2736
Letter
Vol. 44, No. 11 / 1 June 2019 / Optics Letters
Quantitative orbital angular momentum
measurement of perfect vortex beams
JONATHAN PINNELL,
VALERIA RODRÍGUEZ-FAJARDO,
AND
ANDREW FORBES*
School of Physics, University of the Witwatersrand, Johannesburg 2000, South Africa
*Corresponding author: andrew.forbes@wits.ac.za
Received 3 April 2019; revised 30 April 2019; accepted 4 May 2019; posted 6 May 2019 (Doc. ID 364028); published 23 May 2019
Perfect (optical) vortices (PVs) have the mooted ability to
encode orbital angular momentum (OAM) onto the field
within a well-defined annular ring. Although this makes
the near-field radial profile independent of OAM, the
far-field radial profile nevertheless scales with OAM, forming a Bessel structure. As yet, the quantitative measurement
of the OAM of PVs has been elusive, with current detection
protocols opting for more qualitative procedures using interference or mode sorters. Here, we show that the OAM
content of a PV can be measured quantitatively using optical modal decomposition: an already widely utilized technique for decomposing an arbitrary light field into a set of
basis functions. We outline the theory and confirm it by
experiment with holograms written to spatial light modulators, highlighting the care required for accurate decomposition of the OAM content. Our work will be of interest
to the large community who seek to use such structured
light fields in various applications, including optical trapping and tweezing, and optical communications. © 2019
Optical Society of America
https://doi.org/10.1364/OL.44.002736
Complex structured light fields have been demonstrated in a
myriad of forms [1] and found many applications covering both
the classical and quantum regimes [2]. Foremost among these
are helical phase structures that carry orbital angular momentum (OAM) [3,4]. Such modes have an azimuthal phase structure: expilϕ where ϕ is the azimuthal angle and carries lℏ of
OAM per photon, with l an integer. Such OAM modes are
conventionally created by passing a Gaussian beam through
a helical phase, often on a spatial light modulator (SLM)
[5], to create a vortex beam as an approximation to the azimuthal modes in the Laguerre–Gaussian basis. This approach produces hypergeometric modes [6] with little power in the desired
mode of helicity l and radial order p 0 [7], as well as a second moment radius that scales linearly with l. For OAM
modes in the Laguerre–Gaussian
pffiffiffiffiffiffi basis, the ring radius and second moment size scales as jlj. To overcome this, the concept
of a perfect (optical) vortex (PV) was introduced [8]. The PV
has an annular ring whose radius and thickness are independent
of the encoded helicity. Such annular structures are well-known
as the Fourier transform of Bessel beams [9,10] and as such are
0146-9592/19/112736-04 Journal © 2019 Optical Society of America
not propagation invariant. The price to pay for an OAMindependent radial scale in the near-field is an OAMdependent radial scale in the far-field [11].
PVs with OAM have been extensively created [12–19] and
applied [20–25], but measurement techniques are still in their
infancy [26]. In contrast, OAM has been measured qualitatively
using mode sorters for Laguerre–Gaussian and helical beams
[27–30], extended to fractional topological charges [31] and
radial modes [32,33], with adapted approaches to find the
mode indices for Bessel–Gaussian [34–36] and HermiteGaussian modes [37]. These approaches can detect the mode
but cannot return the full information of the field; they cannot
measure inter-modal phases and/or project the field into an
orthonormal basis. Hence, we consider these methods to be
qualitative but useful in their own right. Modal decomposition,
on the other hand, is a general approach to reconstruct any
optical field quantitatively [38–41]. It has been exploited for
the measurement of phase and wavefronts [42], OAM density
[43], and beam quality factors [40,44].
Modal decomposition of OAM beams involves flattening the
azimuthal phase with a spiral phase optic of conjugate helicity
and observing the resulting on-axis intensity at the Fourier
plane of a lens. This method is intended to be independent
of the radial field which carries the vortex. For PVs, however,
there is a misconception that this method will not work
[10,21,26,45]. The reasoning is that a spiral phase optic by itself
cannot transform the PV back into a Gaussian, which is claimed
to be necessary for there to be an on-axis signal. This reasoning
would suggest that a lens, axicon, and spiral phase optic are required to measure the OAM of the PV (essentially performing
the steps for PV generation in reverse).
Here, we demonstrate that the OAM content of a PV can
indeed be measured quantitatively using modal decomposition
and outlining the theory and experiment to do so. We show
accurate reconstruction of the OAM properties and show that
this approach works for superpositions of OAM PVs. We highlight that, in principle at least, the only requirement for quantitative OAM detection of any beam is that its azimuthal phase
is separable from its radial amplitude.
We begin with the generation of PV beams. As stated earlier,
PVs are simply the Fourier transform of the well-studied Bessel
beams. In this Letter, however, we consider the PV to be the
near-field (NF) spatial distribution and the Bessel beam the
Letter
Vol. 44, No. 11 / 1 June 2019 / Optics Letters
2737
Fig. 1. Schematic of the experimental set-up where Li are lenses of focal length f i . The PV (NF) and BG (FF) fields are generated directly on SLM1 .
Example holograms and their corresponding generated PV (NF) and BG (FF) transverse intensities are shown in the upper section with k r 15 mm−1 ,
w0 2 mm, and f 1000 mm. Insets display simulated images. Example camera images at the detection plane are shown in the lower section for
the decomposition of the jl 15i mode. The cross hairs correspond to the optical axis: the region where the on-axis intensity is measured.
far-field (FF) distribution. An ideal PV which is dual to an ideal
Bessel beam would be an annular ring of infinitesimal thickness
and radius R kr f ∕k, where kr is the radial wave number of
the Bessel beam, f is the focal length of the Fourier lens, and k
is the wave number of the light. An ideal Bessel beam cannot be
experimentally realized and so we turn to its finite-energy
approximation: the Bessel–Gauss (BG) beam whose transverse
electric field in cylindrical coordinates r, ϕ at the waist plane
is described as
2
r
E F F r, ϕ ∝ J l k r r exp − 2 expilϕ,
(1)
w0
where J l · is the Bessel function of the first kind of order l,
and w0 is the Gaussian waist radius. The dual PV of a BG beam
is no longer infinitesimally thin but has thickness
T 2f ∕kw0 . The electric field of this experimentally realizable PV at the waist plane is given by [10],
2
w0
r R2
2Rr
exp −
E NF r, ϕ ∝
Il
expilϕ, (2)
T2
T
T2
where I l · is the modified Bessel function. In the upper section of Fig. 1, the theoretical and experimental transverse intensities at the waist plane are shown for PV (NF) and BG (FF)
beams carrying single OAM modes and superpositions. The
correlation coefficient between the theoretical and experimental
intensities is between 97 − 99% in all cases, indicating that the
experimental beams were generated with high fidelity.
We now turn to the quantitative OAM detection of PVs by
modal decomposition. In the general case, suppose that one
wishes to express some initially unknown field Ex in some
basis Φn x, where the components of x are transverse spatial
coordinates. The task is then to find the expansion coefficients
(c n ) in
Ex X
c n Φn x:
(3)
Using the orthogonality of the basis functions, the expansion
coefficients can then be computed from the inner product
Z
(4)
c n d2 x Ex Φn x hΦn xjE xi:
How is this done optically? First, let the field interact with an
optical element whose transmission function is T x Φn x
(such as a SLM displaying a computer generated hologram).
Now consider what happens as this modified field passes
through a Fourier lens of focal length f . At the back focal plane
of the lens,
Z
SLMlens
π
2
Ex ! Ek ∝ d x Ex Φn x exp −i k · x ,
λf
(5)
where λ is the light’s wavelength. If we restrict our gaze to the
on-axis intensity of the beam where k 0, we find
Z
2
2
2
(6)
jE 0j ∝ d x Ex Φn x jc n j2 :
This, then, is an effective way of finding the magnitude of the
expansion coefficients in Eq. (3) optically. The inter-modal
phases can be found by performing two additional optical inner
products with respect to a chosen reference mode [43]; for
simplicity, we do not consider this here.
To reconstruct the OAM properties of the field, we use basis
functions as the OAM eigenstates Φn ϕ expinϕ. Now,
suppose that the azimuthal phase of the field can be factorized
from its radial amplitude, writing Er, ϕ R l r expilϕ
where, in general, the radial amplitude depends on l. It then
follows that
2738
Vol. 44, No. 11 / 1 June 2019 / Optics Letters
Z
jE0j2 ∝ 0
∞
Z
rdrR l r
jγ l j2 δl,n ;
0
2π
2
dϕ expil − nϕ ,
Letter
(7)
(8)
where the last relation holds because the radial integral of the
amplitude evaluates to a constant γ l (which will depend on l if
the radial amplitude does). This result shows that there will be
an on-axis intensity in the Fourier plane if, and only if, the
helical phase of the field has been unwrapped correctly. We
see that PVs are just a special case since, from Eqs. (1) and
(2), the azimuthal phase is indeed separable from the radial amplitude. Further, if the initial field is composed of a superposition of OAM, then Eq. (8) becomes a sum of such terms and
the decomposition is still effective. One can also opt to decompose into the orthonormal Laguerre–Gauss basis, which will
introduce an extra radial factor to the radial integral in
Eq. (7). Lastly, we note that if γ l is identically zero, then
the method clearly breaks down but we don’t know of an
example where this is true.
To confirm the OAM decomposition, we perform the experiment shown schematically in Fig. 1. The beam from a He–Ne
laser is expanded and collimated onto the first SLM, which is
then relayed to the second SLM using a 4f lens system. A camera (Point Grey Firefly) is placed at the Fourier plane of the last
lens for the detection of the on-axis intensity. The two phaseonly SLMs (Holoeye Pluto) carry the bulk of the workload:
the first generates the PV and BG modes directly using complex-amplitude modulation [46]. This was done so as to keep
the experimental system static, thus enabling easy switching between detection in the NF (PV) and FF (BG). The second SLM
scans through a set of forked holograms (one at a time) encoding
Φn . If desired, one can multiplex many holograms onto the detection SLM enabling the full set of OAM measurements to be
performed in a single-shot. The parameters of all experimentally
generated modes were kept constant, and only the topological
charge l was varied. In addition, all generated beams were
normalized to unit amplitude and all measured intensities are
normalized with respect to the maximum camera grayscale for
the particular settings used.
The single mode decomposition results are shown in Fig. 2.
We find excellent agreement with Eq. (8) over a large range of
OAM values: l ∈ −25, 25. The first row of figures display
cross-talk matrices; these are used to quantify the effectiveness
of the detection system. Ideally, one would obtain an identity
matrix indicating that the detection system can successfully decompose what was generated. We see that the OAM mode
cross-talk in the detection system is minimal, owing to the negligible off-diagonal matrix components. Note how the on-axis
signal falls to background level as jlj → 0 when decomposing
in the FF (where the vortex ceases to be “perfect”). This is highlighted in the figure beneath which shows the diagonal part of
the cross-talk matrices or jγ l j2 : the term which governs the
relative magnitude of the on-axis signal. In these measurements, the camera settings were kept fixed to highlight the
variation of the on-axis signal.
Figure 3 shows the decomposition of various symmetric
superpositons of PVs beams, where each row corresponds to
a generated OAM mode of the form j − li jli. As claimed
earlier, the decomposition is still effective because there is no
ambiguity as to which OAM modes contributed to the OAM
Fig. 2. Experimental results of the single mode OAM decomposition of PVs. The first two figures show the cross-talk matrices. The
figure beneath shows the diagonal part of these matrices (bars) and
the theoretical values (dotted lines). Here, the camera settings were
fixed throughout.
spectrum; this is especially true in the NF where the PV exists.
Similar to before, when decomposing in the FF, the on-axis
signal falls to background level as jlj decreases. To combat this,
the camera’s settings (exposure time and gain) were dynamically
adjusted to improve the signal-to noise ratio for decreasing jlj.
For the FF (BG) decompositions, we can interpret the onaxis signal variation as the “cost” for having an OAM dependent
Fig. 3. Experimental decompositions of symmetric superpositions
of PVs. For the FF case, the camera settings were dynamically adjusted
for each row of measurements.
Letter
radial scale. The term that governs this variation is the
factor jγ l j2 in Eq. (7) which inherits it’s l-dependence from
R l r. Note that this factor is not an energy term since the
transverse
energy of theR electric field is proportional to
R∞
2
2
∞
2
0 rdrjR l rj ≠ jγ l j j 0 rdrR l rj . Since cameras have
finite dynamical range and jγ 0 j2 ∕jγ l j2 → 0 for large l, this
makes the OAM decomposition of FF PVs difficult since some
on-axis signals will be orders of magnitude smaller than others.
For the case of NF PVs, the radial amplitude is effectively
l-independent and so the on-axis signal changes negligibly
with l.
Another consequence of an OAM-dependent radial scale is
that one must apply correction factors to the raw OAM spectrum to obtain the correct spectrum. This is also true more
generally when decomposing into a non-orthonormal basis
(the OAM eigenstates are not a basis of the transverse plane).
In other words, one needs to negate the on-axis signal variation,
which is done by dividing the spectrum jc l j by the corresponding set of jγ l j values. These factors can be determined either
numerically (by computing the radial integral in Eq. (7)) or
empirically (from the diagonal part of the cross-talk matrix).
In conclusion, we have examined the theory and shown experimental results demonstrating the detection of the OAM of
PVs with a simple and widely used optical modal decomposition technique, something which was previously thought to be
unviable. The OAM-independence of the ring radius of PVs in
the NF means that the decomposition in this plane is much
more straightforward than in the FF. This is due to the constancy of the on-axis signal in the detection plane (quantified by
jγ l j) and means that no correction factors need be applied post
hoc and also that the camera settings can be fixed. Further, using
NF PVs vastly increases the range of usable OAM modes. If
allowed to propagate to the FF, where the field transforms
to a Bessel beam, then these advantages are lost. Hence, there
is a clear advantage for using PVs as the “carriers” of OAM over
BGs. Similar arguments can be applied in comparison to other
mode sets, therefore making PVs useful in many applications
where regular vortex modes are currently used.
Funding. Council for Scientific and Industrial Research
(CSIR).
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