Optical finite representation of the Lorentz group

Abstract

We present a class of photonic lattices with an underlying symmetry given by a finite-dimensional representation of the 2+1D Lorentz group. In order to construct such a finite-dimensional representation of a non-compact group, we have to design a PT -symmetric optical structure. Thus, the array of coupled waveguides may keep or break PT -symmetry, leading to a device that behaves like an oscillator or directional amplifier, respectively. We show that the so-called linear PT -symmetric dimer belongs to this class of photonic lattices. * bmlara@inaoep.mx 1 arXiv:1508.05419v1 [physics.optics]

Revisiting the Optical PT-Symmetric Dimer

“…In discrete optical couplers, we can consider waveguide permutation and propagation inversion as equivalent to space and time reflection, respectively. Then, in order to recover the standard linear P T -symmetric dimer, we need to work with waveguides that have the same effective refractive indices, (n 1 ) = (n 2 ), such that n − is purely imaginary, and we can write a differential set [48],…”

“…We will keep the standard sign convention used in classical optics, although this Hamiltonian may confuse those working in quantum mechanics, as it implies that a parallel spin-field configuration is the most energetic configuration. In any given case, it is straightforward to find the propagator for this ζ-independent Schrödinger-like equation [48], (12) such that the propagated fields through the device are given in terms of the initial field configuration,…”

“…It is also possible to decouple the dimer differential equation set, Equation (8), into two second order differential equations that have the same form of the one-dimensional wave equation [48],…”

“…Now, we can bring forward a complementary view that can give us asymptotic information of the broken symmetry phases. Let us define instantaneous renormalized fields [48], (30) such that the total renormalized field intensity at each propagation distance is always the unit,…”

“…In Section 4, we will extend the linear P T -symmetric dimer to planar N-waveguide couplers using finite dimensional matrix representations of a complexified version of SU (2). We have previously shown [48] that the P T -symmetric dimer and its extensions to planar N-waveguide couplers possess an SO(2, 1) symmetry realized in a finite dimensional non-unitary irreducible representation. This representation is accomplished through complexification of SU (2), {J x , J y , J z } → {J x , iJ y , iJ z } ≡ {K z , K x , K y }, and it allows us to provide the dispersion relation and a closed form analytic propagator, which have the same regimes and dynamics found for the dimer.…”