Three-dimensional compacted optical waveguide couplers designed by invariant engineerings

Due to the limitations either on the sizes of devices and signal routing channels, the current planar integrated optical waveguide circuits await for the further developments into the three-dimensional (3D) integrations, although their designs and fabrications are still challenges. In this paper we demonstrate an analytical method, basing on the invariant engineering, to overcome the complication in the usual method by numerically solving the relevant 3D coupled-mode equations for designing various 3D optical waveguide devices such as the typical couplers. Our method is based on the quantum-optical analogy, i.e., the Maxwell equation for the electrcomagnetic wave prorogating along the waveguide structure in the spatial domain is formally similar to the Schr\"odinger equation for the evolving quantum state in the time domain. We find that the spatial-domain invariants can be effectively constructed to solve the 3D coupled-mode equations, analogously to solve the dynamical evolutions of quantum systems in the time-domain. As a consequence, as long as appropriately set the coupling parameters between the 3D interconnected waveguides, the 3D three-waveguide couplers could be designed for various desirably power divisions. As the invariant method is a natural shortcut to the adiabaticity, the compacted devices designed by the invariant-based engineerings are robust against the coupling coefficient variations and the coupler lengths.