We demonstrate long-period fiber gratings whose attenuation can be changed by 25 dB over a 48-nm spectral band, with ambient-index changes of only 2.7 x 10(-4). To achieve this, the fiber waveguide is engineered to induce coupling between the core and a highly ambient-sensitive cladding mode with identical group velocities. The device schematic allows arbitrarily high index sensitivities to be achieved, which makes it an attractive platform for realizing sensors and modulators that respond to small index changes.