# Meandering Microstrip Leaky-Wave Antenna with Dual-band Linear–Circular
Polarization and Suppressed Open Stopband

Pratik Vadher, Giulia Sacco, and Denys Nikolayev Manuscript received April 25,
2023.This study was supported in part by the French Agence Nationale de la
Recherche (ANR) through the Project MedWave under Grant ANR-21-CE19-0045; in
part by the European Union’s Horizon 2020 research and innovation program
under the Marie Skłodowska-Curie N∘899546 grant through the REACH-IT project;
and in part by the European Union’s Horizon Europe research and innovation
program through the Marie Skłodowska-Curie IN-SIGHT project N∘101063966\.
(Corresponding author: Denys Nikolayev, denys.nikolayev@deniq.com)P. Vadher,
G. Sacco, and D. Nikolayev are with the Univ Rennes, CNRS, IETR – UMR 6164,
FR-35000 Rennes, France.

###### Abstract

This paper proposes a dual-band frequency scanning meandering microstrip
leaky-wave antenna with linear polarization in the Ku-band and circular
polarization in the K-band. This is achieved by making use of two spatial
harmonics for radiation. The unit cell of the periodic microstrip antenna
contains three meanders with mitred corners. To ensure circular polarization,
a theoretical formulation is developed taking into account the delay caused by
microstrip length intervals. It defines the unit cell geometry by determining
the length of the meanders to ensure that axial ratio remains below 3 dB
throughout the operational band. Moreover, the meanders are used to provide
better control over scanning rate (the ratio of change of angle of maximum
radiation with frequency) and reduce spurious radiation of harmonics by
ensuring single harmonic operation within the operational band. To guarantee
continuous scanning through broadside direction, open stopband is suppressed
using mitered angles. The antenna is designed on a 0.254-mm substrate making
it suitable for conformal applications. The fabricated antenna shows a
backward to forward beam steering range of 72$\degree$ (–42$\degree$ to
30$\degree$) in the K-band (19.4–27.5 GHz) with circular polarization and of
75$\degree$ (–15$\degree$ to 60$\degree$) in the Ku-band (11–15.5 GHz) with
linear polarization.

###### Index Terms:

leaky wave antenna (LWA), Ku-band, K-band, higher spatial order, scanning
rate, meandering microstrip antenna

##  I Introduction

Periodic leaky wave antennas (LWAs) are a class of travelling-wave antennas
that radiate energy at the discontinuities of the guiding medium [1]. Changes
in frequency causes dispersion within the guiding medium, resulting in varying
excitation phases at the discontinuities. This, in turn, alters the main beam
pointing direction in the radiation pattern of the antenna with respect to
frequency. One-dimensional periodic LWAs (1D LWAs) typically radiate a fan-
beam in the E-plane. The direction of maximum radiation changes in the H-plane
with the change in frequency [2].

Many 1D periodic LWAs have been proposed using different guiding media. LWAs
proposed in [3, 4, 5, 6, 7] use substrate integrated waveguide (SIW) or half-
mode SIW to support the travelling wave and usually employ a TEn0 or in
certain cases TM11 mode [8] for radiation. However, to reduce fabrication
complexity, lower manufacturing costs, and to enable the creation of compact
antennas, microstrip-based guiding media LWAs are an appealing solution [9,
10, 11, 12, 13]. Meandering microstrip LWAs radiate due to the discontinuities
created at the edges which results in a net magnetic current responsible for
radiation [14, 13].

Since LWAs are periodic structures, infinite spatial harmonics exist in the
guiding media [15, 2]. Many SIW based LWAs that use higher order Floquet modes
for radiation have been proposed recently [16, 17]. Although the previous
works report scanning due to higher spatial harmonics, the scanning is due to
two or three spatial harmonics at the same time, which leads to more than one
beam radiated by the antenna. However, it is desirable to have a single beam
scanning operation over a high scanning range. This is possible when only a
single spatial harmonic is responsible for radiation. Hence, better separation
of different spatial harmonics is necessary.

To be suitable for on-body conformal applications [18, 19, 20, 21], the
antenna must be flexible and bendable [22]. For such a purpose, antennas based
on microstrip technology are an excellent candidate. Additionally, the
mechanical properties such as tensile modulus of the dielectric substrate also
play an important factor. Hence, the Rogers 3003 substrate
($\varepsilon_{r}~{}=~{}$3.0) is preferred in this work due to its low tensile
modulus ($823\text{\,}\mathrm{MPa}$) making it flexible.

Moreover, it is desirable to have circular polarization for many different
applications where the alignment of the receiving and transmitting antenna may
impact the overall performance of the system such as radars [23], satellite
communications and on-body antenna system [21, 24, 25].

Presence of open stopbands (OSBs) also impacts the scanning of the antenna
through the broadside direction [26, 27]. Several techniques have been
proposed to suppress or completely mitigate OSBs [28, 29, 30]. At the OSB
frequency, the input impedance matching is poor and the Bloch impedance
($Z_{\mathrm{s}}$) has a high imaginary value. Hence, the aim is to minimize
the high imaginary value of impedance to reduce the effects of OSB. The idea
of beam scanning due to higher spatial harmonic, limited to a single-band and
linear polarization, has been presented in [31] with no suppression of OSB.

In this paper, single-layer PCB meandering microstrip based LWA is proposed
with dual-band operation without the use of vias shown in Fig 1(a). The Ku-
band operation of the antenna is due to the $n=-1$ spatial harmonic and it
exhibits linear polarization, while the second band of operation at K-band is
due to $n=-2$ and it depicts circular polarization as described in Fig. 1(b).
The unit cell of the periodic antenna consists of three meanders to ensure
single-beam operation by providing better separation between the different
spatial harmonics.

The structure of the paper is as follows. Section II details the concept of
higher spatial order in a microstrip-based unit cell followed by theoretical
formulation for the dimensions required to have circular polarization for a
unit cell with single meander. In Section III, the unit cell is modified by
adding two smaller meanders to increase the separation of spatial harmonics
and ameliorate performance of circular polarization over the larger frequency
operation range. The smaller meanders also result in better control over
scanning rate (i.e. the ratio between the change in the angle of maximum
radiation and the change in frequency ${\Delta\theta}/{\Delta f}$). A
technique to remove OSB is discussed using Bloch impedance subsequently.
Section IV contains the comparison of simulations and measurements for the
final design. In Section V, the conclusions of the study are presented.

Figure 1: (a) Configuration of the proposed 10 unit cells periodic frequency
scanning antenna with radiation pattern shown in the broadside direction. The
meandering microstrip is etched on top layer while bottom layer is copper. (b)
Beam scanning operation of the proposed LWA as a function of frequency in the
Ku-band and Ka-band. The antenna radiates a fan-beam in the E-plane (X-Z
plane) while in the H-plane (Y-Z plane) the antenna changes the direction of
maximum radiation with change in frequency.

Figure 2: Evolution towards final design of proposed unit cell: (a)
conventional unit cell operating in single band and radiating due to the
$n=-1$ spatial harmonic, (b) dual band operating unit cell radiating due to
$n=-1$ and $n=-2$ spatial harmonics, and (c) modified unit cell with the
introduction of two meanders ($p<p_{0}$) to improve the scanning range and the
circular polarization performance.

##  II Radiation Mechanism and Principle for Antenna Design

Fig. 1(a) shows the design of the proposed microstrip-based LWA with mitred
corners. As detailed in previous works on microstrip based LWAs [13], the
radiation occurs due to the magnetic current at the corners of the meandering
microstrip.

The evolution towards the final design of the unit cell is shown in Fig. 2.
Conventional microstrip-based LWA [9, 32, 13, 33, 34] operate in the radiation
zone due to spatial harmonic of $n=-1$ like the unit cell shown in Fig. 2(a).
This unit cell can be modified to operate in the $n=-2$ spatial harmonic, by
increasing the pathlength at the desired operational frequency. By introducing
mitred corners and properly choosing the length of the interconnecting
microstrip lines between the corners the antennas radiates in circular
polarization (see Fig. 2(b)). The unit cell operates in the radiation zone
associated with $n=-1$ and $n=-2$ spatial harmonics.

To improve the separation of radiation zone due to two harmonics ($n=-1,-2$)
we propose the geometry in Fig. 2(c). The unit cell is modified with two
additional meanders resulting in improvement in scanning range and circular
polarization. The antenna has been optimized for operation in the circularly
polarized K-band using the spatial harmonic of $n=-2$, and all theoretical
formulations for the unit cell dimensions have been performed to ensure
optimal performance in this frequency range.

###  II-A Higher order spatial harmonics in the unit cell based on microstrip
design

Fig. 3(c) shows the microstrip based unit cell with single meander with four
radiating elements due to four mitred corners. Due to the periodicity
($p_{0}$) of the structure, infinite number of space harmonics exist due to
Bloch-Floquet theorem [35, 36]. The phase constant of the $n^{\mathrm{th}}$
space harmonic $\beta_{n}$ satisfies

$\beta_{n}p_{0}=\beta_{0}p_{0}+2n\pi$ (1)

where $n$ ranges from $-\infty$ to $+\infty$. Here $\beta_{0}$ is the zeroth
order propagation constant in the microstrip medium.

Figure 3: $E_{\mathrm{abs}}$ plots at phase $0^{\degree}$ for the frequency at
which broadside radiation occurs for (a) $n=-1$ and (b) $n=-2$. The sinusoidal
graphs above $E_{\mathrm{abs}}$ plots are representative of the electric field
variation across microstrip line throughout the unit cell. (c) Unit cell with
single large meander. Radiation sources for a single large meander are
depicted in the figure. The four radiating magnetic current sources are shown
at points $\mathrm{A}$, $\mathrm{A_{1}}$, $\mathrm{B}$ and $\mathrm{B_{1}}$.
(d) Brillouin diagram for the unit cell with the single meander. (e) Unit cell
design with two additional meanders that results in 12 radiation sources. (f)
Brillouin diagram for the improved unit cell with two additional meanders.

Fig. 3(d) shows the corresponding Brillouin diagram for the unit cell. The
$n^{\mathrm{th}}$ spatial harmonic inside the light cone
($|\beta_{n}|<|k_{0}|$), results in radiation [15]. It is to be noted that
multiple spatial harmonics can exist within the light cone resulting in
multiple beam radiation [16].

From equation (1), the phase difference across the unit cell for the first two
higher harmonics $n=-1$ and $n=~{}-2$ (near the light cone) can be described
in the following manner with respect to fundamental harmonic:

$\displaystyle\beta_{-1}p_{0}$ $\displaystyle=-\pi\text{ to
}\pi\rightarrow\beta_{0}p_{0}=\pi\text{ to }3\pi$ (2a)
$\displaystyle\beta_{-2}p_{0}$ $\displaystyle=-\pi\text{ to
}\pi\rightarrow\beta_{0}p_{0}=3\pi\text{ to }5\pi$ (2b)

Likewise, the direction of maximum radiation $\theta_{n}$ corresponding to
$n^{\mathrm{th}}$ spatial harmonic is given by [36, 37, 16]

$\theta_{n}=\sin^{-1}(\beta_{-n}p_{0}/k_{0}p_{0})$ (3)

where $k_{0}$ is the free space wave number.

According to equation (3), when the phase difference across the unit cell
($\beta_{-n}p_{0}$) is zero, the direction of main beam of the
$n^{\mathrm{th}}$ spatial harmonic is in the broadside direction. Hence, at
the frequencies of broadside radiation for spatial harmonics $n=-1$ and
$n=-2$, the phase difference across the unit cell from equation (1) is:

$\displaystyle\beta_{-1}p_{0|f=f_{\mathrm{a}}}$
$\displaystyle=0\rightarrow\beta_{0}p_{0|f=f_{\mathrm{a}}}=2\pi$ (4a)
$\displaystyle\beta_{-2}p_{0|f=f_{\mathrm{b}}}$
$\displaystyle=0\rightarrow\beta_{0}p_{0|f=f_{\mathrm{b}}}=4\pi\,,$ (4b)

where $f_{\mathrm{a}}$ and $f_{\mathrm{b}}$ are the frequencies corresponding
to the broadside radiation for $n=-1$ and $n=-2$, respectively. Fig. 3(a–b)
depicts the variation of electric field magnitude at the two frequencies
confirming the above two equations. To summarize, the first Ku-band radiation
is due to spatial harmonic $n=-1$, while the second K-band radiation is due to
spatial harmonic $n=-2$.

Eq. (4b) implies that at $f=f_{\mathrm{b}}$, the total phase difference
($\phi_{p\mathrm{{}_{0}}}$) across the unit cell is equal to $4\pi$.
Therefore, the equation to have broadside radiation for second harmonic at the
design frequency $f=f_{\mathrm{b}}$ is,

$\begin{split}4\pi=\phi_{\mathrm{OA}_{1}|f=f_{\mathrm{b}}}+&\phi_{\mathrm{AA}_{1}|f=f_{\mathrm{b}}}+\phi_{\mathrm{A_{1}B}|f=f_{\mathrm{b}}}+\\\
&\phi_{\mathrm{BB_{1}}|f=f_{\mathrm{b}}}+\phi_{\mathrm{B_{1}P}|f=f_{\mathrm{b}}}\end{split}$
(5)

Here, $\phi_{ij}$ is the phase difference across the line segment between the
points $i$ and $j$, where $i,j\in\\{\mathrm{A,A_{1},B,B_{1},O,M}\\}$. As
indicated earlier, the reason for designing the antenna for the second
harmonic is to obtain dual band operation due to spatial harmonics $n=-1$ and
$n=-2$.

###  II-B Analysis for circular polarization in the radiation zone for spatial
harmonic $n=-2$

Figure 4: Normalized RHCP and LHCP gain in the H-plane obtained from
simulation for the LWA formed by connecting 10 unit cells with singular
meander. The main lobe is due to spatial harmonic of $n=-2$. The realized gain
of the main lobe varies from $8\text{\,}\mathrm{dB}$ to
$14\text{\,}\mathrm{dB}$ across the frequency range.

To analyse the circular polarization easily, the axis is rotated equal to the
angle of mitred corners as shown in Fig.3(c). Hence there are two transverse
components ($E_{\mathrm{X^{\prime}}}$ and $E_{\mathrm{Y^{\prime}}}$) radiating
from the mitred corners.

The ratio of the two transverse fields, $E_{\mathrm{X^{\prime}}}$ and
$E_{\mathrm{Y^{\prime}}}$ [14]:

$\frac{E_{\mathrm{Y}^{\prime}}}{E_{\mathrm{X}^{\prime}}}=\mathrm{e}^{\phi_{\mathrm{AA_{1}}}+\phi_{\mathrm{A_{1}B}}}=\mathrm{e}^{-j\beta_{0}(l_{\mathrm{AA_{1}}}+l_{\mathrm{A_{1}B}})}$
(6)

Additionally, $l_{ij}$ represents the length of the line segment between the
points $i$ and $j$, which were indicated earlier. To have accurate length of
line intervals, the model from [9] is chosen for analysis.

To have the circular polarization, the two transverse components of electric
field ($E_{\mathrm{X^{\prime}}}$ and $E_{\mathrm{Y^{\prime}}}$) need to have a
phase difference of odd multiple of $\pi/2$ [2]. Consequently, the condition
to obtain circular polarization for a travelling-wave meandering microstrip
antenna is stated by [14].

$\phi_{\mathrm{AA_{1}}}+\phi_{\mathrm{A_{1}B}}=\beta_{0}(l_{\mathrm{AA_{1}}}+l_{\mathrm{A_{1}B}})=(2k+1)\pi/2$
(7)

where $k=1,2,..$

Since the desire is to operate in spatial harmonic of $n=-2$, from equation
(2b) and (4b), the conclusion can be made to select $k=1$, hence

$\phi_{\mathrm{AA_{1}}}+\phi_{\mathrm{A_{1}B}}=3\pi/2$ (8)

Selecting $k\geq 2$ would lead to similar result with equation (6) still being
satisfied, however this would increase the overall period of the unit cell
resulting in non desirable harmonics in the radiation zone.

Equation (8) imposes a criterion on the the phase difference and hence the
length of microstrip interval between two corners $\mathrm{A}$ and
$\mathrm{B}$ for achieving circular polarization. For initial dimensions, at
the design frequency $f=f_{\mathrm{b}}$, $l_{\mathrm{AA_{1}}}$ is taken such
that $\phi_{\mathrm{AA_{1}}}=\pi$. This implies that
$\phi_{\mathrm{A_{1}B}}=\pi/2$.

From Fig. 3(d), $l_{\mathrm{AA_{1}}}=l_{\mathrm{BB_{1}}}$ which implies
$\phi_{\mathrm{AA_{1}}}~{}=\phi_{\mathrm{BB_{1}}}$. Also, the unit cell is
considered symmetric, hence $l_{\mathrm{OA}}=l_{\mathrm{B_{1}M}}$.

The phase difference across the large meander (consisting of
$\mathrm{AA_{1}}$, $\mathrm{A_{1}B}$ and $\mathrm{BB_{1}}$) is defined as
$\phi_{\mathrm{LargeSeg}}$. Therefore, from equation (8), in order to have the
best circular polarization performance at the desired design frequency of
$f=f_{\mathrm{b}}$,

$\begin{split}\phi_{\mathrm{LargeSeg}|f=f_{\mathrm{b}}}&=\phi_{\mathrm{AA_{1}}|f=f_{\mathrm{b}}}+\phi_{\mathrm{A_{1}B}|f=f_{\mathrm{b}}}+\phi_{\mathrm{BB_{1}}|f=f_{\mathrm{b}}}\\\
&=5\pi/2\end{split}$ (9)

Consequently, there are two design constraints on the length of microstrip
line intervals: 1) Equation (5), to construct a unit cell that has broadside
radiation at $f=f_{\mathrm{b}}$. and 2) Equation (9), to construct a unit cell
with best circular polarization performance at $f=f_{\mathrm{b}}$. Table I
depicts the dimensions of the unit cell based on the theory discussed at
$f_{\mathrm{b}}=$ 23 GHz. The dimensions are mentioned in the form of
$\lambda_{\mathrm{b}}$, corresponding wavelength to frequency $f_{\mathrm{b}}$
in the microstrip medium.

The dielectric is chosen as Rogers 3003 ($\varepsilon_{\mathrm{r}}=3.0$) while
the height of the substrate is $\mathrm{h_{sub}}=$
$0.254\text{\,}\mathrm{mm}$. The width of the microstrip $\mathrm{t_{50}}$ is
equal to $0.5\text{\,}\mathrm{mm}$. It is important to note that in the
microstrip medium the effective $\varepsilon_{\mathrm{r,eff}}(f=0)$ is $2.375$
calculated from [38]. To account for the dispersive nature of microstrip
media, the model proposed by Pramanick and Bhartia from [39] is considered to
calculate $\varepsilon_{\mathrm{r,eff}}(f=f_{\mathrm{b}})$. This parameters
are taken into consideration when calculating $\lambda_{\mathrm{b}}$ in the
microstrip medium.

TABLE I: Dimensions of the unit cell designed to operate in spatial harmonic of $n=-2$ and have circularly polarized fields. Here note that $\lambda_{\mathrm{b}}$ is the corresponding wavelength in the microstrip medium at $f=f_{\mathrm{b}}$ Line interval |  Expressed in $\lambda_{\mathrm{b}}$ |  Phase diff. across line interval |  Physical dim. (mm)
---|---|---|---
$\mathrm{OA}$ |  $3\lambda_{\mathrm{b}}/8$ |  $3\pi/4$ |  3.21
$\mathrm{AA_{1}}$ |  $\lambda_{\mathrm{b}}/2$ |  $\pi$ |  4.44
$\mathrm{A_{1}B}$ |  $\lambda_{\mathrm{b}}/4$ |  $\pi/2$ |  2.47
$\mathrm{BB_{1}}$ |  $\lambda_{\mathrm{b}}/2$ |  $\pi$ |  4.44
$\mathrm{B_{1}M}$ |  $3\lambda_{\mathrm{b}}/8$ |  $3\pi/4$ |  3.21
$p_{0}$ |  $2\lambda_{\mathrm{b}}$ |  $4\pi$ |  5.8
$\mathrm{t}_{50}$ |  - |  - |  0.5
$\mathrm{h}_{\mathrm{sub}}$ |  - |  - |  0.254

Ten such unit cells with single meander are cascaded next to each other to
form a LWA and full-wave simulations are performed in CST and ANSYS HFSS. Fig.
4 shows the radiation pattern indicating the beam scanning with frequency and
circular polarization nature of the antenna. As can be seen, at lower end of
frequency range $f=$ $18.6\text{\,}\mathrm{GHz}$, there is another beam of
spatial harmonic of $n=-1$ limiting the scanning.

The corresponding axial ratio obtained through simulation at the angle of
maximum gain is shown in Fig. 5(a). The antenna has good circular polarization
performance near the design frequency of $f_{\mathrm{b}}=$
$23\text{\,}\mathrm{GHz}$. The frequency range where the axial ratio is lower
than $3\text{\,}\mathrm{dB}$ is from
$20\text{\,}\mathrm{GHz}25.2\text{\,}\mathrm{GHz}$ with the beam scanning from
($-26\degree$ to $+10\degree$).

Figure 5: Axial ratio values in the main beam direction versus frequency
obtained by full-wave simulation of 10 unit cells connected in series for 3
cases: (a) Single meander with $\phi_{\mathrm{AA_{1}}}=\pi$. (b) Single
meander with $\phi_{\mathrm{AA_{1}}}=9\pi/8$ to reduce the period of the unit
cell. (c) Two additional meanders on either side of large meander.

###  II-C Improvement in the scanning range by reducing the period of the unit
cell

In the previous subsection, initial dimensions were taken such that
$\phi_{\mathrm{AA_{1}}}=\pi$ and $\phi_{\mathrm{A_{1}B}}=\pi/2$ at
$f=f_{\mathrm{b}}$. The scanning range and the separation of the harmonics can
be increased by reducing the period of a unit cell. If the vertical length
$l_{\mathrm{AA_{1}}}$ is increased such that $\phi_{\mathrm{AA_{1}}}=9\pi/8$,
then according to (8), to maintain the circular polarization performance
$\phi_{\mathrm{A_{1}B}}=\pi/4$ at the design frequency $f_{\mathrm{b}}$. This
reduces the period of the unit cell since the horizontal interval
($l_{\mathrm{A_{1}B}}$) of the unit cell is shortened leading to better
separation of harmonics. The rest of the dimensions remain the same as
discussed in Table I.

Ten such unit cells are connected in series to form a LWA and full-wave
simulations are performed in CST and Ansys HFSS. However, as can be seen from
Fig. 5(b), there is a negative impact on the circular polarization performance
in the operating frequency band.

In both the discussed cases with only a single meander in the unit cell (see
Fig. 5(a, b)), at the design frequency $f=f_{\mathrm{b}}$ and in the frequency
range close to it, the antenna shows good circular polarization performance.
However, it deteriorates quite fast towards the lower and higher end of
frequencies. Hence, the beam scanning range is pretty limited.

It can be concluded that using a unit cell with only a single large meander,
is extremely challenging and not feasible to obtain large beam scanning range
with good circular polarization performance.

##  III Improvement in the scanning range and circular polarization by
additional meanders

To enhance the beam steering range, improve circular polarization performance
across the operational frequency band, and gain better control over scanning
rate ($\Delta\theta/\Delta f$), two additional smaller meanders are introduced
on either side of larger meander at an equal distance. The following sections
describe the characteristics and steps to design such a unit cell.

###  III-A Constructing the unit cell with three meanders for dual-band
operation

Fig.3(e) depicts the design of improved unit cell. The first additional
meander consists of three microstrip line intervals $\mathrm{CC_{1}}$,
$\mathrm{C_{1}D}$ and $\mathrm{DD_{1}}$, while the third additional meander
consists of line intervals $\mathrm{PP_{1}}$, $\mathrm{P_{1}Q}$ and
$\mathrm{QQ_{1}}$. The lengths of each of these 6 microstrip line intervals is
equal. The phase difference across these three microstrip line interval is
defined to be $\phi_{\mathrm{sect}}$. Additionally, phase difference due to
the first and the third meander is denoted by $\phi_{\mathrm{TrSeg1}}$ and
$\phi_{\mathrm{TrSeg3}}$.

The unit cell is designed to be symmetric, therefore the phase difference
across the line intervals is $\phi_{\mathrm{D_{1}A}}=\phi_{\mathrm{B_{1}P}}$,
$\phi_{\mathrm{OC}}=\phi_{\mathrm{Q_{1}M}}$, and
$\phi_{\mathrm{TrSeg1}}=\phi_{\mathrm{TrSeg3}}$. Furthermore, the phase
difference across the $1^{\mathrm{st}}$ meander and the $3^{\mathrm{rd}}$
meander can be written as

$\displaystyle\phi_{\mathrm{TrSeg1}}$
$\displaystyle=\phi_{\mathrm{CC_{1}}}+\phi_{\mathrm{C_{1}D}}+\phi_{\mathrm{DD_{1}}}=3\times\phi_{\mathrm{sect}}$
(10a) $\displaystyle\phi_{\mathrm{TrSeg3}}$
$\displaystyle=\phi_{\mathrm{PP_{1}}}+\phi_{\mathrm{P_{1}Q}}+\phi_{\mathrm{QQ_{1}}}=3\times\phi_{\mathrm{sect}}$
(10b)

In order to have broadside radiation at frequency of $f=f_{\mathrm{b}}$,
equation (4b) (for this unit cell with period $p$, the equation is
$\beta_{0}p=4\pi$) must be satisfied. Consequently, from the Fig. 3(e), the
design equation for the improved unit cell can be written as:

$\begin{split}4\pi=&\phi_{\mathrm{OC}|f=f_{\mathrm{b}}}+\phi_{\mathrm{TrSeg1}|f=f_{\mathrm{b}}}+\phi_{\mathrm{D_{1}A}|f=f_{\mathrm{b}}}+\\\
&\phi_{\mathrm{LargeSeg}|f=f_{b}}+\phi_{\mathrm{B_{1}P}|f=f_{\mathrm{b}}}+\\\
&\phi_{\mathrm{TrSeg3}|f=f_{\mathrm{b}}}+\phi_{\mathrm{Q_{1}M}|f=f_{\mathrm{b}}}\end{split}$
(11)

Here, $\phi_{ij}$ represents the phase difference between the points $i$ and
$j$ where
$i,j\in\\{\mathrm{C,C_{1},D,D_{1},A,A_{1},B,B_{1},P,P_{1},Q,Q_{1},O,M}\\}$.

To maintain circular polarization, it is necessary that phase difference
across larger meander $\phi_{\mathrm{LargeSeg}|f=f_{\mathrm{b}}}$ equals
$5\pi/2$, as stated in equation (9). Therefore, the following design
constraint can be derived from equation (11):

$6\times\phi_{\mathrm{sect}|f=f_{\mathrm{b}}}+2\times\phi_{\mathrm{D_{1}A}|f=f_{\mathrm{b}}}+2\times\phi_{\mathrm{OC}|f=f_{\mathrm{b}}}=3\pi/2$
(12)

###  III-B Circular polarization analysis for the improved unit cell

It was proved in Section II that a single meander unit with four radiating
elements can produce circular polarization at the desired design frequency.
Hence, for the improved unit cell with 12 radiating currents (see Fig. 3(e)),
circular polarization at $f=f_{\mathrm{b}}$ can be achieved if the effect due
to radiating currents on the first additional meander (at $\mathrm{C}$,
$\mathrm{C_{1}}$, $\mathrm{D}$ and $\mathrm{D_{1}}$) and the third additional
meander (at $\mathrm{P}$, $\mathrm{P_{1}}$, $\mathrm{Q}$ and $\mathrm{Q_{1}}$)
is nullified. This is possible if the direction of current at $\mathrm{C}$ is
opposite to the current at $\mathrm{P}$ resulting in opposing fields at these
corners.

The magnetic currents at the corners $\mathrm{C}$ and $\mathrm{P}$ are
oriented along the ${\mathrm{Y^{\prime}}}$ axis as shown in Fig. 1(b), but
depending on the phase shift introduced by the interconnecting microstrip
lines, they can have the same or opposite directions. Hence, the length of the
microstrip line intervals in between these two corners can be optimized in
such a way that the magnetic currents are in opposite direction at design
frequency $f=f_{\mathrm{b}}$.

The direction of magnetic current at $\mathrm{C}$ will be opposite to the
magnetic current at $\mathrm{P}$ if the following constraint is met at
$f=f_{\mathrm{b}}$ (since $\mathrm{e}^{3\pi}=-1$):

$\displaystyle\phi_{\mathrm{OP}}-\phi_{\mathrm{OC}}$ $\displaystyle=3\pi$
(13a)
$\displaystyle\phi_{\mathrm{TrSeg1}}+\phi_{\mathrm{D_{1}A}}+\phi_{\mathrm{LargeSeg}}+\phi_{\mathrm{B_{1}P}}$
$\displaystyle=3\pi$ (13b)

From equation (9), it is known that to maintain circular polarization at
$f=f_{\mathrm{b}}$, $\phi_{\mathrm{LargeSeg}}=5\pi/2$, hence the following
design equation is obtained

$3\times\phi_{\mathrm{sect}|f=f_{\mathrm{b}}}+2\times\phi_{\mathrm{D_{1}A}|f=f_{\mathrm{b}}}=\pi/2$
(14)

The equations (9), (12) and (14) are the design constraints on the length of
line intervals for the proposed unit cell to maintain broadside radiation and
circular polarization at $f~{}=~{}f_{\mathrm{b}}$.

From equation (9), it is clear that the dimensions for larger meander are
fixed to maintain circular polarization. The other dimensions are governed by
equations (12) and (14). This results in 3 variables (namely
$\phi_{\mathrm{sect}|f=f_{\mathrm{b}}}$,
$\phi_{\mathrm{D_{1}A}|f=f_{\mathrm{b}}}$ and
$\phi_{\mathrm{OC}|f=f_{\mathrm{b}}}$) and 2 equations. In the current
analysis $\phi_{\mathrm{sect}|f=f_{\mathrm{b}}}$ is considered an independent
variable making the other two variables
($\phi_{\mathrm{D_{1}A}|f=f_{\mathrm{b}}}$ and
$\phi_{\mathrm{OC}|f=f_{\mathrm{b}}}$) dependent. Initial design is chosen
such that $\phi_{\mathrm{sect}|f=f_{\mathrm{b}}}=10^{\degree}$. The other two
dimensions can hence be calculated from equation (12) and equation (14) .The
initial lengths of the microstrip sections following this theory are shown in
the table II for $f_{\mathrm{b}}=$ $23\text{\,}\mathrm{GHz}$.

Fig. 6 shows the full-wave simulation of the unit cell at $f=f_{\mathrm{b}}$
designed according to the equations above. As can be seen the direction of
fields at each of the corners in the $1^{\mathrm{st}}$ smaller meander is
opposite to the field at the corresponding corner of the $3^{\mathrm{rd}}$
smaller meander.

10 such unit cells are sequentially connected as shown in Fig. 1(b) to form a
LWA and then simulated in CST. Fig. 5(c) shows the unit cell (in the inset)
designed by the formulated theory described above and simulation respectively,
of axial ratio obtained for different frequencies over the scanning range for
the main fan beam direction emanated from the LWA. Notable improvement in
axial ratio performance over a larger frequency band can be observed following
the inclusion of two smaller meanders.

###  III-C Performance improvement due to the two additional meanders

Due to the two additional smaller meanders the electrical length of the unit
cell has been increased. This results in following improvements for frequency
scanning:

  * •

enhanced scanning range,

  * •

better control over scanning rate by dictating the size of path–length in the
unit cell by reducing or increasing the size of small meanders
($\phi_{\mathrm{sect}}$),

  * •

better separation of spatial harmonics resulting in lower sidelobes due to
non-desired spatial harmonics,

  * •

improvement of circular polarization performance across the band.

By comparing the Brillouin diagrams in Fig. 3(c) and Fig. 3(e) for the unit
cell geometry containing one meander and three meanders, respectively, the
first three conclusions can be easily drawn. With the help of additional two
meanders the rate of dispersion with frequency is controlled within the unit
cell. The impact of the proposed geometry on the circular polarization can be
observed from Fig. 5. The acceptable range for frequency scanning (frequency
range where axial ratio $<$ $3\text{\,}\mathrm{dB}$) has increased from
$20\text{\,}\mathrm{GHz}25.2\text{\,}\mathrm{GHz}$ to
$19.4\text{\,}\mathrm{GHz}27.5\text{\,}\mathrm{GHz}$ resulting in increase of
beam steering from ($-26\degree$ to $+10\degree$) to ($-42\degree$ to
$+30\degree$). The beam steering ($-42\degree$ to $+30\degree$) is discussed
in Section IV.

TABLE II: Dimensions of the unit cell with 3 meanders designed to operate in spatial harmonic of $n=-2$ and have circularly polarized fields. Note that $\lambda_{\mathrm{b}}$ is wavelength corresponding to $f_{\mathrm{b}}$ in microstrip medium Line interval |  Initial Values (From the theory) |  Optimised (mm)
---|---|---
Expressed in $\lambda_{\mathrm{b}}$ |  (mm) |
$\mathrm{CC_{1}}$=$\mathrm{C_{1}D}$=$\mathrm{DD_{1}}$ |  $\phi_{\mathrm{sect}}\times(\lambda_{\mathrm{b}}/2\pi)$ |  0.71 |  0.745
$\mathrm{D_{1}A}$=$\mathrm{B_{1}P}$ |  $(\pi/2-3\phi_{\mathrm{sect}})/2\times(\lambda_{\mathrm{b}}/2\pi)$ |  1.128 |  1.172
$\mathrm{AA_{1}}$ |  $\lambda_{\mathrm{b}}/2$ |  4.265 |  4.343
$\mathrm{A_{1}B}$ |  $\lambda_{\mathrm{b}}/4$ |  2.383 |  3.062
$\mathrm{BB_{1}}$ |  $\lambda_{\mathrm{b}}/2$ |  4.265 |  4.343
$\mathrm{PP_{1}}$=$\mathrm{P_{1}Q}$=$\mathrm{QQ_{1}}$ |  $\phi_{\mathrm{sect}}\times(\lambda_{\mathrm{b}}/2\pi)$ |  0.71 |  0.745
$\mathrm{OC}$=$\mathrm{Q_{1}M}$ |  $(\pi-3\phi_{\mathrm{sect}})/2\times(\lambda_{\mathrm{b}}/2\pi)$ |  1.819 |  0.376
$p$ |  $2\times\lambda_{\mathrm{b}}$ |  9.694 |  8.028
$\mathrm{t}_{50}$ |  - |  0.5 |  0.5
$\mathrm{h}_{\mathrm{sub}}$ |  - |  0.254 |  0.254

Figure 6: Full-wave simulation for the unit cell constructed using the
theoretical formulation. At $f=f_{\mathrm{b}}$, the electric fields at each of
the four mitred corners of $\mathrm{TrSeg1}$ cancels out with corresponding
mitred corner of $\mathrm{TrSeg3}$.

###  III-D Mitigation of OSB

Periodic structures like LWAs have a frequency band near the broadside
frequency ($f_{\mathrm{b}}$ in current case), where there is no conduction of
the travelling-wave, known as OSB [15, 2]. To remove OSB, the parameter of
Bloch impedance ($Z_{\mathrm{s}}$) has to be analysed. The Bloch impedance can
be calculated from the S-parameters extracted from driven-mode full-wave
simulation in terms of circuit parameters $\mathrm{A}$, $\mathrm{B}$,
$\mathrm{C}$ and $\mathrm{D}$ as shown in [38]:

$Z_{\mathrm{s}}=\frac{-2\times\mathrm{B}}{(\mathrm{A}-\mathrm{D}-\sqrt{(\mathrm{A}+\mathrm{D})^{2}-4)}}$
(15)

Fig. 7 shows the Bloch impedance extracted for the K-band operation range. For
the case when OSB is present, at the broadside frequency,
$f_{\mathrm{b}}=$23\text{\,}\mathrm{GHz}$$, there is an abrupt increase in
imaginary $Z_{\mathrm{s}}$, which results in poor transmission and high return
loss. This results in reduced gain at the broadside region.

To mitigate the problem of OSB, a technique similar to [13] is employed. For a
meandering microstrip with mitred corners the effect of bending the microstrip
line (corners) can be modelled as capacitance [40, 41, 42] . This is clearly
evident from Fig. 7 when OSB is present. Hence, the angle for the mitred
corners are changed which introduces an additional inductance [42, 41] that
reduces imaginary impedance at $f=f_{\mathrm{b}}$ as shown in Fig. 8. Fig. 7
shows that imaginary impedance goes to zero which leads to mitigation of OSB
at broadside direction.

Figure 7: The Bloch impedance of the LWA when OSB is present and when OSB is
mitigated by the introduction of additional capacitance. Here the impedance is
normalised to $Z_{0}$ = $50\Omega$.

Figure 8: OSB mitigation by changing the angle of mitred corner. The values of
$\delta_{0}$ and $\delta_{1}$ are $0.15\text{\,}\mathrm{m}\mathrm{m}$ and
$0.06\text{\,}\mathrm{m}\mathrm{m}$ respectively.

Figure 9: Feed Design to match the output impedance ($Z_{0}$) of $50\Omega$
connector to input impedance of of antenna at around $60\Omega$. The values
for matching section is $\mathrm{T_{ms}}=~{}$0.57\text{\,}\mathrm{mm}$$ and
$\mathrm{L_{ms}}=~{}$1.634\text{\,}\mathrm{mm}$$. The $50\Omega$ transmission
line is $\mathrm{t_{Tx}}=~{}$0.64\text{\,}\mathrm{mm}$$.

Figure 10: Fabricated Leaky-Wave Antenna.

Figure 11: Measurements in K-band where the antenna is circularly polarized.
The measured and simulated normalized radiation pattern of the fabricated
antenna in the H-Plane for (a) $f=$
$19.4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. (b) $f=$
$21\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. (c) $f=$
$23\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. (d) $f=$
$26.8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. The antenna scans from
$-42\text{\,}\mathrm{\SIUnitSymbolDegree}30\text{\,}\mathrm{\SIUnitSymbolDegree}$
in the K-band frequency range. (e) The axial ratio of the fabricated antenna
remains below 3 dB throughout the operational frequency range. The measured
realized gain of the antenna in the H-Plane ranges from
$8\text{\,}\mathrm{dB}13.8\text{\,}\mathrm{dB}$. (f) Measured transmission and
return loss parameters. (g) Measurement setup of the antenna.

Figure 12: Measurements in the Ku-band where the antenna depicts linear
polarization. The measured normalized radiation pattern pattern closely
follows the radiation pattern predicted by the simulation for the frequencies
(a) $f=$ $11\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. (b) $f=$
$12\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. (c) $f=$
$14\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. (d) $f=$
$15.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. (e) Measured return loss. (f)
Realized gain in Ku-band with different number of unit cells obtained through
full-wave simulations. (g) Experimental setup for the measurement in Ku-band.
(h) The additional connectors required to connect the antenna to the ports of
VNA.

##  IV Dual-Band Leaky-Wave Antenna

A 10 unit cell dual band LWA, having the layout of Fig. 1(b) and the optimised
dimensions listed in Table II was fabricated. The antenna is based on Rogers
3003 ($\varepsilon_{r}=3.0$ and $\tan\delta=~{}0.001$) with a substrate
thickness of $0.254\text{\,}\mathrm{mm}$.

TABLE III: Comparison with other scanning LWAs operating in similar frequency ranges Ref |  Antenna Type |  Frequency BW |  Scanning Range |  Realized Gain (dB) |  Rel. Permittivity ($\varepsilon_{r}$) |  Pol.
---|---|---|---|---|---|---
[29] |  Microstrip with metallized vias |  $20\text{\,}\mathrm{GHz}29\text{\,}\mathrm{GHz}$ |  $95\text{\,}\mathrm{\SIUnitSymbolDegree}$ ($-50\text{\,}\mathrm{\SIUnitSymbolDegree}$ to $45\text{\,}\mathrm{\SIUnitSymbolDegree}$) |  Approx. 10 |  6.15 |  Circular
[10] |  Microstrip with metallized vias |  $9.8\text{\,}\mathrm{GHz}15\text{\,}\mathrm{GHz}$ |  $72\text{\,}\mathrm{\SIUnitSymbolDegree}$ ($-52\text{\,}\mathrm{\SIUnitSymbolDegree}$ to $20\text{\,}\mathrm{\SIUnitSymbolDegree}$) |  10.0–15.1 |  2.2 |  Circular
[43] |  SIW |  $26\text{\,}\mathrm{GHz}43\text{\,}\mathrm{GHz}$ |  $70\text{\,}\mathrm{\SIUnitSymbolDegree}$ ($-40\text{\,}\mathrm{\SIUnitSymbolDegree}$ to $30\text{\,}\mathrm{\SIUnitSymbolDegree}$) |  13.4–15 |  3.0 |  Linear
[28] |  SIW |  $23\text{\,}\mathrm{GHz}25\text{\,}\mathrm{GHz}$ |  $30\text{\,}\mathrm{\SIUnitSymbolDegree}$ ($-15\text{\,}\mathrm{\SIUnitSymbolDegree}$ to $15\text{\,}\mathrm{\SIUnitSymbolDegree}$) |  15 |  3.66 |  Circular
[44] |  SIW |  $10\text{\,}\mathrm{GHz}14\text{\,}\mathrm{GHz}$ |  $65\text{\,}\mathrm{\SIUnitSymbolDegree}$ ($-40\text{\,}\mathrm{\SIUnitSymbolDegree}$ to $25\text{\,}\mathrm{\SIUnitSymbolDegree}$) |  5.1–13 |  3.0 |  Circular
[31] |  Microstrip |  $20.6\text{\,}\mathrm{GHz}24.6\text{\,}\mathrm{GHz}$ |  $85\text{\,}\mathrm{\SIUnitSymbolDegree}$ ($-40\text{\,}\mathrm{\SIUnitSymbolDegree}$ to $45\text{\,}\mathrm{\SIUnitSymbolDegree}$) |  9.8–13 |  3.0 |  Linear
This Work – K-band |  Microstrip |  $19.4\text{\,}\mathrm{GHz}27.5\text{\,}\mathrm{GHz}$ |  $72\text{\,}\mathrm{\SIUnitSymbolDegree}$ ($-42\text{\,}\mathrm{\SIUnitSymbolDegree}$ to $30\text{\,}\mathrm{\SIUnitSymbolDegree}$) |  7–13 |  3.0 |  Circular
This Work – Ku-band |  Microstrip |  $11\text{\,}\mathrm{GHz}15.5\text{\,}\mathrm{GHz}$ |  $75\text{\,}\mathrm{\SIUnitSymbolDegree}$($-15\text{\,}\mathrm{\SIUnitSymbolDegree}$ to $60\text{\,}\mathrm{\SIUnitSymbolDegree}$) |  2.5–5 |  3.0 |  Linear

###  IV-A Feed design

A Bulgin end-launch connector ($2.4\text{\,}\mathrm{mm}$) with the output
impedance $Z_{0}=50\ \Omega$ is used to feed the antenna. The $Z_{\text{{S}}}$
of the antenna is around $60\ \Omega$ throughout the operating frequency range
as shown in Fig. 7. Hence a matching section is added to match the impedance
depicted in Fig. 9.

###  IV-B Circularly polarized radiation in K-band

In this band, the antenna is right-handed circularly polarized and radiation
is due to spatial harmonic of $n=-2$. The measurements are performed at the
millimeter-wave test facility CAMILL at Institut d’Électronique et de
Télécommunications de Rennes (IETR). The simulated and measured radiation
pattern in the azimuth plane are shown in Fig. 11(a–d) for the
$19.4\text{\,}\mathrm{GHz}27.5\text{\,}\mathrm{GHz}$ band. The measured
radiation pattern at $f=$19.4\text{\,}\mathrm{GHz}$$ shows higher side-lobes
compared to simulations. This is due to the fact that there is reflection from
the connector due to a very tilted beam in the H-plane. The
3-$\text{\,}\mathrm{dB}$ beamwidth of the antenna is about $10\degree$
throughout the operational range. The length of the antenna remains compact at
7.67$\lambda_{0}$.

The efficiency of the antenna in this band varies from 0.5 to 0.75. Measured
and simulated axial ratio are compared in Fig. 11(e). The realized gain plot
of the reconfigurable fan-beam radiation pattern in the H-plane is also shown
in Fig. 11(e). The measured S-parameters show that the reflection coefficient
remains below $-10\text{\,}\mathrm{dB}$ for the operational range.

###  IV-C Linearly polarized radiation in Ku-band

The antenna behaves as a linearly polarized antenna in the Ku-band
$11\text{\,}\mathrm{GHz}15.5\text{\,}\mathrm{GHz}$ and the radiation occurs
due to the spatial harmonic of $n=-1$. The simulated and measured realized
gain are shown in Fig. 12(a)–(d). The antenna scanning range is of
$-15\text{\,}\mathrm{\SIUnitSymbolDegree}60\text{\,}\mathrm{\SIUnitSymbolDegree}$.
The 3-$\text{\,}\mathrm{dB}$ beamwidth of the antenna is less than $17\degree$
throughout the operational range. The measured reflection coefficient is below
$-10\text{\,}\mathrm{dB}$ in the operating frequency range as shown in Fig.
12(e).

Measurement is performed using MVG Starlab measurement system at IETR, as
shown in Fig.12(g). The antenna is fixed on the platform using tapes to avoid
displacement. To avoid the elevated side-lobe levels at high tilted angles
absorbers are added at the either end.

It has to be noted that the antenna has been designed to operate in K-band and
Ku-band, with a special attention to the performance in K-band. As a
consequence, the gain in Ku-band is limited to $5\text{\,}\mathrm{dB}$.
However, the realized gain can be improved by adding unit cells to the LWA as
shown in Fig. 12(f). It is also important to notice that the realized gain in
the Ku-band of the antenna is lower by $12\text{\,}\mathrm{dB}$ in the
measurements than simulations as shown in Fig. 12(f). This is the effect of
the connectors required to properly connect the antenna to the ports of Vector
Network Analyzer (VNA) as depicted in Fig. 12(h).

###  IV-D Literature comparison

Table III shows the comparison with the antennas operating in similar
frequency ranges along with fabrication technology. Circularly polarized LWAs
operating in the range $9.8\text{\,}\mathrm{GHz}15\text{\,}\mathrm{GHz}$ and
$20\text{\,}\mathrm{GHz}29\text{\,}\mathrm{GHz}$ based on microstrip are
reported in [10, 29]. Both designs require via-holes, thus increasing the
complexity for fabrication compared to the proposed design. Ref. [45, 31]
report LWA based on microstrip operating in similar mm-Wave range however they
exhibit linear polarization. Composite right left-handed (CRLH) based
structures such as the ones reported in [46, 5, 3] are extremely sensitive to
dimensions of the slot making them difficult to fabricate with precision at
high frequencies.

Meanwhile, the proposed antenna achieves large beam scanning angles with
simple meandering microstrips, while maintaining circular polarization. The
design can be scaled at lower or higher frequencies very easily without the
increase in complexity of fabrication.

##  V Conclusion

This paper presents a novel approach to designing a compact and single-layer
circularly polarized LWA. The approach is based on utilizing the time delay of
microstrip line intervals to achieve circular polarization. The frequency
scanning LWA operates as circularly polarized antenna in the K-band and as
linearly polarized antenna in the Ku-band. This is possible by the use of
spatial harmonics $n=-2$ and $n=-1$, respectively. To improve the band-
separation and reduce the side-lobes due to unwanted harmonics in the
radiation zone, additional meanders have been introduced. The axial ratio
remains below $3\text{\,}\mathrm{dB}$ over large frequency range
($19.4\text{\,}\mathrm{GHz}27.5\text{\,}\mathrm{GHz}$) in the intended band
for circular polarization. This ensures large scanning range from 72° ($-$42°
to 30°). A novel technique to remove OSB is also explained in the work in
order to have continuous scanning through broadside direction.

The proposed antenna does not make use of metallized vias hence significantly
reducing the complexity of fabrication while maintaining compact size
(7.67$\lambda_{0}$). The antenna is manufactured on a flexible substrate of
Rogers 3003, making it suitable for flexible and conformal purposes at mm-wave
frequencies.

## References

  * [1] C. H. Walter, _Traveling Wave Antennas_. New York: McGraw-Hill, 1965.
  * [2] C. A. Balanis, _Antenna Theory: Analysis and Design_. John Wiley & Sons, Dec. 2015.
  * [3] R. Agarwal, R. L. Yadava, and S. Das, “A multilayered siw-based circularly polarized CRLH leaky wave antenna,” _IEEE Trans. Antennas Propag._ , vol. 69, no. 10, p. 10, May 2021.
  * [4] A. Sarkar and S. Lim, “60 GHz compact larger beam scanning range PCB leaky-wave antenna using HMSIW for millimeter-wave applications,” _IEEE Trans. Antennas Propag._ , vol. 68, no. 8, pp. 5816–5826, Aug. 2020\.
  * [5] M. M. Sabahi, A. A. Heidari, and M. Movahhedi, “A compact CRLH circularly polarized leaky-wave antenna based on substrate-integrated waveguide,” _IEEE Trans. Antennas Propag._ , vol. 66, no. 9, pp. 4407–4414, Sep. 2018\.
  * [6] C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-wave and resonant antennas,” _IEEE Antennas Propag. Mag._ , vol. 50, no. 5, pp. 25–39, Oct. 2008.
  * [7] J. Liu, X. Tang, Y. Li, and Y. Long, “Substrate integrated waveguide leaky-wave antenna with H-shaped slots,” _IEEE Trans. Antennas Propag._ , vol. 60, no. 8, p. 6, May 2012.
  * [8] G. P. Carrara, C. L. Zekios, and S. V. Georgakopoulos, “A TM11 high-order mode leaky wave antenna,” _IEEE Trans. Antennas Propag._ , vol. 71, no. 1, pp. 119–130, Jan. 2023.
  * [9] G. Sacco, O. Caytan, S. Pisa, and H. J. Visser, “Analysis and modelling of rampart line antennas,” _IET Microw , Antennas Propag._ , vol. 15, no. 12, pp. 1605–1617, 2021.
  * [10] S. Zhao and Y. Dong, “Circularly polarized beam-steering microstrip leaky-wave antenna based on coplanar polarizers,” _IEEE Antennas Wireless Propag. Lett._ , vol. 21, no. 11, pp. 2259–2263, Nov. 2022.
  * [11] N. Yang, C. Caloz, and K. Wu, “Full-space scanning periodic phase-reversal leaky-wave antenna,” _IEEE Trans. Microw. Theory Techn._ , vol. 58, no. 10, pp. 2619–2632, Oct. 2010.
  * [12] J. Duan and L. Zhu, “A transversal single-beam EH0-mode microstrip leaky-wave antenna on coupled microstrip lines under differential operation,” _IEEE Antennas Wireless Propag. Lett._ , vol. 20, no. 4, pp. 592–596, Apr. 2021.
  * [13] H. Wang, S. Sun, and X. Xue, “A periodic meandering microstrip line leaky-wave antenna with consistent gain and wide-angle beam scanning,” _Int. J. RF Microw. Comput.-Aided Eng._ , vol. 32, no. 7, p. e23162, 2022.
  * [14] J. R. James, P. S. Hall, and C. Wood, _Microstrip Antenna: Theory and Design_ , ser. (IEE Electromagnetic Waves Series, No.12). London: IET ch 7, sec 7.4.1, pp. 215–218, 1986.
  * [15] J. L. Volakis, _Antenna Engineering Handbook, Fourth Edition_. McGraw-Hill Education, 2007\.
  * [16] M. R. Rahimi, M. S. Sharawi, and K. Wu, “Higher-order space harmonics in substrate integrated waveguide leaky-wave antennas,” _IEEE Trans. Antennas Propag._ , vol. 69, no. 8, pp. 4332–4346, Aug. 2021.
  * [17] D.-F. Guan, Q. Zhang, P. You, Z.-B. Yang, Y. Zhou, and S.-W. Yong, “Scanning rate enhancement of leaky-wave antennas using slow-wave substrate integrated waveguide structure,” _IEEE Trans. Antennas Propag._ , vol. 66, no. 7, pp. 3747–3751, Jul. 2018.
  * [18] P. J. Soh, S. J. Boyes, G. Vandenbosch, Y. Huang, and a. S. L. Ooi, “On-body characterization of dual-band all-textile PIFA,” _Progress In Electromagnetics Research_ , vol. 129, pp. 517–539, 2012.
  * [19] P. J. Soh, G. A. Vandenbosch, M. Mercuri, and D. M.-P. Schreurs, “Wearable wireless health monitoring: Current developments, challenges, and future trends,” _IEEE Microw. Mag._ , vol. 16, no. 4, pp. 55–70, May 2015.
  * [20] N. Chahat, M. Zhadobov, R. Sauleau, and K. Ito, “A compact UWB antenna for on-body applications,” _IEEE Trans. Antennas Propag._ , vol. 59, no. 4, pp. 1123–1131, Apr. 2011.
  * [21] I. V. Soares, P. Vadher, A. K. Skrivervik, G. Sacco, and D. Nikolayev, “Analysis of non-canonical body-conformal arrays with polarization decomposition,” in _Proc. 17th Eur. Conf Antennas and Propag. (EuCAP 2023)_ , Florence, Italy, Mar. 2023.
  * [22] X. Tian, Q. Zeng, D. Nikolayev, and J. S. Ho, “Conformal propagation and near-omnidirectional radiation with surface plasmonic clothing,” _IEEE Trans. Antennas Propag._ , vol. 68, no. 11, pp. 7309–7319, Nov. 2020.
  * [23] J. Hasch, E. Topak, R. Schnabel, T. Zwick, R. Weigel, and C. Waldschmidt, “Millimeter-wave technology for automotive radar sensors in the 77 GHz frequency band,” _IEEE Trans. Microw. Theory Techn._ , vol. 60, no. 3, pp. 845–860, Mar. 2012.
  * [24] J. C. G. Matthews and G. Pettitt, “Development of flexible, wearable antennas,” in _Proc. 3rd Eur. Conf. Antennas Propag. (EuCAP)_ , Mar. 2009, pp. 273–277.
  * [25] P. Vadher, G. Sacco, and D. Nikolayev, “On-body V-band leaky-wave antenna for navigation and safety applications,” in _IEEE Microwaves, Antennas, and Propag. Conf. (MAPCon)_ , Bangalore, Dec. 2022, pp. 1741–1746.
  * [26] R. Henry and M. Okoniewski, “A broadside-scanning half-mode substrate integrated waveguide periodic leaky-wave antenna,” _IEEE Antennas Wireless Propag. Lett._ , vol. 15, pp. 602–605, 2016.
  * [27] D. R. Jackson and A. A. Oliner, “Leaky-Wave Antennas,” in _Modern Antenna Handbook_. John Wiley & Sons, Ltd, 2008, pp. 325–367.
  * [28] S. Otto, Z. Chen, A. Al-Bassam, A. Rennings, K. Solbach, and C. Caloz, “Circular polarization of periodic leaky-wave antennas with axial asymmetry: Theoretical proof and experimental demonstration,” _IEEE Trans. Antennas Propag._ , vol. 62, no. 4, pp. 1817–1829, Apr. 2014.
  * [29] M. H. Rahmani and D. Deslandes, “Backward to forward scanning periodic leaky-wave antenna with wide scanning range,” _IEEE Trans. Antennas Propag._ , vol. 65, no. 7, pp. 3326–3335, Jul. 2017.
  * [30] S. Paulotto, P. Baccarelli, F. Frezza, and D. R. Jackson, “A novel technique for open-stopband suppression in 1-D periodic printed leaky-wave antennas,” _IEEE Trans. Antennas Propag._ , vol. 57, no. 7, pp. 1894–1906, Jul. 2009.
  * [31] P. Vadher, G. Sacco, and D. Nikolayev, “Higher spatial harmonic leaky wave antenna design based on meandering microstrips,” in _Proc. 17th Eur. Conf Antennas and Propag. (EuCAP)_ , Florence, Italy, Mar. 2023.
  * [32] S. Cheng, Y. Li, Z. Liang, S. Zheng, and Y. Long, “An approximate circuit model to analyze microstrip rampart line in osb suppressing,” _IEEE Access_ , vol. 7, pp. 90 412–90 417, 2019.
  * [33] S.-L. Chen, D. K. Karmokar, Z. Li, P.-Y. Qin, R. W. Ziolkowski, and Y. J. Guo, “Circular-polarized substrate-integrated-waveguide leaky-wave antenna with wide-angle and consistent-gain continuous-beam scanning,” _IEEE Trans. Antennas Propag._ , vol. 67, no. 7, pp. 4418–4428, Jul. 2019.
  * [34] C. Wood, “Curved microstrip lines as compact wideband circularly polarised antennas,” _IEE J. Microw. Opt. Acoust._ , vol. 3, no. 1, p. 5, 1979.
  * [35] D. R. Jackson, C. Caloz, and T. Itoh, “Leaky-Wave Antennas,” _Proceedings of the IEEE_ , vol. 100, pp. 2194–2206, 2011.
  * [36] R. E. Collin and F. J. Zucker, _Antenna Theory_. New York: McGraw-Hill, 1969, vol. 2, ch. 19-20.
  * [37] A. Ishimaru, _Electromagnetic Wave Propagation, Radiation, and Scattering: From Fundamentals to Applications_ , 2nd ed., ser. The IEEE Press Series on Electromagnetic Wave Theory. Piscataway, NJ: IEEE Press/Wiley, 2017.
  * [38] D. M. Pozar, _Microwave Engineering_ , 4th ed. Hoboken, NJ: Wiley, Nov. 2011.
  * [39] M. Sadiku, S. Musa, and S. R. Nelatury, “Comparison of Dispersion Formulas for Microstrip Lines,” in _IEEE SoutheastCon, 2004. Proceedings._ , Mar. 2004, pp. 378–382.
  * [40] R. Douville and D. James, “Experimental study of symmetric microstrip bends and their compensation,” _IEEE Trans. Microw. Theory Techn._ , vol. 26, no. 3, pp. 175–182, Mar. 1978.
  * [41] P. Silvester and P. Benedek, “Microstrip discontinuity capacitances for right-angle bends, T junctions, and crossings,” _IEEE Trans. Microw. Theory Techn._ , vol. 21, no. 5, pp. 341–346, May 1973.
  * [42] P. Anders and F. Arndt, “Microstrip discontinuity capacitances and inductances for double steps, mitered bends with arbitrary angle, and asymmetric right-angle bends,” _IEEE Trans. Microwave Theory Techn._ , vol. 28, no. 11, pp. 1213–1217, Nov. 1980.
  * [43] Y. Geng, J. Wang, Y. Li, Z. Li, M. Chen, and Z. Zhang, “A Ka-band leaky-wave antenna array with stable gains based on HMSIW structure,” _IEEE Antennas Wireless Propag. Lett._ , vol. 21, no. 8, pp. 1597–1601, Aug. 2022.
  * [44] Y.-L. Lyu, F.-Y. Meng, G.-H. Yang, D. Erni, Q. Wu, and K. Wu, “Periodic SIW leaky-wave antenna with large circularly polarized beam scanning range,” _IEEE Antennas Wireless Propag. Lett._ , vol. 16, pp. 2493–2496, Jul. 2017.
  * [45] Y.-L. Lyu, F.-Y. Meng, G.-H. Yang, P.-Y. Wang, Q. Wu, and K. Wu, “Periodic leaky-wave antenna based on complementary pair of radiation elements,” _IEEE Trans. Antennas Propag._ , vol. 66, no. 9, pp. 4503–4515, Sep. 2018\.
  * [46] Y. D. Dong and T. Itoh, “Composite right/left-handed substrate integrated waveguide and half-mode substrate integrated waveguide,” in _2009 IEEE MTT-S Int. Microwave Sym. Digest_ , Boston, MA, USA, Jun. 2009, pp. 49–52.