In this paper, we investigate the spectral response of whispering-gallery-mode (WGM) resonators coupled to their access waveguide with a view to design their constitutive waveguides to promote critical-coupling over a wide spectral range and thereby facilitate their use for high-sensitivity sensing or nonlinear frequency conversion applications. The carried-out theoretical analysis is based on the universal response functions of singlemode and unidirectional devices. A coupled-mode treatment of the coupling region enables to derive two sets of favorable designs. The identified resonator/access waveguide systems exploit waveguides with mismatched propagation constants forming a coupling section exhibiting either an achromatic beat-length or an achromatic power-transfer coefficient. This generic model is followed by a numerical case study of vertically-coupled Si3N4 racetrack resonators. The conventional (quasi-)phase-matched configuration, treated as a reference case, is shown to display a critical-coupling bandwidth of 23 nm at a wavelength of 1550nm, whereas the proposed new designs demonstrate critical bandwidths larger than 330nm, i.e. exhibit bandwidths enhanced by more than one order of magnitude.
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