Abstract

Periodic structures consisting of alternating layers of positive-index and negative-index materials have a novel bandgap at the frequency at which the average refractive index is zero. We show that, in the presence of a Kerr nonlinearity, this zero-n gap can switch from low transmission to a perfectly transmitting state, forming a nonlinear resonance or gap soliton in the process. This zero-n gap soliton is omnidirectional, in contrast to the usual Bragg gap soliton of positive-index periodic structures.

© 2005 Optical Society of America

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  1. H. G. Winful, J. H. Marburger, and E. Garmire, Appl. Phys. Lett. 35, 379 (1979).
    [CrossRef]
  2. H. G. Winful, “Optical bistability in periodic structures and in four-wave mixing processes,” Ph.D. dissertation (University of Southern California, 1981).
  3. W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
    [CrossRef] [PubMed]
  4. R. E. Slusher and B. Eggleton, Nonlinear Photonic Crystals (Springer, 2003).
    [CrossRef]
  5. J. Li, L. Zhou, C. T. Chan, and P. Sheng, Phys. Rev. Lett. 90, 083901 (2003).
    [CrossRef]
  6. L. Wu, S. He, and L. Chen, Opt. Lett. 11, 1283 (2003).
  7. L. Wu, S. He, and L. Chen, Phys. Rev. B 67, 235103 (2003).
    [CrossRef]
  8. H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, Appl. Phys. Lett. 83, 5386 (2003).
    [CrossRef]
  9. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, Appl. Phys. Lett. 82, 3820 (2003).
    [CrossRef]
  10. I. V. Shadrivov, N. A. Zharova, A. A. Zharov, and Y. Kivshar, Phys. Rev. E 70, 046615 (2004).
    [CrossRef]
  11. V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  12. M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, Appl. Phys. Lett. 85, 1451 (2004).
    [CrossRef]
  13. R. S. Hedge and H. G. Winful, arXiv.org e-Print archive, physics/0405040, May 9, 2004, http://www.arxiv.org/abs/physics/0405040.
  14. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, Phys. Rev. E 69, 016617 (2004).
    [CrossRef]
  15. U. Trutschel and F. Lederer, J. Opt. Soc. Am. B 5, 2530 (1988).
    [CrossRef]
  16. B. Bobbs, R. Shih, H. Fetterman, and W. Ho, Appl. Phys. Lett. 52, 4 (1988).
    [CrossRef]
  17. R. Shih, H. R. Fetterman, W. W. Ho, and R. McGraw, Phys. Rev. Lett. 65, 579 (1990).
    [CrossRef] [PubMed]

2004

I. V. Shadrivov, N. A. Zharova, A. A. Zharov, and Y. Kivshar, Phys. Rev. E 70, 046615 (2004).
[CrossRef]

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, Appl. Phys. Lett. 85, 1451 (2004).
[CrossRef]

V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, Phys. Rev. E 69, 016617 (2004).
[CrossRef]

2003

J. Li, L. Zhou, C. T. Chan, and P. Sheng, Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef]

L. Wu, S. He, and L. Chen, Opt. Lett. 11, 1283 (2003).

L. Wu, S. He, and L. Chen, Phys. Rev. B 67, 235103 (2003).
[CrossRef]

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, Appl. Phys. Lett. 83, 5386 (2003).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, Appl. Phys. Lett. 82, 3820 (2003).
[CrossRef]

1990

R. Shih, H. R. Fetterman, W. W. Ho, and R. McGraw, Phys. Rev. Lett. 65, 579 (1990).
[CrossRef] [PubMed]

1988

U. Trutschel and F. Lederer, J. Opt. Soc. Am. B 5, 2530 (1988).
[CrossRef]

B. Bobbs, R. Shih, H. Fetterman, and W. Ho, Appl. Phys. Lett. 52, 4 (1988).
[CrossRef]

1987

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

1979

H. G. Winful, J. H. Marburger, and E. Garmire, Appl. Phys. Lett. 35, 379 (1979).
[CrossRef]

1968

V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Boardman, A. D.

V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Bobbs, B.

B. Bobbs, R. Shih, H. Fetterman, and W. Ho, Appl. Phys. Lett. 52, 4 (1988).
[CrossRef]

Chan, C. T.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef]

Chen, H.

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, Appl. Phys. Lett. 83, 5386 (2003).
[CrossRef]

Chen, L.

L. Wu, S. He, and L. Chen, Opt. Lett. 11, 1283 (2003).

L. Wu, S. He, and L. Chen, Phys. Rev. B 67, 235103 (2003).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

Egan, P.

V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Eggleton, B.

R. E. Slusher and B. Eggleton, Nonlinear Photonic Crystals (Springer, 2003).
[CrossRef]

Feise, M. W.

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, Appl. Phys. Lett. 85, 1451 (2004).
[CrossRef]

Fetterman, H.

B. Bobbs, R. Shih, H. Fetterman, and W. Ho, Appl. Phys. Lett. 52, 4 (1988).
[CrossRef]

Fetterman, H. R.

R. Shih, H. R. Fetterman, W. W. Ho, and R. McGraw, Phys. Rev. Lett. 65, 579 (1990).
[CrossRef] [PubMed]

Garmire, E.

H. G. Winful, J. H. Marburger, and E. Garmire, Appl. Phys. Lett. 35, 379 (1979).
[CrossRef]

He, S.

L. Wu, S. He, and L. Chen, Opt. Lett. 11, 1283 (2003).

L. Wu, S. He, and L. Chen, Phys. Rev. B 67, 235103 (2003).
[CrossRef]

Ho, W.

B. Bobbs, R. Shih, H. Fetterman, and W. Ho, Appl. Phys. Lett. 52, 4 (1988).
[CrossRef]

Ho, W. W.

R. Shih, H. R. Fetterman, W. W. Ho, and R. McGraw, Phys. Rev. Lett. 65, 579 (1990).
[CrossRef] [PubMed]

Jiang, H.

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, Appl. Phys. Lett. 83, 5386 (2003).
[CrossRef]

Kivshar, Y.

I. V. Shadrivov, N. A. Zharova, A. A. Zharov, and Y. Kivshar, Phys. Rev. E 70, 046615 (2004).
[CrossRef]

Kivshar, Y. S.

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, Appl. Phys. Lett. 85, 1451 (2004).
[CrossRef]

V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, Phys. Rev. E 69, 016617 (2004).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, Appl. Phys. Lett. 82, 3820 (2003).
[CrossRef]

Lederer, F.

Li, H.

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, Appl. Phys. Lett. 83, 5386 (2003).
[CrossRef]

Li, J.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef]

Marburger, J. H.

H. G. Winful, J. H. Marburger, and E. Garmire, Appl. Phys. Lett. 35, 379 (1979).
[CrossRef]

McGraw, R.

R. Shih, H. R. Fetterman, W. W. Ho, and R. McGraw, Phys. Rev. Lett. 65, 579 (1990).
[CrossRef] [PubMed]

Mills, D. L.

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

Shadrivov, I. V.

I. V. Shadrivov, N. A. Zharova, A. A. Zharov, and Y. Kivshar, Phys. Rev. E 70, 046615 (2004).
[CrossRef]

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, Appl. Phys. Lett. 85, 1451 (2004).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, Appl. Phys. Lett. 82, 3820 (2003).
[CrossRef]

Shadrivov, V.

V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Sheng, P.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef]

Shih, R.

R. Shih, H. R. Fetterman, W. W. Ho, and R. McGraw, Phys. Rev. Lett. 65, 579 (1990).
[CrossRef] [PubMed]

B. Bobbs, R. Shih, H. Fetterman, and W. Ho, Appl. Phys. Lett. 52, 4 (1988).
[CrossRef]

Slusher, R. E.

R. E. Slusher and B. Eggleton, Nonlinear Photonic Crystals (Springer, 2003).
[CrossRef]

Sukhorukov, A. A.

V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, Phys. Rev. E 69, 016617 (2004).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, Appl. Phys. Lett. 82, 3820 (2003).
[CrossRef]

Trutschel, U.

Veselago, V. G.

V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Winful, H. G.

H. G. Winful, J. H. Marburger, and E. Garmire, Appl. Phys. Lett. 35, 379 (1979).
[CrossRef]

H. G. Winful, “Optical bistability in periodic structures and in four-wave mixing processes,” Ph.D. dissertation (University of Southern California, 1981).

Wu, L.

L. Wu, S. He, and L. Chen, Opt. Lett. 11, 1283 (2003).

L. Wu, S. He, and L. Chen, Phys. Rev. B 67, 235103 (2003).
[CrossRef]

Zhang, Y.

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, Appl. Phys. Lett. 83, 5386 (2003).
[CrossRef]

Zharov, A. A.

V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, Phys. Rev. E 69, 016617 (2004).
[CrossRef]

I. V. Shadrivov, N. A. Zharova, A. A. Zharov, and Y. Kivshar, Phys. Rev. E 70, 046615 (2004).
[CrossRef]

Zharova, N. A.

I. V. Shadrivov, N. A. Zharova, A. A. Zharov, and Y. Kivshar, Phys. Rev. E 70, 046615 (2004).
[CrossRef]

Zhou, L.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef]

Zhu, S.

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, Appl. Phys. Lett. 83, 5386 (2003).
[CrossRef]

Appl. Phys. Lett.

H. G. Winful, J. H. Marburger, and E. Garmire, Appl. Phys. Lett. 35, 379 (1979).
[CrossRef]

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, Appl. Phys. Lett. 83, 5386 (2003).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, Appl. Phys. Lett. 82, 3820 (2003).
[CrossRef]

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, Appl. Phys. Lett. 85, 1451 (2004).
[CrossRef]

B. Bobbs, R. Shih, H. Fetterman, and W. Ho, Appl. Phys. Lett. 52, 4 (1988).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

L. Wu, S. He, and L. Chen, Opt. Lett. 11, 1283 (2003).

Phys. Rev. B

L. Wu, S. He, and L. Chen, Phys. Rev. B 67, 235103 (2003).
[CrossRef]

Phys. Rev. E

V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, Phys. Rev. E 69, 016617 (2004).
[CrossRef]

I. V. Shadrivov, N. A. Zharova, A. A. Zharov, and Y. Kivshar, Phys. Rev. E 70, 046615 (2004).
[CrossRef]

Phys. Rev. Lett.

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

J. Li, L. Zhou, C. T. Chan, and P. Sheng, Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef]

R. Shih, H. R. Fetterman, W. W. Ho, and R. McGraw, Phys. Rev. Lett. 65, 579 (1990).
[CrossRef] [PubMed]

Sov. Phys. Usp.

V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Other

R. E. Slusher and B. Eggleton, Nonlinear Photonic Crystals (Springer, 2003).
[CrossRef]

H. G. Winful, “Optical bistability in periodic structures and in four-wave mixing processes,” Ph.D. dissertation (University of Southern California, 1981).

R. S. Hedge and H. G. Winful, arXiv.org e-Print archive, physics/0405040, May 9, 2004, http://www.arxiv.org/abs/physics/0405040.

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Figures (6)

Fig. 1
Fig. 1

Hysteresis behavior of transmittance as a function of a defocusing γ I inc for detuning to the left of the zero-n gap ( f = 3.51 GHz , N = 32 ) for incident angles θ = 0 ° , 15°, and 30°.

Fig. 2
Fig. 2

Zero-n gap soliton: spatial distribution of the E field magnitude (normalized by E inc ) when a defocusing γ I inc = 0.064 results in a near-unity transmittance ( f = 3.51 GHz , N = 32 ) at normal incidence.

Fig. 3
Fig. 3

Spatial distribution of the phase of the E field for conditions as in Fig. 2.

Fig. 4
Fig. 4

Bragg gap soliton observed when a defocusing γ I inc = 0.027 is used ( f = 7.15 GHz , N = 32 ) at normal incidence.

Fig. 5
Fig. 5

Zero-n gap soliton for an incident angle of 30° (solid curve) as compared to the one at normal incidence (dotted curve). γ I inc = 0.064 for both cases and other conditions as in Fig. 2.

Fig. 6
Fig. 6

Zero-n gap soliton for an incident angle of 0° for a lossy structure at two different values of the loss term ( L = 0.01 , 0.001) as compared to a lossless structure (solid curve). The amplitude decreases as the loss term increases.

Equations (9)

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

ε 1 ( ω , z ) = ε 10 ( ω , z ) + γ E 2 .
ε 2 ( f ) = 1.6 + 40 0.81 f 2 i f L ,
μ 2 ( f ) = 1.0 + 40 0.814 f 2 i f L ,
E ( x , z ) = E ( z ) exp [ i k 0 ( β x c t ) ] ,
d 2 E d ζ 2 + p 2 E + μ i γ E 2 = 0 ,
p 2 = μ i ε i ( ω , z ) β 2 ,
E i = E j ,
μ j d E d ζ i = μ i d E d ζ j .
ε 1 ( f ) μ 1 ( f ) d 1 = ε 2 ( f ) μ 2 ( f ) d 2 .

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