Easy-hard phase transition in parameter estimation for optical waveguides

Abstract

The determination of the parameters of cylindrical optical waveguides, e.g. the diameters $$\vec {d}=(d_1,\ldots ,d_r)$$ d → = ( d 1 , … , d r ) of r layers of (semi-) transparent optical fibres, can be executed by inverse evaluation of the scattering intensities that emerge under monochromatic illumination. The inverse problem can be solved by optimising the mismatch $$R(\vec {d})$$ R ( d → ) between the measured and simulated scattering patterns. The global optimum corresponds to the correct parameter values. The mismatch $$R(\vec {d})$$ R ( d → ) can be seen as an energy landscape as a function of the diameters. In this work, we study the structure of the energy landscape for different values of the complex refractive indices $$\vec {n}$$ n → , for $$r=1$$ r = 1 and $$r=2$$ r = 2 layers. We find that for both values of r, depending on the values of $$\vec {n}$$ n → , two very different types of energy landscapes exist, respectively. One type is dominated by one global minimum and the other type exhibits a multitude of local minima. From an algorithmic viewpoint, this corresponds to easy and hard phases, respectively. Our results indicate that the two phases are separated by sharp phase-transition lines and that the shape of these lines can be described by one “critical” exponent b, which depends slightly on r. Interestingly, the same exponent also describes the dependence of the number of local minima on the diameters. Thus, our findings are comparable to previous theoretical studies on easy-hard transitions in basic combinatorial optimisation or decision problems like Travelling Salesperson and Satisfiability. To our knowledge our results are the first indicating the existence of easy-hard transitions for a real-world optimisation problem of technological relevance.