Second order sensitivity analysis for a photonic crystal waveguide bend

Sensitivity analysis provides important information on the tolerance to fabrication imperfections for any designed devices. Standard sensitivity analysis relies on the first order partial derivatives of a response function of the device with respect to the design parameters. These derivatives are also useful in the device optimization process when a gradient based optimization method is used. The first order sensitivity analysis becomes inadequate if the first order derivatives are small or zero. In this paper, an efficient method for computing the second order partial derivatives is developed for idealized two-dimensional photonic crystal devices with circular cylinders, where the response function is the normalized power transmission coefficient in an output waveguide and the design parameters are the radii of the cylinders. Based on that, a second order sensitivity analysis is performed for a 90 photonic crystal waveguide bend at the frequency where the transmission coefficient is close to 1.