This book is a heretical analysis of multiple coupled ring resonators that actually offer an embryonic application as demultiplexing filters in DWDM of fiber optic transmission systems. The rings can be fabricated as integrated optical structures also they can be formed using micro or nano optical fibers. My analytical accession provides detail predictions with minimal computer resources in addition to construct single ring resonator using software which resolves the design and model of multiple ring resonators. In conclusion, I work out the compound ring resonator theory with complex filed equations to account for phase and amplitude.