Fundamentals of Terahertz Devices and Applications. Группа авторов

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approximation of the fields. The fields over the aperture are well‐approximated by the fields transmitted by the lens considering a locally flat surface to derive the transmission coefficients. The dyadic representation of the transmitted field to the incident field on the lens surface is:

(2.12)

(2.13)

where ζ₀ is the free space wave impedance, S(Q) is the spreading factor that ensures the power density within the solid angle of the feed is equal to the power density transmitted by the lens. is the transfer function calculated using the Snell's law. is the radiated field by the feed placed at the lower focus of the ellipse, at an observation point Q (see Figure 2.3b). The fields in (2.12) and (2.13) are propagating in the direction.

      Second is an approximation of the equivalent currents. The equivalent electric and magnetic currents are obtained from these GO fields:

      (2.14)

      (2.15)

      which are defined for ρ < D/2 with D = 2b (twice the minor axis of the ellipse) associated with the GO field domain, and zero outside this domain.

      The third approximation relies on the fact that we assume the incident field on the lens surface to have a spherical wave front, which is correct when the feed phase center is in the lower focus and the lens surface in the feed far field:

      (2.16)

(2.17)

      where is the medium wave impedance. The reference system used to evaluate this field is taken at the lower focus of the elliptical lens.

      Next, we will use these approximations to express the radiation in the far‐field of the elliptical lens, when illuminated from this incident field .

      2.2.2.1 Transmission Function

Let's now analyze an arbitrary point Q over a lens surface. The transmitted magnetic and electric field outside of the lens are related to the incident field through the transmission function as given in (2.17). In general, at each point Q, the incident field can be decomposed into parallel‐TM and perpendicular‐TE components:

      (2.18)

      where and are the parallel and perpendicular polarization vectors (see Figure 2.3b), and can be expressed in spherical coordinates for the on‐focus feed as:

      (2.19)

      (2.20)

      which brings to:

(2.21)

(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)

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