Download Rotating light with light: Generation of helical modes of light by spin

Rotating light with light: Generation of helical modes of light
by spin-to-orbital angular momentum conversion in
inhomogeneous liquid crystals
Lorenzo Marrucci
Copyright 2007 Society of Photo-Optical Instrumentation Engineers (SPIE)
This paper was published in Proceedings of SPIE – The International Society for Optical
Engineering 6587, 658708 (2007) and is made available as an electronic reprint (preprint) with
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Rotating light with light: Generation of helical modes of light
by spin-to-orbital angular momentum conversion in
inhomogeneous liquid crystals
Lorenzo Marrucci
Dipartimento di Scienze Fisiche, Università di Napoli “Federico II”, and CNR-INFM
Coherentia, Complesso di Monte S.Angelo, via Cintia, 80126 Napoli, Italy
ABSTRACT
I review recent results on a novel method for generating helical waves of visible light based on inhomogeneous
birefringent plates made of a suitably patterned liquid crystal. These devices, dubbed “q-plates”, act on the
light wave by converting its spin angular momentum into orbital angular momentum, an optical process not
envisioned before. The output helical wave can be easily and rapidly switched between opposite wavefront
helicities by switching the input polarization with standard electro-optics devices. The process can be cascaded,
so that rapid switching can take place among multiple values of the wavefront helicity. More generally, patterned
liquid-crystal devices similar to those realized for generating helical beams may be used for shaping the optical
wavefront in any prescribed way, with the possibility of dynamical polarization multiplexing between conjugate
wavefronts. This is an application of the Pancharatnam-Berry phase principle, allowing the realization of a novel
kind of optical elements for wavefront shaping. Potential developments in the fields of optical communication
and quantum computation are briefly discussed.
Keywords: Helical modes, orbital angular momentum, optical elements, wavefront shaping, PancharatnamBerry phase, polarization holography, liquid crystal devices, quantum computation
1. INTRODUCTION
The helical modes of an electromagnetic wave are characterized by a helical, or “corkscrew”, shape of the
wavefront and of the energy-transport rays,1–5 and by the presence of a topological phase singularity, an “optical
vortex”, at their beam axis.6–8 Some examples of helical modes are shown in Fig. 1.
These helical waves carry quantized intrinsic angular momentum of an orbital kind, as opposed to the spinlike angular momentum that can be associated with circularly polarized waves.1, 9, 10 Recently, helical modes
of light have attracted a growing interest owing to their possible use for optical trapping and manipulation
of particles11–16 and atoms,17–19 multi-state information encoding for optical communication20 and quantum
computation,21–24 and new spectroscopic techniques in terrestrial25 or astronomical observations.26
Up to last year, spin and orbital angular momentum of light were considered as fairly independent degrees
of freedom, which are not communicating or interacting with each other. One could easily act on the spin
variable alone of a laser beam by means of standard birefringent plates. The creation or manipulation of the
orbital angular momentum of a laser beam was instead possible by means of special phase plates having a
helical-shaped thickness (“spiral phase plates”),27 or by a system of cylindrical lenses acting on beams having an
initial Hermite-Gaussian transverse mode,1 or finally by suitable holograms (either printed in fixed holographic
elements or generated in computer-controlled spatial light modulators).7 All these methods act on the wavefront
alone, leaving the polarization degree of freedom essentially unaffected, apart from secondary effects. Although
the two degrees of freedom are both associated with angular momentum (in this sense, loosely speaking, they
describe two different kinds of “rotation” of the light around its propagation axis), nobody had yet conceived
a way to make them interact, so for example as to transform them into each other in a simple direct way. All
mentioned methods for generating and manipulating the angular momentum of light are based on a transfer of
(a)
(b)
(c)
(d)
Figure 1. Examples of helical waves. Represented are the wavefronts of helical modes for helicities (or vortex charge)
m = +1 (a), m = −1 (b), m = +2 (c), and m = −2 (d).
angular momentum from light to matter or vice versa, but only one species of optical angular momentum could
be involved in each transfer.
To have an interaction between spin and orbital angular momentum means to have an interaction between
polarization and wavefront shape of a light beam. Such an interaction can be one of two kinds (or both): either
the polarization affects the wavefront shape, or the wavefront shape affects the polarization. Of course, it is
conceivable to design a birefringent phase optical element, e.g., a birefringent spiral plate, so as to induce a
specific wavefront shaping only for one linear polarization and not for the orthogonal one. This trivially allows a
direct coupling between the spin and orbital degrees of freedom, but it is an artificial solution, and also difficult
to be realized and likely inefficient.
Instead, a natural and effective solution emerges from an old physical principle: the so-called PancharatnamBerry optical phase.28, 29 When the polarization of an electromagnetic wave undergoes a continuous sequence
of transformations following a closed path in the space of polarization states (e.g., the Poincaré sphere), the
wave acquires a phase shift, known as Pancharatnam-Berry phase, that is determined only by the geometry of
the polarization path. By the same principle, if a wave is subjected to transversely inhomogeneous polarization
transformations with a homogeneous initial and final polarization state, the associated inhomogeneous geometrical phases will induce an overall wavefront reshaping.30, 31 This approach to wavefront shaping is fundamentally
different from the usual optical-path-length approaches of standard lenses, curved mirrors, and gradient-index
(GRIN) elements. It is also conceptually different from holographic approaches, although the two are related,
as I will discuss further below. One important feature of this approach, also for practical purposes, is that the
output wavefront will depend on the input polarization state. In other words, the input polarization controls
the output wavefront: just what we were looking for to set an interaction between spin and orbital angular
momentum of light.
As I will describe in the following, in suitable material geometries, the Pancharatnam-Berry optical phase
gives rise just to the right coupling between spin and orbital angular momentum degrees of freedom of light that
Further author information: E-mail: [email protected]
allows for their direct exchange.32 This approach defines a completely new way of generating helical modes of
light, having important advantages over previously known methods.32, 33 Moreover, this example represents a
proof-of-principle of the general applicability of a patterned liquid crystal technology to polarization-multiplexed
wavefront shaping in the visible domain.33
This paper is organized as follows. In Section 2, the theory of wavefront control based on the PancharatnamBerry phase will be presented. In Section 3, I will illustrate how this method can be applied specifically to the
generation of helical modes of light, with an helicity controlled by the input polarization. Moreover, I will note
that this optical process may correspond for certain geometries to a direct conversion of the spin-form of the
optical angular momentum to the orbital form. In Section 4, I will describe the experimental verification of this
approach. In Section 5, I will argue that the liquid-crystal technology demonstrated in this experiment has the
potential for developing a whole new class of optical phase elements for the visible and near-infrared domain,
based on the Pancharatnam-Berry phase principle. These elements have an output that is controlled by the input
polarization, opening interest prospects, for example in the field of secure optical free-space communication. Some
remarks on the possible applications of these devices to the field of photonic quantum computation are given in
Section 6. Finally, in Section 7, I will conclude by summarizing the main results reviewed in this paper.
2. WAVEFRONT SHAPING BY THE PANCHARATNAM-BERRY PHASE
PRINCIPLE
Let us consider a single uniaxial birefringent plate having a homogeneous phase retardation of π (half-wave) for
light propagation in the longitudinal z direction but a transversely inhomogeneous optical axis n(x, y), lying in
the xy plane. Following Bomzon et al., I will hereafter call such a device “Pancharatnam-Berry phase optical
element” (PBOE).31
Let use denote with Lt the typical transverse length scale of the variations of the optical axis n in the xy
plane, with d the plate thickness and with λ the working optical wavelength. If the condition
λd ≪ L2t
(1)
is satisfied, the phase and polarization effect of the plate on the transmitted wave can be analyzed point-by-point,
neglecting all transverse diffraction effects occurring in the propagation within the plate. In this approximation,
one can use the Jones formalism to fully characterize the propagation through the plate. Let α(x, y) be the angle
between n(x, y) and a fixed reference axis x. The Jones matrix M describing the PBOE action on the field at
each transverse position x, y is the following:
1 0
cos 2α(x, y)
sin 2α(x, y)
M(x, y) = R[−α(x, y)]
R[α(x, y)] =
,
(2)
0 −1
sin 2α(x, y) − cos 2α(x, y)
where
R(α) =
is the two-dimensional rotation matrix.
cos α
− sin α
sin α
cos α
An input left-circular polarized plane wave, described by the Jones electric-field vector
1
Ein (x, y) = E0
i
will be transformed by the action of the PBOE into the following field (up to an overall phase):
1
i2α(x,y)
.
Eout (x, y) = M(x, y) · Ein = E0 e
−i
(3)
(4)
(5)
It is seen from this equation that the output wave is uniformly right-circular polarized, but its wavefront has
acquired a nonuniform phase retardation ∆Φ(x, y) = 2α(x, y). If the input light is right-circular polarized, it is
easy to verify that the output wavefront is the conjugate one, i.e. ∆Φ(x, y) = −2α(x, y).
Figure 2. Optical axis geometry of a PBOE behaving as a polarization-dependent lens. Dashes indicate local optical
axis direction for a selection of radii. A perfect PBOE-lens will actually have a continuous variation of the optical axis.
Therefore, any wavefront shape as specified by the transverse phase retardation ∆Φ(x, y), can be generated
by a suitable PBOE. The needed PBOE optical axis geometry is fixed by the relationship α(x, y) = ±∆Φ(x, y)/2,
where the sign is determined by the circular polarization handedness that will be employed. The same device will
also generate the conjugate wavefront −∆Φ(x, y) if the input polarization handedness is inverted. This flexibility
of PBOEs is akin to that of holographic optical elements, and indeed this approach to wavefront shaping can be
considered as a special case of a form of holography: the so-called “polarization holography”.33, 34
It must be emphasized that, independently of the wavefront shape to be generated, the optical thickness of the
PBOE will be uniform. It can also be very thin, if the material is sufficiently birefringent. Thermotropic nematic
liquid crystals, that have a typical birefringence ∆n ≃ 0.2, if assembled in the planar geometry can yield a halfwave birefringent retardation with a thickness d of about two wavelengths (more precisely, d = λ/(2∆n) ≃ 2.5λ),
i.e., just above a micron for the visible domain. Moreover, despite the small thickness, there is no a priori
limitation set on the relative phase retardation between different parts of the generated wavefront. Phase
retardations of several wavelengths between any two different points of the wavefront can be obtained just by
having the optical axis n make several complete (2π) rotations along the line connecting the two points in the
PBOE plane. The only limitation is in the maximum transverse spatial gradients that can be imposed on the
phase, owing to condition (1).∗
To better appreciate the possible applications of these PBOE phase optical elements, consider, for example,
a plate having a geometry defined by the radial-quadratic law α(r, ϕ) = cr2 , where r, ϕ are the polar coordinates
in the xy plane and c is a constant.30, 35 Such a geometry is illustrated in Fig. 2. This PBOE will induce a
parabolic phase retardation on the input wave, leading to a focusing or defocusing effect, depending on the
retardation sign. In other words, this PBOE will act on a circularly polarized input as a lens having focal length
given by f = ±π/(2cλ), with the sign depending on the input polarization handedness. As mentioned before,
this PBOE-lens will have a uniform thickness, which can be very small. This feature cannot be matched by
standard lenses (including GRIN elements) and will likely find some use in certain specific applications. Only
Fresnel lenses can have a similar thickness uniformity (on the average), but at the price of introducing several
transverse discontinuities of optical properties (either the thickness or the refractive index), thus making the
manufacturing difficult and expensive and imposing unwanted optical losses. In contrast, PBOE are perfectly
continuous and optical losses can be minimal.
∗
Although one can probably overcome in part this limitation by designing the PBOE after taking into account the
effect of diffraction occurring within the plate.
(a)
(b)
(c)
Figure 3. Examples of q-plates. The tangent to the lines shown indicate the local direction of the optical axis n. (a)
q = 1/2 and α0 = 0 (a nonzero α0 is here just equivalent to an overall rigid rotation), which generates helical modes
with m = ±1; (b) q = 1 with α0 = 0 and (c) with α0 = π/2, which can be both used to generate modes with m = ±2.
The last two cases correspond to rotationally symmetric plates, giving rise to perfect spin-to-orbital angular momentum
conversion, with no angular momentum transfer to the plate.
In concluding this Section, we must notice one important limitation of PBOE. Since their birefringent retardation must correspond exactly to a half-wave (π), they will only work correctly for a given “design” wavelength
(i.e., they are “laser-line” and not broad-band optical element). It is however possible to make the PBOE phase
effect fully wavelength-independent by sandwiching the device between two crossed circular polarizers. In this
case, the output will have the desired wavefront shape regardless of the wavelength. However, this will be obtained at a price: the wavelength components that are detuned from the design wavelength of the PBOE will be
attenuated by the polarizers, with losses that increase for larger detunings.
3. HELICAL MODE GENERATION AND ANGULAR MOMENTUM CONVERSION
Let us now consider a generic helical wave having frequency ω and wavenumber k = 2π/λ. Let the beam
propagation axis coincide with the z axis. In the standard complex notation, the electric field of this wave is
given, in the paraxial approximation, by the following expression:
E(r, ϕ, z, t) = E0 (r, z) exp(imϕ) exp(ikz − iωt),
(6)
where r, ϕ are the polar coordinates in the xy plane, transverse to the propagation direction, and m is an
integer. For certain specific choices of the radial profile E0 (r) at a given plane z, Eq. (6) will correspond to
the well known Laguerre-Gaussian modes, which have a particularly simple propagation law in the paraxial
approximation. However, for any choice of the radial profile, Eq. (6) always corresponds to a helical mode, with
a well defined value of the orbital angular momentum (z-component) per photon, given by mh̄ (see, e.g., Ref. 1).
The wavefront of this field is composed of |m| intertwined helical surfaces, with a handedness given by the sign
of m, as shown in Fig. 1. For definiteness, in the following I will refer to m as the orbital-helicity of the beam. In
general, any optical wave can be decomposed in circularly polarized helical modes carrying well defined values
of both spin and orbital angular momentum, such as for example the mentioned Laguerre-Gauss modes.
A transverse phase profile such as that given in Eq. (6) can be generated by a PBOE having the following
geometry:
α(r, ϕ) = qϕ + α0
(7)
where q and α0 are constants. Note that Eq. (7) implies the presence of a defect in the medium localized at the
plane origin, r = 0, similar to the typical defects spontaneously formed by nematic liquid crystals.36 However,
if q is an integer or a semi-integer there will be no discontinuity line in the slab. In the following, we will refer
to PBOE elements having the above specified geometry as q-plates. A few examples of q-plate geometries for
different values of q and α0 are shown in Fig. 3.
To analyze the effect of the q-plate on an input plane-wave, let us consider again Eq. (5) with α(x, y) given
by Eq. (7). For a left-handed circularly polarized input, the q-plate output will be right-circular polarized and
will also have acquired a transverse phase factor given by
ei2qϕ+i2α0 ∝ eimϕ ,
(8)
with m = 2q, i.e. it has been transformed into a helical wave with orbital helicity 2q and orbital angular
momentum 2qh̄ per photon. It is easy to verify that in the case of a right-circular input wave, the orbital helicity
and angular momentum of the outgoing wave are sign-inverted. In other words, the input polarization of the
light controls the sign of the orbital helicity of the output wavefront. Its magnitude |m| is instead fixed by the
PBOE geometry.
In passing through the plate, each photon being converted from left-circular to right-circular changes its spin
z-component angular momentum from +h̄ to −h̄. In the case of a q-plate having q = 1, the orbital z-component
angular momentum of each photon changes instead from zero to 2h̄. Therefore, the total variation of the angular
momentum of light is nil and there is no net transfer of angular momentum to the plate: the plate in this case
acts only as a “coupler” of the two forms of optical angular momentum, allowing their conversion into each
other.32
This exact compensation of the spin and orbital angular momentum exchanges with matter is clearly related
to the circular symmetry (rotation invariance) of the q = 1 plate, as can be proved by general energy arguments
or by a variational approach to the optical angular momentum fluxes.37 If q 6= 1, the plate is not symmetric
and will exchange an angular momentum of ±2h̄(q − 1) with each photon, with a sign depending on the input
polarization. Therefore, in this general case the angular momentum will not be just converted from spin to
orbital, but the spin degree of freedom will still control the “direction” of the angular momentum exchange with
the plate, besides the sign of the output wavefront helicity.32
4. EXPERIMENT
To demonstrate the angular momentum conversion predicted in the previous Section, in the work reported in
Refs. 32, 33, my coworkers and I manufactured q = 1 plates working at the visible wavelength λ = 633 nm
based on a patterned liquid crystal (LC) technology. Nematic LC planar cells were prepared so as to obtain a
birefringence retardation of approximately a half wave. Details on the cell preparation are reported in Ref. 33.
In particular, the q = 1-plate geometry was obtained by a “circular rubbing” procedure, so as to obtain a
surface easy axis for the LC director orientation as that shown in Fig. 3c. A photograph of a LC q-plate held
between crossed polarizers is shown in Fig. 4a.
To test the optical effect of the manufactured q-plate, a circularly-polarized He-Ne laser beam having a
TEM00 transverse mode and a beam-waist radius of about 1 mm was sent through it, taking care of aligning the
beam axis on the q-plate center. The intensity profile of the output beam, shown in Fig. 4b, was found to have
the “doughnut” shape expected for a helical mode.
In order to measure also the wavefront shape of the light emerging from the q-plate, we set up a Mach-Zender
interferometer. A He-Ne laser beam with a TEM00 gaussian profile was split in two beams, namely signal and
reference. The signal beam was first circularly polarized with the desired handedness by means of properly
oriented quarter-wave plate and then was sent through the LC q-plate. The beam emerging from the q-plate was
then sent through another quarter-wave plate and a linear polarizer, arranged for transmitting the polarization
handedness opposite to the initial one, so as to eliminate the residual unchanged circular polarization (this
step is not necessary when using a q-plate having exactly half-wave retardation). Finally, the signal beam was
superimposed with the reference and thus generated an interference pattern directly on the sensing area of a
CCD camera. Two different interference geometries were employed. In the first, the reference beam wavefront
was kept approximately plane (more precisely, it had the same wavefront curvature as the signal beam) but the
two beams were slightly tilted with respect to each other. For non-helical waves, this geometry gives rise to a
regular pattern of parallel straight fringes. If the wavefront of the signal beam is helical, the pattern develops
a dislocation (double, in this case, since q = 1 yields m = ±2), with an orientation depending on the sign of
m and the relative orientation of the two beams. In the second geometry, the reference beam wavefront was
approximately spherical, as obtained by inserting a lens in the reference arm. For non-helical waves, the resulting
interference pattern is made of concentric circular fringes. If the wavefront of the signal beam is helical, the
pattern takes instead the form of a spiral (a double spiral, for m = ±2), with a handedness depending on the
sign of m (counterclockwise outgoing spirals, seen against the propagation direction as in our case, correspond
to a positive m). Figure 4 shows the CCD-acquired images of the interference patterns we obtained in the
(a)
(b)
(c)
(d)
BN
(e)
(f)
Figure 4. Experimental images. (a) A LC q-plate held between crossed polarizers, showing the expected pattern for
q = 1 geometry. (b) “Doughnut” intensity profile of the beam emerging from the q-plate. (c-f) Interference patterns
of helical modes generated by our q-plate. (c-d) panels refer to the plane-wave reference geometry, (e-f) panels to the
spherical-wave reference one. Panels on the left, (c) and (e), are for a left-circular input polarization and those on the
right, (b) and (d), for a right-circular one.
stage 1
Laser
PC
QWP
qplate
stage 2
QWP
PC
QWP
qplate
stage n
QWP
PC
QWP
qplate
Figure 5. A n stages PBOE optical system for generating helical modes of light having an order m which can be
electro-optically switched in the set m ∈ {−2nq, −2(n − 2)q, . . . , +2(n − 2)q, +2nq}. Legend: PC - Pockel cell; QWP quarter-wave plate.
two geometries, respectively for a left-circular [panels (c) and (e)] and right-circular [panels (d) and (f)] input
polarizations. These results show unambiguously that the wavefront of the light emerging from the q-plate is
indeed helical with m = ±2 (i.e. as shown in panels (c) and (d) of Fig. 4), as predicted, and that it carries
an orbital angular momentum just opposite to the variation of spin angular momentum associated with the
polarization occurring in the plate.
5. PANCHARATNAM-BERRY OPTICAL ELEMENTS FOR THE VISIBLE DOMAIN
The polarization-based control of the generated helical wavefront seen in the previous Section is a good example
of the possible advantages of the PBOE approach to wavefront shaping. Indeed, all other existing approaches to
helical mode generation (i.e. cylindrical lenses, spiral phase plates, and holographic methods) have an essentially
fixed output. Of course, by introducing a suitable spatial light modulator, dynamical control becomes possible,
but only at relatively low switching rates. This limitation is particularly important in applications related with
optical information encoding, as it severely limits the communication bandwidth. In contrast, in the approach
described here, based on PBOEs and angular momentum conversion, a simple electro-optical control of the input
polarization allows switching of the generated helical mode at very high rate. By cascading several q-plates in
series with suitable electro-optic devices in between, as shown in Fig. 5, one can obtain fast switching among
several different helical orders. This could be very useful if helical modes are to be used in multi-state optical
information encoding, as recently proposed for classical communication20† and for quantum communication and
computation.23, 24
Although in the proof-of-principle demonstration reviewed here, the employed method for patterning the LC
cell works only for circular-symmetric geometries (as in the q = 1 plate), LC cell patterning has the potential for
obtaining any desired PBOE geometry. Different approaches can be explored, such as micro-rubbing,38 masked
or holographic photo-alignment,39, 40 and silicon-oxide evaporated coatings.41
Finally, I note here (as already anticipated) that the PBOE principle is strictly related to the so-called
polarization holography (PH), in which the holographic material records the information contained in the optical
polarization.34, 42 Typically, in PH one needs a light-sensitive polymer that can align its molecular chains parallel
or perpendicular to the polarization direction.43 In order to memorize a given wavefront in a PH hologram,
one must superimpose the wave carrying that wavefront with a plane-wave reference, taking care that both
waves are circularly polarized, with opposite handedness. The resulting interference field will have uniform
intensity and it will be everywhere linearly polarized, but it will have a nonuniform polarization orientation
which will be imprinted in the hologram. When illuminated with a plane wave, this hologram will reconstruct
the recorded wavefront, or its conjugate, at its ±1 diffraction orders. However, if the hologram is “developed”
into a inhomogeneous birefringent plate having half-wave retardation (for example by using the hologram as a
“command” surface of a LC cell, or if the hologram itself has sufficient birefringence), the zero diffraction order
†
Helical modes have also the advantage of being intrinsically secure against eavesdropping of the scattered light, as the
phase-structure information will be entirely lost in the scattering.20 This possibility is particularly interesting for secure
free-space communication. The availability of a device for inducing a fast switching of the helicity m of a given beam
without affecting its propagation direction is clearly important for this possible application.
will vanish identically and the hologram becomes a PBOE device generating a single optical output with the
recorded wavefront, or its conjugate, when illuminated with a circularly polarized plane wave.
By this polarization holography approach, it is perfectly conceivable the idea of realizing thin free-standing
polymeric liquid crystal films made as PBOE. In this way, one could manufacture ultrathin flat flexible plastic
lenses or other optical phase elements, a rather interesting technological prospect.
6. QUANTUM COMPUTATION WITH SPIN AND ORBITAL ANGULAR
MOMENTUM
Before concluding this paper, I would like to note a few interesting possibilities opened by the q-plate idea for
doing quantum computation with photons.
On a single photon having left or right circular polarization state, |Li or |Ri, and orbital angular momentum
state |mi, the action of the q-plate can be described by a quantum operator Ûq having the following effect (up
to a constant phase shift, that will be equal for the two cases):
Ûq |L, mi
Ûq |R, mi
= |R, m + 2qi
= |L, m − 2qi
(9)
The effect of the q-plate will clearly be optically coherent (although, when analyzing more fully the optical
coherence issues, one must take proper care of the radial mode, which will be modified by the q-plate action).
Therefore, if a single photon having linear (or elliptical) input polarization passes through a q-plate, it will
emerge in a quantum state having an entanglement of the two degrees of freedom of spin and orbital angular
√
momentum. For example, an input linear (e.g., vertical) polarization state given by |V i = (|Li + |Ri)/ 2 will
give rise to the following output:
1
Ûq |V, mi = √ (|R, m + 2qi + |L, m − 2qi) .
2
(10)
The q-plate effect given in Eq. (9) corresponds to a kind of “controlled-shift” quantum gate, operating
on the orbital angular momentum state and controlled by the spin state. In combination with other optical
systems operating separately on the spin or the orbital angular momentum states, this new gate offers interesting
opportunities for doing quantum computation with photons. Importantly, many operations can be done by acting
on a single beam of light, without the need of splitting the beam along different paths, according to its degree
of freedom.
7. CONCLUSIONS
In conclusion, I have reviewed some recent results on a new approach to the manipulation of the angular
momentum of light, introducing an interaction between the spin and the orbital degrees of freedom. In particular,
a specific geometry realizes a perfect all-optical conversion of spin angular momentum into orbital angular
momementum, with no net transfer of angular momentum to matter. This allows for the generation of helical
modes of light carrying orbital angular momentum, with the possibility of fast electro-optic switching among
several helical states.
This approach to the generation of helical modes of light could prove particularly valuable in the foreseen
applications of these modes to the multi-state information encoding for classical and quantum communication and
computation, where the capability for a fast switching of the generated helicity is critical. Moreover, single-beam
quantum computation with spin and orbital angular momentum states becomes possible.
The working principle behind the interaction of optical spin and orbital angular momenta is based on the
Pancharatnam-Berry geometrical phase. This principle is at the root of recently-developed optical elements for
wavefront shaping that are polarization-controlled. Technologies of patterned liquid crystals and of polarizationholography in polymers are available to manufacture these optical elements for the visible domain, with very
interesting prospects for a variety of optical applications.
ACKNOWLEDGMENTS
I thank Domenico Paparo and Carlo Manzo for their contribution to the experimental demonstrations reviewed
in this paper.
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