phononic filter might comprise alternating sections of narrow and wide bars, but simulations indicate that is difficult to achieve large acoustic impedance ratios, and the effect on power transmission is relatively small. More troublesome is the fact that the dominant phonon wavelengths at 100 mK are of order 2 $\mu$m, or shorter, and therefore optical lithography cannot be used easily to define steps that are highly abrupt on a scale size of $\lambda / 4$, diminishing the effectiveness of the filter.

An alternative approach is to make the legs wider and introduce periodic patterns of holes, thereby creating a truly phononic lattice [@PhononWaveBandgap; @ComprehensiveTopology; @HeatGuidingFocusing]. Such phononic crystals have been employed, for example, as support structures for micro-mechanical resonators, to reduce coupling loss due to elastic wave propagation to the substrate [@mohammadi2009high; @hsu2011reducing; @feng2014phononic]. Although this approach produces good filter characteristics, the number of transmission channels available, prior to the filter characteristic being applied, is high. Another way of thinking about this same problem is that the phononic lattice comprises a large number of low-dimensional links, each of which transports at least 4 modes. Thus the filter characteristic must compensate for the large increase in the number of underlying modes 