Abstract

Normal mode is a very fundamental notion in quantum and classical optics. In this paper, we present a method to calculate normal modes by decomposing dyadic Green’s function, where the modes are excited by dipoles. The modes obtained by our method can be directly normalized and their degeneracies can be easily removed. This method can be applied to many theoretical descriptions of cavity electrodynamics and is of interest to nanophotonics.

© 2014 Optical Society of America

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References

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  1. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
    [Crossref] [PubMed]
  2. A. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J 40, 453–488 (1961).
    [Crossref]
  3. K. C. Ho, P. T. Leung, A. Maassen van den Brink, and K. Young, “Second quantization of open systems using quasinormal modes,” Phys. Rev. E 58, 2965–2978 (1998).
    [Crossref]
  4. P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
    [Crossref] [PubMed]
  5. K. M. Lee, P. T. Leung, and K. M. Pang, “Dyadic formulation of morphology-dependent resonances. i. completeness relation,” J. Opt. Soc. Am. B 16, 1409–1417 (1999).
    [Crossref]
  6. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2011).
  7. R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? a theory of single-quasimode laser operation,” Physical Review A 7, 1788 (1973).
    [Crossref]
  8. J. C. Penaforte and B. Baseia, “Quantum theory of a one-dimensional laser with output coupling: Linear approximation,” Phys. Rev. A 30, 1401–1406 (1984).
    [Crossref]
  9. R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
    [Crossref] [PubMed]
  10. S. M. Dutra and P. L. Knight, “Spontaneous emission in a planar fabry-pérot microcavity,” Phys. Rev. A 53, 3587–3605 (1996).
    [Crossref] [PubMed]
  11. K. Ujihara, “Quantum theory of a one-dimensional optical cavity with output coupling. field quantization,” Phys. Rev. A 12, 148–158 (1975).
    [Crossref]
  12. M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multipole interaction between atoms and their photonic environment,” Phys. Rev. A 68, 013822 (2003).
    [Crossref]
  13. P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2–10 (2014).
    [Crossref]
  14. C.-P. Yu and H.-C. Chang, “Yee-mesh-based finite difference eigenmode solver with pml absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Optics Express 12, 6165–6177 (2004).
    [Crossref] [PubMed]
  15. A. Taflove, Computational Electrodynamics: The Finite - Difference Time - Domain Method (Artech House, Incorporated, 1995).
  16. M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and super-radiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
    [Crossref]
  17. Y. Tang, A. Mintairov, J. Merz, V. Tokranov, and S. Oktyabrsky, “Characterization of 2d-photonic crystal nanocavities by polarization-dependent photoluminescence,” in “Nanotechnology, 2005. 5th IEEE Conference on,” (IEEE, 2005), pp. 35–38.

2014 (1)

P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2–10 (2014).
[Crossref]

2004 (2)

C.-P. Yu and H.-C. Chang, “Yee-mesh-based finite difference eigenmode solver with pml absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Optics Express 12, 6165–6177 (2004).
[Crossref] [PubMed]

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and super-radiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
[Crossref]

2003 (1)

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multipole interaction between atoms and their photonic environment,” Phys. Rev. A 68, 013822 (2003).
[Crossref]

1999 (1)

1998 (1)

K. C. Ho, P. T. Leung, A. Maassen van den Brink, and K. Young, “Second quantization of open systems using quasinormal modes,” Phys. Rev. E 58, 2965–2978 (1998).
[Crossref]

1996 (1)

S. M. Dutra and P. L. Knight, “Spontaneous emission in a planar fabry-pérot microcavity,” Phys. Rev. A 53, 3587–3605 (1996).
[Crossref] [PubMed]

1994 (1)

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[Crossref] [PubMed]

1991 (1)

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[Crossref] [PubMed]

1985 (1)

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[Crossref] [PubMed]

1984 (1)

J. C. Penaforte and B. Baseia, “Quantum theory of a one-dimensional laser with output coupling: Linear approximation,” Phys. Rev. A 30, 1401–1406 (1984).
[Crossref]

1975 (1)

K. Ujihara, “Quantum theory of a one-dimensional optical cavity with output coupling. field quantization,” Phys. Rev. A 12, 148–158 (1975).
[Crossref]

1973 (1)

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? a theory of single-quasimode laser operation,” Physical Review A 7, 1788 (1973).
[Crossref]

1961 (1)

A. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J 40, 453–488 (1961).
[Crossref]

Baseia, B.

J. C. Penaforte and B. Baseia, “Quantum theory of a one-dimensional laser with output coupling: Linear approximation,” Phys. Rev. A 30, 1401–1406 (1984).
[Crossref]

Chang, H.-C.

C.-P. Yu and H.-C. Chang, “Yee-mesh-based finite difference eigenmode solver with pml absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Optics Express 12, 6165–6177 (2004).
[Crossref] [PubMed]

Collett, M. J.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[Crossref] [PubMed]

Dutra, S. M.

S. M. Dutra and P. L. Knight, “Spontaneous emission in a planar fabry-pérot microcavity,” Phys. Rev. A 53, 3587–3605 (1996).
[Crossref] [PubMed]

Fox, A.

A. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J 40, 453–488 (1961).
[Crossref]

Gardiner, C. W.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[Crossref] [PubMed]

Glauber, R. J.

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[Crossref] [PubMed]

Ho, K. C.

K. C. Ho, P. T. Leung, A. Maassen van den Brink, and K. Young, “Second quantization of open systems using quasinormal modes,” Phys. Rev. E 58, 2965–2978 (1998).
[Crossref]

Hughes, S.

P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2–10 (2014).
[Crossref]

Joannopoulos, J. D.

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

Johnson, S. G.

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

Knight, P. L.

S. M. Dutra and P. L. Knight, “Spontaneous emission in a planar fabry-pérot microcavity,” Phys. Rev. A 53, 3587–3605 (1996).
[Crossref] [PubMed]

Kristensen, P. T.

P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2–10 (2014).
[Crossref]

Lagendijk, A.

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and super-radiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
[Crossref]

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multipole interaction between atoms and their photonic environment,” Phys. Rev. A 68, 013822 (2003).
[Crossref]

Lamb, W. E.

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? a theory of single-quasimode laser operation,” Physical Review A 7, 1788 (1973).
[Crossref]

Lang, R.

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? a theory of single-quasimode laser operation,” Physical Review A 7, 1788 (1973).
[Crossref]

Lee, K. M.

Leung, P. T.

K. M. Lee, P. T. Leung, and K. M. Pang, “Dyadic formulation of morphology-dependent resonances. i. completeness relation,” J. Opt. Soc. Am. B 16, 1409–1417 (1999).
[Crossref]

K. C. Ho, P. T. Leung, A. Maassen van den Brink, and K. Young, “Second quantization of open systems using quasinormal modes,” Phys. Rev. E 58, 2965–2978 (1998).
[Crossref]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[Crossref] [PubMed]

Lewenstein, M.

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[Crossref] [PubMed]

Li, T.

A. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J 40, 453–488 (1961).
[Crossref]

Liu, S. Y.

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[Crossref] [PubMed]

Maassen van den Brink, A.

K. C. Ho, P. T. Leung, A. Maassen van den Brink, and K. Young, “Second quantization of open systems using quasinormal modes,” Phys. Rev. E 58, 2965–2978 (1998).
[Crossref]

Meade, R. D.

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

Merz, J.

Y. Tang, A. Mintairov, J. Merz, V. Tokranov, and S. Oktyabrsky, “Characterization of 2d-photonic crystal nanocavities by polarization-dependent photoluminescence,” in “Nanotechnology, 2005. 5th IEEE Conference on,” (IEEE, 2005), pp. 35–38.

Mintairov, A.

Y. Tang, A. Mintairov, J. Merz, V. Tokranov, and S. Oktyabrsky, “Characterization of 2d-photonic crystal nanocavities by polarization-dependent photoluminescence,” in “Nanotechnology, 2005. 5th IEEE Conference on,” (IEEE, 2005), pp. 35–38.

Oktyabrsky, S.

Y. Tang, A. Mintairov, J. Merz, V. Tokranov, and S. Oktyabrsky, “Characterization of 2d-photonic crystal nanocavities by polarization-dependent photoluminescence,” in “Nanotechnology, 2005. 5th IEEE Conference on,” (IEEE, 2005), pp. 35–38.

Pang, K. M.

Penaforte, J. C.

J. C. Penaforte and B. Baseia, “Quantum theory of a one-dimensional laser with output coupling: Linear approximation,” Phys. Rev. A 30, 1401–1406 (1984).
[Crossref]

Scully, M. O.

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? a theory of single-quasimode laser operation,” Physical Review A 7, 1788 (1973).
[Crossref]

Suttorp, L. G.

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and super-radiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
[Crossref]

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multipole interaction between atoms and their photonic environment,” Phys. Rev. A 68, 013822 (2003).
[Crossref]

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite - Difference Time - Domain Method (Artech House, Incorporated, 1995).

Tang, Y.

Y. Tang, A. Mintairov, J. Merz, V. Tokranov, and S. Oktyabrsky, “Characterization of 2d-photonic crystal nanocavities by polarization-dependent photoluminescence,” in “Nanotechnology, 2005. 5th IEEE Conference on,” (IEEE, 2005), pp. 35–38.

Tokranov, V.

Y. Tang, A. Mintairov, J. Merz, V. Tokranov, and S. Oktyabrsky, “Characterization of 2d-photonic crystal nanocavities by polarization-dependent photoluminescence,” in “Nanotechnology, 2005. 5th IEEE Conference on,” (IEEE, 2005), pp. 35–38.

Ujihara, K.

K. Ujihara, “Quantum theory of a one-dimensional optical cavity with output coupling. field quantization,” Phys. Rev. A 12, 148–158 (1975).
[Crossref]

Winn, J. N.

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

Wubs, M.

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and super-radiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
[Crossref]

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multipole interaction between atoms and their photonic environment,” Phys. Rev. A 68, 013822 (2003).
[Crossref]

Young, K.

K. C. Ho, P. T. Leung, A. Maassen van den Brink, and K. Young, “Second quantization of open systems using quasinormal modes,” Phys. Rev. E 58, 2965–2978 (1998).
[Crossref]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[Crossref] [PubMed]

Yu, C.-P.

C.-P. Yu and H.-C. Chang, “Yee-mesh-based finite difference eigenmode solver with pml absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Optics Express 12, 6165–6177 (2004).
[Crossref] [PubMed]

ACS Photonics (1)

P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2–10 (2014).
[Crossref]

Bell Syst. Tech. J (1)

A. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J 40, 453–488 (1961).
[Crossref]

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

Optics Express (1)

C.-P. Yu and H.-C. Chang, “Yee-mesh-based finite difference eigenmode solver with pml absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Optics Express 12, 6165–6177 (2004).
[Crossref] [PubMed]

Phys. Rev. A (8)

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and super-radiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
[Crossref]

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[Crossref] [PubMed]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[Crossref] [PubMed]

J. C. Penaforte and B. Baseia, “Quantum theory of a one-dimensional laser with output coupling: Linear approximation,” Phys. Rev. A 30, 1401–1406 (1984).
[Crossref]

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[Crossref] [PubMed]

S. M. Dutra and P. L. Knight, “Spontaneous emission in a planar fabry-pérot microcavity,” Phys. Rev. A 53, 3587–3605 (1996).
[Crossref] [PubMed]

K. Ujihara, “Quantum theory of a one-dimensional optical cavity with output coupling. field quantization,” Phys. Rev. A 12, 148–158 (1975).
[Crossref]

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multipole interaction between atoms and their photonic environment,” Phys. Rev. A 68, 013822 (2003).
[Crossref]

Phys. Rev. E (1)

K. C. Ho, P. T. Leung, A. Maassen van den Brink, and K. Young, “Second quantization of open systems using quasinormal modes,” Phys. Rev. E 58, 2965–2978 (1998).
[Crossref]

Physical Review A (1)

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? a theory of single-quasimode laser operation,” Physical Review A 7, 1788 (1973).
[Crossref]

Other (3)

Y. Tang, A. Mintairov, J. Merz, V. Tokranov, and S. Oktyabrsky, “Characterization of 2d-photonic crystal nanocavities by polarization-dependent photoluminescence,” in “Nanotechnology, 2005. 5th IEEE Conference on,” (IEEE, 2005), pp. 35–38.

A. Taflove, Computational Electrodynamics: The Finite - Difference Time - Domain Method (Artech House, Incorporated, 1995).

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

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Figures (4)

Fig. 1.
Fig. 1. The sketch of photonic crystal slab cavity. The highlighted holes with the red circles are shifted outwards along x from the nominal lattice position. Right: the spatial distribution of y-component of the cavity’s normal mode in the plane z = 0.
Fig. 2.
Fig. 2. Purcell factor corresponding to y-polarized dipole as a function of position x. Purcell factor indicated by the blue solid curve is obtained through the normal mode, while the red circles corresponds to the direct calculations by FDTD.
Fig. 3.
Fig. 3. (a) The y-component of the normalized normal mode of the silicon cuboid as a function of position x. Blue solid line shows the normalized mode. Red solid line in the inset shows the fitting curve. The fitting function is 10.974/(x − 0.364), the shift on x could be neglected at very far field; (b) the y-component of the quasi-normal mode of the silicon cuboid as a function of position x
Fig. 4.
Fig. 4. The spatial distribution of x- and y-component of mode B3u and B1g in the plane z = 0, the resonance frequency is 1.305 × 1015rad/s. (a) f x T for mode B3u, (b) f y T for mode B3u, (c) f x T for mode B1g, (d) f y T for mode B1g.

Equations (7)

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[ × × ω 2 c 2 ε ( r ) ] G ( r , r , ω ) = ω 2 c 2 1 δ ( r r ) ,
G ( r , r , ω ) = n 0 d ν ω 2 ( ν ω i δ ) ( ν + ω + i δ ) × f n T ( ν , r ) [ f n T ( ν , r ) ] * λ f λ L ( r ) [ f λ L ( r ) ] * ,
G ( r , r , ω ) = 1 2 n ( 0 + 0 ) d ν ω 2 ( ν ω i δ ) ( ν + ω + i δ ) × f n T ( ν , r ) [ f n T ( ν , r ) ] * λ f λ L ( r ) [ f λ L ( r ) ] * ,
G ( r , r , ω ) = i π ω 2 n f n T ( ω , r ) f n T ( ω , r ) λ f λ L ( r ) [ f λ L ( r ) ] * 1 2 n C inf + C 0 d ν ω 2 f n T ( ν , r ) [ f n T ( ν , r ) ] * ( ν ω i δ ) ( ν + ω + i δ ) ,
Im { G ( r , r , ω ) } = π ω 2 n f n T ( ω , r ) f n T ( ω , r ) ,
Im { E ( r , ω ) } = π ω 2 ε 0 n , m f n T ( ω , r ) f n T ( ω , r m ) p m ( ω )
f T ( ω , r ) = Im { E ( r , ω ) } π ω p 0 ( ω ) Im { E ( r 0 , ω ) } / 2 ε 0 ,

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