Abstract: A doped barrier region included in an optical phase shifter is disclosed. In one embodiment, an apparatus according to embodiments of the present invention includes a first region of an optical waveguide and a second region of the optical waveguide. The second region of the optical waveguide includes a higher doped region of material and a lower doped region of material. An insulating region disposed between the first and second regions of the optical waveguide is also included. A first portion the higher doped region is disposed proximate to the insulating region. A dopant barrier region is also included and is disposed between the higher and lower doped regions of the second region of the optical waveguide.

Abstract: One embodiment of the present invention determines equivalence between a first circuit and a second circuit. The first circuit represented by a first circuit function is decomposed into first and second decomposition circuits represented by first and second decomposition functions, respectively. The first circuit has a plurality of first primary inputs and first primary outputs. The plurality of the first primary inputs includes first and second primary input subsets. A reducing function of a reducing circuit is selected for the first decomposition function to reduce complexity of a first composition of the first circuit function and the reducing function. Equivalence is determined between the first composition with a second composition of the reducing function and a second circuit function of a second circuit. The second circuit has a plurality of second primary inputs and second primary outputs matching to the plurality of the first primary inputs and the first primary outputs, respectively.