  We theoretically investigate advanced multi-functional integrated photonic
filters formed by three waveguide coupled Sagnac loop reflectors (3WC-SLRs). By
tailoring the coherent mode interference, the spectral response of the 3WC-SLR
resonators is engineered to achieve diverse filtering functions with high
performance. These include optical analogues of Fano resonances that yield
ultrahigh spectral extinction ratios (ERs) and slope rates, resonance mode
splitting with high ERs and low free spectral ranges, and classical
Butterworth, Bessel, Chebyshev, and elliptic filters. A detailed analysis of
the impact of the structural parameters and fabrication tolerances is provided
to facilitate device design and optimization. The requirements for practical
applications are also considered. These results theoretically verify the
effectiveness of using 3WC-SLR resonators as multi-functional integrated
photonic filters for flexible spectral engineering in diverse applications.
