Abstract

We present what we believe to be the first study of deep subwavelength surface modes in binary metal–dielectric metamaterials. By employing anomalous coupling in binary periodicity, peculiar properties of band structure and eigenmode symmetry are obtained. We show that strongly confined plasmonic Tamm-like and Shockley-like surface modes can be formed at the termination of the array. We clarify the character of each surface mode and analyze its unique symmetry with the corresponding band structure.

© 2010 Optical Society of America

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  1. X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
    [CrossRef] [PubMed]
  2. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
    [CrossRef] [PubMed]
  3. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
    [CrossRef] [PubMed]
  4. G. Bartal, G. Lerosey, and X. Zhang, Phys. Rev. B 79, 201103(R) (2009).
    [CrossRef]
  5. S. Feng, J. M. Elson, and P. L. Overfelt, Opt. Express 13, 4113 (2005).
    [CrossRef] [PubMed]
  6. S. G. Davison and M. Steslicka, Basic Theory of Surface States (Oxford Univ. Press, 1996).
  7. N. Malkova and C. Z. Ning, Phys. Rev. B 73, 113113 (2006).
    [CrossRef]
  8. K. Ishizaki and S. Noda, Nature 460, 367 (2009).
    [CrossRef] [PubMed]
  9. P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32104(1978).
    [CrossRef]
  10. M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett. 31, 2332 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  15. A. A. Sukhorukov and Y. S. Kivshar, Opt. Lett. 27, 2112 (2002).
    [CrossRef]
  16. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, Nat. Photonics 2, 496 (2008).
    [CrossRef]
  17. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
  18. εAu=ε∞−ωp2/ω(ω+iγc) where ε∞=10, ωp=1.4×1016, and γc=1.1×1014.
  19. The two eigenvalues for the type H near β/k0=1.5 in Fig. have different physical origins. One is single interface SPP-like mode on the leftmost metal surface, and the other one is associated with a localized mode on the right end due to the strong interaction. Those modes are not the surface modes of interest in this Letter.

2010 (1)

2009 (4)

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

G. Bartal, G. Lerosey, and X. Zhang, Phys. Rev. B 79, 201103(R) (2009).
[CrossRef]

K. Ishizaki and S. Noda, Nature 460, 367 (2009).
[CrossRef] [PubMed]

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, Phys. Rev. A 80, 043806 (2009).
[CrossRef]

2008 (2)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, Nat. Photonics 2, 496 (2008).
[CrossRef]

I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 100, 203904 (2008).
[CrossRef] [PubMed]

2007 (1)

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

2006 (3)

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

N. Malkova and C. Z. Ning, Phys. Rev. B 73, 113113 (2006).
[CrossRef]

M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett. 31, 2332 (2006).
[CrossRef] [PubMed]

2005 (2)

2002 (1)

1978 (1)

P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32104(1978).
[CrossRef]

Angelis, C. D.

Bartal, G.

G. Bartal, G. Lerosey, and X. Zhang, Phys. Rev. B 79, 201103(R) (2009).
[CrossRef]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Bryant, G.

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, Phys. Rev. A 80, 043806 (2009).
[CrossRef]

Catrysse, P. B.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Chan, C. T.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

Chen, Z.

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, Phys. Rev. A 80, 043806 (2009).
[CrossRef]

Cho, A. Y.

P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32104(1978).
[CrossRef]

Conforti, M.

Davison, S. G.

S. G. Davison and M. Steslicka, Basic Theory of Surface States (Oxford Univ. Press, 1996).

Elson, J. M.

Fan, S.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Fan, X.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

Feng, S.

Garanovich, I. L.

I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 100, 203904 (2008).
[CrossRef] [PubMed]

M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett. 31, 2332 (2006).
[CrossRef] [PubMed]

Genov, D. A.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, Nat. Photonics 2, 496 (2008).
[CrossRef]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Hromada, I.

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, Phys. Rev. A 80, 043806 (2009).
[CrossRef]

Hu, X.

Ishizaki, K.

K. Ishizaki and S. Noda, Nature 460, 367 (2009).
[CrossRef] [PubMed]

Jiang, X.

Kivshar, Y. S.

Lee, J. C. W.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

Lerosey, G.

G. Bartal, G. Lerosey, and X. Zhang, Phys. Rev. B 79, 201103(R) (2009).
[CrossRef]

Li, X.

Liu, Y.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Liu, Z.

Locatelli, A.

Malkova, N.

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, Phys. Rev. A 80, 043806 (2009).
[CrossRef]

N. Malkova and C. Z. Ning, Phys. Rev. B 73, 113113 (2006).
[CrossRef]

Modotto, D.

Molina, M. I.

Ning, C. Z.

N. Malkova and C. Z. Ning, Phys. Rev. B 73, 113113 (2006).
[CrossRef]

Noda, S.

K. Ishizaki and S. Noda, Nature 460, 367 (2009).
[CrossRef] [PubMed]

Oulton, R. F.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, Nat. Photonics 2, 496 (2008).
[CrossRef]

Overfelt, P. L.

Pile, D. F. P.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, Nat. Photonics 2, 496 (2008).
[CrossRef]

Sorger, V. J.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, Nat. Photonics 2, 496 (2008).
[CrossRef]

Steslicka, M.

S. G. Davison and M. Steslicka, Basic Theory of Surface States (Oxford Univ. Press, 1996).

Sukhorukov, A. A.

Verslegers, L.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Wang, G. P.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

Wang, X.

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, Phys. Rev. A 80, 043806 (2009).
[CrossRef]

Yang, J.

Yariv, A.

P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32104(1978).
[CrossRef]

Yeh, P.

P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32104(1978).
[CrossRef]

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

Yu, Z.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Zhang, X.

G. Bartal, G. Lerosey, and X. Zhang, Phys. Rev. B 79, 201103(R) (2009).
[CrossRef]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, Nat. Photonics 2, 496 (2008).
[CrossRef]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Zi, J.

Appl. Phys. Lett. (1)

P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32104(1978).
[CrossRef]

Nat. Photonics (1)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, Nat. Photonics 2, 496 (2008).
[CrossRef]

Nature (1)

K. Ishizaki and S. Noda, Nature 460, 367 (2009).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (1)

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, Phys. Rev. A 80, 043806 (2009).
[CrossRef]

Phys. Rev. B (2)

N. Malkova and C. Z. Ning, Phys. Rev. B 73, 113113 (2006).
[CrossRef]

G. Bartal, G. Lerosey, and X. Zhang, Phys. Rev. B 79, 201103(R) (2009).
[CrossRef]

Phys. Rev. Lett. (4)

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 100, 203904 (2008).
[CrossRef] [PubMed]

Other (4)

S. G. Davison and M. Steslicka, Basic Theory of Surface States (Oxford Univ. Press, 1996).

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

εAu=ε∞−ωp2/ω(ω+iγc) where ε∞=10, ωp=1.4×1016, and γc=1.1×1014.

The two eigenvalues for the type H near β/k0=1.5 in Fig. have different physical origins. One is single interface SPP-like mode on the leftmost metal surface, and the other one is associated with a localized mode on the right end due to the strong interaction. Those modes are not the surface modes of interest in this Letter.

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic of 1 D binary array with alternating coupling C + and C , dark gray (blue), waveguides, light gray (yellow) and white, coupling regions. (b) Band diagram of dimensionless propagation constant K for asymmetry coefficient η = 0.5 (dotted curve) and η = 0.5 (solid curve). (c) Corresponding mode symmetries at the band edges A D and A ˜ D ˜ in (b).

Fig. 2
Fig. 2

(a), (b) Top, infinite binary metal– dielectric metamaterial; bottom, four types of surface termination at the left end of the array for positive ( E H ) and negative ( E ˜ H ˜ ) η , respectively. The arrows indicate strength of the couplings and their breaks: ,weak coupling; , strong coupling; and , broken weak and strong couplings, respectively. See the main text for the dimensions. (c), (d) Band structures for the infinite arrays in (a) and (b), respectively. The corresponding β for each surface mode in (a) and (b) are also marked in the diagrams. The dashed lines in the figures indicate the refractive index of the external layer (not isofrequency curves), n ext = 1.5 and n ext = 1.34 , respectively. The type E and E ˜ do not support surface modes.

Fig. 3
Fig. 3

(a), (b) Mode field profile ( E x ) for each surface mode F H and F ˜ H ˜ in Fig. 2, respectively. The fields are normalized to the maximum. The arrows indicate the surface waveguides. The decay length of each surface mode is 4.2, 3.1, 8.0, and 3.5 μm for F, G 1 , G 2 , and H, respectively and 14.4, 7.6, 2.7, and 5.1 μm for F ˜ , G ˜ 1 , G ˜ 2 , and H ˜ , respectively. (c), (d) Propagation of the Shockley-like surface mode (type F ˜ ) and the extended state (type E ˜ in Fig. 2), respectively. Only the surface waveguide at the left end is excited (arrows). The electric field intensity is normalized to the maximum. The color code range is set to 0–0.3, and the metal loss was artificially reduced for the better visibility of propagation.

Equations (1)

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H | ψ = K | ψ , H ( η , κ ) = ( 0 η e i κ + 1 η e i κ + 1 0 ) , | ψ = ( A B ) .

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