to develop and compare manufacturing techniques using optical lithography (OL) and electron beam lithography (EBL); (iii) to investigate whether TESs with phononic filters behave in a conventional way; (iv) to determine whether thermal conductance can in practice be reduced significantly below the few-mode quantised limit; and (v) to investigate uniformity in performance between notionally identical devices. The experimental work was based solely on SiN$_{\rm x}$ membranes, but the results give direct information about the likely behaviour of phononic devices based on crystalline Si membranes.

Theory
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Elastic waves and ballistic thermal power {#subsec:Theory_Elastic}
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The thermal flux through a uniform, low-dimensional, ballistic, dielectric bar can be calculated directly from the dispersion curves of the discrete elastic modes [@DJBallistic]. Here we summarise the calculation because it is central to the subject matter of the paper, and because the ballistic limit will be used later for normalising experimental data.

The classical elastic wave equation is $$\rho\omega^2 u_i +C_{ijkl}\frac{\partial^2 u_k}{\partial x_j \partial x_l} = 0,
\label{eq:ElasticWave}$$ where $u_i$ is the displacement field in Cartesian direction $i$, $C_{ijkl}$ the fourth-rank stiffness tensor, $\rho$ the mass density, $\omega$ the angular frequency, and the Einstein summation notation has been assumed. Equation (\[eq:ElasticWave\]) can be solved by 