# Leaky-Wave Sectorial Antennas for Passive 2-D DoA Applications

Matteo Perrone, Julien Sarrazin, Guido Valerio, Guido Lombardi

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Matteo Perrone, Julien Sarrazin, Guido Valerio, Guido Lombardi. Leaky-Wave Sectorial Antennas for Passive 2-D DoA Applications. IEEE Antennas and Wireless Propagation Letters, 2025, 24 (10), pp.3420-3424. 10.1109/LAWP.2025.3592337. hal-05302624

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# Leaky-Wave Sectorial Antennas for Passive Two-Dimensional DoA Applications

Matteo Perrone, Student Member, IEEE, Julien Sarrazin, Senior Member, IEEE, Guido Valerio, Senior Member, IEEE, Guido Lombardi, Senior Member, IEEE

***Abstract***—A fully metallic sectorial leaky-wave antenna (LWA) is proposed to perform passive two-dimensional direction-of-arrival (DoA) estimation. The LWA consists of a sectorial waveguide with periodic corrugations on the bottom plate and periodic slots on the top plate. The frequency dispersion of the waveguide is enhanced with a suitable choice of the geometric parameters of the corrugations using an analytical model. Two different corrugation shapes are selected: concentric and straight corrugations. The former one produces an isotropic dispersion, while the latter one produces an anisotropic dispersion of the aperture field when slots are etched to open the structure. In the open structure, the sectorial geometry and the circular slots allow for a 2D directional pattern in both azimuth and elevation, and straight corrugations lead to a stronger azimuth-dependent radiation pattern within the field of view. The performance of the LWAs for 2D DoA estimation are evaluated using the Multiple Signal Classification (MUSIC) algorithm. Results show that an anisotropic design of the corrugations enables a correct DoA estimation of both elevation and azimuth angles with a single-port antenna and with a fully passive system, being much more cost- and power-effective than traditional phased arrays.

***Index Terms***—periodic structures, metasurfaces, leaky-wave antennas, direction-of-arrival estimation, MUSIC.

## I. INTRODUCTION

Two-dimensional direction-of-arrival (DoA) estimation is essential in modern sensing and millimeter-wave (mmWave) communication systems, allowing precise spatial detection across both azimuth and elevation angles. Conventional solutions typically employ phased arrays, which offer beam-steering capabilities through active phase shifting across multiple elements. However, such systems are often bulky, expensive, and can lead to a high power consumption due to their reliance on active components [1]–[3].

Alternative solutions are offered by passive metasurfaces, cost-effective structures that enable the spatial modulation of phase and amplitude distributions in guided and radiated waves. They can be used to design leaky wave antennas (LWAs) [4]–[6] such as periodic slotted waveguides with corrugated plates [7]–[14], that enable controlled energy leakage within a desired frequency range allowing tailored frequency-beam scanning. This frequency diversity makes metasurface-based systems ideal for applications such as DoA estimation in complex

M. Perrone and G. Lombardi are with Department of Electronics and Telecommunications (DET), Politecnico di Torino, 10129, Turin, Italy (email: matteo_perrone, guido.lombardi@polito.it).

J. Sarrazin and G. Valerio are with Sorbonne Université, CNRS, Laboratoire de Génie Electrique et Electronique de Paris, 75252, Paris, France and Université Paris-Saclay, CentraleSupélec, CNRS, Laboratoire de Génie Electrique et Electronique de Paris, 91192, Gif-sur-Yvette, France (email: julien.sarrazin, guido.valerio@sorbonne-universite.fr).

<figure><img src="image_2.png" /><figcaption>Fig. 1. Corrugated waveguides studied in this paper with the feeding in red. Sectorial waveguide (a) with concentric corrugations and, (b) with straight corrugations in the bottom plate. The geometric parameters are $\phi_0 = 73^{\circ}$, $L = 210$ mm, $\rho_0 = 4.8$ mm, $a = 7.112$ mm, $b = 2.556$ mm, $h_c = 1.6$ mm, $p_c = 2$ mm, $w_c = 1$ mm ($h = b + h_c$).</figcaption></figure>

scenarios. In this regard, the Multiple Signal Classification (MUSIC) algorithm [15], a widely used high-resolution technique, has proven to be effective for DoA estimation. Using the spatial covariance matrix of the received signals, MUSIC enables precise signal resolution in high-angular-resolution environments. However, LWA DoA two-dimensional estimation currently relies on multiport configurations [16], thus leading to complex front-end architectures.

This letter proposes a passive solution for DoA estimation using LWAs based on sectorial corrugated metasurface. The proposed LWA enables a two-dimensional and an azimuth-dependent directional pattern, scanning the elevation angle with frequency. By exploiting the azimuthal dependence of the frequency dispersion of the radiation pattern with appropriate signal processing, the DoA can be estimated over a 2D FoV using for the first time a single-port antenna, thus resolving with reduced complexity the azimuth ambiguity typical of conventional 2D-periodic LWAs [16]. This passive and low-cost novel structure is well suited for next-generation integrated systems where compactness and power efficiency are critical. The paper is organized as follows. Section II describes the sectorial waveguides with corrugations on the bottom plate, and their dispersion properties. In Section III the LWAs are designed by etching concentric slots on the top plate and simulated results are presented. Section IV performs DoA estimation using the proposed LWAs with MUSIC algorithm. Conclusions are drawn in Section V.

## II. CORRUGATED SECTORIAL WAVEGUIDES

We aim at designing a LWA with an enhanced frequency dispersion in the operating frequency range, having a directional pattern along two principal planes and exhibiting different scanning rates according to the azimuthal observation point. To achieve this goal, the underlying hypothesis in this work is that an azimuth-dependent frequency dispersion of the LWA beam direction can be achieved with a leaky waveguide exhibiting
different frequency dispersion along different propagation di-
rections, i.e. a dispersion anisotropy (DA). In this section, we
describe sectorial waveguides having the desired dispersion
properties, while in the following one, we design the LWA by
opening slots on the top of the closed structures.

The closed sectorial waveguides are shown in Fig. 1. They
have an angular aperture $\phi_0$, a radial length $L$, and are fed
by a rectangular waveguide placed at a distance $\rho_0$ from the
sector center. These waveguides are here completely shielded
with top, bottom, and lateral PEC boundaries [17], [18]. The
angular aperture $\phi_0 = 73^\circ$ is chosen in order to cover a wide
FoV (60$^\circ$ and 100$^\circ$ in azimuth as shown in Section IV), which
requires small apertures, characterized by a noticeable effect
of the DA, achievable with large apertures.

We use a cylindrical coordinate system ($\rho$, $\phi$, $z$) centered in
the sector center. The waveguides have a symmetry plane at
$\phi = 0$. A TM$^z$ polarized wave is excited in the waveguides,
travelling along the $+\rho$ direction with a radial dependence
represented by Hankel functions of the second kind and order
$q$, $H_q^{(2)}(k_\rho \rho)$, and a sinusoidal azimuthal behavior to fulfill
boundary conditions on side plates [18], [19]. $k_\rho = \beta - j\alpha$ is
the radial wavenumber, where $\beta$ and $\alpha$ are the radial phase and
attenuation constants, respectively. The frequency dispersion
of the structure (variation of $\beta$ vs. frequency) is enhanced by
inserting periodic corrugations in the bottom plate and suitably
varying their height. The waveguide is assumed lossless, so
$k_\rho = \beta$. As discussed in [9], we feed the structure with
a rectangular wabeguide port which excites the fundamental
TM$^z_{01}$ propagating mode in the sectorial waveguides.

To increase the frequency dispersion of the waveguides (that
is, to decrease the slope of the dispersion curve in a Brillouin
diagram frequency vs. $\beta$), corrugations can be designed to
exhibit a bandgap close to the operating frequency range.
In our case, for mmWave sensing applications, the chosen
range is [26.5, 29.5] GHz. The increased frequency dispersion
enables a faster frequency scanning in the LWA. By varying
the corrugation height $h_c$, we evaluate the frequency dispersion
of a parallel-plate waveguide having straight corrugations by
using the analytical model for in [20], considering the wave
propagation perpendicular to the corrugations. The analytical
model in [20] is obtained with the assumption that the period
$p_c$ and the width $w_c$ of the corrugations are much smaller than
the operating wavelength [21]. Then, the period $p_c = 2$ mm,
and width $w_c = 1$ mm are chosen. Fig. 2a shows the disper-
sion diagrams obtained varying $h_c$. Increasing $h_c$ makes the
structure more dispersive, enabling faster frequency scanning.
However, large $h_c$ values bring to operating near the bandgap,
increasing metallic losses and worsening input matching,
which reduces antenna gain. Consequently, $h_c = 1.6$ mm is
chosen as a tradeoff between dispersion and efficiency. Two
different corrugated bottom plates are designed: concentric
corrugations in Fig. 1a and straight corrugations in Fig. 1b,
both being symmetric with respect to $\phi = 0$. To analyze the
dispersion using Ansys® HFSS software and the adjacent-
zeros method (AZM) as discussed in [9], a travelling wave
behavior is enforced by terminating both structures with ab-
sorbing boundary conditions and with the end contour of the

<figure><img src="image_3.png" /><figcaption>Fig. 2. Normalized (to $p_c = 2$ mm) dispersion diagrams. (a) Corrugated PPW for different corrugation height $h_c$ using the analytical model [20]. $h_c = 2$ mm (violet), $h_c = 1.8$ mm (yellow), $h_c = 1.6$ mm (orange), $h_c = 1.4$ mm (green), $h_c = 1.2$ mm (light blue) curves. (b) SWCC and SWSC at $\phi = 0$, $3\phi_0/8$ with the AZM and, analytical model [20] for $h_c = 1.6$ mm. Light lines $k_0$ (black lines).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. (a) Normalized (to $p_s = 5.4$ mm) dispersion diagrams for the SWCC and SWSC, at different $\phi$-directions, obtained by (1) and the results in Fig. 2b. Fundamental harmonic $\beta$ of the SWCC along any $\phi$ directions and of the SWSC along $\phi = 0$ (light blue dashed line). Visible harmonic $\beta_{-1}$ of the SWCC along any $\phi$ direction and of the SWSC along $\phi = 0$ (yellow solid line). Harmonics along $\phi = 3\phi_0/8$ of the SWSC: fundamental harmonic (dark blue dashed line) and visible harmonic (orange solid line). Light lines $\pm k_0$ (black lines). (b) Predicted elevation angles of the antennas, obtained from $\beta_{-1}$ (1).</figcaption></figure>

same shape as the corrugations (i.e., circular end for Fig. 1a,
straight end in Fig. 1b). Fig. 2b shows dispersion diagrams
for $h_c = 1.6$ mm, $p_c = 2$ mm, and $w_c = 1$ mm. Results
are computed with the analytical model and using the AZM
for both structures and for two $\phi$-directions, i.e., $\phi = 0$ and
$\phi = 3\phi_0/8$ (in the middle and close to the lateral plate of
the waveguides, respectively). Analytical model is in good
agreement with $\phi = 0$ numerical results. For the sectorial
waveguide with concentric corrugations (SWCC), we have a
superposition of the curves along the two angles (light blue
and green dashed curves). In the sectorial waveguide with
straight corrugations (SWSC), we can observe instead different
dispersions according to the angle of propagation (violet and
dark blue dashed curves). This confirms the evident dispersion
isotropy of SWCC and the DA of the SWSC.

### III. SECTORIAL LEAKY-WAVE ANTENNAS

#### A. Dispersion design

As shown in Fig. 2b, in both corrugated waveguides the guided
modes are in the slow-wave region for any $\phi$ values, i.e., $\beta >
k_0$. In order to radiate, the approach chosen here is to etch
on the top plate a series of concentric slots having the same
 radials) width $w_s$ and the same radial distance $p_s$ between
 them. This creates a radial periodicity and produces higher-
 order harmonics whose phase constant is $\beta_n$, possibly radiating
 (if in the fast region $-k_0 \le \beta_n \le k_0$) at an elevation angle
 $\theta_n$ [4]:
$$ \beta_n = \beta + 2n\pi/p_s, \quad \theta_n = \sin^{-1}(\beta_n/k_0) \quad (1) $$

with $n = 0, \pm 1, \pm 2, \dots$, where $\beta$ is the radial phase constant of the fundamental harmonic, assuming the same as the phase constant of the guided modes in the closed structure. This is a common hypothesis and is confirmed by the following results of radiation properties.

Varying $p_s$, fast (visible) harmonics can be generated in the forward and backward ranges, i.e. $0 < \beta_n < k_0$ and $-k_0 < \beta_n < 0$, respectively. The value of $p_s$ is chosen in order to obtain one visible harmonic in each structure with a pronounced frequency scanning. For a given frequency dependency of $\beta$ and due to the nonlinear shape of the $\sin^{-1}$ function in (1), the frequency scanning is faster far from broadside than around broadside, where the behavior is approximately linear. For this reason, in order to stay far enough away from the broadside, the value $p_s = 5.4$ mm is chosen in both structures.

In Fig. 3a, the dispersion diagram of the fundamental harmonic (dashed lines) and the backward $-1$ visible harmonic (solid lines) obtained using (1) are shown. In particular, while the harmonics in the SWCC do not depend on $\phi$, in the SWSC vary according to the $\phi$ propagation direction (results for $\phi = 0$ and $\phi = 3\phi_0/8$ are shown). Since the propagation along $\phi = 0$ is the same for SWCC and SWSC, we show only one curve in this case.

Following our hypothesis that the DA translates into an anisotropic behavior of the frequency dispersion of the LWA beam direction ($\theta_n$), and further assuming that the $\theta_n$ for a given $\phi$ mainly depends on the phase constant experienced by the guided wave along that same $\phi$, the $\theta_n$ is estimated using (1) and the backward $-1$ visible harmonics of Fig. 3a. Although the radiated pattern along each direction depends on the entire aperture field, it is shown later that this assumption is relevant when predicting the beam direction and its frequency variation. Fig. 3b shows that the SWSC exhibits not only different elevation beam directions, but also different frequency variation depending on $\phi$ ($\Delta\theta = 24.5^\circ$ and $\Delta\theta = 26.5^\circ$, for $\phi = 0$ and $\phi = 3\phi_0/8$, respectively).

## B. Antenna design

The radiating properties of the sectorial LWAs with concentric corrugations (LWACC) and straight corrugations (LWASC) are discussed in this section. 38 slots are etched on the top plates, all with a radial width of $w_s = \lambda_0/5$, where $\lambda_0$ is the free-space wavelength at central frequency of 28 GHz. The geometries of the periodic LWAs are illustrated in Fig. 4.

A radial length $L = 210$ mm was chosen to radiate most of the power before the end of the antennas [4]. This was verified after checking that the structures were well matched in the considered bandwidth, as confirmed by the reflection coefficient $|S_{11}|$ of the LWAs in Fig. 4c, lower than $-15$ dB without the need for initial corrugation or slot tapering. The power reaching the end of the antennas was evaluated by defining a second waveguide port at the end of the LWASC and computing the transmitted power $|S_{21}|$ [22]–[24]. Due to the width of the second waveguide port, five modes were defined

<figure><img src="image_5.png" /><figcaption>Fig. 4. Corrugated LWAs with feeding in red. Sectorial LWAs with concentric slots in the top plate and: (a) concentric corrugations, (b) straight corrugations in bottom plate. The geometric parameters are $\phi_0 = 73^\circ$, $L = 210$ mm, $\rho_0 = 4.8$ mm, $p_s = 5.4$ mm, $w_s = 2.14$ mm. (c) $|S_{11}|$ of the LWAs.</figcaption></figure>

on it, and five transmission coefficients were computed. The coefficient related to the first mode is lower than $-10$ dB in the considered frequency band, while the higher order modes were more strongly attenuated. This confirms that the chosen $L$ is sufficient to radiate most of the power through the slots. The results are not shown here for brevity.

The 2D realized gain for the two LWAs is shown in Fig. 5 for the lower and upper frequencies of the considered bandwidth. To compare the gain with the beam directions predicted in Fig. 3b, the same $\phi$ directions, i.e., $\phi = 0$ and $\phi = 3\phi_0/8 \approx 27.5^\circ$ are selected in Fig. 5. The corresponding $\theta$ of maxima of the gain are in good agreement with the $\theta_{-1}$ predicted in Fig. 3b for structures with concentric and with straight corrugations. The $\theta$ angular scanning within $[26.5, 29.5]$ GHz is $\Delta\theta = 27.5^\circ$ for the LWACC (for any $\phi$ value) and for the LWASC for $\phi = 0^\circ$, and $\Delta\theta = 30.5^\circ$ for the LWASC for $\phi = \pm 27.5^\circ$ (due to the symmetry of the structures at $\phi = 0$). This azimuthal dependence of the $\theta$ frequency-scanning rate is leveraged in the next section for simultaneous $\theta/\phi$ DoA estimation.

# IV. DIRECTION-OF-ARRIVAL ESTIMATION

## A. MUSIC algorithm applied to LWA

The DoA estimation is performed here using the MUSIC algorithm that processes the covariance matrix of signals received by the LWA. The system model assumes $D$ sources using a multicarrier modulation scheme, impinging on the LWA as $D$ ($d = 1, \dots, D$) plane waves. Furthermore, considering only the subcarriers that are modulated by the same symbol [10], the received signal is expressed in the frequency domain as:

$$ \mathbf{x}[k] = \mathbf{A}\mathbf{s}[k] + \mathbf{z}[k], \quad (2) $$

where $\mathbf{x} \in \mathbb{C}^{M \times 1}$ is the received vector, $M$ is the number of frequency samples, $k = 1, \dots, K$ is the $k$th snapshot among $K$, $\mathbf{A} \in \mathbb{C}^{M \times D}$ is the LWA response matrix, $s \in \mathbb{C}^{D \times 1}$ is the source vector, representing the complex amplitudes of the $D$ sources and $\mathbf{z} \in \mathbb{C}^{M \times 1}$ is a complex additive white Gaussian
<figure><img src="image_6.png" /><figcaption>Fig. 5. Realized gain in the H-plane for two frequencies. (a) and (b) gain of the LWACC, (c) and (d) gain of the LWASC.</figcaption></figure>

noise (AWGN) vector with uncorrelated components. Each column of A, i.e., a(θd, φd), is the LWA frequency response for an incident plane wave with (θd, φd) DoA and is obtained with HFSS simulations. It is important to note that M is the number of frequency samples, i.e. subcarriers, as opposed to the number of antenna elements in a traditional linear-array-based MUSIC scenario. To estimate the DoA, the covariance matrix is first estimated from the received signal as:

$$ \hat{\mathbf{R}} = \frac{1}{K} \sum_{k=1}^{K} \mathbf{x}[k] \mathbf{x}^{\mathrm{H}}[k], \quad (3) $$

where (·)H denotes the Hermitian transpose. The null space of the covariance matrix is computed, and the MUSIC pseudospectrum is defined as:

$$ P(\theta, \phi) = \frac{1}{\mathbf{a}^H(\theta_d, \phi_d)\mathbf{E}_N \mathbf{E}_N^H \mathbf{a}(\theta_d, \phi_d)}, \quad (4) $$

where a(θd, φd) is the steering vector for an incoming wave at angles (θd, φd), E_N is the noise subspace of the covariance matrix R. The peak in the pseudo-spectrum corresponds to the estimated DoAs of the incoming signals.

## B. Results

The autocorrelation, defined as $\rho = |\mathbf{a}^H(\theta_d, \phi_d)\mathbf{a}(\theta_d + d\theta, \phi_d + d\phi)|/||\mathbf{a}(\theta_d, \phi_d)||\mathbf{a}(\theta_d + d\theta, \phi_d + d\phi)|$, is shown in Fig. 6a for both LWASC and LWACC with one d = 1 source at $\theta_1 = -40^\circ$, $\phi_1 = -20^\circ$. The autocorrelation represents the gain frequency-variation similarity experienced between sources impinging at two different angles. It can be observed that both antennas exhibit a single $\rho = 1$ peak at the DoA location along $\theta$. In the $\phi$ plane, the autocorrelation behavior is almost flat around $\phi_1$ with the LWACC. This is due to the isotropic behavior of the LWACC pattern along $\phi$ and thus suggests weak $\hat{\phi}$ estimation performance. The LWASC, thanks to its anisotropic behavior, does exhibit a sharper peak at $\phi_1$, which suggests improved $\hat{\phi}$ estimation performance. Both antennas exhibit a symmetric behavior in the $\phi$ plane, which leads to $\pm\hat{\phi}_1$ ambiguity. To assess the DoA estimation

<figure><img src="image_7.png" /><figcaption>Fig. 6. (a) Autocorrelation $\rho$ for one source at $\theta_1 = -40^\circ$, $\phi_1 = -20^\circ$, (b) pseudo-spectrum with 2 sources $\theta_{1,2} = -40^\circ$, $\phi_1 = -20^\circ$, $\phi_2 = -10^\circ$ with K = 25, M = 49, with an SNR (c) for LWACC and (d) for LWASC.</figcaption></figure>

capabilities of the structures, the MUSIC pseudospectrum is shown in Fig. 6b, with the simulation parameters given in the caption. The scenario considers d = 2 sources with identical $\theta_{1,2} = -40^\circ$ and different $\phi_1 = -20^\circ$ and $\phi_2 = -10^\circ$. For the LWA-based MUSIC, each source experiences an SNR = $||\mathbf{a}(\theta_d, \phi_d)||^2 E [|s_d[k]|^2/||\mathbf{z}[k]||^2]$, which depends on the LWA gain, therefore on $(\theta_d, \phi_d)$. Fig. 6c and 6d show the angular distribution of the SNR used in numerical simulations with the LWACC and the LWASC, respectively. The SNR experienced by the $(\theta_1, \phi_1)$ source is 13.1 dB with the LWACC and 12.1 dB with the LWASC while the SNR for the $(\theta_2, \phi_2)$ source is 16.1 dB with the LWACC and 12.4 dB with the LWASC. As anticipated by the $\rho$ behavior, the pseudospectrum cut along $\theta$ exhibits a single sharp peak, leading to accurate estimation of $\theta_1$ with both antennas. Along $\phi$, the LWACC does not show a clear peak at $\phi_1$, and therefore fails to detect one of the two DoAs, owing to the high similarity of the response vectors in this angular range, as already expected from the flatness of $\rho$. The LWASC overcomes this issue thanks to its frequency-dependent radiation pattern along $\phi$.

## V. CONCLUSIONS

A novel fully metallic periodic LWA for passive 2D DoA estimation has been presented. The design employs a sectorial waveguide with corrugated metasurface on the bottom plate and periodic slots on the top plate. Two different corrugated metasurfaces were investigated, i.e. concentric and straight corrugations, resulting in an azimuth-independent ($\phi$-isotropic) and azimuth-dependent ($\phi$-anisotropic) directional pattern of the LWAs, respectively. The performance of the LWAs for 2D DoA estimation are evaluated using the MUSIC algorithm. An anisotropic design of the corrugations enables a correct DoA estimation of both elevation and azimuth angles with a single-port antenna. Future works will focus on breaking the symmetry of the pattern to avoid $\pm\hat{\phi}_d$ ambiguity and on experimental validations.
# VI. ACKNOWLEDGEMENTS

This work was supported by the ANR BeSensiCom project, grant ANR22-CE25-0002 of the French Agence Nationale de la Recherche, Italian PRIN Grant 2017NT5W7Z GREEN TAGS, Next Generation EU within PNRR M4C2, Inv. 1.4-Avv. n.3138 6/12/2021-CN00000013 National Centre for HPC with resources provided by HPC@POLITO (http://hpc.polito.it)

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