Abstract

The nature of guided modes and defect modes in periodic dielectric waveguides is investigated computationally for model systems in two dimensions. It is shown that defect states that exist within the band gap of guided modes can be excited to form tightly localized high-Q resonances.

© 1995 Optical Society of America

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References

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  1. For a review, see the feature issue on the development and applications of materials exhibiting photon band gaps, J. Opt. Soc. Am. B 10, 280–413 (1993).
  2. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 13,772 (1991).
    [CrossRef]
  3. H. A. Haus and C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976).
    [CrossRef]
  4. J. N. Winn, R. D. Meade, and J. D. Joannopoulos, J. Mod. Opt. 41, 257 (1994).
    [CrossRef]
  5. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
    [CrossRef]
  6. J. N. Winn, "Dynamic studies of photonic crystals," M.S. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1994).
  7. B. Engquist and A. Majda, Math. Computat. 31, 629 (1977).
    [CrossRef]
  8. A. Taflove and K. R. Umashankar, J. Electromagn. Waves Appl. 1, 243 (1987).
    [CrossRef]
  9. A. Yariv, Optical Electronics (Saunders, Philadelphia, 1991).
  10. R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, J. Opt. Soc. Am. B 10, 328 (1993).
    [CrossRef]

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 13,772 (1991).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, J. Opt. Soc. Am. B 10, 328 (1993).
[CrossRef]

Engquist, B.

B. Engquist and A. Majda, Math. Computat. 31, 629 (1977).
[CrossRef]

Haus, H. A.

H. A. Haus and C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976).
[CrossRef]

Joannopoulos, J. D.

J. N. Winn, R. D. Meade, and J. D. Joannopoulos, J. Mod. Opt. 41, 257 (1994).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, J. Opt. Soc. Am. B 10, 328 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 13,772 (1991).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

Majda, A.

B. Engquist and A. Majda, Math. Computat. 31, 629 (1977).
[CrossRef]

Meade, R. D.

J. N. Winn, R. D. Meade, and J. D. Joannopoulos, J. Mod. Opt. 41, 257 (1994).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, J. Opt. Soc. Am. B 10, 328 (1993).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 13,772 (1991).
[CrossRef]

Rappe, A. M.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 13,772 (1991).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, J. Opt. Soc. Am. B 10, 328 (1993).
[CrossRef]

Shank, C. V.

H. A. Haus and C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976).
[CrossRef]

Taflove, A.

A. Taflove and K. R. Umashankar, J. Electromagn. Waves Appl. 1, 243 (1987).
[CrossRef]

Umashankar, K. R.

A. Taflove and K. R. Umashankar, J. Electromagn. Waves Appl. 1, 243 (1987).
[CrossRef]

Winn, J. N.

J. N. Winn, R. D. Meade, and J. D. Joannopoulos, J. Mod. Opt. 41, 257 (1994).
[CrossRef]

J. N. Winn, "Dynamic studies of photonic crystals," M.S. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1994).

Yariv, A.

A. Yariv, Optical Electronics (Saunders, Philadelphia, 1991).

Other (10)

For a review, see the feature issue on the development and applications of materials exhibiting photon band gaps, J. Opt. Soc. Am. B 10, 280–413 (1993).

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 13,772 (1991).
[CrossRef]

H. A. Haus and C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976).
[CrossRef]

J. N. Winn, R. D. Meade, and J. D. Joannopoulos, J. Mod. Opt. 41, 257 (1994).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

J. N. Winn, "Dynamic studies of photonic crystals," M.S. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1994).

B. Engquist and A. Majda, Math. Computat. 31, 629 (1977).
[CrossRef]

A. Taflove and K. R. Umashankar, J. Electromagn. Waves Appl. 1, 243 (1987).
[CrossRef]

A. Yariv, Optical Electronics (Saunders, Philadelphia, 1991).

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, J. Opt. Soc. Am. B 10, 328 (1993).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic of the band structures for waveguides with different dielectric-constant contrast along the guide. (a) Uniform dielectric waveguide, with the band structure shown on the right-hand side. (b) Periodic dielectric waveguide with weak index contrast. The origin of the gap is shown on the right-hand side. (c) Periodic dielectric waveguide with strong index contrast. The expected band structure is sketched on the right-hand side.

Fig. 2
Fig. 2

Calculated band diagrams for the (a) TM and the (b) TE modes in a perfect dielectric-rod waveguide structure. The shaded regions represent extended modes, and the solid curves below the light line are guided modes.

Fig. 3
Fig. 3

Contour plots of the TM displacement fields at the zone edge for the (a) first and the (b) second bands in the perfect dielectric-rod waveguide structure. The TM modes have D normal to the plane. The dashed curves represent contours with Dz < 0, while the solid curves represent contours with Dz > 0. The positions of the rods are shown by the shaded areas.

Fig. 4
Fig. 4

Gray-scale plots of the power in the electrical fields of TE states at the zone edge for the (a) first and the (b) second bands in the perfect dielectric-rod structure.

Fig. 5
Fig. 5

Plot of the displacement field of the high-Q TM defect mode. The largest positive and negative peaks of the displacement field lie within the dielectric region of the defect. The other peaks of the displacement field lie within the square dielectric elements of the waveguide.

Fig. 6
Fig. 6

Transient decay of the high-Q TM defect mode. Plotted is the total energy in the defect mode as a function of time. E0 is the initial energy in the fields. T is the period of the high-Q defect mode. Since the mode is not excited completely purely the low-Q modes radiate away before the high-Q exponential decay becomes manifest. The Q value for this defect mode is 12,000.

Fig. 7
Fig. 7

Calculated band diagram for TE modes in a perfect air-column structure. The shaded region represents extended modes, and the solid curves below the light line are guided modes.

Fig. 8
Fig. 8

Gray-scale plots of the power in the electrical fields of TE states at the zone edge for (a) the first and (b) the second bands in the air-column structure.

Fig. 9
Fig. 9

Plot of the magnetic field of the high-Q TE defect mode. The largest peak in the magnetic field lies within the dielectric region of the defect. The other peaks lie within the circular air regions of the waveguide.

Fig. 10
Fig. 10

Transient decay of the high-Q TE defect mode. Plotted is the total energy in the defect mode as a function of time. E0 and T are in the same convention as in Fig. 6. The Q value of this mode is 13,000.

Equations (6)

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

× [ 1 ɛ ( r ) × H ] = ω 2 H .
( G λ ) Θ ( G λ ) , ( G λ ) k h ( G λ ) = ω 2 h ( G λ ) ,
Θ ( G λ ) , ( G λ ) k = [ ( k + G ) × e λ ] · [ ( k + G ) × e λ ] ɛ - 1 ( G , G ) .
2 E ( x , y ) x 2 + 2 E ( x , y ) y 2 = ɛ ( x , y ) 2 E ( x , y ) t 2 .
E i + 1 , j n - 2 E i j n + E i - 1 , j n ( Δ s ) 2 + E i , j + 1 n - 2 E i j n + E i , j - 1 n ( Δ s ) 2 = ( ɛ i j ) E i j n + 1 - 2 E i j n + E i j n - 1 ( Δ t ) 2 .
Q = ω E P = - ω E d E / d t ,

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