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 nonlinear dispersions using the contact broadening matrix Γ(ω), 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 Γ(ω) equals the Geometric Mean of the contact broadenings. This criterion 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 criterion 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.