Abstract

In this paper, we develop spatiotemporal coupled-mode theory to describe optical properties of guided-mode resonant gratings. We derive partial differential equations that describe both spatial and temporal evolution of the field inside the grating. These equations describe the coupling of two counter-propagating grating modes, revealing the structure’s “dark” and “bright” resonances at normal incidence of light. Moreover, the proposed theory allows us to obtain a simple approximation of the transmission and reflection coefficients taking into account both light’s frequency and angle of incidence. This approximation can be considered as the generalization of the Fano line-shape. The approximation is in good agreement with the rigorous computations based on the Fourier modal method. The results of the paper will be useful for design and analysis of guided-mode resonant filters and other photonic devices.

© 2015 Optical Society of America

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References

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  1. W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
    [Crossref]
  2. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
    [Crossref]
  3. S. Collin, “Nanostructure arrays in free-space: optical properties and applications,” Rep. Prog. Phys. 77, 126402 (2014).
    [Crossref] [PubMed]
  4. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, 1984).
  5. H. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
    [Crossref]
  6. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
    [Crossref]
  7. C. Manolatou, M. Khan, S. Fan, P. R. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
    [Crossref]
  8. 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 Electron. 40, 1511–1518 (2004).
    [Crossref]
  9. E. Waks and J. Vuckovic, “Coupled mode theory for photonic crystal cavity-waveguide interaction,” Opt. Express 13, 5064–5073 (2005).
    [Crossref] [PubMed]
  10. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
    [Crossref]
  11. Z. Ruan and S. Fan, “Temporal coupled-mode theory for light scattering by an arbitrarily shaped object supporting a single resonance,” Phys. Rev. A 85, 043828 (2012).
    [Crossref]
  12. L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: A coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108, 083902 (2012).
    [Crossref] [PubMed]
  13. L. Verslegers, Z. Yu, P. B. Catrysse, and S. Fan, “Temporal coupled-mode theory for resonant apertures,” J. Opt. Soc. Am. B 27, 1947–1956 (2010).
    [Crossref]
  14. V. R. Shteeman, I. Nusinsky, and A. A. Hardy, “Time-dependent coupled mode analysis of parallel waveguides,” J. Opt. Soc. Am. B 27, 735–741 (2010).
    [Crossref]
  15. V. Shteeman and A. A. Hardy, “Analysis of advanced photonic devices with time-dependent coupled mode equations,” Opt. Eng. 51, 054001 (2012).
    [Crossref]
  16. E. Smith, V. Shteeman, and A. A. Hardy, “Time-dependent coupled mode analysis of advanced photonic micro devices,” in Proceedings of IEEE 27th Convention of Electrical & Electronics Engineers in Israel (IEEE, 2012).
  17. B. Dana, L. Lobachinsky, and A. Bahabad, “Spatiotemporal coupled-mode theory in dispersive media under a dynamic modulation,” Opt. Commun. 324, 165–167 (2014).
    [Crossref]
  18. D. A. Bykov, L. L. Doskolovich, E. A. Bezus, and V. A. Soifer, “Optical computation of the Laplace operator using phase-shifted Bragg grating,” Opt. Express 22, 25084–25092 (2014).
    [Crossref] [PubMed]
  19. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [Crossref]
  20. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [Crossref]
  21. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
    [Crossref]

2014 (4)

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

S. Collin, “Nanostructure arrays in free-space: optical properties and applications,” Rep. Prog. Phys. 77, 126402 (2014).
[Crossref] [PubMed]

B. Dana, L. Lobachinsky, and A. Bahabad, “Spatiotemporal coupled-mode theory in dispersive media under a dynamic modulation,” Opt. Commun. 324, 165–167 (2014).
[Crossref]

D. A. Bykov, L. L. Doskolovich, E. A. Bezus, and V. A. Soifer, “Optical computation of the Laplace operator using phase-shifted Bragg grating,” Opt. Express 22, 25084–25092 (2014).
[Crossref] [PubMed]

2012 (3)

V. Shteeman and A. A. Hardy, “Analysis of advanced photonic devices with time-dependent coupled mode equations,” Opt. Eng. 51, 054001 (2012).
[Crossref]

Z. Ruan and S. Fan, “Temporal coupled-mode theory for light scattering by an arbitrarily shaped object supporting a single resonance,” Phys. Rev. A 85, 043828 (2012).
[Crossref]

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: A coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref] [PubMed]

2010 (3)

2005 (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 Electron. 40, 1511–1518 (2004).
[Crossref]

2003 (1)

2002 (1)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

1999 (1)

C. Manolatou, M. Khan, S. Fan, P. R. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

1996 (1)

1995 (1)

1991 (1)

H. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[Crossref]

1972 (1)

Bahabad, A.

B. Dana, L. Lobachinsky, and A. Bahabad, “Spatiotemporal coupled-mode theory in dispersive media under a dynamic modulation,” Opt. Commun. 324, 165–167 (2014).
[Crossref]

Bezus, E. A.

Bykov, D. A.

Catrysse, P. B.

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: A coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref] [PubMed]

L. Verslegers, Z. Yu, P. B. Catrysse, and S. Fan, “Temporal coupled-mode theory for resonant apertures,” J. Opt. Soc. Am. B 27, 1947–1956 (2010).
[Crossref]

Chadha, A.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

Chuwongin, S.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

Collin, S.

S. Collin, “Nanostructure arrays in free-space: optical properties and applications,” Rep. Prog. Phys. 77, 126402 (2014).
[Crossref] [PubMed]

Dana, B.

B. Dana, L. Lobachinsky, and A. Bahabad, “Spatiotemporal coupled-mode theory in dispersive media under a dynamic modulation,” Opt. Commun. 324, 165–167 (2014).
[Crossref]

Doskolovich, L. L.

Fan, S.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: A coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref] [PubMed]

Z. Ruan and S. Fan, “Temporal coupled-mode theory for light scattering by an arbitrarily shaped object supporting a single resonance,” Phys. Rev. A 85, 043828 (2012).
[Crossref]

L. Verslegers, Z. Yu, P. B. Catrysse, and S. Fan, “Temporal coupled-mode theory for resonant apertures,” J. Opt. Soc. Am. B 27, 1947–1956 (2010).
[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 Electron. 40, 1511–1518 (2004).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[Crossref]

C. Manolatou, M. Khan, S. Fan, P. R. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

Flach, S.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

Gaylord, T. K.

Gippius, N. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Grann, E. B.

Hardy, A. A.

V. Shteeman and A. A. Hardy, “Analysis of advanced photonic devices with time-dependent coupled mode equations,” Opt. Eng. 51, 054001 (2012).
[Crossref]

V. R. Shteeman, I. Nusinsky, and A. A. Hardy, “Time-dependent coupled mode analysis of parallel waveguides,” J. Opt. Soc. Am. B 27, 735–741 (2010).
[Crossref]

E. Smith, V. Shteeman, and A. A. Hardy, “Time-dependent coupled mode analysis of advanced photonic micro devices,” in Proceedings of IEEE 27th Convention of Electrical & Electronics Engineers in Israel (IEEE, 2012).

Haus, H.

C. Manolatou, M. Khan, S. Fan, P. R. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

H. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[Crossref]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, 1984).

Huang, W.

H. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[Crossref]

Ishihara, T.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Joannopoulos, J.

C. Manolatou, M. Khan, S. Fan, P. R. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

Joannopoulos, J. D.

Khan, M.

C. Manolatou, M. Khan, S. Fan, P. R. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

Kivshar, Y. S.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

Li, L.

Liu, V.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

Lobachinsky, L.

B. Dana, L. Lobachinsky, and A. Bahabad, “Spatiotemporal coupled-mode theory in dispersive media under a dynamic modulation,” Opt. Commun. 324, 165–167 (2014).
[Crossref]

Ma, Z.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

Manolatou, C.

C. Manolatou, M. Khan, S. Fan, P. R. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

Miroshnichenko, A. E.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

Moharam, M. G.

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Nusinsky, I.

Pommet, D. A.

Ruan, Z.

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: A coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref] [PubMed]

Z. Ruan and S. Fan, “Temporal coupled-mode theory for light scattering by an arbitrarily shaped object supporting a single resonance,” Phys. Rev. A 85, 043828 (2012).
[Crossref]

Seo, J.-H.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

Shteeman, V.

V. Shteeman and A. A. Hardy, “Analysis of advanced photonic devices with time-dependent coupled mode equations,” Opt. Eng. 51, 054001 (2012).
[Crossref]

E. Smith, V. Shteeman, and A. A. Hardy, “Time-dependent coupled mode analysis of advanced photonic micro devices,” in Proceedings of IEEE 27th Convention of Electrical & Electronics Engineers in Israel (IEEE, 2012).

Shteeman, V. R.

Shuai, Y.-C.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

Smith, E.

E. Smith, V. Shteeman, and A. A. Hardy, “Time-dependent coupled mode analysis of advanced photonic micro devices,” in Proceedings of IEEE 27th Convention of Electrical & Electronics Engineers in Israel (IEEE, 2012).

Snyder, A. W.

Soifer, V. A.

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 Electron. 40, 1511–1518 (2004).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[Crossref]

Tikhodeev, S. G.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Verslegers, L.

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: A coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref] [PubMed]

L. Verslegers, Z. Yu, P. B. Catrysse, and S. Fan, “Temporal coupled-mode theory for resonant apertures,” J. Opt. Soc. Am. B 27, 1947–1956 (2010).
[Crossref]

Villeneuve, P. R.

C. Manolatou, M. Khan, S. Fan, P. R. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

Vuckovic, J.

Waks, E.

Wang, K. X.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

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 Electron. 40, 1511–1518 (2004).
[Crossref]

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Yang, H.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

Yu, Z.

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: A coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref] [PubMed]

L. Verslegers, Z. Yu, P. B. Catrysse, and S. Fan, “Temporal coupled-mode theory for resonant apertures,” J. Opt. Soc. Am. B 27, 1947–1956 (2010).
[Crossref]

Zhao, D.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

Zhou, W.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

IEEE J. Quantum Electron. (2)

C. Manolatou, M. Khan, S. Fan, P. R. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[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 Electron. 40, 1511–1518 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

B. Dana, L. Lobachinsky, and A. Bahabad, “Spatiotemporal coupled-mode theory in dispersive media under a dynamic modulation,” Opt. Commun. 324, 165–167 (2014).
[Crossref]

Opt. Eng. (1)

V. Shteeman and A. A. Hardy, “Analysis of advanced photonic devices with time-dependent coupled mode equations,” Opt. Eng. 51, 054001 (2012).
[Crossref]

Opt. Express (2)

Phys. Rev. A (1)

Z. Ruan and S. Fan, “Temporal coupled-mode theory for light scattering by an arbitrarily shaped object supporting a single resonance,” Phys. Rev. A 85, 043828 (2012).
[Crossref]

Phys. Rev. B (1)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Phys. Rev. Lett. (1)

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: A coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref] [PubMed]

Proc. IEEE (1)

H. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[Crossref]

Prog. Quantum Electron (1)

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron 38, 1–74 (2014).
[Crossref]

Rep. Prog. Phys. (1)

S. Collin, “Nanostructure arrays in free-space: optical properties and applications,” Rep. Prog. Phys. 77, 126402 (2014).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

Other (2)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, 1984).

E. Smith, V. Shteeman, and A. A. Hardy, “Time-dependent coupled mode analysis of advanced photonic micro devices,” in Proceedings of IEEE 27th Convention of Electrical & Electronics Engineers in Israel (IEEE, 2012).

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Figures (3)

Fig. 1
Fig. 1 Diffraction of a pulse on the guided-mode resonant grating (waveguide with a grating on its top) supporting two quasiguided modes.
Fig. 2
Fig. 2 (a) Two counter-propagating modes in a slab waveguide. (b) Modes excitation by ±m-th diffraction orders. (c) Mode leakage into −m-th diffraction order and mode coupling by means of 2m-th diffraction order.
Fig. 3
Fig. 3 (a) Transmission coefficient of the grating, |T|2, vs. incident light’s kx and ω: rigorous simulation (left part, at kx < 0) and analytical formula (16) (right part, at kx > 0). The dashed lines specify the light lines for θ = 0.2°,1°. (b) Transmission coefficient of the grating, |T|2, for different angles of incidence (θ = 0°,0.2°,1°): rigorous simulation (solid lines) and analytical formula (16) (circles).

Equations (20)

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

ω = ω 0 + v g ( k x k x 0 ) ,
u ( x , t ) = P ( x v g t ) e i ( k x 0 x ω o t ) ,
u t = v g u x + i ( k x 0 v g ω 0 ) u .
{ u t = v g u x + i ( k x 0 v g ω 0 ) u ; v t = v g v x + i ( k x 0 v g ω 0 ) v ,
k x 0 = 2 π m / d .
{ u t = v g u x + i ( k x 0 v g ω 0 ) u α 0 u + q f ( x , t ) e i k x o x ; v t = v g v x + i ( k x 0 v g ω 0 ) u α 0 v + q f ( x , t ) e i k x 0 x ,
{ u t = v g u x α 1 u c 1 u + c 2 e 2 i k x 0 x v + q f ( x , t ) e i k x 0 x ; v t = v g v x α 1 v c 1 v + c 2 e 2 i k x 0 x u + q f ( x , t ) e i k x 0 x ;
f T ( x , t ) = t ˜ f ( x , t ) + γ 1 s ( x , t ) ; f R ( x , t ) = r ˜ f ( x , t ) + γ 2 s ( x , t ) ,
s = u e i k x 0 x + v e i k x 0 x .
{ u ˜ t = v g u ˜ x α u ˜ + c 2 v ˜ + q f ( x , t ) ; v ˜ t = v g v ˜ x α v ˜ + c 2 u ˜ + q f ( x , t ) ,
s t = v g w x ( α c 2 ) s + 2 q f ( x , t ) ;
w t = v g s x ( α + c 2 ) w .
2 s t 2 + 2 α s t = v g 2 2 s x 2 ( α 2 c 2 2 ) s + g ( x , t ) ,
g ( x , t ) = 2 q [ f ( x , t ) t + ( α + c 2 ) f ( x , t ) ] .
F ( k x , ω ) = f ( x , t ) e i ( k x x ω t ) d x d t .
ω 2 S 2 i ω α S = v g 2 k x 2 S ( α 2 c 2 2 ) S + 2 q [ i ω + ( α + c 2 ) ] F .
F T = t ˜ F + γ 1 S ; F R = r ˜ F + γ 2 S .
T = F T F = t ˜ + γ 1 S F ; R = F R F = r ˜ + γ 2 S F ,
S F = 2 i q ω + i ( α + c 2 ) v g 2 k x 2 ( ω + i α ) 2 c 2 2 .
T ( k x , ω ) = t ˜ v g 2 k x 2 ( ω ω t ) ( ω ω p 2 ) v g 2 k x 2 ( ω ω p 1 ) ( ω ω p 2 ) ; R ( k x , ω ) = r ˜ v g 2 k x 2 ( ω ω r ) ( ω ω p 2 ) v g 2 k x 2 ( ω ω p 1 ) ( ω ω p 2 )

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