Cylindrical Conformal Antenna in a Homogeneous Plasma

The cylindrical slotted antenna was first proposed in 1938 by Alan D. Blumlein to use in television broadcasting with horizontal polarization and a circular radiation pattern in a horizontal plane [1]. One of the advantages of slot antennas is that they do not affect the aerodynamics of the objects on which they are established, which later determined their widespread use on aircraft, rockets, submarines, and other movable objects. In addition, slot antennas are widely used as ground antennas [2].

Several articles propose an approach that would make it possible to place an antenna on aircraft, drones or spacecraft without reducing its aerodynamic properties. In other word, to make the antenna more suitable in terms of form and purpose called conformal antenna. Or the antenna should be conformal to a prescribed shape. The shape can be some part of an airplane, high-speed train, or other vehicle. The purpose is to build the antenna so that it becomes integrated with the structure and does not cause extra drag [3]. The purpose can also be that the antenna integration makes the antenna less disturbing, less visible to the human eyes, for instance, in an urban environment. A typical additional requirement in modern defense systems is that the antenna not backscatter microwave radiation when illuminated by an enemy radar transmitter, i.e., it has stealth properties. Specifically, in this case it is better to use plasma as well [4, 5].

In addition, the properties of conformal antennas on a cylindrical surface which allow wide-angle scanning at the desired frequency and polarization for further use. Slot cylindrical antenna with an original device for matching with a feeder is proposed in the study [6]. Such that the original matching device ensures simple and convenient matching and tuning of an antenna with the operating frequency. The antenna is made as a longitudinal slot on a metal pipe with a diameter much smaller than the wavelength. At the same time the length of the slot is less than the wavelength in free space. In the study [7], a multilayered circular cylindrical conformal wide slot antenna show that the operating frequency depends on the size of the slot and to lesser degree to the radius of the cylinder. Such that the slot antenna is in the shape of a hexagon. Broadband characteristics antenna which can be used as communication and radar antenna presented as S band cylindrical waveguide slot omnidirectional antenna [8]. Antenna had an omnidirectional radiation characteristic in the E-plane and in H-plane had an 8-shape radiation characteristic approximately.

In addition to controlling the shape of the antenna according to the flying object, it is also necessary to test its operation in the presence of the plasma. Plasma, the fourth state of matter, has many applications in electromagnetic engineering [9-13]. It has played an important role in military applications. Plasma can transmit, reflect, and absorb the electromagnetic waves depending on the conditions of propagation of the electromagnetic waves. Therefore, there are many studies that investigate in effect of plasma on the work of antennas [14-17]. Reconfigurable slotted antenna using plasma tube in study [18], used to obtain a reconfigurable radiation pattern according to the state of plasma whether it was switched on or switched off. Such that when plasma is switched on, at 1 GHz, antenna have the possibility to radiate, but at 4 GHz, there is no radiation from the antenna. Therefore, when manufacturing an antenna that is intended to work in plasma, it is imperative to take into consideration the difference between plasma frequency and antenna operating frequency. In study [19], a Faraday shield effect using plasma (fluorescent lamp) was presented to show the impact of the plasma on the gain of patch antenna put inside lamp. Plasma behaves like a transparent media or a Faraday shield effect, respectively at switching lamp ON or OFF. In addition, noticed that, at the operating frequency the parameters keep a good matching.

In conformal antenna, transmitters are placed on non-flat surfaces like spherical [20-22], cylindrical [23-25], conical [26, 27], and their combinations [28].

In general, the electrodynamic modeling of conformal antennas in plasma environments involves the application of several established numerical techniques, including the method of auxiliary sources (MAS), surface integral equations (SIE), and volume integral equations (VIE). Each of these methods exhibits unique strengths; however, they are confronted with significant challenges when dealing with complex, multi-layered antenna structures coated with plasma. The MAS, for instance, demonstrates proficiency in handling simple geometries but encounters difficulties in managing the intricacies of conformal antennas with plasma coatings. Conversely, VIE methods, while necessitating volumetric meshing across the entire frequency spectrum, incur substantial computational costs, particularly when high-resolution plasma layers are involved. SIE techniques, which dimensionally reduce the problem, often lead to the formation of large, dense system matrices that complicate computations and may potentially sacrifice precision.

This paper introduces a novel integro-functional equation method as an alternative to the conventional approaches mentioned above. This method offers several notable advantages. It allows for the modeling of external electromagnetic fields without the explicit consideration of the internal plasma field, thereby simplifying the system's complexity and reducing computational demands. The inherent scalar formulation of the method, stemming from the azimuthal symmetry of the magnetic field, contributes to an efficient analysis process. Furthermore, it excels in the treatment of boundary conditions and enables the utilization of Fourier-Bessel expansions for frequency-domain analysis. These characteristics render the proposed approach highly suitable for modeling conformal antennas with two dielectric and plasma layers. The method provides a robust and scalable solution that is superior to traditional numerical techniques in terms of computational efficiency and accuracy.

So, this investigation aims to develop a conformal antenna of cylindrical geometry, encircled by a dielectric material, with an overlying plasma layer. The study employs the ingenious approach of the integrable-functional equation methodology, which significantly streamlines the analytical process by circumventing the computation of internal plasma fields. Moreover, this research scrutinizes the feasibility of maintaining the antenna's optimal performance under such configurations.

The physical system under study is a cylindrical antenna with a slot embedded on a conductive surface, surrounded by two concentric cylindrical layers: an inner dielectric layer and an outer plasma layer as shown in Figure 1, where,

S- The surface of the body,

${{G}_{i1}}$- Green's function of a homogeneous space with body parameters.

${{\nabla }_{s}}$- Gradient operator with respect to coordinates of integration point.

${{\bar{H}}_{e}}$- The desired external field consisting of an unperturbed field of external sources located in $V_{c m e} \subset V_e$.

${{\bar{H}}_{oi}}$- Unperturbed field of external sources located in ${{V}_{i}}$.

Using an integro-functional equation [29, 30], the total field beyond the plasma is given by:

$L\left(l, H_e\right)=\pi \int_{\left.l-\frac{\varepsilon_i}{\varepsilon_e} \frac{\partial\left(\rho_s H_e\right)}{\partial v} G_{i 1}\right\} d l}^{\left\{H_e \frac{\partial\left(\rho_s G_{i 1}\right)}{\partial v}\right.}= \begin{cases}H_{o i} & \left(\vec{r} \in V_e\right) \\ -H_{p i} & \left(\vec{r} \in V_i\right)\end{cases}$                                            (1) 

This equation establishes a connection between the overall field and the source configuration across the body's surface by employing Green's function. The left-hand side of the equation can be separated into two distinct integral components, which correspond to the internal and external regions of the plasma layer.

$L({{l}^{/}},{{H}_{e}})+L({{l}^{//}},{{H}_{e}})=\left\{ \begin{align}& 0 \\& -{{H}_{pi}} \\\end{align} \right.$$\begin{align}& (r\in {{V}_{e}}) \\& (r\in {{V}_{i}}) \\\end{align}$                                            (2)

To simplify the problem due to cylindrical symmetry, we expand the domains in terms of Fourier integrals (in the axial z direction) and Bessel functions (in the radial r direction) such that for ${{R}_{0}}<r<{{R}_{1}}$:

${{H}_{e}}=\int\limits_{-\infty }^{\infty }{\left\{ D(h)H_{1}^{(2)}({{\nu }_{0}}r)+A(h){{J}_{1}}({{\nu }_{0}}r) \right\}{{e}^{ihz}}dh}$                                           (3) 

where, ${{\nu }_{0}}=-i\sqrt{{{h}^{2}}-k_{0}^{2}}$, $D\left( h \right)$ and$~A\left( h \right)$ are unknown spectral densities.

The integral operator on the inner surface becomes:

$L({{l}^{/}},{{H}_{e}})=-\pi \int\limits_{-\infty }^{\infty }{\left\{ {{H}_{e}}\frac{\partial ({{R}_{1}}{{G}_{i1}})}{\partial {{R}_{1}}}-\frac{{{\varepsilon }_{i}}}{{{\varepsilon }_{0}}}\frac{\partial ({{R}_{1}}{{H}_{e}})}{\partial {{R}_{1}}}{{G}_{i1}} \right\}d{{z}_{{{l}^{/}}}}}$                                           (4) 

where, the Green’s function ${{G}_{i1}}$ is expanded for cylindrical geometry (first azimuthal harmonic n=1), identical with Eq. (5),

$\begin{aligned} G_{i 1}=-\frac{i k_i}{2 \pi} \sum_{n=1}^{\infty} \frac{2 n+1}{n(n+1)} \psi_n\left(k_i r^{\prime}\right) \xi_n^{(2)} \\ \left(k_i r\right) P_n^1\left(\cos \theta^{\prime}\right) P_n^1(\cos \theta)\end{aligned}$                                           (5)

and can be obtained from the well-known representation of the Green's function in cylindrical coordinates (1) and written as:

${{G}_{i1}}=\frac{1}{4\pi i}\int\limits_{-\infty }^{\infty }{\left\{ \begin{align} & H_{1}^{(2)}({{\nu }_{i}}r){{J}_{1}}({{\nu }_{i}}{{r}^{/}}) \\& {{J}_{1}}({{\nu }_{i}}r)H_{1}^{(2)}({{\nu }_{i}}{{r}^{/}}) \\\end{align} \right\}}{{e}^{ih(z-{{z}^{/}})}}dh$$\begin{align}& ({{r}^{/}}<r) \\\end{align}$                                           (6)

where, $\left(v_i=-i \sqrt{h^2-k_i^2}\right)$

Assuming linearity, i.e., ${{r}^{/}}={{R}_{1}}$ and ${{z}^{/}}={{z}_{{{l}^{/}}}}$.

Substituting (3) and (6) into (4), the integral operator becomes:

$L({{l}^{/}},{{H}_{e}})=\frac{\pi i}{2}\int\limits_{-\infty }^{\infty }{\left\{ \begin{align} & (D(h){{d}_{h}}+A(h){{a}_{h}}){{J}_{1}}({{\nu }_{i}}r) \\& (D(h){{d}_{j}}+A(h){{a}_{j}})H_{1}^{(2)}({{\nu }_{i}}r) \\\end{align} \right\}}{{e}^{ihz}}dh$$(r<{{R}_{1}});(r>{{R}_{1}})$                                           (7)

where,

$\begin{align}& {{d}_{h}}=H_{1}^{(2)}({{\nu }_{0}}{{R}_{1}})\frac{\partial ({{R}_{1}}H_{1}^{(2)}({{\nu }_{i}}{{R}_{1}}))}{\partial {{R}_{1}}} \\& \text{    }-\frac{{{\varepsilon }_{i}}}{{{\varepsilon }_{0}}}\frac{\partial ({{R}_{1}}H_{1}^{(2)}({{\nu }_{0}}{{R}_{1}}))}{\partial {{R}_{1}}}H_{1}^{(2)}({{\nu }_{i}}{{R}_{1}}), \\\end{align}$                                           (8)

$\begin{align}& {{a}_{h}}={{J}_{1}}({{\nu }_{0}}{{R}_{1}})\frac{\partial ({{R}_{1}}H_{1}^{(2)}({{\nu }_{i}}{{R}_{1}}))}{\partial {{R}_{1}}} \\& \text{    }-\frac{{{\varepsilon }_{i}}}{{{\varepsilon }_{0}}}\frac{\partial ({{R}_{1}}{{J}_{1}}({{\nu }_{0}}{{R}_{1}}))}{\partial {{R}_{1}}}H_{1}^{(2)}({{\nu }_{i}}{{R}_{1}}), \\\end{align}$                                           (9)

$\begin{align}& {{d}_{J}}=H_{1}^{(2)}({{\nu }_{0}}{{R}_{1}})\frac{\partial ({{R}_{1}}{{J}_{1}}({{\nu }_{i}}{{R}_{1}}))}{\partial {{R}_{1}}} \\& \text{    }-\frac{{{\varepsilon }_{i}}}{{{\varepsilon }_{0}}}\frac{\partial ({{R}_{1}}H_{1}^{(2)}({{\nu }_{0}}{{R}_{1}}))}{\partial {{R}_{1}}}{{J}_{1}}({{\nu }_{i}}{{R}_{1}}), \\\end{align}$                                           (10)

$\begin{align}& {{a}_{J}}={{J}_{1}}({{\nu }_{0}}{{R}_{1}})\frac{\partial ({{R}_{1}}{{J}_{1}}({{\nu }_{i}}{{R}_{1}}))}{\partial {{R}_{1}}} \\& \text{    }-\frac{{{\varepsilon }_{i}}}{{{\varepsilon }_{0}}}\frac{\partial ({{R}_{1}}{{J}_{1}}({{\nu }_{0}}{{R}_{1}}))}{\partial {{R}_{1}}}{{J}_{1}}({{\nu }_{i}}{{R}_{1}}). \\\end{align}$                                           (11)

Field in the outer space $({{R}_{2}}<r<\infty )$ has the following integral representation:

${{H}_{e}}=\int\limits_{-\infty }^{\infty }{B(h)}H_{1}^{(2)}({{\nu }_{e}}r){{e}^{ihz}}dh$$({{\nu }_{e}}=-i\sqrt{{{h}^{2}}-k_{e}^{2}})$                                           (12)

and for the integral operator with respect to the outer surface of the plasma layer in (2), we obtain:

$L({{l}^{//}},{{H}_{e}})=-\frac{\pi i}{2}\int\limits_{-\infty }^{\infty }{\left\{ \begin{align}& B(h){{b}_{h}}{{J}_{1}}({{\nu }_{i}}r) \\& B(h){{b}_{J}}H_{1}^{(2)}({{\nu }_{i}}r) \\\end{align} \right\}{{e}^{ihz}}dh}$$(r<{{R}_{2}});(r>{{R}_{2}})$                                           (13)

where:

$\begin{aligned} & b_h=H_1^{(2)}\left(v_e R_2\right) \frac{\partial\left(R_2 H_1^{(2)}\left(v_i R_2\right)\right)}{\partial R_2} \\ & -\frac{\varepsilon_i}{\varepsilon_e} \frac{\partial\left(R_2 H_1^{(2)}\left(v_e R_2\right)\right)}{\partial R_2} H_1^{(2)}\left(v_i R_2\right)\end{aligned}$                                           (14)

$\begin{align}& {{b}_{J}}=H_{1}^{(2)}({{\nu }_{e}}{{R}_{2}})\frac{\partial ({{R}_{2}}{{J}_{1}}({{\nu }_{i}}{{R}_{2}}))}{\partial {{R}_{2}}} \\& \text{   }-\frac{{{\varepsilon }_{i}}}{{{\varepsilon }_{e}}}\frac{\partial ({{R}_{2}}H_{1}^{(2)}({{\nu }_{e}}{{R}_{2}}))}{\partial {{R}_{2}}}{{J}_{1}}({{\nu }_{i}}{{R}_{2}}). \\\end{align}$                                           (15)

These integrals take advantage of well-known properties and derivatives of Bessel functions, such as recurrence relations and asymptotic forms.

Using the known functional relation for Bessel functions of any kind:

$\frac{\partial (z{{Z}_{1}}(z))}{\partial z}=z{{Z}_{0}}(z)$

In the relations Eqs. (8)-(10), (11), (14), (15), the derivatives can be replaced by the Bessel functions themselves.

$\frac{\partial ({{R}_{1}}H_{1}^{(2)}({{\nu }_{i}}{{R}_{1}}))}{\partial {{R}_{1}}}={{\nu }_{i}}{{R}_{1}}H_{0}^{(2)}({{\nu }_{i}}{{R}_{1}}).$

The boundary conditions in the plasma layer are expressed by the integral Green's function (IGF) equations, which are divided into two radial regions as shown in Eq. (16). The IGF formulation applied to a layered cylindrical structure can be written as:

$L({{l}^{/}},{{H}_{e}})+L({{l}^{//}},{{H}_{e}})=0$$({{R}_{0}}<r<{{R}_{1}});({{R}_{2}}<r<\infty )$                                           (16)

The two constituents of Eq. (16) represent integral operators that are applied to the magnetic field components situated within the dielectric and plasma layers. It is essential to note that the total magnetic field, or its Green's function representation, is subject to the rigorous adherence of physical boundary conditions at the interfaces between these regions. These boundary conditions typically necessitate the continuity of both the tangential electric and magnetic fields across the dielectric-plasma boundaries. In addition, this equation, considering the representations Eqs. (7), (13), leads to a system of equations for the spectral densities. For the first condition in Eq. (16), we obtain:

$D(h){{d}_{h}}+A(h){{a}_{h}}-B(h){{b}_{h}}=0$                                           (17)

and for the second, respectively:

$D(h){{d}_{J}}+A(h){{a}_{J}}-B(h){{b}_{J}}=0$                                           (18)

D(h), A(h), B(h) are spectral response functions or integral kernels, depending on the geometry and material parameters.

It is convenient to solve this linear system with respect to A(h) and B(h) assuming a temporarily known spectral density D(h):

$A(h)=\sigma D(h)$                                           (19)

$B(h)=\gamma D(h)$                                           (20)

For the coefficients s and g we have the system of equations:

$\begin{align}& \sigma {{a}_{h}}-\gamma {{b}_{h}}=-{{d}_{h}} \\& \sigma {{a}_{J}}-\gamma {{b}_{J}}=-{{d}_{J}} \\\end{align}$                                           (21)

The solution of which has the form:

$\sigma =\frac{{{d}_{J}}{{b}_{h}}-{{d}_{h}}{{b}_{J}}}{{{a}_{h}}{{b}_{J}}-{{a}_{J}}{{b}_{h}}}$                                           (22)

$\gamma =\frac{{{d}_{J}}{{a}_{h}}-{{d}_{h}}{{a}_{J}}}{{{a}_{h}}{{b}_{J}}-{{a}_{J}}{{b}_{h}}}$                                           (23)

Using the boundary condition of the electric field tangent to the surface (voltage-excited slot). Unknown spectral density D(h) can be found from the boundary conditions on the surface of the cylinder. The tangential component of the electric field ${{E}_{ez}}$ can be determined from the magnetic field:

${{E}_{ez}}=-\frac{i}{\omega {{\varepsilon }_{e}}}\frac{1}{r}\frac{\partial (r{{H}_{e}})}{\partial r}$                                           (24)

Using representation (12) and formulas for the derivatives of Bessel functions, we obtain the following representation for the tangential component of the electric field on the cylinder surface:

$\left.E_{e x}\right|_{r=R_0}=-\frac{i}{\omega \varepsilon_e} \int_{-\infty}^{\infty} v_0 D(h)\left\{H_0^{(2)}\left(v_0 R_0\right)+\sigma(h) J_0\left(v_0 R_0\right)\right\} e^{i h c} d h$                                           (25)

On the other hand, for slot on a conductive cylinder with an applied voltage ${{U}_{s}}$ and coordinate ${{Z}_{s}}$ (slot):

${{E}_{ez}}={{U}_{s}}\delta (z-{{z}_{s}})=\frac{{{U}_{s}}}{2\pi }\int\limits_{-\infty }^{\infty }{{{e}^{ih(z-{{z}_{s}})}}dh}$                                           (26)

where, for the spectral density for Fourier transform of the slot voltage profile D(h):

$D(h)=\frac{i{{U}_{s}}\omega {{\varepsilon }_{e}}{{e}^{-ih{{z}_{s}}}}}{2\pi {{\nu }_{0}}[H_{0}^{(2)}({{\nu }_{0}}{{R}_{0}})+\sigma (h){{J}_{0}}({{\nu }_{0}}{{R}_{0}})]}.$                                           (27)

To find the radiation pattern (far field), we can use directly the integral representation Eq. (12) to estimate the value of the integral by the saddle-point method. The electric field in the far field has a component of$~{{E}_{{}}}$, where,$~{{E}_{{}}}={{W}_{e}}~{{H}_{e}}$ and ${{w}_{e}}=\sqrt{\frac{_{e}}{{{\varepsilon }_{e}}}}$ - wave resistance of the medium in the external space. In the far zone, it can be represented in the form:

${{E}_{\theta }}={{G}_{e}}(r){{F}_{\theta }}(\theta )$                                           (28)

where, ${{G}_{e}}=\frac{{{e}^{-i{{k}_{e}}r}}}{4\pi r}.$

The resulting radiation pattern is:

${{F}_{\theta }}(\theta )={{\left. 4{{W}_{e}}\gamma (h)D\left( h \right) \right|}_{h=-{{k}_{e}}\cos \theta }}$                                           (29)

This model efficiently captures the complex interaction of the cylindrical antenna with the surrounding dielectric layers and neighboring plasma, resulting in radiation properties that are significantly influenced by the particular plasma conditions and the antenna's design across a broad spectrum of frequencies.

This study presents an efficient electrodynamic model employing the integra-functional equation method for evaluating cylindrical slotted antennas with dielectric and plasma coverings. Unlike conventional techniques, such as auxiliary sources, surface integral equations, and volume integral equations, it provides accurate external field computations without explicit internal plasma modeling, reducing computational complexity, especially in axial symmetry. The Drude model's incorporation reveals the antenna's radiation characteristics are significantly influenced by the relationship between the operating frequency and the plasma frequency. The radiation efficiency is maximized when the frequency is equal to or exceeds the plasma frequency, as this minimizes wave attenuation. Conversely, lower frequencies cause wave reflection and absorption, degrading performance. The research confirms that antenna curvature, determined by the cylinder radius, impacts radiation intensity and pattern, with larger radii leading to increased radiation levels due to a larger effective aperture. This insight is crucial for enhancing antenna designs in aerospace and defense, particularly for conformal antennas operating in plasma-rich environments. The model represents a substantial advancement for modern communication systems and stealth technologies which are contingent upon the effective functioning of such antennas.