Abstract

We derive tight upper and lower bounds of the ratio between decay rates to two ports from a single resonance exhibiting Fano interference, based on a general temporal coupled-mode theory formalism. The photon transport between these two ports involves both direct and resonance-assisted contributions, and the bounds depend only on the direct process. The bounds imply that, in a lossless system, full reflection is always achievable at Fano resonance, even for structures lacking mirror symmetries, while full transmission can only be seen in a symmetric configuration where the two decay rates are equal. The analytic predictions are verified against full-field electromagnetic simulations.

© 2013 Optical Society of America

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References

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  1. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  2. 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).
  3. S. Fan, Optical Fiber Telecommunications V Vol. A: Components and Subsystems (Elsevier, 2008),  Chap. 12, p. 431.
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    [CrossRef]
  5. Z. Yu, A. Raman, and S. Fan, Proc. Natl. Acad. Sci. USA 107, 17491 (2010).
    [CrossRef]
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    [CrossRef]
  7. H. Lu, X. Liu, Y. Gong, D. Mao, and L. Wang, Opt. Express 19, 12885 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. M. Sumetskii and M. Felshtyn, J. Exp. Theor. Phys. Lett. 53, 24 (1991).

2012 (4)

Z. Ruan and S. Fan, Phys. Rev. A 85, 043828 (2012).
[CrossRef]

Y. Zhang, T. Mei, and D. Zhang, Appl. Opt. 51, 504 (2012).
[CrossRef]

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
[CrossRef]

V. Liu and S. Fan, Comput. Phys. Commun. 183, 2233 (2012).
[CrossRef]

2011 (2)

M. Ghebrebrhan, P. Bermel, Y. X. Yeng, I. Celanovic, M. Soljacic, and J. D. Joannopoulos, Phys. Rev. A 83, 033810 (2011).
[CrossRef]

H. Lu, X. Liu, Y. Gong, D. Mao, and L. Wang, Opt. Express 19, 12885 (2011).
[CrossRef]

2010 (2)

Z. Yu, A. Raman, and S. Fan, Proc. Natl. Acad. Sci. USA 107, 17491 (2010).
[CrossRef]

T. Lepetit, E. Akmansov, J.-P. Ganne, and J.-M. Lourtioz, Phys. Rev. B 82, 195307 (2010).
[CrossRef]

2009 (1)

2007 (1)

Y. Kanamori, T. Kitani, and K. Hane, Appl. Phys. Lett. 90, 031911 (2007).
[CrossRef]

2006 (1)

M. Laroche, R. Carminati, and J.-J. Greffet, Phys. Rev. Lett. 96, 123903 (2006).
[CrossRef]

2005 (1)

2004 (2)

Y. Ding and R. Magnusson, Opt. Express 12, 5661 (2004).
[CrossRef]

W. Suh, Z. Wang, and S. Fan, IEEE J. Quantum Electron. 40, 1511 (2004).
[CrossRef]

2003 (1)

2002 (2)

S. Fan and J. D. Joannopoulos, Phys. Rev. B 65, 235112 (2002).
[CrossRef]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

1999 (1)

1991 (1)

M. Sumetskii and M. Felshtyn, J. Exp. Theor. Phys. Lett. 53, 24 (1991).

Akmansov, E.

T. Lepetit, E. Akmansov, J.-P. Ganne, and J.-M. Lourtioz, Phys. Rev. B 82, 195307 (2010).
[CrossRef]

Astratov, V. N.

Berggren, J.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
[CrossRef]

Bermel, P.

M. Ghebrebrhan, P. Bermel, Y. X. Yeng, I. Celanovic, M. Soljacic, and J. D. Joannopoulos, Phys. Rev. A 83, 033810 (2011).
[CrossRef]

Bussmann, K.

Carminati, R.

M. Laroche, R. Carminati, and J.-J. Greffet, Phys. Rev. Lett. 96, 123903 (2006).
[CrossRef]

Carter, M.

Casey, J.

Celanovic, I.

M. Ghebrebrhan, P. Bermel, Y. X. Yeng, I. Celanovic, M. Soljacic, and J. D. Joannopoulos, Phys. Rev. A 83, 033810 (2011).
[CrossRef]

Chuwongin, S.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
[CrossRef]

Culshaw, I. S.

De La Rue, R. M.

Ding, Y.

Eddy, C.

Fan, S.

Z. Ruan and S. Fan, Phys. Rev. A 85, 043828 (2012).
[CrossRef]

V. Liu and S. Fan, Comput. Phys. Commun. 183, 2233 (2012).
[CrossRef]

Z. Yu, A. Raman, and S. Fan, Proc. Natl. Acad. Sci. USA 107, 17491 (2010).
[CrossRef]

W. Suh, Z. Wang, and S. Fan, IEEE J. Quantum Electron. 40, 1511 (2004).
[CrossRef]

S. Fan, W. Suh, and J. D. Joannopoulos, J. Opt. Soc. Am. A 20, 569 (2003).
[CrossRef]

S. Fan and J. D. Joannopoulos, Phys. Rev. B 65, 235112 (2002).
[CrossRef]

S. Fan, Optical Fiber Telecommunications V Vol. A: Components and Subsystems (Elsevier, 2008),  Chap. 12, p. 431.

Felshtyn, M.

M. Sumetskii and M. Felshtyn, J. Exp. Theor. Phys. Lett. 53, 24 (1991).

Ganne, J.-P.

T. Lepetit, E. Akmansov, J.-P. Ganne, and J.-M. Lourtioz, Phys. Rev. B 82, 195307 (2010).
[CrossRef]

Ghebrebrhan, M.

M. Ghebrebrhan, P. Bermel, Y. X. Yeng, I. Celanovic, M. Soljacic, and J. D. Joannopoulos, Phys. Rev. A 83, 033810 (2011).
[CrossRef]

Gippius, N. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Gong, Y.

Greffet, J.-J.

M. Laroche, R. Carminati, and J.-J. Greffet, Phys. Rev. Lett. 96, 123903 (2006).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Hammar, M.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
[CrossRef]

Hane, K.

Y. Kanamori, T. Kitani, and K. Hane, Appl. Phys. Lett. 90, 031911 (2007).
[CrossRef]

Haus, H. A.

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

Henry, R.

Holm, R.

Ishihara, T.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Joannopoulos, J. D.

M. Ghebrebrhan, P. Bermel, Y. X. Yeng, I. Celanovic, M. Soljacic, and J. D. Joannopoulos, Phys. Rev. A 83, 033810 (2011).
[CrossRef]

S. Fan, W. Suh, and J. D. Joannopoulos, J. Opt. Soc. Am. A 20, 569 (2003).
[CrossRef]

S. Fan and J. D. Joannopoulos, Phys. Rev. B 65, 235112 (2002).
[CrossRef]

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).

Kanamori, Y.

Y. Kanamori, T. Kitani, and K. Hane, Appl. Phys. Lett. 90, 031911 (2007).
[CrossRef]

Kim, M.

Kim, S.

Kitani, T.

Y. Kanamori, T. Kitani, and K. Hane, Appl. Phys. Lett. 90, 031911 (2007).
[CrossRef]

Krauss, T. F.

Laroche, M.

M. Laroche, R. Carminati, and J.-J. Greffet, Phys. Rev. Lett. 96, 123903 (2006).
[CrossRef]

Lepetit, T.

T. Lepetit, E. Akmansov, J.-P. Ganne, and J.-M. Lourtioz, Phys. Rev. B 82, 195307 (2010).
[CrossRef]

Liu, V.

V. Liu and S. Fan, Comput. Phys. Commun. 183, 2233 (2012).
[CrossRef]

Liu, X.

Lourtioz, J.-M.

T. Lepetit, E. Akmansov, J.-P. Ganne, and J.-M. Lourtioz, Phys. Rev. B 82, 195307 (2010).
[CrossRef]

Lu, H.

Ma, Z.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
[CrossRef]

Magnusson, R.

Mao, D.

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).

Mei, T.

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Prather, D.

Raman, A.

Z. Yu, A. Raman, and S. Fan, Proc. Natl. Acad. Sci. USA 107, 17491 (2010).
[CrossRef]

Rosenberg, A.

Ruan, Z.

Z. Ruan and S. Fan, Phys. Rev. A 85, 043828 (2012).
[CrossRef]

Seo, J.-H.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
[CrossRef]

Shamamian, V.

Shi, S.

Shuai, Y.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
[CrossRef]

Skolnick, M. S.

Soljacic, M.

M. Ghebrebrhan, P. Bermel, Y. X. Yeng, I. Celanovic, M. Soljacic, and J. D. Joannopoulos, Phys. Rev. A 83, 033810 (2011).
[CrossRef]

Song, H. Y.

Stevenson, R. M.

Suh, W.

W. Suh, Z. Wang, and S. Fan, IEEE J. Quantum Electron. 40, 1511 (2004).
[CrossRef]

S. Fan, W. Suh, and J. D. Joannopoulos, J. Opt. Soc. Am. A 20, 569 (2003).
[CrossRef]

Sumetskii, M.

M. Sumetskii and M. Felshtyn, J. Exp. Theor. Phys. Lett. 53, 24 (1991).

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Tikhodeev, S. G.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Wang, L.

Wang, Z.

W. Suh, Z. Wang, and S. Fan, IEEE J. Quantum Electron. 40, 1511 (2004).
[CrossRef]

Whittaker, D. M.

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).

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Yang, H.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
[CrossRef]

Yang, W.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
[CrossRef]

Yeng, Y. X.

M. Ghebrebrhan, P. Bermel, Y. X. Yeng, I. Celanovic, M. Soljacic, and J. D. Joannopoulos, Phys. Rev. A 83, 033810 (2011).
[CrossRef]

Yu, Z.

Z. Yu, A. Raman, and S. Fan, Proc. Natl. Acad. Sci. USA 107, 17491 (2010).
[CrossRef]

Zhang, D.

Zhang, Y.

Zhao, D.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
[CrossRef]

Zhou, W.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Y. Kanamori, T. Kitani, and K. Hane, Appl. Phys. Lett. 90, 031911 (2007).
[CrossRef]

Comput. Phys. Commun. (1)

V. Liu and S. Fan, Comput. Phys. Commun. 183, 2233 (2012).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. Suh, Z. Wang, and S. Fan, IEEE J. Quantum Electron. 40, 1511 (2004).
[CrossRef]

J. Exp. Theor. Phys. Lett. (1)

M. Sumetskii and M. Felshtyn, J. Exp. Theor. Phys. Lett. 53, 24 (1991).

J. Lightwave Technol. (1)

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

Nat. Photonics (1)

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, Nat. Photonics 6, 615 (2012).
[CrossRef]

Opt. Express (4)

Phys. Rev. A (2)

M. Ghebrebrhan, P. Bermel, Y. X. Yeng, I. Celanovic, M. Soljacic, and J. D. Joannopoulos, Phys. Rev. A 83, 033810 (2011).
[CrossRef]

Z. Ruan and S. Fan, Phys. Rev. A 85, 043828 (2012).
[CrossRef]

Phys. Rev. B (3)

T. Lepetit, E. Akmansov, J.-P. Ganne, and J.-M. Lourtioz, Phys. Rev. B 82, 195307 (2010).
[CrossRef]

S. Fan and J. D. Joannopoulos, Phys. Rev. B 65, 235112 (2002).
[CrossRef]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

M. Laroche, R. Carminati, and J.-J. Greffet, Phys. Rev. Lett. 96, 123903 (2006).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

Z. Yu, A. Raman, and S. Fan, Proc. Natl. Acad. Sci. USA 107, 17491 (2010).
[CrossRef]

Other (4)

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

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).

S. Fan, Optical Fiber Telecommunications V Vol. A: Components and Subsystems (Elsevier, 2008),  Chap. 12, p. 431.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

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Figures (3)

Fig. 1.
Fig. 1.

Schematic of a single resonance coupled to two ports. The amplitude of an output electromagnetic wave, either reflected or transmitted, is the sum of an indirect (resonance-assisted) component and a direct (background) component.

Fig. 2.
Fig. 2.

Asymmetric photonic crystal slab and its transmission spectrum. (a) Asymmetric photonic crystal slab structure consisting of two layers. The top layer has a square lattice of air holes of radius 0.3 a introduced into a slab with dielectric constant 12 and thickness of 0.1 a , where a is the lattice constant. The bottom layer is a uniform slab of the same material and thickness. (b) Transmission spectra of the photonic crystal slab at normal incidence. The temporal coupled-mode theory results agree well with the simulation.

Fig. 3.
Fig. 3.

Scatterplot of the r and τ 1 / τ 2 values of 50 different asymmetric photonic crystal slabs. (Each of the data points is reflected with respect to the line of τ 1 = τ 2 because the subscripts are interchangeable, and there are in total 100 scatter points in the plot.) r is obtained by fitting the simulated spectrum with a Fabry–Perot background characteristic of the direct process. τ 1 / τ 2 is obtained by comparing the electromagnetic flux leaking above and below the structure.

Equations (14)

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

1 r 1 + r τ 1 τ 2 1 + r 1 r ,
d u d t = ( j ω 0 1 τ 1 1 τ 2 ) u + ( d 1 d 2 ) ( s 1 + s 2 + ) ,
( s 1 s 2 ) = C ( s 1 + s 2 + ) + ( d 1 d 2 ) u ,
C = e j ϕ ( r j t j t r ) ,
d 1 * d 1 = 2 τ 1 , d 2 * d 2 = 2 τ 2 ,
C ( d 1 d 2 ) * = ( d 1 d 2 ) .
d 1 = e j ϕ ( r d 1 * + j t d 2 * ) ,
d 2 = e j ϕ ( j t d 1 * + r d 2 * ) .
d 2 * = 1 j t ( e j ϕ 2 τ 1 1 d 1 * + r d 1 * ) .
2 τ 2 = 1 j t ( e j ϕ 2 τ 1 1 d 1 * + r d 1 * ) e j ϕ [ j t d 1 * r j t ( e j ϕ 2 τ 1 1 d 1 * + r d 1 * ) ] .
τ 1 τ 2 = 1 + 2 ( r t ) 2 + 2 r t 2 cos ϕ ,
R = [ r ( ω ω 0 ) ± 2 τ 1 2 + 2 τ 2 2 ( r τ ) 2 ( 1 r σ ) 2 ] 2 + ( 1 r σ ) 2 ( ω ω 0 ) 2 + ( 1 τ ) 2 ,
ω = ω 0 ± r 1 r 2 2 τ 1 2 + 2 τ 2 2 r 2 ( 1 τ 1 + 1 τ 2 ) 2 1 r 2 ( 1 τ 1 1 τ 2 ) 2 .
ω = ω 0 ± t r ( 1 τ 1 + 1 τ 2 )

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