Shaped Omnidirectional Reflector Fed by a Dielectric Lens Associated with a Coaxial Feed Horn

Te paper explores an omnidirectional antenna confguration composed of a refector fed by a shaped dielectric lens associated with a coaxial TEM horn. A simple formula describes the lens shape obtained by applying Fermat’s principle to control the rays’ caustic emerging from the dielectric interface. Based on geometrical optics (GO) principles, a synthesis technique defnes the refector shape to control the antenna radiation pattern in the vertical plane. Concatenated conic describes the refector generatrix. Te study employs a full-wave analysis to validate the designs and explores the proposed confguration to attend two distinct far-feld specifcations.


Introduction
For operation in millimeter waves, shaped omnidirectional refector antennas can provide efcient anikd broadband communications with compact confgurations. Design examples with a single and dual refector have been explored for omnidirectional coverages [1][2][3][4][5]. Tese antenna design examples are usually fed by a small aperture horn that generates radiation patterns with large beamwidth [1][2]. Consequently, a single refector confguration requires refectors with large diameters to minimize the spillover effects. Dual classical geometries successfully control peak radiation patterns and provide compact solutions [4,5]. One of the major features of a dual omnidirectional confguration is the possibility of adjusting the position of the real or virtual caustic generated by the subrefector to minimize the main refector diameter, leading to more compact designs when compared with the single case with a similar aperture size [3]. Based on geometrical optics principles, the authors in [4,5] show a shaping procedure for omnidirectional dual refector antennas where an axis-displaced conical section describes desubrefector and the main refector GO shape controls the radiation pattern in the vertical plane. Tey show confgurations that difer by the ray structure emerging from the main refector, presenting a real or a virtual caustic. For both cases, the value for grazing incidence on the main refector for rays emerging from the subrefector and the conditions for main refector blockage limit the large-angle coverages. As observed for the virtual caustic case, the coverage angle and the shaping increase the main refector dimensions.
Dielectric lenses associated with a primary feed ofer the advantages of mechanical rigidity, wide-band capabilities, and low dissipative loss. For single surface-lenses, their profle can be either canonical or shaped to satisfy far-feld radiation pattern specifcations [6][7][8]. Te usual shaping techniques based on geometric optics (GO) principles are formulated by imposing a power conservation on a ray tube and the Snell refraction law to control the lens radiation pattern [7][8][9]. Alternatively, the lens shape can also be obtained by applying Fermat's principle to control the rays' caustic emerging from the dielectric interface [10].
For operation in local multipoint distribution services in millimeter waves at 30 GHz, here, we explore an alternative confguration for omnidirectional antennas where a circularly symmetric dielectric lens illuminates a circularly symmetric main refector, as illustrated in Figure 1. A TEM coaxial horn feeds the lens and provides vertical polarization. Te antenna radiation pattern in the vertical plane is obtained by applying a shaping technique based on GO assumptions and considering the system horn-lens as a point source at a virtual focus. Te rays emerging from the shaped lens interface creates a virtual focus behind the feed, narrows the lens radiation pattern that illuminates the refector, and reduces the required refector diameter and, consequently, the antenna's overall volume. Compared with the dual confgurations, the refector on the top of the lens avoids the grazing incidence and allows a large coverage angle without blockage. Te material is presented as follows: Section 2 ofers a simple formula to describe the lens generatrix based on Fermat's principle and the lens radiation pattern. Section 3 presents the GO refector shaping method assuming the new virtual feed focus. Te resulting refector generatrix is represented by concatenated local conic sections [11]. Section 4 presents two design examples that are validated by employing the full-wave analysis provided by the CST Studio Suite. Figure 2 shows the basic dimensions of the circularly symmetric dielectric lens fed by a point source at the origin O of the Cartesian system. Te base of the dielectric lens is planar and coincides with the plane z � 0. Te lens dimensions are larger than the wavelength, and we used GO approximations to shape the lens and produce a spherical wavefront with a center at the virtual focus at P(0, −Z 0 ). By applying Fermat's principle to the ray path from the source point at the origin to the dielectric interface, the distance r 0 (θ) has to satisfy the following equation:

Lens Design
where r 0 (θ) e r p (θ) are the distance of the point S on the interface to the points O and P, respectively, n is the refraction index of the dielectric, Z A defnes the size of the lens along the symmetry axis (z axis), and c � Z A (n − 1) − Z 0 . Te solution of (1) leads to the following expression for the generatrix of the dielectric-air interface (see Appendix 1): Te lens surface is circularly symmetric and obtained by rotating the generatrix around the Z-axis, the antenna symmetry axis. Te direction of the incident ray and refracted rays is related as follows: Te desire for a compact lens requires the reduction of Z A , bringing, as a consequence, the increase of the surface curvature and total refections of the rays incident at a larger angle (θ > θ C ). To avoid this limitation, Z A has to satisfy the following condition Z A > Z 0 /(n 1).
Te GO refector synthesis supposes the refector is illuminated by a spherical wave with a phase center at P and requires an analytical representation of the radiation intensity G L (α). By applying the power conservation principle along an elementary ray tube, G L (α) can be expressed by comparing the transmitted power from the source point to an elementary external surface of the lens in the direction θ with the power radiated in the direction α, yielding:  2 International Journal of Antennas and Propagation where T(θ) is ratio of the transmitted power to the incident one on the interface and R is the Fresnel local refection coefcient for the parallel component of the incident electric feld [8].
To illustrate the lens performance, we chose a TEM coaxial horn with dimensions similar to the design described in [12]. We considered it immersed in the dielectric material employed for the lens (dielectric constant is ε � 2.56 and n � 1.56), as shown in Figure 3. Consequently, the horn has internal and external aperture radii r a � 0.45λ d � 0.2815 cm and r b � 0.9λ d � 0.5625 cm, respectively, where λ d is the wavelength inside the dielectric (see Figure 2 in [12]). As the aperture dimension is relatively small, the phase center of the horn's far-zone radiation is very close to the origin O, at the plane z � 0, as supposed in the formulation. Te horn radiation pattern model is expressed as follows: where G F is the normalization factor, J 0 (.) is the Bessel of order zero, and k d is the wave number inside the dielectric (see Appendix 2). Figure 4 shows the CST horn radiation pattern in the absence of the lens interface and compares it with the model described by (6). Te lens design supposes the rays emerging from the lens interface with a virtual focal point at Z 0 � 3.5 cm on the symmetry axis. To avoid any critical incidence on the interface, the lens size Z A � 6 cm is adjusted to satisfy the condition Z A > Z 0 /(n 1). Tese lens dimensions make the rays radiated in the semispace z > 0 and concentrate in the solid angle defned by α < 39.6°, reducing the width of the main lobe, as illustrated the GO radiation in Figure 5. Te diameter of the metallic base in the plane z � 0 is adjusted to minimize the radiation in the semispace z < 0 and avoid interference with the refector radiation. Figure 5 compares the lens radiation patterns given by the CST and the GO approximation calculated from (4). As observed, the main lobe shows good agreement, and the minor diferences are due to higher-order modes generated by refections on the dielectric interface. Due to a small radiation phase error in the main lobe region, we consider a displacement of 0.1 cm in the lens focus towards the negative z-axis. Figure 6 compares the return loss at the horn 50 ohms port with and without the lens.

Reflector Shaping
Te circularly symmetric refector surface is illuminated by a spherical wavefront with a phase center at P with a radiation pattern G L (α), as described by (4), and α ∈ [0, α Ν ] defnes the tube of rays incident on the main refector. Te refector generatrix is shaped to radiate a prescribed vertical pattern G B (β) in the far-feld region of the antenna, where β is the direction of observation relative to the z-axis, and β ∈ [β 0 , β Μ ] defnes the tube of rays refected by the main refector.
Tis work follows the GO shaping technique employed in [11], where the generatrix is described by a series of concatenated conical sections M m (m � 1,...,M), as illustrated in Figure 7. Te sections are sequentially concatenated to each other and all of them with one focus at P 0 . Te angles α m−1 and α m limit the conic section M m , and its axis has an elevation angle c m with respect to the z axis. r S represents the distance between P and a point at M m and is expressed as follows: and 2c m and e m are the interfocal and eccentricity of M m , respectively. By applying the Snells Law of refection on the polar equation of M m , one derives the relation between the incident and refected directions of the ray [11].
Te coefcients a m , b m , and d m defne the section M m , and the following iterative procedure obtains them. By using (9), the coefcients b m and d m are obtained by imposing the known mapping condition to the rays at the extremes of the section m, (β n−1 , α n−1 ), and (β m , α n ). To ensure the continuity of the surface, the value of a m is obtained from values of r Sm−1 and α n−1 determined in the previous step (m − 1).

International Journal of Antennas and Propagation
For the frst section, the refector vertex V 0 defnes the initial parameter a 1 . As observed, the iterative technique requires the previous defnition of the relation between the angle α m and β m , the angles of the incident, and refracted rays at the extremes of each section, respectively. Tis relation is obtained by applying the conservation of energy principle within a ray tube that relates the radiated power density incident at the dielectric on the interface and the refracted power density specifed in the far-feld.
N 0 is the normalization factor to impose energy conservation within the tube of rays emerging from the lens and the far-feld coverage. As described in [11], the synthesis technique is fast, and 25 sections can accurately describe the surface for the examples shown in the design.

Design Examples
To illustrate the performance of the proposed omnidirectional antenna confguration, we explore the feed lens design described in Section 2 to illuminate two types of circularly symmetric refectors. Case I considers the refector obtained by spinning a parabola section around the symmetry axis (z-axis). Te parabola focal point is at P at the symmetry axis and coincides with the virtual focus of the rays emerging from the lens interface, as shown in Figure 2. It transforms the spherical wavefront emanating from the point P into a conical wavefront at the antenna aperture (see Figures 1 and 2) and axis pointing at β 0 � 102°.   To illuminate the refector, we consider the rays within the cones with semiangle θ C � 55°and α C � 31.4°, which provide a refector edge illumination lower than −10 dB and minimize the spillover above the horizon. For comparative purposes, the vertex distance from the origin is V 0 � 7.4 cm and is adjusted to make the volume of the truncated cone that circumscribes the antenna close to the volume of the Option I design presented in [4]. As illustrated in Figure 8, it yields a parabola with a focal distance of F � 6.644 cm, antenna dimensions D M � 20.58 cm, H � 13.344 cm, and an aperture width of W A � 7.95 cm, larger than the designs presented in [4]. Table 1 lists the main antenna parameters together with the design options in [4], where is possible to observe that the use of a dielectric lens to illuminate the refector leads to a smaller high (H) and a larger refector radius (D M ).
From the results of the CST full-wave analysis, Figure 9 shows the antenna radiation patterns in the vertical plane, and Figure 10 shows a near-feld map that helps to understand the radiative behavior of the antenna and estimate the interaction with nearby objects [13]. In Figure 10, it is possible to observe the conical wavefront emerging from the refector towards the β � 102°direction. Te main lobe has an 11.12 dBi peak radiation pattern at 102.25°and includes the refection losses at the dielectric interface. Compared to the dual refector cases in [4], it shows a gain close to the provided by Option II and 1.2 dB higher than Option I design and compactness similar to Option I. Te main diferences appear in the sidelobe region where the lens  International Journal of Antennas and Propagation 5 refector radiation pattern shows lower sidelobe levels close to the z-axis negative, while displaying higher refector spillover above the horizon. Case II employs the feed lens system described in Section 2 to illuminate a refector shaped to generate a cosecant squared radiation pattern G B (β) in the elevation plane. For the refector synthesis, G B (β) is expressed as follows: As observed in Figure 11, the rays emerging from the refector are concentrated within the space region defned by 95°<β < 135°in the vertical plane and have a virtual caustic. One critical point of the design is the ray incident at the refector vertex and refected in the direction β � 135°,  International Journal of Antennas and Propagation passing close to the dielectric lens and the metallic base. To avoid lens and metallic base blockage in the optical sense and minimize the refector lens interaction, we move the refector vertex upwards by making V 0 � 7.9 cm, including the horn-lens phase center displacement. Te synthesis procedure considers the rays within the cone with a semiangle θ C � 55°to illuminate the refector, as explained in the previous example. Figure 11 shows the resulting shaped refector with dimensions D R � 20.57 cm and H � 13.36 cm.    International Journal of Antennas and Propagation and compares them with the dimensions of Case IIA described in [5].
From the CST analysis, Figure 12 shows the near-feld map and Figure 13 shows the antenna radiation pattern. It is observed a peak directivity of 9.16 dBi at β � 99°and small oscillations around the desired cosecant-squared pattern due to the interaction of the lens felds with those emerging from the refector. Diferent from the previous case, the wavefront emerging from the refector shows a curvature to generate the cosecant-squared radiation pattern. Compared with the omnidirectional dual refector shaped for a cosecant-squared radiation pattern, using a lens allows the design for compacter confguration [5], as observed in Table 2, and it also avoids the grazing incidence when the main refector is placed below the feed-subrefector system, as observed in [5].

Conclusions
Te paper has explored an omnidirectional antenna confguration composed of a refector fed by a shaped dielectric lens associated with a coaxial TEM horn for vertical polarization. For LMDS applications around 30 GHz, two successful design examples were presented and analyzed using a full-wave approach. Tey showed results comparable to other types of omnidirectional antenna confgurations. For each case, geometrical limitations were outlined, such as compactness and the conditions to avoid lens blockage.

B. Horn Radiation Pattern
Te expression (6) can be obtained by computing the radiated feld of an annular aperture with inner radius r a and outer radius r b , and illuminated by the TEM mode (see Figure 2 in [13]). For the task, the aperture felds are replaced by equivalent magnetic current on the annular aperture: Te electric vector potential in the far-feld region is calculated by integrating the magnetic aperture currents  Te electric vector potential has an azimuthal component given by

Data Availability
All data supporting the results were generated during the study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.