  We extend concepts from microwave engineering to thermal interfaces and
explore the principles of impedance matching in 1D. The extension is based on
the generalization of acoustic impedance to non linear dispersions using the
contact broadening matrix $\Gamma(\omega)$, extracted from the phonon self
energy. For a single junction, we find that for coherent and incoherent phonons
the optimal thermal conductance occurs when the matching $\Gamma(\omega)$
equals the Geometric Mean (GM) of the contact broadenings. This criteria favors
the transmission of both low and high frequency phonons by requiring that (1)
the low frequency acoustic impedance of the junction matches that of the two
contacts by minimizing the sum of interfacial resistances; and (2) the cut-off
frequency is near the minimum of the two contacts, thereby reducing the
spillage of the states into the tunneling regime. For an ultimately scaled
single atom/spring junction, the matching criteria transforms to the arithmetic
mean for mass and the harmonic mean for spring constant. The matching can be
further improved using a composite graded junction with an exponential varying
broadening that functions like a broadband antireflection coating. There is
however a trade off as the increased length of the interface brings in
additional intrinsic sources of scattering.
