Abstract

Both rod and air-hole types of photonic crystal waveguide gratings are proposed and their coupling coefficients and transmission characteristics are effectively investigated by using a simple coupled-mode theory combined with a finite-element method. The results obtained are compared with the results obtained by using a full numerical simulation method. A new definition for unperturbed waveguides is introduced to obtain accurate coupling coefficients. It is shown that, by using a π-phase-shifted waveguide structure in the case of an air-hole type of photonic crystal waveguide grating, the coupling coefficient is strongly enhanced. The accuracy of the method is discussed through numerical examples of high-index-contrast waveguide gratings.

© 2006 Optical Society of America

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2005 (2)

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

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]

2004 (2)

2002 (1)

2001 (2)

K. Yamada, H. Morita, A. Shinya, and M. Notomi, "Improved line-defect structures for photonic-crystal waveguides with high group velocity," Opt. Commun. 198, 395-402 (2001).
[CrossRef]

M. Koshiba, "Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers," J. Lightwave Technol. 19, 1970-1975 (2001).
[CrossRef]

2000 (1)

1997 (2)

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

K. Watanabe and K. Yasumoto, "Coupled-mode analysis of wavelength filtering in a grating-assisted asymmetric three-waveguide directional coupler," J. Opt. Soc. Am. A 14, 2944-3000 (1997).
[CrossRef]

1987 (1)

F. Favre, "Theoretical analysis of external optical feedback on DFB semiconductor lasers," IEEE J. Quantum Electron. QE-23, 81-88 (1987).
[CrossRef]

1975 (1)

W. D. R. Streifer, D. R. Scifres, and R. D. Burnham, "Coupling coefficients for distributed feedback single- and double-heterostructure diode lasers," IEEE J. Quantum Electron. QE-11, 867-873 (1975).
[CrossRef]

Almeida, V. R.

Burnham, R. D.

W. D. R. Streifer, D. R. Scifres, and R. D. Burnham, "Coupling coefficients for distributed feedback single- and double-heterostructure diode lasers," IEEE J. Quantum Electron. QE-11, 867-873 (1975).
[CrossRef]

Fan, S.

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

Favre, F.

F. Favre, "Theoretical analysis of external optical feedback on DFB semiconductor lasers," IEEE J. Quantum Electron. QE-23, 81-88 (1987).
[CrossRef]

Fujisawa, T.

Fukuda, H.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Hikari, M.

Itabashi, S.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Joannopoulos, J. D.

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

Koshiba, M.

Lipson, M.

Morita, H.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

K. Yamada, H. Morita, A. Shinya, and M. Notomi, "Improved line-defect structures for photonic-crystal waveguides with high group velocity," Opt. Commun. 198, 395-402 (2001).
[CrossRef]

Notomi, M.

K. Yamada, H. Morita, A. Shinya, and M. Notomi, "Improved line-defect structures for photonic-crystal waveguides with high group velocity," Opt. Commun. 198, 395-402 (2001).
[CrossRef]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

Scifres, D. R.

W. D. R. Streifer, D. R. Scifres, and R. D. Burnham, "Coupling coefficients for distributed feedback single- and double-heterostructure diode lasers," IEEE J. Quantum Electron. QE-11, 867-873 (1975).
[CrossRef]

Shinya, A.

K. Yamada, H. Morita, A. Shinya, and M. Notomi, "Improved line-defect structures for photonic-crystal waveguides with high group velocity," Opt. Commun. 198, 395-402 (2001).
[CrossRef]

Shoji, T.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Streifer, W. D. R.

W. D. R. Streifer, D. R. Scifres, and R. D. Burnham, "Coupling coefficients for distributed feedback single- and double-heterostructure diode lasers," IEEE J. Quantum Electron. QE-11, 867-873 (1975).
[CrossRef]

Takahashi, J.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Takahashi, M.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Tamechika, E.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Tsuchizawa, T.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Tsuji, Y.

Villeneuve, P. R.

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

Watanabe, K.

K. Watanabe and K. Yasumoto, "Coupled-mode analysis of wavelength filtering in a grating-assisted asymmetric three-waveguide directional coupler," J. Opt. Soc. Am. A 14, 2944-3000 (1997).
[CrossRef]

Watanabe, T.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Yamada, K.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

K. Yamada, H. Morita, A. Shinya, and M. Notomi, "Improved line-defect structures for photonic-crystal waveguides with high group velocity," Opt. Commun. 198, 395-402 (2001).
[CrossRef]

Yasumoto, K.

K. Watanabe and K. Yasumoto, "Coupled-mode analysis of wavelength filtering in a grating-assisted asymmetric three-waveguide directional coupler," J. Opt. Soc. Am. A 14, 2944-3000 (1997).
[CrossRef]

IEEE J. Quantum Electron. (2)

W. D. R. Streifer, D. R. Scifres, and R. D. Burnham, "Coupling coefficients for distributed feedback single- and double-heterostructure diode lasers," IEEE J. Quantum Electron. QE-11, 867-873 (1975).
[CrossRef]

F. Favre, "Theoretical analysis of external optical feedback on DFB semiconductor lasers," IEEE J. Quantum Electron. QE-23, 81-88 (1987).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

J. Lightwave Technol. (5)

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

K. Watanabe and K. Yasumoto, "Coupled-mode analysis of wavelength filtering in a grating-assisted asymmetric three-waveguide directional coupler," J. Opt. Soc. Am. A 14, 2944-3000 (1997).
[CrossRef]

Nature (1)

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

Opt. Commun. (1)

K. Yamada, H. Morita, A. Shinya, and M. Notomi, "Improved line-defect structures for photonic-crystal waveguides with high group velocity," Opt. Commun. 198, 395-402 (2001).
[CrossRef]

Opt. Lett. (1)

Other (1)

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

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Figures (13)

Fig. 1
Fig. 1

Periodic waveguide with input and output waveguides.

Fig. 2
Fig. 2

(a) Grating waveguide based on Si wire waveguides, (b) one period of the grating waveguide, (c) unperturbed waveguide for the grating waveguide.

Fig. 3
Fig. 3

(a) Dispersion curves of the grating waveguides based on Si wire waveguides, (b) intermodal coupling coefficients as a function of index difference Δn, (c) reflection spectra of the grating waveguide for Δn = 0.2.

Fig. 4
Fig. 4

Reflection spectra of the grating waveguide for different index modulations Δn.

Fig. 5
Fig. 5

(a) PC grating waveguide, (b) one period of the PC grating waveguide, (c) unperturbed waveguide for the PC grating waveguide.

Fig. 6
Fig. 6

(Color online) (a) Dispersion curves and (b) transmission characteristics of the PC grating waveguide with d 1 = 0.4a and N = 20.

Fig. 7
Fig. 7

Intermodal coupling coefficients of PC grating waveguides as a function of (a) d 1d for n = 3.5 and (b) n for d 1d = 0.9.

Fig. 8
Fig. 8

(Color online) (a) Transmission characteristics and (b) dispersion curves of the PC grating waveguide with d 1 = 0.7a and N = 10.

Fig. 9
Fig. 9

(Color online) (a) Air-hole-type PC grating waveguides and (b) unperturbed waveguides.

Fig. 10
Fig. 10

(Color online) (a) Dispersion curves and (b) transmission characteristics of PC grating waveguides with d 1 = 0.4a and N = 20.

Fig. 11
Fig. 11

Intermodal coupling coefficients as a function of d 1d.

Fig. 12
Fig. 12

(Color online) (a) π-phase-shifted grating waveguides and (b) unperturbed waveguides.

Fig. 13
Fig. 13

(Color online) (a) Dispersion curves and (b) transmission characteristics of π-phase-shifted grating waveguides with d 1 = 0.55a and N = 20. (c) Field distributions of the unperturbed waveguide with d 2 = 0.5652a and a∕λ = 0.29.

Equations (7)

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d A d z = j ( δ + κ 11 ) A j κ 12 B ,
d B d z = j κ 12 * A + j ( δ + κ 11 ) B ,
κ 11 = π Λ [ β ( ω 1 ) + β ( ω 2 ) ] 2 ,
| κ 12 | = | β ( ω 1 ) β ( ω 2 ) | 2 ,
| ϕ | { | sin ( π z Λ θ 2 ) | at   ω = ω 1 | cos ( π z Λ θ 2 ) | at   ω = ω 2 ,
R = j κ 12 * sinh ( γ N Λ ) γ cosh ( γ N Λ ) + j ( δ + κ 11 ) sinh ( γ N Λ ) ,
γ = { j ( δ + κ 11 ) 2 | κ 12 | 2 for   ω ω 1 | κ 12 | 2 ( δ + κ 11 ) 2   for   ω 2 ω ω 1 j ( δ + κ 11 ) 2 | κ 12 | 2 for   ω 2 ω .

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