Spectral behavior of integrated distributed-feedback resonators utilizing a distributed phase shift

Bragg-grating-based distributed-feedback waveguide resonators, with a discrete phase shift introduced inside the Bragg grating, exhibit within their grating reflection band a Lorentzian-shaped resonance line with an ultranarrow linewidth. If the phase shift is pi/2, the resonance is located at the center of the reflection band, i.e., at the Bragg wavelength, where the grating reflectivity is maximum, hence the resonance linewidth is minimum. Alternatively, the required pi/2 phase shift is often introduced by a distributed change in effective refractive index, e.g. by adiabatically widening the waveguide. Despite careful design and fabrication, the experimentally observed resonance wavelength deviates from the designed one. Besides deviations owing to fabrication errors, a fundamental, systematic shift towards shorter wavelengths occurs. We show theoretically and experimentally that the decay of light intensity during propagation from the phase-shift center into both sides of the Bragg grating due to (i) reflection by the periodic grating and (ii) the adiabatic refractive-index change causes an incomplete accumulation of designed phase shift by the oscillating light, thereby systematically shifting the resonance to a shorter wavelength. Calculations are performed based on the characteristic-matrix approach. Experimental studies are carried out in distributed-feedback channel-waveguide resonators in an amorphous aluminum oxide thin film on silicon with a distributed phase shift introduced by adiabatic widening of the waveguide according to a sin(2) function. Calculations and experiments show good agreement. Considering in the design the overlap integral between distributed phase shift and light intensity provides a performance that is much closer to the desired value.