HAL Id: hal-01941207
https://hal.archives-ouvertes.fr/hal-01941207
Submitted on 30 Nov 2018
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Multispectral Management of the Photon Orbital
Angular Momentum
Mikaël Ghadimi Nassiri, Etienne Brasselet
To cite this version:
Mikaël Ghadimi Nassiri, Etienne Brasselet. Multispectral Management of the Photon Orbital Angular
Momentum. Physical Review Letters, American Physical Society, 2018, 121 (21), pp.213901 (1-6).
�10.1103/PhysRevLett.121.213901�. �hal-01941207�
Multispectral Management of the Photon Orbital Angular Momentum
Mikaël Ghadimi Nassiri and Etienne Brasselet
*
Universit´e de Bordeaux, CNRS, LOMA, UMR 5798, F-33400 Talence, France
(Received 21 August 2018; published 20 November 2018)
We report on a programmable liquid crystal spatial light modulator enabling independent orbital angular
momentum state control on multiple spectral channels. This is done by using electrically controllable
“topological pixels" that independently behave as geometric phase micro-optical elements relying on self-
engineered liquid crystal defects. These results open interesting opportunities in optical manipulation,
sensing, imaging, and communications, as well as information processing. In particular, spectral vortex
modulation allows considering singular spatiote mporal shaping of ultrashort pulses which may find
applications in many areas such as material processing, spectroscopy, or elementary particles acceleration.
DOI: 10.1103/PhysRevLett.121.213901
Photonics technologies provide our societies with tools
to better see and understand our world and explore others,
but also to improve health care, manufacturing, and ways to
inform and communicate. For more than two decades, the
sought-after management of the photon orbital angular
momentum has continued to foster many opportunities
[1,2]. Spatial light modulators (SLMs) refer to optical
devices enabling spatiotemporal modulation of the phase,
polarization, or amplitude of light fields. Many technolo-
gies have been developed since the 1980s, liquid crystals
(LCs) are prime choice materials owing to their great
sensitivity to external fields and their high birefringence.
Electrically controlled liquid crystal spatial light modula-
tors (LC-SLMs) imposed themselves in many areas when-
ever structured light is needed, see, for instance, [3,4],
which nowadays covers a broad range of topics [2].
A conventional LC-SLM typically consists of a two-
dimensional matrix made of up to several megapixels,
each of them behaving as an electrically controlled optical
retarder characterized by a uniform optical axis orientation
angle ψ
0
, see Fig. 1(a). Depending on the incident polari-
zation state and postpolarization filtering, versatile modula-
tion of light is, thus, obtained.
Another option towards structuring light fields consists
of exploiting the geometric phase spatial modulations
associated with inhomogeneous anisotropic optical ele-
ments [5]. Noting that LC-based materials are widely used
to fabricate geometric phase optical elements [6,7], what if
the independent assets of SLMs and geometric phase are
combined? This would give access to spatial light modu-
lation at the level of a pixel, hence, opening novel beam
shaping possibilities, since geometric phase is basically
wavelength independent. To do so, one would need pixels
enabling controllable phase modulation. Such a device does
not yet exist, and here, we propose a step forward to this
aim, by using a LC-SLM whose pixel action on light is of a
geometric nature. Here, we choose a specific pixel design
that consists of in-planeoptical axis orientation givenby ψ ¼
qϕ with q ∈ Z=2, see Fig. 1(b), which is the microscopic
counterpart of macroscopic so-called q plates [8]. The key
feature of such elements is to enable isomorphic mapping
from the two-dimensional polarization space to the two-
dimensional subspace defined by a pair of opposite orbital
angular momentum eigenvalues l ¼2q, which proved to
be a powerful tool both in classical and quantum optics [9].
Of course, it could be argued that choosing a hard-coded
design for individual pixels has an inherent limitation.
However, as shown here, such a step already provides for
spectrally agile beam shaping capabilities that cannot be
achieved with existing optical systems. By doing so, this
work not only brings a technical advance in the field of optics
and photonics, but also offers a novel generic approach for
manipulating the spatial degrees of freedom of waves.
Considering an array of electrically controlled q plates,
the multispectral management of the photon orbital angular
momentum is possible, as depicted in Fig. 2(a). The working
(a) (b)
FIG. 1. (a) Illustration of a usual optica l device made of a two-
dimensional matrix of electrically controlled liquid crystal pixels
having a uniform molecular orientation, ψ ¼ ψ
0
. (b) Topological
counterpart where every pixel acts as an electrically controlled
microscopic q plate (q ¼þ1 for the enlarged pixel).
PHYSICAL REVIEW LETTERS 121, 213901 (2018)
0031-9007=18=121(21)=213901(6) 213901-1 © 2018 American Physical Society
principle of the device is obtained by generalizing the
monochromatic behavior of a q plate [8] to a polychromatic
field that consists of discrete wavelengths λ
n
. For this, we
introduce the circular polarization basis ðc
þ
; c
−
Þwhere c
σ
¼
ðx þ iσyÞ=
ffiffiffi
2
p
with σ ¼1 the photon helicity. Neglecting
the propagation factor, we express the input light field as
E
in
¼
P
λ
n
E
n
ða
n
c
þ
þ b
n
c
−
Þand, discarding the diffraction
inside each element, the output field is expressed as
E
out
¼
X
λ
n
E
n
e
iΦ
n
cos
Δ
n
2
i sin
Δ
n
2
e
−il
n
ϕ
i sin
Δ
n
2
e
þil
n
ϕ
cos
Δ
n
2
a
n
b
n
;
ð1Þ
where Φ
n
¼ πðL=λ
n
Þðn
k
þ n
⊥
Þ is the dynamic phase and
Δ
n
¼ 2πðL=λ
n
Þðn
k
− n
⊥
Þ is the birefringence phase retar-
dation, with n
k
and n
⊥
being the refractive indices along and
perpendicular to the optical axis, and l
n
¼ 2q
n
.Notethat
Eq. (1) is only slightly perturbed by the finite practical width
δλ
n
of the channels provided that δλ
n
=λ
n
≪ 1. The orbital
angular state of the σ-polarized channel n is, therefore,
modified by an increment of 2σq
n
when Δ
n
¼ð2m þ 1Þπ,
m ∈ N, while it remains unaffected when Δ
n
¼ 2mπ.
The realization of a topological LC-SLM raises serious
manufacturing challenges although LC structuring tech-
nologies are mature at the macroscopic scale for the
production of electrically tunable single [10] or arrays
[11] of electrically controlled q plates. Indeed, the exten-
sion of existing machining techniques to the creation of
structured pixels with lateral size of the order of ∼10 μm,
which, nowadays, is the typical pixel size of LC-SLM, is
yet to be validated. We address this issue by implementing
a nature-assisted approach originally developed using
LC droplets [12] and further extended to LC films [13].
Following a preliminary demonstration of a linear array of
self-engineered microscopic þ1 plates without individual
control [14], here, we fabricate a device that consists of
a 10 × 127 matrix of 50 × 50 μm square-shaped q pixels,
where the 10 lines can be electrically controlled independ-
ently. The device is prepared by sandwiching a thin film
(typical thickness of the order of 10 μm) of nematic LC
with negative dielectric anisotropy between two transparent
and conductive indium-tin-oxide coated glass slabs treated
to ensure perpendicular anchoring of the LC molecules
using cetyl-trimethyl-ammonium bromide surfactant.
Electrode patterning is achieved by direct laser ablation
[15] performed before assembling the LC cell. Without
applied voltage, the LC molecules remain aligned perpen-
dicularly to the plane of the sample (Δ ¼ 0 for all pixels).
The sample used in Fig. 3 has a thickness of ≃13 μm and is
made of the nematic liquid crystal mixture MLC-6608
(Merck) and the sample used in Fig. 4 has a thickness of
≃7 μm and is made of the nematic liquid crystal mixture
DFLNC (Beam Co).
Above a threshold voltage of the order of a few Volts, the
LC spontaneously forms one umbilical defect per pixel
which can be assessed by observation between crossed
linear polarizers, as shown in Fig. 2(c). There, the twisted
extinction crosses, whose right or left handedness is
randomly selected as the defect is spontaneously generated
from the unperturbed initial LC state, are a generic
manifestation of localized umbilical nematic LC defects
with q ¼þ1 resulting from the elastic anisotropy of the LC
[16,17]. This modifies the wavefront curvature without
altering the orbital angular momentum control operated by
every topological pixel. This is ascertained by interfero-
metric analysis of the output light field that reveals a
structured phase factor expð2iσϕ
Þ, see Fig. 2(d),asex-
pected from Eq. (1).
As a first demonstration, we report on ultrabroadband
scalar and vectorial vortex beam shaping. This is done by
choosing Δ
n
¼ π for all n, hence, obtaining pure vortex
generation independently of the incident polarization state
according to Eq. (1). The experimental implementation is
depicted in Fig. 3(a). In practice, we use a supercontinuum
laser source dispersed by a blazed grating and recollimated
onto an array of optical fibers that define the spectral
(a) (b) (c) (d)
FIG. 2. (a) Multispectral modulation of the orbital angular momentum content of a discrete spectral channel λ
n
by an array of q plates
of order q
n
. (b) Fabricated LC-SLM based on self-engineered electrically controlled LC topological defects with q ¼þ1. The ruler is in
centimeter units. (1) 2 × 5 electrical cables supplying applied voltage; (2) ground; (3) Teflon holder. Top Teflon holder not shown here.
(c) Typical observation of a matrix of 49 q plates observed between crossed linear polarizers and white light incoherent illumination.
(d) Interferometric demonstration of the orbital ang ular momentum state change by an increment of two units performed by an
individual structured pixel, here, at 550 nm wavelength. The forked pattern results from the far-field superposition of a processed
circularly polarized Gaussian beam with a reference Gaussian beam.
PHYSICAL REVIEW LETTERS 121, 213901 (2018)
213901-2
channels. Multispectral scalar vortex shaping is achieved
by using an incident circular polarization state, namely,
ða
n
¼ 1;b
n
¼ 0Þ or ða
n
¼ 0;b
n
¼ 1Þ for all n, which,
respectively, gives E
out
∝ e
2iσϕ
c
−σ
. The results are shown
in Fig. 3(b). On the other hand, vectorial beam shaping is
achieved by using an incident linear polarization state,
namely, ja
n
j¼jb
n
j for all n, which gives a polychromatic
nonseparable spin-orbit state E
out
∝ e
−2iϕ
c
þ
þ e
þ2iϕ
c
−
.
The results are shown in Fig. 3(c) where observations
are made by placing a linear polarizer at the output of the
device in order to reveal the vectorial nature of the proceed
light beam. These experiments emphasize both the broad-
band and geometric phase features of our topological
LC-SLM. Importantly, the present approach allows one-
to-one mapping between the Poincar´e sphere of polariza-
tion and the high-order Poincar´e sphere of order l [18] for
all wavelengths. This solves previous optical state
dispersion drawbacks associated with the global proces-
sing of the whole spectrum using “Bragg-Berry” optical
elements [19].
Note that broadband optical vortex generation based on
geometric phase has been reported using various approaches
[20–28]; however, none of them enables spectral agility of
the associated photon orbital angular momentum. Here,
wavelength-dependent orbital angular momentum control
is also achieved, as illustrated in Fig. 4, where on-demand
activation or deactivation of on-axis intensity of any spectral
component of a polychromatic field is demonstrated. In the
context of optical nanoscopy [29], this allows us to consider
spectrally agile stimulation-emission-depletion (STED)
super-resolved imaging. Indeed, the wavelength(s) of the
on-axis reading light can be adapted to the fluorescence
characteristics of the scrutinized sample. Straightforwardly,
this can also be deployed in STED-inspired optical tech-
niques, for instance, in the field of optical nanolithography
where the spatial resolution of three-dimensional direct laser
writing techniques can be improved [30]. We also propose
another important application, that is, the four-dimensional
shaping of ultrafast optical pulses. Optical pulse shaping is a
mature technology having a huge range of applications, for
instance, in spectroscopy and light wave communications
[31] that still continue to be developed, as illustrated, for
instance, by recent works on spatiotemporal focusing
[32–34]. Situations involving the optical orbital angular
momentum are also attractive in high-energy optics where
techniques are developed towards space-time manipulation
of optical pulses [35–37]. Hereafter, we discuss a case study
illustrating how spectralvortex modulation is likelytoimpact
this field.
The basic scheme relies on Fourier-transform pulse
shaping. It consists in decomposing an incident pulse into
its constituent spectral components that are individually
modulated. Once recombined, the spectral components
reconstruct an output waveform given by the Fourier
transform of the reshaped spectrum. For the purpose of
demonstration, we consider an incident chirped Gaussian
pulse described by the temporal waveform amplitude
(a)
(b)
(c)
FIG. 3. (a) Sketch of the experimental setup for ultrabroadba nd
scalar and vectorial vortex beam shaping. The optical fiber array
consists of 12 multimode step-index fibers with numerical
aperture 0.2 and core radius 50 μm. The detection system is
either a camera or a fiber spectrometer. Scalar vortex (b) and
vectorial (c) beam shaping when incident light is, respectively,
circularly and linearly polarized. In both cases, the observation is
made by placing an output polarizer that selects the polarization
state orthogonal to the incident one. The image on the left
corresponds to the seven spectral channels λ
n
¼ 485, 511, 533,
555, 575, 595, and 617 nm with ∼10 nm full width at half
maximum in the plane of the LC-SLM and the image on the right
refers to the far field of the polychromatic structured light field.
(a) (b)
(c)
FIG. 4. Demonstration of on-demand polychromatic super-
position of orbital angular momentum states. (a) Spectrum of
the incident light that consists of five spectral channels at
λ
n
¼ 485, 515, 546, 578, and 614 nm. (b) Chart of the on-axis
power P collected with a fiber spectrometer having a clear
aperture of diameter D ¼ 200 μm, see leftmost panel (c) for the
five chosen configurations labeled #m, with m ¼ð1; 2; 3; 4; 5Þ.
The vortex states of every channel is optimized for Δ
vortex
n
¼ 7π,
7π, 5 π, 5π, and 5π, respectively. The nonvortex states correspond
to Δ
nonvortex
n
¼ Δ
vortex
n
− π. The collected power values are nor-
malized to the reference power P
ref
ðλÞ associated to the non-
vortex state. (c) Total far-field intensity profiles for the five
configurations.
PHYSICAL REVIEW LETTERS 121, 213901 (2018)
213901-3
E
in
ðr; tÞ¼E
0
e
−r
2
=r
2
0
þiω
0
t−αt
2
with α ¼ 1=t
2
0
− iδω=ð2t
0
Þ,
where ω
0
is the central angular frequency, and r
0
and t
0
are, respectively, the spatial and temporal characteristic
pulse widths, see Fig. 5(a). In addition, δω characterizes the
instantaneous angular frequency shift at t
0
, noting that
ωðtÞ¼ω
0
þðt=t
0
Þδω. The corresponding intensity spec-
trum jF
T
½E
in
j
2
is shown as the black curve in Fig. 5(b),
where F
T
refers to the temporal Fourier transform. Then,
we consider the ideal hyperspectral (i.e., spectrally con-
tinuous) situation in the simple case of a retardance having
a linear spectral dependence of the form ΔðωÞ¼π þ
βðω − ω
0
Þ. This implies that the central frequency ω
0
is
fully converted into a vortex state while others experience
partial vortex transformation according to the frequency-
dependent purity parameter ηðωÞ¼sin
2
½ΔðωÞ=2. The
latter characterizes the fraction of the incident energy of
a σ-circularly polarized incident pulse that is transformed
into an orbital state l ¼ 2σq. By doing so, it is possible to
create the pulsed analog of continuous-wave bottle beams
[38]—an optical bottle pulse. The demonstration is shown
in Fig. 5(c), where an optical pulse with a dark focus
surrounded by regions of higher intensity can be created
for an appropriate value of the parameter β, here, in the
particular case q ¼þ1. The calculations are made account-
ing for Eq. (1) taking a σ-polarized incident pulse. This
gives rise to a superposition of a σ-polarized field
emerging from the topological pulse shaper. Namely,
E
out
ðr; tÞ¼F
−1
T
½F
T
½E
in
cosðΔðωÞ=2Þc
σ
þ iF
−1
T
½F
T
½E
in
sinðΔðωÞ=2Þe
2iσqϕ
c
−σ
, where F
−1
refers to the inverse
Fourier transform. The far field then gives the optical bottle
pulse described by F
S
½E
out
ðr; tÞ, where F
S
refers to the
spatial Fourier transform. Note that the latter calculation
implies that the dynamic phase factor in Eq. (1) does not
play a role, which can be met in principle using compen-
sation techniques before topological processing or by
elaborating dynamic phase independent geometric phase
elements following a recent approach [39]. Interestingly,
the optical bottle pulse is associated with a nonstationary
spin and orbital angular momentum, which can find uses in
contactless optorheological studies based on light-matter
angular momentum transfer. This is illustrated in Fig. 5(d)
where the respective contributions of the field components
carrying different orbital contents are depicted as distinct
colors. This, obviously, opens further possibilities for four-
dimensional pulse shaping once other configurations are
considered.
Finally, we note that the quality of the geometric phase
shaping provided by the topological liquid crystal pixels
inherently benefits from the high-purity performances of
umbilical defects, as demonstrated early in Ref. [13]. The
limitation comes from the finite size of the pixel, and a key
practical parameter is the ratio pixel size/defect core size.
In the present case, the purity typically ranges from 0.8 up
to more than 0.95 depending on the operating conditions.
In addition, the efficiency of the process (i.e., the fraction
of incident photons that can actually be processed) also
depends on the unwanted Fresnel reflections of the sub-
strates, which could be optimized by appropriate broad-
band coatings. On the other hand, the speed of the proposed
liquid crystal device is basically that of a usual liquid crystal
pixel in displays and may reach up to the kHz frequency for
thin pixel thickness using conventional nematics.
The programmable manipulation of light, both in space
and time, via the spectral management of the photon orbital
angular momentum provides for a very large set of
prospective applications, from continuous light waves to
ultrashort optical pulses. Moreover, its general principle
can be formally adapted to any wavelength range. Spectral
vortex modulation is likely to impact many research areas
such as optical imaging, quantum optical information,
optical communications, optical spectroscopy, optical
manipulation, or high-energy physics. Also, since optical
phase singularities are a generic feature of wave physics,
the proposed approach can be further extended to other
(a)
(c)
(d)
(b)
FIG. 5. Four-dimensional optical pulse shaping by spectral
vortex modulation. (a) Electric field temporal waveform of a
chirped ultrashort optical Gaussian pulse characterized by central
wavelength λ
0
¼ 500 nm, spatial waist r
0
¼ 500 μm, temporal
waist t
0
¼ 21 fs, and chirp parameter δω ¼ 0.60 rad=fs, whose
intensity spectrum is shown in panel (b) (black curve). The
spectral dependence of the optical vortex purity for three cases, η
i
with i ¼ð1; 2; 3Þ, is shown in panel (b) (colored curves). (c) Total
far field intensity profile of the spatiot emporally shape d pulse for
the three cases, which are associated with parameters β
i
, in the
case q ¼þ1. An optical bottle pulse is obtained for i ¼ 2 while
only transverse and longitudinal spatial modulation is obtained
for i ¼ 1 and i ¼ 3, respectively. κ refers to the spatial frequen-
cies in the transverse plane (arbitrary units). (d) Intensity profiles
of the field component carrying distinct orbital angular momen-
tum per photon, here 0ℏ (red color) and 2σℏ (green color) per
photon.
PHYSICAL REVIEW LETTERS 121, 213901 (2018)
213901-4
