Abstract

We present a theory of the Fano resonance for optical resonators, based on a temporal coupled-mode formalism. This theory is applicable to the general scheme of a single optical resonance coupled with multiple input and output ports. We show that the coupling constants in such a theory are strongly constrained by energy-conservation and time-reversal symmetry considerations. In particular, for a two-port symmetric structure, Fano-resonant line shape can be derived by using only these symmetry considerations. We validate the analysis by comparing the theoretical predictions with three-dimensional finite-difference time-domain simulations of guided resonance in photonic crystal slabs. Such a theory may prove to be useful for response-function synthesis in filter and sensor applications.

© 2003 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  11. S. M. Norton, T. Erdogan, G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14, 629–639 (1997).
    [CrossRef]
  12. T. Tamir, S. Zhang, “Resonant scattering by multilayered dielectric gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
    [CrossRef]
  13. G. Levy-Yurista, A. A. Friesem, “Very narrow spectral filters with multilayered grating waveguide structures,” Appl. Phys. Lett. 77, 1596–1598 (2000).
    [CrossRef]
  14. S. Fan, “Sharp asymmetric lineshapes in side-coupled waveguide-resonator systems,” Appl. Phys. Lett. 80, 910–912 (2002).
    [CrossRef]
  15. M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
    [CrossRef]
  16. V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Kraus, R. M. De La Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2057 (1999).
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  17. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  18. S. Fan, J. D. Joannopoulos, “Analysis of guided resonance in photonic crystal slabs,” Phys. Rev. B 65, art. no. 235112 (2002).
    [CrossRef]
  19. P. Vincent, “Singularity expansions for cylinders of finite conductivity,” Appl. Phys. 17, 239–248 (1978).
    [CrossRef]
  20. J. U. Nöckel, A. D. Stone, “Resonance line shapes in quasi-one-dimensional scattering,” Phys. Rev. B 50, 17415–17432 (1994).
    [CrossRef]

2002 (2)

S. Fan, “Sharp asymmetric lineshapes in side-coupled waveguide-resonator systems,” Appl. Phys. Lett. 80, 910–912 (2002).
[CrossRef]

S. Fan, J. D. Joannopoulos, “Analysis of guided resonance in photonic crystal slabs,” Phys. Rev. B 65, art. no. 235112 (2002).
[CrossRef]

2000 (1)

G. Levy-Yurista, A. A. Friesem, “Very narrow spectral filters with multilayered grating waveguide structures,” Appl. Phys. Lett. 77, 1596–1598 (2000).
[CrossRef]

1999 (1)

1997 (3)

S. M. Norton, T. Erdogan, G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14, 629–639 (1997).
[CrossRef]

T. Tamir, S. Zhang, “Resonant scattering by multilayered dielectric gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
[CrossRef]

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

1996 (2)

1995 (1)

1994 (1)

J. U. Nöckel, A. D. Stone, “Resonance line shapes in quasi-one-dimensional scattering,” Phys. Rev. B 50, 17415–17432 (1994).
[CrossRef]

1992 (1)

R. Magnuson, S. Wang, “New principles for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

1990 (1)

1989 (1)

H. L. Bertoni, L.-H. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

1978 (1)

P. Vincent, “Singularity expansions for cylinders of finite conductivity,” Appl. Phys. 17, 239–248 (1978).
[CrossRef]

1965 (1)

1941 (1)

1902 (1)

R. W. Wood, “On the remarkable case of uneven distribution of a light in a diffractived grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Astratov, V. N.

Bagby, J. S.

Bertoni, H. L.

H. L. Bertoni, L.-H. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Busch, A.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Cheo, L.-H. S.

H. L. Bertoni, L.-H. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Culshaw, I. S.

De La Rue, R. M.

Engel, H.

Erdogan, T.

Fan, S.

S. Fan, “Sharp asymmetric lineshapes in side-coupled waveguide-resonator systems,” Appl. Phys. Lett. 80, 910–912 (2002).
[CrossRef]

S. Fan, J. D. Joannopoulos, “Analysis of guided resonance in photonic crystal slabs,” Phys. Rev. B 65, art. no. 235112 (2002).
[CrossRef]

Fano, U.

Friesem, A. A.

G. Levy-Yurista, A. A. Friesem, “Very narrow spectral filters with multilayered grating waveguide structures,” Appl. Phys. Lett. 77, 1596–1598 (2000).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
[CrossRef] [PubMed]

Haus, H. A.

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

Hessel, A.

Joannopoulos, J. D.

S. Fan, J. D. Joannopoulos, “Analysis of guided resonance in photonic crystal slabs,” Phys. Rev. B 65, art. no. 235112 (2002).
[CrossRef]

Johnson, S. R.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Kanskar, M.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Kraus, T. F.

Levy-Yurista, G.

G. Levy-Yurista, A. A. Friesem, “Very narrow spectral filters with multilayered grating waveguide structures,” Appl. Phys. Lett. 77, 1596–1598 (2000).
[CrossRef]

MacKenzie, J.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Magnuson, R.

Maystre, D.

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, Chichester, UK, 1982).

Moharam, M. G.

Morin, R.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Morris, G. M.

Nevière, M.

Nöckel, J. U.

J. U. Nöckel, A. D. Stone, “Resonance line shapes in quasi-one-dimensional scattering,” Phys. Rev. B 50, 17415–17432 (1994).
[CrossRef]

Norton, S. M.

Oliner, A. A.

Pacradouni, V.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Paddon, P.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Peng, S.

Popov, E.

Reinisch, R.

Rosenblatt, D.

Sharon, A.

Skolnick, M. S.

Steingrueber, R.

Stevenson, R. M.

Stone, A. D.

J. U. Nöckel, A. D. Stone, “Resonance line shapes in quasi-one-dimensional scattering,” Phys. Rev. B 50, 17415–17432 (1994).
[CrossRef]

Tamir, T.

T. Tamir, S. Zhang, “Resonant scattering by multilayered dielectric gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
[CrossRef]

H. L. Bertoni, L.-H. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Tiedje, T.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Vincent, P.

P. Vincent, “Singularity expansions for cylinders of finite conductivity,” Appl. Phys. 17, 239–248 (1978).
[CrossRef]

Wang, S.

R. Magnuson, S. Wang, “New principles for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

Wang, S. S.

Weber, H. G.

Whittaker, D. M.

Wood, R. W.

R. W. Wood, “On the remarkable case of uneven distribution of a light in a diffractived grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Young, J. F.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Zhang, S.

Appl. Opt. (1)

Appl. Phys. (1)

P. Vincent, “Singularity expansions for cylinders of finite conductivity,” Appl. Phys. 17, 239–248 (1978).
[CrossRef]

Appl. Phys. Lett. (4)

R. Magnuson, S. Wang, “New principles for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

G. Levy-Yurista, A. A. Friesem, “Very narrow spectral filters with multilayered grating waveguide structures,” Appl. Phys. Lett. 77, 1596–1598 (2000).
[CrossRef]

S. Fan, “Sharp asymmetric lineshapes in side-coupled waveguide-resonator systems,” Appl. Phys. Lett. 80, 910–912 (2002).
[CrossRef]

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

H. L. Bertoni, L.-H. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Lett. (1)

Philos. Mag. (1)

R. W. Wood, “On the remarkable case of uneven distribution of a light in a diffractived grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Phys. Rev. B (2)

J. U. Nöckel, A. D. Stone, “Resonance line shapes in quasi-one-dimensional scattering,” Phys. Rev. B 50, 17415–17432 (1994).
[CrossRef]

S. Fan, J. D. Joannopoulos, “Analysis of guided resonance in photonic crystal slabs,” Phys. Rev. B 65, art. no. 235112 (2002).
[CrossRef]

Other (2)

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, Chichester, UK, 1982).

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

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Figures (2)

Fig. 1
Fig. 1

Schematic of an optical resonator system coupled with multiple ports. The arrows indicate the incoming and outgoing waves. The dashed lines are reference planes for the wave amplitudes in the ports.

Fig. 2
Fig. 2

(a) Photonic crystal structure consisting of a square lattice of air holes of radius 0.2a in a dielectric slab with dielectric constant 12 and a thickness of 0.5a. The arrow indicates the direction of the incident light. (b) The intensity transmission spectrum through such a structure. The circles are the results from the finite-difference time-domain simulations. The solid curve is determined from analytic theory as represented by Eq. (16). (c) The same plot as in (b), except that the frequency range is now restricted to [0.36(c/a), 0.42(c/a)] to exhibit further details of the resonance line shape.

Equations (22)

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dadt=jω0-1τa+(κ|*)|s+,
|s-=C|s++a|d,
|s+=s1+s2+sm+
κ|*=κ1κ2κm.
|s-=s1-s2-sm-
|d=d1d2dm.
|s-S|s+=C+|dκ|*j(ω-ω0)+1/τ|s+.
|dκ|*=|κd|*.
a=(κ|*)|s+j(ω-ω0)+1/τ.
d|a|2dt=-2τ|a|2=-s-|s-=-|a|2d|d,
d|d=2/τ.
a*=(κ|s-)*2/τ=(κ|da)*2/τ,
κ|d=2/τ=(κ|d)*.
|κ=|d.
0=C|s-*+a*|d=a*C|d*+a*|d,
C|d*=-|d.
SS+=CC++(2/τ)|dd|(ω-ω0)2+(1/τ)2+C|d*d|-j(ω-ω0)+(1/τ)+|dd|*C+j(ω-ω0)+(1/τ).
d|*C+=(d|CT)*=(C|d*)+=-(|d)+=-d|,
SS+=CC++(2/τ)|dd|(ω-ω0)2+τ2+-|dd|-j(ω-ω0)+(1/τ)+-|dd|j(ω-ω0)+(1/τ)=CC+=I.
C=exp(jϕ)rjtjtr,
S=exp(jϕ)rjtjtr+1/τj(ω-ω0)+1/τ-(r±jt)(r±jt)(r±jt)-(r±jt).
R=r2(ω-ω0)2+t2(1/τ)22rt(ω-ω0)(1/τ)(ω-ω0)2+(1/τ)2.

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