A tapered fiber with a depressed-index outer ring is fabricated and dispersion engineered to generate a widely tunable $(1250–1650\mathrm{nm})$ fundamental-mode leakage loss with a high cutoff slope $(-1.2\mathrm{dB}∕\mathrm{nm})$ and a high attenuation for stop band $(>50\mathrm{dB})$ by modification of both waveguide and material dispersions. The higher cutoff slope is achieved with a larger cross angle between the two refractive index dispersion curves of the tapered fiber and surrounding optical liquids through the use of depressed-index outer ring structures in double-cladding fibers.