Optical multilayers as a tool for visualizing special relativity (Q2746724)

Multilayer optics is a common topic in undergraduate courses but, it is usually approached from a practical point of view, in direct connection with antireflection or high-reflectance coatings. In this paper, with an obvious didactic character, the matter is also the stratified planar structures, but the investigation takes place on another path, that of the analogy between the multilayer optics and Lorentz transformations. The interest in this new viewpoint is twofold. First, optical properties of multilayers appear in complete equivalence with special relativity, and then the curious composition law for reflection and transmission coefficients emerges as a direct consequence of the abstract composition law of the Lorentz group.NEWLINENEWLINESecond, the appearance of phase shifts in these coefficients can be interpreted in a natural way, in terms of geometrical phase, confirming the general belief that such phases is common place in physics.NEWLINENEWLINEThe main results in this paper: NEWLINENEWLINE-- It is shown that the matrix representing the action of a general multilayer belongs to the group \(\text{SL}(2,\mathbb{C})\), which is locally isomorphic to the \((3+1)\)-dimensional Lorentz group \(\text{SO}(3,1)\);NEWLINENEWLINE-- In the special case of lossless multilayers, the subgroup \(\text{SU}(1,1)\) is locally isomorphic to the \((2+ 1)\)-dimensional Lorentz group \(\text{SO}(2,1)\), and the parameters of the equivalent Lorentz transformation are found;NEWLINENEWLINE-- It is shown that multilayers can be used to visualize special relativity. The authors propose a very simple and feasible optical experiment that allows for the direct measurement of Wigner angle (Thomas rotation), and for its interpretation as a geometric phase.