Fundamentals of Terahertz Devices and Applications. Группа авторов

(2.22)

      For a feed placed in the center of the coordinate system, the propagation vector see Figure 2.3b inside and outside of the lens are defined as:

      (2.23)

      (2.24)

      The normal vector to the lens surface defined in cylindrical coordinates for an elliptical lens is:

      (2.25)

      Then, the transmitted field can be expressed as:

(2.26)

being τ^⊥(Q), τ^∥(Q) the Fresnel transmission coefficients which are evaluated assuming a locally flat boundary (see Figure 2.4):

(2.27)

(2.28)

Figure 2.4 (a) Scheme of the critical angle calculation on an elliptical lens. (b) Subtended critical angle vs permittivity of the elliptical lens.

where . Using (2.27) and (2.28) in the equation (2.26), and considering that and , we finally arrive at the expression of the transmitted fields at the equivalent aperture:

(2.29)

(2.30)

The spatial domain of (2.29 and 2.30) is determined by the GO approximation. In case of a full angular lens as depicted in Figure 2.4a, the GO transmitted fields will be zero after the critical angle. For a flat interface between a dense medium and free space, the angle of refraction θ_t increases as the incident angle θ_i increases (θ_t > θ_i). Total internal reflection (Γ = 1) occurs when the angle of refraction is 90° and the incident angle is known as the critical angle . After this angle there is no transmission, and therefore, this angle defines the domain of the equivalent aperture. As this angle lies at the center distance between the two focuses, the radius of the equivalent aperture is therefore the minor axis b. Thus, the fields in (2.29 and 2.30) are zero for a region outside the lens antenna aperture, i.e. ρ > D/2 with D = 2b in the case of a full angular lens. In the case of a smaller angular lens such as in [33], the spatial domain of the fields will be given by edge angle of the lens instead. Using the definition of the critical angle and that of the normal vector, , we can evaluate the subtended critical angle, θ₀, seen from the lens feeder in Figure 2.4a at the critical angle:

      (2.31)

      As an example, Figure 2.4b shows θ₀ as a function of the relative permittivity of the elliptical lens. It can be observed how the lower the permittivity, the smaller the angle is, and therefore, a more directive feed will be required to efficiently illuminate the lens above the critical angle.

      2.2.2.2 Spreading Factor S(Q)

Let's now analyze the power balance in an elliptical lens antenna. We can evaluate the incident power crossing an area A (see Figure 2.5a) by considering each ray incident on a point Q as a plane wave. The orthogonal and parallel components of the incident power are defined as follows:

      (2.32)

      (2.33)

      where S_i represents the amplitude of the Poynting vector. The same can be done for the reflected power towards inside of the lens:

      (2.34)

      (2.35)

      and for the transmitted power:

      (2.36)

      (2.37)

Figure 2.5 (a) Incident, reflected, and transmitted power density in a dielectric lens. (b) Referenced geometrical parameters.

      The relation between the incident, reflected and the transmitted power density is represented in Figure