Locally parity-time-symmetric and globally parity-symmetric systems

Abstract

We introduce a new class of systems holding Parity Time (PT)-symmetry locally whereas being globally Psymmetric. The potential is globally symmetric, U=U(|r|), and fulfills PT-symmetry with respect to periodically distributed points r 0 : U(|r 0 +r|)=U*(|r 0 -r|) being r 0 ∫ 0. We show that such systems hold novel properties arising from the merging of the two different symmetries, leading to a strong field localization and enhancement at the double-symmetry center, r=0, when the coupling of outward to inward propagating waves is favored. We explore such general potentials in 1D and 2D, which could have actual realizations in different fields, in particular in optics, combining gain/loss and index modulations in nanophotonic structures. As a direct application, we show how to render a broad aperture VCSEL into a bright and narrow beam source.PACS numbers: 78.20. Bh, 42.25.Bs PT-symmetric systems, introduced as a curiosity in quantum mechanics [1,2], are recently being explored in the field of optics, acoustics, plasmonics or Bose-Einstein condensates [3][4][5][6][7][8]. A necessary condition for a system to be PT-symmetric is that the complex potential fulfills U(r)=U* (-r). Such complex systems with real spectra may support novel unexpected properties [9][10][11].Most PT-symmetric systems can be regarded as belonging to two limiting situations of complex periodic potentials. On one extreme, there are purely real-valued potentials holding real periodic modulations in space; which potential, in the simplest harmonic modulation case, reads: U(r) = n Re cos(qx), being q the spatial period of the modulation and n Re its amplitude. On the other extreme, we find purely imaginary potentials only exhibiting gain-loss modulations, which in the simplest harmonic case may be expressed as: U(r) = n Im cos(qx). Both limits lead to a symmetric coupling of resonant modes, i.e. the two counter-propagating modes with wavevector |k|, exp( ) ikx  and exp( ), ikx are coupled symmetrically at resonance, for 2 q k  . The most peculiar situation arises when both the real and imaginary parts of the potential are simultaneously modulated, with a / 2   phase shift: U(r) = n Re cos(qx) + n Im sin(qx). When both modulations are balanced, n Re = n Im , the complex potential can be simply expressed as: U(r) = n exp(±iqx), which evidences that the coupling becomes strongly unidirectional. E.g. for such a complex modulation the left-propagating mode exp(, is efficiently coupled to the right propagating mode, exp( ), ikx  but not vice versa. The point n Re = n Im is precisely the so-called phase transition, separating two extreme situations. Mathematically, the coupling between left/right propagating modes is conveniently described via linear coupling matrices, M = {{0,n Re +n Im },{n Re -n Im ,0}} which at the PT-phase transition point degenerate to M ={{0,2n},{0,0}}. Generally, in the presence of several modes (or mode continuum) the situation becomes more engaged, however the phase transition separating the two extreme limits of ...

PT-axisymmetry for extraordinary field confinement

“…Broad aperture lasers, especially the semiconductor lasers, are known to result in spatial pattern formation [15,16], which is usually an undesirable effect from the technological viewpoint. Different techniques to regularize the spatial structure of the emitted field are being studied, in particular the spatial modulation of the amplifying media [17,18]. The PT-like modulation proposed here could serve as advanced technique to regularize the spatial structure of output radiation.…”

“…In (c), due to the strong exponential decay, the Fourier transform does not provide the half-integer dominant modes; for a deeper analysis using the Laplace transform, see Ref. [18]. …”

Scattering properties of a PT dipole

“…This means that the bulk is uniformly filled by a continuous PT media (i.e., by PT lattices). Perhaps a single exception within this context is the work in [21], where the local-PT concept has been introduced, providing the PT effect in different directions. This results in PT systems with nontrivial flows, i.e., for instance axisymmetric flows toward some focus as suggested in an original proposal [21].…”

“…Perhaps a single exception within this context is the work in [21], where the local-PT concept has been introduced, providing the PT effect in different directions. This results in PT systems with nontrivial flows, i.e., for instance axisymmetric flows toward some focus as suggested in an original proposal [21]. Moreover, it could be extended to build a system with arbitrary PT flows, like closed loops of currents, chiral objects, and any other flow configuration on demand [22].…”

PT-axisymmetric VCSELs with linear central defect

“…Unidirectional light coupling may be achieved by periodic PT-symmetric materials in which the index and gain-loss modulations are dephased a quarter of the spatial period of the modulation introducing an asymmetric coupling between light transverse modes. By an analogous periodic axial PT-symmetric potential, light fields may be expected to be localized at a desired point, the symmetry axis [17]. In this case the inwards radial coupling would be responsible of directing to the central point of the structure obtaining important localization effects (see Fig.…”