Fundamentals of Terahertz Devices and Applications. Группа авторов

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.

      2.2.2.3 Equivalent Current Distribution and Far‐field Calculation

The equivalent current distributions on the aperture, using the incident field is defined as in (2.21 and 2.22) and the phase relation derived in (2.1), can then be written as follows:

      (2.50)

      (2.51)

      where r ′ (θ′) is given by (2.5) and tan θ′ = ρ′/z′ where . The vectorial components of these equivalent current distributions can be also expressed in Cartesian coordinates using and .

With these equations, we have determined the surface currents on the lens aperture, and thus, the far field patterns in the reference system used in Figure 2.6 can be obtained using the following expressions:

      (2.52)

Figure 2.6 Reference system for the evaluation of the far fields radiated by the elliptical lens antenna.

      (2.53)

      where and ; and η₀ is