We discuss the design of a compact ring resonator (RR) and Mach-Zehnder interferometer (MZI) in a low-refractive-index-contrast waveguide material system through the use of air trenches. A narrow air trench at the intersection of one input and two output waveguides can function as a high-efficiency splitter, while wider air trenches operate as waveguide bends. We first discuss the design of individual splitters and bends and then show how they can be used to realize a compact MZI and RR. The RR has a footprint of only , and its optical efficiency at the drop wavelengths is . The free spectral range and full width at half maximum are 7.2 and , while the factor is . The MZI occupies only , and its calculated optical efficiency is 90%.