A recently proposed concept suggests that a matched periodic modulation of both the refractive index and the gain/loss of the media breaks the coupling symmetry of the two co-propagating modes and allows only a unidirectional coupling from the i-th mode to j-the mode but not the opposite. This concept has been used to design a ring resonator coupled through a complex grating composed of both real (index) and imaginary (loss/gain) parts according to Euler relation: n = n0 exp(-jkx) = n0 (cos(kx) - j sin(kx)). Such asymmetrical coupling allows light to be coupled into the ring without letting it out. We present a detailed theoretical analysis of the ring resonator in the linear regime, and we investigate its linear temporal dynamics. Three possible states of the complex grating leads to the possibility of developing a dynamic optical memory cell where, for example, a data modulated train of optical pulses can be stored. This data can be accessed without destroying it, and can also be erased thus permitting the storage of a new bit. Finally, the ring can be used for pulse retiming.