Matrix Lie Group-Theoretic Design of Coupled Linear Optical Waveguide Devices

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The linear mapping effected by a coupled linear waveguide system on its input electromagnetic field can be defined by the system’s transfer matrix and this paper addresses the systematic design of the system’s internal dimensions to achieve any desired transfer matrix within a broad class. Manufacturing constraints limit the set of all realisable couplers and their transfer matrices can be used to define a Lie group, whose Lie algebra can be used to study the limitations imposed by such constraints. After having defined the group, the paper presents a design procedure that is guaranteed to realise any transfer matrix in the group by a finite concatenation of basic building-block coupler sections. This guarantee is afforded by a simple manipulation of canonical coordinate systems for suitable neighbourhoods of the group’s identity. The design procedure is illustrated for the group $\mathfrak{G}\mathfrak{U}( 2 )$, where the homomorphism with $\mathfrak{G}\mathfrak{D}( 3 )$ yields geometrical insight in terms of rigid rotations of the Poincaré sphere. A further example uses the procedure to realise a fully planar implementation of a $3 \times 3$ coupler of importance in optical sensing and signal processing.