Abstract

We extend the coupled-wave-theory (CWT) framework to a supercell lattice photonic crystal (PC) structure to model the radiation of high-Q resonances under structural fluctuations since they are inevitable in realistic devices. The comparison of CWT results and the finite-element-method (FEM) simulations confirm the validity of CWT. It is proved that the supercell model approaches a realistic finite-size PC device when the supercell size is large enough. The Q factors within fluctuated structures are constraint owing to the appearance of fractional orders of radiative waves, which are induced by structural fluctuations. For a large enough footprint size, the upper bound of the Q factor is determined by the fabrication precision, and further increasing the device size will no longer benefit the Q factor.

© 2017 Optical Society of America

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References

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  23. L. Ni, Z. Wang, C. Peng, and Z. Li, “Tunable optical bound states in the continuum beyond in-plane symmetry protection,” Phys. Rev. B 94, 245148 (2016).
    [Crossref]
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    [Crossref]
  25. H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79, 085112 (2009).
    [Crossref]
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    [Crossref]
  27. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011).
    [Crossref] [PubMed]
  28. Z. Wang, H. Zhang, L. Ni, W. Hu, and C. Peng, “Analytical perspective of interfering resonances in high-index-contrast periodic photonic structures,” IEEE J. Quantum Electron. 52, 1–9 (2016).
    [Crossref]
  29. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic crystal surface emitting lasers with transverse-electric polarization: finite-size effects,” Opt. Express 20, 15945–15961 (2012).
    [Crossref] [PubMed]
  30. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
    [Crossref]
  31. Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, “Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization,” Opt. Express 21, 565–580 (2013).
    [Crossref] [PubMed]
  32. Y. Yang, C. Peng, and Z. Li, “Semi-analytical approach for guided mode resonance in high-index-contrast photonic crystal slab: TE polarization,” Opt. Express 21, 20588–20600 (2013).
    [Crossref] [PubMed]
  33. Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Three-dimensional coupled-wave theory for the guided mode resonance in photonic crystal slabs: TM-like polarization,” Opt. Lett. 39, 4498–4501 (2014).
    [Crossref] [PubMed]

2017 (1)

E. Bulgakov and A. Sadreev, “Trapping of light with angular orbital momentum above the light cone,” Advanced Electromagnetics 6, 1–10 (2017).
[Crossref]

2016 (3)

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

L. Ni, Z. Wang, C. Peng, and Z. Li, “Tunable optical bound states in the continuum beyond in-plane symmetry protection,” Phys. Rev. B 94, 245148 (2016).
[Crossref]

Z. Wang, H. Zhang, L. Ni, W. Hu, and C. Peng, “Analytical perspective of interfering resonances in high-index-contrast periodic photonic structures,” IEEE J. Quantum Electron. 52, 1–9 (2016).
[Crossref]

2015 (1)

A. Liu, W. Hofmann, and D. Bimberg, “Integrated high-contrast-grating optical sensor using guided mode,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

2014 (3)

E. N. Bulgakov and A. F. Sadreev, “Bloch bound states in the radiation continuum in a periodic array of dielectric rods,” Phys. Rev. A 90, 053801 (2014).
[Crossref]

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Three-dimensional coupled-wave theory for the guided mode resonance in photonic crystal slabs: TM-like polarization,” Opt. Lett. 39, 4498–4501 (2014).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113, 257401 (2014).
[Crossref]

2013 (4)

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188–191 (2013).
[Crossref] [PubMed]

Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, “Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization,” Opt. Express 21, 565–580 (2013).
[Crossref] [PubMed]

Y. Yang, C. Peng, and Z. Li, “Semi-analytical approach for guided mode resonance in high-index-contrast photonic crystal slab: TE polarization,” Opt. Express 21, 20588–20600 (2013).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

2012 (4)

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108, 070401 (2012).
[Crossref] [PubMed]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic crystal surface emitting lasers with transverse-electric polarization: finite-size effects,” Opt. Express 20, 15945–15961 (2012).
[Crossref] [PubMed]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
[Crossref]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

2011 (3)

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B 84, 195119 (2011).
[Crossref]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011).
[Crossref] [PubMed]

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref] [PubMed]

2009 (2)

K. Ishizaki, M. Okano, and S. Noda, “Numerical investigation of emission in finite-sized, three-dimensional photonic crystals with structural fluctuations,” J. Opt. Soc. Am. B 26, 1157–1161 (2009).
[Crossref]

H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79, 085112 (2009).
[Crossref]

2008 (2)

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100, 183902 (2008).
[Crossref] [PubMed]

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78, 075105 (2008).
[Crossref]

2007 (2)

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1, 119–122 (2007).
[Crossref]

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (2007).
[Crossref]

2004 (1)

1997 (1)

D. Rosenblatt, A. Sharon, and A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[Crossref]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

1985 (3)

H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum,” Phys. Rev. A 32, 3231–3242 (1985).
[Crossref]

H. Friedrich and D. Wintgen, “Physical realization of bound states in the continuum,” Phys. Rev. A 31, 3964–3966 (1985).
[Crossref]

R. Kazarinov and C. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
[Crossref]

1976 (1)

W. Streifer, D. Scifres, and R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides - I,” IEEE J. Quantum Electron. 12, 422–428 (1976).
[Crossref]

1929 (1)

J. Von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z. 30, 467–470 (1929).

Asano, T.

H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79, 085112 (2009).
[Crossref]

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (2007).
[Crossref]

Bimberg, D.

A. Liu, W. Hofmann, and D. Bimberg, “Integrated high-contrast-grating optical sensor using guided mode,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

Borisov, A. G.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100, 183902 (2008).
[Crossref] [PubMed]

Bulgakov, E.

E. Bulgakov and A. Sadreev, “Trapping of light with angular orbital momentum above the light cone,” Advanced Electromagnetics 6, 1–10 (2017).
[Crossref]

Bulgakov, E. N.

E. N. Bulgakov and A. F. Sadreev, “Bloch bound states in the radiation continuum in a periodic array of dielectric rods,” Phys. Rev. A 90, 053801 (2014).
[Crossref]

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78, 075105 (2008).
[Crossref]

Burnham, R.

W. Streifer, D. Scifres, and R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides - I,” IEEE J. Quantum Electron. 12, 422–428 (1976).
[Crossref]

Chang-Hasnain, C. J.

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1, 119–122 (2007).
[Crossref]

Chua, S.-L.

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188–191 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Ding, Y.

Dreisow, F.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref] [PubMed]

Friedrich, H.

H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum,” Phys. Rev. A 32, 3231–3242 (1985).
[Crossref]

H. Friedrich and D. Wintgen, “Physical realization of bound states in the continuum,” Phys. Rev. A 31, 3964–3966 (1985).
[Crossref]

Friesem, A.

D. Rosenblatt, A. Sharon, and A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[Crossref]

Fujita, M.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (2007).
[Crossref]

Hagino, H.

H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79, 085112 (2009).
[Crossref]

Heinrich, M.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref] [PubMed]

Henry, C.

R. Kazarinov and C. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
[Crossref]

Hofmann, W.

A. Liu, W. Hofmann, and D. Bimberg, “Integrated high-contrast-grating optical sensor using guided mode,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

Hsu, C. W.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113, 257401 (2014).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188–191 (2013).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

Hu, W.

Z. Wang, H. Zhang, L. Ni, W. Hu, and C. Peng, “Analytical perspective of interfering resonances in high-index-contrast periodic photonic structures,” IEEE J. Quantum Electron. 52, 1–9 (2016).
[Crossref]

Huang, M. C. Y.

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1, 119–122 (2007).
[Crossref]

Ishizaki, K.

Iwahashi, S.

Joannopoulos, J. D.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188–191 (2013).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Johnson, S. G.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188–191 (2013).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

Kazarinov, R.

R. Kazarinov and C. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
[Crossref]

Kitamura, K.

Kivshar, Y. S.

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108, 070401 (2012).
[Crossref] [PubMed]

Lee, J.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188–191 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Li, Z.

Liang, Y.

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Three-dimensional coupled-wave theory for the guided mode resonance in photonic crystal slabs: TM-like polarization,” Opt. Lett. 39, 4498–4501 (2014).
[Crossref] [PubMed]

Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, “Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization,” Opt. Express 21, 565–580 (2013).
[Crossref] [PubMed]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
[Crossref]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic crystal surface emitting lasers with transverse-electric polarization: finite-size effects,” Opt. Express 20, 15945–15961 (2012).
[Crossref] [PubMed]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B 84, 195119 (2011).
[Crossref]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011).
[Crossref] [PubMed]

Liu, A.

A. Liu, W. Hofmann, and D. Bimberg, “Integrated high-contrast-grating optical sensor using guided mode,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

Lu, L.

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113, 257401 (2014).
[Crossref]

Magnusson, R.

Marinica, D. C.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100, 183902 (2008).
[Crossref] [PubMed]

Miroshnichenko, A. E.

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108, 070401 (2012).
[Crossref] [PubMed]

Molina, M. I.

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108, 070401 (2012).
[Crossref] [PubMed]

Ni, L.

Z. Wang, H. Zhang, L. Ni, W. Hu, and C. Peng, “Analytical perspective of interfering resonances in high-index-contrast periodic photonic structures,” IEEE J. Quantum Electron. 52, 1–9 (2016).
[Crossref]

L. Ni, Z. Wang, C. Peng, and Z. Li, “Tunable optical bound states in the continuum beyond in-plane symmetry protection,” Phys. Rev. B 94, 245148 (2016).
[Crossref]

Noda, S.

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Three-dimensional coupled-wave theory for the guided mode resonance in photonic crystal slabs: TM-like polarization,” Opt. Lett. 39, 4498–4501 (2014).
[Crossref] [PubMed]

Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, “Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization,” Opt. Express 21, 565–580 (2013).
[Crossref] [PubMed]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
[Crossref]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic crystal surface emitting lasers with transverse-electric polarization: finite-size effects,” Opt. Express 20, 15945–15961 (2012).
[Crossref] [PubMed]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B 84, 195119 (2011).
[Crossref]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011).
[Crossref] [PubMed]

H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79, 085112 (2009).
[Crossref]

K. Ishizaki, M. Okano, and S. Noda, “Numerical investigation of emission in finite-sized, three-dimensional photonic crystals with structural fluctuations,” J. Opt. Soc. Am. B 26, 1157–1161 (2009).
[Crossref]

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (2007).
[Crossref]

Nolte, S.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref] [PubMed]

Okano, M.

Peleg, O.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref] [PubMed]

Peng, C.

L. Ni, Z. Wang, C. Peng, and Z. Li, “Tunable optical bound states in the continuum beyond in-plane symmetry protection,” Phys. Rev. B 94, 245148 (2016).
[Crossref]

Z. Wang, H. Zhang, L. Ni, W. Hu, and C. Peng, “Analytical perspective of interfering resonances in high-index-contrast periodic photonic structures,” IEEE J. Quantum Electron. 52, 1–9 (2016).
[Crossref]

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Three-dimensional coupled-wave theory for the guided mode resonance in photonic crystal slabs: TM-like polarization,” Opt. Lett. 39, 4498–4501 (2014).
[Crossref] [PubMed]

Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, “Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization,” Opt. Express 21, 565–580 (2013).
[Crossref] [PubMed]

Y. Yang, C. Peng, and Z. Li, “Semi-analytical approach for guided mode resonance in high-index-contrast photonic crystal slab: TE polarization,” Opt. Express 21, 20588–20600 (2013).
[Crossref] [PubMed]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic crystal surface emitting lasers with transverse-electric polarization: finite-size effects,” Opt. Express 20, 15945–15961 (2012).
[Crossref] [PubMed]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
[Crossref]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B 84, 195119 (2011).
[Crossref]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011).
[Crossref] [PubMed]

Plotnik, Y.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref] [PubMed]

Qiu, W.

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Rosenblatt, D.

D. Rosenblatt, A. Sharon, and A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[Crossref]

Sadreev, A.

E. Bulgakov and A. Sadreev, “Trapping of light with angular orbital momentum above the light cone,” Advanced Electromagnetics 6, 1–10 (2017).
[Crossref]

Sadreev, A. F.

E. N. Bulgakov and A. F. Sadreev, “Bloch bound states in the radiation continuum in a periodic array of dielectric rods,” Phys. Rev. A 90, 053801 (2014).
[Crossref]

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78, 075105 (2008).
[Crossref]

Sakai, K.

Scifres, D.

W. Streifer, D. Scifres, and R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides - I,” IEEE J. Quantum Electron. 12, 422–428 (1976).
[Crossref]

Segev, M.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref] [PubMed]

Shabanov, S. V.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100, 183902 (2008).
[Crossref] [PubMed]

Shapira, O.

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Sharon, A.

D. Rosenblatt, A. Sharon, and A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[Crossref]

Soljacic, M.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113, 257401 (2014).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188–191 (2013).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Stone, A. D.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113, 257401 (2014).
[Crossref]

Streifer, W.

W. Streifer, D. Scifres, and R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides - I,” IEEE J. Quantum Electron. 12, 422–428 (1976).
[Crossref]

Szameit, A.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref] [PubMed]

Takahashi, Y.

H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79, 085112 (2009).
[Crossref]

Tanaka, Y.

Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, “Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization,” Opt. Express 21, 565–580 (2013).
[Crossref] [PubMed]

H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79, 085112 (2009).
[Crossref]

Von Neumann, J.

J. Von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z. 30, 467–470 (1929).

Wang, Z.

L. Ni, Z. Wang, C. Peng, and Z. Li, “Tunable optical bound states in the continuum beyond in-plane symmetry protection,” Phys. Rev. B 94, 245148 (2016).
[Crossref]

Z. Wang, H. Zhang, L. Ni, W. Hu, and C. Peng, “Analytical perspective of interfering resonances in high-index-contrast periodic photonic structures,” IEEE J. Quantum Electron. 52, 1–9 (2016).
[Crossref]

Wigner, E.

J. Von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z. 30, 467–470 (1929).

Wintgen, D.

H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum,” Phys. Rev. A 32, 3231–3242 (1985).
[Crossref]

H. Friedrich and D. Wintgen, “Physical realization of bound states in the continuum,” Phys. Rev. A 31, 3964–3966 (1985).
[Crossref]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

Yang, Y.

Zhang, H.

Z. Wang, H. Zhang, L. Ni, W. Hu, and C. Peng, “Analytical perspective of interfering resonances in high-index-contrast periodic photonic structures,” IEEE J. Quantum Electron. 52, 1–9 (2016).
[Crossref]

Zhen, B.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113, 257401 (2014).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188–191 (2013).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Zhou, Y.

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1, 119–122 (2007).
[Crossref]

Advanced Electromagnetics (1)

E. Bulgakov and A. Sadreev, “Trapping of light with angular orbital momentum above the light cone,” Advanced Electromagnetics 6, 1–10 (2017).
[Crossref]

IEEE J. Quantum Electron. (5)

W. Streifer, D. Scifres, and R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides - I,” IEEE J. Quantum Electron. 12, 422–428 (1976).
[Crossref]

R. Kazarinov and C. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
[Crossref]

A. Liu, W. Hofmann, and D. Bimberg, “Integrated high-contrast-grating optical sensor using guided mode,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

D. Rosenblatt, A. Sharon, and A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[Crossref]

Z. Wang, H. Zhang, L. Ni, W. Hu, and C. Peng, “Analytical perspective of interfering resonances in high-index-contrast periodic photonic structures,” IEEE J. Quantum Electron. 52, 1–9 (2016).
[Crossref]

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

Light Sci. Appl. (1)

C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

Nat. Photonics (2)

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (2007).
[Crossref]

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1, 119–122 (2007).
[Crossref]

Nat. Rev. Mater. (1)

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

Nature (1)

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188–191 (2013).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. A (3)

E. N. Bulgakov and A. F. Sadreev, “Bloch bound states in the radiation continuum in a periodic array of dielectric rods,” Phys. Rev. A 90, 053801 (2014).
[Crossref]

H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum,” Phys. Rev. A 32, 3231–3242 (1985).
[Crossref]

H. Friedrich and D. Wintgen, “Physical realization of bound states in the continuum,” Phys. Rev. A 31, 3964–3966 (1985).
[Crossref]

Phys. Rev. B (5)

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78, 075105 (2008).
[Crossref]

L. Ni, Z. Wang, C. Peng, and Z. Li, “Tunable optical bound states in the continuum beyond in-plane symmetry protection,” Phys. Rev. B 94, 245148 (2016).
[Crossref]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
[Crossref]

H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79, 085112 (2009).
[Crossref]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic crystal lasers with transverse electric polarization: a general approach,” Phys. Rev. B 84, 195119 (2011).
[Crossref]

Phys. Rev. Lett. (6)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113, 257401 (2014).
[Crossref]

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108, 070401 (2012).
[Crossref] [PubMed]

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100, 183902 (2008).
[Crossref] [PubMed]

Phys. Z. (1)

J. Von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z. 30, 467–470 (1929).

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Figures (7)

Fig. 1
Fig. 1 (a)(b) The schematics of a typical 1D-PC slab. It is periodic in the x direction, extends infinitely in the y direction, and is three layered in the z direction. The permittivity of the upper/lower cladding layers are εu and εl, respectively. The PC layer is patterned with two materials εa and εb with a period of a and a filling factor of f0 = w/a. (c) Two types of structural fluctuations concerned in this work: (i) fluctuations of the center positions of the holes, (ii) fluctuations of the filling factors. (d) A typical band structure near the 2nd order Γ point matched with mode TE0. We denote the modes as A and B in increasing frequency.
Fig. 2
Fig. 2 A schematic of coupling paths indicated in Eq. (21), by assuming the basic wave set V = {−1, 1}. The big dots represent the Bloch waves of mZ, whereas the small ones depict the fractional Bloch waves of mZ. The coupling paths indicated by the coupling matrices C0 z , C0 d , Cz, Cd are illustrated in the figure.
Fig. 3
Fig. 3 The amplitude distribution in the k-space for mode A on SOI under a random fluctuation pattern, with supercell size N = 300, (a) near m = 0 (b) near m = 1, (c) near m = −1, and (d) large scale. The red area in (d) represents the Bloch waves above the light cone. Data is obtained from CWT.
Fig. 4
Fig. 4 1/Q of mode A on an SOI supercell of N = 20. As a Monte Carlo simulation, 25 randomly generated fluctuation patterns are plotted with (a) σρ = 1/550, σξ = 0, and (b) σρ = 0, σξ = 1/550. The results of both CWT (blue circles) and FEM (red plus sign) are presented. σρ and σξ are determined by assuming the fabrication process precision of 1nm. The green dashed lines show the expected value of 1/Q calculated by CWT, which agrees well with the average value of FEM results.
Fig. 5
Fig. 5 Dependence of 1/Q of mode A on σ2 under five random fluctuation patterns on an SOI supercell structure of N = 20. Results of both CWT and FEM are presented when fluctuations exist in (a) only Δρ, (b) only Δξ, and (c) both Δρ and Δξ. Each of them shows a clear quadratic dependence of 1/Q on σ.
Fig. 6
Fig. 6 1/Q of mode A on an SOI supercell structure under random fluctuation patterns versus varying supercell size N, with σρ = 1/550 and σξ = 1/550. Results calculated by both CWT (blue circles) and FEM (red plus signs) are presented for individual patterns. The gray dashed line indicates the critical supercell size N = 67 of switching non-degenerate perturbation to degenerate perturbation. The purple line shows the asymptotic value of 1/Q.
Fig. 7
Fig. 7 (a) The CWT and FEM calculated 1/Q of mode A on SOI supercell structure under 25 random fluctuation patterns, with supercell size N = 300, and the FEM results for the corresponding finite-size structure with the same fluctuation patterns. (b) The asymptotic 1/Q versus the structural fluctuation σ obtained from CWT using Monte-Carlo method. Data is obtained by setting the supercell size N = 300.

Tables (1)

Tables Icon

Table 1 Structural Parameters

Equations (34)

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

ε z , m = ε a ε b m π sin ( m π f 0 )
Δ ε m = 2 ( ε a ε b ) N [ cos ( m π f 0 ) l = l m l M Δ ρ l e i 2 π m l + i sin ( m π f ) l = l m l M Δ ξ l e i 2 π m l ]
( 2 z 2 + ε 0 ( z ) k 2 m 2 β 0 2 ) E m = k 2 m m ε m m ( z ) E m ( z )
( 2 z 2 + ε 0 ( z ) k m 2 m 2 β 0 2 ) Θ m ( z ) = 0
( 2 z 2 + ε 0 ( z ) k 2 m 2 β 0 2 ) V m Θ m ( z ) = k 2 m V ε m m ( z ) V m Θ m ( z ) k 2 m A ε m m ( z ) A m Θ m ( z )
( k 2 k m 2 ) h m V m = k 2 m V ε m m O m m V m k 2 m A ε m m O m m A m
k 2 V m = k m 2 δ m m V m k 2 h m m V ε m m O m m V m k 2 h m m A ε m m O m m A m
k 2 V = ( D + L V ) V + L A A
( 2 z 2 + ε 0 k 2 m 2 β 0 2 ) G m ( z ; z ) = δ ( z z )
A m Θ m ( z ) = k 2 m V ε m m V m G ^ m Θ m ( z ) k 2 m A ε m m A m G ^ m Θ m ( z )
A m = k 2 m V ε m m G m m V m k 2 m A ε m m G m m A m
A = C 0 V + C A
A = T V
k 2 V = ( D + L V + L A T ) V
Θ m ( z ) = Θ m ( 0 ) ( z ) / ( P . C . | Θ m ( 0 ) ( z ) | 2 d z ) 1 2
T = [ I + ( I C ) 1 C ] C 0
C z , m m = k c 2 ε z , m m G m m ( 1 δ m m ) , m , m A
C d , m m = k c 2 Δ ε m m G m m ( 1 δ m m ) , m , m A
C 0 z , m m = k c 2 ε z , m m G m m , m A , m V
C 0 d , m m = k c 2 Δ ε m m G m m , m A , m V
T [ I + ( I C z ) 1 C ] C 0 = ( C 0 z + C 0 d ) V + ( I C z ) 1 ( C z C 0 z + C z C 0 d + C d C 0 z + C d C 0 d ) ( C 0 z + C 0 d ) V + ( I C z ) 1 ( C z C 0 z + C z C 0 d + C d C 0 z ) = ( I C z ) 1 ( C 0 z + C 0 d + C d C 0 z )
T ( i , j ) = ( I C z ( i ) ) 1 ( C 0 z ( i ) δ ( i , j ) + C 0 d ( i , j ) + C d ( i , j ) C 0 z ( j ) )
P m = ( 1 2 ε 0 , u | Θ m , u | 2 cos θ m , u + 1 2 ε 0 , l | Θ m , l | 2 cos θ m , l ) | A m | 2 = B m | A m | 2
A = ( I C z ) 1 ( C 0 d + C d C 0 z ) V
A i = l f i l Δ ρ l + l g i l Δ ξ l
E | A i | 2 = σ ρ 2 l | f i l | 2 + N 1 N σ ξ 2 l | g i l | 2
| A m | 2 = l 1 l 2 [ f m l 1 * f m l 2 Δ ρ l 1 Δ ρ l 2 + g m l 1 * g m l 2 Δ ξ l 1 Δ ξ l 2 + 2 R e ( f m l 1 * g m l 2 Δ ρ l 1 Δ ξ l 2 ) ] = σ 2 σ 0 2 l 1 l 2 [ f m l 1 * f m l 2 Δ ρ l 1 ( 0 ) Δ ρ l 2 ( 0 ) + g m l 1 * g m l 2 Δ ξ l 1 ( 0 ) Δ ξ l 2 ( 0 ) + 2 R e ( f m l 1 * g m l 2 Δ ρ l 1 ( 0 ) Δ ξ l 2 ( 0 ) ) ]
A i = j k l B i j ( a j k , l Δ ρ l + b j k , l Δ ξ l ) V k + j k l B i j ( c j k , l Δ ρ l + d j k , l Δ ξ l ) V z , k = l Δ ρ l ( j k B i j a j k , l V k + j k B i j c j k , l V z , k ) + l Δ ξ l ( j k B i j b j k , l V k + j k B i j d j k , l V z , k ) = l f i l Δ ρ l + l g i l Δ ξ l
C 0 d , m m = k 2 G m m Δ ε m m = l a m m , l Δ ρ l + b m m , l Δ ξ l
a m m , l = k 2 G m m 2 ( ε a ε b ) N cos [ ( m m ) π f 0 ] e i 2 π ( m m ) l
b m m , l = k 2 G m m 2 i ( ε a ε b ) N sin [ ( m m ) π f 0 ] e i 2 π ( m m ) l
C d , m m = ( 1 δ m m ) k 2 G m m Δ ε m m = ( 1 δ m m ) l c m m , l Δ ρ l + d m m , l Δ ξ l
c m m , l = k 2 G m m 2 ( ε a ε b ) N cos [ ( m m ) π f 0 ] e i 2 π ( m m ) l
d m m , l = k 2 G m m 2 i ( ε a ε b ) N sin [ ( m m ) π f 0 ] e i 2 π ( m m ) l

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