quarter wavelength super‐layer (Source: Modified from Llombart et al. [25]; John Wiley & Sons.
Another advantage of this configuration is that the leaky‐wave mode shows the same frequency behavior as the quartz super‐layer, even if a more directive feed is radiated inside the dielectric, and therefore will have similar antenna impedances and bandwidth. Figure 2.14 shows the simulated reflection coefficient using a full‐wave simulator of a square waveguide loaded by a double‐slot iris in the presence of the resonant air‐cavity. Several cases are shown in the figure, all of them with the same cavity and feed dimensions: an infinite silicon dielectric medium, a quartz quarter wavelength super‐layer, and a silicon lens on top of the cavity with and without coating. The infinite silicon medium and quartz quarter wavelength super‐layer present very similar reflection coefficients since they both present very similar cavity load impedances Zl. A comparison in the reflection coefficient with and without an anti‐reflection layer of εr = 3.45 is shown in Figure 2.14. Note the strong effect of the multiple reflections on the reflection coefficient when an antireflective coating is not used.
All in all, when employing LWA, one should consider different dielectric combinations, e.g. infinite quartz layer, quartz cavity, and silicon lens, depending on the trade‐offs between the bandwidth and the directivity [45].
2.4.2 Primary Fields Radiated by a Leaky‐wave Antenna Feed on an Infinite Medium
This part will explain how to calculate the radiated fields into an infinite medium of a certain permittivity starting from a known aperture distribution. These fields can be then used as the incident fields in the previously explained lens analysis. We will use the example of a leaky‐wave feed over silicon but it can be applied to any source or lens dielectric material.
Figure 2.15 Sketch of the leaky‐wave feed with its main parameters. The red grid represents the field aperture grid.
The basic geometry of the feed is shown in Figure 2.15. A waveguide exciting the fundamental mode mounted under an infinite ground plane with an iris. Above the iris, there is a resonant air cavity and a silicon slab that extends infinitely in z and the xy plane.
Using Love's Equivalence Principle, the electric fields computed at the aperture plane (z = h) can be used to compute the equivalent magnetic and electric currents outside of the surface where
The lens can be in the radiative near field or far field of the feed. In the case of the leaky‐wave feed, due to its high directivity, the lens will be in most cases in the radiative near field. In the radiative region of the near‐field, the field can still be represented by a local spherical wave with an angular distribution that depends on the distance, and therefore the same PO explained previously can still be used. The near and far‐field effect is illustrated in Figure 2.16, where the fields are computed over a sphere of two certain radius ρ = 4.5λ and ρ = 20λ. The far field is calculated independently of the lens geometry, as the dependency in r is eliminated (i.e. the term e−jkr/r). This dependency is added when the equivalent currents are computed over the lens. From the phase shown in the radiation patterns, we can see that the phase center is not in the plane of the iris ground plane, as shown in Figure 2.15, it is below and it varies with frequency. Thus, in the contrary to the design equations on the previous sections, the lens extension height L and radius R will need adjustment to maximize the aperture efficiency of the overall antenna. This aperture efficiency will be computed over an aperture of diameter D (see Fig. 2.17a), which is smaller than the lens radius R because of the high directivity of the leaky‐wave feed.
Figure 2.16 Amplitude and phase of the electric centered at a central frequency 550 GHz in the (a) far‐field and (b) near‐field at ρ = 4.5 λ of the leaky‐wave feed radiating over an infinite silicon medium.
2.4.3 Shallow‐lens Geometry Optimization
This part will cover the design of a leaky‐wave feed with an integrated silicon lens in the order of 4 − 20λ as described in [33]. We will start by choosing a leaky‐wave feed that is matched over 15% bandwidth and provides the highest directivity by using a resonant Fabry–Perot air‐cavity. For a desired aperture diameter, the procedure to design the overall integrated silicon lens dimensions, shown in Figure 2.17a is, as follows:
1 The waveguide, iris, and air cavity are optimized at the central frequency. After setting the air cavity thickness to λ0/2,the other dimensions of the waveguide and iris can be optimized with a full‐wave simulator for maximum radiation efficiency. The dimensions and reflection coefficient are shown in Figure 2.14 for a central frequency of 550 GHz. Next, the electric field components in the aperture plane are exported from the 3D simulator in order to perform the optimization of the lens geometry using the formulation previously explained.Figure 2.17 (a) Drawing of the basic parameters of the shallow lens antenna geometry. (b) Taper angle θf as a function of ρ for a shallow lens of diameter D = 2.5 mm and D = 5 mm, calculated at the central frequency of 550 GHz.Figure 2.18 Optimum (a) taper angle θf and (b) lens thickness W as a function of each diameter D that maximize the lens antenna performance using the procedure described.
2 Obtain the
