Abstract

Complex metamaterials with multiple optical resonances in constituent elements possess many similarities with open quantum systems that can be described by non-Hermitian Hamiltonian. By analogy with a two-state open quantum system, we show that a classic analogue of exceptional points can be observed in the transmission spectra of dual subwavelength metallic gratings. Anti-crossing (crossing) between the two branches λR of extraordinary optical transmission, with crossing (anti-crossing) of the corresponding widths ΓR, is observed in the parameter space spanned by the lateral displacement L and the angle of incidence φ0. Exchanges of field patterns and phases, and the variation of field profile when circling the exceptional point are discussed. This work highlights the potential to transfer the concepts and applications from open quantum systems to optical metamaterials.

© 2013 Optical Society of America

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
    [Crossref]
  2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [Crossref] [PubMed]
  3. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
    [Crossref] [PubMed]
  4. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
    [Crossref] [PubMed]
  5. W. D. Heiss and A. L. Sannino, “Avoided level crossing and exceptional points,” J. Phys. A: Math. Gen. 23, 1167–1178 (1990).
    [Crossref]
  6. W. D. Heiss, “Phases of wave functions and level repulsion,” Eur. Phys. J. D 7, 1–4 (1999).
    [Crossref]
  7. W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
    [Crossref]
  8. N. Moiseyev, Non-Hermitian Quantum Mechanics(Cambridge University Press, 2011).
    [Crossref]
  9. Q. H. Song and H. Cao, “Improving optical confinement in nanostructures via external mode coupling,” Phys. Rev. Lett. 105, 053902 (2010).
    [Crossref] [PubMed]
  10. W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
    [Crossref]
  11. A. A. Mailybaev, O. N. Kirillov, and A. P. Seyranian, “Geometric phase around exceptional points,” Phys. Rev. A 72, 014104 (2005).
    [Crossref]
  12. S. Y. Lee, J. W. Ryu, S. W. Kim, and Y. Chung, “Geometric phase around multiple exceptional points,” Phys. Rev. A 85, 064103 (2012).
    [Crossref]
  13. C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
    [Crossref] [PubMed]
  14. J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
    [Crossref]
  15. B. Dietz, T. Friedrich, J. Metz, M. Miski-Oglu, A. Richter, F. Schäfer, and C. A. Stafford, “Rabi oscillations at exceptional points in microwave billiards,” Phys. Rev. E 75, 027201 (2007).
    [Crossref]
  16. J. Wiersig, S. W. Kim, and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 78, 053809 (2008).
    [Crossref]
  17. J. W. Ryu, S. Y. Lee, and S. W. Kim, “Coupled nonidentical microdisks: avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858 (2009).
    [Crossref]
  18. B. Dietz, H. L. Harney, O. N. Kirillov, M. Miski-Oglu, A. Richter, and F. Schäfer, “Exceptional points in a microwave billiard with time-reversal invariance violation,” Phys. Rev. Lett. 106, 150403 (2011).
    [Crossref] [PubMed]
  19. J. Wiersig, A. Eberspächer, J. B. Shim, J. W. Ryu, S. Shinohara, M. Hentschel, and H. Schomerus, “Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities,” Phys. Rev. A 84, 023845 (2011).
    [Crossref]
  20. 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]
  21. X. Yin and X. Zhang, “Unidirectional light propagation at exceptional points,” Nat. Mater. 12, 175–177 (2013).
    [Crossref] [PubMed]
  22. C. Cheng, J. Chen, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, and H. T. Wang, “Controllable electromagnetic transmission based on dual-metallic grating structures composed of subwavelength slits,” Appl. Phys. Lett. 91, 111111 (2007).
    [Crossref]
  23. C. Cheng, J. Chen, D. J. Shi, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, J. P. Ding, and H. T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78, 075406 (2008).
    [Crossref]
  24. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
    [Crossref] [PubMed]
  25. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
    [Crossref] [PubMed]

2013 (1)

X. Yin and X. Zhang, “Unidirectional light propagation at exceptional points,” Nat. Mater. 12, 175–177 (2013).
[Crossref] [PubMed]

2012 (2)

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]

S. Y. Lee, J. W. Ryu, S. W. Kim, and Y. Chung, “Geometric phase around multiple exceptional points,” Phys. Rev. A 85, 064103 (2012).
[Crossref]

2011 (2)

B. Dietz, H. L. Harney, O. N. Kirillov, M. Miski-Oglu, A. Richter, and F. Schäfer, “Exceptional points in a microwave billiard with time-reversal invariance violation,” Phys. Rev. Lett. 106, 150403 (2011).
[Crossref] [PubMed]

J. Wiersig, A. Eberspächer, J. B. Shim, J. W. Ryu, S. Shinohara, M. Hentschel, and H. Schomerus, “Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities,” Phys. Rev. A 84, 023845 (2011).
[Crossref]

2010 (1)

Q. H. Song and H. Cao, “Improving optical confinement in nanostructures via external mode coupling,” Phys. Rev. Lett. 105, 053902 (2010).
[Crossref] [PubMed]

2009 (2)

J. W. Ryu, S. Y. Lee, and S. W. Kim, “Coupled nonidentical microdisks: avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858 (2009).
[Crossref]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[Crossref] [PubMed]

2008 (3)

C. Cheng, J. Chen, D. J. Shi, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, J. P. Ding, and H. T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78, 075406 (2008).
[Crossref]

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

J. Wiersig, S. W. Kim, and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 78, 053809 (2008).
[Crossref]

2007 (2)

B. Dietz, T. Friedrich, J. Metz, M. Miski-Oglu, A. Richter, F. Schäfer, and C. A. Stafford, “Rabi oscillations at exceptional points in microwave billiards,” Phys. Rev. E 75, 027201 (2007).
[Crossref]

C. Cheng, J. Chen, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, and H. T. Wang, “Controllable electromagnetic transmission based on dual-metallic grating structures composed of subwavelength slits,” Appl. Phys. Lett. 91, 111111 (2007).
[Crossref]

2006 (2)

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
[Crossref]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

2005 (1)

A. A. Mailybaev, O. N. Kirillov, and A. P. Seyranian, “Geometric phase around exceptional points,” Phys. Rev. A 72, 014104 (2005).
[Crossref]

2003 (1)

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[Crossref] [PubMed]

2001 (3)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
[Crossref]

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref] [PubMed]

2000 (1)

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[Crossref]

1999 (1)

W. D. Heiss, “Phases of wave functions and level repulsion,” Eur. Phys. J. D 7, 1–4 (1999).
[Crossref]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

1990 (1)

W. D. Heiss and A. L. Sannino, “Avoided level crossing and exceptional points,” J. Phys. A: Math. Gen. 23, 1167–1178 (1990).
[Crossref]

Cao, H.

Q. H. Song and H. Cao, “Improving optical confinement in nanostructures via external mode coupling,” Phys. Rev. Lett. 105, 053902 (2010).
[Crossref] [PubMed]

Cerjan, A.

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]

Chen, J.

C. Cheng, J. Chen, D. J. Shi, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, J. P. Ding, and H. T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78, 075406 (2008).
[Crossref]

C. Cheng, J. Chen, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, and H. T. Wang, “Controllable electromagnetic transmission based on dual-metallic grating structures composed of subwavelength slits,” Appl. Phys. Lett. 91, 111111 (2007).
[Crossref]

Cheng, C.

C. Cheng, J. Chen, D. J. Shi, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, J. P. Ding, and H. T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78, 075406 (2008).
[Crossref]

C. Cheng, J. Chen, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, and H. T. Wang, “Controllable electromagnetic transmission based on dual-metallic grating structures composed of subwavelength slits,” Appl. Phys. Lett. 91, 111111 (2007).
[Crossref]

Chung, Y.

S. Y. Lee, J. W. Ryu, S. W. Kim, and Y. Chung, “Geometric phase around multiple exceptional points,” Phys. Rev. A 85, 064103 (2012).
[Crossref]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Dembowski, C.

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref] [PubMed]

Dietz, B.

B. Dietz, H. L. Harney, O. N. Kirillov, M. Miski-Oglu, A. Richter, and F. Schäfer, “Exceptional points in a microwave billiard with time-reversal invariance violation,” Phys. Rev. Lett. 106, 150403 (2011).
[Crossref] [PubMed]

B. Dietz, T. Friedrich, J. Metz, M. Miski-Oglu, A. Richter, F. Schäfer, and C. A. Stafford, “Rabi oscillations at exceptional points in microwave billiards,” Phys. Rev. E 75, 027201 (2007).
[Crossref]

Ding, J. P.

C. Cheng, J. Chen, D. J. Shi, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, J. P. Ding, and H. T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78, 075406 (2008).
[Crossref]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Eberspächer, A.

J. Wiersig, A. Eberspächer, J. B. Shim, J. W. Ryu, S. Shinohara, M. Hentschel, and H. Schomerus, “Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities,” Phys. Rev. A 84, 023845 (2011).
[Crossref]

Fan, Y. X.

C. Cheng, J. Chen, D. J. Shi, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, J. P. Ding, and H. T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78, 075406 (2008).
[Crossref]

C. Cheng, J. Chen, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, and H. T. Wang, “Controllable electromagnetic transmission based on dual-metallic grating structures composed of subwavelength slits,” Appl. Phys. Lett. 91, 111111 (2007).
[Crossref]

Fleischhauer, M.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[Crossref] [PubMed]

Friedrich, T.

B. Dietz, T. Friedrich, J. Metz, M. Miski-Oglu, A. Richter, F. Schäfer, and C. A. Stafford, “Rabi oscillations at exceptional points in microwave billiards,” Phys. Rev. E 75, 027201 (2007).
[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]

Genov, D. A.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Giessen, H.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[Crossref] [PubMed]

Gräf, H. D.

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref] [PubMed]

Halas, N. J.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[Crossref] [PubMed]

Harney, H. L.

B. Dietz, H. L. Harney, O. N. Kirillov, M. Miski-Oglu, A. Richter, and F. Schäfer, “Exceptional points in a microwave billiard with time-reversal invariance violation,” Phys. Rev. Lett. 106, 150403 (2011).
[Crossref] [PubMed]

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
[Crossref]

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref] [PubMed]

Heine, A.

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref] [PubMed]

Heiss, W. D.

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref] [PubMed]

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
[Crossref]

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[Crossref]

W. D. Heiss, “Phases of wave functions and level repulsion,” Eur. Phys. J. D 7, 1–4 (1999).
[Crossref]

W. D. Heiss and A. L. Sannino, “Avoided level crossing and exceptional points,” J. Phys. A: Math. Gen. 23, 1167–1178 (1990).
[Crossref]

Hentschel, M.

J. Wiersig, A. Eberspächer, J. B. Shim, J. W. Ryu, S. Shinohara, M. Hentschel, and H. Schomerus, “Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities,” Phys. Rev. A 84, 023845 (2011).
[Crossref]

J. Wiersig, S. W. Kim, and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 78, 053809 (2008).
[Crossref]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Kästel, J.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[Crossref] [PubMed]

Kim, S. W.

S. Y. Lee, J. W. Ryu, S. W. Kim, and Y. Chung, “Geometric phase around multiple exceptional points,” Phys. Rev. A 85, 064103 (2012).
[Crossref]

J. W. Ryu, S. Y. Lee, and S. W. Kim, “Coupled nonidentical microdisks: avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858 (2009).
[Crossref]

J. Wiersig, S. W. Kim, and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 78, 053809 (2008).
[Crossref]

Kirillov, O. N.

B. Dietz, H. L. Harney, O. N. Kirillov, M. Miski-Oglu, A. Richter, and F. Schäfer, “Exceptional points in a microwave billiard with time-reversal invariance violation,” Phys. Rev. Lett. 106, 150403 (2011).
[Crossref] [PubMed]

A. A. Mailybaev, O. N. Kirillov, and A. P. Seyranian, “Geometric phase around exceptional points,” Phys. Rev. A 72, 014104 (2005).
[Crossref]

Langguth, L.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[Crossref] [PubMed]

Lee, S. Y.

S. Y. Lee, J. W. Ryu, S. W. Kim, and Y. Chung, “Geometric phase around multiple exceptional points,” Phys. Rev. A 85, 064103 (2012).
[Crossref]

J. W. Ryu, S. Y. Lee, and S. W. Kim, “Coupled nonidentical microdisks: avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858 (2009).
[Crossref]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Liertzer, M.

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]

Liu, M.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

Liu, N.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[Crossref] [PubMed]

Mailybaev, A. A.

A. A. Mailybaev, O. N. Kirillov, and A. P. Seyranian, “Geometric phase around exceptional points,” Phys. Rev. A 72, 014104 (2005).
[Crossref]

Metz, J.

B. Dietz, T. Friedrich, J. Metz, M. Miski-Oglu, A. Richter, F. Schäfer, and C. A. Stafford, “Rabi oscillations at exceptional points in microwave billiards,” Phys. Rev. E 75, 027201 (2007).
[Crossref]

Miski-Oglu, M.

B. Dietz, H. L. Harney, O. N. Kirillov, M. Miski-Oglu, A. Richter, and F. Schäfer, “Exceptional points in a microwave billiard with time-reversal invariance violation,” Phys. Rev. Lett. 106, 150403 (2011).
[Crossref] [PubMed]

B. Dietz, T. Friedrich, J. Metz, M. Miski-Oglu, A. Richter, F. Schäfer, and C. A. Stafford, “Rabi oscillations at exceptional points in microwave billiards,” Phys. Rev. E 75, 027201 (2007).
[Crossref]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Moiseyev, N.

N. Moiseyev, Non-Hermitian Quantum Mechanics(Cambridge University Press, 2011).
[Crossref]

Nordlander, P.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[Crossref] [PubMed]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Pfau, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[Crossref] [PubMed]

Prodan, E.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[Crossref] [PubMed]

Radloff, C.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[Crossref] [PubMed]

Rehfeld, H.

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref] [PubMed]

Ren, F. F.

C. Cheng, J. Chen, D. J. Shi, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, J. P. Ding, and H. T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78, 075406 (2008).
[Crossref]

C. Cheng, J. Chen, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, and H. T. Wang, “Controllable electromagnetic transmission based on dual-metallic grating structures composed of subwavelength slits,” Appl. Phys. Lett. 91, 111111 (2007).
[Crossref]

Richter, A.

B. Dietz, H. L. Harney, O. N. Kirillov, M. Miski-Oglu, A. Richter, and F. Schäfer, “Exceptional points in a microwave billiard with time-reversal invariance violation,” Phys. Rev. Lett. 106, 150403 (2011).
[Crossref] [PubMed]

B. Dietz, T. Friedrich, J. Metz, M. Miski-Oglu, A. Richter, F. Schäfer, and C. A. Stafford, “Rabi oscillations at exceptional points in microwave billiards,” Phys. Rev. E 75, 027201 (2007).
[Crossref]

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref] [PubMed]

Rotter, S.

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]

Ryu, J. W.

S. Y. Lee, J. W. Ryu, S. W. Kim, and Y. Chung, “Geometric phase around multiple exceptional points,” Phys. Rev. A 85, 064103 (2012).
[Crossref]

J. Wiersig, A. Eberspächer, J. B. Shim, J. W. Ryu, S. Shinohara, M. Hentschel, and H. Schomerus, “Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities,” Phys. Rev. A 84, 023845 (2011).
[Crossref]

J. W. Ryu, S. Y. Lee, and S. W. Kim, “Coupled nonidentical microdisks: avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858 (2009).
[Crossref]

Sannino, A. L.

W. D. Heiss and A. L. Sannino, “Avoided level crossing and exceptional points,” J. Phys. A: Math. Gen. 23, 1167–1178 (1990).
[Crossref]

Schäfer, F.

B. Dietz, H. L. Harney, O. N. Kirillov, M. Miski-Oglu, A. Richter, and F. Schäfer, “Exceptional points in a microwave billiard with time-reversal invariance violation,” Phys. Rev. Lett. 106, 150403 (2011).
[Crossref] [PubMed]

B. Dietz, T. Friedrich, J. Metz, M. Miski-Oglu, A. Richter, F. Schäfer, and C. A. Stafford, “Rabi oscillations at exceptional points in microwave billiards,” Phys. Rev. E 75, 027201 (2007).
[Crossref]

Schomerus, H.

J. Wiersig, A. Eberspächer, J. B. Shim, J. W. Ryu, S. Shinohara, M. Hentschel, and H. Schomerus, “Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities,” Phys. Rev. A 84, 023845 (2011).
[Crossref]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Seyranian, A. P.

A. A. Mailybaev, O. N. Kirillov, and A. P. Seyranian, “Geometric phase around exceptional points,” Phys. Rev. A 72, 014104 (2005).
[Crossref]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Shi, D. J.

C. Cheng, J. Chen, D. J. Shi, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, J. P. Ding, and H. T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78, 075406 (2008).
[Crossref]

Shim, J. B.

J. Wiersig, A. Eberspächer, J. B. Shim, J. W. Ryu, S. Shinohara, M. Hentschel, and H. Schomerus, “Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities,” Phys. Rev. A 84, 023845 (2011).
[Crossref]

Shinohara, S.

J. Wiersig, A. Eberspächer, J. B. Shim, J. W. Ryu, S. Shinohara, M. Hentschel, and H. Schomerus, “Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities,” Phys. Rev. A 84, 023845 (2011).
[Crossref]

Smith, D. R.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Song, Q. H.

Q. H. Song and H. Cao, “Improving optical confinement in nanostructures via external mode coupling,” Phys. Rev. Lett. 105, 053902 (2010).
[Crossref] [PubMed]

Stafford, C. A.

B. Dietz, T. Friedrich, J. Metz, M. Miski-Oglu, A. Richter, F. Schäfer, and C. A. Stafford, “Rabi oscillations at exceptional points in microwave billiards,” Phys. Rev. E 75, 027201 (2007).
[Crossref]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Stone, A. D.

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]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Türeci, H. E.

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]

Wang, H. T.

C. Cheng, J. Chen, D. J. Shi, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, J. P. Ding, and H. T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78, 075406 (2008).
[Crossref]

C. Cheng, J. Chen, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, and H. T. Wang, “Controllable electromagnetic transmission based on dual-metallic grating structures composed of subwavelength slits,” Appl. Phys. Lett. 91, 111111 (2007).
[Crossref]

Wang, Y.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

Weiss, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[Crossref] [PubMed]

Wiersig, J.

J. Wiersig, A. Eberspächer, J. B. Shim, J. W. Ryu, S. Shinohara, M. Hentschel, and H. Schomerus, “Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities,” Phys. Rev. A 84, 023845 (2011).
[Crossref]

J. Wiersig, S. W. Kim, and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 78, 053809 (2008).
[Crossref]

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
[Crossref]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Wu, Q. Y.

C. Cheng, J. Chen, D. J. Shi, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, J. P. Ding, and H. T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78, 075406 (2008).
[Crossref]

C. Cheng, J. Chen, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, and H. T. Wang, “Controllable electromagnetic transmission based on dual-metallic grating structures composed of subwavelength slits,” Appl. Phys. Lett. 91, 111111 (2007).
[Crossref]

Xu, J.

C. Cheng, J. Chen, D. J. Shi, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, J. P. Ding, and H. T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78, 075406 (2008).
[Crossref]

C. Cheng, J. Chen, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, and H. T. Wang, “Controllable electromagnetic transmission based on dual-metallic grating structures composed of subwavelength slits,” Appl. Phys. Lett. 91, 111111 (2007).
[Crossref]

Yin, X.

X. Yin and X. Zhang, “Unidirectional light propagation at exceptional points,” Nat. Mater. 12, 175–177 (2013).
[Crossref] [PubMed]

Zhang, S.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

Zhang, X.

X. Yin and X. Zhang, “Unidirectional light propagation at exceptional points,” Nat. Mater. 12, 175–177 (2013).
[Crossref] [PubMed]

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

C. Cheng, J. Chen, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, and H. T. Wang, “Controllable electromagnetic transmission based on dual-metallic grating structures composed of subwavelength slits,” Appl. Phys. Lett. 91, 111111 (2007).
[Crossref]

Eur. Phys. J. D (2)

W. D. Heiss, “Phases of wave functions and level repulsion,” Eur. Phys. J. D 7, 1–4 (1999).
[Crossref]

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
[Crossref]

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

W. D. Heiss and A. L. Sannino, “Avoided level crossing and exceptional points,” J. Phys. A: Math. Gen. 23, 1167–1178 (1990).
[Crossref]

Nat. Mater. (2)

X. Yin and X. Zhang, “Unidirectional light propagation at exceptional points,” Nat. Mater. 12, 175–177 (2013).
[Crossref] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[Crossref] [PubMed]

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Phys. Rev. A (5)

A. A. Mailybaev, O. N. Kirillov, and A. P. Seyranian, “Geometric phase around exceptional points,” Phys. Rev. A 72, 014104 (2005).
[Crossref]

S. Y. Lee, J. W. Ryu, S. W. Kim, and Y. Chung, “Geometric phase around multiple exceptional points,” Phys. Rev. A 85, 064103 (2012).
[Crossref]

J. Wiersig, S. W. Kim, and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 78, 053809 (2008).
[Crossref]

J. W. Ryu, S. Y. Lee, and S. W. Kim, “Coupled nonidentical microdisks: avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858 (2009).
[Crossref]

J. Wiersig, A. Eberspächer, J. B. Shim, J. W. Ryu, S. Shinohara, M. Hentschel, and H. Schomerus, “Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities,” Phys. Rev. A 84, 023845 (2011).
[Crossref]

Phys. Rev. B (1)

C. Cheng, J. Chen, D. J. Shi, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, J. P. Ding, and H. T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78, 075406 (2008).
[Crossref]

Phys. Rev. E (2)

B. Dietz, T. Friedrich, J. Metz, M. Miski-Oglu, A. Richter, F. Schäfer, and C. A. Stafford, “Rabi oscillations at exceptional points in microwave billiards,” Phys. Rev. E 75, 027201 (2007).
[Crossref]

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[Crossref]

Phys. Rev. Lett. (6)

Q. H. Song and H. Cao, “Improving optical confinement in nanostructures via external mode coupling,” Phys. Rev. Lett. 105, 053902 (2010).
[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]

B. Dietz, H. L. Harney, O. N. Kirillov, M. Miski-Oglu, A. Richter, and F. Schäfer, “Exceptional points in a microwave billiard with time-reversal invariance violation,” Phys. Rev. Lett. 106, 150403 (2011).
[Crossref] [PubMed]

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref] [PubMed]

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
[Crossref]

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

Science (3)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[Crossref] [PubMed]

Other (1)

N. Moiseyev, Non-Hermitian Quantum Mechanics(Cambridge University Press, 2011).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Schematic view of the unit cell of a dual metallic grating. The first (second) grating, meta-atom 1 (2), has a thickness of h1(h2). The two meta-atoms are longitudinal displaced by G in the z direction, and lateral displaced by L in the x direction. The angle of incidence is φ0.

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Fig. 2
Fig. 2 Transmission spectra of DMGs at (a) φ0 = 4°, (b) φ0 = 5°, (c) φ0 = 9°, and (d) φ0 = 12°, respectively. Plot (e) is a simple diagram of the topological structure of the EOT peaks in the 2D plane of parameters (L, φ0).

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Fig. 3
Fig. 3 Wavelengths λR and widths ΓR of the transmission branches at (a) φ0 = 4.5° < φc and (b) φ0 = 10° > φc, respectively.

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Fig. 4
Fig. 4 Distributions of field intensity I in the two transmission branches at φ0 = 4.5° for different L values. Upper (lower) row is for the branch with smaller (larger) widths ΓR as shown in Fig. 3(a).

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Fig. 5
Fig. 5 Distributions of field intensity I in the two branches at φ0 = 10° for different L values. Upper (lower) row is for the branch with larger (shorter) wavelengths λR as shown in Fig. 3(b).

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Fig. 6
Fig. 6 Phase difference Δθ versus L for (a) φ0 = 4.5° and (b) φ0 = 10°. Inset shows the definition of phases θ1 and θ2, from which we can calculate Δθ = θ1 − θ2.

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Equations (6)

Equations on this page are rendered with MathJax. Learn more.

( ω − ω α − i γ α ) E α = g α E 0 ,
t 0 = g α E α E 0 = g α 2 ω − ω α − i γ α .
( ω − ω 1 − i γ 1 ) E 1 − κ 12 E 2 = g 1 E 0 ,
− κ 21 E 1 + ( ω − ω 2 − i γ 2 ) E 2 = 0 ,
t 0 = g 1 g 2 κ 21 ( ω − ω 1 − i γ 1 ) ( ω − ω 2 − i γ 2 ) − κ 12 κ 21 .
( ω 1 + i γ 1 κ 12 κ 21 ω 2 + i γ 2 ) ( ψ 1 ψ 2 ) = ω 0 ( ψ 1 ψ 2 ) ,

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