  The properties of the square array of coupled Microring Resonators (MRRs)
with interstitial rings are studied. Dispersion behavior of the interstitial
square coupled MRRs is obtained through the transfer matrix method with the
Floquet-Bloch periodic condition. Analytical formulas of the eigen wave
vectors, band gaps and eigen mode vectors are derived for the special cases of
the interstitial square coupled MRRs array with identical couplers and the
regular square coupled MRRs array without the interstitial rings. Then, the
eigen modes' field distribution are calculated for each of the four eigen wave
vectors for a given frequency through the secular equation. Finally, numerical
simulation is performed for an interstitial square coupled MRRs array with
identical couplers and a regular square coupled MRRs array. The simulation
result verifies the analytical analysis. Finally, the loaded quality factors of
the interstitial 5-ring configuration, the regular 4-ring configuration and the
1-ring configuration are obtained. It is found that the loaded quality factor
of the interstitial 5-ring configuration is up to 20 times and 8 times as high
as those of the 1-ring configuration and the regular 4-ring configuration
respectively, mainly due to the degenerated eigen modes at the resonant
frequency. Thus, the interstitial square coupled MRRs array has the great
potential to form high-quality integrated photonics components, including
filters and resonance based sensing devices like the parity-time symmetric
sensors.
