Revisit the Poynting vector in P T-symmetric coupled waveguides

Abstract

We show that the time-averaged Poynting vector of S → = E → × H → ∗ / 2 in parity-time ( P T ) symmetric coupled waveguides is always positive and cannot explain the stopped light at exceptional points (EPs). In order to solve this paradox, we must accept the fact that the fields E → and H → and the Poynting vector in non-Hermitian systems are in general complex. Based on the original definition of the instantaneous Poynting vector S → = E → × H → , a formula on the group velocity is proposed, which agrees perfectly well with that calculated directly from the dispersion curves. It explains not only the stopped light at EPs, but also the fast-light effect near it. This investigation bridges a gap between the classic electrodynamics and the non-Hermitian physics, and highlights the novelty of non-Hermitian optics.

Non-Hermitian guided modes and exceptional points using loss-free negative-index materials

“…Note that the stopped light at EP demonstrated here is different from that in PT symmetric WGs [12][13][14]. Here the stopped light is associated with the negative flux in NIM, and the whole structure is still Hermitian because no loss or gain is presented.…”

“…The coalescence of dispersion at a critical value of a c shown in Fig. 2 resembles EPs in PT symmetry very much [12][13][14][15][16][17][18][19][20]. We check the eigenmodes around this point (see Fig.…”

“…Assuming the direction of the wavevector k is z, the dispersion and distribution of the eigenmodes can be numerically found by using Maxwell's equations and boundary conditions [14,21,22]. For example, considering the transverse-electric (TE) polarized eigenmodes with non-vanishing E y component…”

“…The exact phase difference is determined by the sign of ∆/γ. Generally γ is determined by the overlap integral of fields in the gap between the two WGs and its sign is fixed [14,15,28], but sign(∆/γ) can be tuned by changing the resonant conditions of k NIM, PIM in the two WGs, e.g. by changing the index of refraction inside or the thicknesses of WGs.…”

“…The simplest configuration in studying PT symmetric optics is two parallel waveguides (WGs). Such a kind configuration can be readily studied by using Maxwell's equations, and it supports some intriguing optical effects especially the stopped light at exceptional points (EPs) [12][13][14][15] that separate the conserved and broken PT phases [16][17][18][19][20]. However, for an optical wave propagating inside a WG, it contains multiple important physical parameters especially the wavevector k charactering the propagation of phase front and the Poynting vector S presenting the propagation of energy flux.…”