Abstract

Open quantum and wave systems can exhibit non-Hermitian degeneracies called exceptional points, where both the eigenvalues and the corresponding eigenstates coalesce. Previously, such exceptional points have been investigated in dielectric microcavities in terms of optical modes which are well confined inside the cavity. However, beside these so-called “internal modes” with a relatively high quality factor, there exists another kind of mode called “external modes,” which have a large decay rate and almost zero intensity inside the cavity. In the present paper, we demonstrate the physical significance of the external modes via the occurrence of exceptional points of internal–external mode pairs for transverse electric polarization. Our numerical studies show that these exceptional points can be achieved by either a boundary deformation of the microdisk or by introducing absorption into a circular cavity.

© 2019 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2018 (1)

C.-H. Yi, J. Kullig, and J. Wiersig, “Pair of exceptional points in a microdisk cavity under an extremely weak deformation,” Phys. Rev. Lett. 120, 093902 (2018).
[Crossref]

2017 (3)

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

S. Richter, T. Michalsky, C. Sturm, B. Rosenow, M. Grundmann, and R. Schmidt-Grund, “Exceptional points in anisotropic planar microcavities,” Phys. Rev. A 95, 023836 (2017).
[Crossref]

2016 (5)

J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
[Crossref]

J. Kullig and J. Wiersig, “Perturbation theory for asymmetric deformed microdisk cavities,” Phys. Rev. A 94, 043850 (2016).
[Crossref]

J. Kullig and J. Wiersig, “Q spoiling in deformed optical microdisks due to resonance-assisted tunneling,” Phys. Rev. E 94, 022202 (2016).
[Crossref]

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Nat. Acad. Sci. USA 113, 6845–6850 (2016).
[Crossref]

2015 (2)

N. Zhang, S. Liu, K. Wang, G. Z. L. Meng, N. Yi, S. Xiao, and Q. H. Song, “Single nanoparticle detection using far-field emission of photonic molecule around an exceptional point,” Sci. Rep. 5, 11912 (2015).
[Crossref]

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
[Crossref]

2014 (2)

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[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 points,” Nat. Commun. 5, 4034 (2014).
[Crossref]

2012 (4)

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]

J. Wiersig, “Perturbative approach to optical microdisks with a local boundary deformation,” Phys. Rev. A 85, 063838 (2012).
[Crossref]

E. Bogomolny and R. Dubertrand, “Trace formula for dielectric cavities. III. TE modes,” Phys. Rev. E 86, 026202 (2012).
[Crossref]

F. Vollmer and L. Yang, “Review label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 1, 267–291 (2012).
[Crossref]

2011 (6)

M. Benyoucef, J.-B. Shim, J. Wiersig, and O. Schmidt, “Quality-factor enhancement of supermodes in coupled microdisks,” Opt. Lett. 36, 1317–1319 (2011).
[Crossref]

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
[Crossref]

J. Wiersig, “Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles,” Phys. Rev. A 84, 063828 (2011).
[Crossref]

J.-W. Ryu and S.-Y. Lee, “Quasiscarred modes and their branching behaviour at an exceptional point,” Phys. Rev. E 83, 015203(R) (2011).
[Crossref]

J. Cho, S. Rim, and C.-M. Kim, “Dynamics of morphology-dependent resonances by openness in dielectric disks for TE polarization,” Phys. Rev. A 83, 043810 (2011).
[Crossref]

B. Alfassi, O. Peleg, N. Moiseyev, and M. Segev, “Diverging Rabi oscillations in subwavelength photonic lattices,” Phys. Rev. Lett. 106, 073901 (2011).
[Crossref]

2010 (3)

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref]

J. Cho, I. Kim, S. Rim, G.-S. Yim, and C.-M. Kim, “Outer resonances and effective potential analogy in two-dimensional dielectric cavities,” Phys. Lett. A 374, 1893–1899 (2010).
[Crossref]

J. Zhu, Ş. K. Özdemir, L. He, and L. Yang, “Controlled manipulation of mode splitting in an optical microcavity by two Rayleigh scatterers,” Opt. Express 18, 23535–23543 (2010).
[Crossref]

2009 (3)

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

C. P. Dettmann, G. V. Morozov, M. Sieber, and H. Waalkens, “Internal and external resonances of dielectric disks,” Euro. Lett. 87, 34003 (2009).
[Crossref]

S. Bittner, B. Dietz, M. Miski-Oglu, P. O. Iriarte, A. Richter, and F. Schäfer, “Experimental test of a two-dimensional approximation for dielectric microcavities,” Phys. Rev. A 80, 023825 (2009).
[Crossref]

2008 (2)

R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A 77, 013804 (2008).
[Crossref]

E. Bogomolny, R. Dubertrand, and C. Schmit, “Trace formula for dielectric cavities: general properties,” Phys. Rev. E 78, 056202 (2008).
[Crossref]

2007 (2)

M. Lebental, C. Djellali, N. Arnaud, J. S. Lauret, J. Zyss, R. Dubertrand, C. Schmit, and E. Bogomolny, “Inferring periodic orbits from spectra of simply shaped microlasers,” Phys. Rev. A 76, 023830 (2007).
[Crossref]

H. Cartarius, J. Main, and G. Wunner, “Exceptional points in atomic spectra,” Phys. Rev. Lett. 99, 173003 (2007).
[Crossref]

2006 (1)

S. V. Boriskina, T. M. Benson, P. D. Sewell, and A. I. Nosich, “Directional emission, increased free spectral range, and mode Q-factors in 2-D wavelength-scale optical microcavity structures,” IEEE J. Sel. Top. Quantum Electron. 12, 1175–1182 (2006).
[Crossref]

2004 (1)

M. V. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54, 1039–1047 (2004).
[Crossref]

2003 (2)

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53–60 (2003).
[Crossref]

C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Observation of a chiral state in a microwave cavity,” Phys. Rev. Lett. 90, 034101 (2003).
[Crossref]

2002 (1)

M. Hentschel and H. Schomerus, “Fresnel laws at curved dielectric interfaces of microresonators,” Phys. Rev. E 65, 045603(R) (2002).
[Crossref]

2001 (1)

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]

2000 (1)

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

1990 (1)

B. R. Bennett, R. A. Soref, and J. A. Del Alamo, “Carrier-induced change in refractive index of InP, GaAs and InGaAsP,” IEEE J. Quantum Electron. 26, 113–122 (1990).
[Crossref]

1989 (1)

S. Adachi, “Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, AlxGa1-xAs, and In1-xGaxAsyPy-1,” J. Appl. Phys. 66, 6030–6040 (1989).
[Crossref]

1986 (1)

D. Aspnes, S. Kelso, R. Logan, and R. Bhat, “Optical properties of AlxGa1-xAs,” J. Appl. Phys. 60, 754–767 (1986).
[Crossref]

Adachi, S.

S. Adachi, “Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, AlxGa1-xAs, and In1-xGaxAsyPy-1,” J. Appl. Phys. 66, 6030–6040 (1989).
[Crossref]

Alfassi, B.

B. Alfassi, O. Peleg, N. Moiseyev, and M. Segev, “Diverging Rabi oscillations in subwavelength photonic lattices,” Phys. Rev. Lett. 106, 073901 (2011).
[Crossref]

An, K.

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref]

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

Arnaud, N.

M. Lebental, C. Djellali, N. Arnaud, J. S. Lauret, J. Zyss, R. Dubertrand, C. Schmit, and E. Bogomolny, “Inferring periodic orbits from spectra of simply shaped microlasers,” Phys. Rev. A 76, 023830 (2007).
[Crossref]

Aspnes, D.

D. Aspnes, S. Kelso, R. Logan, and R. Bhat, “Optical properties of AlxGa1-xAs,” J. Appl. Phys. 60, 754–767 (1986).
[Crossref]

Bennett, B. R.

B. R. Bennett, R. A. Soref, and J. A. Del Alamo, “Carrier-induced change in refractive index of InP, GaAs and InGaAsP,” IEEE J. Quantum Electron. 26, 113–122 (1990).
[Crossref]

Benson, T. M.

S. V. Boriskina, T. M. Benson, P. D. Sewell, and A. I. Nosich, “Directional emission, increased free spectral range, and mode Q-factors in 2-D wavelength-scale optical microcavity structures,” IEEE J. Sel. Top. Quantum Electron. 12, 1175–1182 (2006).
[Crossref]

Benyoucef, M.

Berry, M. V.

M. V. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54, 1039–1047 (2004).
[Crossref]

Bhat, R.

D. Aspnes, S. Kelso, R. Logan, and R. Bhat, “Optical properties of AlxGa1-xAs,” J. Appl. Phys. 60, 754–767 (1986).
[Crossref]

Bittner, S.

S. Bittner, B. Dietz, M. Miski-Oglu, P. O. Iriarte, A. Richter, and F. Schäfer, “Experimental test of a two-dimensional approximation for dielectric microcavities,” Phys. Rev. A 80, 023825 (2009).
[Crossref]

Bogomolny, E.

E. Bogomolny and R. Dubertrand, “Trace formula for dielectric cavities. III. TE modes,” Phys. Rev. E 86, 026202 (2012).
[Crossref]

E. Bogomolny, R. Dubertrand, and C. Schmit, “Trace formula for dielectric cavities: general properties,” Phys. Rev. E 78, 056202 (2008).
[Crossref]

R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A 77, 013804 (2008).
[Crossref]

M. Lebental, C. Djellali, N. Arnaud, J. S. Lauret, J. Zyss, R. Dubertrand, C. Schmit, and E. Bogomolny, “Inferring periodic orbits from spectra of simply shaped microlasers,” Phys. Rev. A 76, 023830 (2007).
[Crossref]

Böhm, J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Boriskina, S. V.

S. V. Boriskina, T. M. Benson, P. D. Sewell, and A. I. Nosich, “Directional emission, increased free spectral range, and mode Q-factors in 2-D wavelength-scale optical microcavity structures,” IEEE J. Sel. Top. Quantum Electron. 12, 1175–1182 (2006).
[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 points,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Brown, J. W.

J. W. Brown and R. V. Churchill, Complex Variables and Applications (McGraw-Hill Higher Education, 2009).

Cao, H.

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
[Crossref]

Cartarius, H.

H. Cartarius, J. Main, and G. Wunner, “Exceptional points in atomic spectra,” Phys. Rev. Lett. 99, 173003 (2007).
[Crossref]

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]

Chen, W.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Nat. Acad. Sci. USA 113, 6845–6850 (2016).
[Crossref]

Cho, J.

J. Cho, S. Rim, and C.-M. Kim, “Dynamics of morphology-dependent resonances by openness in dielectric disks for TE polarization,” Phys. Rev. A 83, 043810 (2011).
[Crossref]

J. Cho, I. Kim, S. Rim, G.-S. Yim, and C.-M. Kim, “Outer resonances and effective potential analogy in two-dimensional dielectric cavities,” Phys. Lett. A 374, 1893–1899 (2010).
[Crossref]

Choi, Y.

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref]

Christodoulides, D. N.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Churchill, R. V.

J. W. Brown and R. V. Churchill, Complex Variables and Applications (McGraw-Hill Higher Education, 2009).

Del Alamo, J. A.

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M. Benyoucef, J.-B. Shim, J. Wiersig, and O. Schmidt, “Quality-factor enhancement of supermodes in coupled microdisks,” Opt. Lett. 36, 1317–1319 (2011).
[Crossref]

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

Shinohara, S.

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
[Crossref]

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C. P. Dettmann, G. V. Morozov, M. Sieber, and H. Waalkens, “Internal and external resonances of dielectric disks,” Euro. Lett. 87, 34003 (2009).
[Crossref]

Song, Q. H.

N. Zhang, S. Liu, K. Wang, G. Z. L. Meng, N. Yi, S. Xiao, and Q. H. Song, “Single nanoparticle detection using far-field emission of photonic molecule around an exceptional point,” Sci. Rep. 5, 11912 (2015).
[Crossref]

Soref, R. A.

B. R. Bennett, R. A. Soref, and J. A. Del Alamo, “Carrier-induced change in refractive index of InP, GaAs and InGaAsP,” IEEE J. Quantum Electron. 26, 113–122 (1990).
[Crossref]

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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]

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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 points,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Sturm, C.

S. Richter, T. Michalsky, C. Sturm, B. Rosenow, M. Grundmann, and R. Schmidt-Grund, “Exceptional points in anisotropic planar microcavities,” Phys. Rev. A 95, 023836 (2017).
[Crossref]

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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 points,” Nat. Commun. 5, 4034 (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]

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 points,” Nat. Commun. 5, 4034 (2014).
[Crossref]

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F. Vollmer and L. Yang, “Review label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 1, 267–291 (2012).
[Crossref]

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C. P. Dettmann, G. V. Morozov, M. Sieber, and H. Waalkens, “Internal and external resonances of dielectric disks,” Euro. Lett. 87, 34003 (2009).
[Crossref]

Wang, K.

N. Zhang, S. Liu, K. Wang, G. Z. L. Meng, N. Yi, S. Xiao, and Q. H. Song, “Single nanoparticle detection using far-field emission of photonic molecule around an exceptional point,” Sci. Rep. 5, 11912 (2015).
[Crossref]

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C.-H. Yi, J. Kullig, and J. Wiersig, “Pair of exceptional points in a microdisk cavity under an extremely weak deformation,” Phys. Rev. Lett. 120, 093902 (2018).
[Crossref]

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
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J. Kullig and J. Wiersig, “Perturbation theory for asymmetric deformed microdisk cavities,” Phys. Rev. A 94, 043850 (2016).
[Crossref]

J. Kullig and J. Wiersig, “Q spoiling in deformed optical microdisks due to resonance-assisted tunneling,” Phys. Rev. E 94, 022202 (2016).
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B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Nat. Acad. Sci. USA 113, 6845–6850 (2016).
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H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
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J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
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J. Wiersig, “Perturbative approach to optical microdisks with a local boundary deformation,” Phys. Rev. A 85, 063838 (2012).
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M. Benyoucef, J.-B. Shim, J. Wiersig, and O. Schmidt, “Quality-factor enhancement of supermodes in coupled microdisks,” Opt. Lett. 36, 1317–1319 (2011).
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J. Wiersig, “Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles,” Phys. Rev. A 84, 063828 (2011).
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J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53–60 (2003).
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H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
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N. Zhang, S. Liu, K. Wang, G. Z. L. Meng, N. Yi, S. Xiao, and Q. H. Song, “Single nanoparticle detection using far-field emission of photonic molecule around an exceptional point,” Sci. Rep. 5, 11912 (2015).
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Yang, J.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
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W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
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B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Nat. Acad. Sci. USA 113, 6845–6850 (2016).
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F. Vollmer and L. Yang, “Review label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 1, 267–291 (2012).
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J. Zhu, Ş. K. Özdemir, L. He, and L. Yang, “Controlled manipulation of mode splitting in an optical microcavity by two Rayleigh scatterers,” Opt. Express 18, 23535–23543 (2010).
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C.-H. Yi, J. Kullig, and J. Wiersig, “Pair of exceptional points in a microdisk cavity under an extremely weak deformation,” Phys. Rev. Lett. 120, 093902 (2018).
[Crossref]

Yi, N.

N. Zhang, S. Liu, K. Wang, G. Z. L. Meng, N. Yi, S. Xiao, and Q. H. Song, “Single nanoparticle detection using far-field emission of photonic molecule around an exceptional point,” Sci. Rep. 5, 11912 (2015).
[Crossref]

Yilmaz, H.

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Nat. Acad. Sci. USA 113, 6845–6850 (2016).
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N. Zhang, S. Liu, K. Wang, G. Z. L. Meng, N. Yi, S. Xiao, and Q. H. Song, “Single nanoparticle detection using far-field emission of photonic molecule around an exceptional point,” Sci. Rep. 5, 11912 (2015).
[Crossref]

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W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
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T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
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F. Vollmer and L. Yang, “Review label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 1, 267–291 (2012).
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Nat. Commun. (1)

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 points,” Nat. Commun. 5, 4034 (2014).
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W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
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H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
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Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. A (1)

J. Cho, I. Kim, S. Rim, G.-S. Yim, and C.-M. Kim, “Outer resonances and effective potential analogy in two-dimensional dielectric cavities,” Phys. Lett. A 374, 1893–1899 (2010).
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Phys. Rev. A (9)

J. Cho, S. Rim, and C.-M. Kim, “Dynamics of morphology-dependent resonances by openness in dielectric disks for TE polarization,” Phys. Rev. A 83, 043810 (2011).
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J. Wiersig, “Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles,” Phys. Rev. A 84, 063828 (2011).
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J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
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J. Kullig and J. Wiersig, “Perturbation theory for asymmetric deformed microdisk cavities,” Phys. Rev. A 94, 043850 (2016).
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S. Richter, T. Michalsky, C. Sturm, B. Rosenow, M. Grundmann, and R. Schmidt-Grund, “Exceptional points in anisotropic planar microcavities,” Phys. Rev. A 95, 023836 (2017).
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C.-H. Yi, J. Kullig, and J. Wiersig, “Pair of exceptional points in a microdisk cavity under an extremely weak deformation,” Phys. Rev. Lett. 120, 093902 (2018).
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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).
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S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

Proc. Nat. Acad. Sci. USA (1)

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Nat. Acad. Sci. USA 113, 6845–6850 (2016).
[Crossref]

Rev. Mod. Phys. (1)

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
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Sci. Rep. (1)

N. Zhang, S. Liu, K. Wang, G. Z. L. Meng, N. Yi, S. Xiao, and Q. H. Song, “Single nanoparticle detection using far-field emission of photonic molecule around an exceptional point,” Sci. Rep. 5, 11912 (2015).
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Figures (12)

Fig. 1.
Fig. 1. (a) Real and (b) imaginary part of scaled frequency kR of the modes in the microdisk obtained by Eq. (1) for TE polarization with n=3.14, as a function of azimuthal mode number m. Dots (·), crosses (+), and open circles (∘) mark internal modes, external modes, and kmBR corresponding to the Brewster angle, respectively.
Fig. 2.
Fig. 2. Degree of degeneracy of mode pairs Re(kmBR) and Re(kmIR) given by Eq. (3) as a function of azimuthal mode number m and refractive index n in the circular cavity. The arrowed curve separates the weak and strong coupling regimes. Δ1 and the color code are in log scale from (black) min=101 to (white) max=105. The refractive index is sampled with 500 points from n=1.5 to n=4.
Fig. 3.
Fig. 3. (a) Real and (b) imaginary part of kR in the circular cavity as a function of refractive index n with a fixed azimuthal mode number m=10 undergoing strong (i, ii) and weak [(iii, iv) and (v, vi)] coupling between k10BR and k10IR with different radial mode numbers l=3, 4, and 5. Solid-black and dashed-orange curves are k10BR and k10IR, respectively. Solid-blue arrows in (a) and (b) guide the trajectory of kmIR for increasing n. Dotted-blue vertical arrows indicate the change of a radial mode number l of kmIR (frequency of the nearest internal mode to the Brewster mode) from 3 to 4. The right panels show intensities |ψ(x,y)|2 of the modes marked by the same labels as in (a) and (b). The white-quarter circular and sky-blue oscillating curves superimposed on the right panels are the cavity boundaries and |ψ(x,0)|2, respectively.
Fig. 4.
Fig. 4. (a) Real and (b) imaginary part of kmBR and kmIR as a function of refractive index n with a fixed azimuthal mode number m=50. The short segmented orange curves with a steeper slope in (a) and upper fluctuating orange curves in (b) belong to k50IR with different radial mode number l. The mode number l increases from left to right. The black curves with the more gentle slope in (a) and the lower fluctuating black curves in (b) belong to k50BR. Two examples l=17 and 18 of the internal mode with frequency k50IR are indicated in (a). Thin solid-green and dashed-red curves connect the values of k50IR and k50BR at which the real or the imaginary part of them crosses. The vertical-blue arrow separates the regions of strong and weak coupling at n2.22.
Fig. 5.
Fig. 5. Illustration of the cavity boundary in Eq. (6) with ε=0.1 and N=20. The gray shaded region bounded by the corrugated black curve depicts the deformed cavity, while the region bounded by the dashed red curve is the undeformed circle (ε=0). n1 and n2 are the refractive indices of the interior and exterior of the cavity, respectively.
Fig. 6.
Fig. 6. (a) Real and (b) imaginary part of the frequencies kR of the modes in the microflower cavity as a function of deformation parameter ε and refractive index n with fixed (l,m)=(4,10). Note that ε decreases from left to right. Labels iii and iv are the same as in Fig. 3. Orange curves connecting the points marked by numbers from 1 to 12 show the Riemann surface topology around the EP. Blue curves are the branch-cut in Eq. (7).
Fig. 7.
Fig. 7. Intensity mode pattern |ψ(x,y)|2 in the microflower cavity corresponding to the marked points in Fig. 6 with the same labels. The white circular and corrugated curves are the cavity boundaries.
Fig. 8.
Fig. 8. Intensity mode pattern |ψ(x,y)|2 in the microflower cavity corresponding to the marked points in Fig. 6 with the same labels. The white corrugated circular and sky-blue oscillating curves, superimposed on the figures, are the cavity boundaries and |ψ(x,0)|2, respectively. The red dashed horizontal line in the middle panel is the x axis.
Fig. 9.
Fig. 9. (a) Real and (b) imaginary part of the frequency relative to kR¯=(kmBR+kmIR)/2 of the selected internal and external modes’ frequencies kR as a function of deformation parameter ε. The EPs marked by vertical arrows are at εl,mEP=0.00127, 0.00582, and 0.0136 for the modes (l,m)=(4,12), (4,10), and (3,7), respectively.
Fig. 10.
Fig. 10. (a),(c) and (b),(d) show the real and the imaginary parts of kR in the complex n plane, respectively. The (blue and gray) dashed curves belong to kR of the external mode and the (orange and gray) solid curves correspond to the internal mode. In (a) and (b), the cross (empty circle) marks the initial frequency of the internal (external) mode. The EP is marked by a black dot. In (c) and (d), the vicinity of the EP is shown via Δn=nnEP. The outer thick red curve corresponds to a two-fold encircling of the EP. Thin gray dots represent the Riemann sheets of kR.
Fig. 11.
Fig. 11. (a) Real (left/red axis/dashed curves) and imaginary (right/blue axis/solid curves) parts of the wave number kR relative to kR¯=(kIR+kEXR)/2 along the parameter curve in the complex n plane. The parameter curve is shown in (b), where the EP is marked as a thick black dot. τ parameterizes this curve starting at τ=0 for real n. τ=3 is marked as a black cross. The corresponding mode patterns at τ=0 (i), τ=τEP1.411 (EP), and τ=3 (f) are shown. The color map of the intensities ranging from black to red is truncated outside the cavity at twice the maximum value inside the cavity.
Fig. 12.
Fig. 12. In (a), Re(kR) is shown for varying real refractive index n for internal (black thick curves) and external (magenta thin curves) modes with m=12 in a circular cavity. In (b)–(e), the paths in the complex n plane are shown for which two modes (one internal and one external) with m=12 have the same Re(kR). The end point of each curve marks an EP. Colored dots in (a) mark the intersections as starting points for the parameter curves in (b)–(e).

Equations (8)

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

ηJmJm(nkR)HmHm(kR)=0,
dRe(kmR)dn1n2,
dRe(kmBR)dn1n2n2+1.
Δ1|[Re(kmBR)Re(kmIR)]1|,
H=(e1vve2),
2v=|Im(e1e2)|,
ρ(θ)=ρ0(ε)[1+εcos(Nθ)],
Re[kmB(n,ε)R]=Re[kmI(n,ε)R]Im[kmB(n,ε)R]=Im[kmI(n,ε)R]forε>εEPε<εEP,

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