscattering.
U.D. Jentschura V.G. Serbo
Generation of High-Energy Photons with Large Orbital Angular Momentum by
Compton Backscattering.
Phys.Rev.Lett. 103 (2011) 013001, e-Print: arXiv:1008.4788;
Compton Upconversion of Twisted Photons: Backscattering of Particles with NonPlanar Wave Functions. Eur.Phys.J. C71 (2011) 1571, arXiv:1101.1206.
. Theoretically demonstrated that OAM properties of twisted photons are
preserved in Compton backscattering.
. If holds, it provides a new tool in nuclear, hadronic and high-energy physics,
that are photon beams with pre-selected OAM along their direction of
propagation.
QCD Evolution Workshop, Jefferson Lab, May 9, 2013
Summary
. Twisted photons carry large orbital angular momenta along the axis of
.
.
propagation
Can be applied at widely different scales, from dust particles, to
nanoparticles, molecules, (Rydberg) atoms, and nuclei
. Atomic photoexcitation considered here
. Results in excitation of states with a range of quantum numbers,
different from plane waves
. Predicted parity-conserving helicity asymmetry in the central
region of an optical vortex: flipping the helicity results in a
different photon state
Accelerator-based light source are most efficient for generating twisted
X-rays and gamma-rays
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Outlook
.
Is it possible to generate twisted photons with GeV energies to probe
the structure of hadrons?
.
Are the cross sections or spin asymmetries of basic QED processes
involving twisted photons affected by their additional angular
momentum?
.
Can (inelastic) polarized electron scattering on a twisted photon
provide an information on its angular momentum? (=polarization
structure functions of a twisted photon, TMDs, GPDs, etc)
.
Extension to non-Abelian fields=> “twisted gluons”
QCD Evolution Workshop, Jefferson Lab, May 9, 2013