Analytical description of photonic waveguides with multilayer claddings: Towards on-chip generation of entangled photons and Bell states

We present a theoretical description of slab and ridge waveguides with multilayer claddings, such as Bragg reflection waveguides (BRWs), using analytical expressions based on Fresnel and Airy formulas. This approach simplifies the waveguide calculations in comparison with more traditional techniques such as the transfer matrix method or direct stitching of plane-wave solutions at layer interfaces. Approximate but simple and straightforward analytic relations describing the effective indices and group velocities of guided modes in arbitrary 1D BRWs are derived. The formalism is then extended to 2D ridge waveguides using the effective index method. The approach is employed to engineer BRWs where several types of phase matching are present simultaneously at the same wavelength, as well as BRWs where photons generated by spontaneous parametric downconversion have the maximal degree of polarization entanglement. These results promote the use of BRWs as on-chip entangled photon sources, and facilitate on-chip generation of multiple optical Bell states. The designs are based on the AlGaAs fabrication platform and are within reach of experimental realization.