  It has been demonstrated that dynamic refractive index modulation, which
breaks time-reversal symmetry, can be used to create on-chip non-reciprocal
photonic devices. In order to achieve amplitude non-reciprocity, all such
devices moreover require modulations that break spatial symmetries, which adds
complexity in implementations. Here we introduce a modal circulator, which
achieves amplitude non-reciprocity through a circulation motion among three
modes. We show that such a circulator can be achieved in a
dynamically-modulated structure that preserves mirror symmetry, and as a result
can be implemented using only a single standing-wave modulator, which
significantly simplifies the implementation of dynamically-modulated
non-reciprocal device. We also prove that in terms of the number of modes
involved in the transport process, the modal circulator represents the minimum
configuration in which complete amplitude non-reciprocity can be achieved while
preserving spatial symmetry.
