Double-material lenses were shown in [1] to be very good solutions for high permittivity
integrated scanning lenses. In that paper an example of this type of lenses was
demonstrated skipping the systematic lens design guidelines. This is the objective of the
present paper. There are two well know Geometrical Optics based methods to design
axial-symmetric scanning lenses: the Bifocal approach [2]-[3] and the Abbe sine
condition approach [4]. Both methods can be modified to design a double material
integrated scanning lens [1]. In an integrated lens an array of compact uniplanar printed
feeds is directly printed at the planar base of the lens instead of the classic single material
two-surface scanning lens configuration where the feed is located well outside the lens, in
air [2]-[4].
The Bifocal design approach is based on a marching procedure that starts from the lens
axis and successively calculates lens profile points until the lens edge is reached. The
resulting separation between consecutive points in each surface is dictated only by the
lens parameters. Unfortunately, for the specific case of double-material integrated lens
configuration, the Bifocal design procedure provides too few profile points of the lens to
allow acceptable accuracy of the surface shape definition. Conversely, in the Abbe design
approach, the lens profile can be calculated with arbitrary accuracy. Although the Abbe
implementation is the preferred approach from the calculation point of view, the
important relation between the feed off-axis position and the off-set output angle is
absent from the lens formulation. Therefore, once the lens profile is determined, an
analysis has to be performed to find this relation a posteriori and then redesign it if
needed. This inconvenient trial and error process is generally used.
The study presented in this paper allows to establish closed-form approximate
expressions that relate all the parameters that are relevant for the design of double
material axial symmetric integrated scanning lenses, thus enabling the design according
to the Abbe sine formulation in a single run.