Topological defects, geometric phases, and the angular momentum of light
Topological defects, geometric phases, and the angular momentum of light
S. C. Tiwari
Institute of Natural Philosophy
c/o 1 Kusum Kutir Mahamanapuri,Varanasi 221005, India
Recent reports on the intriguing features of vector vortex bearing beams are analyzed using
geometric phases in optics. It is argued that the spin redirection phase induced circular birefringence
is the origin of topological phase singularities arising in the inhomogeneous polarization patterns.
A unified picture of recent results is presented based on this proposition. Angular momentum
shift within the light beam (OAM) has exact equivalence with the angular momentum holonomy
associated with the geometric phase consistent with our conjecture.
PACS numbers: 42.25.-p, 41.20.Jb, 03.65.Vf
Topological defects in continuous field theoretic frame-
work are usually associated with field singularities, how-
ever in analogy with crystal defects wavefront disloca-
tions for scalar waves [1] and disclinations for vector
waves [2] have been discussed in the literature. An im-
portant advancement was the realization that topological
charge was related with the orbital angular momentum
(OAM) of finite sized (transverse) light beams: typically
for the Laguerre-Gaussian (LG) beams helicoidal spatial
structure of the wavefront with azimuthal phase exp(ilφ)
gives rise to OAM per photon of lh̄ where l is the topolog-
ical charge, see review [3]. Adopting the fluid dynamical
paradigm topological defects in optics are termed vor-
tices; singularities in the polarization patterns are called
vector vortices [4].
The aim of this Letter is to present a unified picture
of the underlying physics of intriguing aspects of recent
reports [5, 6, 7] in terms of the transformation of topo-
logical charges due to spin redirection phase (SRP) such
that OAM is exchanged within the beams [8]. The role
of Pancharatnam phase (PP) invoked in [4, 6, 7] is also
critically examined.
For the sake of clarity we briefly review the essentials
of geometric phases (GP) in optics which are primarily
of two types, see [8, 9] for details and original references.
Rytov-Vladimirskii phase rediscovered by Chiao and Wu
in 1986 (inspired by the Berry phase) arises in the wave
vector or momentum space of light. The unit wave vector
κ and polarization vector ǫ(κ) describe the intrinsic prop-
erties of the light wave. A plane wave propagating along
a slowly varying path in the real space can be mapped on
to the surface of a unit sphere in the wave vector space,
and under parallel transport along a curve in this space
preserving the spin helicity, ǫ.κ, the polarization vector is
found to be rotated after the completion of a closed cycle
on the sphere. The magnitude of the rotation is given
by the solid angle enclosed by the cycle, and the sign is
determined by the initial polarization state. Since left
circular |L > and right circular |R > polarization states
acquire equal but opposite geometric phases, Berry terms
this effect as geometric circular birefringence [10].
A polarized light wave propagating in a fixed direction
passed through optical elements traversing a polariza-
tion cycle on the Poincare sphere acquires Pancharatnam
phase equal to half of the solid angle of the cycle. Berry
points out that [11] Pancharatnam actually made two
important contributions. One,a notion of Pancharatnam
connection was introduced for the phase difference be-
tween two arbitrary nonorthogonal polarizations which
can be written as arg(E1
∗.E2) for complex electric field
vectors. Secondly this connection is nontransitive result-
ing into the Pancharatnam phase for a geodesic triangle
on the sphere. Note that a parallel transport on the
Poincare sphere is made with fixed direction of propaga-
tion for the occurrence of PP.
In the case of space varying polarization patterns,
defining a direction of propagation is not easy, however
Nye [12] has used Pancharatnam connection to define
propagation vector kδ as a gradient of phase difference
between fields at spatial locations r1 and r2 given by
dδ = Im(E∗.dE)/|E|2 (1)
In [4] authors correctly use Pancharatnam connection to
obtain the phase difference of light at two locations in
the space varying polarization plane, however the GP
involved is not Pancharatnam phase as one cannot com-
plete polarization cycle without changing the wave vec-
tor direction. As discussed above we have to construct
appropriate wave vector space for the case of vector vor-
tices. For an initial beam propagating along z-axis, at
each point (r, φ) on the inhomogeneous polarization plane
there will correspond a k-space, and spin helicity preserv-
ing parallel transport will give SRP for closed cycles. It
is known that for a linearly polarized plane wave repre-
sented by
|P >= e−iα|R > +eiα|L > (2)
the SRP corresponding to a cycle with solid angle Ω re-
sults into [10]
|P t >= e
−i(α+Ω)|R > +ei(α+Ω)|L > (3)
For the optical vortices this equation has to be general-
ized: we suggest a spatially evolving GP embodied in the
solid angle as a function of (r, φ). This is one of the main
contributions of this Letter leading to Eq.(4) below. For
http://arxiv.org/abs/0704.0137v1
the special case in which only the azimuthal angle de-
pendence matters we obtain Ω in the following way. In
the transverse plane consider a point A on a circle spec-
ified by φ, then the area enclosed by the great circles
in k-space corresponding to this point and the reference
point O specified by φ = 0 would subtend a solid an-
gle which varies linearly with φ; the solid angle will vary
from 0 to 4π as φ varies from 0 to 2π. Thus we obtain
the generalization of Eq.(3) to
|P v >= e
−i(α+2φ)|R > +ei(α+2φ|L > (4)
Spatially evolving SRP [13] is crucial to understand in-
teresting features observerd in vector vortices; we state
our second principal contribution in the form of a propo-
sition.
Proposition: Geometric phase induced circular bire-
fringence is the origin of topological charge transforma-
tion in vector vortex carrying beams, and angular mo-
mentum holonomy is manifested as OAM.
We demonstrate in the following that this proposition
offers transparent physical mechanism to explain the re-
cent reports on inhomogeneous polarization patterns.
Backscattered polarization [5] : Theory and experi-
ments on the backscattered light for linearly polarized
light from random media have been of current interest.
Interesting features for the backscattering geometry have
been observed. Authors of [5] give insightful treatment
of the observations invoking GP in wave vector space,
and this is in agreement with our proposition. Note that
backscattered light wave vector could be treated similar
to the discrete transformation for reflection from a mir-
ror, see discussion in [9] one can envisage an adiabatic
path in a modified k-space. It may be noted that essen-
tially spatially evolving SRP is used in [5]. It seems the
term ’geometrical phase vortex’ introduced by them for
scalar vortices appearing in space varying polarization
pattern is quite revealing.
The q-plate experiment [6]: An inhomogeneous
anisotropic optical element called q-plate has novel addi-
tion to HWP : inhomogeneity is introduced orienting the
fast (or slow) optical axis making an angle of α with
the x-axis in the xy-plane for a planar slab given by
α(r, φ) = qφ+α0. Jones calculus applied at each point of
the q-plate shows that the output beam for an incident
|L > state is not only converted to |R > state but it also
acquires an azimuthal phase factor of exp(2iqφ). Simi-
lar to the LG beams this phase is interpreted as OAM
of 2qh̄ per photon in the output wave. Experiment is
carried out using nematic liquid crystal planar cell for
q = 1 and the measurements on the interference pattern
formed by the superposition of the output beam with the
reference beam display the wave front singularities and
helical modes in the output beam.
We argue that in the light of our proposition SRP in-
duced circular birefringence is the origin of topological
phase singularity and OAM in q-plates. We picture he-
licity preserving transformation in the wave vector space
defined by kδ. Since the polarization variation is confined
in the transverse plane for the q-plates the constraint of
the spin helicity preserving process in the q-plate with
the azimuthal dependence of α would lead to a spiral
path for the wave vector. In q=1 plate the circular plus
linear propagation along z-axis will result into a helical
path and the width of the plate ensures that the input
and output ends of the helix are parallel. On the unit
sphere in the wave vector space this will correspond to a
great circle, and the solid angle would be 2π. Since the
SRP equals the solid angle for the evolving paths, our
Eq.(4) above is in agreement with Eq.(3) of [6]. The im-
portant observation emphasized by the authors that the
incident polarization controls the sign of the orbital helic-
ity or topological charge is easily explained in view of the
property of the geometric birefingence in which handed-
ness decides the sign of the phase. Thus both magnitude
and sign of the azimuthal phase have been explained in
accordance with our proposition.
Tightly focused beams [7]: Analysis of the light field
at the focal plane of a high numerical aperture lens for
the incoming circularly polarized plane wave shows the
existence of inhomogeneous polarization pattern. Pan-
charatnam connection at two different points on the cir-
cle around the focus shows φ-dependence of the phase
difference. The field can be decomposed into |L > and
|R > states, Eq.(7) in [7], and it is found that the compo-
nent with the spin same as that in the object plane does
not change phase while the one with opposite spin ac-
quires an azimuthal phase of 2φ, i. e. topological charge
2 and OAM of 2h̄ per photon. Application of Eq. (4)
immediately leads to this result in conformity with our
proposition. We may remark that the construction used
by the authors to derive PP, namely the geodesic trian-
gle on the Poincare sphere formed by the pole, E(r, 0),
and E(r, φ) cannot be completed with a fixed direction of
propagation for space varying polarization pattern, and
therefore it is SRP not PP that arises.
Having established first part of the proposition, we dis-
cuss the role of angular momentum (AM) holonomy con-
jecture [8, 9]. Transfer of spin AM of light to matter
was measured long ago by Beth [14], and there are many
reports of OAM transfer to particles in recent years [3].
Since polarization cycle for PP requires spin exchange
with optical elements, it is natural to envisage a role of
AM in GP; however it would be trivial. In the AM holon-
omy conjecture, we argued AM level shifts within the
light beams as physical mechanism for GP. This implies
exact equivalence between AM shift and GP. Indirectly
the backscattered light experiments and their interpre-
tation in terms of SRP supports our conjecture. In the
context of the AM conservation [5] the redistribution of
total AM within the beam is also indicative of AM level
shifts. The q=1 plate is a special optical element in which
no transfer of AM to the crystal takes place, and total
AM is conserved within the light beam. We argue that
spin is intrinsic, and the OAM is a manifestation of the
GP with exact equivalence between them in this case.
The counter-intuitive interpretation in terms of spin to
OAM conversion claimed in [6] is clearly ruled out. In
fact, Marrucci et al experiment offers first direct evidence
in support of our conjecture. It is remarkable that the
light field structure calculated for tightly focused beams
shows strong resemblence with the action of q-plates on
light wave, and offers another setting to test our propo-
sition.
To conclude the Letter we make few observations. First
let us note that even without the existence of phase singu-
larities it should be possible to exchange AM within the
light beam accompanied with GP: as argued earlier trans-
verse shifts in the beam would account for the change in
OAM [9]. Secondly the interplay of evolving GP in space
and time domains could be of interest. A simple rotat-
ing q-plate experiment is suggested: polarized light beam
after traversing the q-plate is made to pass through a ro-
tating HWP. Another variant with nonintegral q for this
arrangement, i.e. q-plate plus rotating HWP, is also sug-
gested. Analysis of the emerging beams may delineate
the role of SRP and PP as well as provide further test to
AM holonomy conjecture.
The physical mechanism proposed here for space vary-
ing polarization pattern of light could find important
application in ’all optical information processing’. The
angular momentum holonomy associated with GP, and
the strong evidence of its proof discussed here will have
significance in the context of the controversy surround-
ing ’the hidden momentum’ and Aharanov-Bohm effect.
We believe present ideas also hold promise to address
some fundamental questions in physics. An important
recent example is that of birefringence of the vacuum
in quantum electrodynamics in strong external magnetic
field. Though this has been known since long, last year
PVLAS experiment reported polarization rotation [15]
apparently very much in excess than the expected one.
This has led to a controversy on the interpretation of
QED birefringence in external rotating magnetic field,
see [16] for a short review. As remarked by Adler essen-
tially it involves light wave propagation in a nontrivial
refracive media, and he finds that to first order there
should be no rotation of the polarization of light. Could
there be a role of GP in this case? It would be interesting
to use Pancharatnam connection to calculate the phase of
propagating light, and see if evolving GP in time domain
will arise due to rotating magnetic field. It is interesting
to note that the magnetic field direction rotates in the
plane transverse to the direction of the propagation of
the light. Obviously it would give additional polariza-
tion rotation. This problem is being investigated, and
will be reported elewhere.
The Library facility at Banaras Hindu University is
acknowledged.
[1] J. F. Nye and M. V. Berry, Dislocations in wave trains,
Proc. R. Soc. Lond. A 336,165 (1974).
[2] J. F. Nye, Polarization effects in the diffraction of electro-
magnetic waves: the role of disclinations, Proc. R. Soc.
Lond. A 387, 105 (1983).
[3] L. Allen, M. J. Padgett, and M. Babiker, The orbital
angular momentum of light, Prog. Opt. 39,291 (1999).
[4] A. Niv, G. Biener, V. Kleiner, and E. Hasman, Manip-
ulation of the Pancharatnam phase in vectorial vortices,
Opt. Express, 14, 4208 (2006).
[5] C. Schwartz and A. Dogariu, Backscattered polarization
patterns, optical vortices, and the angular momentum of
light, Opt. Lett. 31,1121(2006).
[6] L. Marrucci, C. Manzo, and D. Paparo, Optical spin-to-
orbital angular momentum conversion in inhomogeneous
anisotropic media, Phys. Rev. Lett. 96,163905(2006).
[7] Z. Bomzon, M. Gu, and J. Shamir, Angular momentum
and geometrical phases in tight-focused circularly polar-
ized plane waves, Appl. Phys. Lett. 89,241 (2006).
[8] S. C. Tiwari, Geometric phase in optics: quantal or clas-
sical?, J. Mod. Opt. 39,1097(1992).
[9] S. C. Tiwari, Geometric phase in optics and angular mo-
mentum of light, J.Mod. Opt. 51,2297(2004).
[10] M. V. Berry, Quantum adiabatic anholonomy, Lectures
Ferrara School on Anomalies, defects, phases..., June
1989.
[11] M. V. Berry, The adiabatic phase and Pancharatnam’s
phase for polarized light, J. Mod. Opt. 34, 1401 (1987).
[12] J. F. Nye, Phase gradient and crystal-like geometry in
electromagnetic and elastic wavefields, in Sir Charles
Frank OBE, FRS:An eightieth birthday tribute (IOP,UK
1991)pp220-231.
[13] S. C. Tiwari, Nature of the angular momentum of
light: rotational energy and geometric phase, arxiv.org
: quant-ph/0609015.
[14] R. A. Beth, Direct detection of the angular momentum
of light, Phys. Rev. 48, 471 (1935).
[15] E. Zavattini et al, Experimental observation of optical
rotation generated in vacuum by a magnetic field, Phys.
Rev. Lett. 96, 110406 (2006).
[16] S. L. Adler, Vacuum birefringence in a rotating magnetic
field, J. Phys. A: Math. Theor. 40, F143 (2007).
http://arxiv.org/abs/quant-ph/0609015
