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 second-order partial derivatives is developed for idealized two-dimensional photonic crystal (PhC) devices with circular cylinders. 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° PhC waveguide bend at the frequency where the transmission coefficient is close to 1.