BREAD

# First Results from a Broadband Search for Dark Photon Dark Matter in the 44
to 52 $\mu$eV range with a coaxial dish antenna

Stefan Knirck  Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Gabe Hoshino  Department of Physics, University of Chicago, Chicago, IL 60637,
USA  Mohamed H. Awida  Fermi National Accelerator Laboratory, Batavia, IL
60510, USA  Gustavo I. Cancelo  Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA  Martin Di Federico  Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  Universidad Nacional del Sur, IIIE-
CONICET, Argentina  Benjamin Knepper  Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA  Enrico Fermi Institute, University of Chicago,
Chicago, IL 60637, USA  Alex Lapuente  Department of Physics, University of
Chicago, Chicago, IL 60637, USA  Mira Littmann  Department of Physics,
University of Chicago, Chicago, IL 60637, USA  David W. Miller  Department of
Physics, University of Chicago, Chicago, IL 60637, USA  Enrico Fermi
Institute, University of Chicago, Chicago, IL 60637, USA  Kavli Institute for
Cosmological Physics, University of Chicago, Chicago IL 60637, USA  Donald V.
Mitchell  Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Derrick Rodriguez  Department of Physics, University of Chicago, Chicago, IL
60637, USA  Mark K. Ruschman  Fermi National Accelerator Laboratory, Batavia,
IL 60510, USA  Matthew A. Sawtell  Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA  Leandro Stefanazzi  Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  Andrew Sonnenschein  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Enrico Fermi Institute,
University of Chicago, Chicago, IL 60637, USA  Gary W. Teafoe  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Daniel Bowring  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  G. Carosi  Lawrence Livermore
National Laboratory, Livermore, California 94550, USA  Aaron Chou  Fermi
National Accelerator Laboratory, Batavia, IL 60510, USA  Clarence L. Chang
Argonne National Laboratory, Lemont, IL 60439, USA  Department of Astronomy &
Astrophysics, University of Chicago, Chicago, IL 60637, USA  Kavli Institute
for Cosmological Physics, University of Chicago, Chicago IL 60637, USA
Kristin Dona  Department of Physics, University of Chicago, Chicago, IL 60637,
USA  Rakshya Khatiwada  Fermi National Accelerator Laboratory, Batavia, IL
60510, USA  Department of Physics, Illinois Institute of Technology, Chicago,
IL 60616, USA  Noah A. Kurinsky  SLAC National Accelerator Laboratory/Kavli
Institute for Particle Astrophysics and Cosmology, Menlo Park, Stanford
University, Stanford, CA 94025, USA  Jesse Liu  Cavendish Laboratory,
University of Cambridge, Cambridge CB3 0HE, UK  Cristián Pena  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Chiara P. Salemi  SLAC
National Accelerator Laboratory/Kavli Institute for Particle Astrophysics and
Cosmology, Menlo Park, Stanford University, Stanford, CA 94025, USA  Christina
W. Wang  California Institute of Technology, Pasadena, CA 91125, USA  Jialin
Yu  Department of Physics, Illinois Institute of Technology, Chicago, IL
60616, USA

###### Abstract

We present first results from a dark photon dark matter search in the mass
range from 44 to 52 $\mu{\rm eV}$ ($10.7-12.5\,{\rm GHz}$) using a room-
temperature dish antenna setup called GigaBREAD. Dark photon dark matter
converts to ordinary photons on a cylindrical metallic emission surface with
area $0.5\text{\,}{\mathrm{m}}^{2}$ and is focused by a novel parabolic
reflector onto a horn antenna. Signals are read out with a low-noise receiver
system. A first data taking run with 24 days of data does not show evidence
for dark photon dark matter in this mass range, excluding dark photon – photon
mixing parameters $\chi\gtrsim 10^{-12}$ in this range at 90 % confidence
level. This surpasses existing constraints by about two orders of magnitude
and is the most stringent bound on dark photons in this range below 49
$\mu$eV.

wave-like dark matter, dark photon, dish antenna, room-temperature pilot, low-
noise amplifier

††preprint: FERMILAB-PUB-23-625-PPD-STUDENT Figure 1:  Experimental Setup.
Dark photon dark matter converts to photons emitted perpendicularly from the
cylindrical barrel (center). This signal is focused on a coaxial horn antenna
and amplified using a low-noise receiver chain (center right) before it is
down-converted and digitized using a custom real-time FPGA-based broadband DAQ
(bottom).

_Introduction –_ Dark matter (DM), about 85 % of the mass content of the
Universe, remains one of the most elusive enigmas of modern physics [1, 2, 3,
4, 5, 6]. Bosonic particles with masses below $\sim$\mathrm{eV}$$ (wave-like
DM) are well motivated by theory [7, 8, 9, 10, 11, 12, 13, 14, 15], and there
are a multitude of experimental efforts aiming to detect them, for a review
see [15, 16, 17]. Notably the resonant cavity experiments ADMX [18] and
CAPP-12TB [19] have reached the highly-motivated DFSZ threshold for detecting
QCD axions around about $2-5\,\mu$eV. Despite these efforts, many orders of
magnitude in mass range are still unprobed due to the narrowband sensitivity
and poor higher-mass scalability of resonant cavity methods. Wave-like DM such
as dark photons (DPs) can convert to photons emitted perpendicularly to a
metallic surface (dish antenna), which can then be focused onto a detector
[20]. This conversion happens independent of mass, overcoming the narrowband
limitation of resonant setups. The converted photons carry a power of

$\displaystyle P\,=$ $\displaystyle\
$3\text{\times}{10}^{-21}\text{\,}\mathrm{W}$\,\bigg{(}\frac{\eta}{0.5}\bigg{)}\bigg{(}\frac{A}{$0.5\text{\,}{\mathrm{m}}^{2}$}\bigg{)}$
(1)
$\displaystyle\times\bigg{(}\frac{\chi}{${10}^{-12}$}\bigg{)}^{2}\bigg{(}\frac{\rho_{\rm
DM}}{$0.45\text{\,}\mathrm{GeV}\text{\,}{\mathrm{cm}}^{-3}$}\bigg{)}\bigg{(}\frac{\alpha^{2}}{{1/3}}\bigg{)}$

at a frequency, $f$, set by the DP mass, $m_{\rm DP}$, and a narrow line shape
due to the non-relativistic DM velocity distribution, $f=m_{\rm
DP}c^{2}/h+\mathcal{O}(10^{-6})$ [21, 22]. $\eta$ is a detection efficiency,
$A$ is the emission area of the dish, $\chi$ is the kinetic mixing between
photons and DPs, $\rho_{\rm DM}$ is the local DM halo density and $\alpha$
accounts for the dark photon’s polarization with $\alpha=\sqrt{1/3}$ for an
experiment sensitive to a single polarization and randomly polarized DM.

Several prototype dish antennas have recently been demonstrated [23, 24, 25,
26, 27, 28, 29], but are difficult to combine with high-field magnets in order
to make them sensitive to axion DM, as well. We recently proposed BREAD
(Broadband Reflector Experiment for Axion Detection) [30], a dish antenna with
a novel focusing reflector geometry shown in the center of figure 1. It allows
placement of the emission surface parallel to the field of a solenoid magnet
in a future upgrade. In this paper we present first results from a room
temperature pilot experiment, GigaBREAD, to search for DPs in the range from
$44-52\,\mu$eV ($10.7-$12.5\text{\,}\mathrm{GHz}$$).

_Experimental Setup –_

Figure 2: Focal spot measurements of the GigaBREAD dish antenna as a function
of vertical horn antenna position $z$ (vertical axes) and frequency
(horizontal axes). (a) Expected dark photon signal strength from a full-wave
Comsol® simulation. (b) Measured system reflectivity $|S_{11}|^{2}$. (c)
Measured system noise temperature. The focus is symmetric around $z=0$ (bold
dotted line). Standing wave resonances between barrel and horn antenna can be
swept over the full frequency range by adjusting the horn antenna position
between $z\approx$10\text{\,}\mathrm{mm}$$ and
$z\approx$-10\text{\,}\mathrm{mm}$$ (faint dotted lines).

Figure 1 shows a schematic of the experimental setup. DPs convert to photons
emitted perpendicularly from the inner area of the cylindrical aluminum
barrel. They are focused by an inner parabolic reflector and the outer
cylinder onto a focal spot at the vertex of the parabolic shape where a
coaxial horn antenna [31, 32] is placed. This custom-design horn receives
radiation impinging from the side with radially outward-pointing polarization
matched to our geometry, unlike conventional horns typically matched to
linearly polarized radiation impinging head on. The horn can be moved
vertically in and out of the focal spot. It is connected to a low-noise
amplifier chain with a heterodyne real-time data acquisition (DAQ) scheme. A
pin antenna inserted in front of the conversion surface allows us to inject
test signals as an experimental control. The dish antenna (cylindrical barrel
with reflector and horn antenna), as well as the amplifier chain are placed in
an RF-shielded Faraday cage with additional foam absorbers to shield them from
external sources in the signal band.

The signal from the dish antenna has been simulated with full-wave
azimuthally-symmetric Comsol® [33] simulations of the modified Maxwell’s
equations accounting for the wave-like DM field using a space-filling current
density [34, 35]. The simulations include surface deformations of the inner
reflector and outer barrel at the level of a few 100 $\mu$m measured with a
coordinate measuring machine. They shift the focus by a few millimeters
axially with otherwise little impact on sensitivity. Figure 2 (a) shows the
simulated received power compared to an ideal dish of same size, $\eta$, as a
function of antenna position and frequency of a putative DM signal. The signal
vanishes when moving the antenna by about a wavelength out of the focal spot.
It is weakly resonantly enhanced on a comb of frequencies corresponding to
standing waves between the antenna and barrel. Moving the antenna vertically
adjusts the corresponding optical length and shifts these frequencies.
Measurements of the reflectivity seen with the horn antenna, fig. 2 (b), as
well as the thermal noise, fig. 2 (c), reveal the same features and match
corresponding simulations. When measuring reflectivity, the signal emitted
from the antenna couples back into the antenna when the antenna is at the
focal spot. However, when the antenna is moved out of focus the signal is
refocused away from the antenna. Hence, the reflectivity is maximized on
focus. Conversely, thermal radiation from the setup is minimized on focus,
because the absorption coefficient equals the thermal emission coefficient
according to Kirchhoff’s law, or in other words, thermal radiation from the
absorbers couples into the antenna only at the off-focal position. On-
resonance photons bounce between the antenna and emission surface, making them
more likely to be lost before recoupling to the antenna, leading to dips in
the reflectivity and higher thermal noise. For clarity we removed effects from
the cable between the horn antenna and amplifier in fig. 2. These effects are
taken into account in our data analysis.

The gain and added noise from the low-noise receiver chain were characterized
with the $y$-factor method [36] by connecting it to a noise source with
$6\text{\,}\mathrm{dB}$ excess-noise-ratio and measuring noise with the noise
source on and off. The added noise from amplification is $120\pm
10.\text{\,}\mathrm{K}$ compatible with the low-noise amplifier’s data sheet
[37].

After about $80\text{\,}\mathrm{dB}$ of amplification the radio frequency (RF)
signal passes a $10.7-$12.7\text{\,}\mathrm{GHz}$$ bandpass filter and is
down-converted to a so-called intermediate frequency (IF) of $f_{\rm
IF}=|f_{\rm RF}-f_{\rm LO}|<$2\text{\,}\mathrm{GHz}$$ using a broadband mixer
and local oscillator (LO) frequency of $f_{\rm
LO}\approx$10.6\text{\,}\mathrm{GHz}$$. The readout system is based on the
Fermilab Open Source platform QICK [38]. The signal is digitized using a
Xilinx RFSoC FPGA [39] (DAQ board) with an ADC (analog-to-digital converter)
at sampling rate $4\text{\,}\mathrm{GHz}$. Custom firmware on the DAQ board
Fourier transforms and squares the digitized voltage signals to obtain power
spectra, and finally adds them to previous data in real time. After a batch of
$10,000$ spectra this sum is transferred to the Pynq [40] logic on the board
where the data is stored to an internal hard drive. The resolution is $\Delta
f=$7.8\text{\,}\mathrm{kHz}$$, approximately matching the line width of the
expected signal. This choice minimizes the noise competing with a potential
signal, while still allowing us to digitize over the maximum bandwidth
available with the board. Low-frequency radio interference (RFI) backgrounds
in the IF band can pose a significant challenge for such a DAQ scheme, since
it can mimic a DP signal. To avoid this, we employed a custom frequency
hopping scheme: the LO frequency is randomly shifted by an integer number of
frequency bins within a range of a few $\mathrm{MHz}$ ($\sim 1,000$ bins)
about once every second. When averaging the acquired power spectra are
accordingly shifted relative to each other in RF frequency domain. This causes
any RFI in the IF band to spread across multiple RF frequency bins and average
out after sufficiently many acquisitions without affecting signals in the RF
domain. In addition, eight high-power single bin RFI signals were identified
in the IF band and digitally masked before data taking. The frequency hopping
also causes these masks to smear out in the RF domain such that their impact
on sensitivity becomes negligible.

_Data Taking Run –_ The experiment was operated at 41° 47’ 31.6098” N, 87° 36’
6.141” W in the basement of the William Eckhardt Research Center at the
University of Chicago. The setup is sensitive to DP polarizations
perpendicular to Earth’s surface, since the focal spot antenna only couples to
vertically polarized radiation emitted from the barrel. During data taking the
horn antenna position was regularly swept from $1\text{\,}\mathrm{cm}$ to
$-1\text{\,}\mathrm{cm}$ around the focus (as indicated in fig. 2) with a
$0.2\text{\,}\mathrm{mm}$ step size. This tunes the comb of resonances over
the full frequency range and gives approximately uniform sensitivity in DP
mass. Each sweep took $\sim 4$ hours, much longer than the timescale on which
frequency hopping occurs, but much shorter than a day to be clearly
distinguishable from possible daily signal modulations [22, 41]. Data taking
took place from June 16 to July 17, 2023. It was interrupted deliberately
around once per week, in addition to three pauses due to power outages,
leading to 24 days of science data. The interruptions were used to remeasure
reflectivity as a function of antenna position and to monitor receiver gain by
measuring noise from a matched load. No significant changes were observed
except slow gain variations $<$0.1\text{\,}\mathrm{d}\mathrm{B}\mathrm{/}{\rm
day}$$. Test signals were injected from June 16 – June 20 at
$11.45\text{\,}\mathrm{GHz}$ and July 14 – July 17 at
$11.53\text{\,}\mathrm{GHz}$ to confirm the performance of the setup in situ.
The injections were blind, i.e., the person first analyzing the data did not
know the set frequency.

Figure 3: (a) Observed Excess Power $P_{0}$ as a function of RF frequency,
normalized with the receiver gain and detection efficiency. No signal exceeds
a global significance of 3$\sigma$. (b) Distribution of the observed excess
powers $P_{0}$ divided by the expected standard deviation from thermal noise
$\sigma$, following a normal distribution. (c) Observed excess power for the
test signal injected between June 16 and June 20 as a function of horn antenna
position showing the expected weak resonant enhancement.

_Data Analysis –_ The data were analyzed using standard procedures analogous
to other haloscope experiments [42]. All acquired power spectra were averaged
for each axial horn antenna position $z$, giving $P_{z}(f)$. We searched for a
power excess in a single bin. To this end, we subtracted the baseline $P_{{\rm
bl},z}(f)$ obtained with a fourth-order Savitzky-Golay filter [43], removing
features on scales $\gtrsim$1\text{\,}\mathrm{MHz}$$ much larger than the
signal line width. The standard deviation of residual noise around the
baseline is expected to be $\sigma_{z}=P_{\rm bl}/\sqrt{N_{\rm av}}$, where
$N_{\rm av}$ is the total number of averages taken at this position. The
baseline is related to system noise temperature $T_{\rm sys}$ as $P_{\rm
bl}=k_{B}T_{\rm sys}\Delta f$ with Boltzmann’s constant $k_{B}$ and resolution
$\Delta f$. The vertical average along $z$ was calculated using an optimally
weighted average [44, 45],

$P_{0}~{}\equiv~{}\left\langle\frac{P}{\eta}\right\rangle_{z}=\frac{\sum_{z}\left({\eta_{z}}/{\sigma_{z}}\right)^{2}P_{z}/\eta_{z}}{\sum_{z}\left({\eta_{z}}/{\sigma_{z}}\right)^{2}}.$
(2)

After applying this procedure to the sub-runs with injected signals, both were
identified at the correct frequencies with $30\sigma$ and $40\sigma$
significance, respectively, and bins with injected signals subsequently
removed from the analysis. Fig. 3 (a) shows the obtained excess noise $P_{0}$
for the full data set. In the absence of a signal $P_{0}$ is expected to be
normal distributed with variance
$\sigma^{2}=1/\sum_{z}(\eta_{z}/\sigma_{z})^{2}$. In fig. 3 (b) we histogram
excess powers normalized to the expected standard deviation. They are normal
distributed, verifying that long-term averaging and background subtraction
work as expected. Fig. 3 (c) illustrates the dependence of the test signal
injected from June 16 to Jun 20 on the vertical antenna position showing the
expected weak resonant enhancement around a certain antenna position.

Signals are searched using a cross-correlation of $P_{0}$ with the expected
Maxwell-Boltzmann line shape [21]. The largest excess is observed at a local
significance of $5.2\sigma$ at $10.829\text{\,}\mathrm{GHz}$. Its global
significance is less than $2\sigma$ as estimated using Monte Carlo
simulations, too small to claim a detection. We derive a 90 % confidence level
(CL) upper limit on the received DM power based on the observed excess powers
greater than zero. Applying eq. 1 gives the limit on the mixing parameter
$\chi$.

Table 1:  Frequency-averaged systematic uncertainties on the DP signal power.  Effect | Uncertainty on $P_{0}$
---|---
Non-zero DM velocity | $<$1\text{\,}\mathrm{\char 37\relax}$$
Detection Efficiency Simulation | $20\text{\,}\mathrm{\char 37\relax}$
System Noise Temperature | $9\text{\,}\mathrm{\char 37\relax}$
Gain Variations | $5\text{\,}\mathrm{\char 37\relax}$
Baseline Removal | $3\text{\,}\mathrm{\char 37\relax}$
Total | $23\text{\,}\mathrm{\char 37\relax}$

Table 1 summarizes systematic uncertainties. Momentum-transfer from DM to the
converted photons can shift the focus [46, 47, 48]. This effect is negligible
since our setup is much smaller than the DP de Broglie wavelength
$\lambda_{\rm DB}\sim$30\text{\,}\mathrm{m}$$. The largest systematic arises
from uncertainties in the simulated efficiency $\eta$. They are estimated
based on the differences in the measured and simulated reflectivity $S_{11}$
and the corresponding impact on signal power. The differences may be
attributed to geometrical imperfections not taken into account in the
simulation. Another important systematic uncertainty arises from the
$y$-factor calibration of the low-noise amplifier’s added noise, besides small
gain drifts discussed above. The baseline removal in the analysis can
attenuate signals [45, 49]. This effect is smaller than
$3\text{\,}\mathrm{\char 37\relax}$ for our filter parameters. The total
systematic uncertainty is obtained by adding the individual uncertainties in
quadrature. The most conservative $P_{0}$ within this uncertainty is adopted
for the limit.

Figure 4: Dark photon dark matter parameter space excluded in this work
(GigaBREAD-0), compared to cosmological constraints, other dish antennas like
SHUKET [24], QUALIPHIDE [26], BRASS-p [28] and DOSUE [27, 29], as well as
other haloscopes [50] over about two decades in mass (left) and magnified
(right).

_Conclusion –_ We obtain an upper bound of $\chi\lesssim 10^{-12}$ for the
mass range of $44-52\,\mu$eV at a 90 % confidence level as shown in fig. 4.
This constraint surpasses cosmological bounds by more than two orders of
magnitude. Our search is the most sensitive experimental DPDM search between
$44$ and $49\,\mu$eV. Furthermore, our setup is optimized for inclusion in a
solenoid magnet and a first run in a $4\text{\,}\mathrm{T}$ field [51] is
under preparation allowing to search for axion-like particles. Our setup
establishes a testbed for future research-and-development directions, e.g.,
cryogenic quantum-limited amplifiers [26], other photo-sensing technologies
and larger dish sizes. This also includes additional signal enhancement by
improving quality factors of the standing wave resonances or larger volume
efficiency with dielectric stacks [52, 53, 54, 55].

_Acknoledgements –_ We acknowledge Abigail Vieregg and her group for providing
access to the RF shielded room. We thank Dan Zhang and the ADMX collaboration
for helpful discussions about analysis. We thank all BREAD collaborators for
inspiring and helpful discussions. This work is funded by the by the
Department of Energy through the resources of the Fermi National Accelerator
Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP
User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA),
acting under Contract No. DE-AC02-07CH11359. J. Liu is supported by a Junior
Research Fellowship at Trinity College, University of Cambridge. C. P. Salemi
is supported by the Kavli Institute for Particle Astrophysics and Cosmology
Porat Fellowship.

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# On the Schwarzschild-de Sitter metric of nonlocal de Sitter gravity

Ivan Dimitrijevic<EMAIL_ADDRESS>Branko Dragovich<EMAIL_ADDRESS>Zoran Rakic<EMAIL_ADDRESS>Jelena Stankovic<EMAIL_ADDRESS>Faculty
of Mathematics, University of Belgrade, Studentski trg 16, Belgrade, Serbia
Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade,
Serbia  Mathematical Institute, Serbian Academy of Sciences, Kneza Mihaila 36,
Belgrade, Serbia  Teacher Education Faculty, University of Belgrade, Kraljice
Natalije 43, Belgrade, Serbia

###### Abstract

Earlier constructed a simple nonlocal de Sitter gravity model has a
cosmological solution in a very good agreement with astronomical observations.
In this paper, we continue the investigation of the nonlocal de Sitter model
of gravity, focusing on finding an appropriate solution for the Schwarzschild-
de Sitter metric. We succeeded to solve the equations of motion in a certain
approximation. The obtained approximate solution is of particular interest for
examining the possible role of non-local de Sitter gravity in describing the
effects in galactic dynamics that are usually attributed to dark matter.

##  1 Introduction

For more than 100 years, General Relativity (GR) has been considered as one of
the most beautiful and successful physical theories – from a phenomenological
point of view, several remarkable predictions have been confirmed ellis . The
Standard Model of Cosmology (SMC) assumes that GR is valid and applicable not
only in the Solar system but also at the galactic and cosmological scale.
However, GR has not been verified at the galactic and cosmological scales
without the use of Dark Matter (DM) and Dark Energy (DE). Furthermore, the SMC
assumes that the universe contains about 68 % of DE, 27 % of DM and only 5 %
of ordinary matter. Also, GR solutions for the black holes as well as for the
beginning of the universe contain singularity and it means that GR should be
modified in the vicinity of these singularities. It should be also mentioned
that GR is nonrenormalizable theory from the quantum point of view. Hence, in
spite of significant successes of GR, it is reasonable to doubt in its
validity in description and understanding of all astrophysical and
cosmological gravitational phenomena. Keeping all this in mind, it follows
that general relativity is not a final theory of gravitation and that its
expansion should be considered.

Since, there is still no physical principle that could suggest us in which
direction we should search for an extension of GR, there are many approaches
to its modification, see faraoni ; nojiri ; clifton ; nojiri1 ; capozziello ;
dragovich0  as some reviews. Despite many attempts, there is not yet generally
accepted modification of general relativity. One of the current and attractive
approaches is nonlocal modified gravity, see, e.g. biswas1 ; biswas3 ; biswas4
; biswas5 ; deser . In an analytic nonlocal modification, the Einstein-Hilbert
action is extended by a term that contains all higher order degrees of
d’Alembert-Beltrami operator
$\Box=\nabla_{\mu}\nabla^{\mu}=\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}).$
Usually, $\Box$ is used in the form of an analytic expression
$F(\Box)=\sum_{n=0}^{+\infty}f_{n}\ \Box^{n},$ see koshelev2 ; koshelev3 ;
eliz ; koivisto ; capozziello2009 ; dimitrijevic10 . Recently, we used
nonlocal operator in the form
$F(\Box)=1+\mathcal{F}(\Box)=1+\sum_{n=1}^{+\infty}f_{n}\
\Box^{n}+\sum_{n=1}^{+\infty}f_{-n}\ \Box^{-n}$ dimitrijevic13 . Some other
typical nonlocal models can be seen in woodard ; maggiore ; barvinsky2012 ;
modesto2 .

So far we have considered the nonlocal de Sitter models with analytic
nonlocality based on the action given by

$\displaystyle S=\frac{1}{16\pi G}\int d^{4}x\ \sqrt{-g}\
\big{(}R-2\Lambda+P(R)\ \mathcal{F}(\Box)\ Q(R)\big{)},$ (1)

where $\Lambda$ is the cosmological constant, $P(R)$ and $Q(R)$ are some
differentiable functions of the Ricci scalar $R$, see dimitrijevic1 ;
dimitrijevic2 ; dimitrijevic3 ; dimitrijevic4 ; dimitrijevic5 ; dimitrijevic6
; dimitrijevic7 ; dimitrijevic8 ; dimitrijevic9 ; dimitrijevic10 ;
dimitrijevic11 ; dimitrijevic12  and references therein. The case
$P(R)=Q(R)=R$ and $\Lambda=0$ has attracted a lot of attention, e.g. nonlocal
$R^{2}$ gravity, see biswas1 ; koshelev2  and references therein. In
dimitrijevic10  a very special case is $P(R)=Q(R)=\sqrt{R-2\Lambda}$ was
considered:

$\displaystyle S=\frac{1}{16\pi G}\int d^{4}x\ \sqrt{-g}\
\big{(}R-2\Lambda+\sqrt{R-2\Lambda}\ \mathcal{F}(\Box)\
\sqrt{R-2\Lambda}\big{)},$ (2)

where $\mathcal{F}(\Box)=\sum_{n=1}^{+\infty}f_{n}\Box^{n}.$ Importance of
this model is not only in its simple form but also in its cosmological
solutions in flat space-time: (i)
$a_{1}(t)=At^{\frac{2}{3}}e^{\frac{\Lambda}{14}t^{2}}$ and (ii)
$a_{2}(t)=Ae^{\frac{\Lambda}{6}t^{2}}.$ Solution $a_{1}(t)$ mimics interplay
of dark matter and dark energy in a good agreement with cosmological
observations dimitrijevic10 . Solution $a_{2}(t)$ is the nonsingular bounce
one. Nonlocal gravity model (2) also contains several other vacuum
cosmological solutions in flat, closed and open space, not only with
$\mathcal{F}(\Box)=\sum_{n=1}^{+\infty}f_{n}\Box^{n}$ but also when
$\mathcal{F}(\Box)=\sum_{n=1}^{+\infty}\big{(}f_{n}\Box^{n}+f_{-n}\Box^{-n}\big{)},$
see dimitrijevic13 . After very successful application of model (2) with
respect to dark energy and dark matter at the cosmological scale, it is
natural to be interested how this model works at smaller scales.

In this paper we investigate model (2) with static metric around spherically
symmetric body (stellar, galactic). In other words, we are interested in the
Schwarzschild-type metric in nonlocal de Sitter gravity given by (2). The
corresponding nonlocal operator has the form

$\displaystyle\mathcal{F}(\Box)=\sum_{n=1}^{+\infty}\big{(}f_{n}\Box^{n}+f_{-n}\Box^{-n}\big{)}.$
(3)

Here we want to find metric solution around a massive, non-rotating, without
charge, and spherically symmetric object. It is known that in GR with the
cosmological constant $\Lambda$ (de Sitter gravity) such solution is related
to the Schwarzschild-de Sitter metric,

$\mathrm{d}s^{2}=-A(r)\mathrm{d}t^{2}+A(r)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\varphi^{2},$
(4)

where $A(r)=1-\frac{2MG}{r}-\frac{\Lambda r^{2}}{3}$ (the speed of light in
the vacuum is taken $c=1$).

Structure of the paper is as follows. Section 2 contains an introduction to
our very simple nonlocal de Sitter gravity model,
$P(R)=Q(R)=\sqrt{R-2\Lambda}.$ In Section 3, we look for the Schwarzschild-
type metric and we find a solution using the corresponding eigenvalue problem.
In Section 4, we discuss obtained solution of this nonlocal de Sitter gravity
and give some concluding remarks with our plan for future investigation. There
is also an appendix which contains derivation of some useful formulas.

##  2 On nonlocal de Sitter gravity model

Let us mention some facts of our nonlocal gravity model (see dimitrijevic13 )
which is given by the action

$\displaystyle S=\frac{1}{16\pi G}\int d^{4}x\ \sqrt{-g}\ \sqrt{R-2\Lambda}\
F(\Box)\ \sqrt{R-2\Lambda},$ (5)

where $F(\Box)$ is the following formal expansion in terms of the d’Alemberian
$\Box$:

$\displaystyle
F(\Box)=1+\mathcal{F}(\Box)=1+\mathcal{F}_{+}(\Box)+\mathcal{F}_{-}(\Box),\quad\mathcal{F}_{+}(\Box)=\sum_{n=1}^{+\infty}f_{n}\
\Box^{n},\ \mathcal{F}_{-}(\Box)=\sum_{n=1}^{+\infty}f_{-n}\ \Box^{-n}.$ (6)

When $F(\Box)=1$, i.e. $\mathcal{F}(\Box)=0$, then model (5) becomes local and
coincides with Einstein-Hilbert action with cosmological constant $\Lambda$:

$\displaystyle S_{0}=\frac{1}{16\pi G}\int d^{4}x\ \sqrt{-g}\
\sqrt{R-2\Lambda}\ \sqrt{R-2\Lambda}=\frac{1}{16\pi G}\int d^{4}x\ \sqrt{-g}\
(R-2\Lambda).$ (7)

It is worth pointing out that action (5) can be obtained in a very simple and
natural way from action (7) by embedding nonlocal operator (6) in symmetric
product form of $R-2\Lambda$, that is $\sqrt{R-2\Lambda}\ \sqrt{R-2\Lambda}.$
Action (5) does not contain matter term and this is intentionally done to
better see possible role of this nonlocal model in effects usually assigned to
dark matter and dark energy.

The equations of motion (EoM) for model (1), for $P(R)=Q(R)$ are given by (for
more detail, see dimitrijevic9 ; dimitrijevic13 ):

$\displaystyle\hskip 28.45274ptG_{\mu\nu}+\Lambda
g_{\mu\nu}-\frac{g_{\mu\nu}}{2}P(R)\mathcal{F}(\Box)P(R)+R_{\mu\nu}W-K_{\mu\nu}W+\frac{1}{2}\Omega_{\mu\nu}=0,$
(8) $\displaystyle\hskip 28.45274ptW=2P^{\prime}(R)\ \mathcal{F}(\Box)\
P(R),\quad K_{\mu\nu}=\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box,$ (9)
$\displaystyle\hskip
28.45274pt\Omega_{\mu\nu}=\sum_{n=1}^{+\infty}f_{n}\sum_{\ell=0}^{n-1}S_{\mu\nu}(\Box^{\ell}P,\Box^{n-1-\ell}P)-\sum_{n=1}^{+\infty}f_{-n}\sum_{\ell=0}^{n-1}S_{\mu\nu}(\Box^{-(\ell+1)}P,\Box^{-(n-\ell)}P).$
(10)

If $P(R)$ is an eigenfunction of the corresponding d’Alembert-Beltrami
operator $\Box$, and consequently also of its inverse $\Box^{-1}$, i.e. holds

$\displaystyle\Box
P(R)=qP(R),\qquad\Box^{-1}P(R)=q^{-1}P(R),\qquad\mathcal{F}(\Box)P(R)=\mathcal{F}(q)P(R),\qquad
q\neq 0,$ (11)

then

$\displaystyle\hskip
28.45274ptW=2\mathcal{F}(q)P^{\prime}P,\qquad\mathcal{F}(q)=\sum_{n=1}^{+\infty}f_{n}\
q^{n}+\sum_{n=1}^{+\infty}f_{-n}\ q^{-n},$ (12) $\displaystyle\hskip
28.45274ptS_{\mu\nu}(\Box^{\ell}P,\Box^{(n-1-\ell)}P)=q^{n-1}S_{\mu\nu}(P,P),$
(13) $\displaystyle\hskip
28.45274ptS_{\mu\nu}(\Box^{-(\ell+1)}P,\Box^{-(n-\ell)}P)=q^{-n-1}S_{\mu\nu}(P,P),$
(14) $\displaystyle\hskip
28.45274ptS_{\mu\nu}(P,P)=g_{\mu\nu}\big{(}\nabla^{\alpha}P\
\nabla_{\alpha}P+P\Box P\big{)}-2\nabla_{\mu}P\ \nabla_{\nu}P,$ (15)
$\displaystyle\hskip
28.45274pt\Omega_{\mu\nu}=\mathcal{F}^{\prime}(q)S_{\mu\nu}(P,P),\qquad\mathcal{F}^{\prime}(q)=\sum_{n=1}^{+\infty}n\
f_{n}\ q^{n-1}-\sum_{n=1}^{+\infty}n\ f_{-n}\ q^{-n-1},$ (16)

and

$G_{\mu\nu}+\Lambda
g_{\mu\nu}-\frac{g_{\mu\nu}}{2}\mathcal{F}(q)P^{2}+2\mathcal{F}(q)R_{\mu\nu}PP^{\prime}-2\mathcal{F}(q)K_{\mu\nu}PP^{\prime}+\frac{1}{2}\mathcal{F}^{\prime}(q)S_{\mu\nu}(P,P)=0.$
(17)

The last equation becomes

$\displaystyle\left(G_{\mu\nu}+\Lambda
g_{\mu\nu}\right)\left(1+2\mathcal{F}(q)PP^{\prime}\right)+\mathcal{F}(q)g_{\mu\nu}\left(-\frac{1}{2}P^{2}+PP^{\prime}(R-2\Lambda)\right)-2\mathcal{F}(q)K_{\mu\nu}PP^{\prime}+\frac{1}{2}\mathcal{F}^{\prime}(q)S_{\mu\nu}(P,P)=0.$
(18)

If $P(R)=\sqrt{R-2\Lambda}$, then $P(R)P^{\prime}(R)=\frac{1}{2}$ and

$\Box\sqrt{R-2\Lambda}=q\,\sqrt{R-2\Lambda}=\eta\,\Lambda\sqrt{R-2\Lambda},\qquad\eta\,\Lambda\neq
0,$ (19)

where $q=\eta\,\Lambda$ and $q^{-1}=\eta^{-1}\Lambda^{-1}$ ($\eta$ –
dimensionless) follows from dimensionality of equalities (19). Since
$P(R)=\sqrt{R-2\Lambda}$, EoM (18) simplify to

$\left(G_{\mu\nu}+\Lambda
g_{\mu\nu}\right)\left(1+\mathcal{F}(q)\right)+\frac{1}{2}\mathcal{F}^{\prime}(q)S_{\mu\nu}(\sqrt{R-2\Lambda},\sqrt{R-2\Lambda})=0.$
(20)

It is clear that EoM (20) are satisfied if

$\displaystyle\mathcal{F}(q)=-1\qquad\qquad\text{and}\qquad\qquad\mathcal{F}^{\prime}(q)=0.$
(21)

It is worth pointing out that not only nonlocal de Sitter model (5) is very
simple and natural but also such are corresponding EoM (20) with respect to
all other models and their EoM that can be derived from (1) with $\Lambda\neq
0$.

Let us remark that nonlocal operator $\mathcal{F}(\Box)$, which satisfies
conditions (21) in model (5), can be taken in the symmetric form

$F(\Box)=1+\mathcal{F}(\Box),\qquad\mbox{where}\qquad\mathcal{F}(\Box)=\sum_{n=1}^{+\infty}\tilde{f}_{n}\Big{[}\Big{(}\frac{\Box}{q}\Big{)}^{n}+\Big{(}\frac{q}{\Box}\Big{)}^{n}\Big{]},$
(22)

where $\tilde{f}_{n}$ are dimensionless coefficients. It is easy to prove that
$\mathcal{F}(\Box)$ presented in the following symmetric form:

$\displaystyle\mathcal{F}(\Box)=\sum_{n=1}^{+\infty}\tilde{f}_{n}\Big{[}\Big{(}\frac{\Box}{q}\Big{)}^{n}+\Big{(}\frac{q}{\Box}\Big{)}^{n}\Big{]}=-\frac{1}{2e}\,\Big{(}\frac{\Box}{q}e^{\frac{\Box}{q}}+\frac{q}{\Box}e^{\frac{q}{\Box}}\Big{)},\qquad
q\neq 0,$ (23)

satisfies conditions (21). We can take the eigenvalue $q$ of d’Alembertian
$\Box$ to be proportional to $\Lambda$, i.e. $q=\eta\,\Lambda$, where
$\eta\,\neq 0$ (since $\Lambda\neq 0$) is a definite dimensionless constant.
Moreover, it has to be $q=\eta\,\Lambda$, since there is no other parameter
than $\Lambda$ which is of the same dimension as $\Box$ in this nonlocal
gravity model. Hence, nonlocal operator (23) can be rewritten as

$\displaystyle\mathcal{F}(\Box)=\sum_{n=1}^{+\infty}\tilde{f}_{n}\Big{[}\Big{(}\frac{\Box}{\eta\Lambda}\Big{)}^{n}+\Big{(}\frac{\eta\Lambda}{\Box}\Big{)}^{n}\Big{]}=-\frac{1}{2e}\Big{(}\frac{\Box}{\eta\Lambda}e^{\frac{\Box}{\eta\Lambda}}+\frac{\eta\Lambda}{\Box}e^{\frac{\eta\Lambda}{\Box}}\Big{)},\qquad\eta\,\Lambda\neq
0,$ (24)

where for some specific $\,\Box\,$ holds
$\,\Box\sqrt{R-2\Lambda}=\eta\,\Lambda\sqrt{R-2\Lambda}.$

##  3 Schwarzschild-de Sitter-type metric

We want to investigate our model outside the spherically symmetric massive
body. Since this model is a nonlocal generalization of general relativity with
the cosmological constant $\Lambda$, it is natural to consider a
generalization of the Schwarzschild-de Sitter (SdS) metric starting from the
standard Schwarzschild expression

$\mathrm{d}s^{2}=-A(r)\mathrm{d}t^{2}+B(r)\mathrm{d}r^{2}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\varphi^{2}.$
(25)

The corresponding scalar curvature $R$ of above metric (25) is

$\displaystyle
R=\frac{2}{r^{2}}-\frac{2}{r^{2}B(r)}-\frac{2A^{\prime}(r)}{rA(r)B(r)}+\frac{A^{\prime}(r)^{2}}{2A(r)^{2}B(r)}+\frac{2B^{\prime}(r)}{rB^{2}(r)}+\frac{A^{\prime}(r)B^{\prime}(r)}{2A(r)B(r)^{2}}-\frac{A^{\prime\prime}(r)}{A(r)B(r)}\,$
(26)

where ’ denotes derivative with respect to $r.$ In this paper we will consider
the case $B(r)=A(r)^{-1}$, and formula (26) becomes

$\displaystyle
R=\frac{2-2A(r)-4rA^{\prime}(r)-r^{2}A^{\prime\prime}(r)}{r^{2}}.$ (27)

Note that (27) can be rewritten in the more compact form

$\displaystyle R(r)=\frac{1}{r^{2}}\frac{\partial^{2}}{\partial
r^{2}}\big{[}r^{2}\big{(}1-A(r)\big{)}\big{]}.$ (28)

According to Section 2, to find a solution of equations of motion it is
necessary to solve an eigenvalue problem (19), that is
$\Box\sqrt{R-2\Lambda}=q\,\sqrt{R-2\Lambda}$. Note that here d’Alembertian
$\Box$ acts in the following way:

$\displaystyle\Box
u(r)=A(r)\,u^{\prime\prime}(r)+(A^{\prime}(r)+\frac{2}{r}\;A(r))\,u^{\prime}(r)=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\big{[}r^{2}A(r)\frac{\partial u}{\partial r}\big{]},$ (29)

where $u(r)$ is any differentiable scalar function.

Let us now consider function $A(r)$ in the form

$\displaystyle
A(r)=1-\frac{\mu}{r}-\frac{\nu}{r^{2}}-\frac{\Lambda}{3}r^{2}-f(r),$ (30)

where $\mu$ and $\nu$ are some parameters to be discussed later. Then one can
show that for $A(r)$ given by (30) holds

$\displaystyle R(r)=\frac{1}{r^{2}}\frac{\partial^{2}}{\partial
r^{2}}\big{[}r^{2}\big{(}1-A(r)\big{)}\big{]}=4\Lambda+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial
r^{2}}\big{[}r^{2}f(r)\big{]}.$ (31)

Denoting $u(r)=\sqrt{R-2\Lambda}$ and using expression (29) for d’Alembertian,
the corresponding equation $\Box\sqrt{R-2\Lambda}=q\,\sqrt{R-2\Lambda}$
becomes

$\displaystyle\Box\sqrt{R-2\Lambda}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\big{[}r^{2}A(r)\frac{\partial}{\partial
r}\sqrt{R-2\Lambda}\big{]}=q\,\sqrt{R-2\Lambda}.$ (32)

Since unknown function $A(r)$ is contained also in the scalar curvature
$R(r)$, equation (32) is very complicated and difficult to find exact
solution. To get an approximative solution we take $A(r)\thickapprox 1$ in
(32), what is applicable when

$\displaystyle\big{|}\frac{\mu}{r}\big{|}\ll
1,\quad\big{|}\frac{\nu}{r^{2}}\big{|}\ll 1,\quad|\Lambda r^{2}|\ll
1,\quad|f(r)|\ll 1,$ (33)

in units $c=1$. Under conditions (33), equation (32) becomes

$\displaystyle\triangle\sqrt{R-2\Lambda}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\big{[}r^{2}\frac{\partial}{\partial
r}\sqrt{R-2\Lambda}\big{]}=q\,\sqrt{R-2\Lambda},$ (34)

where

$\displaystyle\triangle=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\big{[}r^{2}\frac{\partial}{\partial r}\big{]}=\frac{\partial^{2}}{\partial
r^{2}}+\frac{2}{r}\frac{\partial}{\partial r}$ (35)

is the Laplace operator (Laplacian) in spherical coordinates.

One can easily check that the general solution of equation

$\displaystyle\triangle u(r)=\frac{\partial^{2}u(r)}{\partial
r^{2}}+\frac{2}{r}\frac{\partial u(r)}{\partial r}=q\ u(r)$ (36)

is

$\displaystyle u(r)=\frac{C_{1}}{r}\ e^{\sqrt{q}\ r}+\frac{C_{2}}{r}\
e^{-\sqrt{q}\ r}\,.$ (37)

Since the metric should tend to the Minkowski one at large distances, in the
sequel we will use only solution

$\displaystyle u(r)=\frac{C_{2}}{r}\ e^{-\sqrt{q}\ r}.$ (38)

According to equation (31) we have

$\displaystyle
R(r)-2\Lambda=2\Lambda+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial
r^{2}}\big{[}r^{2}f(r)\big{]}=u^{2}(r).$ (39)

Equation (39) can be rewritten in the form

$\displaystyle r^{2}f^{{}^{\prime\prime}}(r)+4rf^{\prime}(r)+2f(r)=-2\Lambda
r^{2}+C_{2}^{2}\ e^{-2\sqrt{q}\ r}.$ (40)

General solution of equation (40) is

$\displaystyle f(r)=-\frac{\Lambda}{6}r^{2}+\frac{C_{2}^{2}}{4q}\
\frac{1}{r^{2}}\ e^{-2\sqrt{q}\ r}+\frac{C_{3}}{r}+\frac{C_{4}}{r^{2}}.$ (41)

Replacing $f(r)$ in (30) by expression (41) one obtains

$\displaystyle A(r)=1-\frac{\mu}{r}-\frac{\nu}{r^{2}}-\frac{\Lambda
r^{2}}{6}-\frac{C_{2}^{2}}{4qr^{2}}\ e^{-2\sqrt{q}\
r}-\frac{C_{3}}{r}-\frac{C_{4}}{r^{2}}.$ (42)

Since all parameters $\mu,\nu,C_{3},C_{4}$ have been so far arbitrary we can
take $C_{3}=C_{4}=0$ and maintain only $\mu$ and $\nu$, i. e. in the sequel we
have

$\displaystyle A(r)=1-\frac{\mu}{r}-\frac{\nu}{r^{2}}-\frac{\Lambda
r^{2}}{6}-\frac{C_{2}^{2}}{4qr^{2}}\ e^{-2\sqrt{q}\ r}.$ (43)

Then the corresponding scalar curvature becomes

$\displaystyle R(r)=2\Lambda+\frac{C_{2}^{2}}{r^{2}}\ e^{-2\sqrt{q}\ r}.$ (44)

##  4 Discussion and conclusion

In the previous sections we considered nonlocal de Sitter gravity model
defined by its action (5) at scales smaller than the cosmological one, i.e.
related to stars, galaxies, and clusters of galaxies. Cosmological solutions
of model (5) are presented in dimitrijevic10 ; dimitrijevic13 , and it was
shown that the cosmological solution in flat space-time,
$a_{1}(t)=At^{\frac{2}{3}}e^{\frac{\Lambda}{14}t^{2}}$ gives good agreement
with observational data. Consequently, it is natural to see how this nonlocal
model works at the smaller scales.

It is well known that $A(r)$ of the standard Schwarzschild-de Sitter metric,
i.e. in the case of local de Sitter gravity, is

$\displaystyle A_{\ell}(r)=1-\frac{2GM}{c^{2}r}-\frac{\Lambda
r^{2}}{3c^{2}},\qquad r\geq r_{0},$ (45)

where $r_{0}$ is the radius of spherically symmetric massive body ($M$–mass),
and $G$ is the Newton gravitational constant. Note that (45) is written in the
international system of units (SI). The nonlocal version of $A(r)$ (42) can be
rewritten as

$\displaystyle A_{n\ell}(r)=1-\frac{2GM}{c^{2}r}-\frac{\Lambda
r^{2}}{6c^{2}}+\frac{\delta^{2}}{qr^{2}}\big{(}1-e^{-2\sqrt{q}\
r}\big{)},\qquad q=\frac{\eta\Lambda}{c^{2}},$ (46)

where we joined terms $\frac{\nu}{r^{2}}$ and $\frac{C_{2}^{2}}{4qr^{2}}\
e^{-2\sqrt{q}\ r}$ taking
$\delta^{2}=\frac{C_{2}^{2}}{4},\nu=-\frac{\delta^{2}}{q}$. It is worth noting
that $\delta$ and $\eta$ are dimensionless parameters, and their values should
be determined by experiments (astronomical observations). Comparing (45) and
(46) follows formula

$\displaystyle A_{n\ell}(r)-A_{\ell}(r)=\frac{\Lambda
r^{2}}{6c^{2}}+\frac{\delta^{2}}{qr^{2}}\big{(}1-e^{-2\sqrt{q}\ r}\big{)}.$
(47)

Testing formula (46) in the Solar and other astronomical systems is one of our
main next tasks, and the results will be presented elsewhere.

At the end, it is worth noting that in paper dragovich2  was found connection
between cosmological constant $\Lambda$ and mass of a scalar particle which
has its origin in $p$-adic string theory dragovich1 . About cosmological
research with application of nonlocal fields in the matter sector of general
relativity one can see arefeva2 ; koshelev2011  and references therein.

## Appendix

In the appendix we present curvature tensors for the Schwarzschild-de Sitter-
type metric (25)

$\mathrm{d}s^{2}=-A(r)\mathrm{d}t^{2}+B(r)\mathrm{d}r^{2}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\varphi^{2}.$
(48)

The Christoffel symbols are given as follows (unlisted ones are equal to zero
and ′ denotes derivative with respect to $r$):

$\displaystyle\Gamma_{01}^{0}$
$\displaystyle=\frac{1}{2}\frac{A^{\prime}}{A},$
$\displaystyle\Gamma_{00}^{1}$
$\displaystyle=\frac{1}{2}\frac{A^{\prime}}{B},$
$\displaystyle\Gamma_{11}^{1}$
$\displaystyle=\frac{1}{2}\frac{B^{\prime}}{B},$
$\displaystyle\Gamma_{22}^{1}$ $\displaystyle=-\frac{r}{B},$
$\displaystyle\Gamma_{33}^{1}$ $\displaystyle=-\frac{r\sin^{2}\theta}{B},$
(49) $\displaystyle\Gamma_{12}^{2}$ $\displaystyle=\frac{1}{r},$
$\displaystyle\Gamma_{33}^{2}$ $\displaystyle=-\sin\theta\cos\theta,$
$\displaystyle\Gamma_{13}^{3}$ $\displaystyle=\frac{1}{r},$
$\displaystyle\Gamma_{23}^{3}$ $\displaystyle=\cot\theta.$

Curvature tensor components with three or four different indices are all equal
to zero. Only nonzero components are with two different indices and they are
listed below:

$\displaystyle R_{0101}$
$\displaystyle=\frac{A}{4}\left(-\left(\frac{A^{\prime}}{A}\right)^{2}-\frac{A^{\prime}}{A}\frac{B^{\prime}}{B}+2\frac{A^{\prime\prime}}{A}\right),$
$\displaystyle R_{0202}$ $\displaystyle=\frac{r}{2}\frac{A^{\prime}}{B},$
$\displaystyle R_{0303}$
$\displaystyle=\frac{r}{2}\frac{A^{\prime}}{B}\sin^{2}\theta,$ (50)
$\displaystyle R_{1212}$ $\displaystyle=\frac{r}{2}\frac{B^{\prime}}{B},$
$\displaystyle R_{1313}$
$\displaystyle=\frac{r}{2}\frac{B^{\prime}}{B}\sin^{2}\theta,$ $\displaystyle
R_{2323}$ $\displaystyle=r^{2}\frac{B-1}{B}\sin^{2}\theta.$

The Ricci tensor is diagonal and its components are:

$\displaystyle R_{00}$
$\displaystyle=\frac{A^{\prime\prime}}{2B}-\frac{A^{\prime}B^{\prime}}{4B^{2}}-\frac{A^{\prime
2}}{4AB}+\frac{A^{\prime}}{rB},$ $\displaystyle R_{11}$
$\displaystyle=-\frac{A^{\prime\prime}}{2A}+\frac{A^{\prime}B^{\prime}}{4A(r)B(r)}+\frac{A^{\prime
2}}{4A^{2}}+\frac{B^{\prime}}{r},$ (51) $\displaystyle R_{22}$
$\displaystyle=-\frac{rA^{\prime}}{2AB}+\frac{rB^{\prime}}{2B^{2}}-\frac{1}{B}+1,$
$\displaystyle R_{33}$
$\displaystyle=\left(-\frac{rA^{\prime}}{2AB}+\frac{rB^{\prime}}{2B^{2}}-\frac{1}{B}+1\right)\sin^{2}\theta.$

The scalar curvature is

$R=-\frac{A^{\prime\prime}}{AB}+\frac{A^{\prime}B^{\prime}}{2AB^{2}}+\frac{A^{\prime
2}}{2A^{2}B}-\frac{2A^{\prime}}{rAB}+\frac{2B^{\prime}}{rB^{2}}-\frac{2}{r^{2}B}+\frac{2}{r^{2}}.$
(52)

The Einstein tensor is diagonal and components are presented as

$\displaystyle G_{00}$
$\displaystyle=\frac{AB^{\prime}}{rB^{2}}-\frac{A}{r^{2}B}+\frac{A}{r^{2}},$
$\displaystyle G_{11}$
$\displaystyle=\frac{A^{\prime}}{rA}-\frac{B}{r^{2}}+\frac{1}{r^{2}},$ (53)
$\displaystyle G_{22}$
$\displaystyle=\frac{r^{2}A^{\prime\prime}}{2AB}-\frac{r^{2}A^{\prime}B^{\prime}}{4AB^{2}}-\frac{r^{2}A^{\prime
2}}{4A^{2}B}+\frac{rA^{\prime}}{2AB}-\frac{rB^{\prime}}{2B^{2}},$
$\displaystyle G_{33}$
$\displaystyle=\left(\frac{r^{2}A^{\prime\prime}}{2AB}-\frac{r^{2}A^{\prime}B^{\prime}}{4AB^{2}}-\frac{r^{2}A^{\prime
2}}{4A^{2}B}+\frac{rA^{\prime}}{2AB}-\frac{rB^{\prime}}{2B^{2}}\right)\sin^{2}\theta.$

###  Case $B=\frac{1}{A}$.

In particular, for $B=\frac{1}{A}$ we have

$\mathrm{d}s^{2}=-A(r)\mathrm{d}t^{2}+\frac{1}{A(r)}\mathrm{d}r^{2}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\varphi^{2}.$
(54)

The Christoffel symbols are:

$\displaystyle\Gamma_{01}^{0}$
$\displaystyle=\frac{1}{2}\frac{A^{\prime}}{A},$
$\displaystyle\Gamma_{00}^{1}$ $\displaystyle=\frac{1}{2}AA^{\prime},$
$\displaystyle\Gamma_{11}^{1}$
$\displaystyle=-\frac{1}{2}\frac{A^{\prime}}{A},$
$\displaystyle\Gamma_{22}^{1}$ $\displaystyle=-rA,$
$\displaystyle\Gamma_{33}^{1}$ $\displaystyle=-rA\sin^{2}\theta,$ (55)
$\displaystyle\Gamma_{12}^{2}$ $\displaystyle=\frac{1}{r},$
$\displaystyle\Gamma_{33}^{2}$ $\displaystyle=-\sin\theta\cos\theta,$
$\displaystyle\Gamma_{13}^{3}$ $\displaystyle=\frac{1}{r},$
$\displaystyle\Gamma_{23}^{3}$ $\displaystyle=\cot\theta.$

Curvature tensor components are:

$\displaystyle R_{0101}$ $\displaystyle=\frac{1}{2}A^{\prime\prime},$
$\displaystyle R_{0202}$ $\displaystyle=\frac{r}{2}AA^{\prime},$
$\displaystyle R_{0303}$ $\displaystyle=\frac{r}{2}AA^{\prime}\sin^{2}\theta,$
(56) $\displaystyle R_{1212}$
$\displaystyle=-\frac{r}{2}\frac{A^{\prime}}{A},$ $\displaystyle R_{1313}$
$\displaystyle=-\frac{r}{2}\frac{A^{\prime}}{A}\sin^{2}\theta,$ $\displaystyle
R_{2323}$ $\displaystyle=r^{2}(1-A)\sin^{2}\theta.$

The Ricci tensor is diagonal and its components are:

$\displaystyle R_{00}$
$\displaystyle=\frac{1}{2}AA^{\prime\prime}+\frac{1}{r}AA^{\prime},$
$\displaystyle R_{11}$
$\displaystyle=-\frac{1}{2}\frac{A^{\prime\prime}}{A}-\frac{1}{r}\frac{A^{\prime}}{A},$
$\displaystyle R_{22}$ $\displaystyle=1-A-rA^{\prime},$ $\displaystyle R_{33}$
$\displaystyle=\left(1-A-rA^{\prime}\right)\sin^{2}\theta.$ (57)

The scalar curvature is

$R=-A^{\prime\prime}-\frac{4}{r}A^{\prime}-\frac{2}{r^{2}}A+\frac{2}{r^{2}}.$
(58)

The Einstein tensor is presented as follows:

$\displaystyle G_{00}$
$\displaystyle=-\frac{A(r)A^{\prime}(r)}{r}-\frac{A(r)^{2}}{r^{2}}+\frac{A(r)}{r^{2}},$
$\displaystyle G_{11}$
$\displaystyle=\frac{A^{\prime}(r)}{rA(r)}-\frac{1}{r^{2}A(r)}+\frac{1}{r^{2}},$
(59) $\displaystyle G_{22}$
$\displaystyle=\frac{1}{2}r^{2}A^{\prime\prime}(r)+rA^{\prime}(r),$
$\displaystyle G_{33}$
$\displaystyle=\left(\frac{1}{2}r^{2}A^{\prime\prime}(r)+rA^{\prime}(r)\right)\sin^{2}\theta.$

## Acknowledgement

This research was partially funded by the Ministry of Education, Science and
Technological Developments of the Republic of Serbia: grant number
451-03-68/2022-14/200104 with Faculty of Mathematics, and grant number
451-03-1/2022-14/4 with Teacher Education Faculty, University of Belgrade. It
is also partially supported by the COST Action: CA21136 – Addressing
observational tensions in cosmology with systematics and fundamental physics
(CosmoVerse).

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# Near-Field Localization with an Exact Propagation Model in Presence of
Mutual Coupling ††thanks: This work was supported by Leverhulme Trust under
Research Leadership Award under Grant RL-2019-019 and British Telecom under
UKRI Future Leaders Fellowship grant MR/T019980/1.

Zohreh Ebadi1, Amir Masoud Molaei1, Muhammad Ali Babar Abbasi1, Simon Cotton1,
Anvar Tukmanov2,
and Okan Yurduseven1 Email<EMAIL_ADDRESS>1 CWI, EEECS, Queen’s
University Belfast, BT3 9DT Belfast, U.K.  2 BT Labs, Adastral Park, BT Group,
Ipswich, U.K.

###### Abstract

Localizing near-field sources considering practical arrays is important in
wireless communications. Array-based apertures exhibit mutual coupling between
the array elements, which can significantly degrade the performance of the
localization method. In this paper, we propose two methods to localize near-
field sources by direction of arrival (DOA) and range estimations in the
presence of mutual coupling. The first method utilizes a two-dimensional
search to estimate DOA and the range of the source. Therefore, it suffers from
a high computational load. The second method reduces the two-dimensional
search to one-dimensional, thus decreasing the computational complexity while
offering similar DOA and range estimation performance. Besides, our second
method reduces computational time by over 50% compared to the multiple signal
classification (MUSIC) algorithm.

###### Index Terms:

DOA and range estimations, exact model, mutual coupling, near-field
localization

##  I introduction

Source localization is crucial in sonar, radar, and wireless communication
applications [1, 2]. Conventionally, localization algorithms can be classified
based on source location; far-field and near-field localization algorithms [3,
4]. If the source is in the near-field region, known as the Fresnel region,
then the far-field localization algorithms will not be able to estimate the
location of the source. This is due to the fact that the received wavefront
from the far-field source becomes planar. Therefore, the far-field wavefront
will be characterized by only the direction of arrival (DOA), while both DOA
and range characterize the near-field wavefront because the received signal
from the near-field source has a spherical shape[5, 6].

Mutual coupling refers to the electromagnetic interaction between the antenna
elements of an array. This interaction can reduce the accuracy of the
localization algorithm [7, 8, 9]. Based on the type of elements in the antenna
array, mutual coupling can be categorized as either direction-independent or
direction-dependent [10, 11, 12]. For arrays consisting of omnidirectional
antennas, the mutual coupling can be considered direction-independent.
However, for practical arrays with non-omnidirectional antennas, the mutual
coupling is direction-dependent [12, 11].

Numerous algorithms have been developed over the decades to estimate the
location of a source in the presence of mutual coupling [11, 13, 14, 15, 16,
17, 18, 7]. These algorithms can be categorized into active calibration [11]
and self-calibration[14]. The active calibration algorithms rely on prior
knowledge of the locations of the sources. However, the accuracy of these
methods significantly reduces in the presence of localization errors [11]. On
the other hand, the self-calibration methods estimate the mutual coupling and
location of the source jointly. For example, the method in [14] is an
iterative algorithm that estimates DOA and mutual coupling. However, these
methods only consider the far-field sources.

For near-field DOA and range estimations in the presence of mutual coupling,
methods such as those presented in [15, 16, 17, 18, 7] were proposed. Methods
presented in [15, 16] are developed for direction-independent mutual coupling.
On the other hand, the methods in [17, 18, 7] consider mixed far-field and
near-field sources. In the mixed sources scenarios, the far-field sources’ DOA
and mutual coupling are first estimated. Then, the estimated mutual coupling
is used to compensate for the received signal. This compensated received
signal is then used to predict the near-field source location by estimating
the DOA and the range of the near-field sources. However, the mutual coupling,
DOA, and range must all be estimated simultaneously in scenarios where only
the near-field sources are considered. It will cause a three-dimensional
search algorithm to estimate the three mentioned unknowns, which can
drastically increase the computational complexity.

In this work, we consider near-field source localization in the presence of
the direction-dependent mutual coupling between the receiver array elements.
We use the exact system model [19, 20] for the near-field sources because
using the approximated model [21, 22] causes an error in DOA and range
estimations. This is due to the fact that the exact model, unlike the
approximated model, relies on the behavior of electromagnetic waves in the
near-field [20]. We propose two methods to estimate the location of the source
in the near-field. The first method is inspired by the one described in [23].
The main difference between our first method and method in [23] is that the
method in [23] is designed for the far-field source location estimation, while
our method is for estimating source location in the near-field. Since our
first method relies on two-dimensional searches, we propose a second method,
which employs one-dimensional searches. Therefore, the main contribution of
this work is presenting two algorithms for near-field source localization in
the presence of direction-dependent mutual coupling. In addition, we reduce
the search to two-dimensional and one-dimensional problems, minimizing the
computational complexity.

The rest of the paper is organized as follows. In Section II, we present the
system model. Section III explains the multiple signal classification (MUSIC)
algorithm for the localization of the source in the near-field. Section IV
presents our methods for near-field source localization. Section V provides
simulation results. Finally, conclusions are drawn in Section VI.

##  II system model

Suppose that $N$ narrowband signals impinging on a uniform linear array (ULA),
as shown in Fig. 1.

Figure 1: The considered system model with the sources in the near-field.

The ULA consists of $M$ elements, where $M$ is odd. The inter-element spacing
of elements is $d={\lambda}/{2}$, where $\lambda$ is the signal wavelength.
The output signal of the ULA can be expressed as [10]

$\displaystyle\mathbf{Y}=\mathbf{\tilde{A}}(\boldsymbol{\theta},\mathbf{r})\mathbf{S}+\mathbf{W},$
(1)

where $\mathbf{S}=[s_{1}(t)\ s_{2}(t)\ \cdots\ s_{N}(t)]^{T}$ represents the
vector of the transmitted non-coherent signals, $\mathbf{W}=[w_{1}(t)\
w_{2}(t)\ \cdots\ w_{M}(t)]^{T}$ is white Gaussian noise vector with a zero
mean and variance $\sigma^{2}$, and $t=1,2,\cdots,L$. $L$ is the number of the
snapshots. $\boldsymbol{\theta}=[\theta_{1}\ \theta_{2}\ \cdots\ \theta_{N}]$
is the DOA vector of the received signals at the reference element, i.e.,
zeroth element in the ULA, and $\mathbf{r}=[r_{0,1}\ r_{0,2}\ \cdots\
r_{0,N}]$ is the vector of range of sources where the range is defined as the
distance between the $n$-th source and the reference element, where
$n=1,2,\cdots,N$. We assume that all sources are in the near-field (Fresnel)
region of the antenna array,
$r_{0,n}\in(0.62(D^{3}/\lambda)^{\frac{1}{2}},2D^{2}/\lambda)$, where $D$ is
array aperture [5, 2]. Besides,
$\mathbf{\tilde{A}}(\boldsymbol{\theta},\mathbf{r})=[\mathbf{\tilde{a}}(\theta_{1},r_{0,1})\
\mathbf{\tilde{a}}(\theta_{2},r_{0,2})\ \cdots\
\mathbf{\tilde{a}}(\theta_{N},r_{0,N})]$ is $M\times N$ matrix. Each vector in
$\mathbf{\tilde{A}}(\boldsymbol{\theta},\mathbf{r})$ can be written as [24]

$\displaystyle\mathbf{\tilde{a}}(\theta_{n},r_{0,n})=\mathbf{C}(\theta_{n})\mathbf{a}(\theta_{n},r_{0,n}),$
(2)

where $\mathbf{C}(\theta_{n})\in\mathbb{C}^{M\times M}$ is a direction-
dependent mutual coupling matrix of ULA at the $n$-th angle as follows [10,
12]:

$\displaystyle\mathbf{C}(\theta_{n})=\text{Toeplitz}\\{\mathbf{z}(\theta_{n})\\},$
(3)

where $\mathbf{z}=[\mathbf{c}^{T}(\theta_{n}),\mathbf{0}]^{T}$.
$\mathbf{c}(\theta_{n})=[c_{1}(\theta_{n}),c_{2}(\theta_{n}),\cdots,c_{P}(\theta_{n})]^{T}$
is mutual coupling coefficients vector with dimension of $P\times 1$, and
$P<M$. This is because the mutual coupling coefficients decrease as the
distance between elements increases. Beyond $P$ inter-element spacings,
coupling coefficients become negligible. Furthermore, $\mathbf{0}$ is vector
with dimension of $1\times(M-P)$. Besides, $\mathbf{a}(\theta_{n},r_{0,n})$ is
$M\times 1$ steering vector, where each element in the steering vector based
on the exact model is given by [20]

$\displaystyle\mathbf{a}(\theta_{n},r_{0,n})=\frac{r_{0,n}}{r_{m,n}}e^{-j\tau_{m,n}},$
(4)

where $m=-\frac{M-1}{2},\ \cdots,\ -1,\ 0,\ 1,\ \cdots,\ \frac{M-1}{2}$.
Furthermore, $\tau_{m,n}$ is the phase difference between the $n$-th source
and $m$-th element in the array as follows:

$\displaystyle\tau_{m,n}=\frac{2\pi}{\lambda}(r_{m,n}-r_{0,n}),$ (5)

where $r_{m,n}$ is the distance between the $n$-th source and the array’s
$m$-th element and is given by [20]

$\displaystyle
r_{m,n}=\sqrt{r^{2}_{0,n}+m^{2}d^{2}-2mdr_{0,n}\cos{\theta_{n}}}~{}.$ (6)

In most literature [25, 2, 26], (4) is simplified using the Fresnel
approximation and neglecting the magnitude as follows:

$\displaystyle\mathbf{a}^{{}^{\prime}}(\theta_{n},r_{0,n})=e^{j(\gamma_{n}m+\eta_{n}m^{2})},$
(7)

where

$\displaystyle\begin{split}&\gamma_{n}=-\frac{2\pi
d}{\lambda}\cos{\theta_{n}},\\\ &\eta_{n}=\frac{\pi d^{2}}{\lambda
r_{0,n}}\sin^{2}{\theta_{n}}.\end{split}$ (8)

The model in (7) is the approximated model.

With the array output signal (1), this paper aims to estimate
$\boldsymbol{\theta}$ and $\mathbf{r}$ of non-coherent sources in the presence
of unknown mutual coupling.

##  III doa and range estimation

One standard way to estimate the DOA and range is using the MUSIC algorithm
[27]. To do this, the covariance matrix of the received signal can be written
as

$\displaystyle\mathbf{R}_{y}=\frac{1}{L}\sum_{t=1}^{L}\mathbf{y}(t)\mathbf{y}^{H}(t).$
(9)

where $\mathbf{y}(t)$ is $M\times 1$ vector and denotes the received signal at
$t$-th snapshot. The eigendecomposition of (9) is given by

$\displaystyle\mathbf{R}_{y}=\mathbf{U}\mathbf{\Sigma}\mathbf{U}^{H}.$ (10)

where $\boldsymbol{\Sigma}$ is eigenvalues of $R_{y}$ and
$\mathbf{U}=[\mathbf{U}_{s}\ \mathbf{U}_{w}]$ is eigenvector matrix.
$\mathbf{U}_{s}\in\mathbb{C}^{M\times N}$ and
$\mathbf{U}_{w}\in\mathbb{C}^{M\times(M-N)}$ denote signal and noise
subspaces, respectively. Then, the unknowns can be jointly obtained by
searching $N$ peaks of the MUSIC spatial spectrum as follows:

$\displaystyle
P(\theta_{n},r_{0,n})=\frac{1}{\textbf{a}^{H}(\theta_{n},r_{0,n})\mathbf{C}^{H}(\theta_{n})\textbf{U}_{w}\textbf{U}_{w}^{H}\mathbf{C}(\theta_{n})\textbf{a}(\theta_{n},r_{0,n})}.$
(11)

However, (11) cannot be used to estimate the unknowns including
$\mathbf{C}(\theta_{n})$, DOA, and range. Due to the complex values in
$\mathbf{C}(\theta_{n})$, defining the search area for these values may be
infeasible. On the other hand, $\theta_{n}\in[-\frac{\pi}{2},\frac{\pi}{2}]$
and $r_{0,n}\in(0.62(D^{3}/\lambda)^{\frac{1}{2}},2D^{2}/\lambda)$. Thus,
defining the search area for $\theta_{n}$ and $r_{0,n}$ in (11) is feasible.

##  IV proposed algorithms

In this section, we propose two methods to estimate the DOA and the range in
the presence of mutual coupling.

###  IV-A First Algorithm

In this subsection, we first examine whether the transformation matrix,
commonly used in the literature, can be applied in near-field scenarios. To do
this, according to (2), $\mathbf{\tilde{a}}(\theta_{n},r_{0,n})$ can be
written as follows:

$\displaystyle\mathbf{\tilde{a}}(\theta_{n},r_{0,n})=$
$\displaystyle\sum_{m=1}^{P}\mathbf{E}_{m}\mathbf{a}(\theta_{n},r_{0,n})c_{m}(\theta_{n})=$
$\displaystyle[\mathbf{E}_{1}\mathbf{a}(\theta_{n},r_{0,n})\
\mathbf{E}_{2}\mathbf{a}(\theta_{n},r_{0,n})\ \cdots\
\mathbf{E}_{P}\mathbf{a}(\theta_{n},r_{0,n})]$
$\displaystyle[c_{1}(\theta_{n})\ c_{2}(\theta_{n})\ \cdots\
c_{P}(\theta_{n})]^{T}=$
$\displaystyle\mathbf{X}(\theta_{n},r_{0,n})\mathbf{c}(\theta_{n}),$ (12)

where $\mathbf{X}(\theta_{n},r_{0,n})$ is a ${M\times P}$ matrix. Besides,
$\mathbf{E}_{m}$ is defined as

$\displaystyle[\mathbf{E}_{m}]_{ij}=\begin{cases}1,&[\mathbf{C}]_{ij}(\theta_{n})=z_{m}(\theta_{n})\\\
0,&\text{otherwise}.\end{cases}$ (13)

where $[\cdot]_{ij}$ indicates ${ij}$-th element of the corresponding matrix.
Note that the transformation matrix in (IV-A) differs from that in [23, 10,
28, 12] because $\mathbf{X}$ in (IV-A) is a function of both $\theta$ and $r$,
while in [23, 10, 28, 12], $\mathbf{X}$ is only a function of $\theta$.

By inserting (IV-A) into (11), (11) can be written as

$\displaystyle\\{\hat{\theta}_{n},\hat{r}_{0,n}\\}_{n=1}^{N}=\arg\max_{\theta_{n},r_{0,n},\mathbf{c}}\frac{1}{\textbf{c}^{H}(\theta_{n})\boldsymbol{\Omega}(\theta_{n},r_{0,n})\textbf{c}(\theta_{n})}.$
(14)

where
$\boldsymbol{\Omega}(\theta_{n},r_{0,n})=\mathbf{X}^{H}(\theta_{n},r_{0,n})\textbf{U}_{w}\textbf{U}_{w}^{H}\mathbf{X}(\theta_{n},r_{0,n})$.
To find the unknowns in (14), we solve the following optimization problem for
given $\theta$ and $r$:

$\displaystyle\min_{\mathbf{c}}\textbf{c}^{H}(\theta_{n})\boldsymbol{\Omega}(\theta_{n},r_{0,n})\textbf{c}(\theta_{n})$
$\displaystyle\text{subject to}\ \mathbf{e}_{1}^{H}\mathbf{c}(\theta_{n})=1.$
(15)

where $\mathbf{e}_{1}$ is the first column of the unitary matrix. The detailed
solution of (IV-A) is provided in [23]. To solve this optimization problem, we
form the Lagrangian function as follows

$\displaystyle\mathcal{L}(\mathbf{c},\beta)=\mathbf{c}^{H}(\theta_{n})\boldsymbol{\Omega}(\theta_{n},r_{0,n})\mathbf{c}(\theta_{n})-\beta(\mathbf{e}_{1}^{H}\mathbf{c}(\theta_{n})-1)$
(16)

where $\beta$ is Lagrange multiplier. By setting
$\frac{\partial\mathcal{L}(\mathbf{c},\beta)}{\partial\mathbf{c}(\theta_{n})}$
(first derivative of $\mathcal{L}(\mathbf{c},\beta)$ with respect to
$\mathbf{c}(\theta_{n})$) to zero and using the constraint
$\mathbf{e}_{1}^{H}\mathbf{c}(\theta_{n})=1$, the estimated DOA and range are
given by

$\displaystyle\\{\hat{\theta}_{n},\hat{r}_{0,n}\\}_{n=1}^{N}=\arg\max_{\theta_{n},r_{0,n}}\textbf{e}^{H}_{1}\boldsymbol{\Omega}^{-1}(\theta_{n},r_{0,n})\textbf{e}_{1}.$
(17)

However, this method requires a two-dimensional search. Therefore, the
computational complexity of this method is high. In the next method, we
propose a second algorithm with less computational complexity than the first.

###  IV-B Second Algorithm

This method also utilizes the exact model of the steering vector. The
algorithm steps are as follows:

1) Initialize: Since this method needs an initial value for one of the
unknowns, we use the approximated model to find the initial value for DOA.
$\mathbf{C}(\theta_{n})\mathbf{a}^{{}^{\prime}}(\theta_{n},r_{0,n})$ can be
written as follows

$\displaystyle\mathbf{C}(\theta_{n})\mathbf{a}^{{}^{\prime}}(\theta_{n},r_{0,n})=$
$\displaystyle\begin{bmatrix}g_{1}&c_{1}g_{2}&c_{2}g_{3}&\cdots&c_{M-1}g_{M}\\\
c_{1}g_{1}&g_{2}&c_{1}g_{3}&\cdots&c_{M-2}g_{M}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
c_{M-1}g_{1}&c_{M-2}g_{2}&c_{M-3}g_{3}&\cdots&g_{M}\end{bmatrix}\begin{bmatrix}v_{1}\\\
v_{2}\\\ \vdots\\\ v_{M}\end{bmatrix}=$
$\displaystyle\mathbf{B}(\theta_{n})\mathbf{v}(\theta_{n},r_{0,n}),$ (18)

where

$\displaystyle\begin{split}{v}_{m}(\theta_{n},r_{0,n})=e^{j\eta_{n}m^{2}}\\\
{g}_{m}(\theta_{n})=e^{j\gamma_{n}m}.\end{split}$ (19)

Therefore, by inserting (IV-B) into (11), (11) can be written as

$\displaystyle\\{\hat{\theta}_{n},\hat{r}_{0,n}\\}_{n=1}^{N}=$
$\displaystyle\arg\min_{\theta_{n},r_{0,n}}\textbf{v}^{H}(\theta_{n},r_{0,n})\mathbf{B}^{H}(\theta_{n})\textbf{U}_{w}\textbf{U}^{H}_{w}\mathbf{B}(\theta_{n})\textbf{v}^{H}(\theta_{n},r_{0,n}).$
(20)

To estimate the initial value for $\theta_{n}$, (IV-B) can be approximated by

$\displaystyle\\{\hat{\theta}_{n}\\}_{n=1}^{N}=\arg\min_{\theta_{n}}\mathbf{B}^{H}(\theta_{n})\textbf{U}_{w}\textbf{U}^{H}_{w}\mathbf{B}(\theta_{n}).$
(21)

Using the same procedure as (IV-A),
$\mathbf{B}(\theta_{n})=\mathbf{X}(\theta_{n})\mathbf{c}(\theta_{n})$.
Therefore, (21) can be written as follows

$\displaystyle\\{\hat{\theta}_{n}\\}_{n=1}^{N}=\arg\min_{\theta_{n}}\textbf{c}^{H}(\theta_{n})\mathbf{X}^{H}(\theta_{n})\textbf{U}_{w}\textbf{U}^{H}_{w}\mathbf{X}(\theta_{n})\textbf{c}(\theta_{n}).$
(22)

Therefore, to estimate DOA, the following optimization problem needs to be
solved for given $\theta_{n}$

$\displaystyle\min\mathbf{c}^{H}(\theta_{n})\boldsymbol{\Omega}^{{}^{\prime}}(\theta_{n})\mathbf{c}(\theta_{n})$
$\displaystyle\text{subject to}\ \mathbf{e}^{H}_{1}\mathbf{c}(\theta_{n})=1.$
(23)

where
$\boldsymbol{\Omega}^{{}^{\prime}}(\theta_{n})=\mathbf{X}^{H}(\theta_{n})\textbf{U}_{w}\textbf{U}^{H}_{w}\mathbf{X}(\theta_{n})$.
Therefore, the initial value of DOA is as follows

$\displaystyle\hat{\theta}^{(i)}_{n}=\arg\max_{\theta_{n}}\mathbf{e}^{H}_{1}{\boldsymbol{\Omega}^{{}^{\prime}}}^{-1}(\theta_{n})\mathbf{e}_{1}.$
(24)

where $i$ denotes the iteration index.

2) Estimate the range of the source: $\hat{r}_{0,n}$ can be found using

$\displaystyle\hat{r}_{0,n}=\arg\max_{r_{0,n}}\mathbf{e}^{H}_{1}\boldsymbol{\Omega}^{-1}(\hat{\theta}_{n},r_{0,n})\mathbf{e}_{1}.$
(25)

3) Estimate the DOA of the source: since the initial estimate
$\hat{\theta}_{n}$ in (24) relies on an approximated model and certain
assumptions, the resulting $\hat{r}_{0,n}$ can be utilized to modify the DOA.

$\displaystyle\hat{\theta}^{(i+1)}_{n}=\arg\max_{\theta_{n}}\mathbf{e}^{H}_{1}\boldsymbol{\Omega}^{-1}(\theta_{n},\hat{r}_{0,n})\mathbf{e}_{1}.$
(26)

4) Check the constraint: If $|\hat{\theta}^{i+1}_{n}-\hat{\theta}^{i}_{n}|<q$,
then terminate; otherwise, return step 2. In the simulations, $q$ is a small
positive number.

Figure 2: Second algorithm for DOA and range estimation of sources in the
near-field.

Based on the above steps, the second algorithm is summarized in Fig. 2. This
method requires one-dimensional searches to estimate the DOA and the range of
the near-field source.

##  V simulation results

In this section, simulations are conducted to validate the performance of the
proposed methods. In these simulations, we consider a ULA consisting of five
elements operating at a frequency of $5$ GHz. The number of snapshots equals
$200$. We consider one source in different locations within the Fresnel
region, $[1.75\lambda,8\lambda]$, with DOA of
$[30^{\circ},40^{\circ},50^{\circ},60^{\circ}]$ and range of $3.3\lambda$. We
assume that the mutual coupling is direction-dependent. So, for each DOA, the
mutual coupling coefficients are generated randomly [12]. The number of
nonzero mutual coupling coefficients, $P$, is considered to be equal to $3$.
All simulation results are obtained based on the average of $K=1000$
independent trials. As a performance metric for the accuracy of the
estimations, the root mean square error (RMSE) is used. RMSE is defined as [2]

$\displaystyle\text{RMSE}_{\alpha}=\sqrt{\frac{1}{KN}\sum_{k=1}^{K}\sum_{n=1}^{N}(\hat{\alpha}_{n,k}-\alpha_{n})^{2}}.$
(27)

where $\alpha$ and $\hat{\alpha}$ denote the true and estimated values of DOA
or range, respectively. In all simulations, $q$ for the second method is
$0.1^{\circ}$.

(a)

(b)

Figure 3: RMSE of estimated (a) DOA, (b) range versus SNR.

Example 1: This simulation investigates the RMSE of DOA and range versus
signal-to-noise ratio (SNR), as shown in Fig. 3. The SNR varies from $0$ dB to
$30$ dB in steps of $5$ dB. The MUSIC algorithm is used as a baseline to
compare the accuracy of the proposed methods with it. The mutual coupling is
assumed to be known for the MUSIC algorithm.

As seen in Fig. 3(a), the first and second algorithms’ accuracy is almost the
same for all locations. Furthermore, the RMSE of the estimated DOA using MUSIC
is less than the RMSE of the estimated DOA using the first and second methods.
For example, for a DOA of $30^{\circ}$, the RMSE of the estimated DOA using
MUSIC is $0.69^{\circ}$ less than the RMSE of the estimated DOA using the
second method in SNR of $0$ dB. This difference decreases as SNR increases,
such that at an SNR of $30$ dB, the difference reaches $0.03^{\circ}$.

Fig. 3(b) shows that the RMSE of the estimated range using the MUSIC algorithm
is lower than the RMSE of the estimated range using the first and second
methods for all locations. For instance, for DOA of $30^{\circ}$, the RMSE of
the estimated range using MUSIC is $0.46\lambda$ and $0.01\lambda$ less than
the RMSE of the estimated range using the second method in SNR of $0$ dB and
$30$ dB, respectively.

It should be noted that the MUSIC algorithm outperforms the proposed methods
since it assumes the knowledge of mutual coupling.

Example 2: In this simulation, the computational load of the methods is
compared using the computational time. In this simulation, SNR and $K$ are
$10$ dB and $1$, respectively. The simulations are conducted on the same
machine with Intel(R) Core(TM) i7-1265U processor and $16$ GB of RAM. MATLAB
version is R2022b.

The computational time for the first method, second method, and MUSIC
algorithm are $0.90$ sec, $0.12$ sec, and $0.30$ sec, respectively. The second
method’s computational time is less than the other two methods because it
estimates DOA and the range of sources using one-dimensional searches.

##  VI conclusion

In this paper, we proposed two methods to estimate the DOA and range of a
source in the near-field, in the presence of mutual coupling. The first method
is based on a two-dimensional search that simultaneously estimates the DOA and
range of the source. On the other hand, the second algorithm is an iterative
method based on a one-dimensional search. In the latter method, each unknown
is estimated separately using the other estimated unknown from the previous
step.

The significance of this work is twofold. Firstly, our proposed methods
estimate DOA and the range of the source in near-field without estimating
mutual coupling coefficients. As a result, our methods reduce the
computational load by avoiding the need to search through a large search area
to estimate mutual coupling. Secondly, our second method yields performance
similar to our first method while having a lower computational load compared
to the first method. In addition, our second method reduces the computational
time by more than $50\%$ in comparison to the MUSIC algorithm.

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