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The determination of the parameters of cylindrical optical waveguides, e.g. the diameters    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\vec {d}=(d_1,\ldots ,d_r)$$\end{document}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    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$R(\vec {d})$$\end{document}    R  (   d  →   )between the measured and simulated scattering patterns. The global optimum corresponds to the correct parameter values. The mismatch    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$R(\vec {d})$$\end{document}    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    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\vec {n}$$\end{document}    n  →, for    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$r=1$$\end{document}    r  =  1and    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$r=2$$\end{document}    r  =  2layers. We find that for both values of  r , depending on the values of    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\vec {n}$$\end{document}    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.

Easy-hard phase transition in parameter estimation for optical waveguides