Higher-order numerical derivatives for photonic applications

Often during research and development a precise knowledge on derivatives is required. In many cases it is very difficult or impossible to obtain derivatives analytically. This usually occurs in situations when the data to be processed are from an experiment and, therefore, is discrete and with a mixture of noise. The same situation is observed when data to be processed are obtained from numerical simulations. Here we present a detailed comparison of four methods to obtain higher-order derivatives from digital/discrete data. Finite differences method, complex step method, Richardson's extrapolation method, and complex integration method are compared to get an accurate higher-order derivative approximation. Each of them has different properties which make them reliable for a variety of applications and can be easily implemented using software tools.