<p><b>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.</b></p>