Abstract

Two numerical techniques for analysis of defect modes in photonic crystals are presented. Based on the finite-difference time-domain method (FDTD), we use plane wave incidences and point sources for excitation and analysis. Using a total-field/scattered-field scheme, an ideal plane wave incident at different angles is implemented; defect modes are selectively excited and mode symmetries are probed. All modes can be excited by an incident plane wave along a non-symmetric direction of the crystal. Degenerate modes can also be differentiated using this method. A proper arrangement of point sources with positive and negative amplitudes in the cavity flexibly excites any chosen modes. Numerical simulations have verified these claims. Evolution of each defect mode is studied using spectral filtering. The quality factor of the defect mode is estimated based on the field decay. The far-field patterns are calculated and the Q values are shown to affect strongly the sharpness of these patterns. Animations of the near-fields of the defect modes are presented to give an intuitive image of their oscillating features.

© 2003 Optical Society of America

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References

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  1. P R Villeneuve et al, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phy. Rev. B 54, 7837 (1996)
    [Crossref]
  2. Shanhui Fan et al, “Channel drop filters in photonic crystals,” Opt. Express, 3, 4 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4
    [Crossref]
  3. K. M. Ho et al, “Existence of a photonic gap in periodic dielectric structures,” Phy. Rev. Lett. 65, 3152 (1990)
    [Crossref]
  4. Min Qiu et al, “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phy. Rev. B 61, 12871 (2000)
    [Crossref]
  5. Kazuaki Sakoda et al, “Optical response of three-dimensional photonic lattices: solution of inhomgeneous Maxwell’s equations and their applications,” Phy. Rev. B 54, 5732 (1996)
    [Crossref]
  6. Vladimir Kuzmiak et al, “Localized defect modes in a two-dimensional triangular photonic crystal,” Phy. Rev. B 57, 15242 (1998)
    [Crossref]
  7. Kazuaki Sakoda et al, “Numerical method for localized defect modes in photonic lattices,” Phy. Rev. B 56, 4830 (1997)
    [Crossref]
  8. Allen Taflove, Computational electrodynamics, the finite difference time domain method (Artech House1995)
  9. K. S Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenna and Propagation 14, 302 (1966)
    [Crossref]
  10. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computational Physics, 185, (1994)
    [Crossref]
  11. S. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express 11, 167 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167
    [Crossref] [PubMed]
  12. J. D. Joannopoulos et al, Photonic crystals - Molding the flow of light (Princeton University Press1995)

2003 (1)

2000 (1)

Min Qiu et al, “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phy. Rev. B 61, 12871 (2000)
[Crossref]

1998 (2)

Shanhui Fan et al, “Channel drop filters in photonic crystals,” Opt. Express, 3, 4 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4
[Crossref]

Vladimir Kuzmiak et al, “Localized defect modes in a two-dimensional triangular photonic crystal,” Phy. Rev. B 57, 15242 (1998)
[Crossref]

1997 (1)

Kazuaki Sakoda et al, “Numerical method for localized defect modes in photonic lattices,” Phy. Rev. B 56, 4830 (1997)
[Crossref]

1996 (2)

P R Villeneuve et al, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phy. Rev. B 54, 7837 (1996)
[Crossref]

Kazuaki Sakoda et al, “Optical response of three-dimensional photonic lattices: solution of inhomgeneous Maxwell’s equations and their applications,” Phy. Rev. B 54, 5732 (1996)
[Crossref]

1990 (1)

K. M. Ho et al, “Existence of a photonic gap in periodic dielectric structures,” Phy. Rev. Lett. 65, 3152 (1990)
[Crossref]

1966 (1)

K. S Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenna and Propagation 14, 302 (1966)
[Crossref]

Albin, S.

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computational Physics, 185, (1994)
[Crossref]

Fan, Shanhui

Shanhui Fan et al, “Channel drop filters in photonic crystals,” Opt. Express, 3, 4 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4
[Crossref]

Guo, S.

Ho, K. M.

K. M. Ho et al, “Existence of a photonic gap in periodic dielectric structures,” Phy. Rev. Lett. 65, 3152 (1990)
[Crossref]

Joannopoulos, J. D.

J. D. Joannopoulos et al, Photonic crystals - Molding the flow of light (Princeton University Press1995)

Kuzmiak, Vladimir

Vladimir Kuzmiak et al, “Localized defect modes in a two-dimensional triangular photonic crystal,” Phy. Rev. B 57, 15242 (1998)
[Crossref]

Qiu, Min

Min Qiu et al, “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phy. Rev. B 61, 12871 (2000)
[Crossref]

Sakoda, Kazuaki

Kazuaki Sakoda et al, “Numerical method for localized defect modes in photonic lattices,” Phy. Rev. B 56, 4830 (1997)
[Crossref]

Kazuaki Sakoda et al, “Optical response of three-dimensional photonic lattices: solution of inhomgeneous Maxwell’s equations and their applications,” Phy. Rev. B 54, 5732 (1996)
[Crossref]

Taflove, Allen

Allen Taflove, Computational electrodynamics, the finite difference time domain method (Artech House1995)

Villeneuve, P R

P R Villeneuve et al, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phy. Rev. B 54, 7837 (1996)
[Crossref]

Yee, K. S

K. S Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenna and Propagation 14, 302 (1966)
[Crossref]

IEEE Trans. Antenna and Propagation (1)

K. S Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenna and Propagation 14, 302 (1966)
[Crossref]

Opt. Express (1)

Opt. Express, (1)

Shanhui Fan et al, “Channel drop filters in photonic crystals,” Opt. Express, 3, 4 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4
[Crossref]

Phy. Rev. B (5)

Min Qiu et al, “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phy. Rev. B 61, 12871 (2000)
[Crossref]

Kazuaki Sakoda et al, “Optical response of three-dimensional photonic lattices: solution of inhomgeneous Maxwell’s equations and their applications,” Phy. Rev. B 54, 5732 (1996)
[Crossref]

Vladimir Kuzmiak et al, “Localized defect modes in a two-dimensional triangular photonic crystal,” Phy. Rev. B 57, 15242 (1998)
[Crossref]

Kazuaki Sakoda et al, “Numerical method for localized defect modes in photonic lattices,” Phy. Rev. B 56, 4830 (1997)
[Crossref]

P R Villeneuve et al, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phy. Rev. B 54, 7837 (1996)
[Crossref]

Phy. Rev. Lett. (1)

K. M. Ho et al, “Existence of a photonic gap in periodic dielectric structures,” Phy. Rev. Lett. 65, 3152 (1990)
[Crossref]

Other (3)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computational Physics, 185, (1994)
[Crossref]

J. D. Joannopoulos et al, Photonic crystals - Molding the flow of light (Princeton University Press1995)

Allen Taflove, Computational electrodynamics, the finite difference time domain method (Artech House1995)

Supplementary Material (4)

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» Media 3: GIF (221 KB)     
» Media 4: GIF (279 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

Setup of the FDTD simulation. Dotted line represents the near-to-far field virtual plane, and the dashed line represents the total-field/scattered-field interface. D1, D2 and D3 are detectors.

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Fig. 2.
Fig. 2.

The spectral information obtained from the three detectors using a plane wave incident at an angle of 60°. The y-axis is the spectral amplitude of the field.

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Fig. 3.
Fig. 3.

Spectra of defect modes using different point source arrangements. The solid line and dotted line are the spectra obtained from detector 2 and 3, respectively. The y-axis is the spectral amplitude of the field. The last graph in the right is for Mode 3 by taking Fourier transform after 10,000 time steps, verifying the non-Lorentzian oscillations in the spectra are due to the transient effect.

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Fig. 4.
Fig. 4.

Far field pattern of the four defect modes. The far field pattern is normalized to its maximum. Mode 4(1) and Mode 4(2) are obtained using 0° and 90° plane wave incidence, respectively. The mixed Mode 4 is obtained using 45° plane wave incidence.

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Fig. 5.
Fig. 5.

The dependence of far-field pattern on the Q values of the defect mode. The red, green and blue colored lines are for 3×3, 5×5 and 7×7 supercells, respectively.

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Fig. 6.
Fig. 6.

Field (Ez) evolutions of Mode 1, Mode 2, Mode 3 and Mode 4. A 3rd order Butterworth digital filter is used to get the signal. The y-axis is the normalized field amplitude. Small ripples in Mode 4 are due to the filtering.

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Fig. 7.
Fig. 7.

(266kb, 271kb, 227kb, 286kb) The animation of the near-field of Ez for the four defect modes in the micro-cavity. (a–d) are Mode 1–4 respectively. Only one mode of the doubly-degenerate hexapole is shown. The spatial coordinate in these pictures is the same as in Fig. 1.

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Tables (3)

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Table 1. Resonant defect modes supported in the micro-cavity

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Table 2. Excited defect modes in the micro-cavity by plane waves with different incident angles

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Table 3. Excitation of all or individual modes by arrangement of point sources. The coordinates are the offset from the center grid (100,100) in Fig. 1.

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