Exploring the fundamental limits of integrated beam splitters with arbitrary phase via topology optimization

Optical beam splitters are essential for classical and quantum photonic on -chip systems. In integrated optical technology, a beam splitter can be implemented as a beam coupler with two input and two output ports. The output phases are constrained by the conservation of energy. In lossless beam splitters, the phase shift between the output fields is pi and zero for excitation from the first and second input ports, respectively. Therefore, for excitation from both inputs, the phase between the output fields, defined as beam splitter phase (BSP), is pi. The BSP leads to several phenomena, such as the quantum interference between two photons, known as the Hong-Ou-Mandel effect. By introducing losses, BSP values different than pi become theoretically possible, but the design of 2 x 2 beam couplers with an arbitrary phase is elusive in integrated optics. Inspired by the growing interest on fundamental limits in electromagnetics and inverse design, here we explore the theoretical limits of symmetrical integrated beam splitters with an arbitrary BSP via adjoint-based topology optimization. Optimized 2D designs accounting for fabrication constraints are obtained for several combinations of loss and phase within the theoretical design space. Interestingly, the algorithm does not converge for objectives outside of the theoretical limits. Designs of beam splitters with arbitrary phase may find use in integrated optics for quantum information processing.