Fig. 2: Implementation of signal-combining elements (SCEs) with closed-loop phase stabilization.

a Conceptual setup of an SCE relying on a 120^∘ optical hybrid (OH), which is implemented as 3 × 3 multi-mode interference (MMI) coupler with one unused input port. The MMI superimposes the input signals with relative phases of −2π/3, 0, and 2π/3 at the various output ports. Two overlapping spectral slices \({\underline{a}}_{{\rm{in}},1}(t)\) and \({\underline{a}}_{{\rm{in}},2}(t)\exp \left({\rm{j}}\Delta \varphi (t)\right)\) with phase error \(\Delta\varphi(t)\) are combined at the “zero-phase” port, leading to the optical output signal \({\underline{a}}_{{\rm{out}}}(t)\). The error signal \(U_{\rm{err}}(t)\) is generated by connecting the remaining two output ports of the 120^∘ OH, which are associated with relative phases of −2π/3 and 2π/3 between the input signals, to a low-speed balanced photodetector (BPD). Upon low-pass filtering, the resulting BPD output signal \(U_{\rm{err}}(t)\) is essentially proportional to phase error \(\Delta\varphi(t)\) for a linear approximation close to the desired operating point \(\Delta\varphi=0\) see Eq. (6) and Section S1.1 in Supplementary information 1. A proportional-integral (PI) controller is used to drive the phase shifter (PS) and to compensate for the measured phase error \(\Delta\varphi(t)\). b Alternative SCE implementation using a 90^∘ OH as passive combiner, e.g., implemented as a 4 × 4 MMI with two unused input ports. The targeted output signal is found again at the “zero-phase” port, and the error signals are derived by a BPD connected to the “ − π/2” and the “π/2” ports. The remaining output port “π” is unused in this implementation