Analytical design of X-couplers

Symmetric X-couplers are analyzed using the supermode approach. Constraints for lossless propagation are presented. A simple analytic approximation for the total accumulated phase difference of the supermodes is found that enables the calculation of the coupling ratio of arbitrary symmetric X-couplers. Good agreement is found between results obtained with the proposed method and data calculated using a beam-propagation method. The impact of several design parameters on the coupling ratio is discussed. Due to the simplicity of the structure, X-couplers are well suited for being realized by direct light-writing techniques.