Читать книгу Fundamentals of Terahertz Devices and Applications - Группа авторов - Страница 17

In the following example, we will derive the canonical geometry (elliptical lens) that achieves a planar wave front at the aperture, starting from a spherical wave at its focus. This demonstration is based on a geometrical optics (GO) approach, or also called ray tracing. We start by imposing that all the rays in Figure 2.2a have the same electrical length:

(2.1)

Figure 2.2 Geometrical parameters of an elliptical lens.

where k₀ and are the wave numbers in free space and the dielectric mediums, respectively. Equation (2.1) can be rewritten as:

(2.2)

And we can also equate the projection in the z‐axis of these rays:

(2.3)

using (2.1) and (2.2) we conclude on the following equation:

(2.4)

On the other hand, we know that an elliptical lens geometry can be described in polar coordinates using the following expression:

(2.5)

where a and e are the semi‐major axis and eccentricity, respectively (see Figure 2.2b).

By recognizing in (2.5) that:

(2.6)

(2.7)

we can conclude that a spherical wave front produced by an antenna at the focus point of a lens of with an ellipsoidal shape will be transformed into a planar waveform. The antenna is placed at the second focus of an ellipse defined by the equation:

(2.8)

where b is the semi-minor axis of a ellipse. The eccentricity of the lens e, relates the geometric focus ellipse to the optical focus of the lens with the following relationship:

(2.9)

using Eqs. (2.7) and (2.8) we can derive the foci of the ellipse, defined as c as:

(2.10)

and the semi‐minor axis b as:

(2.11)

The radiation pattern obtained by elliptical lenses is the one that reaches the highest possible directivity when illuminated with a spherical phase front generated by the feeding antenna.