  We propose a general method for designing wideband matching tapers in
inhomogeneous media where phase velocity is coupled to the taper impedance
profile. Such tapers are used to match wideband gradient index (GRIN) lens
antennas. To simplify fabrication tapers are often constrained to physically
uniform layers wherein commensurate line theory cannot predict the frequency
response. Therefore, we present a new design algorithm which derives an
effective permittivity $\epsilon_{eff}$ to equalize the electrical length of
commensurate and non-commensurate line tapers. The algorithm provides a
systematic design method with predictable frequency response for
non-commensurate line tapers. Nevertheless, there are several unavoidable
nonidealities present in such discretized tapers which we discuss and provide
recommendations for mitigation. The algorithm is used to design a Klopfenstein
taper with return loss better than 15 dB from 8 to 78 GHz. The design is
fabricated and measurements agree with simulation across the WR90, WR28, and
WR12 bands. An approximate efficiency formula is proposed which predicts
aperture efficiency of taper-matched lenses without the need for time-consuming
full-wave simulations. Various lenses are designed and compared to highlight
the advantages of Klopfenstein tapers in GRIN lens design. The results
demonstrate the usefulness of the proposed design method.
