## Abstract

We demonstrate the Fourier relationship between angular position and angular momentum for a light mode. In particular we measure the distribution of orbital angular momentum states of light that has passed through an aperture and verify that the orbital angular momentum distribution is given by the complex Fourier-transform of the aperture function. We use spatial light modulators, configured as diffractive optical components, to define the initial orbital angular momentum state of the beam, set the defining aperture, and measure the angular momentum spread of the resulting beam. These measurements clearly confirm the Fourier relationship between angular momentum and angular position, even at light intensities corresponding to the single photon level.

© 2006 Optical Society of America

## 1. Introduction

The spin angular momentum of a light beam is manifest as circular polarisation. By
contrast, the orbital angular momentum of a light beam is manifest in helical phase
fronts, with an azimuthal phase term exp(*iℓφ*), which carry an
associated orbital angular momentum (OAM) of *ℓh̄* per photon [1]. Both the spin and orbital angular momentum of
light can be transferred to solid objects, causing them to rotate about their own axis
or around the beam axis respectively [2].
Quantized spin and OAM have been measured for single photons [3]. Just like linear momentum and linear position, angular momentum
and angular position are related by a Fourier relationship [4], linking the standard deviations of the measurements. This is a
purely classical phenomenon, but the Fourier relation also holds for quantum observables
and in the quantum regime the Fourier relation is associated with the Heisenberg
uncertainty principle. Although this concept forms the basis of various calculations
[5, 6]
and experiments [7] its validity has never been
directly tested. Here we present measurements testing the Fourier relation between the
orbital angular momentum of light and its azimuthal probability (i.e. intensity)
distribution.

## 2. Fourier conjugate pairs

Linear momentum and position are both unbounded and continuous variables of a physical
system and are related by a continuous Fourier transform. For angular momentum and
angular position the 2*π* periodic nature of the angle variable means
that the relationship is a Fourier-series leading to discrete values of the angular
momentum. Assuming a Fourier relationship between the distribution of angular momenta,
*ψℓ*, and the angular distribution, Ψ(*φ*), we can
express one observable as the generating function of the other [10],

When light passes through an aperture or mask with an angular dependance given by ψ
_{Mask}(*φ*) its phase and/or intensity profile is modified
such that

where for simplicity, we have omitted the normalisation factor. If the incident light is
in a pure OAM state, defined by a single value of *ℓ*, this simplifies
to

Note that as with the light beam in (2), the complex transmission function of the mask
can be expressed in terms of its angular harmonics with Fourier coefficients
*A*_{n} ,

where ${\sum}_{n=-\infty}^{+\infty}$|*A*_{n} |^{2} is the total intensity transmission of the mask. Upon transmission,
each OAM component of the incident light acquires OAM sidebands shifted by
δℓ=*n*, where the amplitude of each component is given by the
corresponding Fourier coefficient of the mask,

In the experiments presented here we have used hard-edge aperture segments of width Θ,
i. e. Ψ_{Mask}(*φ*)=1 for-Θ/2<*φ*≤Θ/2 and 0
elsewhere. A single-segment mask can be expressed in terms of its Fourier coefficients
as

hence giving OAM sidebands with amplitudes

More generally, any azimuthal intensity distribution with *m*-fold
symmetry only has angular harmonics at multiples of *m*. Extending the
design of the masks to comprise *m* identical equi-spaced apertures with
Fourier components

we obtain OAM sidebands with amplitudes

where *N* is an integer. Consequently, in apertures with two-fold
symmetry, only every second OAM sideband is present, and in three-fold symmetric
apertures only every third. In our experiments we use masks comprising
*m* hard-edge segments so that only every *m*
^{th} sideband within the sinc envelope is present.

The complex transmission function Ψ_{Mask} may also include phase information.
Specifically we consider the situation where each of the *m* hard-edged
aperture segments has a definite and non-zero relative phase, Φ _{q} . It is instructive to consider the *m*-fold symmetric composite
mask as a superposition of m single-segment apertures, each giving rise to its own set
of OAM sidebands which may constructively or destructive interfere. This interference
between the individual OAM sidebands constitutes a test of the Fourier relationship
between angle and angular momentum. In our experiments we investigate one representative
case, when the phase of the *m* segments advances in discrete steps so
that Φ _{q} =α2*πq*/*m*. The Fourier components and hence OAM
sidebands can then be calculated according to

If α is an integer, the central OAM component will be shifted by *δℓ*=α,
constructive interference will generate OAM sidebands at multiples of
*m*, and destructive interference will cancel any other sidebands. If
*α* is not integer, interference between light passing through the
different segments will modulate the sidebands. Note that if *m* tends to
infinity only the central peak at *δℓ*=α remains, turning the mask
effectively into a spiral phase plate with optical step height αλ [11]. It is worth pointing out that these considerations still hold
for pure phase masks by setting *mAδℓ*=1. Such masks generate OAM modes
of α modulo *m*, where *m* is given by the rotational
symmetry, and the integer α shifts between the different sets.

## 3. Experimental configuration

We generate a low intensity laser beam in a pure *ℓ*-state by
transforming a collimated He-Ne laser (*ℓ*=0) with a spatial light
modulator (see figure 1). Spatial light
modulators act as reconfigurable phase gratings, or holograms, giving control over the
complex amplitude of the diffracted beams. As is standard practice, our modulator is
programmed with a diffraction grating containing a fork dislocation to produce a beam
with helical phase front in the first diffraction order [8]. A second spatial light modulator is used to analyse the
*ℓ*-state. If the index of the analysing hologram is opposite to that
of the incoming beam, it makes the helical phase fronts of the incoming beam planar
again. A significant fraction of the resulting beam can then be coupled into a
single-mode optical fibre. If the beam and analysing hologram do not match, the
diffracted beam has helical phase fronts and therefore no on-axis intensity, resulting
in virtually no coupling into the fibre. To deduce the *ℓ*-state, the
analysing hologram is switched between various indices whilst monitoring the power
transmitted through the fibre. It should be emphasised that the cycling of the hologram
index makes the detection process inherently inefficient, where even with perfect
optical components, the quantum detection efficiency cannot exceed the reciprocal of the
number of different *ℓ*-states to be examined [9]. In principle an amplitude and/or phase mask could be introduced
at any position between the two spatial light modulators. However, combining aperture
and analysing hologram on a single spatial light modulator eases alignment and improves
optical efficiency. We achieve this combination by a modulo 2*π* addition
of the two holograms. We measure the light coupled into the single mode fibre with an
avalanche photodiode which enables photon counting. Inserting a neutral density filter
immediately after the laser restricts the maximum count rate to less than 100kHz so that
at any one time there is on average less than one photon within the apparatus.

To investigate the relation between the angular aperture function and the orbital
angular momentum states we adopt a family of aperture functions comprising
*m* equi-spaced segments of defined width and phase. For each aperture
function the transmitted photons are analysed for the orbital angular momentum
states-18<*ℓ*+*δℓ*<18. One complication is that
the manufacturing limitations of the spatial light modulators result in deviation from
optical flatness by three or four lambda over the full aperture. This degrades the
point-spread function of the diffracted beam and hence the efficiency of the mode
coupling into the fibre, spoiling the discrimination between different
*ℓ*-states. Therefore, prior to their use within this experiment, we
optimise each of the spatial light modulators by applying a hologram of the Zernike
terms compansating for astigmatism to give the smallest point spread function of the
diffracted *HG*_{00} mode.

## 4. Experimental results & discussion

For a single hard-edge aperture of uniform phase and width Θ, the resulting integer
angular momentum distribution has a sinc function envelope centred on the
*ℓ* of the incident mode, as given by (8). Figure 2 shows the measured OAM sidebands for a hard-edge aperture
of width Θ=*π*/4. We find almost perfect agreement between the observed
distribution and that predicted from the Fourier-relation. However, as discussed, a more
subtle test of the Fourier-relation is when the aperture function is multi-peaked and
when these peaks are offset in phase. Introducing an aperture comprising two segments of
the same width generates an OAM sideband distribution with the same envelope function,
but if the Fourier relationship holds true the sidebands can interfere either
constructively or destructively depending on the relative phase of the individual
components. Figure 3 compares the angular
momentum distribution as predicted from (11) with the one observed in the experiment for
the case of two (i.e. *m*=2) diametrically opposed hard-edge apertures,
each of angular width 2*π*/9. During the experimental sequence their
relative phase δΦ=Φ_{2}-Φ_{1} is varied from 0 to 2*π*(α
is varied from 0 to 1). As discussed, the OAM sidebands of the two segments differ in
their phase by (*δℓ*+*α*)×*π* . When α=0,
the OAM sidebands with odd *δℓ* interfere destructively. As the relative
phase increases, the light intensity in the odd modes rises at the expense of the even
modes until all the even modes disappear when δΦ=*π*. At intermediate
positions when δΦ=*π*/2 or 3*π*/2 even and odd sidebands
have equal weights. The width of the aperture Θ changes the width of the sinc
distribution but not the underlying interference effects. Figure 4 shows the results for four (i.e. *m*=4)
equi-spaced apertures, at a phase difference of *απ*/2. Increasing
*α* from 0 to 4 gives OAM sideband distributions in excellent
agreement to that predicted by (11).

We have shown that angle and angular momentum states are related as conjugate variables
by a Fourier transformation, and that this relationship holds for both amplitude and
phase. In doing so we have, in effect, provided a physical test of a mathematical
relationship: that between the angular coordinate *ϕ* and its Fourier
conjugate variable *ℓ*. Fourier relationships of this type give rise to
uncertainty relations between the standard deviations of the conjugate variables.
However, the 2*π* cyclic nature of angular measurement raises
difficulties in the formulation of an angular uncertainty relation and the definition of
a suitable angle operator. An angle operator should yield results defined within a
chosen 2*π* radian range [12].
This approach gives an uncertainty relation which limits the accuracy of possible
measurements to Δ*ϕ*Δ*L*_{z} ≥(*h̄*/2)|1-2*πP*(*ϕ*
_{0})|, where *P*(*ϕ*
_{0})=*P*(*ϕ*
_{0}+2*π*) is the normalised probability at the limit of the
angle range [10]. This uncertainty relation may
be seen as a consequence of the Fourier-relationship, directly demonstrated in this
paper. Throughout our investigations, we used low light intensities corresponding to
single photon flux rates. Although all our measurements were classical in nature, the
results and Fourier-relationship should also hold at the single photon or quantum level.
Furthermore, while demonstrated in the optical regime, the Fourier-relation is expected
to be valid for any system having a wave nature including superfluids or BECs.

Hard-edge apertures as used in this investgation, to shape the azimuthal distribution of a light beam, can be used to generate sidebands of the orbital angular momentum or indeed controlled superpositions of particular orbital angular momentum states. The presence of sidebands may create ambiguities if measured using holographic techniques but such modes are completely compatible with mode sorters based on the rotational symmetry of the modes [13].

This work was supported by the UK’s Engineering and Physical Sciences Research Council, JC and SFA are supported by the Royal Society.

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