Abstract
Propagation of rays in 2D and 3D corrugated waveguides is performed in the general framework of stability indicators. The analysis of stability is based on the Lyapunov and reversibility error. It is found that the error growth follows a power law for regular orbits and an exponential law for chaotic orbits. A relation with the Shannon channel capacity is devised and an approximate scaling law found for the capacity increase with the corrugation depth.
Highlights
The equivalence between geometrical optics and mechanics was established in a variational form by the principles of Fermat and Maupertuis
We determine the horizontal component of the velocity vxn+1 of the ray outgoing from Pn+1 which is equal to vx∗, where the velocity v∗ of the ray from Qn to Pn+1 is determined by the reflection condition v∗ = vn − 2 ν (ν · vn) and the normal of the guide at Qn is given by (6)
The time τ between a reflection on the lower and upper plane is τvz = 1 so that, choosing (x, y, vx, vy) as phase space coordinates, the map between two consecutive reflections reads vx n+1 = vx n vyn+1 = vyn
Summary
For the 2D wave guide the phase portrait of the corresponding 2D map allows to detect the regions of regular and chaotic motion Finite time indicators such as the Fast Lyapunov Indicator (FLI)[16,17] have been first proposed to analyze the orbital stability. The variational indicators, computed for the orbits issued from the points of a regular grid in a 2D phase plane for the 4D map, and visualized with a colour plot, allow to determine the stability properties just as for the 2D map. We have analyzed a model of 2D waveguide for different values of the corrugation amplitude, showing that the LE, RE, REM provide comparable results, which describe the orbits sensitivity to a small initial random displacement, to a noise along the orbit and to round off.
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