Abstract

Parity–time (PT) symmetry has been demonstrated in the frame of classic optics. Its applications in laser science have resulted in unconventional control and manipulation of resonant modes. PT-symmetric periodic circular Bragg lasers were previously proposed. Analyses with a transfer-matrix method have shown their superior properties of reduced threshold and enhanced modal discrimination between the radial modes. However, the properties of the azimuthal modes were not analyzed, which restricts further development of circular Bragg lasers. Here, we adopt the coupled-mode theory to design and analyze chirped circular Bragg lasers with radial PT symmetry. The new structures possess more versatile modal control with further enhanced modal discrimination between the azimuthal modes. We also analyze azimuthally modulated circular Bragg lasers with radial PT symmetry, which are shown to achieve even higher modal discrimination.

© 2018 Chinese Laser Press

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References

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  1. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
    [Crossref]
  2. T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
    [Crossref]
  3. T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
    [Crossref]
  4. C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
    [Crossref]
  5. X. M. Gong, A. K. Chan, and H. F. Taylor, “Lateral mode discrimination in surface emitting DBR lasers with cylindrical symmetry,” IEEE J. Quantum Electron. 30, 1212–1218 (1994).
    [Crossref]
  6. A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
    [Crossref]
  7. J. Scheuer, “Radial Bragg lasers: optimal design for minimal threshold levels and enhanced mode discrimination,” J. Opt. Soc. Am. B 24, 2178–2184 (2007).
    [Crossref]
  8. J. Scheuer and A. Yariv, “Coupled-waves approach to the design and analysis of Bragg and photonic crystal annular resonators,” IEEE J. Quantum Electron. 39, 1555–1562 (2003).
    [Crossref]
  9. X. K. Sun and A. Yariv, “Modal properties and modal control in vertically emitting annular Bragg lasers,” Opt. Express 15, 17323–17333 (2007).
    [Crossref]
  10. X. K. Sun and A. Yariv, “Surface-emitting circular DFB, disk-, and ring-Bragg resonator lasers with chirped gratings. II: nonuniform pumping and far-field patterns,” Opt. Express 17, 1–6 (2009).
    [Crossref]
  11. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
  12. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
    [Crossref]
  13. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [Crossref]
  14. C. E. Ruter, 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]
  15. 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]
  16. J. H. Gu, X. Xi, J. W. Ma, Z. J. Yu, and X. K. Sun, “Parity-time-symmetric circular Bragg lasers: a proposal and analysis,” Sci. Rep. 6, 37688 (2016).
    [Crossref]
  17. Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of parity-time-symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22, 5–11 (2016).
    [Crossref]
  18. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
    [Crossref]
  19. M. Kulishov, B. Kress, and R. Slavík, “Resonant cavities based on Parity-Time-symmetric diffractive gratings,” Opt. Express 21, 9473–9483 (2013).
    [Crossref]
  20. H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
    [Crossref]
  21. K.-H. Kim, M.-S. Hwang, H.-R. Kim, J.-H. Choi, Y.-S. No, and H.-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
    [Crossref]
  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. Commun. 5, 4034 (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]
  24. 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]
  25. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13, 3068–3078 (2005).
    [Crossref]
  26. 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]
  27. M. Fujita and T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253–1258 (2001).
    [Crossref]
  28. www.lumerical.com/tcad-products/fdtd/ .

2016 (5)

J. H. Gu, X. Xi, J. W. Ma, Z. J. Yu, and X. K. Sun, “Parity-time-symmetric circular Bragg lasers: a proposal and analysis,” Sci. Rep. 6, 37688 (2016).
[Crossref]

Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of parity-time-symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22, 5–11 (2016).
[Crossref]

K.-H. Kim, M.-S. Hwang, H.-R. Kim, J.-H. Choi, Y.-S. No, and H.-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

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]

H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
[Crossref]

2014 (3)

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. Commun. 5, 4034 (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]

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

2013 (1)

2012 (1)

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]

2011 (1)

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]

2010 (1)

C. E. Ruter, 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 (2)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

X. K. Sun and A. Yariv, “Surface-emitting circular DFB, disk-, and ring-Bragg resonator lasers with chirped gratings. II: nonuniform pumping and far-field patterns,” Opt. Express 17, 1–6 (2009).
[Crossref]

2008 (1)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

2007 (2)

2005 (1)

2003 (2)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref]

J. Scheuer and A. Yariv, “Coupled-waves approach to the design and analysis of Bragg and photonic crystal annular resonators,” IEEE J. Quantum Electron. 39, 1555–1562 (2003).
[Crossref]

2001 (1)

M. Fujita and T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253–1258 (2001).
[Crossref]

1999 (1)

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

1994 (1)

X. M. Gong, A. K. Chan, and H. F. Taylor, “Lateral mode discrimination in surface emitting DBR lasers with cylindrical symmetry,” IEEE J. Quantum Electron. 30, 1212–1218 (1994).
[Crossref]

1993 (1)

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[Crossref]

1992 (1)

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
[Crossref]

1990 (1)

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Azaña, J.

Baba, T.

M. Fujita and T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253–1258 (2001).
[Crossref]

Bélanger, N.

Bender, C. M.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Benisty, H.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[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. Commun. 5, 4034 (2014).
[Crossref]

Cao, 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]

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]

Chan, A. K.

X. M. Gong, A. K. Chan, and H. F. Taylor, “Lateral mode discrimination in surface emitting DBR lasers with cylindrical symmetry,” IEEE J. Quantum Electron. 30, 1212–1218 (1994).
[Crossref]

Choi, J.-H.

K.-H. Kim, M.-S. Hwang, H.-R. Kim, J.-H. Choi, Y.-S. No, and H.-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

Christodoulides, D. N.

H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
[Crossref]

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

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. Ruter, 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]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

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. Commun. 5, 4034 (2014).
[Crossref]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

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. Ruter, 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]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Erdogan, T.

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
[Crossref]

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[Crossref]

Fallahi, M.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[Crossref]

Fan, J.

Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of parity-time-symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22, 5–11 (2016).
[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]

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]

Fujita, M.

M. Fujita and T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253–1258 (2001).
[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]

Glinski, J.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[Crossref]

Gong, X. M.

X. M. Gong, A. K. Chan, and H. F. Taylor, “Lateral mode discrimination in surface emitting DBR lasers with cylindrical symmetry,” IEEE J. Quantum Electron. 30, 1212–1218 (1994).
[Crossref]

Gu, J. H.

J. H. Gu, X. Xi, J. W. Ma, Z. J. Yu, and X. K. Sun, “Parity-time-symmetric circular Bragg lasers: a proposal and analysis,” Sci. Rep. 6, 37688 (2016).
[Crossref]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Hall, D. G.

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
[Crossref]

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[Crossref]

Hassan, A. U.

Hayenga, W. E.

Hegarty, J.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[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]

Hodaei, H.

H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
[Crossref]

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

Houdré, R.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[Crossref]

Hwang, M.-S.

K.-H. Kim, M.-S. Hwang, H.-R. Kim, J.-H. Choi, Y.-S. No, and H.-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

Khajavikhan, M.

H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
[Crossref]

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

Kim, H.-R.

K.-H. Kim, M.-S. Hwang, H.-R. Kim, J.-H. Choi, Y.-S. No, and H.-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

Kim, K.-H.

K.-H. Kim, M.-S. Hwang, H.-R. Kim, J.-H. Choi, Y.-S. No, and H.-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

Kip, D.

C. E. Ruter, 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]

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. Commun. 5, 4034 (2014).
[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.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[Crossref]

Kress, B.

Kulishov, M.

Labilloy, D.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[Crossref]

Laniel, J. M.

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. 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]

Lin, Z.

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]

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]

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]

Ma, J. W.

J. H. Gu, X. Xi, J. W. Ma, Z. J. Yu, and X. K. Sun, “Parity-time-symmetric circular Bragg lasers: a proposal and analysis,” Sci. Rep. 6, 37688 (2016).
[Crossref]

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]

Maciejko, R.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[Crossref]

Makino, T.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[Crossref]

Makris, K. G.

C. E. Ruter, 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]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

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]

Miri, M. A.

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]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Najafi, S. I.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[Crossref]

No, Y.-S.

K.-H. Kim, M.-S. Hwang, H.-R. Kim, J.-H. Choi, Y.-S. No, and H.-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

Oesterle, U.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[Crossref]

Park, H.-G.

K.-H. Kim, M.-S. Hwang, H.-R. Kim, J.-H. Choi, Y.-S. No, and H.-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

Plant, D. V.

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]

Rotter, S.

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. 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]

Roycroft, B.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[Crossref]

Ruter, C. E.

C. E. Ruter, 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]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Scheuer, J.

J. Scheuer, “Radial Bragg lasers: optimal design for minimal threshold levels and enhanced mode discrimination,” J. Opt. Soc. Am. B 24, 2178–2184 (2007).
[Crossref]

J. Scheuer and A. Yariv, “Coupled-waves approach to the design and analysis of Bragg and photonic crystal annular resonators,” IEEE J. Quantum Electron. 39, 1555–1562 (2003).
[Crossref]

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. Commun. 5, 4034 (2014).
[Crossref]

Segev, M.

C. E. Ruter, 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]

Shaw, A.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[Crossref]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Slavík, R.

Smith, C. J. M.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[Crossref]

Stanley, R.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[Crossref]

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]

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. Commun. 5, 4034 (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]

Sun, X. K.

Svilans, M.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[Crossref]

Taylor, H. F.

X. M. Gong, A. K. Chan, and H. F. Taylor, “Lateral mode discrimination in surface emitting DBR lasers with cylindrical symmetry,” IEEE J. Quantum Electron. 30, 1212–1218 (1994).
[Crossref]

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

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[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]

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]

Weisbuch, C.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[Crossref]

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]

Wu, C.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[Crossref]

Xi, X.

J. H. Gu, X. Xi, J. W. Ma, Z. J. Yu, and X. K. Sun, “Parity-time-symmetric circular Bragg lasers: a proposal and analysis,” Sci. Rep. 6, 37688 (2016).
[Crossref]

Yariv, A.

Yu, Z. J.

J. H. Gu, X. Xi, J. W. Ma, Z. J. Yu, and X. K. Sun, “Parity-time-symmetric circular Bragg lasers: a proposal and analysis,” Sci. Rep. 6, 37688 (2016).
[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]

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]

Zhao, Y.

Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of parity-time-symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22, 5–11 (2016).
[Crossref]

Zhu, L.

Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of parity-time-symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22, 5–11 (2016).
[Crossref]

Zhu, Y.

Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of parity-time-symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22, 5–11 (2016).
[Crossref]

Appl. Phys. Lett. (1)

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdré, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[Crossref]

IEEE J. Quantum Electron. (5)

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
[Crossref]

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[Crossref]

X. M. Gong, A. K. Chan, and H. F. Taylor, “Lateral mode discrimination in surface emitting DBR lasers with cylindrical symmetry,” IEEE J. Quantum Electron. 30, 1212–1218 (1994).
[Crossref]

J. Scheuer and A. Yariv, “Coupled-waves approach to the design and analysis of Bragg and photonic crystal annular resonators,” IEEE J. Quantum Electron. 39, 1555–1562 (2003).
[Crossref]

M. Fujita and T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253–1258 (2001).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of parity-time-symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22, 5–11 (2016).
[Crossref]

J. Appl. Phys. (1)

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[Crossref]

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

Nat. Commun. (2)

K.-H. Kim, M.-S. Hwang, H.-R. Kim, J.-H. Choi, Y.-S. No, and H.-G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[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. Commun. 5, 4034 (2014).
[Crossref]

Nat. Phys. (1)

C. E. Ruter, 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 (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. Lett. (5)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[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]

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]

Sci. Rep. (1)

J. H. Gu, X. Xi, J. W. Ma, Z. J. Yu, and X. K. Sun, “Parity-time-symmetric circular Bragg lasers: a proposal and analysis,” Sci. Rep. 6, 37688 (2016).
[Crossref]

Science (3)

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

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]

Other (1)

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Figures (3)

Fig. 1.
Fig. 1. (a) Schematic of the PT-symmetric circular Bragg laser. (b) Schematic of the azimuthally modulated circular Bragg laser with radial PT symmetry. (c) Radial profile of the refractive index of the two types of lasers, showing the real ( Δ n r ) and imaginary ( Δ n i ) refractive index modulations that satisfy the PT symmetry requirement.
Fig. 2.
Fig. 2. (a) Normalized threshold gain ( α r 0 ) for even-order azimuthal modes of PT-symmetric and conventional radially chirped circular Bragg lasers. Inset shows the E field profile of the targeted 8th-order azimuthal mode. (b) Effect of a small deviation away from the exceptional point on the normalized threshold gain values for the PT-symmetric radially chirped circular Bragg lasers.
Fig. 3.
Fig. 3. (a) Output spectra for two circular Bragg laser structures with radial PT symmetry. The upper corresponds to the one without azimuthal modulation, while the lower corresponds to the one with azimuthal modulation. (b)  E field profiles of the modes at the three peaks in the laser output spectra. Modes (i) and (ii) from the upper spectrum are of the 8th and 1st azimuthal order, respectively. Mode (iii) from the lower spectrum is of the 8th azimuthal order.

Equations (10)

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[ r 2 2 r 2 + r r + 2 φ 2 + r 2 ϵ ( r , φ ) k 0 2 ] E z = 0 .
d A l d r ( 2 i H l ( 1 ) ) d B l d r ( 2 i H l ( 2 ) ) + k 0 Δ ϵ n av ( A l H l ( 1 ) + B l H l ( 2 ) ) = 0 ,
{ Δ ϵ r ( r ) = Δ ϵ r cos [ 2 phase ( H l ( 1 ) ( k de r ) ) + θ 1 ] Δ ϵ i ( r ) = Δ ϵ i cos [ 2 phase ( H l ( 1 ) ( k de r ) ) + θ 2 ] ,
{ d A l d r = i Δ ϵ r exp ( i θ 1 ) + i Δ ϵ i exp ( i θ 2 ) 4 n av k 0 B l ( r ) exp ( 2 i δ r ) d B l d r = i Δ ϵ r exp ( i θ 1 ) + i Δ ϵ i exp ( i θ 2 ) 4 n av k 0 A l ( r ) exp ( 2 i δ r ) .
{ d R l d r = i δ R l ( r ) + i ( κ exp ( i θ 1 ) + i g exp ( i θ 2 ) ) S l ( r ) d S l d r = i ( κ exp ( i θ 1 ) + i g exp ( i θ 2 ) ) R l ( r ) i δ S l ( r ) .
n = n av + Δ n r sign { cos [ 2 phase ( H 8 ( 1 ) ( k de r ) ) + θ 1 ] } + i Δ n i sign { cos [ 2 phase ( H 8 ( 1 ) ( k de r ) ) + θ 2 ] } ,
[ A 8 ( r w ) B 8 ( r w ) ] = [ M 11 e i δ ( r w r 0 ) M 12 e i δ ( r w + r 0 ) M 21 e i δ ( r w + r 0 ) M 22 e i δ ( r w r 0 ) ] [ A 8 ( r 0 ) B 8 ( r 0 ) ]
{ M 11 = cosh ( i γ L ) + δ sinh ( i γ L ) / γ M 12 = ( i e i θ 1 κ e i θ 2 g ) sinh ( i γ L ) / i γ M 21 = ( i e i θ 1 κ + e i θ 2 g ) sinh ( i γ L ) / i γ M 22 = cosh ( i γ L ) δ sinh ( i γ L ) / γ .
n = n av Δ n r sign { sin [ 2 phase ( H 8 ( 1 ) ( k de r ) ) ] } Θ ( φ ) i Δ n i sign { cos [ 2 phase ( H 8 ( 1 ) ( k de r ) ) ] } Θ ( φ )
Θ ( φ ) = { 1 , cos ( q φ ) > 0 0 , cos ( q φ ) < 0 ,