††thanks: Publication of the U.S. government, not subject to U.S. copyright.

# Determining the Angle-of-Arrival of an Radio-Frequency Source with a Rydberg
Atom-Based Sensor

Amy K. Robinson  Nikunjkumar Prajapati  Depart. of Electr. Engin., University
of Colorado, Boulder, CO 80305, USA  Damir Senic  ANSYS, Inc., Boulder, CO,
USA  Matthew T. Simons  Christopher L. Holloway<EMAIL_ADDRESS>National Institute of Standards and Technology, Boulder, CO 80305, USA

###### Abstract

In this work, we demonstrate the use of a Rydberg atom-based sensor for
determining the angle-of-arrival of an incident radio-frequency (RF) wave or
signal. The technique uses electromagnetically induced transparency in Rydberg
atomic vapor in conjunction with a heterodyne Rydberg atom-based mixer. The
Rydberg atom mixer measures the phase of the incident RF wave at two different
locations inside an atomic vapor cell. The phase difference at these two
locations is related to the direction of arrival of the incident RF wave. To
demonstrate this approach, we measure phase differences of an incident 19.18
GHz wave at two locations inside a vapor cell filled with cesium atoms for
various incident angles. Comparisons of these measurements to both full-wave
simulation and to a plane-wave theoretical model show that these atom-based
sub-wavelength phase measurements can be used to determine the angle-of-
arrival of an RF field.

The ability to measure angle-of-arrival (AoA) is of great importance to radar
and advanced communications applications. Here we present a method of
determining AoA based on Rydberg-atom sensors. Atom-based sensors have
garnered a lot of attention in the past several years because of their many
possible advantages over other conventional technologies. Measurement
standards have evolved towards atom-based measurements over the last couple
decades; most notably length (m), frequency (Hz), and time (s) standards.
Recently there has been a great interest in extending this to magnetic and
electric (E) field sensors. In particular, since the initiation and completion
of DARPA’s QuASAR program, NIST and other groups have made great progress in
the development of Rydberg atom-based radio-frequency (RF) E-field sensors
gor1 ; sed1 ; holl1 ; holl2 ; holl3 ; sed2 ; tan1 ; gor2 ; fan ; sim1 ; sim2 ;
anderson1 . The Rydberg atom-based sensors now have the capability of
measuring amplitude, polarization, and phase of the RF field. As such, various
applications are beginning to emerge. These include SI-traceable E-field
probes holl1 ; holl2 , power-sensorsholl5 , receivers for communication
signals (AM/FM modulated and digital phase modulation signals) song1 ; meyer1
; holl6 ; cox1 ; holl4 ; anderson2 , and even recording musical
instrumentsholl7 . In this paper, we investigate the capability of a Rydberg
atom-based sensor for determining AoA of an incident RF field.

The majority of the work on Rydberg atom-based E-field sensors uses on-
resonant electromagnetically induced transparency (EIT) and Autler-Townes (AT)
splitting techniques sed2 ; holl2 ; holl3 . The concept uses a vapor of alkali
atoms placed in a glass cell (referred to as a “vapor cell”) as a means of
detecting and receiving the RF E-field or signal. The EIT technique involves
using two lasers. One laser (called a “probe” laser) is used to monitor the
optical response of the medium in the vapor cell and a second laser (called a
“coupling” laser) is used to establish a coherence in the atomic system. When
the RF E-field is applied, it alters the susceptibility of the atomic vapor
seen by the probe laser. By detecting the power in the probe laser propagating
through the cell, the RF E-field strength can be determined. This approach has
shown to be very successful for determining the magnitude of an RF E-field.
However, an alternative approach is required to measure phase, which is
necessary to determine AoA. Recently, we developed a heterodyne technique
using a Rydberg atom-based mixer sim3 . In this approach, a reference RF field
is applied to the atoms. This reference RF field is on-resonance with the
Rydberg-atom transition, and acts as a local oscillator (LO). The LO field
causes the EIT/AT effect in the Rydberg atoms which is used to down-convert a
second, co-polarized RF field (referred to as SIG and is the field for which
the phase is desired). The SIG field is detuned (by a few kHz) from the LO
field. The frequency difference between the LO and the SIG is an intermediate
frequency (IF) and the IF is detected by optically probing the Rydberg atoms.
This IF is essentially the beat-note between the LO and SIG frequencies. The
phase of the IF signal corresponds directly to the relative phase between the
LO and SIG signals. In effect, the atoms down-convert the SIG to the IF, and
the phase of SIG is obtained by the probe laser propagating through the atomic
vapor.

In order to determine the AoA, the phase ($\phi$) of SIG is needed at two
different locations, see Fig. 1. Once the phase of SIG is determined at the
two different locations, the relationship between AoA (defined as $\theta$ in
Fig. 1) and the phase difference at the two locations (location 1 and 2 in the
figure) can be calculated. Assuming SIG is a plane wave, the relationship
between $\theta$ and $\phi$ is:

$\displaystyle\Delta\phi_{2,1}=\phi_{2}-\phi_{1}\approx
k\,d\,\,\sin(\theta):\theta\approx\sin^{-1}\left(\frac{\Delta\phi_{2,1}}{k\,d}\right)$
(1)

where $d$ is the separation between the two locations, $\phi_{1,2}$ are the
phases of SIG at the two locations, $k=2\pi/\lambda$, and $\lambda$ is the
wavelength of SIG. This expression assumes that the line formed by locations 1
and 2 is perpendicular to the line for which the angle $\theta$ is measured.
If the two locations (say locations 1 and 3 in Fig. 1) form a line that is not
perpendicular to the line that determine $\theta$, then the phase difference
between locations 1 and 3 are given by

$\displaystyle\Delta\phi_{3,1}$ $\displaystyle=\phi_{3}-\phi_{1}$
$\displaystyle\approx
k\sqrt{d^{2}+t^{2}}\sin\left[\theta+\tan^{-1}\left(t/d\right)\right]$ (2)
$\displaystyle\theta\approx\sin^{-1}\left(\frac{\Delta\phi_{3,1}}{k\,\sqrt{d^{2}+t^{2}}}\right)\,-\tan^{-1}\left(t/d\right)\,\,,$
(3)

where $t$ is defined in Fig. 1. Eqs. (1)-(3) relate AoA to the measured phase
of the SIG and LO signals at two locations in the cell, assuming that the AoA
is defined in a plane orthogonal to the probe laser propagation. Future work
will include the measurement AoA in two dimensions, see discussion below.

Figure 1: Incident plane wave (SIG) onto three locations separated by $d$ and
offset by $t$.

To measure the phase at any two different locations, we generate EIT in two
locations inside a vapor cell filled with 133Cs, see Fig. 2(a). The probe
laser is split with a beam cube and passed through the vapor cell at two
locations. The full power of the coupling laser is passed through each of the
two locations, see Fig. 2(a). The beam directions are chosen to ensure that at
both locations in the cell, the probe and coupling lasers are counter-
propagating. To generate EIT at the two locations in the cell, we tune the
probe laser to the ${\rm D}_{2}$ transition for 133Cs ($6S_{1/2}$-$6P_{3/2}$
or wavelength of $\lambda_{p}=852.35$ nm) focused to a full-width at half
maximum (FWHM) of 390$~{}\mu$m, with a power of 96 $\mu$W. To produce an EIT
signal, we couple to the 133Cs $6P_{3/2}$-$58S_{1/2}$ states by applying a
counter-propagating coupling laser at $\lambda_{c}=509.26$ nm with a power of
60 mW, focused to a FWHM of 450 $\mu$m.

(a) Laser field schematic

(b) antenna arrangement

Figure 2: (a) Schematic of the orientation of the optical fields. The probe
beam is split in two by a beam cube, and one coupling field is re-circulated
using a dichroic mirror to counter-propagate along each probe beam. (b) The LO
antenna is suspended above the cell, such that the LO field is incident nearly
perpendicular to the line between the two optical beams. The SIG is held by an
adjustable arm to vary the angle of incidence, which is measured using an
electronic compass attached to the horn mount.

The LO and SIG are applied to the vapor cell as shown in Fig. 2(b), where the
LO is at a fixed position and the SIG is rotated to different incident
directions ($\theta$). We use a signal generator (SG) to apply a continuous
wave (CW) LO field at 19.18 GHz to couple states $58S_{1/2}$ and $59P_{3/2}$.
While we use 19.18 GHz in these experiments, this approach can work at
carriers from 100 MHz to 1 THz (because of the broadband nature of the EIT/AT
approach holl1 ; holl2 ). A second SG is used to generate a CW SIG field at
19.18 GHz+$f_{IF}$ (where the $f_{IF}$=50 kHz)). The output from the two SG
are connected to two standard gain horn antennas via RF cables. The LO horn is
mounted directly above the vapor cell and is stationary, whereas the SIG horn
sits on a rotating arm which sets the incident angle ($\theta$).

Two different photodetectors are used to monitor the two probe beams that
travel through the vapor cell. The output of the photodetectors are sent to an
oscilloscope and a lock-in amplifier. Fig 3(a) shows the beam position at the
two locations inside the vapor cell. These beam positions correspond to
locations 1 and 3 as defined in Fig. 1, and the phase relationship is given in
eq. (3). In our experiments, $d=2.6$ mm and $t=0.3$ mm. The lock-in is
referenced to a 50 kHz signal from a mixer that is fed by the two signal
generators. The Rydberg atoms automatically down-convert the CW carrier (i.e.,
SIG) to the IF (the amplitude of the probe laser transmission) and the phase
of SIG is determined.

(a) (b)

Figure 3: (a)The $x$-$y$ location of lasers inside vapor cell, where the
origin is the center of the cell, and (b) Beat-note for two locations inside
the vapor cell.

The Rydberg-atom sensor and the photodetectors act like a mixer and low pass
filter in a classic RF heterodyne setup. The LO and SIG create a beat-note and
the atoms respond directly to this beat-note, which is detected by the probe
laser transmission measured on the photodetectors. At each location inside the
vapor cell, the total electric field ($E_{atoms}$) is the sum of the LO and
SIG fields ($E_{LO}$ and $E_{SIG}$). The atoms demodulate the high-frequency
$\omega_{LO}$ field and the probe transmission as a function of time at
locations $i$ and $j$ (1 and 3 as defined in Fig. 1) is given by sim3 ; gor3

$T_{(i,j)}\propto|E_{atoms}|\approx
E_{LO}+E_{SIG}\,\cos\left(\Delta\omega\,t+\phi_{i,j}\right)\,\,,$ (4)

where $\phi_{i,j}$ corresponds to the phase of SIG at locations i and j, and
$\Delta\omega=\omega_{LO}-\omega_{SIG}$. Once $\phi_{i}$ and $\phi_{j}$ are
determined from the probe laser transmissions measured on the two different
photodetectors, the phase difference ($\Delta\phi$) between the two locations
is given by

$\Delta\phi=\phi_{j}-\phi_{i}~{}~{}~{}\textrm{.}$ (5)

To be more exact, $\phi_{i,j}$ is actually the phase difference (at each
location) between the LO and SIG sim3 . In these experiments, LO is at a fixed
location such that a measurement of $\Delta\phi$ is a measurement of the phase
change of SIG between the two locations.

For a given incident angle $\theta$, the beat-notes as measured from the two
photodetectors are shown in Fig. 3(b). From the figure, we see the “cosine”
behavior as predicted by eq. (4) with a period of 20 $\mu$s (or the IF
frequency of 50 kHz used in the experiments). In this figure we see that the
two beat-notes are shifted in phase. This is the phase difference for the
given incident angles that is defined in eq. (3).

Using the setup shown in Fig. 2(b), the SIG antenna is scanned from
$\theta=\pm 40^{o}$. The phase difference ($\Delta\phi$) at each $\theta$
position was determined and the measured $\Delta\phi$ for each incident angle
is shown in Fig. 4(a). The error bars correspond to the standard deviation of
5 data sets. The uncertainties of Rydberg atom based measurements in general
are discussed in Ref.emcconf  and it is shown in Ref.ieeeaccess  that the
heterodyne Rydberg atom-based mixer approach can measure the phase to within
$1^{o}$. Also shown in this figure are the theoretical results given in eq.
(3). Upon comparing the experimental results to the theoretical results, we
see that while the standard deviation for the phase measurement for each
incident angle is small (i.e., small error bars), the measurements do not lie
exactly on the theoretical results. The reason why the data does not exactly
follow the theoretical model is twofold. First, from Fig. 2(b) we see there
are several objects in the apparatus used to rotate the SIG antenna. These
objects cause scattering which are not accounted for in the theoretical
results. The second reason is due to the vapor cell itself. Because the vapor
cell is a dielectric, the RF fields can exhibit multi-reflections inside the
cell and RF standing waves (or resonances) in the field strength can develop
in the cellholl2 ; holl3 ; sim5 ; fan5 . Thus, for a given location inside the
cell, the RF field can be larger or smaller than the incident field and the
phase of the field at a given location will be perturbed as well. Hence, the
standing wave can generates differences in the measured AoA when compared to
the expected sinusoidal relationship as given in eqs. (1) and (3). Numerical
models can be used to investigate this effect. While modeling the entire
structure used to support the SIG antenna is difficult, we can use full-wave
numerical tools to simulate the vapor cell effects.

We use ANSYS HFSS (High Frequency Structure Simulator)hfss  to simulate only
the SIG antenna and the vapor cell (including the plastic vapor-cell holder),
see Fig. 5. HFSS convergence criteria was based on the energy of a plane wave
to 0.01 W, and the mesh around the cell was seeded using curvilinear
approximation, and inside the cell using length-restriction to 1 mm, with
first order polynomial solving. With this model, we determine the phase at
location 1 and 3 (as defined in Fig. 1) and the $\Delta\phi_{3,1}$ obtained
from HFSS are shown in Fig. 4(a). To ensure that the phases are being
calculated correctly with the HFSS simulation, we first determine
$\Delta\phi_{3,1}$ with no vapor cell present. These results are shown in Fig.
4(a) and match the theoretical calculation closely, as expected. Now that we
have confirmed that the HFSS is implemented correctly, the result from the
HFSS for the case when the vapor cell is included are shown in Fig. 4(a). We
see that the HFSS results (including the vapor cell) correspond well to the
measured data for angles $>$-25o. As with the experimental results, the HFSS
results indicate that the vapor cell does perturb the phase measurement and
causes deviation from the theoretical results. We see that the HFSS results do
not correspond exactly to the measured data over all the angles, but do show
the same trends. The deviations between the measured data and HFSS are
twofold. First, the exact permittivity ($\epsilon_{r}$) of the glass is not
known, $\epsilon_{r}$ ranges from 3 to 6tropf  (in this numerical model we
assume $\epsilon_{r}=5$). Secondly, upon comparing the photo of the
experimental setup in Fig. 2(b) and the HFSS model in Fig. 5, we see that not
all the objects used to rotate the SIG antenna are included in the HFSS model.
With that said, the measured and HFSS model compare well and follow the same
trends, especially for angles $>$-25o. There are asymmetries in the apparatus
use in the experimenters. For angles $<$-25o degrees, the apparatus used to
support the SIG antenna and vapor cell in the experiment begin to influence
the phase of the measured data. The additional scattering caused by this
apparatus is not included in the HFSS numerical model. While the vapor cell
does perturb the measurement of $\Delta\phi_{3,1}$, the results in Fig. 4(a)
show that the Rydberg-atom based sensor can detect the relative phase
difference between two locations inside the vapor cell and work as a AoA
detector.

(a) (b)

Figure 4: (a) Experimental and HFSS data for $\Delta\phi$. The error bars
correspond to the standard deviation of 5 data sets, and (b) AoA from the
experimental data.

Figure 5: HFSS model for the cell and horn antenna. The horn is rotated by an
angle $\Theta$ around the $x-$axis, such that it points towards the cell. The
model also shows the vapor cell holder.

With the measured $\Delta\phi$, the AoA can be determined from eq. (3). Fig.
4(b) shows the AoA from the measured $\phi_{3,1}$ for incident angles ranging
from $\theta=\pm 40^{o}$. The solid line in this figure represents the one-to-
one correspondence of the incident angle and AoA. The measured AoA should lay
on the line. While the measured AoA follows this line, it does not exactly lay
on the line. Also, in this figure we show the AoA obtained by using the HFSS
results for $\Delta\phi$ in eq. (3). Here again, we see that the HFSS results
deviate from the solid line. As with the measured $\Delta\phi$, the deviation
in the measured AoA (and the HFSS results for AoA) is due to the vapor cell
perturbation and due to the supporting apparatus used to experimental
equipment (SIG antenna and vapor cell). This demonstrates that the Rydberg
atom-based sensor can be used to determine AoA of an incident RF signal.

While the cell does perturb the AoA measurement, two approaches can be pursued
to mitigate this effect. One approach is to design a vapor cell that can
minimize and even eliminate the vapor cell perturbations. Various groups are
investigating different approaches to modify the vapor cell used for these
Rydberg atom-based sensors. Two examples include the use of vapor cells with
honeycomb sides schafer  or the use of metamaterials on the sides of the vapor
cells meta . A second approach is to use the HFSS results to calibrate the
vapor cell to reduce the perturbation effects. This is done by defining a
calibration factor as

${\cal C}={\rm AoA}_{HFSS}-{\rm AoA}_{theory}$ (6)

and subtracting this from the measured AoA

${\rm AoA}_{cal}={\rm AoA}_{meas}-{\cal C}$ (7)

where ${\rm AoA}_{HFSS}$, ${\rm AoA}_{theory}$, and ${\rm AoA}_{meas}$ are the
AoA obtained from the HFSS results, theory, and experimental results,
respectively. Fig. 4(b) shows ${\rm AoA}_{cal}$. While there is not a perfect
correlation to the solid line with the calibration based on the HFSS results,
we do see that the calibration did improve the AoA measurement, especially for
angles $>$-25o. Once again, the deviations from the theory and HFSS simulation
for angles $<$-25o is due to the asymmetries in associated with the apparatus
to support the experiments. The larger discrepancies could be handled with a
more accurate models of the experimental setup.

This paper demonstrates that it is possible to determine the angle of arrival
of an RF signal using an atom-based sub-wavelength phase measurement method.
While the vapor cell perturbs this measurement, we see that this effect can be
mostly accounted for or at least explained, and these Rydberg atom-based
sensors have the capability of measuring the AoA of a incident RF signal.
Future iterations of this experiment will explore the perimittivity of the
glass for more precise modeling of the standing wave in the glass cell. We are
also investigating different types of vapor cell designs and beam orientations
in order to minimize or eliminate the vapor cell effect on the AoA
measurements.

Now that we have demonstrated that it is possible to determine the AoA with a
Rydberg atom-based sensor, one can envision (1) developing arrays of these
Rydberg atom sensors (2) or sampling the phase at numerous locations inside
one vapor cell in order to detect the AoA of more general incidence angles, or
for the purpose of simultaneously detecting the AoA of several sources at
once. We are currently developing these two types of sensors and these will be
the topic of a future publication.

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