Current Optics and Photonics

Optical Society of Korea (한국광학회)
권호 표지
DOI QR코드

DOI QR Code

Chirality in Non-Hermitian Photonics

  • Yu, Sunkyu (Photonic Systems Laboratory, Department of Electrical and Computer Engineering, Seoul National University) ;
  • Piao, Xianji (Photonic Systems Laboratory, Department of Electrical and Computer Engineering, Seoul National University) ;
  • Park, Namkyoo (Photonic Systems Laboratory, Department of Electrical and Computer Engineering, Seoul National University)
  • Received : 2019.04.26
  • Accepted : 2019.05.24
  • Published : 2019.08.25
Download PDF
⟨ Previous Next ⟩

Abstract

Chirality is ubiquitous in physics and biology from microscopic to macroscopic phenomena, such as fermionic interactions and DNA duplication. In photonics, chirality has traditionally represented differentiated optical responses for right and left circular polarizations. This definition of optical chirality in the polarization domain includes handedness-dependent phase velocities or optical absorption inside chiral media, which enable polarimetry for measuring the material concentration and circular dichroism spectroscopy for sensing biological or chemical enantiomers. Recently, the emerging field of non-Hermitian photonics, which explores exotic phenomena in gain or loss media, has provided a new viewpoint on chirality in photonics that is not restricted to the traditional polarization domain but is extended to other physical quantities such as the orbital angular momentum, propagation direction, and system parameter space. Here, we introduce recent milestones in chiral light-matter interactions in non-Hermitian photonics and show an enhanced degree of design freedom in photonic devices for spin and orbital angular momenta, directionality, and asymmetric modal conversion.

Keywords

KGHHD@_2019_v3n4_275_f0002.png 이미지

FIG. 6. Chiral geometric phase in non-Hermitian potentials. (a,b) Chiral encircling around the EP [83]: (a) Evolution of two eigenmodes with starting points on different Riemann sheets for a CCW loop and (b) the same as that for a CW loop. (c) Asymmetric mode switching based on the dynamical encircling [83]. (d) Silicon photonics platform for the encircling of the EP in the on-chip photonic device [86]. Figure adapted from (a-c), ref. [83], Springer Nature; and (d), ref. [86], Springer Nature, with permission.

KGHHD@_2019_v3n4_275_f0003.png 이미지

FIG. 1. Chirality in different physical domains of non-Hermitian photonics. For the optical field, E = eA(r,R)exp (iωt - ik(R) · r), where R is the system parameter vector and r is the position vector, the extended definition of optical chirality in non-Hermitian photonics can be classified according to each physical quantity: polarization e for SAM, wavefront A(r,R) for OAM, canonical momentum k(R) for wave propagation, and the geometry of state evolution in the system parameter space R. The system parameter R represents the complex optical potential that determines the condition of PT symmetry, including on-site and hopping constants defined by structural and material parameters.

KGHHD@_2019_v3n4_275_f0004.png 이미지

FIG. 2. Chiral polarizations in non-Hermitian potentials. (a-e) Evolution of eigenpolarizations according to the phase of PT symmetry [36]: (a) Hermitian, (b) unbroken, (c) EP, and (d,e) non-Hermitian states. (f) The convergence of polarizations to the LCP state, showing spin black hole [36]. (g,h) The experimental platform for PT-symmetry-protected chirality: (g) lattice structures [36] and (h) photonic molecules [35]. Figure adapted from (a-g), ref. [36], OSA; and (h), ref. [35], APS, with permission.

KGHHD@_2019_v3n4_275_f0005.png 이미지

FIG. 3. Chiral wavefronts in non-Hermitian potentials. (a-c) OAM microlaser [52]: (a) schematic, (b) OAM wavefront, and (c) fabricated device. (d,e) Broadband OAM laser using a tapered structure [53]: (d) schematic and (e) fabricated device. Figure adapted from (a-c), ref. [52], AAAS; and (d,e), ref. [53], with permission.

KGHHD@_2019_v3n4_275_f0006.png 이미지

FIG. 4. Chiral propagations in non-Hermitian potentials. (a) A schematic of a WGM resonator for observing the chiral wave propagation [56]. (b,c) Asymmetric wave propagation for chiral absorption: (b) CCW and (c) CW wave propagation [57]. (d,e) Operation principle of the EP sensor: (d) square root response and (e) the physical origin from the large backscattering induced by the unperturbed system [67]. Figure adapted from (a), ref. [56], NAS; (b,c), ref. [57], APS; and (d,e), ref. [67], Springer Nature, with permission.

KGHHD@_2019_v3n4_275_f0007.png 이미지

FIG. 5. Roles of degeneracy and disorders in chirality in non-Hermitian potentials. (a) The absence of PT-symmetric transitions: (b) modal profiles [69]. (c) The emergence of PT-symmetric transitions with discrete spatial symmetry: (d) modal profiles [69]. (e-g) Realizations of the chiral wave evolution in disordered photonic networks with different topological charges [70]. Figure adapted from (a-d), ref. [69], APS; and (e-g), ref. [70], AAAS, with permission.

References

  1. M. K. Gaillard, P. D. Grannis, and F. J. Sciulli, "The standard model of particle physics," Rev. Mod. Phys. 71, S96 (1999). https://doi.org/10.1103/RevModPhys.71.S96
  2. K. Soai, T. Shibata, H. Morioka, and K. Choji, "Asymmetric autocatalysis and amplification of enantiomeric excess of a chiral molecule," Nature 378, 767-768 (1995). https://doi.org/10.1038/378767a0
  3. P. L. Polavarapu, "Optical rotation: recent advances in determining the absolute configuration," Chirality 14, 768-781 (2002). https://doi.org/10.1002/chir.10145
  4. Y. Tang and A. E. Cohen, "Optical chirality and its interaction with matter," Phys. Rev. Lett. 104, 163901 (2010). https://doi.org/10.1103/PhysRevLett.104.163901
  5. K. Y. Bliokh and F. Nori, "Characterizing optical chirality," Phys. Rev. A 83, 021803(R) (2011). https://doi.org/10.1103/PhysRevA.83.021803
  6. A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of bi-anisotropic materials: Theory and applications (Gordon and Breach Science, 2001).
  7. A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, "Optical manifestations of planar chirality," Phys. Rev. Lett. 90, 107404 (2003). https://doi.org/10.1103/PhysRevLett.90.107404
  8. Y. Tang and A. E. Cohen, "Enhanced enantioselectivity in excitation of chiral molecules by superchiral light," Science 332, 333-336 (2011). https://doi.org/10.1126/science.1202817
  9. J. B. Pendry, "A chiral route to negative refraction," Science 306, 1353-1355 (2004). https://doi.org/10.1126/science.1104467
  10. X. Piao, S. Yu, J. Hong, and N. Park, "Spectral separation of optical spin based on antisymmetric Fano resonances," Sci. Rep. 5, 16585 (2015). https://doi.org/10.1038/srep16585
  11. A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, "Photonic topological insulators," Nat. Mater. 12, 233-239 (2013). https://doi.org/10.1038/nmat3520
  12. H.-E. Lee, H.-Y. Ahn, J. Mun, Y. Y. Lee, M. Kim, N. H. Cho, K. Chang, W. S. Kim, J. Rho, and K. T. Nam, "Amino-acid-and peptide-directed synthesis of chiral plasmonic gold nanoparticles," Nature 556, 360-365 (2018). https://doi.org/10.1038/s41586-018-0034-1
  13. L. Feng, R. El-Ganainy, and L. Ge, "Non-Hermitian photonics based on parity-time symmetry," Nat. Photonics 11, 752-762 (2017). https://doi.org/10.1038/s41566-017-0031-1
  14. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, "Non-Hermitian physics and PT symmetry," Nat. Phys. 14, 11-19 (2018). https://doi.org/10.1038/nphys4323
  15. M.-A. Miri and A. Alu, "Exceptional points in optics and photonics," Science 363, eaar7709 (2019). https://doi.org/10.1126/science.aar7709
  16. S. Longhi, "Parity-time symmetry meets photonics: A new twist in non-Hermitian optics," Europhys. Lett. 120, 64001 (2018). https://doi.org/10.1209/0295-5075/120/64001
  17. R. El-Ganainy, M. Khajavikhan, D. N. Christodoulides, and S. K. Ozdemir, "The dawn of non-Hermitian optics," Commun. Phys. 2, 37 (2019). https://doi.org/10.1038/s42005-019-0130-z
  18. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, "Parity-time synthetic photonic lattices," Nature 488, 167-171 (2012). https://doi.org/10.1038/nature11298
  19. L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, "Single-mode laser by parity-time symmetry breaking," Science 346, 972-975 (2014). https://doi.org/10.1126/science.1258479
  20. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, "Parity-time-symmetric microring lasers," Science 346, 975-978 (2014). https://doi.org/10.1126/science.1258480
  21. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, "Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators," Nat. Photonics 8, 524-529 (2014). https://doi.org/10.1038/nphoton.2014.133
  22. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, "Optical solitons in PT periodic potentials," Phys. Rev. Lett. 100, 030402 (2008). https://doi.org/10.1103/PhysRevLett.100.030402
  23. C. M. Bender and S. Boettcher, "Real spectra in non-Hermitian Hamiltonians having PT symmetry," Phys. Rev. Lett. 80, 5243 (1998). https://doi.org/10.1103/PhysRevLett.80.5243
  24. C. M. Bender, D. C. Brody, and H. F. Jones, "Complex extension of quantum mechanics," Phys. Rev. Lett. 89, 270401 (2002). https://doi.org/10.1103/PhysRevLett.89.270401
  25. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, "Observation of parity-time symmetry in optics," Nat. Phys. 6, 192-195 (2010). https://doi.org/10.1038/nphys1515
  26. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, "Observation of PT-symmetry breaking in complex optical potentials," Phys. Rev. Lett. 103, 093902 (2009). https://doi.org/10.1103/PhysRevLett.103.093902
  27. S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. Makris, M. Segev, M. Rechtsman, and A. Szameit, "Topologically protected bound states in photonic parity-time-symmetric crystals," Nat. Mater. 16, 433-438 (2017). https://doi.org/10.1038/nmat4811
  28. S. Assawaworrarit, X. Yu, and S. Fan, "Robust wireless power transfer using a nonlinear parity-time-symmetric circuit," Nature 546, 387-390 (2017). https://doi.org/10.1038/nature22404
  29. R. Fleury, D. L. Sounas, and A. Alu, "Negative refraction and planar focusing based on parity-time symmetric metasurfaces," Phys. Rev. Lett. 113, 023903 (2014). https://doi.org/10.1103/PhysRevLett.113.023903
  30. W. Heiss and H. Harney, "The chirality of exceptional points," Eur. Phys. J. D 17, 149-151 (2001). https://doi.org/10.1007/s100530170017
  31. W. D. Heiss, "The physics of exceptional points," J. Phys. A 45, 444016 (2012). https://doi.org/10.1088/1751-8113/45/44/444016
  32. W. D. Heiss, M. Müller, and I. Rotter, "Collectivity, phase transitions, and exceptional points in open quantum systems," Phys. Rev. E 58, 2894 (1998). https://doi.org/10.1103/PhysRevE.58.2894
  33. W. Heiss, "Repulsion of resonance states and exceptional points," Phys. Rev. E 61, 929-932 (2000). https://doi.org/10.1103/PhysRevE.61.929
  34. S. Yu, X. Piao, D. R. Mason, S. In, and N. Park, "Spatiospectral separation of exceptional points in PT-symmetric optical potentials," Phys. Rev. A 86, 031802 (2012). https://doi.org/10.1103/PhysRevA.86.031802
  35. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, "Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces," Phys. Rev. Lett. 113, 093901 (2014). https://doi.org/10.1103/PhysRevLett.113.093901
  36. S. Yu, H. S. Park, X. Piao, B. Min, and N. Park, "Low-dimensional optical chirality in complex potentials," Optica 3, 1025-1032 (2016). https://doi.org/10.1364/OPTICA.3.001025
  37. M. Kang and Y. D. Chong, "Coherent optical control of polarization with a critical metasurface," Phys. Rev. A 92, 043826 (2015). https://doi.org/10.1103/PhysRevA.92.043826
  38. M. Kang, J. Chen, and Y. D. Chong, "Chiral exceptional points in metasurfaces," Phys. Rev. A 94, 033834 (2016). https://doi.org/10.1103/PhysRevA.94.033834
  39. S. Yu, X. Piao, and N. Park, "Acceleration toward polarization singularity inspired by relativistic E $\times$ B drift," Sci. Rep. 6, 37754 (2016). https://doi.org/10.1038/srep37754
  40. S. Yu, X. Piao, and N. Park, "Designing non-Hermitian dynamics for conservative state evolution on the Bloch sphere," Phys. Rev. A 97, 033805 (2018). https://doi.org/10.1103/PhysRevA.97.033805
  41. A. Cerjan and S. Fan, "Achieving arbitrary control over pairs of polarization states using complex birefringent metamaterials," Phys. Rev. Lett. 118, 253902 (2017). https://doi.org/10.1103/PhysRevLett.118.253902
  42. B. Baum, M. Lawrence, D. Barton III, J. Dionne, and H. Alaeian, "Active polarization control with a parity-time-symmetric plasmonic resonator," Phys. Rev. B 98, 165418 (2018). https://doi.org/10.1103/PhysRevB.98.165418
  43. X. Piao, S. Yu, and N. Park, "Design of transverse spinning of light with globally unique handedness," Phys. Rev. Lett. 120, 203901 (2018). https://doi.org/10.1103/PhysRevLett.120.203901
  44. H. Kuratsuji and S. Kakigi, "Maxwell-Schrodinger equation for polarized light and evolution of the Stokes parameters," Phys. Rev. Lett. 80, 1888 (1998). https://doi.org/10.1103/PhysRevLett.80.1888
  45. D. L. Andrews and M. Babiker, The angular momentum of light (Cambridge University Press, Cambridge, UK, 2012).
  46. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, "Terabit-scale orbital angular momentum mode division multiplexing in fibers," Science 340, 1545-1548 (2013). https://doi.org/10.1126/science.1237861
  47. D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, "Controlled generation of higher-order Poincare sphere beams from a laser," Nat. Photonics 10, 327-332 (2016). https://doi.org/10.1038/nphoton.2016.37
  48. R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, "Arbitrary spin-to-orbital angular momentum conversion of light," Science 358, 896-901 (2017). https://doi.org/10.1126/science.aao5392
  49. A. Mock, D. Sounas, and A. Alu, "Tunable orbital angular momentum radiation from angular-momentum-biased microcavities," Phys. Rev. Lett. 121, 103901 (2018). https://doi.org/10.1103/PhysRevLett.121.103901
  50. N. C. Zambon, P. St-Jean, M. Milicevic, A. Lemaitre, A. Harouri, L. Le Gratiet, O. Bleu, D. D. Solnyshkov, G. Malpuech, and I. Sagnes, S. Ravets, A. Amo, and J. Bloch, "Optically controlling the emission chirality of microlasers," Nat. Photonics 13, 283-288 (2019). https://doi.org/10.1038/s41566-019-0380-z
  51. C. Dembowski, B. Dietz, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, "Observation of a chiral state in a microwave cavity," Phys. Rev. Lett. 90, 034101 (2003). https://doi.org/10.1103/PhysRevLett.90.034101
  52. P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, "Orbital angular momentum microlaser," Science 353, 464-467 (2016). https://doi.org/10.1126/science.aaf8533
  53. W. E. Hayenga, J. Ren, M. Parto, F. Wu, M. P. Hokmabadi, C. Wolff, R. El-Ganainy, N. A. Mortensen, D. N. Christodoulides, and M. Khajavikhan, "Direct generation of tunable orbital angular momentum beams in microring lasers with broadband exceptional points," arXiv preprint arXiv:1903.10108 (2019). https://doi.org/10.1021/acsphotonics.9b00779
  54. J. M. Lee, S. Factor, Z. Lin, I. Vitebskiy, F. M. Ellis, and T. Kottos, "Reconfigurable directional lasing modes in cavities with generalized PT symmetry," Phys. Rev. Lett. 112, 253902 (2014). https://doi.org/10.1103/PhysRevLett.112.253902
  55. F.-J. Shu, C.-L. Zou, X.-B. Zou, and L. Yang, "Chiral symmetry breaking in a microring optical cavity by engineered dissipation," Phys. Rev. A 94, 013848 (2016). https://doi.org/10.1103/PhysRevA.94.013848
  56. B. Peng, S. K. Ozdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, "Chiral modes and directional lasing at exceptional points," Proc. Natl. Acad. Sci. 113, 6845-6850 (2016). https://doi.org/10.1073/pnas.1603318113
  57. W. R. Sweeney, C. W. Hsu, S. Rotter, and A. D. Stone, "Perfectly absorbing exceptional points and chiral absorbers," Phys. Rev. Lett. 122, 093901 (2019). https://doi.org/10.1103/PhysRevLett.122.093901
  58. D. A. B. Miller, "Are optical transistors the logical next step?," Nat. Photonics 4, 3-5 (2010). https://doi.org/10.1038/nphoton.2009.240
  59. D. L. Sounas and A. Alu, "Non-reciprocal photonics based on time modulation," Nat. Photonics 11, 774-783 (2017). https://doi.org/10.1038/s41566-017-0051-x
  60. S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popovic, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, "Comment on "Nonreciprocal light propagation in a silicon photonic circuit"," Science 335, 38 (2012).
  61. D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, "What is - and what is not - an optical isolator," Nat. Photonics 7, 579-582 (2013). https://doi.org/10.1038/nphoton.2013.185
  62. Y. Chong, L. Ge, H. Cao, and A. D. Stone, "Coherent perfect absorbers: time-reversed lasers," Phys. Rev. Lett. 105, 053901 (2010). https://doi.org/10.1103/PhysRevLett.105.053901
  63. Y. D. Chong, L. Ge, and A. D. Stone, "PT-symmetry breaking and laser-absorber modes in optical scattering systems," Phys. Rev. Lett. 106, 093902 (2011). https://doi.org/10.1103/PhysRevLett.106.093902
  64. S. Yu, X. Piao, J. Hong, and N. Park, "Progress toward high-Q perfect absorption: A Fano antilaser," Phys. Rev. A 92, 011802 (2015). https://doi.org/10.1103/PhysRevA.92.011802
  65. J. Wiersig, "Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection," Phys. Rev. Lett. 112, 203901 (2014). https://doi.org/10.1103/PhysRevLett.112.203901
  66. J. Wiersig, "Sensors operating at exceptional points: general theory," Phys. Rev. A 93, 033809 (2016). https://doi.org/10.1103/PhysRevA.93.033809
  67. W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, "Exceptional points enhance sensing in an optical microcavity," Nature 548, 192-196 (2017). https://doi.org/10.1038/nature23281
  68. J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, "On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator," Nat. Photonics 4, 46-49 (2010). https://doi.org/10.1038/nphoton.2009.237
  69. L. Ge and A. D. Stone, "Parity-time symmetry breaking beyond one dimension: the role of degeneracy," Phys. Rev. X 4, 031011 (2014).
  70. S. Yu, X. Piao, J. Hong, and N. Park, "Metadisorder for designer light in random systems," Sci. Adv. 2, e1501851 (2016). https://doi.org/10.1126/sciadv.1501851
  71. S. Yu, X. Piao, and N. Park, "Target decoupling in coupled systems resistant to random perturbation," Sci. Rep. 7, 2139 (2017). https://doi.org/10.1038/s41598-017-01241-1
  72. S. Yu, X. Piao, and N. Park, "Bohmian photonics for independent control of the phase and amplitude of waves," Phys. Rev. Lett. 120, 193902 (2018). https://doi.org/10.1103/PhysRevLett.120.193902
  73. K. G. Makris, Z. H. Musslimani, D. N. Christodoulides, and S. Rotter, "Constant-intensity waves and their modulation instability in non-Hermitian potentials," Nat. Commun. 6, 7257 (2015). https://doi.org/10.1038/ncomms8257
  74. E. Rivet, A. Brandstötter, K. G. Makris, H. Lissek, S. Rotter, and R. Fleury, "Constant-pressure sound waves in non-Hermitian disordered media," Nat. Phys. 14, 942-947 (2018). https://doi.org/10.1038/s41567-018-0188-7
  75. K. G. Makris, A. Brandstötter, P. Ambichl, Z. H. Musslimani, and S. Rotter, "Wave propagation through disordered media without backscattering and intensity variations," Light Sci. Appl. 6, e17035 (2017). https://doi.org/10.1038/lsa.2017.35
  76. S. Pancharatnam, "Generalized theory of interference and its applications. Part. II. Partially coherent pencils," Proc. Indiana Acad. Sci. 44, 398-417 (1956). https://doi.org/10.1007/BF03046095
  77. M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. Lond. A 392, 45-57 (1984). https://doi.org/10.1098/rspa.1984.0023
  78. Y. Aharonov and J. Anandan, "Phase change during a cyclic quantum evolution," Phys. Rev. Lett. 58, 1593 (1987). https://doi.org/10.1103/PhysRevLett.58.1593
  79. W. D. Heiss, "Phases of wave functions and level repulsion," Eur. Phys. J. D 7, 1-4 (1999). https://doi.org/10.1007/s100530050339
  80. C. Dembowski, B. Dietz, H.-D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, "Encircling an exceptional point," Phys. Rev. E 69, 056216 (2004). https://doi.org/10.1103/PhysRevE.69.056216
  81. R. Lefebvre, O. Atabek, M. Sindelka, and N. Moiseyev, "Resonance coalescence in molecular photodissociation," Phys. Rev. Lett. 103, 123003 (2009). https://doi.org/10.1103/PhysRevLett.103.123003
  82. C. Dembowski, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, "Experimental observation of the topological structure of exceptional points," Phys. Rev. Lett. 86, 787-790 (2001). https://doi.org/10.1103/PhysRevLett.86.787
  83. J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, "Dynamically encircling an exceptional point for asymmetric mode switching," Nature 537, 76-79 (2016). https://doi.org/10.1038/nature18605
  84. X.-L. Zhang, S. Wang, B. Hou, and C. T. Chan, "Dynamically encircling exceptional points: in situ control of encircling loops and the role of the starting point," Phys. Rev. X 8, 021066 (2018). https://doi.org/10.1103/PhysRevX.8.021066
  85. A. U. Hassan, B. Zhen, M. Soljacic, M. Khajavikhan, and D. N. Christodoulides, "Dynamically encircling exceptional points: Exact evolution and polarization state conversion," Phys. Rev. Lett. 118, 093002 (2017). https://doi.org/10.1103/PhysRevLett.118.093002
  86. J. W. Yoon, Y. Choi, C. Hahn, G. Kim, S. H. Song, K.-Y. Yang, J. Y. Lee, Y. Kim, C. S. Lee, J. K. Shin, H.-S. Lee, and P. Berini, "Time-asymmetric loop around an exceptional point over the full optical communications band," Nature 562, 86-90 (2018). https://doi.org/10.1038/s41586-018-0523-2
  87. H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, "Topological energy transfer in an optomechanical system with exceptional points," Nature 537, 80-83 (2016). https://doi.org/10.1038/nature18604
  88. X.-L. Zhang, T. Jiang, H.-B. Sun, and C. T. Chan, "Dynamically encircling an exceptional point in anti-PT-symmetric systems: asymmetric mode switching for symmetry-broken states," arXiv preprint arXiv:1806.07649 (2018).
  89. T. E. Lee, "Anomalous edge state in a non-Hermitian lattice," Phys. Rev. Lett. 116, 133903 (2016). https://doi.org/10.1103/PhysRevLett.116.133903
  90. Q. Zhong, M. Khajavikhan, D. N. Christodoulides, and R. El-Ganainy, "Winding around non-Hermitian singularities," Nat. Commun. 9, 4808 (2018). https://doi.org/10.1038/s41467-018-07105-0
  91. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, "Parity-time synthetic photonic lattices," Nature 488, 167-171 (2012). https://doi.org/10.1038/nature11298
  92. E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, "Photonic topological insulator in synthetic dimensions," Nature 567, 356-360 (2019). https://doi.org/10.1038/s41586-019-0943-7
  93. K. Y. Bliokh, D. Smirnova, and F. Nori, "Quantum spin Hall effect of light," Science 348, 1448-1451 (2015). https://doi.org/10.1126/science.aaa9519
  94. D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, "Edge modes, degeneracies, and topological numbers in non-Hermitian systems," Phys. Rev. Lett. 118, 040401 (2017). https://doi.org/10.1103/PhysRevLett.118.040401
  95. S. Lieu, "Topological phases in the non-Hermitian Su-Schrieffer-Heeger model," Phys. Rev. B 97, 045106 (2018). https://doi.org/10.1103/PhysRevB.97.045106
  96. D. S. Wiersma, "Disordered photonics," Nat. Photonics 7, 188-196 (2013). https://doi.org/10.1038/nphoton.2013.29
  97. H. H. Sheinfux, Y. Lumer, G. Ankonina, A. Z. Genack, G. Bartal, and M. Segev, "Observation of Anderson localization in disordered nanophotonic structures," Science 356, 953-956 (2017). https://doi.org/10.1126/science.aah6822
  98. S. Yu, X. Piao, and N. Park, "Disordered potential landscapes for anomalous delocalization and superdiffusion of light," ACS Photonics 5, 1499-1505 (2018). https://doi.org/10.1021/acsphotonics.7b01532
  99. M.-A. Miri, M. Heinrich, R. El-Ganainy, and D. N. Christodoulides, "Supersymmetric optical structures," Phys. Rev. Lett. 110, 233902 (2013). https://doi.org/10.1103/PhysRevLett.110.233902
  100. M. P. Hokmabadi, N. S. Nye, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, "Supersymmetric laser arrays," Science 363, 623-626 (2019). https://doi.org/10.1126/science.aav5103
  101. M. Heinrich, M. A. Miri, S. Stutzer, R. El-Ganainy, S. Nolte, A. Szameit, and D. N. Christodoulides, "Supersymmetric mode converters," Nat. Commun. 5, 3698 (2014). https://doi.org/10.1038/ncomms4698
  102. S. Yu, X. Piao, J. Hong, and N. Park, "Bloch-like waves in random-walk potentials based on supersymmetry," Nat. Commun. 6, 8269 (2015). https://doi.org/10.1038/ncomms9269
  103. S. Yu, X. Piao, and N. Park, "Controlling random waves with digital building blocks based on supersymmetry," Phys. Rev. Appl. 8, 054010 (2017). https://doi.org/10.1103/PhysRevApplied.8.054010