Читать книгу Fundamentals of Terahertz Devices and Applications - Группа авторов - Страница 23

While elliptical lenses reach the highest possible directivity, extended hemispherical lenses are often employed to ease their fabrication. This alternative configuration can have a close performance to the elliptical when the primary feed is designed in such a way to excite only the upper part of the dielectric lens and a dense material is used. The dimensions of the lens are chosen to approximate the focusing properties of the elliptical lens when the feed is placed at the foci. The synthesis of the ellipse is achieved using a semi‐hemisphere of unit radius defined as:

(2.55)

and an extension defined by L (see Fig. 2.9). The distance b + c should be equal to L + 1 so both lenses are overlaid. Thus, the extension height L can be defined as:

(2.56)

It can also be synthesized the other way around, from an extended semi‐hemispherical lens defined with the R and L, we can obtain the equivalent ellipse with the following expressions:

(2.57)

Figure 2.9 Synthesis of an elliptical lens from an extended hemispherical lens for silicon, fused silica and PTFE. Note that the extended hemisphere is a good approximation of the elliptical lens at high dielectric constants.

(2.58)

(2.59)

Depending on the dielectric constant, the fitting of the elliptical lens and the extended hemispherical lens varies. Three examples are shown in Figure 2.9 where both lenses of silicon (ε_r ≈ 11.9), quartz (ε_r ≈ 4.5) and polytetrafluoroethylene (PTFE) (ε_r ≈ 2). The rays from a broadside plane wave incident onto the extended semi‐hemispherical lens, as opposed to the elliptical lens, do not come to a point focus, thus, which means that the lens introduces an aberration to the radiated fields. But as we will see in the following sections, even if it does not couple well to a planar equi‐phase front, the extended hemispherical lens will couple well to a Gaussian‐beam system.