Читать книгу Fundamentals of Terahertz Devices and Applications - Группа авторов - Страница 20

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 2.5a. The power transmitted by the ellipse Pa and propagating in parallel to the z‐axis through the aperture area of dA_a = ρ′dρ′dϕ, is equal to the transmitted power, P_t, by the ellipse through the area ; where r_i is the distances between the first focus and the lens surface (see Figure 2.5b). Moreover, the transmitted power is related to the power incident, on the lens surface, P_i (i.e. neglecting the reflected power):

(2.38)

where P_a = S_adA_a and P_i = S_idA_i; S_a and S_i are the amplitude of the Poynting vectors of the aperture, and incident fields, respectively. Moreover, τ is the Fresnel coefficient in transmission, θ_i and θ_t are the incident and transmitted angles on the lens surface, respectively. By using the relations ρ′ = r_i sin θ, (2.38) can be simplified as:

(2.39)

The amplitude of the aperture and incident Poynting vectors can be expressed as:

(2.40)

(2.41)

where |E_a| is the amplitude of the electric field on the equivalent aperture of the lens, and |E_i| is the amplitude of the far‐field pattern of the incident electric field. Moreover, the term can be simplified as:

(2.42)

where . The term can be calculated as:

(2.43)

by substituting (2.43) in (2.42):

(2.44)

by substituting in (2.39):

(2.45)

For an elliptical lens with , one can represent cos θ_i and cos θ_t as:

(2.46)

(2.47)

by substituting (2.46) and (2.47) in (2.45), the amplitude of the field at equivalent aperture of the lens is related to the amplitude of the incident field as:

(2.48)

Therefore, the spreading factor can be defined as:

(2.49)

This is because, in the considered geometry (feed at the focus), the output GO phase front is planar and consequently has no power spreading.