A formalism is presented that unifies coherence, phase modulation, and dispersive propagation in Sagnac interferometers and is valid for large amplitude phase modulation. Several effects have been found as a result. For the fiber optic gyroscope, parasitic amplitude modulation effects that are maximum at the loop frequency have been found, in contrast to several effects that vanish when operating at the loop frequency. A correction to the gyroscope scale factor that is dependent on modulation amplitude and source bandwidth has also been found. In addition, there exists a maximum rotation rate limited by coherence effects, or maximum sensed phase shift for the Sagnac acoustic sensor. Phase noise effects caused by environmental or other disturbances can cause signal fade out and loss of information for the Sagnac data ring, and the magnitude of these effects are calculated. The Fm/Am conversion effect and its relevance for Sagnac interferometers is also treated. Finally, the use of a frequency shifting approach or various types of modulation to reduce these effects is discussed.