Abstract

We consider the coupling into a slow mode that appears near an inflection point in the band structure of a photonic crystal waveguide. Remarkably, the coupling into this slow mode, which has a group index ng>1000, can be essentially perfect without any transition region. We show that this efficient coupling occurs thanks to an evanescent mode in the slow medium, which has appreciable amplitude and helps satisfy the boundary conditions but does not transport any energy.

© 2008 Optical Society of America

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References

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  1. Y. A. Vlasov, M. O'Booyle, H. F. Hamann, and S. J. McNab, Nature 438, 65 (2005).
    [CrossRef] [PubMed]
  2. T. D. Happ, M. Kamp, and A. Forchel, Opt. Lett. 26, 1102 (2001).
    [CrossRef]
  3. P. Velha, J. P. Hugonin, and P. Lalanne, Opt. Express 15, 6102 (2007).
    [CrossRef] [PubMed]
  4. C. M. de Sterke, J. Walker, K. B. Dossou, and L. C. Botten, Opt. Express 15, 10984 (2007).
    [CrossRef] [PubMed]
  5. J. P. Hugonin, P. Lalanne, T. P. White, and T. F. Krauss, Opt. Lett. 32, 2638 (2007).
    [CrossRef] [PubMed]
  6. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
    [CrossRef]
  7. J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy, Phys. Rev. E 71, 036612 (2005).
    [CrossRef]
  8. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).
  9. At negative k[Re(k)d/(2π)=−0.38] there are also two evanescent modes, one of which decays with z. It has a field in which the f+/- are switched, and it therefore does not match the forward-propagating slow mode well; we therefore ignore it in our qualitative description, and indeed it plays virtually no role in the coupling.

2007

2005

Y. A. Vlasov, M. O'Booyle, H. F. Hamann, and S. J. McNab, Nature 438, 65 (2005).
[CrossRef] [PubMed]

J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy, Phys. Rev. E 71, 036612 (2005).
[CrossRef]

2004

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

2001

Asatryan, A. A.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Ballato, A.

J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy, Phys. Rev. E 71, 036612 (2005).
[CrossRef]

Ballato, J.

J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy, Phys. Rev. E 71, 036612 (2005).
[CrossRef]

Botten, L. C.

C. M. de Sterke, J. Walker, K. B. Dossou, and L. C. Botten, Opt. Express 15, 10984 (2007).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

de Sterke, C. M.

C. M. de Sterke, J. Walker, K. B. Dossou, and L. C. Botten, Opt. Express 15, 10984 (2007).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Dossou, K. B.

Figotin, A.

J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy, Phys. Rev. E 71, 036612 (2005).
[CrossRef]

Forchel, A.

Hamann, H. F.

Y. A. Vlasov, M. O'Booyle, H. F. Hamann, and S. J. McNab, Nature 438, 65 (2005).
[CrossRef] [PubMed]

Happ, T. D.

Hugonin, J. P.

Kamp, M.

Krauss, T. F.

Lalanne, P.

Langtry, T. N.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

McNab, S. J.

Y. A. Vlasov, M. O'Booyle, H. F. Hamann, and S. J. McNab, Nature 438, 65 (2005).
[CrossRef] [PubMed]

McPhedran, R. C.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

O'Booyle, M.

Y. A. Vlasov, M. O'Booyle, H. F. Hamann, and S. J. McNab, Nature 438, 65 (2005).
[CrossRef] [PubMed]

Velha, P.

Vitebskiy, I.

J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy, Phys. Rev. E 71, 036612 (2005).
[CrossRef]

Vlasov, Y. A.

Y. A. Vlasov, M. O'Booyle, H. F. Hamann, and S. J. McNab, Nature 438, 65 (2005).
[CrossRef] [PubMed]

Walker, J.

White, T. P.

J. P. Hugonin, P. Lalanne, T. P. White, and T. F. Krauss, Opt. Lett. 32, 2638 (2007).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Nature

Y. A. Vlasov, M. O'Booyle, H. F. Hamann, and S. J. McNab, Nature 438, 65 (2005).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. E

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy, Phys. Rev. E 71, 036612 (2005).
[CrossRef]

Other

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

At negative k[Re(k)d/(2π)=−0.38] there are also two evanescent modes, one of which decays with z. It has a field in which the f+/- are switched, and it therefore does not match the forward-propagating slow mode well; we therefore ignore it in our qualitative description, and indeed it plays virtually no role in the coupling.

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Figures (4)

Fig. 1
Fig. 1

Schematic of the PC geometry. The light is coupled from PC1 into PC2, which supports a slow mode.

Fig. 2
Fig. 2

Part of the band structure of PC1 and PC2 for TE polarization. Propagating modes are indicated by solid curves, evanescent modes with Im ( k ) d < 0.51 by dashed curves. The dotted line indicates the inflection point frequency.

Fig. 3
Fig. 3

Normalized magnetic fields of the slow (dots) and the weakly evanescent (solid curves) modes in PC2 (cross section at the top interface), showing (a) the real part and (b) the imaginary part. The slow mode can be scaled to be real, while the normalized evanescent mode has been divided by 0.706 such that its maximum amplitude is real and equals that of the slow mode.

Fig. 4
Fig. 4

Square modulus of the electric field when the light efficiently couples into the slow waveguide from the left. The vertical dashed line indicates the interface between PC1 and PC2. The inset shows the transmittance (solid curve) and the group velocity (dashed curve) versus frequency.

Equations (1)

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1 Z z ̂ ( E t m × H t l * ) d x = [ f + l ( x ) + f l ( x ) ] * [ ( γ f + m ) ( x ) ( γ f m ) ( x ) ] d x = { δ l m p for propagating states l , m ± i δ l m e for evanescent states l , m 0 otherwise }

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