Abstract

Using both the tight-binding approximation and the finite-difference time domain method, we analyze two types of coupled-resonator optical waveguide (CROW), a coupled-microdisks waveguide and a waveguide composed of coupled defect cavities in a two-dimensional photonic crystal. We find that the dispersion relation of the CROW band can be simply described by a small coupling parameter κ, and the spatial characteristics of the CROW modes remain the same as those of the single-resonator high Q modes. As applications of these unique properties, we demonstrate that CROW’s can be utilized in constructing waveguides without cross talk and enhance the efficiency of second-harmonic generation.

© 2000 Optical Society of America

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  2. P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
    [CrossRef]
  3. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
    [CrossRef]
  4. R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
    [CrossRef]
  5. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
    [CrossRef]
  6. O. Painter, J. Vuckovic, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16, 275–285 (1999).
    [CrossRef]
  7. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef] [PubMed]
  8. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  9. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University, Princeton, N.J., 1995).
  10. S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. L. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
    [CrossRef]
  11. N. C. Frateschi and A. F. Levi, “The spectrum of microdisk lasers,” J. Appl. Phys. 80, 644–653 (1996).
    [CrossRef]
  12. See, for example, N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, Philadelphia, Pa., 1976).
  13. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  14. A. Taflove, ed. Advances in Computational Electromagnetics: the Finite Difference Time Domain Method (Artech House, Boston, Mass., 1998).
  15. K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
    [CrossRef]
  16. K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742–5749 (1996).
    [CrossRef]
  17. M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
    [CrossRef]
  18. J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
    [CrossRef]
  19. J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
    [CrossRef]
  20. J. Martorell and R. Corbalan, “Enhancement of second harmonic generation in a periodic structure with a defect,” Opt. Commun. 108, 319–323 (1994).
    [CrossRef]
  21. J. Trull, R. Vilaseca, J. Martorell, and R. Corbalan, “Second-harmonic generation in local modes of a truncated periodic structure,” Opt. Lett. 20, 1746–1748 (1995).
    [CrossRef] [PubMed]
  22. T. Hattori, N. Tsurumachi, and H. Nakatsuka, “Analysis of optical nonlinearity by defect states in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 14, 348–355 (1997).
    [CrossRef]
  23. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
    [CrossRef]
  24. K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Expr. 4, 167–176 (1999).
    [CrossRef]
  25. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
    [CrossRef]
  26. X. Feng and Y. Arakawa, “Off-plane angle dependence of photonic band gap in a two-dimensional photonic crystal,” IEEE J. Quantum Electron. 32, 535–542 (1996).
    [CrossRef]
  27. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
    [CrossRef]
  28. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  29. S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
    [CrossRef]
  30. See, for example, J. Mathews and R. L. Walker, Mathematical Methods of Physics (Wiley, New York, 1970).
  31. S. G. Johnson, C. Manolatou, S. Fan, P. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
    [CrossRef]
  32. A. Yariv, Optical Electronics in Modern Communications (Oxford U. Press, New York, 1997).
  33. See, for example, C. Cohen-Tannoudji, B. Piu, and F. Laloe, Quantum Mechanics (Wiley, New York, 1977).

1999 (4)

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
[CrossRef]

O. Painter, J. Vuckovic, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16, 275–285 (1999).
[CrossRef]

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[CrossRef]

K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Expr. 4, 167–176 (1999).
[CrossRef]

1998 (2)

S. G. Johnson, C. Manolatou, S. Fan, P. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
[CrossRef]

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[CrossRef]

1997 (2)

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

T. Hattori, N. Tsurumachi, and H. Nakatsuka, “Analysis of optical nonlinearity by defect states in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 14, 348–355 (1997).
[CrossRef]

1996 (7)

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

X. Feng and Y. Arakawa, “Off-plane angle dependence of photonic band gap in a two-dimensional photonic crystal,” IEEE J. Quantum Electron. 32, 535–542 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742–5749 (1996).
[CrossRef]

N. C. Frateschi and A. F. Levi, “The spectrum of microdisk lasers,” J. Appl. Phys. 80, 644–653 (1996).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

1995 (1)

1994 (4)

J. Martorell and R. Corbalan, “Enhancement of second harmonic generation in a periodic structure with a defect,” Opt. Commun. 108, 319–323 (1994).
[CrossRef]

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

1992 (1)

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. L. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

1987 (2)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

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

1981 (1)

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

1976 (1)

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Alerhand, O. L.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Arakawa, Y.

X. Feng and Y. Arakawa, “Off-plane angle dependence of photonic band gap in a two-dimensional photonic crystal,” IEEE J. Quantum Electron. 32, 535–542 (1996).
[CrossRef]

Bendickson, J. M.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bloemer, M. J.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Bowden, C. M.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Cole, J. D.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[CrossRef]

Corbalan, R.

J. Trull, R. Vilaseca, J. Martorell, and R. Corbalan, “Second-harmonic generation in local modes of a truncated periodic structure,” Opt. Lett. 20, 1746–1748 (1995).
[CrossRef] [PubMed]

J. Martorell and R. Corbalan, “Enhancement of second harmonic generation in a periodic structure with a defect,” Opt. Commun. 108, 319–323 (1994).
[CrossRef]

Devenyi, A.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Dowling, J. P.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Fan, S.

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[CrossRef]

S. G. Johnson, C. Manolatou, S. Fan, P. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

Feng, X.

X. Feng and Y. Arakawa, “Off-plane angle dependence of photonic band gap in a two-dimensional photonic crystal,” IEEE J. Quantum Electron. 32, 535–542 (1996).
[CrossRef]

Flynn, R. J.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

Fork, R. L.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

Frateschi, N. C.

N. C. Frateschi and A. F. Levi, “The spectrum of microdisk lasers,” J. Appl. Phys. 80, 644–653 (1996).
[CrossRef]

Gedney, S. D.

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

Hattori, T.

Haus, H. A.

Haus, J. W.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Ippen, E. P.

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[CrossRef]

Joannopoulos, J. D.

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[CrossRef]

S. G. Johnson, C. Manolatou, S. Fan, P. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Johnson, S. G.

Kalocsai, A. G.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[CrossRef]

Kash, K.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Leavitt, R. P.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

Ledbetter, H. S.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

Lee, R. K.

Levi, A. F.

N. C. Frateschi and A. F. Levi, “The spectrum of microdisk lasers,” J. Appl. Phys. 80, 644–653 (1996).
[CrossRef]

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. L. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Logan, R. L.

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. L. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Manolatou, C.

Martorell, J.

J. Trull, R. Vilaseca, J. Martorell, and R. Corbalan, “Second-harmonic generation in local modes of a truncated periodic structure,” Opt. Lett. 20, 1746–1748 (1995).
[CrossRef] [PubMed]

J. Martorell and R. Corbalan, “Enhancement of second harmonic generation in a periodic structure with a defect,” Opt. Commun. 108, 319–323 (1994).
[CrossRef]

McCall, S. L.

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. L. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Meade, R. D.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Mur, G.

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

Nakatsuka, H.

Ohtaka, K.

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742–5749 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
[CrossRef]

Painter, O.

Pearton, S. J.

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. L. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Reinhardt, S. B.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

Sakoda, K.

K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Expr. 4, 167–176 (1999).
[CrossRef]

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742–5749 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
[CrossRef]

Scalora, M.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Scherer, A.

Slusher, R. E.

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. L. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Smith, D. A.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Theimer, J.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[CrossRef]

Tocci, M. D.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

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Tsurumachi, N.

Vilaseca, R.

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Villeneuve, P. R.

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

Viswanathan, R.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Vuckovic, J.

Winn, J. N.

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[CrossRef]

Xu, Y.

Yablonovitch, E.

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

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Yeh, P.

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Appl. Phys. Lett. (1)

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. L. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

X. Feng and Y. Arakawa, “Off-plane angle dependence of photonic band gap in a two-dimensional photonic crystal,” IEEE J. Quantum Electron. 32, 535–542 (1996).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

IEEE Trans. Electromagn. Compat. (1)

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

J. Appl. Phys. (3)

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

N. C. Frateschi and A. F. Levi, “The spectrum of microdisk lasers,” J. Appl. Phys. 80, 644–653 (1996).
[CrossRef]

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

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

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[CrossRef]

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[CrossRef]

Opt. Expr. (1)

K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Expr. 4, 167–176 (1999).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (2)

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[CrossRef]

Phys. Rev. B (4)

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[CrossRef]

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742–5749 (1996).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

Phys. Rev. E (1)

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, 1078–1081 (1996).
[CrossRef]

Phys. Rev. Lett. (2)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

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

Other (7)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University, Princeton, N.J., 1995).

See, for example, N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, Philadelphia, Pa., 1976).

A. Taflove, ed. Advances in Computational Electromagnetics: the Finite Difference Time Domain Method (Artech House, Boston, Mass., 1998).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

A. Yariv, Optical Electronics in Modern Communications (Oxford U. Press, New York, 1997).

See, for example, C. Cohen-Tannoudji, B. Piu, and F. Laloe, Quantum Mechanics (Wiley, New York, 1977).

See, for example, J. Mathews and R. L. Walker, Mathematical Methods of Physics (Wiley, New York, 1970).

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Figures (16)

Fig. 1
Fig. 1

Two examples of CROW. (a) A CROW composed of coupled defect cavities embedded in a 2D triangular lattice photonic crystal. R is the size of a unit cell, and ex is the direction of the periodicity for the coupled resonators. (b) A CROW realized by coupling of the microdisks along the ex direction. R is defined in the same way as for (a).

Fig. 2
Fig. 2

FDTD algorithm to calculate the eigenmodes and the eigenfrequencies of a given dielectric structure.

Fig. 3
Fig. 3

FDTD computational domain. (a) For the calculation of a single defect cavity in a 2D photonic crystal. The photonic crystal is characterized by a, the distance between the nearest air holes, and r, the radius of the air hole. The mirror boundary condition is used at the bottom y boundary. For the other three boundaries the first-order Mur absorbing boundary is used. (b) For the calculation of CROW composed of coupled defect cavities with four air holes between them. The PML absorbing boundary condition and the mirror boundary condition, respectively, are used for the top and bottom y boundaries. At both of the x boundaries, x=0 and x=R, the Bloch boundary condition is used.

Fig. 4
Fig. 4

High-Q modes of a single defect cavity surrounded by five layers of air holes. Even and odd single defect modes are shown in (a) and (b), respectively.

Fig. 5
Fig. 5

Even waveguide modes of the CROW with different intercavity hole spacings. The spacing is two holes in (a) and four holes in (b). The crystal momentum K of both the waveguide modes is K=0.6π/R.

Fig. 6
Fig. 6

Dispersion diagram of the even CROW band of coupled defect cavities. The frequencies, calculated with the FDTD algorithm, are shown for intercavity spacings of two, three and four air holes. Solid curves are least-squares fits of the numerical results with Eq. (9).

Fig. 7
Fig. 7

Odd waveguide mode of the CROW with intercavity hole spacing of three air holes. The crystal momentum K of the waveguide mode is 0.6π/R.

Fig. 8
Fig. 8

Dispersion diagram of the odd CROW band of coupled defect cavities. The frequencies, calculated with FDTD algorithm, are shown for intercavity spacings of two, three and four air holes. Solid curves are least-squares fits of the numerical results with Eq. (9).

Fig. 9
Fig. 9

CROW intersection without cross talk. The CROW’s contain two branches, an X branch and a Y branch. The darker shaded regions refer to the single resonators that compose the CROW’s. Two types of single resonator mode, an X mode and a Y mode, are supported by the X branch and the Y branch, respectively, represented by the lighter shaded regions. The X mode is antisymmetric with respect to the xz plane and symmetric with respect to the yz plane. The Y mode has exactly the opposite symmetry properties.

Fig. 10
Fig. 10

TE(7, 1) whispering-gallery modes in a single microdisk cavity. The mode with even mirror reflection symmetry is shown in (a), and the one with odd mirror reflection symmetry is shown in (b).

Fig. 11
Fig. 11

Even waveguide mode of coupled microdisks, which is formed by coupling of the even TE(7, 1) modes together. R/2r, the ratio of intermicrodisk spacing to the microdisk diameter, is 1.1, and K=0.5π/R.

Fig. 12
Fig. 12

Effective Q factors of the even CROW modes of the coupled microdisks for parameter R/2r=1.17. Qeff is calculated according to Eq. (20).

Fig. 13
Fig. 13

Dispersion relations of the even CROW band of coupled microdisks. The numerically calculated results are represented by asterisks. The error bars refer to the frequency error caused by the finite decay rate of the CROW modes and are estimated to be ω/2Qeff. Solid curves are least-squares fits of the numerical results with Eq. (9).

Fig. 14
Fig. 14

Odd waveguide mode of coupled microdisks, which is formed by coupling of the odd TE(7, 1) modes together. R/2r, the ratio of intermicrodisk spacing to the microdisk diameter, is 1.1, and K=0.5π/R.

Fig. 15
Fig. 15

Dispersion relations of the odd CROW band of coupled microdisks. The numerically calculated results are represented by asterisks. The error bars refer to the frequency error caused by the finite decay rate of the CROW modes and are estimated to be ω/2Qeff. Solid curves are least-squares fits of the numerical results with Eq. (9).

Fig. 16
Fig. 16

SHG in the CROW. The fundamental frequency modes can propagate by hopping from one defect cavity to another along the x axis. The second-harmonic mode can either propagate in the CROW along the x axis or leak out of the CROW along the z axis.

Tables (1)

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Table 1 Coupling Coefficient κl of the Coupled Defect Cavities

Equations (72)

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×[×El(r)]=0(r) Ωl2c2El(r),
 dr0(r)El(r)·Em(r)=δl,m,
EK(r, t)=A exp(iωKt) n,l exp(-inKR) blEl(r-nRex).
×[×EK]=(r) ωK2c2EK,
l blΩl2δm,l+n0 exp(-inKR)βm,ln
=ωK2 l blδm,l+Δαm,l+n0 exp(-inKR)αm,ln,
αm,ln= dr(r)Em(r)·El(r-nRex),n0,
βm,ln= dr0(r-nRex)Em(r)·El(r-nRex),
n0,
Δαm,l= dr[(r)-0(r)]Em(r)·El(r).
lblΩl2[δm,l+2βm,l1 cos(KR)]
=ωK2 l bl[δm,l+Δαm,l+2αm,l1 cos(KR)].
Ex(x, y, z)=PEx(x, -y, z),
Ey(x, y, z)=-PEy(x, -y, z),
Ez(x, y, z)=PEz(x, -y, z),
Bx(x, y, z)=-PBx(x, -y, z),
By(x, y, z)=PBy(x, -y, z),
Bz(x, y, z)=-PBz(x, -y, z),
ωKl=Ω1-Δαl,l2+κl cos(KR),
κl=βl,l1-αl,l1= dr[0(r-Rex)-(r-Rex)]El(r)·El(r-Rex).
vgl(K)=dωKldK=-ΩRκl sin(KR),
EK(r, t)=A exp(-ΓKt) exp(iωKt) n,l×exp(-inKR) blEl(r-nRex),
×[×EK]=-(r)c22EKt2-4π σ(r)c2EKt,
ΓK=2π  drσ(r)EK*·EK dr(r)EK*·EK.
1cB(r, t)t=-×E(r, t),
(r)cE(r, t)t=×B(r, t)
E(r, ω)=0T dt exp(-iωt) E(r, t),
B(r, ω)=0T dt exp(-iωt) B(r, t).
×1(r)×B(r)=ωc2B(r),
×[×E(r)]=ωc2(r)E(r).
ω2=c2 V drB*(r)·×1(r)×B(r)+E*(r)·{×[×E(r)]}V dr[B*(r)·B(r)+(r)E*(r)·E(r)],
E(x=R, y)=exp(-iKR) E(x=0, y),
B(x=R, y)=exp(-iKR) B(x=0, y),
Q=EmodeavgEmode(0)-Emode(T)ΩmodeT,
Qeff (K)=ωK2ΓK,
EK(r, t)=exp[(-Γω+iω)t] exp(-iKex·r)uK(r),
×[×EK]+4πc2σ(r) EKt=-(r)c22EKt2,
Γω=2π uc drσ(r)uK*(r)·uK(r).
vω,g=c22ωuc druK*·{-2Kex×[ex×uK]-iex×[×uK]-i×[ex×uK]}.
E1(r, t)=½{E1 exp(iωt) exp[-iK1(ω)x]uK1(ω)(r)+c.c.},
E1(r, t)=½{E1 exp(iωt) exp[-iK2(ω)x]uK2(ω)(r)+c.c.},
E2(r, t)=½{E2(x)exp(i2ωt) exp[-iK(2ω)x]uK(2ω)(r)+c.c.}.
×[×E2]+4πc2σ(r) E2t+(r)c22E2t2
=-1c22t2PNL(r, t).
PNL(r, t)=12(E12 exp(i2ωt)exp{-i[K1(ω)+K2(ω)]x}
×d˜2ω(r)uK1(ω)(r)uK2(ω)(r)+c.c.),
v2ω,g dE2(x)dx+Γ2ωE2(x)
=-iωE12 uc dr exp{-i[K1(ω)+K2(ω)-K(2ω)]x}
×uK(2ω)*·d˜uK1(ω)uK2(ω),
v2ω,g dE2(x)dx+Γ2ωE2(x)
=-iωDnE12 exp(-iΔKnx),
ΔKn=K1(ω)+K2(ω)-K(2ω)+n 2πR,
n=0,±1,±2 ,
Dn=uc dr exp [in(2π/R)x]uK(2ω)*(r)·d˜(r)
×uK1(ω)(r)uK2(ω)(r).
K(2ω)=K1(ω)+K2(ω)+n 2πR,
n=0,±1,±2.
E2(x)=-i ωE12D0Γ2ω[1-exp(-Γ2ω x/v2ω,g)].
E2(x)=-i ωE12D0v2ω,gx,xLs,
E2(x)=-i ωE12D0Γ2ω,xLs.
E2=-i ωE12D0Γ2ω.
P=18πvgR|E|2.
P=18πNΓ|E|2,
ηSHG=P2ωPω=1v2ω,gvω,g2ω2|D0|28πRPω L2.
ηSHG=P2ωPω=1Γ2ωvω,g2ω2|D0|28πRPω L.
H(K)uK(r)=λ(K)(r)uK(r).
V dr(r)uK*·uK=1.
V dr(r)duK*dK·uK+uK*·duKdK=0.
dλ(K)dK=V drduK*dK·H(K)uK+uK*·H(K) duKdK+uK*·H(K)dKuK.
dλ(K)dK=V druK*·H(K)dKuK.
HKuK(r)=ω2c2(r)uK(r),
HKuK=×[×uK]-iK{×[ex×uK]+ex×[×uK]}-K2ex×[ex×uK].

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