Fig. 12: Design flow of the proposed discrete phase filtering based dispersive element.

a A continuous quadratic spectral phase variation of a 31.12-km SMF. b The resultant discretized and bounded within [0, 2\({\rm{\pi }}\)) phase levels with frequency resolution \({{\rm{\nu }}}_{r}\,\)= 10 GHz. Note, the amplitude response of the filter is assumed to be constant over the operation bandwidth (i.e., all-pass response)³⁷. The target spectral transfer function is implemented using a waveguide Bragg grating (WBG) structure. c The complex coupling coefficient profile (\({\rm{\kappa }}=\left|{\rm{\kappa }}({\rm{z}})\right|\exp \left(j{{\rm{\phi }}}_{{\rm{\kappa }}}(z)\right)\)) of the WBG; magnitude \(\left|{\rm{\kappa }}\left(z\right)\right|\) on the left and phase \({{\rm{\phi }}}_{{\rm{\kappa }}}\left(z\right)\) on the right. An inverse layer peeling algorithm is employed to calculate the complex coupling coefficient \(\left({\rm{\kappa }}\right)\) profile, \({\rm{\kappa }}\left(z\right)=\left|{\rm{\kappa }}\left(z\right)\right|\exp \left(j{{\rm{\phi }}}_{k}\left(z\right)\right)\) i.e., the strength and phase of the coupling induced between the forward and backward propagating modes per unit length along the WBG, that is required to achieve the target spectral response²²². A phase modulated apodization technique is employed to practically implement the target coupling coefficient²²³. d A phase modulated grating based apodization is employed to physically realize the target coupling coefficient by modulating the distances (\({d}_{i}\)) between adjacent corrugations while keeping the corrugation width (\(\Delta W\)) constant. \({\Lambda }_{0}\) is the nominal grating period. \(H\) and \({W}\) represent the waveguide height and width, respectively. e Schematic of the on-chip layout utilized for coupling light in and out of the WBG-based phase filter. The zoomed-in view shows the SEM image of one of the fabricated WBGs. The cross-sectional schematic of the fully etched silicon waveguide on top of the buried oxide is also shown. A Y-splitter collects the reflected signal from the WBG. A 20-µm linear adiabatic taper connects the input single-mode waveguide (W = 0.5 µm) with the 2-µm wide multimode waveguide. The transmitted signal from the WBG is terminated using a taper. Figures adapted from^37,222,223.