Fundamentals of Terahertz Devices and Applications. Группа авторов

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      (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 the intrinsic impedance of free space.

      2.2.2.4 Lens Reflection Efficiency

One of the main differences of lenses w.r.t. reflectors is the fact that in a reflector all the incident field is transformed into a transmitted field as the surface can be modeled as a perfect electric conductor (PEC). However, in the case of lenses, the radiation principle is based on the refraction law; thus some incident energy is transmitted to the air but some part is reflected inside of the lens toward the top focus (see Figure 2.7a) [34]. GO/PO field approximation only includes the effect in the far field of the first transmitted rays. Part of the energy is actually reflected in the lens interface. This energy will eventually be radiated to the far field via multiple reflections or it will be lost in the lens material. In the GO/PO field analysis, this reflected energy is simply considered a loss in efficiency. That is typically a good assumption since the multiple reflections do not usually contribute to the radiated field main beam, as they have a random phase, but to the far side‐lobes. It means that the broadside gain of the antenna can be related as the directivity multiply by the radiation efficiency:

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