Abstract

Degenerate band edges (DBEs) of a photonic bandgap have the form (ωωD)k2m for integers m>1, with ωD the frequency at the band edge. We show theoretically that DBEs lead to efficient coupling into slow-light modes without a transition region, and that the field strength in the slow mode can far exceed that in the incoming medium. A method is proposed to create a DBE of arbitrary order m by coupling m optical modes with multiple superimposed gratings. The enhanced coupling near a DBE occurs because of the presence of one or more evanescent modes, which are absent at conventional quadratic band edges. We furthermore show that the coupling can be increased or suppressed by varying the number of excited evanescent waves.

© 2011 Optical Society of America

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References

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  1. T. Baba, Nat. Photon. 2, 465 (2008).
    [CrossRef]
  2. K. Sakoda, Optical Properties of Photonic Crystal(Springer, 2001).
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    [CrossRef] [PubMed]
  6. T. D. Happ, M. Kamp, and A. Forchel, Opt. Lett. 26, 1102(2001).
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  7. A. A. Sukhorukov, C. J. Handmer, C. M. de Sterke, and M. J. Steel, Opt. Express 15, 17954 (2007).
    [CrossRef] [PubMed]
  8. H. Kogelnik, in Topics in Applied Physics: Integrated Optics, T.Tamir, ed. (Springer, 1979), pp. 13–81.
  9. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, Nat. Phys. 2, 775 (2006).
    [CrossRef]
  10. V. B. Lidskii, USSR Comput. Math. Math. Phys. 6, 73 (1966).
    [CrossRef]
  11. A. Figotin and I. Vitebskiy, Waves Random Complex Media 16, 293 (2006).
    [CrossRef]
  12. A. Figotin and I. Vitebskiy, Phys. Rev. E 74, 17 (2006).
    [CrossRef]

2009 (1)

2008 (1)

T. Baba, Nat. Photon. 2, 465 (2008).
[CrossRef]

2007 (2)

2006 (3)

A. Figotin and I. Vitebskiy, Waves Random Complex Media 16, 293 (2006).
[CrossRef]

A. Figotin and I. Vitebskiy, Phys. Rev. E 74, 17 (2006).
[CrossRef]

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, Nat. Phys. 2, 775 (2006).
[CrossRef]

2005 (1)

2001 (2)

K. Sakoda, Optical Properties of Photonic Crystal(Springer, 2001).

T. D. Happ, M. Kamp, and A. Forchel, Opt. Lett. 26, 1102(2001).
[CrossRef]

1979 (1)

H. Kogelnik, in Topics in Applied Physics: Integrated Optics, T.Tamir, ed. (Springer, 1979), pp. 13–81.

1966 (1)

V. B. Lidskii, USSR Comput. Math. Math. Phys. 6, 73 (1966).
[CrossRef]

Baba, T.

T. Baba, Nat. Photon. 2, 465 (2008).
[CrossRef]

Botten, L. C.

de Sterke, C. M.

Dossou, K. B.

Eggleton, B. J.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, Nat. Phys. 2, 775 (2006).
[CrossRef]

Figotin, A.

A. Figotin and I. Vitebskiy, Waves Random Complex Media 16, 293 (2006).
[CrossRef]

A. Figotin and I. Vitebskiy, Phys. Rev. E 74, 17 (2006).
[CrossRef]

Forchel, A.

Handmer, C. J.

Happ, T. D.

Hugonin, J. P.

Joannopoulos, J. D.

Johnson, S. G.

Kamp, M.

Kogelnik, H.

H. Kogelnik, in Topics in Applied Physics: Integrated Optics, T.Tamir, ed. (Springer, 1979), pp. 13–81.

Lalanne, P.

Lidskii, V. B.

V. B. Lidskii, USSR Comput. Math. Math. Phys. 6, 73 (1966).
[CrossRef]

Littler, I. C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, Nat. Phys. 2, 775 (2006).
[CrossRef]

McPhedran, R. C.

Mok, J. T.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, Nat. Phys. 2, 775 (2006).
[CrossRef]

Povinelli, M. L.

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystal(Springer, 2001).

Steel, M. J.

Sukhorukov, A. A.

Velha, P.

Vitebskiy, I.

A. Figotin and I. Vitebskiy, Waves Random Complex Media 16, 293 (2006).
[CrossRef]

A. Figotin and I. Vitebskiy, Phys. Rev. E 74, 17 (2006).
[CrossRef]

White, T. P.

Nat. Photon. (1)

T. Baba, Nat. Photon. 2, 465 (2008).
[CrossRef]

Nat. Phys. (1)

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, Nat. Phys. 2, 775 (2006).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. E (1)

A. Figotin and I. Vitebskiy, Phys. Rev. E 74, 17 (2006).
[CrossRef]

USSR Comput. Math. Math. Phys. (1)

V. B. Lidskii, USSR Comput. Math. Math. Phys. 6, 73 (1966).
[CrossRef]

Waves Random Complex Media (1)

A. Figotin and I. Vitebskiy, Waves Random Complex Media 16, 293 (2006).
[CrossRef]

Other (2)

K. Sakoda, Optical Properties of Photonic Crystal(Springer, 2001).

H. Kogelnik, in Topics in Applied Physics: Integrated Optics, T.Tamir, ed. (Springer, 1979), pp. 13–81.

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Figures (4)

Fig. 1
Fig. 1

Schematic of the approach to create DBEs. For z < 0 the multimode fiber has m incident and m reflected fiber modes. For z > 0 multiple gratings with different periodicities couple a single forward mode of the uniform fiber to all backward modes and vice versa.

Fig. 2
Fig. 2

Complex band structure near an upper band edge ( ω > ω D ): (a) quadratic, (b) quartic, and (c) sextic. Each has one forward (dotted black) and one backward (dotted gray) propagating mode. (b), (c) DBEs also posses forward (black) and backward (gray) decaying modes.

Fig. 3
Fig. 3

(a) Reflection from a fiber with quartic band edge for different incident fields [Eq. (3)] for v g = 0.045 c . (b) Coupling efficiency versus v g . White line in (a) and gray line in (b) represent maximum coupling. In both (a) and (b), solid and dashed black lines represent coupling only to the propagating and evanescent modes, respectively. In (b), dashed gray line represents average coupling over all α and β values.

Fig. 4
Fig. 4

(a) Coupling efficiency versus group velocity close to a sextic DBE when exciting only: evanescent modes (dashed gray), propagating mode (dashed black) multiplied by 5 to fit the scale, propagating mode and one evanescent mode (gray), and propagating mode with both evanescent modes (black); (b) coupling efficiency for quadratic (dashed gray), quartic (dashed black), sextic (gray), and octic (black) DBE.

Equations (3)

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k = ξ Δ ω 2 m e i ( 2 π / 2 m ) j j = 0 , , ( 2 m 1 ) ,
i 1 = cos ( α ) e i β / V 1 , i 2 = sin ( α ) / V 2 ,
η v g 2 m 1 .

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