Abstract

We present a general theory of spontaneous emission at exceptional points (EPs)—exotic degeneracies in non-Hermitian systems. Our theory extends beyond spontaneous emission to any light–matter interaction described by the local density of states (e.g., absorption, thermal emission, and nonlinear frequency conversion). Whereas traditional spontaneous-emission theories imply infinite enhancement factors at EPs, we derive finite bounds on the enhancement, proving maximum enhancement of 4 in passive systems with second-order EPs and significantly larger enhancements (exceeding 400×) in gain-aided and higher-order EP systems. In contrast to non-degenerate resonances, which are typically associated with Lorentzian emission curves in systems with low losses, EPs are associated with non-Lorentzian lineshapes, leading to enhancements that scale nonlinearly with the resonance quality factor. Our theory can be applied to dispersive media, with proper normalization of the resonant modes.

© 2017 Optical Society of America

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  1. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
  2. H. Yokoyama and K. Ujihara, Spontaneous Emission and Laser Oscillation in Microcavities (CRC Press, 1995), Vol. X.
  3. S. V. Gaponenko, Introduction to Nanophotonics (Cambridge University, 2010).
    [Crossref]
  4. W. J. Firth and A. M. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. 95, 073903 (2005).
    [Crossref] [PubMed]
  5. M. V. Berry, “Mode degeneracies and the Petermann excess-noise factor for unstable lasers,” J. Mod. Opt. 50, 63–81 (2003).
    [Crossref]
  6. S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, “Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity,” Phys. Rev. A 78, 015805 (2008).
    [Crossref]
  7. K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quant. Elect. 15, 566–570 (1979).
    [Crossref]
  8. M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314 (1996).
    [Crossref] [PubMed]
  9. A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357 (1997).
    [Crossref]
  10. A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
    [Crossref]
  11. A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
    [Crossref]
  12. N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University, 2011).
  13. W. D. Heiss, “The physics of exceptional points,” J. Phys. A - Math. Theo. 45, 444016 (2012).
    [Crossref]
  14. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref]
  15. B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
    [Crossref]
  16. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
    [Crossref]
  17. J. Doppler, A. A. Mailybaev, J. Böhm, A. G. U. Kuhl, 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).
    [Crossref] [PubMed]
  18. 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).
    [Crossref] [PubMed]
  19. D. Heiss, “Mathematical physics: Circling exceptional points,” Nat. Phys. 12, 823–824 (2016).
    [Crossref]
  20. 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).
    [Crossref] [PubMed]
  21. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science 346, 975–978 (2014).
    [Crossref] [PubMed]
  22. M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. comm. 5, 1 (2014).
    [Crossref]
  23. M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
    [Crossref] [PubMed]
  24. B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
    [Crossref] [PubMed]
  25. G. Yoo, H.-S. Sim, and H. Schomerus, “Quantum noise and mode nonorthogonality in non-Hermitian PT-symmetric optical resonators,” Phys. Rev. A 84, 063833 (2011).
    [Crossref]
  26. W. D. Heiss and G. Wunner, “Fano-Feshbach resonances in two-channel scattering around exceptional points,” Eur. Phys. J. D 68, 1–6 (2014).
    [Crossref]
  27. A. M. van den Brink, K. Young, and M. H. Yung, “Eigenvector expansion and Petermann factor for ohmically damped oscillators,” J. Phys A: Math. Gen. 39, 3725 (2006).
    [Crossref]
  28. M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, “Conversion of broadband to narrowband thermal emission through energy recycling,” Nat. Photon. 6, 535–539 (2012).
    [Crossref]
  29. X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. 107, 045901 (2011).
    [Crossref] [PubMed]
  30. M. Laroche, R. Carminati, and J.-J. Greffet, “Coherent thermal antenna using a photonic crystal slab,” Phys. Rev. Lett. 96, 123903 (2006).
    [Crossref] [PubMed]
  31. C. Khandekar, W. Jin, O. D. Miller, A. Pick, and A. W. Rodriguez, “Giant frequency-selective near-field heat transfer in PT-symmetric structures,” Phys. Rev. B 94, 115402 (2016).
    [Crossref]
  32. T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Comm. 3, 969 (2012).
    [Crossref]
  33. W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
    [Crossref] [PubMed]
  34. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
    [Crossref] [PubMed]
  35. Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with pt phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
    [Crossref] [PubMed]
  36. H. Linnenbank, Y. Grynko, J. Förstner, and S. Linden, “Second harmonic generation spectroscopy on hybrid plasmonic/dielectric nanoantennas,” Light Sci. Appl. 5, e16013 (2016).
    [Crossref]
  37. B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
    [Crossref] [PubMed]
  38. T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
    [Crossref] [PubMed]
  39. A. P. Seyranian and A. A. Mailybaev, Multiparameter Stability Theory With Mechanical Applications (World Scientific Publishing, 2003), Vol. XIII.
    [Crossref]
  40. D. C. Brody and E.-M. Graefe, “Information geometry of complex hamiltonians and exceptional points,” Entropy 15, 3361–3378 (2013).
    [Crossref]
  41. A. Taflove, A. Oskooi, p., and S. G. Johnson, Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology (Artech House, 2013), http://arxiv.org/ftp/arxiv/papers/1301/1301.5366.pdf .
  42. G. W. Hanson, A. I. Nosich, and E. M. Kartchevski, “Green’s function expansions in dyadic root functions for shielded layered waveguides,” Prog. Electromagn. Res. 39, 61–91 (2003).
    [Crossref]
  43. E. Hernàndez, A. Jàuregui, and A. Mondragòn, “Jordan blocks and Gamow-Jordan eigenfunctions associated with a degeneracy of unbound states,” Phys. Rev. A 67, 022721 (2003).
    [Crossref]
  44. E. Hernandez, A. Jauregui, and A. Mondragon, “Degeneracy of resonances in a double barrier potential,” J. Phys. A 33, 4507–4523 (2000).
    [Crossref]
  45. J. D. Jackson, Classical Electrodynamics, Edition III (John Wiley and Sons, 1999).
  46. L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (Princeton University, 2005).
  47. S. Longhi, “Enhanced excess noise in laser cavities with tilted mirrors,” Opt. Lett. 25, 811–813 (2000).
    [Crossref]
  48. G. D’Alessandro and C. B. Laforet, “Giant noise amplification in synchronously pumped optical parametric oscillators,” Opt. Lett. 34, 614–616 (2009).
    [Crossref]
  49. G. D’Alessandro and F. Papoff, “Giant subthreshold amplification in synchronously pumped optical parametric oscillators,” Phys. Rev. A 80, 023804 (2009).
    [Crossref]
  50. F. Wijnands, J. B. Pendry, F. J. Garcia-Vidal, P. M. Bell, L. M. N. Moreno, and P. J. Roberts, “Green’s functions for Maxwell’s equations: Application to spontaneous emission,” Opt. Quant. Elect. 29, 199–216 (1997).
    [Crossref]
  51. Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, 033807 (2000).
    [Crossref]
  52. K. Sacked, Optical Properties of Photonic Crystals (Springer Science & Business Media, 2004), Vol. LXXX.
  53. A. Lagendijk and B. A. V. Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
    [Crossref]
  54. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Elsevier Academic Press, 2006).
  55. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).
  56. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. I.
  57. A. E. Siegman, Lasers (University Science Books, 1986).
  58. H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
    [Crossref]
  59. R. G. Newton, Scattering Theory of Waves and Particles (Springer Science & Business Media, 2013).
  60. S. H. Weintraub, “Jordan canonical form: Application to differential equations,” Synth. Lect. Math. Statist. 1, 1–85 (2008).
    [Crossref]
  61. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Science & Business Media, 2007).
  62. 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).
    [Crossref]
  63. A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microw. Theory Techn. 35, 688–696 (1987).
    [Crossref]
  64. W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [Crossref]
  65. K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2010).
  66. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).
  67. R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
    [Crossref]
  68. F. Capolino, D. R. Jackson, and a. B. F. D. R. Wilton, “Comparison of methods for calculating the field excited by a dipole near a 2-d periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
    [Crossref]
  69. A. P. F. Hernàndez and S. G. Johnson, “Scalable computation of jordan chains,” arXiv:1704.05837 (2017).
  70. H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” App. Phys. Lett. 76, 532–534 (2000).
    [Crossref]
  71. S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. 77, 2444 (1996).
    [Crossref] [PubMed]
  72. E. M. Graefe, H. J. Korsch, and A. E. Niederle, “A non-hermitian symmetric bose? hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A. Math. Theor. 41, 255206 (2008).
    [Crossref]
  73. J.-W. Ryu, S.-Y. Lee, and S. W. Kim, “Analysis of multiple exceptional points related to three interacting eigenmodes in a non-hermitian hamiltonian,” Phys. Rev. A 85, 042101 (2012).
    [Crossref]
  74. W. D. Heiss and G. Wunner, “Resonance scattering at third-order exceptional points,” J. Phys. A. Math. Theor. 48, 345203 (2015).
    [Crossref]
  75. K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
  76. Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
    [Crossref] [PubMed]
  77. N. Colthup, Introduction to Infrared and Raman Spectroscopy (Elsevier, 2012).
  78. J. G. Fujimoto, C. Pitris, S. A. Boppart, and M. E. Brezinski, “Optical coherence tomography: An emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
    [Crossref] [PubMed]
  79. J. D. McKinney, M. A. Webster, K. J. Webb, and A. M. Weiner, “Characterization and imaging in optically scattering media by use of laser speckle and a variable-coherence source,” Opt. Let. 25, 4–6 (2000).
    [Crossref]
  80. 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).
    [Crossref] [PubMed]
  81. A. Pick, Z. Lin, W. Jin, and A. W. Rodriguez, “Enhanced frequency conversion and nonlinear purcell effect at exceptional points,” (In preparation).
  82. D. R. Jackson and A. A. Oliner, Leaky-Wave Antennas (Wiley Online Library, 2008).
  83. F. Monticone and A. Alù, “Leaky-wave theory, techniques, and applications: From microwaves to visible frequencies,” Proc. IEEE 103, 793–821 (2015).
    [Crossref]
  84. O. D. Miller, A. G. Polimeridis, M. T. H. Reid, C. H. H. Chia, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Fundamental limits to optical response in absorptive systems,” Opt. express 24, 3329–3364 (2016).
    [Crossref] [PubMed]
  85. A. E. Siegman, “Excess quantum noise in nonnormal oscillators,” Frontiers of Laser Physics and Quantum Optics (Springer, 2000).
    [Crossref]
  86. B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
    [Crossref]
  87. M. Perrin, “Eigen-energy effects and non-orthogonality in the quasi-normal mode expansion of maxwell equations,” Opt. Express 24, 27137–27151 (2016).
    [Crossref] [PubMed]
  88. C. Sauvan, J.-P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
    [Crossref] [PubMed]
  89. J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1, 58–106 (2009).
    [Crossref]
  90. R. E. Collin, Field theory of guided waves (McGraw-Hill, 1960).
  91. P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994).
    [Crossref] [PubMed]
  92. W. C. Chew and W. H. Weedon, “A 3d perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
    [Crossref]
  93. A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
    [Crossref]

2016 (10)

J. Doppler, A. A. Mailybaev, J. Böhm, A. G. U. Kuhl, 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).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

D. Heiss, “Mathematical physics: Circling exceptional points,” Nat. Phys. 12, 823–824 (2016).
[Crossref]

C. Khandekar, W. Jin, O. D. Miller, A. Pick, and A. W. Rodriguez, “Giant frequency-selective near-field heat transfer in PT-symmetric structures,” Phys. Rev. B 94, 115402 (2016).
[Crossref]

H. Linnenbank, Y. Grynko, J. Förstner, and S. Linden, “Second harmonic generation spectroscopy on hybrid plasmonic/dielectric nanoantennas,” Light Sci. Appl. 5, e16013 (2016).
[Crossref]

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

M. Perrin, “Eigen-energy effects and non-orthogonality in the quasi-normal mode expansion of maxwell equations,” Opt. Express 24, 27137–27151 (2016).
[Crossref] [PubMed]

O. D. Miller, A. G. Polimeridis, M. T. H. Reid, C. H. H. Chia, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Fundamental limits to optical response in absorptive systems,” Opt. express 24, 3329–3364 (2016).
[Crossref] [PubMed]

2015 (5)

A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

F. Monticone and A. Alù, “Leaky-wave theory, techniques, and applications: From microwaves to visible frequencies,” Proc. IEEE 103, 793–821 (2015).
[Crossref]

W. D. Heiss and G. Wunner, “Resonance scattering at third-order exceptional points,” J. Phys. A. Math. Theor. 48, 345203 (2015).
[Crossref]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

2014 (8)

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref] [PubMed]

W. D. Heiss and G. Wunner, “Fano-Feshbach resonances in two-channel scattering around exceptional points,” Eur. Phys. J. D 68, 1–6 (2014).
[Crossref]

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).
[Crossref] [PubMed]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. comm. 5, 1 (2014).
[Crossref]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
[Crossref]

Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with pt phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref] [PubMed]

2013 (3)

C. Sauvan, J.-P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref] [PubMed]

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

D. C. Brody and E.-M. Graefe, “Information geometry of complex hamiltonians and exceptional points,” Entropy 15, 3361–3378 (2013).
[Crossref]

2012 (5)

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Comm. 3, 969 (2012).
[Crossref]

W. D. Heiss, “The physics of exceptional points,” J. Phys. A - Math. Theo. 45, 444016 (2012).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

J.-W. Ryu, S.-Y. Lee, and S. W. Kim, “Analysis of multiple exceptional points related to three interacting eigenmodes in a non-hermitian hamiltonian,” Phys. Rev. A 85, 042101 (2012).
[Crossref]

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, “Conversion of broadband to narrowband thermal emission through energy recycling,” Nat. Photon. 6, 535–539 (2012).
[Crossref]

2011 (4)

X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. 107, 045901 (2011).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

G. Yoo, H.-S. Sim, and H. Schomerus, “Quantum noise and mode nonorthogonality in non-Hermitian PT-symmetric optical resonators,” Phys. Rev. A 84, 063833 (2011).
[Crossref]

2010 (2)

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

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).
[Crossref]

2009 (3)

G. D’Alessandro and C. B. Laforet, “Giant noise amplification in synchronously pumped optical parametric oscillators,” Opt. Lett. 34, 614–616 (2009).
[Crossref]

G. D’Alessandro and F. Papoff, “Giant subthreshold amplification in synchronously pumped optical parametric oscillators,” Phys. Rev. A 80, 023804 (2009).
[Crossref]

J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1, 58–106 (2009).
[Crossref]

2008 (3)

S. H. Weintraub, “Jordan canonical form: Application to differential equations,” Synth. Lect. Math. Statist. 1, 1–85 (2008).
[Crossref]

E. M. Graefe, H. J. Korsch, and A. E. Niederle, “A non-hermitian symmetric bose? hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A. Math. Theor. 41, 255206 (2008).
[Crossref]

S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, “Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity,” Phys. Rev. A 78, 015805 (2008).
[Crossref]

2007 (1)

F. Capolino, D. R. Jackson, and a. B. F. D. R. Wilton, “Comparison of methods for calculating the field excited by a dipole near a 2-d periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

2006 (3)

M. Laroche, R. Carminati, and J.-J. Greffet, “Coherent thermal antenna using a photonic crystal slab,” Phys. Rev. Lett. 96, 123903 (2006).
[Crossref] [PubMed]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

A. M. van den Brink, K. Young, and M. H. Yung, “Eigenvector expansion and Petermann factor for ohmically damped oscillators,” J. Phys A: Math. Gen. 39, 3725 (2006).
[Crossref]

2005 (1)

W. J. Firth and A. M. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. 95, 073903 (2005).
[Crossref] [PubMed]

2004 (1)

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[Crossref]

2003 (3)

M. V. Berry, “Mode degeneracies and the Petermann excess-noise factor for unstable lasers,” J. Mod. Opt. 50, 63–81 (2003).
[Crossref]

G. W. Hanson, A. I. Nosich, and E. M. Kartchevski, “Green’s function expansions in dyadic root functions for shielded layered waveguides,” Prog. Electromagn. Res. 39, 61–91 (2003).
[Crossref]

E. Hernàndez, A. Jàuregui, and A. Mondragòn, “Jordan blocks and Gamow-Jordan eigenfunctions associated with a degeneracy of unbound states,” Phys. Rev. A 67, 022721 (2003).
[Crossref]

2000 (6)

E. Hernandez, A. Jauregui, and A. Mondragon, “Degeneracy of resonances in a double barrier potential,” J. Phys. A 33, 4507–4523 (2000).
[Crossref]

S. Longhi, “Enhanced excess noise in laser cavities with tilted mirrors,” Opt. Lett. 25, 811–813 (2000).
[Crossref]

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” App. Phys. Lett. 76, 532–534 (2000).
[Crossref]

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, 033807 (2000).
[Crossref]

J. G. Fujimoto, C. Pitris, S. A. Boppart, and M. E. Brezinski, “Optical coherence tomography: An emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
[Crossref] [PubMed]

J. D. McKinney, M. A. Webster, K. J. Webb, and A. M. Weiner, “Characterization and imaging in optically scattering media by use of laser speckle and a variable-coherence source,” Opt. Let. 25, 4–6 (2000).
[Crossref]

1997 (2)

F. Wijnands, J. B. Pendry, F. J. Garcia-Vidal, P. M. Bell, L. M. N. Moreno, and P. J. Roberts, “Green’s functions for Maxwell’s equations: Application to spontaneous emission,” Opt. Quant. Elect. 29, 199–216 (1997).
[Crossref]

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357 (1997).
[Crossref]

1996 (3)

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314 (1996).
[Crossref] [PubMed]

A. Lagendijk and B. A. V. Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[Crossref]

S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. 77, 2444 (1996).
[Crossref] [PubMed]

1994 (3)

W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[Crossref]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994).
[Crossref] [PubMed]

W. C. Chew and W. H. Weedon, “A 3d perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[Crossref]

1989 (2)

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[Crossref]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[Crossref]

1987 (1)

A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microw. Theory Techn. 35, 688–696 (1987).
[Crossref]

1979 (1)

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quant. Elect. 15, 566–570 (1979).
[Crossref]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Alù, A.

F. Monticone and A. Alù, “Leaky-wave theory, techniques, and applications: From microwaves to visible frequencies,” Proc. IEEE 103, 793–821 (2015).
[Crossref]

An, K.

S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, “Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity,” Phys. Rev. A 78, 015805 (2008).
[Crossref]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Elsevier Academic Press, 2006).

Asano, T.

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, “Conversion of broadband to narrowband thermal emission through energy recycling,” Nat. Photon. 6, 535–539 (2012).
[Crossref]

Asatryan, A. A.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[Crossref]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).

Barnett, S. M.

S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. 77, 2444 (1996).
[Crossref] [PubMed]

Beermann, J.

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Comm. 3, 969 (2012).
[Crossref]

Bell, P. M.

F. Wijnands, J. B. Pendry, F. J. Garcia-Vidal, P. M. Bell, L. M. N. Moreno, and P. J. Roberts, “Green’s functions for Maxwell’s equations: Application to spontaneous emission,” Opt. Quant. Elect. 29, 199–216 (1997).
[Crossref]

Bender, C. M.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Benisty, H.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” App. Phys. Lett. 76, 532–534 (2000).
[Crossref]

Beraud, A.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” App. Phys. Lett. 76, 532–534 (2000).
[Crossref]

Berry, M. V.

M. V. Berry, “Mode degeneracies and the Petermann excess-noise factor for unstable lasers,” J. Mod. Opt. 50, 63–81 (2003).
[Crossref]

Bliokh, K. Y.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Böhm, J.

J. Doppler, A. A. Mailybaev, J. Böhm, A. G. U. Kuhl, 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).
[Crossref] [PubMed]

Boppart, S. A.

J. G. Fujimoto, C. Pitris, S. A. Boppart, and M. E. Brezinski, “Optical coherence tomography: An emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
[Crossref] [PubMed]

Botten, L. C.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[Crossref]

Bozhevolnyi, S. I.

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Comm. 3, 969 (2012).
[Crossref]

Brandstetter, M.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. comm. 5, 1 (2014).
[Crossref]

Brezinski, M. E.

J. G. Fujimoto, C. Pitris, S. A. Boppart, and M. E. Brezinski, “Optical coherence tomography: An emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
[Crossref] [PubMed]

Brodbeck, S.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Brody, D. C.

D. C. Brody and E.-M. Graefe, “Information geometry of complex hamiltonians and exceptional points,” Entropy 15, 3361–3378 (2013).
[Crossref]

Cao, H.

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Capolino, F.

F. Capolino, D. R. Jackson, and a. B. F. D. R. Wilton, “Comparison of methods for calculating the field excited by a dipole near a 2-d periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

Carminati, R.

M. Laroche, R. Carminati, and J.-J. Greffet, “Coherent thermal antenna using a photonic crystal slab,” Phys. Rev. Lett. 96, 123903 (2006).
[Crossref] [PubMed]

Cassagne, D.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” App. Phys. Lett. 76, 532–534 (2000).
[Crossref]

Cerjan, A.

A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

Chan, C. T.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

Chen, H.

Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with pt phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref] [PubMed]

Chen, Y.-F.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3d perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[Crossref]

Chia, C. H. H.

Chong, Y.

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Chong, Y. D.

A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Christ, A.

A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microw. Theory Techn. 35, 688–696 (1987).
[Crossref]

Christodoulides, D. N.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

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).
[Crossref]

Chua, S.-L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Collier, B.

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

Collin, R. E.

R. E. Collin, Field theory of guided waves (McGraw-Hill, 1960).

Colthup, N.

N. Colthup, Introduction to Infrared and Raman Spectroscopy (Elsevier, 2012).

Commandré, M.

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
[Crossref]

D’Alessandro, G.

G. D’Alessandro and C. B. Laforet, “Giant noise amplification in synchronously pumped optical parametric oscillators,” Opt. Lett. 34, 614–616 (2009).
[Crossref]

G. D’Alessandro and F. Papoff, “Giant subthreshold amplification in synchronously pumped optical parametric oscillators,” Phys. Rev. A 80, 023804 (2009).
[Crossref]

Dall, R. G.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

de Sterke, C. M.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[Crossref]

DeLacy, B. G.

Deutsch, C.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. comm. 5, 1 (2014).
[Crossref]

Ding, K.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

Doppler, J.

J. Doppler, A. A. Mailybaev, J. Böhm, A. G. U. Kuhl, 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).
[Crossref] [PubMed]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

El-Ganainy, R.

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).
[Crossref]

Embree, M.

L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (Princeton University, 2005).

Eriksen, R. L.

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Comm. 3, 969 (2012).
[Crossref]

Estrecho, E.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Exter, M. P. van

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357 (1997).
[Crossref]

Fan, S.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Fegadolli, W. S.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Feng, L.

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).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. I.

Firth, W. J.

W. J. Firth and A. M. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. 95, 073903 (2005).
[Crossref] [PubMed]

Förstner, J.

H. Linnenbank, Y. Grynko, J. Förstner, and S. Linden, “Second harmonic generation spectroscopy on hybrid plasmonic/dielectric nanoantennas,” Light Sci. Appl. 5, e16013 (2016).
[Crossref]

Fraser, M. D.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Fujimoto, J. G.

J. G. Fujimoto, C. Pitris, S. A. Boppart, and M. E. Brezinski, “Optical coherence tomography: An emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
[Crossref] [PubMed]

Gao, T.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Gaponenko, S. V.

S. V. Gaponenko, Introduction to Nanophotonics (Cambridge University, 2010).
[Crossref]

Garcia-Vidal, F. J.

F. Wijnands, J. B. Pendry, F. J. Garcia-Vidal, P. M. Bell, L. M. N. Moreno, and P. J. Roberts, “Green’s functions for Maxwell’s equations: Application to spontaneous emission,” Opt. Quant. Elect. 29, 199–216 (1997).
[Crossref]

Ge, L.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Gianfreda, M.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Graefe, E. M.

E. M. Graefe, H. J. Korsch, and A. E. Niederle, “A non-hermitian symmetric bose? hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A. Math. Theor. 41, 255206 (2008).
[Crossref]

Graefe, E.-M.

D. C. Brody and E.-M. Graefe, “Information geometry of complex hamiltonians and exceptional points,” Entropy 15, 3361–3378 (2013).
[Crossref]

Greffet, J.-J.

M. Laroche, R. Carminati, and J.-J. Greffet, “Coherent thermal antenna using a photonic crystal slab,” Phys. Rev. Lett. 96, 123903 (2006).
[Crossref] [PubMed]

Grynko, Y.

H. Linnenbank, Y. Grynko, J. Förstner, and S. Linden, “Second harmonic generation spectroscopy on hybrid plasmonic/dielectric nanoantennas,” Light Sci. Appl. 5, e16013 (2016).
[Crossref]

Han, Z.

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Comm. 3, 969 (2012).
[Crossref]

Hanson, G. W.

G. W. Hanson, A. I. Nosich, and E. M. Kartchevski, “Green’s function expansions in dyadic root functions for shielded layered waveguides,” Prog. Electromagn. Res. 39, 61–91 (2003).
[Crossref]

Harris, J. G. E.

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).
[Crossref] [PubMed]

Hartnagel, H. L.

A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microw. Theory Techn. 35, 688–696 (1987).
[Crossref]

Heinrich, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Heiss, D.

D. Heiss, “Mathematical physics: Circling exceptional points,” Nat. Phys. 12, 823–824 (2016).
[Crossref]

Heiss, W. D.

W. D. Heiss and G. Wunner, “Resonance scattering at third-order exceptional points,” J. Phys. A. Math. Theor. 48, 345203 (2015).
[Crossref]

W. D. Heiss and G. Wunner, “Fano-Feshbach resonances in two-channel scattering around exceptional points,” Eur. Phys. J. D 68, 1–6 (2014).
[Crossref]

W. D. Heiss, “The physics of exceptional points,” J. Phys. A - Math. Theo. 45, 444016 (2012).
[Crossref]

Hernandez, E.

E. Hernandez, A. Jauregui, and A. Mondragon, “Degeneracy of resonances in a double barrier potential,” J. Phys. A 33, 4507–4523 (2000).
[Crossref]

Hernàndez, A. P. F.

A. P. F. Hernàndez and S. G. Johnson, “Scalable computation of jordan chains,” arXiv:1704.05837 (2017).

Hernàndez, E.

E. Hernàndez, A. Jàuregui, and A. Mondragòn, “Jordan blocks and Gamow-Jordan eigenfunctions associated with a degeneracy of unbound states,” Phys. Rev. A 67, 022721 (2003).
[Crossref]

Hodaei, H.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Hofling, S.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Holmgaard, T.

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Comm. 3, 969 (2012).
[Crossref]

Hsu, C. W.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Hu, J.

J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1, 58–106 (2009).
[Crossref]

Huang, W. P.

Hugonin, J.-P.

C. Sauvan, J.-P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref] [PubMed]

Igarashi, Y.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Inoue, T.

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, “Conversion of broadband to narrowband thermal emission through energy recycling,” Nat. Photon. 6, 535–539 (2012).
[Crossref]

Jackson, D. R.

F. Capolino, D. R. Jackson, and a. B. F. D. R. Wilton, “Comparison of methods for calculating the field excited by a dipole near a 2-d periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

D. R. Jackson and A. A. Oliner, Leaky-Wave Antennas (Wiley Online Library, 2008).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, Edition III (John Wiley and Sons, 1999).

Jauregui, A.

E. Hernandez, A. Jauregui, and A. Mondragon, “Degeneracy of resonances in a double barrier potential,” J. Phys. A 33, 4507–4523 (2000).
[Crossref]

Jàuregui, A.

E. Hernàndez, A. Jàuregui, and A. Mondragòn, “Jordan blocks and Gamow-Jordan eigenfunctions associated with a degeneracy of unbound states,” Phys. Rev. A 67, 022721 (2003).
[Crossref]

Jiang, L.

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).
[Crossref] [PubMed]

Jin, W.

C. Khandekar, W. Jin, O. D. Miller, A. Pick, and A. W. Rodriguez, “Giant frequency-selective near-field heat transfer in PT-symmetric structures,” Phys. Rev. B 94, 115402 (2016).
[Crossref]

A. Pick, Z. Lin, W. Jin, and A. W. Rodriguez, “Enhanced frequency conversion and nonlinear purcell effect at exceptional points,” (In preparation).

Joannopoulos, J. D.

O. D. Miller, A. G. Polimeridis, M. T. H. Reid, C. H. H. Chia, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Fundamental limits to optical response in absorptive systems,” Opt. express 24, 3329–3364 (2016).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Johnson, S. G.

O. D. Miller, A. G. Polimeridis, M. T. H. Reid, C. H. H. Chia, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Fundamental limits to optical response in absorptive systems,” Opt. express 24, 3329–3364 (2016).
[Crossref] [PubMed]

A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

A. P. F. Hernàndez and S. G. Johnson, “Scalable computation of jordan chains,” arXiv:1704.05837 (2017).

Jokerst, N. M.

X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. 107, 045901 (2011).
[Crossref] [PubMed]

Jouanin, C.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” App. Phys. Lett. 76, 532–534 (2000).
[Crossref]

Kaminer, I.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Kamp, M.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Kartchevski, E. M.

G. W. Hanson, A. I. Nosich, and E. M. Kartchevski, “Green’s function expansions in dyadic root functions for shielded layered waveguides,” Prog. Electromagn. Res. 39, 61–91 (2003).
[Crossref]

Khajavikhan, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Khandekar, C.

C. Khandekar, W. Jin, O. D. Miller, A. Pick, and A. W. Rodriguez, “Giant frequency-selective near-field heat transfer in PT-symmetric structures,” Phys. Rev. B 94, 115402 (2016).
[Crossref]

Kim, S. W.

J.-W. Ryu, S.-Y. Lee, and S. W. Kim, “Analysis of multiple exceptional points related to three interacting eigenmodes in a non-hermitian hamiltonian,” Phys. Rev. A 85, 042101 (2012).
[Crossref]

S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, “Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity,” Phys. Rev. A 78, 015805 (2008).
[Crossref]

Kip, D.

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).
[Crossref]

Kivshar, Y. S.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Klang, P.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. comm. 5, 1 (2014).
[Crossref]

Korsch, H. J.

E. M. Graefe, H. J. Korsch, and A. E. Niederle, “A non-hermitian symmetric bose? hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A. Math. Theor. 41, 255206 (2008).
[Crossref]

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Krauss, T. F.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” App. Phys. Lett. 76, 532–534 (2000).
[Crossref]

Kuhl, A. G. U.

J. Doppler, A. A. Mailybaev, J. Böhm, A. G. U. Kuhl, 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).
[Crossref] [PubMed]

Labilloy, D.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” App. Phys. Lett. 76, 532–534 (2000).
[Crossref]

Laforet, C. B.

Lagendijk, A.

A. Lagendijk and B. A. V. Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[Crossref]

Lalanne, P.

C. Sauvan, J.-P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref] [PubMed]

Laroche, M.

M. Laroche, R. Carminati, and J.-J. Greffet, “Coherent thermal antenna using a photonic crystal slab,” Phys. Rev. Lett. 96, 123903 (2006).
[Crossref] [PubMed]

Lee, R. K.

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, 033807 (2000).
[Crossref]

Lee, S.-B.

S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, “Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity,” Phys. Rev. A 78, 015805 (2008).
[Crossref]

Lee, S.-Y.

J.-W. Ryu, S.-Y. Lee, and S. W. Kim, “Analysis of multiple exceptional points related to three interacting eigenmodes in a non-hermitian hamiltonian,” Phys. Rev. A 85, 042101 (2012).
[Crossref]

S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, “Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity,” Phys. Rev. A 78, 015805 (2008).
[Crossref]

Lei, F.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Leung, P. T.

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994).
[Crossref] [PubMed]

Li, H.-Q.

Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with pt phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref] [PubMed]

Li, J.

Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with pt phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref] [PubMed]

Libisch, F.

J. Doppler, A. A. Mailybaev, J. Böhm, A. G. U. Kuhl, 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).
[Crossref] [PubMed]

Liertzer, M.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. comm. 5, 1 (2014).
[Crossref]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref] [PubMed]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

Liew, T. C. H.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Lin, Z.

Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

A. Pick, Z. Lin, W. Jin, and A. W. Rodriguez, “Enhanced frequency conversion and nonlinear purcell effect at exceptional points,” (In preparation).

Lindberg, Å. M.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357 (1997).
[Crossref]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314 (1996).
[Crossref] [PubMed]

Linden, S.

H. Linnenbank, Y. Grynko, J. Förstner, and S. Linden, “Second harmonic generation spectroscopy on hybrid plasmonic/dielectric nanoantennas,” Light Sci. Appl. 5, e16013 (2016).
[Crossref]

Linnenbank, H.

H. Linnenbank, Y. Grynko, J. Förstner, and S. Linden, “Second harmonic generation spectroscopy on hybrid plasmonic/dielectric nanoantennas,” Light Sci. Appl. 5, e16013 (2016).
[Crossref]

Litchinitser, N. M.

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).
[Crossref] [PubMed]

Liu, D.

A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

Liu, S. Y.

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994).
[Crossref] [PubMed]

Liu, X.

X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. 107, 045901 (2011).
[Crossref] [PubMed]

Loncar, M.

Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

Long, G. L.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Longhi, S.

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).
[Crossref] [PubMed]

S. Longhi, “Enhanced excess noise in laser cavities with tilted mirrors,” Opt. Lett. 25, 811–813 (2000).
[Crossref]

Loudon, R.

S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. 77, 2444 (1996).
[Crossref] [PubMed]

Lu, L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Lu, M.-H.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Ma, G.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

Ma, R.-M.

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).
[Crossref] [PubMed]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Science & Business Media, 2007).

Mailybaev, A. A.

J. Doppler, A. A. Mailybaev, J. Böhm, A. G. U. Kuhl, 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).
[Crossref] [PubMed]

A. P. Seyranian and A. A. Mailybaev, Multiparameter Stability Theory With Mechanical Applications (World Scientific Publishing, 2003), Vol. XIII.
[Crossref]

Makris, K. G.

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).
[Crossref]

Maksymov, I. S.

C. Sauvan, J.-P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref] [PubMed]

Mason, D.

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).
[Crossref] [PubMed]

McKinney, J. D.

J. D. McKinney, M. A. Webster, K. J. Webb, and A. M. Weiner, “Characterization and imaging in optically scattering media by use of laser speckle and a variable-coherence source,” Opt. Let. 25, 4–6 (2000).
[Crossref]

McOrist, J.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[Crossref]

McPhedran, R. C.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Menyuk, C. R.

J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1, 58–106 (2009).
[Crossref]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).

Miao, P.

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).
[Crossref] [PubMed]

Mieremet, A. L.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357 (1997).
[Crossref]

Milburn, T. J.

J. Doppler, A. A. Mailybaev, J. Böhm, A. G. U. Kuhl, 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).
[Crossref] [PubMed]

Miller, O. D.

C. Khandekar, W. Jin, O. D. Miller, A. Pick, and A. W. Rodriguez, “Giant frequency-selective near-field heat transfer in PT-symmetric structures,” Phys. Rev. B 94, 115402 (2016).
[Crossref]

O. D. Miller, A. G. Polimeridis, M. T. H. Reid, C. H. H. Chia, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Fundamental limits to optical response in absorptive systems,” Opt. express 24, 3329–3364 (2016).
[Crossref] [PubMed]

Miri, M.-A.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Mochizuki, K.

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, “Conversion of broadband to narrowband thermal emission through energy recycling,” Nat. Photon. 6, 535–539 (2012).
[Crossref]

Moiseyev, N.

J. Doppler, A. A. Mailybaev, J. Böhm, A. G. U. Kuhl, 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).
[Crossref] [PubMed]

N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University, 2011).

Mondragon, A.

E. Hernandez, A. Jauregui, and A. Mondragon, “Degeneracy of resonances in a double barrier potential,” J. Phys. A 33, 4507–4523 (2000).
[Crossref]

Mondragòn, A.

E. Hernàndez, A. Jàuregui, and A. Mondragòn, “Jordan blocks and Gamow-Jordan eigenfunctions associated with a degeneracy of unbound states,” Phys. Rev. A 67, 022721 (2003).
[Crossref]

Monifi, F.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref] [PubMed]

Monticone, F.

F. Monticone and A. Alù, “Leaky-wave theory, techniques, and applications: From microwaves to visible frequencies,” Proc. IEEE 103, 793–821 (2015).
[Crossref]

Moreno, L. M. N.

F. Wijnands, J. B. Pendry, F. J. Garcia-Vidal, P. M. Bell, L. M. N. Moreno, and P. J. Roberts, “Green’s functions for Maxwell’s equations: Application to spontaneous emission,” Opt. Quant. Elect. 29, 199–216 (1997).
[Crossref]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. I.

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (Springer Science & Business Media, 2013).

Nicolet, A.

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
[Crossref]

Nicorovici, N. A.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[Crossref]

Niederle, A. E.

E. M. Graefe, H. J. Korsch, and A. E. Niederle, “A non-hermitian symmetric bose? hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A. Math. Theor. 41, 255206 (2008).
[Crossref]

Noda, S.

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, “Conversion of broadband to narrowband thermal emission through energy recycling,” Nat. Photon. 6, 535–539 (2012).
[Crossref]

Noh, H.

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Nori, F.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Nosich, A. I.

G. W. Hanson, A. I. Nosich, and E. M. Kartchevski, “Green’s function expansions in dyadic root functions for shielded layered waveguides,” Prog. Electromagn. Res. 39, 61–91 (2003).
[Crossref]

Novikov, S. M.

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Comm. 3, 969 (2012).
[Crossref]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2010).

Oliner, A. A.

D. R. Jackson and A. A. Oliner, Leaky-Wave Antennas (Wiley Online Library, 2008).

Oliveira, J. E.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Oskooi, A.

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, “Conversion of broadband to narrowband thermal emission through energy recycling,” Nat. Photon. 6, 535–539 (2012).
[Crossref]

Ostrovskaya, E. A.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Özdemir, S. K.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Padilla, W. J.

X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. 107, 045901 (2011).
[Crossref] [PubMed]

Papoff, F.

G. D’Alessandro and F. Papoff, “Giant subthreshold amplification in synchronously pumped optical parametric oscillators,” Phys. Rev. A 80, 023804 (2009).
[Crossref]

Pedersen, K.

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Comm. 3, 969 (2012).
[Crossref]

Pendry, J. B.

F. Wijnands, J. B. Pendry, F. J. Garcia-Vidal, P. M. Bell, L. M. N. Moreno, and P. J. Roberts, “Green’s functions for Maxwell’s equations: Application to spontaneous emission,” Opt. Quant. Elect. 29, 199–216 (1997).
[Crossref]

Peng, B.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Perrin, M.

Petermann, K.

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quant. Elect. 15, 566–570 (1979).
[Crossref]

Pick, A.

C. Khandekar, W. Jin, O. D. Miller, A. Pick, and A. W. Rodriguez, “Giant frequency-selective near-field heat transfer in PT-symmetric structures,” Phys. Rev. B 94, 115402 (2016).
[Crossref]

Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

A. Pick, Z. Lin, W. Jin, and A. W. Rodriguez, “Enhanced frequency conversion and nonlinear purcell effect at exceptional points,” (In preparation).

Pitris, C.

J. G. Fujimoto, C. Pitris, S. A. Boppart, and M. E. Brezinski, “Optical coherence tomography: An emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
[Crossref] [PubMed]

Polimeridis, A. G.

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Rabl, P.

J. Doppler, A. A. Mailybaev, J. Böhm, A. G. U. Kuhl, 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).
[Crossref] [PubMed]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Reid, M. T. H.

Roberts, P. J.

F. Wijnands, J. B. Pendry, F. J. Garcia-Vidal, P. M. Bell, L. M. N. Moreno, and P. J. Roberts, “Green’s functions for Maxwell’s equations: Application to spontaneous emission,” Opt. Quant. Elect. 29, 199–216 (1997).
[Crossref]

Rodriguez, A. W.

Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

C. Khandekar, W. Jin, O. D. Miller, A. Pick, and A. W. Rodriguez, “Giant frequency-selective near-field heat transfer in PT-symmetric structures,” Phys. Rev. B 94, 115402 (2016).
[Crossref]

A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

A. Pick, Z. Lin, W. Jin, and A. W. Rodriguez, “Enhanced frequency conversion and nonlinear purcell effect at exceptional points,” (In preparation).

Rotter, S.

J. Doppler, A. A. Mailybaev, J. Böhm, A. G. U. Kuhl, 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).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref] [PubMed]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. comm. 5, 1 (2014).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

Rüter, C. E.

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).
[Crossref]

Ryu, J.-W.

J.-W. Ryu, S.-Y. Lee, and S. W. Kim, “Analysis of multiple exceptional points related to three interacting eigenmodes in a non-hermitian hamiltonian,” Phys. Rev. A 85, 042101 (2012).
[Crossref]

S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, “Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity,” Phys. Rev. A 78, 015805 (2008).
[Crossref]

Sacked, K.

K. Sacked, Optical Properties of Photonic Crystals (Springer Science & Business Media, 2004), Vol. LXXX.

Sauvan, C.

C. Sauvan, J.-P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref] [PubMed]

Scherer, A.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Schneider, C.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Schöberl, J.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. comm. 5, 1 (2014).
[Crossref]

Schomerus, H.

G. Yoo, H.-S. Sim, and H. Schomerus, “Quantum noise and mode nonorthogonality in non-Hermitian PT-symmetric optical resonators,” Phys. Rev. A 84, 063833 (2011).
[Crossref]

Segev, M.

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).
[Crossref]

Seyranian, A. P.

A. P. Seyranian and A. A. Mailybaev, Multiparameter Stability Theory With Mechanical Applications (World Scientific Publishing, 2003), Vol. XIII.
[Crossref]

Shim, J.-B.

S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, “Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity,” Phys. Rev. A 78, 015805 (2008).
[Crossref]

Siegman, A. E.

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[Crossref]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[Crossref]

A. E. Siegman, Lasers (University Science Books, 1986).

A. E. Siegman, “Excess quantum noise in nonnormal oscillators,” Frontiers of Laser Physics and Quantum Optics (Springer, 2000).
[Crossref]

Sim, H.-S.

G. Yoo, H.-S. Sim, and H. Schomerus, “Quantum noise and mode nonorthogonality in non-Hermitian PT-symmetric optical resonators,” Phys. Rev. A 84, 063833 (2011).
[Crossref]

Smith, C. J. M.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” App. Phys. Lett. 76, 532–534 (2000).
[Crossref]

Soljacic, M.

O. D. Miller, A. G. Polimeridis, M. T. H. Reid, C. H. H. Chia, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Fundamental limits to optical response in absorptive systems,” Opt. express 24, 3329–3364 (2016).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Søndergaard, T.

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Comm. 3, 969 (2012).
[Crossref]

Starr, A. F.

X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. 107, 045901 (2011).
[Crossref] [PubMed]

Starr, T.

X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. 107, 045901 (2011).
[Crossref] [PubMed]

Stone, A. D.

A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

Strasser, G.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. comm. 5, 1 (2014).
[Crossref]

Sun, J.

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).
[Crossref] [PubMed]

Sun, Y.

Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with pt phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref] [PubMed]

Tan, W.

Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with pt phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref] [PubMed]

Thijssen, M. S.

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314 (1996).
[Crossref] [PubMed]

Tiggelen, B. A. V.

A. Lagendijk and B. A. V. Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[Crossref]

Trefethen, L. N.

L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (Princeton University, 2005).

Truscott, A. G.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Türeci, H. E.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. comm. 5, 1 (2014).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

Tyler, T.

X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. 107, 045901 (2011).
[Crossref] [PubMed]

Ujihara, K.

H. Yokoyama and K. Ujihara, Spontaneous Emission and Laser Oscillation in Microcavities (CRC Press, 1995), Vol. X.

Unterrainer, K.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. comm. 5, 1 (2014).
[Crossref]

van den Brink, A. M.

A. M. van den Brink, K. Young, and M. H. Yung, “Eigenvector expansion and Petermann factor for ohmically damped oscillators,” J. Phys A: Math. Gen. 39, 3725 (2006).
[Crossref]

van der Lee, A. M.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357 (1997).
[Crossref]

van Druten, N. J.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357 (1997).
[Crossref]

van Eijkelenborg, M. A.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357 (1997).
[Crossref]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314 (1996).
[Crossref] [PubMed]

Vial, B.

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
[Crossref]

Walasik, W.

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).
[Crossref] [PubMed]

Wan, W.

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Wang, Y.

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).
[Crossref] [PubMed]

Webb, K. J.

J. D. McKinney, M. A. Webster, K. J. Webb, and A. M. Weiner, “Characterization and imaging in optically scattering media by use of laser speckle and a variable-coherence source,” Opt. Let. 25, 4–6 (2000).
[Crossref]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Elsevier Academic Press, 2006).

Webster, M. A.

J. D. McKinney, M. A. Webster, K. J. Webb, and A. M. Weiner, “Characterization and imaging in optically scattering media by use of laser speckle and a variable-coherence source,” Opt. Let. 25, 4–6 (2000).
[Crossref]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3d perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[Crossref]

Weiner, A. M.

J. D. McKinney, M. A. Webster, K. J. Webb, and A. M. Weiner, “Characterization and imaging in optically scattering media by use of laser speckle and a variable-coherence source,” Opt. Let. 25, 4–6 (2000).
[Crossref]

Weintraub, S. H.

S. H. Weintraub, “Jordan canonical form: Application to differential equations,” Synth. Lect. Math. Statist. 1, 1–85 (2008).
[Crossref]

Weisbuch, C.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” App. Phys. Lett. 76, 532–534 (2000).
[Crossref]

Wijnands, F.

F. Wijnands, J. B. Pendry, F. J. Garcia-Vidal, P. M. Bell, L. M. N. Moreno, and P. J. Roberts, “Green’s functions for Maxwell’s equations: Application to spontaneous emission,” Opt. Quant. Elect. 29, 199–216 (1997).
[Crossref]

Wilton, a. B. F. D. R.

F. Capolino, D. R. Jackson, and a. B. F. D. R. Wilton, “Comparison of methods for calculating the field excited by a dipole near a 2-d periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Woerdman, J. P.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357 (1997).
[Crossref]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314 (1996).
[Crossref] [PubMed]

Wong, Z. J.

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).
[Crossref] [PubMed]

Wunner, G.

W. D. Heiss and G. Wunner, “Resonance scattering at third-order exceptional points,” J. Phys. A. Math. Theor. 48, 345203 (2015).
[Crossref]

W. D. Heiss and G. Wunner, “Fano-Feshbach resonances in two-channel scattering around exceptional points,” Eur. Phys. J. D 68, 1–6 (2014).
[Crossref]

Xiao, M.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

Xu, H.

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).
[Crossref] [PubMed]

Xu, Y.

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, 033807 (2000).
[Crossref]

Xu, Y.-L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Yamamoto, Y.

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Yang, L.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref] [PubMed]

Yao, A. M.

W. J. Firth and A. M. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. 95, 073903 (2005).
[Crossref] [PubMed]

Yariv, A.

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, 033807 (2000).
[Crossref]

Yilmaz, H.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref] [PubMed]

Yokoyama, H.

H. Yokoyama and K. Ujihara, Spontaneous Emission and Laser Oscillation in Microcavities (CRC Press, 1995), Vol. X.

Yoo, G.

G. Yoo, H.-S. Sim, and H. Schomerus, “Quantum noise and mode nonorthogonality in non-Hermitian PT-symmetric optical resonators,” Phys. Rev. A 84, 063833 (2011).
[Crossref]

Young, K.

A. M. van den Brink, K. Young, and M. H. Yung, “Eigenvector expansion and Petermann factor for ohmically damped oscillators,” J. Phys A: Math. Gen. 39, 3725 (2006).
[Crossref]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994).
[Crossref] [PubMed]

Yung, M. H.

A. M. van den Brink, K. Young, and M. H. Yung, “Eigenvector expansion and Petermann factor for ohmically damped oscillators,” J. Phys A: Math. Gen. 39, 3725 (2006).
[Crossref]

Zhang, X.

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).
[Crossref] [PubMed]

Zhang, Z.

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).
[Crossref] [PubMed]

Zhang, Z. Q.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

Zhen, B.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Zolla, F.

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
[Crossref]

Zoysa, M. D.

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, “Conversion of broadband to narrowband thermal emission through energy recycling,” Nat. Photon. 6, 535–539 (2012).
[Crossref]

Adv. Opt. Photonics (1)

J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1, 58–106 (2009).
[Crossref]

App. Phys. Lett. (1)

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” App. Phys. Lett. 76, 532–534 (2000).
[Crossref]

Entropy (1)

D. C. Brody and E.-M. Graefe, “Information geometry of complex hamiltonians and exceptional points,” Entropy 15, 3361–3378 (2013).
[Crossref]

Eur. Phys. J. D (1)

W. D. Heiss and G. Wunner, “Fano-Feshbach resonances in two-channel scattering around exceptional points,” Eur. Phys. J. D 68, 1–6 (2014).
[Crossref]

IEEE J. Quant. Elect. (1)

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quant. Elect. 15, 566–570 (1979).
[Crossref]

IEEE Trans. Antennas Propag. (1)

F. Capolino, D. R. Jackson, and a. B. F. D. R. Wilton, “Comparison of methods for calculating the field excited by a dipole near a 2-d periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

IEEE Trans. Microw. Theory Techn. (1)

A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microw. Theory Techn. 35, 688–696 (1987).
[Crossref]

J. Mod. Opt. (1)

M. V. Berry, “Mode degeneracies and the Petermann excess-noise factor for unstable lasers,” J. Mod. Opt. 50, 63–81 (2003).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Phys A: Math. Gen. (1)

A. M. van den Brink, K. Young, and M. H. Yung, “Eigenvector expansion and Petermann factor for ohmically damped oscillators,” J. Phys A: Math. Gen. 39, 3725 (2006).
[Crossref]

J. Phys. A (1)

E. Hernandez, A. Jauregui, and A. Mondragon, “Degeneracy of resonances in a double barrier potential,” J. Phys. A 33, 4507–4523 (2000).
[Crossref]

J. Phys. A - Math. Theo. (1)

W. D. Heiss, “The physics of exceptional points,” J. Phys. A - Math. Theo. 45, 444016 (2012).
[Crossref]

J. Phys. A. Math. Theor. (2)

E. M. Graefe, H. J. Korsch, and A. E. Niederle, “A non-hermitian symmetric bose? hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A. Math. Theor. 41, 255206 (2008).
[Crossref]

W. D. Heiss and G. Wunner, “Resonance scattering at third-order exceptional points,” J. Phys. A. Math. Theor. 48, 345203 (2015).
[Crossref]

Light Sci. Appl. (1)

H. Linnenbank, Y. Grynko, J. Förstner, and S. Linden, “Second harmonic generation spectroscopy on hybrid plasmonic/dielectric nanoantennas,” Light Sci. Appl. 5, e16013 (2016).
[Crossref]

Microw. Opt. Technol. Lett. (1)

W. C. Chew and W. H. Weedon, “A 3d perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[Crossref]

Nat. Comm. (1)

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Comm. 3, 969 (2012).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. comm. 5, 1 (2014).
[Crossref]

Nat. Mater. (1)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Nat. Photon. (1)

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, “Conversion of broadband to narrowband thermal emission through energy recycling,” Nat. Photon. 6, 535–539 (2012).
[Crossref]

Nat. Phys. (3)

D. Heiss, “Mathematical physics: Circling exceptional points,” Nat. Phys. 12, 823–824 (2016).
[Crossref]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

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).
[Crossref]

Nature (4)

J. Doppler, A. A. Mailybaev, J. Böhm, A. G. U. Kuhl, 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).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
[Crossref] [PubMed]

Neoplasia (1)

J. G. Fujimoto, C. Pitris, S. A. Boppart, and M. E. Brezinski, “Optical coherence tomography: An emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
[Crossref] [PubMed]

Opt. express (1)

Opt. Let. (1)

J. D. McKinney, M. A. Webster, K. J. Webb, and A. M. Weiner, “Characterization and imaging in optically scattering media by use of laser speckle and a variable-coherence source,” Opt. Let. 25, 4–6 (2000).
[Crossref]

Opt. Lett. (2)

Opt. Quant. Elect. (1)

F. Wijnands, J. B. Pendry, F. J. Garcia-Vidal, P. M. Bell, L. M. N. Moreno, and P. J. Roberts, “Green’s functions for Maxwell’s equations: Application to spontaneous emission,” Opt. Quant. Elect. 29, 199–216 (1997).
[Crossref]

Phys. Rep. (1)

A. Lagendijk and B. A. V. Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[Crossref]

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Phys. Rev. A (12)

S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, “Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity,” Phys. Rev. A 78, 015805 (2008).
[Crossref]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[Crossref]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[Crossref]

G. Yoo, H.-S. Sim, and H. Schomerus, “Quantum noise and mode nonorthogonality in non-Hermitian PT-symmetric optical resonators,” Phys. Rev. A 84, 063833 (2011).
[Crossref]

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, 033807 (2000).
[Crossref]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

G. D’Alessandro and F. Papoff, “Giant subthreshold amplification in synchronously pumped optical parametric oscillators,” Phys. Rev. A 80, 023804 (2009).
[Crossref]

E. Hernàndez, A. Jàuregui, and A. Mondragòn, “Jordan blocks and Gamow-Jordan eigenfunctions associated with a degeneracy of unbound states,” Phys. Rev. A 67, 022721 (2003).
[Crossref]

J.-W. Ryu, S.-Y. Lee, and S. W. Kim, “Analysis of multiple exceptional points related to three interacting eigenmodes in a non-hermitian hamiltonian,” Phys. Rev. A 85, 042101 (2012).
[Crossref]

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
[Crossref]

A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994).
[Crossref] [PubMed]

Phys. Rev. B (1)

C. Khandekar, W. Jin, O. D. Miller, A. Pick, and A. W. Rodriguez, “Giant frequency-selective near-field heat transfer in PT-symmetric structures,” Phys. Rev. B 94, 115402 (2016).
[Crossref]

Phys. Rev. E (1)

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[Crossref]

Phys. Rev. Lett. (12)

S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. 77, 2444 (1996).
[Crossref] [PubMed]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. 107, 045901 (2011).
[Crossref] [PubMed]

M. Laroche, R. Carminati, and J.-J. Greffet, “Coherent thermal antenna using a photonic crystal slab,” Phys. Rev. Lett. 96, 123903 (2006).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with pt phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref] [PubMed]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314 (1996).
[Crossref] [PubMed]

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357 (1997).
[Crossref]

W. J. Firth and A. M. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. 95, 073903 (2005).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

C. Sauvan, J.-P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref] [PubMed]

Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

Phys. Rev. X (1)

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).

Proc. IEEE (1)

F. Monticone and A. Alù, “Leaky-wave theory, techniques, and applications: From microwaves to visible frequencies,” Proc. IEEE 103, 793–821 (2015).
[Crossref]

Prog. Electromagn. Res. (1)

G. W. Hanson, A. I. Nosich, and E. M. Kartchevski, “Green’s function expansions in dyadic root functions for shielded layered waveguides,” Prog. Electromagn. Res. 39, 61–91 (2003).
[Crossref]

Science (5)

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).
[Crossref] [PubMed]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Synth. Lect. Math. Statist. (1)

S. H. Weintraub, “Jordan canonical form: Application to differential equations,” Synth. Lect. Math. Statist. 1, 1–85 (2008).
[Crossref]

Other (22)

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Science & Business Media, 2007).

K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2010).

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).

A. P. F. Hernàndez and S. G. Johnson, “Scalable computation of jordan chains,” arXiv:1704.05837 (2017).

A. Taflove, A. Oskooi, p., and S. G. Johnson, Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology (Artech House, 2013), http://arxiv.org/ftp/arxiv/papers/1301/1301.5366.pdf .

J. D. Jackson, Classical Electrodynamics, Edition III (John Wiley and Sons, 1999).

L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (Princeton University, 2005).

R. G. Newton, Scattering Theory of Waves and Particles (Springer Science & Business Media, 2013).

K. Sacked, Optical Properties of Photonic Crystals (Springer Science & Business Media, 2004), Vol. LXXX.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Elsevier Academic Press, 2006).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. I.

A. E. Siegman, Lasers (University Science Books, 1986).

A. P. Seyranian and A. A. Mailybaev, Multiparameter Stability Theory With Mechanical Applications (World Scientific Publishing, 2003), Vol. XIII.
[Crossref]

N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University, 2011).

H. Yokoyama and K. Ujihara, Spontaneous Emission and Laser Oscillation in Microcavities (CRC Press, 1995), Vol. X.

S. V. Gaponenko, Introduction to Nanophotonics (Cambridge University, 2010).
[Crossref]

A. Pick, Z. Lin, W. Jin, and A. W. Rodriguez, “Enhanced frequency conversion and nonlinear purcell effect at exceptional points,” (In preparation).

D. R. Jackson and A. A. Oliner, Leaky-Wave Antennas (Wiley Online Library, 2008).

N. Colthup, Introduction to Infrared and Raman Spectroscopy (Elsevier, 2012).

A. E. Siegman, “Excess quantum noise in nonnormal oscillators,” Frontiers of Laser Physics and Quantum Optics (Springer, 2000).
[Crossref]

R. E. Collin, Field theory of guided waves (McGraw-Hill, 1960).

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Figures (7)

Fig. 1
Fig. 1

Plasmonic system with exceptional points. (a) Left: Two plasmonic resonators with silver cores and a silica coating, with gain and loss (red/blue) added to the outer sides of the coating. Right: Eigenvalue trajectories in the complex plane upon increasing gain/loss. The trajectories merge at the EP (orange dot). (b) The Petermann factor, ( E n R , E n R ) ( E n L , E n L ) / | ( E n L , E n R ) | 2, diverges at the EP (left) while the LDOSx [evaluated using Eq. (2)] remains finite (right), since the giant contributions of the terms in Eq. (2) have opposite signs (blue/red curves, scaled by 10−3).

Fig. 2
Fig. 2

Spectral properties of LDOS at EPs. (a) LDOS peak vs. quality factor Q for five coupling values κ, showing quadratic (linear) scaling for large (small) κ values. Q is varied by adding gain to the coating, while κ is varied by changing the rod–rod separation. Inset: Red and blue points from the main plot, on a log-log scale. (b) Normalized linewidth (FWHM) at the EP, Γ/Γ0, vs. normalized coupling, κ/γ, computed using: 1. FDFD-discretization of Maxwell’s equations (dots), and 2. the CMT-based linewidth formula, Eq. (16) (black line). The limit as κ/γ → ∞ is shown in purple (dashed line). (Γ0 ≡ 2γ is the FWHM of a Lorentzian curve, see text.) Inset: LDOSx at the EP, computed via FDFD (black) and via CMT [Eq. (14), red dashed line].

Fig. 3
Fig. 3

Passive periodic waveguides with EPs. (a) Top left: Periodic waveguide with outgoing boundary condition in the transverse direction. Bottom left: Field patterns of TM modes at k = 0 (only the real part is shown). Top right: Degenerate mode and Jordan vector at kEP. Bottom right: Resonance [ωn (k)] vs. k-vector, showing an EP (orange dot) at kEPa/2π ≈ 1.46 × 10−4. (b) Top plot, positive y-axis: LDOSk at r0 for four k-values (kEP, k1, k2, k, marked on the lower plot). Negative y-axis: Resonances [ωn (k)] in the complex plane. Bottom: Normalized LDOSk peak (Mk/M) vs. deviation from the EP (Δk) for four structures with different Q values, showing 4-fold enhancement at the EP. (Mk is the LDOS peak at k.)

Fig. 4
Fig. 4

LDOSk enhancement in active structures. (a) Resonances with different passive loss rates γp (hollow symbols) and equal active loss rates γa (full symbols). Inset: Structure from Fig. 3(a) with gain/loss added to the waveguide. (b) Normalized LDOSk peak (Mk =M) vs. Δk, evaluated at r0 [marked in Fig. 3(a)], after adding gain/loss. The enhancement at the EP increases with the overall gain.

Fig. 5
Fig. 5

Leaky modes in one dimension. (a) A slab in air with PML at the cell boundary. (b) The intensity of the leaky mode increases exponentially in the air region and is attenuated in the PML.

Fig. 6
Fig. 6

Forcing EPs in the periodic waveguides. (a) Real and (b) imaginary parts of the eigenvalues of Maxwell’s operator, as a function of the wavevector kx and the permittivity contrast δεε2ε1, for the geometry from Fig. 3(a). The EP is found by numerically minimizing the distance between the two eigenvalue sheets (red and blue surfaces). Cyan dots: Eigenvalues for fixed kx = kEP and varying δε. Black dots: Eigenvalues for fixed δε = δεEP and varying kx.

Fig. 7
Fig. 7

LDOS enhancement in active periodic waveguides. Middle: LDOS peak (Mk) vs. Δk for increasing amounts of gain in the waveguide from Fig. 3(a). Left: log10(MEP) vs. log10(Qa) for the data from the middle panel near kEP, showing quadratic scaling with Qa. Right: log10(M) vs. log10(Qa) for the data from the middle panel at k, showing linear scaling with Qa. Inset: Eigenvalues move vertically in the complex plane upon adding gain.

Equations (55)

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LDOS μ ( x , ω ) = 2 ω π Im [ G μ μ ( x , x , ω ) ] .
G μ μ ( ω , x , x ) = n E n μ R ( x ) E n μ L ( x ) ( ω 2 ω n 2 ) ( E n L , E n R ) .
LDOS μ ( x , ω ) n 1 π γ n ( ω Ω n ) 2 + γ n 2 Re [ E n μ R ( x ) E n μ L ( x ) ( E n L , E n R ) ] .
A ^ 0 E 0 R = λ EP E 0 R , A ^ 0 J 0 R = λ EP J 0 R + E 0 R ,
A ^ ( p ) E ± R = λ ± E ± R ,
λ ± = λ 0 ± p 1 2 λ 1 + p λ 2 ± p 3 2 λ 3 + E ± R = E 0 R ± p 1 2 E 1 R + p E 2 R ± p 3 2 E 3 R +
λ ± = λ 0 ± p 1 2 Δ + O ( p ) , E ± R = E 0 R ± p 1 2 Δ J 0 R + O ( p ) ,
G μ μ EP ( ω , x , x ) lim p 0 [ E + μ R ( x ) E + μ L ( x ) ( λ λ + ) ( E + L , E + R ) + E μ R ( x ) E μ L ( x ) ( λ λ ) ( E L , E R ) ] .
G μ μ EP ( ω , x , x ) 1 ( ω 2 ω EP 2 ) 2 E 0 μ R ( x ) E 0 μ L ( x ) ( E 0 L , J 0 R ) + 1 ω 2 ω EP 0 E 0 μ R ( x ) J 0 μ L ( x ) + J 0 μ R ( x ) E 0 μ L ( x ) ( E 0 L , J 0 R ) .
LDOS μ EP ( x , ω ) Ω n 2 π ( γ n ( ω Ω n ) 2 + γ n 2 ) 2 × [ 1 2 Im ( E 0 μ R ( x ) E 0 μ L ( x ) ( E n 0 , J n 0 ) ) ω Ω n γ n Re ( E 0 μ R ( x ) E 0 μ L ( x ) ( E n 0 , J n 0 ) ) ] .
A ^ = V T A ^ U = ( [ ω EP i η ] 2 2 Ω EP κ 2 Ω EP κ [ ω EP + i η ] 2 ) .
G EP ( r , r , ω ) ( A ^ 0 ω 2 1 ) 1 U ( A ^ 0 ω 2 1 ) 1 V T .
( A ^ 0 ω 2 1 ) [ 1 , 1 ] 1 = 1 ω 2 ω EP 2 + 2 Ω EP κ [ ω 2 ω EP 2 ] 2 .
LDOS x ( r 0 , ω ) E 1 R ( r 0 ) E 1 L ( r 0 ) 2 Ω EP [ γ ( ω Ω EP ) 2 + γ 2 + κ [ γ 2 ( ω Ω EP ) 2 ] [ γ 2 ( ω Ω EP ) 2 ] 2 + [ 2 γ ( ω Ω EP ) ] 2 ] .
M EP max ω { LDOS ( r 0 , ω ) } 1 γ + κ γ 2 .
Γ = Γ 0 γ 2 + 2 γ κ + 5 κ 2 2 κ γ + κ .
A ^ = ( [ Ω EP ± i η ] 2 2 Ω EP ν g k 2 Ω EP ν g k [ Ω EP 2 i γ p ± i η ] 2 ) ,
κ | ω EP | max | Im ε | .
( A ^ r ω 2 1 ) E ( r ) = i ω J ( r ) .
( A ^ r ω 2 1 ) G ( r , r ) = δ ( r r ) 1 .
( A ^ r ω n 2 1 ) E n R ( r ) = 0 ( A ^ r T ω n 2 1 ) E n L ( r ) = 0 .
δ ( r r ) 1 = n E n R ( r ) E n L ( r ) ,
G ( r , r ) = n E n R ( r ) a n ( r ) .
n ( A r ω 2 ) E n R ( r ) a n ( r ) = n E n R ( r ) E n L ( r ) .
d r E m L ( r ) E n R ( r ) = δ m , n ,
G ( r , r , ω ) = n 1 ω 2 ω n 2 E n R ( r ) E n L ( r ) d r E n R ( r ) E n L ( r ) .
G ( r , r 0 , ω ) = m E m R ( r ) α m ( ω , r 0 ) ,
[ × × ω 2 ε ( r , ω ) ] G ( r , r 0 , ω ) = δ ( r r 0 ) 1
E n L ( r 0 ) = d r E n L ( r ) [ ω 2 ε ( r , ω ) ω n 2 ε ( r , ω n ) ] G ( r , r 0 , ω )
E n L ( r 0 ) = m d r E n L ( r ) [ ω 2 ε ( ω ) ω n 2 ε ( ω n ) ] E m R ( r ) α m ( ω , r 0 ) m ( E m , E n ) α m ( ω , r 0 ) .
G ( r , r 0 , ω ) = n m ( O 1 ) n m E n R ( r ) E m L ( r 0 )
O i j = d r E i L [ ω 2 ε ( r , ω ) ω i 2 ε ( r , ω i ) ] E j R .
ω 2 ε ( ω ) ω ± 2 ε ( ω ± ) = ( ω ω ± ) d ( ω 2 ε ) d ω |   ω 0 ( ω ω ± ) ( ω 2 ε ) 0 ,
O 1 = ( 1 ω ω + ( ω 2 ε ) 0 E L E R N 1 ω ω + ( ω 2 ε ) 0 E + L E R N 1 ω ω ( ω 2 ε ) 0 E L E + R N 1 ω ω ( ω 2 ε ) 0 E + L E + R N ) ,
G ( r , r 0 , ω ) = E + R E + L ω ω + ( ω 2 ε ) 0 E L E R N + E R E L ω ω ( ω 2 ε ) 0 E + L E + R N E R E + L ω ω + ( ω 2 ε ) 0 E + L E R N E + R E L ω ω ( ω 2 ε ) 0 E L E + R N .
ω ± ω 0 ± p Δ , E ± E 0 ± p Δ J 0 .
( ω 2 ε ) 0 E ± L E ± R = ( ω 2 ε ) 0 E 0 L E 0 R ± 2 p Δ ( ω 2 ε ) 0 E 0 L J 0 R + p Δ 2 ( ω 2 ε ) 0 J 0 L J 0 R + O ( p 3 / 2 ) ( ω 2 ε ) 0 E ± L E ± R = ( ω 2 ε ) 0 E 0 L E 0 R p Δ 2 ( ω 2 ε ) 0 J 0 L J 0 R + O ( p 3 / 2 ) N = 4 p Δ 2 ( ω 2 ε ) 0 E 0 L J 0 R + O ( p 3 / 2 )
G EP ( r , r 0 , ω ) = E 0 R ( r ) E 0 L ( r 0 ) ( ω ω 0 ) 2 + J 0 R ( r ) E 0 L ( r 0 ) + E 0 R ( r ) J 0 L ( r 0 ) ω ω 0 .
x ˜ = x + i σ 0 ( x L 2 ) 2 2 ω d .
E n = { A n e i ω n x ( σ 0 / 2 d ) ( x + L / 2 ) 2 + B n e i ω n x + ( σ 0 / 2 d ) ( x + L / 2 ) 2 N / 2 < x < L / 2 A n e i ω n x + B n e i ω n x L / 2 < x < a / 2 e i ω n n x + e i ω n n x a / 2 < x < a / 2 A n e i ω n x + B n e i ω n x a / 2 < x < L / 2 A n e i ω n x ( σ 0 / 2 d ) ( x L / 2 ) 2 + B n e i ω n x + ( σ 0 / 2 d ) ( x L / 2 ) 2 L / 2 < x < N / 2 ,
A n = e i n k a / 2 + e i n k a / 2 e i k a / 2 C n e i k a / 2 , B n = A n C n ,
J ( x ) x d x ε ( A n e i ω n x + B n e i ω n x ) 2 ,
( d ) all d x ε E 2 = Cav d x ε E 2 + 2 [ J ( L 2 ) J ( a 2 ) ] + 2 { J [ x ˜ ( N 2 ) ] J [ x ˜ ( L 2 ) ] } .
[ ω n ( α + 2 ) + σ 0 ] d 1 .
lim d ( d ) = 2 n ω n [ sin ( n ω n a ) + n ω n a ] + 4 i ω n cos ( n ω n a 2 ) .
A ^ E 1 ε × × E = ω 2 E .
1 ε × × E R = ω 2 E R ,
E j L 1 ε × × E j R E j L 1 ε j × × E j R ω j 2 .
E i L 1 ε × × E j R E i L ε j ε 1 ε j × × E j R = ω j 2 E i L ε Δ ε j ε E j R = 2 ω j ( ω j 2 E i L Δ ε j ε E j R ) 2 ω j κ i j
A ^ = V T A ^ U = ( ( ω EP i η ) 2 2 Ω EP κ 2 Ω EP κ ( ω EP + i η ) 2 ) .
E 0 R ( E + R + E R ) / 2
J 0 R ( E + R E R ) / ( 2 λ 1 p 1 2 ) .
E 0 R , E 0 R E + , E + / 2 J 0 R , J 0 R E + , E + / 2 | λ 1 | 2 p .
E 0 R , E 0 R J 0 R , J 0 R | λ 1 | 2 p | ( E 0 L , A ^ , E 0 R ) ( J 0 L , E 0 R ) , |
| ( E 0 L , A ^ , E 0 R ) | | ω EP 2 E 0 L ( Im ε / ε ) E 0 R | = | ω EP 2 Im ε E 0 R 2 | | ω EP 2 Im ε | E 0 R | 2 | | ω EP | 2 max | Im ε | ,