How do phase-shifting algorithms work?

Interference is an important phenomenon that plays a key role in understanding the properties of light. However, novice students have difficulties with optical interference experiments because fringe-pattern processing algorithms are not a topic of optics courses. The difficulty increases taking into account that the phase shift is usually induced manually and advanced phase-shifting algorithms must be applied. In this paper, the fundamental principles behind seven well-established phase-shifting algorithms are presented. First, five basic algorithms are analyzed: the three-, four-, and n-steps as well as two least-squares algorithms. Then, the advanced iterative algorithm and the generalized phase-shifting are studied. The operation of the algorithms is illustrated by a typical experiment on the Michelson interferometer. This work could be useful as an introductory lesson to the optics laboratory.