Determination of optical fiber layer parameters by inverse evaluation of lateral scattering patterns

Easy-hard phase transition in parameter estimation for optical waveguides

“…The diameters d constitute the set of parameters of the scattering setup that are yet to be estimated. The actual scattering intensities can be calculated according to Lorenz-Mie theory 34 , and this scheme is not only the topic of multiple textbooks on optics 35,36 , but also has been used and outlined in an earlier article on algorithms actually dedicated to parameter estimation 9 . Thus, instead of recounting these expressions yet again, we instead leave it with a reference to these sources and the supplementary material, where the respective equations are given nevertheless.…”

“…For parameter determination, we use the iteratively regularised Gauss-Newton algorithm, which iteratively calculates the best matching parameters for a given scattering intensity pattern u ∞ 9,11 . Since this algorithm has also been researched exhaustively in the past, we again refer to the original sources and the supplementary material for a formal introduction of this algorithm.…”

“…Thus, the algorithm is highly efficient in lowering this quantity, but also prone to wrongful termination in local minima. But as we explicitly seek for local minima here, we are not required to apply any of the modifications proposed earlier 9 and instead may use the algorithm in its basic form.…”

“…For just evaluating the energy landscapes, we only have to evaluate R( d) for a grid of many values of d . When actually trying to find the "best" parameters in a practical application with, e.g., the IRGN algorithm, one seeks to minimise such a norm, which hence could also be used as the objective function for the DiRect algorithm 9 . Therefore, we need a meaningful definition of the norm || • || .…”

“…In a previous work 9 it could be pointed out easily that for transparent cylinders ( Im(n s ) ≡ 0 ) this landscape exhibits local minima, which let the IRGN algorithm terminate wrongly. It could be observed that these minima are usually separated systematically by a distance in d-space that only depends on the relative refractive index m s = n s n s+1 or m s = n s n m for the outermost layer, and thus with this knowledge it is feasible to utilise this property in order to allow the IRGN algorithm to traverse these local minima and terminate at the global minimum 9 . Concurrently, the Dividing Rectangles (DiRect) algorithm was also considered, and with this being an algorithm dedicated to global optimisation, i.e., minimising an objective function from the onset 12 , it appeared that the treatment of our task as an optimisation problem of R(•) as an objective function is more than suitable.…”