  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.
