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Due to the limitations either on the sizes of devices and signal routingchannels, the current planar integrated optical waveguide circuits await forthe further developments into the three-dimensional (3D) integrations, althoughtheir designs and fabrications are still challenges. In this paper wedemonstrate an analytical method, basing on the invariant engineering, toovercome the complication in the usual method by numerically solving therelevant 3D coupled-mode equations for designing various 3D optical waveguidedevices such as the typical couplers. Our method is based on thequantum-optical analogy, i.e., the Maxwell equation for the electrcomagneticwave prorogating along the waveguide structure in the spatial domain isformally similar to the Schr\"odinger equation for the evolving quantum statein the time domain. We find that the spatial-domain invariants can beeffectively constructed to solve the 3D coupled-mode equations, analogously tosolve the dynamical evolutions of quantum systems in the time-domain. As aconsequence, as long as appropriately set the coupling parameters between the3D interconnected waveguides, the 3D three-waveguide couplers could be designedfor various desirably power divisions. As the invariant method is a naturalshortcut to the adiabaticity, the compacted devices designed by theinvariant-based engineerings are robust against the coupling coefficientvariations and the coupler lengths.

Three-dimensional compacted optical waveguide couplers designed by invariant engineerings