Efficient sorting of orbital angular momentum states of light
Berkhout, G.C.G.; Lavery, L.A.; Courtial, J.; Beijersbergen, M.W.; Padgett, M.J.
Citation
Berkhout, G. C. G., Lavery, L. A., Courtial, J., Beijersbergen, M. W., & Padgett, M. J. (2010).
Efficient sorting of orbital angular momentum states of light. Physical Review Letters,
105(15), 153601. doi:10.1103/PhysRevLett.105.153601
Version: Not Applicable (or Unknown)
License: Leiden University Non-exclusive license
Downloaded from: https://hdl.handle.net/1887/63550
Note: To cite this publication please use the final published version (if applicable).
Efficient Sorting of Orbital Angular Momentum States of Light
Gregorius C. G. Berkhout,
1,2,
*
Martin P. J. Lavery,
3,†
Johannes Courtial,
3
Marco W. Beijersbergen,
1,2
and Miles J. Padgett
3
1
Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands
2
cosine Science & Computing BV, Niels Bohrweg 11, 2333 CA Leiden, The Netherlands
3
School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom
(Received 19 July 2010; published 4 October 2010)
We present a method to efficiently sort orbital angular momentum (OAM) states of light using two
static optical elements. The optical elements perform a Cartesian to log-polar coordinate transformation,
converting the helically phased light beam corresponding to OAM states into a beam with a transverse
phase gradient. A subsequent lens then focuses each input OAM state to a different lateral position. We
demonstrate the concept experimentally by using two spatial light modulators to create the desired optical
elements, applying it to the separation of eleven OAM states.
DOI: 10.1103/PhysRevLett.105.153601 PACS numbers: 42.50.Tx, 42.40.Jv, 42.79.Sz
Nearly twenty years ago it was recognized by Allen
et al. that helically phased light beams, described by a
phase cross section of expði‘Þ, carry an orbital angular
momentum (OAM) of ‘@ per photon, where ‘ can take any
integer value [1–3]. The unlimited range of ‘ gives an
unbounded state space, and hence a large potential infor-
mation capacity [4,5]. At the level of single photons, OAM
entanglement is a natural consequence of the conservation
of angular momentum in nonlinear optics [6]. This entan-
glement makes OAM a potential variable for increased-
bandwidth quantum cryptography [7–9], but only if a
single photon can be measured to be in one of many
different states.
Generation of helically phased beams with OAM is most
usually accomplished using a diffractive optical element,
i.e., hologram, the design of which is a diffraction grating
containing an ‘-fold fork dislocation on the beam axis
[10,11]. If the hologram is illuminated by the output
from a laser, or single-mode fiber, the first-order diffracted
beam has the required helical phase structure. The same
setup, when used in reverse, couples light in one particular
OAM state into the fiber. In this case, the hologram acts as
a mode specific detector, working even for single photons
[12]. However, such a hologram can only test for one state
at a time. Testing for a large number of possible states
requires a sequence of holograms, thereby negating the
potential advantage of the large OAM state space. More
sophisticated holograms can test for multiple states, but
only with an efficiency approximately equal to the recip-
rocal of the number of states [5,13]. For classical light
beams, the OAM state can be readily inferred by the
interference of the beam with a plane wave and counting
the number of spiral fringes in the resulting pattern [14].
One can also use the diffraction pattern behind specific
apertures to determine the OAM state of the incoming light
beams [15,16]. All of these approaches again require many
photons to be in the same mode so as to produce a well-
defined pattern.
The symmetry of helically phased beams means that
their rotation about the beam axis induces a frequency
shift, each OAM component inducing a separate frequency
sideband [17], which could, in principle, be used to mea-
sure OAM [4]. However, spinning a beam about its own
axis at a rate sufficient to measure its frequency shift is not
technically possible. This technical challenge is lessened
by using a static beam rotation to introduce an ‘-dependent
phase shift within a Mach-Zehnder interferometer; a cas-
cade of N 1 interferometers can measure N different
states [ 18]. Although establishing the principle for single-
photon measurement of OAM, for large N, this cascaded
interferometric approach remains technically demanding
for inclusion into larger systems.
In this Letter we are motivated by the simple example of
the discrimination of plane waves within direction space. A
lens is all that is required to focus a plane wave to a spot in
its focal plane, the transverse position depending on the
transverse phase gradient of the plane wave. This allows
multiple plane waves to be distinguished from each other
using a detector array. A requirement for the separation of
any two plane waves is an additional phase change of 2
across the aperture of the lens, resulting in a difference in
spot positions comparable to the Rayleigh resolution limit.
This suggests an approach for separating OAM states, for
which a change in mode index of
‘ ¼ 1 corresponds to an
increment in the azimuthal phase change of 2. The key
optical component in this approach is one that transforms
azimuthal position in the input beam into a transverse
position in the output beam, i.e., an optical element that
transforms a helically phased beam into a transverse phase
gradient. This corresponds to the transformation of an
input image comprising concentric circles into an output
image of parallel lines. Mapping each input circle onto an
output line gives the required deviation in ray direction and
hence the phase profile of the transforming optical element.
However, the resulting variation in optical path length
means that the transformation introduces a phase distortion
PRL 105, 153601 (2010)
PHYSICAL REVIEW LETTERS
week ending
8 OCTOBER 2010
0031-9007=10=105(15)=153601(4) 153601-1 Ó 2010 The American Physical Society
that needs to be corrected by a second element. The trans-
forming system therefore comprises two custom optical
elements, one to transform the image and a second, posi-
tioned in the Fourier plane of the first, to correct for the
phase distortion. This transformation is an example of an
optical geometric transformation which has been previ-
ously studied in the context of optical image processing
[19]. It was shown that a geometric transformation can
only be implemented by a single optical element if the
mapping is conformal. The optical element performs a
mapping ðx; yÞ ° ðu; vÞ, where (x, y) and (u, v) are the
Cartesian coordinate systems in the input and output plane,
respectively. In our approach v ¼ a arctanðy=xÞ and the
conformal mapping requires u ¼a lnð
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
þ y
2
p
=bÞ,
similar to [20,21]. The phase profile of the transforming
optical element is then given by
1
ðx;yÞ¼
2a
f
yarctan
y
x
xln
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
þy
2
p
b
þx
; (1)
where is the wavelength of the incoming beam, and f is
the focal length of the Fourier-transforming lens. The
parameter a scales the transformed image and a ¼ d=2,
where d is the length of the transformed beam. b translates
the transformed image in the u direction and can be chosen
independently of a.
The required phase correction can be calculated by the
stationary phase approximation [21] and is given by
2
ðu; vÞ¼
2ab
f
exp
u
a
cos
v
a
; (2)
where u and v are the Cartesian coordinates in the Fourier
plane of the first element. Figures 1(a) and 1(b) show the
phase profiles of the transforming and phase-correcting
optical element, respectively. One can see that the trans-
forming optical element contains a line discontinuity. The
end of this line, i.e., the center of the phase profile, defines
the axis around which the OAM is measured.
A lens is inserted after the phase-correcting element to
focus the transformed beam, which now has a 2‘ phase
gradient, to a spot in its focal plane. In the plane of this
lens, the transformed beam is rectangular, meaning that the
diffraction limited focal spot is elongated in the direction
orthogonal to the direction in which the spot moves. The
transverse position of the spot changes as a function of ‘
and is given by
t
‘
¼
f
d
‘: (3)
We use diffractive spatial light modulators (SLMs) to
create the desired phase profiles. For monochromatic light,
an SLM can be programmed such that any desired phase
profile is applied to the first-order diffracted beam, limited
in complexity only by the spatial resolution of the SLM.
Figure 1(c) shows a schematic overview of the optical
system. We use Laguerre-Gaussian (LG) beams as our
OAM states. The first SLM, programmed with both phase
and intensity information [9], is used to generate any
superposition of LG modes. Using relay optics and an
iris to select the first-order diffracted beam, this input state
is directed onto the transforming element, displayed on the
second SLM, which performs the required geometrical
transformation in the back focal plane of the Fourier-
transforming lens. We choose d such that the transformed
beam fills 80% of the width of the phase-corrector element
in order to avoid diffraction effects at its edges. A third
SLM is used to project the phase-correcting element. The
diffracted beam from this SLM has a transverse phase
gradient dependent on the input OAM state. These direc-
tion states are focused onto a CCD array by a lens and, as
discussed above, the lateral position, t
‘
, of the resulting
elongated spots is proportional to the OAM state of the
incident beam.
Figure 2 shows modeled and observed phase and inten-
sity profiles at various places in the optical system for a
range of OAM states. The modeled data are calculated by
plane wave decomposition. In the second column, one can
see that an input beam with circular intensity profile is
unfolded to a rectangular intensity profile with a 2‘ phase
FIG. 1. Phase profiles of (a) the transforming and (b) the
phase-correcting optical element. d is the length of the trans-
formed beam. In (b) only that part of the phase-correcting
element is shown, that is illuminated by the transformed beam.
In the experiment, the phase profiles are displayed on the spa-
tial light modulators (SLMs) with 2 phase modulation.
(c) Schematic overview of the setup. We use SLMs to both
generate Laguerre-Gaussian beams (SLM1) and create the de-
sired phase profiles for the transforming and phase-correcting
optical elements (SLM2 and SLM3, respectively). L1 is the
Fourier-transforming lens and L2 focuses the transformed
beams. We use beam splitters to ensure perpendicular incidence
on the SLMs.
PRL 105, 153601 (2010)
PHYSICAL REVIEW LETTERS
week ending
8 OCTOBER 2010
153601-2
gradient. As predicted, the position of the elongated spot
changes with the OAM input state. We recorded the output
of the mode sorter for input states between ‘ ¼5 and
‘ ¼ 5. The experimentally observed spot positions are in
good agreement with the model prediction. Our system
further allows us to identify a superposition of OAM states,
as can be seen in the final row of Fig. 2, where an equal
superposition of ‘ ¼1 and ‘ ¼ 2 gives two separate
spots in the detector plane at the position of ‘ ¼1 and
‘ ¼ 2. We note that observed spots are slightly broader
than the modeled ones, which is due to aberrations intro-
duced by the optical system.
To directly measure the state of any input beam, we
define eleven, equally sized, rectangular regions in the
detector plane, all centered around one of the expected
spot position for the eleven input modes used in the ex-
periment. By measuring the total intensity in each of these
regions, we can determine the relative fraction of a specific
OAM state in the input beam. Figures 3(a) and 3(b) show
the results for eleven pure input states, both modeled and
observed, as shown in the third and fourth column of Fig. 2,
respectively. Since the spots for two neighboring states
slightly overlap, some of the light in a state leaks into
neighboring regions; i.e., there is some cross talk between
different states. This cross talk shows up as the off-
diagonal elements in Fig. 3. As described before, our
experimental results show slightly broader spots than the
modeled data and hence the off-diagonal elements are
slightly larger. It is clearly possible to determine the input
state of the light beam from the position of the output spot
in the detector plane.
A commonly used measure to quantify the amount of
cross talk between channels is the channel capacity, which
is the maximum amount of information that can be reliably
transmitted by an information carrier [22]. In an optical
system, this channel capacity can be quoted as ‘‘bits per
photon.’’ If a photon can be in one of N input states and its
state can be measured perfectly at the output, the channel
capacity takes the theoretical maximum value of log
2
N.
Table I presents the channel capacity of the system for
the modeled and observed results, calculated from the data
shown in Fig. 3. A generic approach to minimize cross talk
is to increase the separation between channels. We there-
fore consider the cases where we use only every other state,
‘ ¼ 2, and every third state, ‘ ¼ 3. This approach
gives fewer states, but less overlap between different spots.
In all cases, due to the experimental imperfections, the
channel capacity for the observed data is slightly lower
than the modeled one, but for l ¼ 3 it approaches the
model very closely.
We note that the optical transformation is only perfect
for rays which are normally incident on the transforming
element. Helically phased beams are inherently not of this
type, the skew angle of the rays being ‘=kr [23]. Although
this skew angle is small when compared to the angles
introduced by the transforming element, it might introduce
a slight transformation error which increases with ‘. If the
input is a ringlike intensity profile, the skew angle leads to
a sinusoidal distortion from the expected rectangular
output. This potential skew ray distortion is reduced by
FIG. 2 (color). Modeled and observed phase and intensity
profiles at various planes in the optical system. From left to
right, the images show the modeled phase and intensity distri-
bution of the input beam just before the transforming optical
element and just after the phase-correcting element, and the
modeled and observed images in the CCD plane for five different
values of ‘. The final row shows the results for an equal
superposition of ‘ ¼1 and ‘ ¼ 2. The last two columns are
6 magnified with respect to the first two columns.
TABLE I. Channel capacity calculated from the results shown
in Fig. 3. The first three columns show the separation between
the channels, l, the number of states taken into account, N, and
the theoretical maximum value, log
2
N. The last two columns
correspond to the data shown in Figs. 3(a) and 3(b), respectively.
‘ N log
2
N Modeled Observed
1 11 3.46 2.36 1.96
2 6 2.59 2.10 1.93
3 4 2.00 1.70 1.68
PRL 105, 153601 (2010)
PHYSICAL REVIEW LETTERS
week ending
8 OCTOBER 2010
153601-3
decreasing the propagation distance over which the trans-
formation occurs, i.e., reducing f.
In its present form, our approach is limited by the fact
that the resulting spots are slightly overlapping. This is
because our transformation discards the periodic nature of
the angular variable, using instead only a single angular
cycle and producing an inclined plane wave of finite width,
and similarly a finite width of spot. One option for im-
provement is to modify the transformation to give multiple
transverse cycles, which results in larger phase gradient
and thus a larger separation between the spots, albeit at the
expense of increased optical complexity. One approach to
implementing this improvement would be to add a binary
phase grating to the transforming elements, producing both
positive and negative diffraction orders. By adjusting the
pitch of the grating appropriately, two identical, adjoining
copies of the reformatted image are created in the plane of
the phase corrector.
We further recognize that there is a 70% light loss
associated with the two SLMs that comprise the mode
sorter. This loss could, however, be eliminated by replacing
the SLMs with the equivalent custom-made refractive
optical elements.
In conclusion, we have described a novel system com-
prising of two bespoke optical elements that can be used to
efficiently measure the OAM state of light. We have shown
numerical and observed data to support our method. The
method has a limitation due to the overlap of the spots for
different states that could be reduced by applying an addi-
tional diffraction grating to the first surface. The system
opens the way to many interesting investigations ranging
from experiments in multiport quantum entanglement [7],
to applications in astrophysics [24] and microscopy [25],
all of which make use of the OAM state basis.
We acknowledge Richard Bowman, Robert Boyd, Dan
Gauthier, and Eric Eliel for helpful discussions. G. C. G. B.
acknowledges the Pieter Langerhuizen Lambertuszoon-
fonds. M. J. P. is supported by the Royal Society. We would
like to thank Hamamatsu for their support. We acknowl-
edge the financial support of the Future and Emerging
Technologies (FET) program within the Seventh
Framework Programme for Research of the European
Commission, under the FET Open grant agreement
HIDEAS No. FP7-ICT-221906.
*j.berkhout@cosine.nl
†
m.lavery@physics.gla.ac.uk
[1] L. Allen et al., Phys. Rev. A 45, 8185 (1992).
[2] G. Molina-Terriza, J. P. Torres, and L. Torner, Nature
Phys. 3, 305 (2007).
[3] S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon.
Rev. 2, 299 (2008).
[4] G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev.
Lett. 88, 013601 (2001).
[5] G. Gibson et al., Opt. Express 12, 5448 (2004).
[6] N. Bloembergen, J. Opt. Soc. Am. 70, 1429 (1980).
[7] A. Vaziri, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 89,
240401 (2002).
[8] J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, Nature Phys. 4,
282 (2008).
[9] J. Leach et al., Science 329, 662 (2010).
[10] V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, J.
Mod. Opt. 39, 985 (1992).
[11] N. R. Heckenberg et al., Opt. Quantum Electron. 24, S951
(1992).
[12] A. Mair et al., Nature (London) 412, 313 (2001).
[13] S. N. Khonina et al., Opt. Commun. 175, 301 (2000).
[14] M. Padgett et al., Am. J. Phys. 64, 77 (1996).
[15] G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev.
Lett. 101, 100801 (2008).
[16] J. M. Hickmann et al.,
Phys. Rev. Lett. 105, 053904
(2010).
[17] J. Courtial et al., Phys. Rev. Lett. 81, 4828 (1998).
[18] J. Leach et al., Phys. Rev. Lett. 88, 257901 (2002).
[19] O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974).
[20] Y. Saito, S. Komatsu, and H. Ohzu, Opt. Commun. 47,8
(1983).
[21] W. J. Hossack and A. M. Darling, and A. Dahdouh, J.
Mod. Opt. 34, 1235 (1987).
[22] C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948).
[23] M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[24] M. Harwit, Astrophys. J. 597, 1266 (2003).
[25] S. Fu
¨
rhapter et al., Opt. Express 13, 689 (2005).
FIG. 3. Total intensities in all detector regions for pure input
OAM states from ‘ ¼5 to ‘ ¼ 5, for both the (a) modeled and
(b) observed results. The regions all have the same size and are
chosen such that they fill the entire aperture. The intensities are
shown as a fraction of the total intensity in the input beam.
PRL 105, 153601 (2010)
PHYSICAL REVIEW LETTERS
week ending
8 OCTOBER 2010
153601-4
