Читать книгу Advanced Antenna Array Engineering for 6G and Beyond Wireless Communications - Richard W. Ziolkowski - Страница 26

A number of different types of lens antennas operating in the mm‐wave and THz bands have been reported [21–25]. These include the elliptical lens, extended hemispherical lens, and Fresnel zone lens. Each has its own unique physical and performance characteristics.

A homogeneous elliptical lens has two focal points. It can transform the radiation pattern of a feed placed at one focal point into a plane wave exterior to it propagating in the direction of the second focal point. Assuming a represents the major semiaxis, b represents the minor semiaxis, L represents the distance between the focal point of the feed to the centre of the ellipsoid, and n is the index of refraction of the dielectric from which the lens is fabricated, one has the following relationships:

(1.1)

(1.2)

An integrated elliptical lens antenna is obtained by cutting off the part of the dielectric below the bottom focal point and placing the feeding antenna beneath it. As depicted in Figure 1.10, only rays that hit the surface of the elliptical lens above the plane of its maximum diameter, denoted herein as its waist, are collimated. The portion of the radiated fields intersecting the lens below its waist is not collimated, but rather propagates along undesired directions or excites surface wave modes, thus giving rise to side lobes or other perturbations in the lens’ radiation pattern [20]. One solution to solve this problem is to control the beamwidth of the feed in order that the majority of its radiated energy falls within the angular range above the waist of the lens.

Figure 1.10 Illustration of (a) an integrated elliptical lens antenna and (b) an extended hemispherical lens antenna.

Another issue arising from the internal reflections at the surface of an elliptical lens is the matching of the feed. One inherent characteristic of elliptical lenses is that all of the reflected rays that pass through the second focal point are reflected back to the first focal point. This reflected power causes a substantial mismatch to the feed impedance. A classical method to address this issue is to enclose the elliptical length with a matching shell that is a quarter‐wave thick. The shell dimensions are specified according to the following equations:

(1.3)

(1.4)

where n₁, n₂, and n_match represent the refraction indexes of the lens, the air, and the matching shell. The main drawback of this approach is that the improved matching performance can only be maintained within a relatively narrow bandwidth. To improve the bandwidth, one can incorporate multiple consecutive matching layers to perform a gradual transition between the two dielectric constants across each interface.

Since the collimation from an elliptical lens only occurs for the portion of the wave front that impinges on its front surface, the part below its waist can be replaced with a cylinder. Furthermore, the top elliptical part, the hemi‐ellipse, can be approximated by a hemisphere. This modification significantly reduces the fabrication complexity. The difference in the height of the hemi‐ellipse and the hemisphere can be compensated by the height of the cylindrical extension. This new lens is known as an extended hemispherical lens. It turns out to be a rather good approximation to a true elliptical lens, although it tends to present a slightly lower directivity compared to one having the same diameter. The relationship between the radius of the hemisphere, R; the height of the cylinder under it, L; and the refraction index of the lens material, n, is given by

(1.5)

A lens similar to the extended hemispherical lens is known as a hyper hemispherical lens. In contrast to Eq. (1.5), the cylindrical extension length is now given by [23]:

(1.6)

The rays at the output of the hyper hemispherical lens are not collimated. Therefore, the beam that it generates is much broader than that of the extended hemispherical lens. Nevertheless, it does sharpen the beam radiated by the feed antenna and increases its gain by a factor of n². However, unlike the collimating lenses, the directivity of this lens does not increase with the lens size, i.e., its aperture size. The hyper hemispherical length satisfies the Abbe sine condition so the lens itself is free from coma aberration when the feed is transversely displaced from the lens axis [25]. Therefore, it is well suited for beam steering.

It must be noted that Eqs. (1.5) and (1.6) are based simply on geometrical optics and point source assumptions. For actual antenna designs at mm‐wave and THz frequencies, optimization of the cylindrical extension would be required to achieve the optimal radiation performance [26, 27].

Another type of lenses for mm‐wave and THz operations is the Fresnel lens [28]. A Fresnel lens consists of a number of alternately transparent and opaque half‐wave zones. The source of the antenna is placed at its focal point. The opaque zones are attained by covering the corresponding portions of the lens with conducting or absorbing materials. Figure 1.11a shows a circular Fresnel lens. It consists of a series of zonal openings in a finite conducting sheet. To increase the antenna efficiency and reduce the sidelobe level, one can introduce phase correcting elements into the zones as depicted in Figure 1.11b [29]. The resulting radiator is effectively a transmit array.

Figure 1.11 Illustration of Fresnel lenses. (a) Original Fresnel lens. (b) Circular phase correcting version [29].

Source: Modified from [25] / IEEE.

One salient advantage of the Fresnel lens is its low profile. On the other hand, its main disadvantage is its relatively narrow bandwidth. Nevertheless, substantial progress has been made to increase the bandwidth of transmit arrays in recent years [30]. It should be pointed out that although a Fresnel lens can be made flat, it still needs a feed typically placed many wavelengths away from the lens. If the required beamwidth is not too narrow, one can achieve a completely flat version, i.e., a metasurface‐based antenna that is created by placing a metasurface above an antenna backed by a ground plane [31].