Abstract

Apodized photonic crystal (PC) waveguide gratings are proposed to suppress sidelobes which appear in reflection spectra of usual PC waveguide gratings. By using specific functions (Gauss and Gauss-cosine functions) for the longitudinal refractive index distribution, it is possible to suppress sidelobes in the reflection spectra of PC waveguide gratings efficiently. The apodization is realized by simply changing diameters of dielectric pillars adjacent to the PC waveguide core. It is shown that by using Gauss-cosine functions for the apodization, Bragg frequency of the waveguide grating becomes insensitive to the magnitude of perturbation leading to the possibility of designing waveguide gratings with arbitrarily Bragg frequency and bandwidth by modulating geometrical parameters only.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist of light," Nature 386, 143-149 (1997).
    [CrossRef]
  2. M. Koshiba, "Wavelength division demultiplexing and multiplexing with photonic crystal waveguide couplers," J. Lightwave Technol. 19, 1970-1975 (2001).
    [CrossRef]
  3. E. A. Camargo, H. M. H. Chong, and R. M. E. L. Rue, "2D photonic crystal thermo-optic switch based on AlGaAs/GaAs epitaxial structure," Opt. Express 12, 588-592 (2004).
    [CrossRef] [PubMed]
  4. M. Soljaèiæ, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R), (2002).
  5. Y. Sugimoto, H. Nakamura, U. Tanaka, N. Ikeda, and K. Asakawa, "High-precision optical interference in Mach-Zehnder-type photonic crystal waveguide," Opt. Express 13, 96-105 (2005).
    [CrossRef] [PubMed]
  6. T. Fujisawa and M. Koshiba, "Finite-element mode-solver for nonlinear periodic optical waveguides and its application to photonic crystal circuits," J. Lightwave Technol. 23, 382-387 (2005).
    [CrossRef]
  7. T. Fujisawa and M. Koshiba, "An analysis of photonic crystal waveguide gratings using coupled-mode theory and finite-element method," Appl. Opt.to be published.
  8. T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
    [CrossRef]
  9. M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightwave Technol. 18, 102-110 (2000).
    [CrossRef]
  10. T. Fujisawa and M. Koshiba, "Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits," J. Lightwave Technol. 22, 684-691 (2004).
    [CrossRef]
  11. M. Tokushima, H. Yamada, and Y. Arakawa, "1.5-μm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab," Appl. Phys. Lett. 84, 4298-4300 (2004).
    [CrossRef]
  12. S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004).
    [CrossRef]
  13. C.-C. Chen, C.-Y. Chen, W.-K. Wang, F.-H. Fluang, C.-K. Lin, W.-Y. Chiu, and Y.-J. Chan, "Photonic crystal directional couplers formed by InAlGaAs nano-rods," Opt. Express 13, 38-43 (2005).
    [CrossRef] [PubMed]
  14. M. Soljaèiæ, S. G. Johnson, S. Fan, M. Ibanscu, E. Ippen, and J. D. Joannopoulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Am. B 19, 2052-2059 (2002).
    [CrossRef]

2005 (3)

2004 (4)

M. Tokushima, H. Yamada, and Y. Arakawa, "1.5-μm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab," Appl. Phys. Lett. 84, 4298-4300 (2004).
[CrossRef]

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004).
[CrossRef]

E. A. Camargo, H. M. H. Chong, and R. M. E. L. Rue, "2D photonic crystal thermo-optic switch based on AlGaAs/GaAs epitaxial structure," Opt. Express 12, 588-592 (2004).
[CrossRef] [PubMed]

T. Fujisawa and M. Koshiba, "Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits," J. Lightwave Technol. 22, 684-691 (2004).
[CrossRef]

2002 (1)

2001 (1)

2000 (1)

1997 (2)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist of light," Nature 386, 143-149 (1997).
[CrossRef]

T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

Arakawa, Y.

M. Tokushima, H. Yamada, and Y. Arakawa, "1.5-μm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab," Appl. Phys. Lett. 84, 4298-4300 (2004).
[CrossRef]

Asakawa, K.

Assefa, S.

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004).
[CrossRef]

Bienstman, P.

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004).
[CrossRef]

Camargo, E. A.

Chan, Y.-J.

Chen, C.-C.

Chen, C.-Y.

Chiu, W.-Y.

Chong, H. M. H.

Erdogan, T.

T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

Fan, S.

Fluang, F.-H.

Fujisawa, T.

Hikari, M.

Ibanscu, M.

Ikeda, N.

Ippen, E.

Ippen, E. P.

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004).
[CrossRef]

Joannopoulos, J. D.

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004).
[CrossRef]

M. Soljaèiæ, S. G. Johnson, S. Fan, M. Ibanscu, E. Ippen, and J. D. Joannopoulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Am. B 19, 2052-2059 (2002).
[CrossRef]

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist of light," Nature 386, 143-149 (1997).
[CrossRef]

Johnson, S. G.

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004).
[CrossRef]

M. Soljaèiæ, S. G. Johnson, S. Fan, M. Ibanscu, E. Ippen, and J. D. Joannopoulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Am. B 19, 2052-2059 (2002).
[CrossRef]

Kolodziejski, L. A.

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004).
[CrossRef]

Koshiba, M.

Lin, C.-K.

Nakamura, H.

Petrich, G. S.

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004).
[CrossRef]

Rakich, P. T.

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004).
[CrossRef]

Rue, R. M. E. L.

Smith, H. I.

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004).
[CrossRef]

Soljaèiæ, M.

Sugimoto, Y.

Tanaka, U.

Tokushima, M.

M. Tokushima, H. Yamada, and Y. Arakawa, "1.5-μm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab," Appl. Phys. Lett. 84, 4298-4300 (2004).
[CrossRef]

Tsuji, Y.

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist of light," Nature 386, 143-149 (1997).
[CrossRef]

Wang, W.-K.

Yamada, H.

M. Tokushima, H. Yamada, and Y. Arakawa, "1.5-μm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab," Appl. Phys. Lett. 84, 4298-4300 (2004).
[CrossRef]

Appl. Opt. (1)

T. Fujisawa and M. Koshiba, "An analysis of photonic crystal waveguide gratings using coupled-mode theory and finite-element method," Appl. Opt.to be published.

Appl. Phys. Lett. (2)

M. Tokushima, H. Yamada, and Y. Arakawa, "1.5-μm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab," Appl. Phys. Lett. 84, 4298-4300 (2004).
[CrossRef]

S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004).
[CrossRef]

J. Lightwave Technol. (5)

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

Nature (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist of light," Nature 386, 143-149 (1997).
[CrossRef]

Opt. Express (3)

Other (1)

M. Soljaèiæ, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R), (2002).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1.
Fig. 1.

PC waveguide gratings with Gaussian apodization composed of dielectric pillars on square array.

Fig. 2.
Fig. 2.

Longitudinal dn variation of PC waveguide gratings with Gaussian apodization for dc =0.05a, ω 0=0.25NΛ, and N=52.

Fig. 3.
Fig. 3.

Reflection spectra of the PC waveguide grating. Solid and dashed lines are reflection spectra of the PC waveguide grating with Gaussian apodization for dc =0.05a, ω 0=0.25NΛ. The numbers of periods are 104 and 52, respectively. Dash-dot line shows reflection spectrum of PC waveguide gratings without apodization for dn =0.45a, N=52.

Fig. 4.
Fig. 4.

Transmission spectra of the PC waveguide grating with Gaussian apodization (dc =0.05a and N=104), for different values of ω 0.

Fig. 5.
Fig. 5.

Dispersion curves of the PC waveguide grating with dn =0.45a.

Fig. 6.
Fig. 6.

PC waveguide gratings with Gauss-cosine apodization composed of dielectric pillars on square array.

Fig. 7.
Fig. 7.

Longitudinal di variation of PC waveguide gratings with Gauss-cosine apodization for dc =0.025a, ω 0=0.5NΛ, and N=26.

Fig. 8.
Fig. 8.

Solid, dashed, and dash-dot lines are reflection spectra of PC waveguide gratings with Gauss-cosine apodization (dc =0.025a, ω 0=0.5NΛ, and N=104), with Gaussian apodization (dc =0.05a, ω 0=0.25NΛ, and N=104), and without apodization (d2j -1=0.5a, d2j =0.45a, j=1, 2, ⋯, N, and N=52), respectively.

Fig. 9.
Fig. 9.

Transmission spectra of PC waveguide gratings with Gauss-cosine apodization (dc =0.025a and N=104) for different values of ω 0.

Fig. 10.
Fig. 10.

The tolerance of the spot size for the transmission coefficient at the Bragg frequency.

Fig. 11.
Fig. 11.

Transmission spectra of the PC waveguide grating without apodization for different values of dc (N=52).

Fig. 12.
Fig. 12.

Transmission spectra of the PC waveguide grating with Gaussian apodization for different values of dc (ω0 =0.5NΛ and N=104).

Fig. 13.
Fig. 13.

Transmission spectra of the PC waveguide grating with Gauss-cosine apodization for different values of dc (ω 0=0.25NΛ and N=104).

Fig. 14.
Fig. 14.

(a) One period of the PC waveguide grating with Gauss-cosine apodization, and (b) the corresponding dispersion curves of the input PC waveguide (solid line), waveguide A with d i-1=0.525a (dashed line), and waveguide B with di =0.475a (dash-dot line).

Fig. 15.
Fig. 15.

Bragg frequency and bandwidth of PC waveguide gratings as a function of dc .

Equations (4)

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

d n = d d c exp [ { ( n N 2 ) Λ ω 0 } 2 ]
d i = d + d c exp [ { ( i N ) Λ ω 0 } 2 ] cos ( i π ) .
β A a + β B a = π
2 β a + ( Δ β A Δ β B ) a = π .

Metrics