Abstract

We study the coupling of mode in time for non-Hermitian cavities. Using variational principle, we provide a self-consistent approach to study the mode hybridization in non-Hermitian cavities from the first-principle of Maxwell’s equations. We first extend the reaction concept for time reversal adjoint system using the scalar inner product. We apply our theory to the non-Hermitian parity-time symmetric cavities, and obtain excellent agreement with results obtained by finite element fullwave simulations. In contrast, the conventional coupled mode theory using complex inner product fails to capture the bifurcation of the dispersion of parity-time symmetric cavities, as non-Hermicity increases. Our theory may have potential applications in non-Hermitian optical systems.

© 2016 Optical Society of America

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References

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  1. A. D. Berk, “Variational principles for electromagnetic resonators and waveguides,” IRE IEEE Trans. Antennas Propag. 4, 104–111 (1956).
    [Crossref]
  2. C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE Trans. Microw. Theory Tech. 28, 878–886 (1980).
    [Crossref]
  3. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
    [Crossref]
  4. H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
    [Crossref]
  5. S.-L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987).
    [Crossref]
  6. W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
    [Crossref]
  7. A. M. S.-Z.-Amiri, S. S.-Naeini, S. K. Chaudhuri, and R. Sabry, “Generalized reaction and unrestricted variational formulation of cavity resonators-part 1: basic theory,” IEEE Trans. Microw. Theory Tech. 50, 2480–2490 (2002).
    [Crossref]
  8. G. Zhu, “Pseudo-Hermitian Hamiltonian formalism of electromagnetic wave propagation in a dielectric medium-application to the nonorthogonal coupled-mode theory,” J. Lightwave Technol. 29, 905–911 (2011).
    [Crossref]
  9. R. F. Harrington, Time-Harmonic Electromagnetic Fields (Join Wiley & Sons, 2001).
    [Crossref]
  10. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Join Wiley & Sons, 2002).
  11. F. Monticone and A. Alù, “Leaky-wave theory, techniques, and applications: from microwaves to visible frequencies,” Proc. IEEE 103, 793–821 (2015).
    [Crossref]
  12. J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photon. 1, 58–106 (2009).
    [Crossref]
  13. E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
    [Crossref]
  14. P. T. Kristensen, J. Rosenkrantz de Lasson, and N. Gregersen, “Calculation, normalization, and perturbation of quasinormal modes in coupled cavity-waveguide systems,” Opt. Lett. 39, 6359–6362 (2014).
    [Crossref] [PubMed]
  15. P. T. Kristensen, “Normalization of quasinormal modes in leaky optical cavities and plasmonic resonators,” Phys. Rev. A 92, 053810 (2015).
    [Crossref]
  16. P. T. Kristensen, C. V. Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett. 37, 1649–1651 (2012).
    [Crossref] [PubMed]
  17. B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89, 023829 (2014).
    [Crossref]
  18. C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
    [Crossref] [PubMed]
  19. H. A. Haus and M. N. Islam, “Application of a variational principle to systems with radiation loss,” IEEE J. Quantum Elect. 19, 106–117 (1983).
    [Crossref]
  20. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Elect. 40, 1511–1518 (2004).
    [Crossref]
  21. J. Xu and Y. Chen, “General coupled mode theory in non-Hermitian waveguides,” Opt. Express 23, 22619–22627 (2015).
    [Crossref] [PubMed]
  22. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).
  23. C. Altman and K. Suchy, Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics, 2nd ed. (Springer, 2011).
    [Crossref]
  24. V. H. Rumsey, “Reaction concept in electromagnetic theory,” Phys. Rev. 94, 1483–1491 (1954).
    [Crossref]
  25. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
  26. www.comsol.com

2015 (3)

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

P. T. Kristensen, “Normalization of quasinormal modes in leaky optical cavities and plasmonic resonators,” Phys. Rev. A 92, 053810 (2015).
[Crossref]

J. Xu and Y. Chen, “General coupled mode theory in non-Hermitian waveguides,” Opt. Express 23, 22619–22627 (2015).
[Crossref] [PubMed]

2014 (2)

P. T. Kristensen, J. Rosenkrantz de Lasson, and N. Gregersen, “Calculation, normalization, and perturbation of quasinormal modes in coupled cavity-waveguide systems,” Opt. Lett. 39, 6359–6362 (2014).
[Crossref] [PubMed]

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

2013 (1)

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

2012 (1)

2011 (1)

2009 (1)

2004 (1)

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Elect. 40, 1511–1518 (2004).
[Crossref]

2002 (1)

A. M. S.-Z.-Amiri, S. S.-Naeini, S. K. Chaudhuri, and R. Sabry, “Generalized reaction and unrestricted variational formulation of cavity resonators-part 1: basic theory,” IEEE Trans. Microw. Theory Tech. 50, 2480–2490 (2002).
[Crossref]

1998 (1)

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

1987 (3)

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

S.-L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987).
[Crossref]

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

1985 (1)

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[Crossref]

1983 (1)

H. A. Haus and M. N. Islam, “Application of a variational principle to systems with radiation loss,” IEEE J. Quantum Elect. 19, 106–117 (1983).
[Crossref]

1980 (1)

C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE Trans. Microw. Theory Tech. 28, 878–886 (1980).
[Crossref]

1956 (1)

A. D. Berk, “Variational principles for electromagnetic resonators and waveguides,” IRE IEEE Trans. Antennas Propag. 4, 104–111 (1956).
[Crossref]

1954 (1)

V. H. Rumsey, “Reaction concept in electromagnetic theory,” Phys. Rev. 94, 1483–1491 (1954).
[Crossref]

Altman, C.

C. Altman and K. Suchy, Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics, 2nd ed. (Springer, 2011).
[Crossref]

Alù, A.

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

Berk, A. D.

A. D. Berk, “Variational principles for electromagnetic resonators and waveguides,” IRE IEEE Trans. Antennas Propag. 4, 104–111 (1956).
[Crossref]

Chaudhuri, S. K.

A. M. S.-Z.-Amiri, S. S.-Naeini, S. K. Chaudhuri, and R. Sabry, “Generalized reaction and unrestricted variational formulation of cavity resonators-part 1: basic theory,” IEEE Trans. Microw. Theory Tech. 50, 2480–2490 (2002).
[Crossref]

Chen, C. H.

C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE Trans. Microw. Theory Tech. 28, 878–886 (1980).
[Crossref]

Chen, Y.

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

Ching, E. S. C.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

Chuang, S.-L.

S.-L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987).
[Crossref]

Commandré, M.

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

Fan, S.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Elect. 40, 1511–1518 (2004).
[Crossref]

Gregersen, N.

Hardy, A.

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[Crossref]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (Join Wiley & Sons, 2001).
[Crossref]

Haus, H. A.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

H. A. Haus and M. N. Islam, “Application of a variational principle to systems with radiation loss,” IEEE J. Quantum Elect. 19, 106–117 (1983).
[Crossref]

Hu, J.

Huang, W. P.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

Hughes, S.

Hugonin, J. P.

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

Islam, M. N.

H. A. Haus and M. N. Islam, “Application of a variational principle to systems with radiation loss,” IEEE J. Quantum Elect. 19, 106–117 (1983).
[Crossref]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Join Wiley & Sons, 2002).

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Kawakami, S.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

Kristensen, P. T.

Lalanne, P.

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

Leung, P. T.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

Lien, C.-D.

C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE Trans. Microw. Theory Tech. 28, 878–886 (1980).
[Crossref]

Maassen van den Brink, A.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

Maksymov, I. S.

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

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Menyuk, C. R.

Monticone, F.

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

Nicolet, A.

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

Osinski, M.

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

Rosenkrantz de Lasson, J.

Rumsey, V. H.

V. H. Rumsey, “Reaction concept in electromagnetic theory,” Phys. Rev. 94, 1483–1491 (1954).
[Crossref]

S.-Naeini, S.

A. M. S.-Z.-Amiri, S. S.-Naeini, S. K. Chaudhuri, and R. Sabry, “Generalized reaction and unrestricted variational formulation of cavity resonators-part 1: basic theory,” IEEE Trans. Microw. Theory Tech. 50, 2480–2490 (2002).
[Crossref]

S.-Z.-Amiri, A. M.

A. M. S.-Z.-Amiri, S. S.-Naeini, S. K. Chaudhuri, and R. Sabry, “Generalized reaction and unrestricted variational formulation of cavity resonators-part 1: basic theory,” IEEE Trans. Microw. Theory Tech. 50, 2480–2490 (2002).
[Crossref]

Sabry, R.

A. M. S.-Z.-Amiri, S. S.-Naeini, S. K. Chaudhuri, and R. Sabry, “Generalized reaction and unrestricted variational formulation of cavity resonators-part 1: basic theory,” IEEE Trans. Microw. Theory Tech. 50, 2480–2490 (2002).
[Crossref]

Sauvan, C.

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

Streifer, W.

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[Crossref]

Suchy, K.

C. Altman and K. Suchy, Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics, 2nd ed. (Springer, 2011).
[Crossref]

Suen, W. M.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

Suh, W.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Elect. 40, 1511–1518 (2004).
[Crossref]

Tong, S. S.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

Vial, B.

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

Vlack, C. V.

Wang, Z.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Elect. 40, 1511–1518 (2004).
[Crossref]

Whitaker, N. A.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Xu, J.

Young, K.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

Zhu, G.

Zolla, F.

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

Adv. Opt. Photon. (1)

IEEE J. Quantum Elect. (2)

H. A. Haus and M. N. Islam, “Application of a variational principle to systems with radiation loss,” IEEE J. Quantum Elect. 19, 106–117 (1983).
[Crossref]

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Elect. 40, 1511–1518 (2004).
[Crossref]

IEEE Trans. Microw. Theory Tech. (2)

C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE Trans. Microw. Theory Tech. 28, 878–886 (1980).
[Crossref]

A. M. S.-Z.-Amiri, S. S.-Naeini, S. K. Chaudhuri, and R. Sabry, “Generalized reaction and unrestricted variational formulation of cavity resonators-part 1: basic theory,” IEEE Trans. Microw. Theory Tech. 50, 2480–2490 (2002).
[Crossref]

IRE IEEE Trans. Antennas Propag. (1)

A. D. Berk, “Variational principles for electromagnetic resonators and waveguides,” IRE IEEE Trans. Antennas Propag. 4, 104–111 (1956).
[Crossref]

J. Lightwave Technol. (5)

G. Zhu, “Pseudo-Hermitian Hamiltonian formalism of electromagnetic wave propagation in a dielectric medium-application to the nonorthogonal coupled-mode theory,” J. Lightwave Technol. 29, 905–911 (2011).
[Crossref]

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[Crossref]

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

S.-L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987).
[Crossref]

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. (1)

V. H. Rumsey, “Reaction concept in electromagnetic theory,” Phys. Rev. 94, 1483–1491 (1954).
[Crossref]

Phys. Rev. A (2)

P. T. Kristensen, “Normalization of quasinormal modes in leaky optical cavities and plasmonic resonators,” Phys. Rev. A 92, 053810 (2015).
[Crossref]

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

Phys. Rev. Lett. (1)

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

Proc. IEEE (1)

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

Rev. Mod. Phys. (1)

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

Other (6)

R. F. Harrington, Time-Harmonic Electromagnetic Fields (Join Wiley & Sons, 2001).
[Crossref]

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Join Wiley & Sons, 2002).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

www.comsol.com

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

C. Altman and K. Suchy, Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics, 2nd ed. (Springer, 2011).
[Crossref]

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Figures (2)

Fig. 1
Fig. 1 Coupled single mode cavities in photonic crystal structures. Each individual point-defect can support a monopole state. (a) The 3D sketch of photonic crystal. The rods are periodic along x and y directions, and assumed to extend indefinitely in the z direction. (b) The structure parameters. The crystal constant is a and the rods radii are 0.2a. The radius of each defect rod is reduced to 0.05a.
Fig. 2
Fig. 2 Real part (a, c, e, f) and imaginary part (b, d) of eigen-frequency versus Δε. The relative permittivities of the point-defects are given as follows: r,1 = r,0 + iΔε, r,2 = r,0iΔε in (a), (b), (c) and (d); r,1 = r,0 + Δε, r,2 = r,0 − Δε in (e) and r,1 = r,0 − Δε, r,2 = r,0 − Δε in (f). The structure parameters are given by Fig. 1(b), and the original relative permittivity of the rods r,0 = 8.9. We compare the results obtained by four different approaches with the results obtained by fullwave simulations. CTCMT (GTCMT) is derived from the first priciple based on complex (scalar) inner product. PMTPF (PM-FPF) is the phenomenological model which we need to fit two (four) parameters based on fullwave simulations or experimental measurements.

Tables (1)

Tables Icon

Table 1 Comparisons between the CMT derived from the first principle and two-band PMs

Equations (14)

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

( T ¯ F b , S a ) = d V [ T ¯ E b T ¯ H b ] σ [ J a M a ] ,
( T ¯ F b , S a ) + ( T ¯ S b , F a ) = d S [ E a × H b E b × H a ] ,
( T ¯ F b , S a ) + ( T ¯ S b , F a ) = 0 .
( T ¯ ψ , H ¯ ϕ ) + ( T ¯ ( H ¯ # ψ ) , ϕ ) = 0 ,
Σ n a n [ b m n i ( k k 0 , m ) p m n i k κ m n ] = 0 ,
b m n = i ( k 0 , n k 0 , m ) p m n .
Σ n a n [ ( k 0 , n k ) p m n k κ m n ] = 0 .
( ψ * , H ¯ ϕ ) + ( ( H ¯ # ψ ) * , ϕ ) = 0 ,
Σ n a n [ b m n i ( k 0 , m r , 0 k * r , 0 * ) e m n i ( k 0 , m μ r , 0 k * μ r , 0 * ) h m n + i k * κ m n ] = 0 ,
b m n = i [ ( k 0 , n * r , 0 * k 0 , m r , 0 ) e m n + ( k 0 , n * μ r , 0 * k 0 , m μ r , 0 ) h m n ] .
Σ n a n [ ( k 0 , n * k * ) p m n k * κ m n ] = 0 ,
b m n = i ( k 0 , n R k 0 , m R ) Q m n 2 k 0 , n R ( r , 0 I e m n + μ r , 0 I h m n ) ,
H ¯ a = k W ¯ a ,
H ¯ a = k * W ¯ a ,

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