Synthesis of a 4 $\times$ 4 Walsh–Hadamard Transformer Using Long-Period Waveguide Grating Arrays

Synthesis of discrete unitary transformations using long-period grating arrays are analyzed. Matrix analysis of the coupled-mode equations is used to derive the conditions for the generation of unitary transformations on coherent spatial optical signals. General expressions for the transform matrix are derived. In particular, the grating parameters for the synthesis of a 4 times 4 Walsh-Hadamard transformer are derived. A design example with device parameters is also presented.