# Rapid readout of terahertz orbital angular momentum beams using atom-based
imaging

Lucy A. Downes*, Daniel J. Whiting, Charles S. Adams, Kevin J. Weatherill
Department of Physics, Durham University, DH1 3LE
<EMAIL_ADDRESS>
###### Abstract

We demonstrate the rapid readout of terahertz (THz) orbital angular momentum
(OAM) beams using an atomic-vapour based imaging technique. OAM modes with
both azimuthal and radial indices are created using phase-only transmission
plates. The beams undergo terahertz to optical conversion in an atomic vapour,
before being imaged in the far field using an optical CCD camera. In addition
to the spatial intensity profile, we also observe the self-interferogram of
the beams by imaging through a tilted lens, allowing the sign and magnitude of
the azimuthal index to be read out directly. Using this technique, we can
reliably read out the OAM mode of low-intensity beams with high fidelity in 10
$\mathrm{ms}$. Such a demonstration is expected to have far-reaching
consequences for proposed applications of terahertz OAM beams in
communications and microscopy.

##  I Introduction

The orbital angular momentum (OAM) of light has been of fundamental interest
in optics research for the past thirty years [1, 2] and OAM beams have found
many applications, notably in free-space information transfer [3, 4] and
encryption [5]. Recently, there has been growing interest in OAM beams outside
of the optical frequency range. For example, Bragg mirrors have been used to
create OAM X-ray beams from a free-electron laser [6] and microwave OAM beams
have been applied to radar imaging techniques [7, 8]. There has also been
considerable interest in the creation, manipulation and detection of OAM beams
in the THz frequency range [9] for applications as diverse as short-range
communications [10], super resolution imaging [11] and astronomy [12]. The
creation of THz OAM beams has been demonstrated using numerous methods
including spiral phase-plates [13, 14], helical axicons [15], V-shaped antenna
structures [16], polarizing optical elements [17], THz spatial light
modulators [18] and soft-aperture difference frequency generation [19].
However, the detection and readout of these beams remains challenging, largely
due to the lack of fast and sensitive terahertz detectors [20].

In this work we demonstrate the rapid (10 ms) readout of terahertz OAM beam
modes using a recently-developed, atom-based THz imaging technology [21, 22],
and show that we can distinguish between beams of varying azimuthal and radial
phase index. This work offers a way forward in realising the demanding
applications of THz OAM beams.

##  II Experiment

The experimental setup is shown in Fig. 1a). The THz field is a 20
$\mathrm{\SIUnitSymbolMicro W}$, 0.55 $\mathrm{THz}$ free space beam produced
by an amplifier multiplier chain and diagonal horn antenna. The beam is
predominantly a TEM00 mode with 84% Gaussian mode content [23] and is
collimated using a Teflon lens with focal length $f=75\,$\mathrm{mm}$$ to give
a $1/e^{2}$ radius of $w_{0}=5.55\,$\mathrm{mm}$$. Teflon phase plates,
manufactured by CNC machining, are then placed in the centre of the THz beam,
at a position approximately $75\,$\mathrm{mm}$$ from the first lens. The
resulting beam passes though a second $f=75\,$\mathrm{mm}$$ Teflon lens and is
imaged in the focal plane. The final Teflon lens can be positioned parallel to
the phase plate (solid lines, Fig. 1a)) or tilted with respect to the phase
plate (dashed lines, Fig. 1a)). We refer to the image observed in the case of
the parallel lens as the intensity pattern and the image produced by the
tilted lens as the auto-interference pattern [24]. The theory behind the
differences in the structure of these patterns is described in detail in [25].
Photographs of example phase plates are shown in Fig. 1b) along with plots
showing the thickness of the Teflon and the resulting phase profile of the
plates. Both azimuthal phase, $\ell$, and radial phase, $p$, can be applied.

Figure 1: Experimental setup and phase plate design a) The phase plates are
placed at the centre of a collimated THz beam (purple dotted line) which is
focused onto an atomic vapour using a second Teflon lens. This lens can be
tilted with respect to the phase plate at an angle $\theta$, where
$\theta=0$\mathrm{\SIUnitSymbolDegree}$$ corresponds to the lens being
parallel to the phase plate. The resulting fluorescence from the atomic vapour
is then imaged using an optical camera. b) Photographs and thickness/phase
profiles of the 50 $\mathrm{mm}$ diameter Teflon plates used to create THz OAM
beams, designed using the method described in [26]. The design frequency is
550 $\mathrm{GHz}$, equal to that of the THz field used in the atomic imaging
system. Note that the constant minimum thickness of 2 $\mathrm{mm}$ is a
result of manufacturing constraints and does not alter the phase profile.

The method used to image the THz field is the same as described in [22] and
utilises the concept of THz-to-optical conversion by an atomic vapour [21].
Briefly, three resonant infrared lasers, stabilised to atomic transitions [27,
28, 29], are used to excite caesium atoms to a high-lying energy state, a so-
called Rydberg state. All Rydberg transition frequencies are calculated using
the ARC Python package [30]. The laser beams are shaped such that, at the
position of the glass cell containing Cs vapour, they overlap and form a light
sheet of approximately $10\,$\mathrm{mm}$\times 10\,$\mathrm{mm}$\times
100\,$\mathrm{\SIUnitSymbolMicro m}$$ in the $x,y$ and $z$ directions
respectively. The Rydberg atoms decay back to the electronic ground state
through the emission of visible fluorescence, the wavelength of which depends
on the pathway taken. Once excited, interaction with a resonant THz field
transfers atomic population from the initial Rydberg state to another nearby
Rydberg state, from which there is one dominant decay pathway leading to the
emission of green (535 $\mathrm{nm}$) photons. Therefore, by looking at the
intensity of the 535 $\mathrm{nm}$ fluorescence emitted by the vapour we can
infer the spatial distribution of the incident THz field. The THz beam
propagates perpendicular to the IR lasers used to create the light sheet and
the atomic fluorescence is imaged from the opposite side of the vapour cell to
that on which the THz is incident, as shown in Fig. 1a). Throughout this work
we use an Andor iXon EMCCD to image the optical fluorescence from the vapour
with a $(535\pm 6)\,$\mathrm{nm}$$ bandpass filter to eliminate unwanted
background light. All images are taken using an exposure time of 100
$\mathrm{ms}$ unless otherwise stated, and an image of the vapour in the
absence of the THz field is subtracted to remove background fluorescence and
increase contrast.

##  III Results and Discussion

Figure 2: Intensity and auto-interference patterns of THz OAM beams created
using spiral phase plates. Intensity patterns of beams with
$p=0,\>|\ell|=1,2,3$. The top row shows the images produced when the second
lens is parallel to the phase plate ($\theta=0$\mathrm{\SIUnitSymbolDegree}$$)
while the second and third rows show the resulting intensity pattern for a
lens tilt of $\theta\approx 30$\mathrm{\SIUnitSymbolDegree}$$. Since the
donut-like patterns are identical for $\ell=-\ell$, only the images for
positive $l$ are shown here. The second and third rows show the auto-
interference patterns for positive and negative values of $\ell$ respectively.

Images of the intensity patterns produced for beams with $p=0,\,\ell=1,2,3$
are shown in Fig. 2. The top row shows the beam intensity imaged with the lens
parallel to the phase plate ($\theta=0$\mathrm{\SIUnitSymbolDegree}$$) for
increasing values of $\ell$. As expected the intensity patterns for beams with
$p=0,\ell\neq 0$ have a ‘donut-like’ structure; a central vortex surrounded by
a single ring, the radius of which is determined by the value of $\ell$. Since
these intensity patterns are identical for $\ell=-\ell$ only the images for
beams with positive values of $\ell$ are shown. The lower two rows show the
auto-interference patterns resulting from a lens tilt of $\theta\approx
30$\mathrm{\SIUnitSymbolDegree}$$. These patterns not only display a number of
dark fringes equal to the value of $\ell$, but also exhibit a
$45$\mathrm{\SIUnitSymbolDegree}$$ rotation in the imaging plane. The
direction of this rotation depends on the sign of $\ell$, allowing this to be
determined from the pattern. This rotation is not affected by the direction of
tilt of the final lens which remained unchanged between rows two and three.

Figure 3 shows the intensity (top row) and auto-interference patterns (bottom
row) for beams with $p=1,\>\ell=1,2$. The intensity patterns of beams with
both $\ell\neq 0$ and $p\neq 0$ have a central vortex surrounded by a number
of rings equal to $p+1$. The radius of the rings is again determined by the
value of $\ell$. Their auto-interference patterns exhibit orthogonal dark
fringes, where the number of fringes corresponds to the values of $p$ and
$p+|\ell|$ (shown in the second row of Fig. 3 by the white dashed and black
dotted lines respectively). Again the rotation of these fringe patterns
depends solely on the sign of the index $\ell$. As $p$ increases the beams
produced become larger and no longer fit within our $1\,$\mathrm{cm}$^{2}$
imaging area. As a result we see interference patterns at the edges of the
image where the THz field is reflected from the walls of the glass cell
containing the atomic vapour. These interference effects can be seen as
vertical stripes at the edges of the images in Fig. 3, and is the reason why
we were unable to image beams with $p>1$ clearly.

Figure 3: Intensity and auto-interference patterns for beams with
$\mathbf{\bm{\mathit{\ell}}\neq 0}$ and $\mathbf{\bm{\mathit{p}}\neq 0}$.
Intensity (top) and auto-interference patterns (bottom) for beams with
$p=1,\>\ell=1,2$ (left and right columns respectively). The white dashed and
black dotted lines have been added to the auto-interference patterns to
highlight the dark fringes from which we can extract the values of $p$ and
$p+|\ell|$. Only the auto-interference patterns for positive $\ell$ values are
shown here, those for negative $\ell$ values would be tilted in the opposite
direction with respect to the vertical.

By using phase plates made from a material with a low absorption coefficient
in this frequency range such as Teflon ($\alpha=0.2\,$\mathrm{cm}$^{-1}$ at
550 $\mathrm{GHz}$ [31]) we achieve over 90% transmission, even for the
thickest plates used in this work ($\approx 5\,$\mathrm{mm}$$). Because of
these low losses we can combine phase plates additively to form beams with the
sum of the indices of the plates used. For example placing the $\ell=1,\>p=0$
and the $\ell=2,\>p=0$ plates sequentially in the beam creates an intensity
pattern equivalent to the $\ell=3,\>p=0$ case. This is shown in the left hand
column of Figure 4. Similarly reversing the direction of the $\ell=1,\>p=0$
plate creates a beam with intensity pattern equivalent to the $\ell=1,\>p=0$
case, as shown on the right hand side of Figure 4. In both cases the images
created using multiple plates are not as clear as those created using a single
plate of equivalent topological charge. This is due to unwanted aberrations
induced by each plate.

Figure 4: Combinations of phase plates. Intensity (top) and auto-interference
patterns (bottom) for OAM beams produced by combining phase plates with
$p=0,\>\ell=1$ and $p=0,\>\ell=2$. In the left hand column the plates were
placed in the same direction, hence the resulting beam has $\ell=3$. In the
second column the $\ell=1$ phase plate was reversed resulting in the final
beam having $\ell=1$.

From Figures 2 and 3 we observe that the radius of the rings scales
approximately linearly with the value of $|\ell|$. Measuring the radius of
maximum intensity ($r_{\ell}$) for the intensity patterns produced by the
spiral phase plates for the $p=0$ case (top row, Fig. 2) we find that they
scale as $r_{\ell=2}=1.92\times r_{\ell=1}$ and $r_{\ell=3}=2.85\times
r_{\ell=1}$. For a pure Laguerre-Gauss mode we would expect the radius of
maximum intensity to scale as $\sqrt{|\ell|}$ [32] however, a linear scaling
has previously been observed in optical vortices created using spiral phase
plates [33, 34, 35]. This discrepancy in scaling is because the beams created
by the phase plates are not pure LG modes but are instead a superposition of
modes with different $\ell$ and $p$, meaning that the resulting beams do not
share the same scaling properties.

Figure 5: Fringe visibility at reduced exposure times. a) Fitted fringe
amplitude as a function of exposure time for the $p=0,\>\ell=3$ auto-
interference pattern. Below $10~{}$\mathrm{ms}$$ the fitted amplitude is lower
than its associated uncertainty so we say that the fringes are
indistinguishable at this point. Insets show data (dark purple) and
corresponding fits (light purple) for the shortest (top left) and longest
(bottom right) exposures. b) Fitted fringe spacing as a function of exposure
time for $p=0,\>\ell=1,2,3$. Again below $10~{}$\mathrm{ms}$$ the fitted
fringe spacing is not consistent, and hence the value of $\ell$ cannot be
determined.

To demonstrate how the fast readout capabilities of the imaging technique
presented in [22] could be applied to OAM beams, we image auto-interference
patterns as the exposure time is reduced. For each image a cross-section is
taken along the diagonal and a sine function fitted to the fringe pattern.
Initially the fringe spacing and position are fixed in the fitting, the only
free parameters are the amplitude of the fringes and the background level.
This is shown in Fig. 5a) for beams with $p=0,\ell=3$. From this analysis it
can be seen that the fringes are distinguishable at exposure times greater
than $10~{}$\mathrm{ms}$$. At shorter exposure times the fitted amplitude of
the fringes is less than the associated error and hence the fringes cannot be
distinguished from the noise. Fig. 5b) shows the results of letting both the
fringe amplitude and spacing vary for cross-sections of the $p=0,\>\ell=1,2,3$
auto-interference patterns. Again, at exposure times greater than
$10~{}$\mathrm{ms}$$ the fitting can reliably distinguish between the
different orders of $\ell$, shown by the consistency of the fitted spacing. We
note that the limit of $10~{}$\mathrm{ms}$$ presented here is specific to the
camera used and may be reduced by using a camera better suited to shorter
exposures.

While phase plates present an easy way to create and manipulate arbitrary OAM
beams in the THz frequency range, in order to use these beams for information
transfer a method of quickly switching between values of $\ell$ and $p$ would
be needed to achieve reasonable data transfer rates. Fast switching could be
realised by using a spatial light modulator (SLM) [36] to rapidly modify the
helicity of the THz beam, or by using a programmable reflectarray antenna
[37]. Combining these techniques with the imaging method presented here would
enable fast information transfer in the THz regime. Furthermore, the use of an
atomic detection system allows for the calibrated electric field measurement
of such beams [21, 38].

##  IV Conclusion

We have demonstrated high-speed readout of THz OAM beams created using Teflon
phase plates with both varying $\ell$ and $p$ indices. We have also
demonstrated that the sign of the phase can be determined by imaging the beam
through a tilted lens, and that this can be done at exposure times down to 10
$\mathrm{ms}$. We note that these phase plates do not create pure Laguerre-
Gauss modes as demonstrated by the fact that the vortex radius scales with
$\ell$ and not with $\sqrt{\ell}$ as would be expected for a pure mode. This
high-speed method of determining the characteristics of THz OAM beams could
enable their use in free-space data transfer.

Acknowledgements: We thank Stephen Lishman, Lee Bainbridge and Andrew Davies
for the manufacture of the phase plates, Neal Radwell, Andrew Forbes and Ifan
Hughes for fruitful discussions and Mike Tarbutt for the loan of equipment.

Funding: Engineering and Physical Sciences Research Council (EP/R002061/1,
EP/S015973/1). European Commission Quantum Technology Flagship project
macQsimal (820393)

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