A spectrometer includes an interferometer having a first interference arm and a second interference arm to produce interference patterns from incident light. At least one of the interference arms includes a series of cascaded optical switches connected by two (or more) waveguides of different lengths. Each optical switch directs the incident light into one waveguide or another, thereby changing the optical path length difference between the first interference arm and the second interference arm. This approach can be extended to multi-mode incident light by placing parallel interferometers together, each of which performs spectroscopy of one single mode in the multi-mode incident light. To maintain the compactness of the spectrometer, adjacent interferometers can share one interference arm.