Abstract
The determination of the parameters of cylindrical optical waveguides, e.g. the diameters vec {d}=(d_1,ldots ,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}) between the measured and simulated scattering patterns. The global optimum corresponds to the correct parameter values. The mismatch R(vec {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}, for r=1 and r=2 layers. We find that for both values of r, depending on the values of vec {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.
Highlights
The determination of the parameters of cylindrical optical waveguides, e.g. the diameters d = (d1, . . . , dr ) of r layers of transparent optical fibres, can be executed by inverse evaluation of the scattering intensities that emerge under monochromatic illumination
In this work we have studied the determination of diameters of r layers fortransparent cylinders, e.g., optical waveguides, from lateral diffraction patterns
This is an inverse problem which can be stated as an optimisation problem for a norm R measuring the difference between a pattern resulting from a numerical simulation and the measured pattern
Summary
One thing appeared striking right from the beginning: When considering a solid weak-transparent cylinder, the energy landscape given by R(d) over a range of diameters d exposes no local minima, but only one global minimum. Having practical applications in mind, one would probably not use starting values d0 for the IRGN algorithm which are very far from the actual diameter, so studying the number of observed minima Nmin within a rather large range is sufficient to obtain an understanding of the energy landscapes.
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