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Efficient Sorting of Orbital Angular Momentum States of Light

Gregorius C.G. Berkhout,1,2,*Martin P.J. Lavery,3,†Johannes Courtial,3Marco W. Beijersbergen,1,2and Miles J. Padgett3

1Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands

2cosine Science & Computing BV, Niels Bohrweg 11, 2333 CA Leiden, The Netherlands

3School 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 discriminationof planewaves 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

whichachangeinmodeindexof?‘ ¼ 1correspondstoan

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

hencethephaseprofileofthetransformingopticalelement.

However, the resulting variation in optical path length

means that the transformation introduces a phase distortion

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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 aarctanðy=xÞ and the

conformal mapping requires u ¼ ?alnð

similar to [20,21]. The phase profile of the transforming

optical element is then given by

?

x

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2þ y2

p

=bÞ,

?1ðx;yÞ¼2?a

?f

yarctan

?y

?

?xln

?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b

x2þy2

p

?

þx

?

;

(1)

?2ðu;vÞ ¼ ?2?ab

?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 profilewith a 2?‘ phase

FIG. 1.

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.

Phase profiles of (a) the transforming and (b) the

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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 ‘ ¼ ?5 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 ‘ ¼ ?1 and ‘ ¼ 2 gives two separate

spots in the detector plane at the position of ‘ ¼ ?1 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, asshown inthe thirdand fourthcolumnof 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 log2N.

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 wherewe 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).

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 ‘ ¼ ?1 and ‘ ¼ 2. The last two columns are

6? magnified with respect to the first two columns.

Modeled and observed phase and intensity

TABLE I.

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, log2N. The last two columns

correspond to the data shown in Figs. 3(a) and 3(b), respectively.

Channel capacity calculated from the results shown

?‘N log2N

ModeledObserved

1

2

3

11

6

4

3.46

2.59

2.00

2.36

2.10

1.70

1.96

1.93

1.68

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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 planewave of finitewidth,

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 losscould,however,be eliminated byreplacing

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. Wewould

like to thank Hamamatsu for their support. We acknowl-

edge the financial support of the Future and Emerging

Technologies (FET)program

Framework Programme for Research of the European

Commission, under the FET Open grant agreement

HIDEAS No. FP7-ICT-221906.

withintheSeventh

*j.berkhout@cosine.nl

†m.lavery@physics.gla.ac.uk

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FIG. 3.

OAM states from ‘ ¼ ?5 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.

Total intensities in all detector regions for pure input

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