One-way optical modal transition based on causality in momentum space.

The concept of parity-time (PT) symmetry has been used to identify a route toward unidirectional dynamics in optical k-space: imposing asymmetry on the flow of light. Although PT-symmetric potentials have been implemented under the requirement of V(x) = V*(-x), this precondition has only been interpreted within the mathematical framework for the symmetry of Hamiltonians and has not been directly linked to unidirectionality induced by PT symmetry. In this paper, within the context of light-matter interactions, we develop an alternative route toward unidirectionality in k-space by employing the concept of causality. We demonstrate that potentials with real and causal momentum spectra produce unidirectional transitions of optical modes inside the k-continuum, which corresponds to an exceptional point on the degree of PT symmetry. Our analysis reveals a critical link between non-Hermitian problems and spectral theory and also enables multi-dimensional designer manipulation of optical modes, in contrast to the one-dimensional approach that used a Schrödinger-like equation in previous PT-symmetric optics.

[1]  Andrea Alù,et al.  Negative refraction and planar focusing based on parity-time symmetric metasurfaces. , 2014, Physical review letters.

[2]  C. Bender,et al.  Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry , 1997, physics/9712001.

[3]  Vincenzo Galdi,et al.  Tunneling of obliquely incident waves through PT -symmetric epsilon-near-zero bilayers , 2014, 1401.1619.

[4]  Shiyue Hua,et al.  Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators , 2014, Nature Photonics.

[5]  A. Mostafazadeh.  Pseudo-supersymmetric quantum mechanics and isospectral pseudo-Hermitian Hamiltonians , 2002, math-ph/0203041.

[6]  Zongfu Yu,et al.  Comment on “Nonreciprocal Light Propagation in a Silicon Photonic Circuit” , 2012, Science.

[7]  Steven G. Johnson,et al.  Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  H. Yilmaz,et al.  Loss-induced suppression and revival of lasing , 2014, Science.

[9]  Vilson R. Almeida,et al.  Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies. , 2013, Nature materials.

[10]  N. Park,et al.  Spatiospectral separation of exceptional points inPT-symmetric optical potentials , 2012 .

[11]  Stefano Longhi,et al.  PT-symmetric laser absorber , 2010, 1008.5298.

[12]  Yeshaiahu Fainman,et al.  Nonreciprocal Light Propagation in a Silicon Photonic Circuit , 2011, Science.

[13]  M Heinrich,et al.  Parity-time-symmetric coupled microring lasers operating around an exceptional point. , 2015, Optics letters.

[14]  J. Senovilla.  Super-energy tensors , 1999, gr-qc/9906087.

[15]  Limin Tong,et al.  Asymmetric light propagation in composition-graded semiconductor nanowires , 2012, Scientific Reports.

[16]  W. Cai,et al.  Plasmonics for extreme light concentration and manipulation. , 2010, Nature materials.

[17]  T. Gaylord,et al.  Rigorous coupled-wave analysis of planar-grating diffraction , 1981 .

[18]  Vincenzo Galdi,et al.  Analytical study of subwavelength imaging by uniaxial epsilon-near-zero metamaterial slabs , 2012, 1205.1620.

[19]  U. Peschel,et al.  Parity–time synthetic photonic lattices , 2012, Nature.

[20]  D. A. Dunnett.  Classical Electrodynamics , 2020, Nature.

[21]  Y. Chong,et al.  Optical resonator analog of a two-dimensional topological insulator. , 2012, Physical review letters.

[22]  Seung-Man Yang,et al.  Flexible, Angle‐Independent, Structural Color Reflectors Inspired by Morpho Butterfly Wings , 2012, Advanced materials.

[23]  Andreas Bräuer,et al.  Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays , 1999 .

[24]  Zhaowei Liu,et al.  Far-Field Optical Hyperlens Magnifying Sub-Diffraction-Limited Objects , 2007, Science.

[25]  Seung Hoon Lee,et al.  Mode junction photonics with a symmetry-breaking arrangement of mode-orthogonal heterostructures. , 2011, Optics express.

[26]  M. Segev,et al.  Observation of parity–time symmetry in optics , 2010 .

[27]  Matthias Heinrich,et al.  Supersymmetry-generated complex optical potentials with real spectra , 2013, 1305.1689.

[28]  Zongfu Yu,et al.  What is — and what is not — an optical isolator , 2013, Nature Photonics.

[29]  Shanhui Fan,et al.  Parity–time-symmetric whispering-gallery microcavities , 2013, Nature Physics.

[30]  Carl M. Bender,et al.  Twofold Transition in PT-Symmetric Coupled Oscillators , 2013, 1305.7107.

[31]  S. Linden,et al.  Bloch oscillations in plasmonic waveguide arrays , 2014, Nature Communications.

[32]  H. Ramézani,et al.  Engineering wavefront caustics trajectories in PT-symmetric lattices. , 2015, Optics letters.

[33]  Xiang Zhang,et al.  One-way invisible cloak using parity-time symmetric transformation optics. , 2013, Optics letters.

[34]  R. Morandotti,et al.  Observation of PT-symmetry breaking in complex optical potentials. , 2009, Physical review letters.

[35]  Jie Yao,et al.  Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws , 2012, Nature Photonics.

[36]  S. Hahn,et al.  Multidimensional complex signals with single-orthant spectra , 1992, Proc. IEEE.

[37]  Michael Unser,et al.  Splines: a perfect fit for signal and image processing , 1999, IEEE Signal Process. Mag..

[38]  I. Smolyaninov,et al.  Magnifying Superlens in the Visible Frequency Range , 2006, Science.

[39]  Z. Musslimani,et al.  Beam dynamics in PT symmetric optical lattices. , 2008, Physical review letters.

[40]  Thomas Bülow,et al.  Hypercomplex signals-a novel extension of the analytic signal to the multidimensional case , 2001, IEEE Trans. Signal Process..

[41]  Y. Wang,et al.  Single-mode laser by parity-time symmetry breaking , 2014, Science.

[42]  Dorje C Brody,et al.  Complex extension of quantum mechanics. , 2002, Physical review letters.