simply to break even.

We have taken a different approach based on few-mode elastic interferometers and ring resonators: An interferometer is formed by dividing a leg into two paths, one of which is longer than the other. Simulations based on multimode travelling wave calculations [@DJThesis] indicate that flux reductions of 25-75 % are possible, depending on the number of interferometers used in series. We have measured the thermal elastic attenuation length in SiN$_{\rm x}$ to be 20$\mu$m, and given that this is much larger than a typical wavelength, individual filters behave in a phase coherent way. Large series arrays of interferometers, however, become comparable to the attenuation length, and so operate in the diffusive to ballistic transition. To some extent, absorption isolates the effects of one interferometer on another. In other words, locally the structure behaves as a phase-coherent few-element filter, but globally, the structure conducts diffusively and behaves phase incoherently.

For the purposes of this paper, we define interferometers to be two-path elements that divide and recombine the travelling waves gradually. Another option is a ring resonator design, in which case one benefits from the modal scattering that takes place at the junctions, as well as the interferometric effects 