Figure of merit for photonic differentiators

We introduce a universal figure of merit to evaluate the processing speed (operation bandwidth) performance of arbitrary-order optical differentiators. In particular, we define the maximum-to-minimum bandwidth ratio (MMBR) as a main figure of merit of these devices, which essentially informs about the broadness of the acceptable input pulse bandwidth range. We derive and numerically confirm a general analytical expression for the MMBR of an arbitrary optical differentiator, showing that this can be expressed simply as a function of the differentiator's amplitude resonance depth. The device MMBR can be improved by increasing the filter's resonance depth, depending also on the differentiation order; in particular, the MMBR quickly deteriorates as the differentiator order is increased. In our analysis, photonic differentiators are considered in two main groups, namely (i) non-minimum phase and (ii) minimum phase optical filtering implementations. The derived analytical expression for the device MMBR is generalized for these two different solutions, and the validity of the obtained analytical estimates is verified through numerical simulations, including results for the cases of 1st-, 2nd-, and 3rd-order differentiators.