Abstract

Abstract

We study light coupling between two photonic crystal waveguides, one of which supports slow light. We show theoretically that a short photonic crystal waveguide between the two that need to be coupled, can lead to a vanishingly small reflectivity. The design relies on the analogy with a λ/4 anti-reflection layer in thin-film optics. We find that some of the usual relationships between the Fresnel coefficients at an interface no longer hold.

© 2007 Optical Society of America

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  1. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
    [CrossRef] [PubMed]
  2. R. S. Jacobsen,  et al., "Strained silicon as a new electro-optic material," Nature 441, 199-202 (2006).
    [CrossRef] [PubMed]
  3. J. B. Khurgin, "Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis," J. Opt. Soc. Am. B 22, 1062-1074 (2005).
    [CrossRef]
  4. T. D. Happ, M. Kamp, and A. Forchel, "Photonic crystal tapers for ultracompact mode conversion," Opt. Lett. 26, 1102-1104 (2001).
    [CrossRef]
  5. K. Dossou, L. C. Botten, S. Chen, J. Brnovic, R. C. McPhedran, and C. M. de Sterke, "Efficient couplers for photonic crystal waveguides," Opt. Commun. 265, 207-219 (2006).
    [CrossRef]
  6. P. Pottier, M. Gnan, and R. M. De La Rue, "Efficient coupling into slow-light photonic crystal channel guides using photonic crystal tapers," Opt. Express 15, 6560-6575 (2007).
    [CrossRef]
  7. H. A. Macleod, Thin Film Optical Filters, 3rd Edition (Institute of Physics Pub., Bristol, 2001), Chap. 3.
  8. P. Velha, J. P. Hugonin, and P. Lalanne, "Compact and efficient injection of light into band-edge slow-modes," Opt. Express 15, 6102-6112 (2007).
    [CrossRef] [PubMed]
  9. N. Ozaki, Y. Kitagawa, Y. Takata, N. Ikeda, Y. Watanabe, A. Mizutani, Y. Sugimoto, and K. Asakawa, "High transmission recovery of slow light in a photonic crystal waveguide uing a hetero group velocity waveguide," Opt. Express 13, 7974-7982 (2007).
    [CrossRef]
  10. L. C. Botten, T. P. White, C.M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, "Photonic crystal devices modelled as grating stacks," Opt. Express 12, 1592-1604 (2004).
    [CrossRef] [PubMed]
  11. L. C. Botten, T. P. White, A. A. Asatryan, and T. N. Langtry, C. M. de Sterke, R. C. McPhedran, "Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part 1: Theory," Phys. Rev. E 70, 056606:1-13 (2004).
    [CrossRef]
  12. K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comp. Phys. 219, 120-143 (2006).
    [CrossRef]
  13. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C.M. de Sterke and R. C. McPhedran, "Photonic crystal devices modelled as grating stacks: matrix generalizations of thin film optics," Opt. Express 12, 1592-1604 (2004).
    [CrossRef] [PubMed]

2007 (3)

P. Pottier, M. Gnan, and R. M. De La Rue, "Efficient coupling into slow-light photonic crystal channel guides using photonic crystal tapers," Opt. Express 15, 6560-6575 (2007).
[CrossRef]

N. Ozaki, Y. Kitagawa, Y. Takata, N. Ikeda, Y. Watanabe, A. Mizutani, Y. Sugimoto, and K. Asakawa, "High transmission recovery of slow light in a photonic crystal waveguide uing a hetero group velocity waveguide," Opt. Express 13, 7974-7982 (2007).
[CrossRef]

P. Velha, J. P. Hugonin, and P. Lalanne, "Compact and efficient injection of light into band-edge slow-modes," Opt. Express 15, 6102-6112 (2007).
[CrossRef] [PubMed]

2006 (3)

R. S. Jacobsen,  et al., "Strained silicon as a new electro-optic material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

K. Dossou, L. C. Botten, S. Chen, J. Brnovic, R. C. McPhedran, and C. M. de Sterke, "Efficient couplers for photonic crystal waveguides," Opt. Commun. 265, 207-219 (2006).
[CrossRef]

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comp. Phys. 219, 120-143 (2006).
[CrossRef]

2005 (2)

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

J. B. Khurgin, "Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis," J. Opt. Soc. Am. B 22, 1062-1074 (2005).
[CrossRef]

2004 (3)

2001 (1)

Asakawa, K.

N. Ozaki, Y. Kitagawa, Y. Takata, N. Ikeda, Y. Watanabe, A. Mizutani, Y. Sugimoto, and K. Asakawa, "High transmission recovery of slow light in a photonic crystal waveguide uing a hetero group velocity waveguide," Opt. Express 13, 7974-7982 (2007).
[CrossRef]

Asatryan, A. A.

Botten, L. C.

K. Dossou, L. C. Botten, S. Chen, J. Brnovic, R. C. McPhedran, and C. M. de Sterke, "Efficient couplers for photonic crystal waveguides," Opt. Commun. 265, 207-219 (2006).
[CrossRef]

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comp. Phys. 219, 120-143 (2006).
[CrossRef]

L. C. Botten, T. P. White, C.M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, "Photonic crystal devices modelled as grating stacks," Opt. Express 12, 1592-1604 (2004).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C.M. de Sterke and R. C. McPhedran, "Photonic crystal devices modelled as grating stacks: matrix generalizations of thin film optics," Opt. Express 12, 1592-1604 (2004).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, A. A. Asatryan, and T. N. Langtry, C. M. de Sterke, R. C. McPhedran, "Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part 1: Theory," Phys. Rev. E 70, 056606:1-13 (2004).
[CrossRef]

Brnovic, J.

K. Dossou, L. C. Botten, S. Chen, J. Brnovic, R. C. McPhedran, and C. M. de Sterke, "Efficient couplers for photonic crystal waveguides," Opt. Commun. 265, 207-219 (2006).
[CrossRef]

Byrne, M. A.

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comp. Phys. 219, 120-143 (2006).
[CrossRef]

Chen, S.

K. Dossou, L. C. Botten, S. Chen, J. Brnovic, R. C. McPhedran, and C. M. de Sterke, "Efficient couplers for photonic crystal waveguides," Opt. Commun. 265, 207-219 (2006).
[CrossRef]

De La Rue, R. M.

P. Pottier, M. Gnan, and R. M. De La Rue, "Efficient coupling into slow-light photonic crystal channel guides using photonic crystal tapers," Opt. Express 15, 6560-6575 (2007).
[CrossRef]

de Sterke, C. M.

K. Dossou, L. C. Botten, S. Chen, J. Brnovic, R. C. McPhedran, and C. M. de Sterke, "Efficient couplers for photonic crystal waveguides," Opt. Commun. 265, 207-219 (2006).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, and T. N. Langtry, C. M. de Sterke, R. C. McPhedran, "Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part 1: Theory," Phys. Rev. E 70, 056606:1-13 (2004).
[CrossRef]

de Sterke, C.M.

Dossou, K.

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comp. Phys. 219, 120-143 (2006).
[CrossRef]

K. Dossou, L. C. Botten, S. Chen, J. Brnovic, R. C. McPhedran, and C. M. de Sterke, "Efficient couplers for photonic crystal waveguides," Opt. Commun. 265, 207-219 (2006).
[CrossRef]

Forchel, A.

Gnan, M.

P. Pottier, M. Gnan, and R. M. De La Rue, "Efficient coupling into slow-light photonic crystal channel guides using photonic crystal tapers," Opt. Express 15, 6560-6575 (2007).
[CrossRef]

Hamann, H. F.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Happ, T. D.

Hugonin, J. P.

Ikeda, N.

N. Ozaki, Y. Kitagawa, Y. Takata, N. Ikeda, Y. Watanabe, A. Mizutani, Y. Sugimoto, and K. Asakawa, "High transmission recovery of slow light in a photonic crystal waveguide uing a hetero group velocity waveguide," Opt. Express 13, 7974-7982 (2007).
[CrossRef]

Jacobsen, R. S.

R. S. Jacobsen,  et al., "Strained silicon as a new electro-optic material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Kamp, M.

Khurgin, J. B.

Kitagawa, Y.

N. Ozaki, Y. Kitagawa, Y. Takata, N. Ikeda, Y. Watanabe, A. Mizutani, Y. Sugimoto, and K. Asakawa, "High transmission recovery of slow light in a photonic crystal waveguide uing a hetero group velocity waveguide," Opt. Express 13, 7974-7982 (2007).
[CrossRef]

Lalanne, P.

Langtry, T. N.

McNab, S. J.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

McPhedran, R. C.

K. Dossou, L. C. Botten, S. Chen, J. Brnovic, R. C. McPhedran, and C. M. de Sterke, "Efficient couplers for photonic crystal waveguides," Opt. Commun. 265, 207-219 (2006).
[CrossRef]

L. C. Botten, T. P. White, C.M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, "Photonic crystal devices modelled as grating stacks," Opt. Express 12, 1592-1604 (2004).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, A. A. Asatryan, and T. N. Langtry, C. M. de Sterke, R. C. McPhedran, "Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part 1: Theory," Phys. Rev. E 70, 056606:1-13 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C.M. de Sterke and R. C. McPhedran, "Photonic crystal devices modelled as grating stacks: matrix generalizations of thin film optics," Opt. Express 12, 1592-1604 (2004).
[CrossRef] [PubMed]

Mizutani, A.

N. Ozaki, Y. Kitagawa, Y. Takata, N. Ikeda, Y. Watanabe, A. Mizutani, Y. Sugimoto, and K. Asakawa, "High transmission recovery of slow light in a photonic crystal waveguide uing a hetero group velocity waveguide," Opt. Express 13, 7974-7982 (2007).
[CrossRef]

O’Boyle, M.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Ozaki, N.

N. Ozaki, Y. Kitagawa, Y. Takata, N. Ikeda, Y. Watanabe, A. Mizutani, Y. Sugimoto, and K. Asakawa, "High transmission recovery of slow light in a photonic crystal waveguide uing a hetero group velocity waveguide," Opt. Express 13, 7974-7982 (2007).
[CrossRef]

Pottier, P.

P. Pottier, M. Gnan, and R. M. De La Rue, "Efficient coupling into slow-light photonic crystal channel guides using photonic crystal tapers," Opt. Express 15, 6560-6575 (2007).
[CrossRef]

Sugimoto, Y.

N. Ozaki, Y. Kitagawa, Y. Takata, N. Ikeda, Y. Watanabe, A. Mizutani, Y. Sugimoto, and K. Asakawa, "High transmission recovery of slow light in a photonic crystal waveguide uing a hetero group velocity waveguide," Opt. Express 13, 7974-7982 (2007).
[CrossRef]

Takata, Y.

N. Ozaki, Y. Kitagawa, Y. Takata, N. Ikeda, Y. Watanabe, A. Mizutani, Y. Sugimoto, and K. Asakawa, "High transmission recovery of slow light in a photonic crystal waveguide uing a hetero group velocity waveguide," Opt. Express 13, 7974-7982 (2007).
[CrossRef]

Velha, P.

Vlasov, Y. A.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Watanabe, Y.

N. Ozaki, Y. Kitagawa, Y. Takata, N. Ikeda, Y. Watanabe, A. Mizutani, Y. Sugimoto, and K. Asakawa, "High transmission recovery of slow light in a photonic crystal waveguide uing a hetero group velocity waveguide," Opt. Express 13, 7974-7982 (2007).
[CrossRef]

White, T. P.

J. Comp. Phys. (1)

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comp. Phys. 219, 120-143 (2006).
[CrossRef]

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

Nature (2)

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

R. S. Jacobsen,  et al., "Strained silicon as a new electro-optic material," Nature 441, 199-202 (2006).
[CrossRef] [PubMed]

Opt. Commun. (1)

K. Dossou, L. C. Botten, S. Chen, J. Brnovic, R. C. McPhedran, and C. M. de Sterke, "Efficient couplers for photonic crystal waveguides," Opt. Commun. 265, 207-219 (2006).
[CrossRef]

Opt. Express (5)

P. Pottier, M. Gnan, and R. M. De La Rue, "Efficient coupling into slow-light photonic crystal channel guides using photonic crystal tapers," Opt. Express 15, 6560-6575 (2007).
[CrossRef]

N. Ozaki, Y. Kitagawa, Y. Takata, N. Ikeda, Y. Watanabe, A. Mizutani, Y. Sugimoto, and K. Asakawa, "High transmission recovery of slow light in a photonic crystal waveguide uing a hetero group velocity waveguide," Opt. Express 13, 7974-7982 (2007).
[CrossRef]

P. Velha, J. P. Hugonin, and P. Lalanne, "Compact and efficient injection of light into band-edge slow-modes," Opt. Express 15, 6102-6112 (2007).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C.M. de Sterke and R. C. McPhedran, "Photonic crystal devices modelled as grating stacks: matrix generalizations of thin film optics," Opt. Express 12, 1592-1604 (2004).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, C.M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, "Photonic crystal devices modelled as grating stacks," Opt. Express 12, 1592-1604 (2004).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. E (1)

L. C. Botten, T. P. White, A. A. Asatryan, and T. N. Langtry, C. M. de Sterke, R. C. McPhedran, "Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part 1: Theory," Phys. Rev. E 70, 056606:1-13 (2004).
[CrossRef]

Other (1)

H. A. Macleod, Thin Film Optical Filters, 3rd Edition (Institute of Physics Pub., Bristol, 2001), Chap. 3.

Supplementary Material (1)

» Media 1: MOV (1103 KB)     

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Figures (3)

Fig. 1.
Fig. 1.

Fig. 1. (a) (1.14 MB) Schematic of the geometry and field animation for the perfectly matched structure: light from above in the fundamental mode of PC 1 (vg=0.157 c) is incident on the lower semi-infinite PC 3 (vg=0.01 c). The transmission is perfect thanks to the inclusion of an appropriately chosen PC 2 which acts as an anti-reflection coating. (b) Associated band structure for PC 1 (red) and PC 3 (blue); both waveguides have single mode operation for 0.211918≤d/λ≤0.214481 indicated by the horizontal dashed lines. [Media 1]

Fig. 2.
Fig. 2.

(a) Illustration of the interpretation of ϕ 12 through the use of alternative phase origins. (b) Interface reflectances ρ 21 and ρ 23 versus the width w 2 of the waveguide in PC2.

Fig. 3.
Fig. 3.

(a) Reflectance ρ of the entire structure versus m, the length of PC 2, for a waveguide width 0.90356√3d. (b) Reflectance versus group velocity (bottom scale) in PC 3 and normalized frequency (top scale) for w 2=0.90356√3d and m=8.

Equations (5)

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

R = R 12 + T 21 Λ 2 m R 23 Λ 2 m ( I R 21 Λ 2 m R 23 Λ 2 m ) 1 T 12 ,
r = r 12 + f 12 r 23 μ 2 m 1 r 21 r 23 μ 2 n , where f 12 = t 12 t 21 r 12 r 21 ,
S = ( r 12 t 21 t 12 r 21 ) ,
r = e i ϕ 12 r ˜ 12 + r ˜ 23 e 2 i ψ 1 r ˜ 21 r ˜ 23 e 2 i ψ , t = e j ( ϕ 12 + ϕ 23 ) 2 t ˜ 12 t ˜ 23 e i ψ 1 r ˜ 21 r ˜ 23 e 2 i ψ .
ρ 21 = ρ 23 ,

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