Abstract

We developed a novel universal eigenvalue analysis for 2D arbitrary nanostructures comprising dispersive and lossy materials. The complex dispersion relation (or complex Bloch band structure) of a metallic grating is rigorously calculated by the proposed algorithm with the finite-difference implementation. The abnormally large group velocity is observed at a plasmonic band edge with a large attenuation constant. Interestingly, we found the abnormal group velocity is caused by the leaky (radiation) loss, not by metallic absorption (ohmic) loss. The periodically modulated surface of the grating significantly modifies the original dispersion relation of the semi-infinite dielectric-metal structure and induces the extraordinarily large group velocity, which is different from the near-zero group velocity at photonic band edge. The work is fundamentally important to the design of plasmonic nanostructures.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2013

W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, Nat. Nanotechnol. 8, 506 (2013).
[CrossRef]

2012

2011

D. de Ceglia, M. A. Vincenti, M. Scalora, N. Akozbek, and M. J. Bloemer, AIP Adv. 1, 032151 (2011).
[CrossRef]

C. Fietz, Y. Urzhumov, and G. Shvets, Opt. Express 19, 19027 (2011).
[CrossRef]

2010

M. Luo and Q. H. Liu, J. Opt. Soc. Am. A 27, 1878 (2010).
[CrossRef]

A. Raman and S. H. Fan, Phys. Rev. Lett. 104, 087401 (2010).
[CrossRef]

2008

T. Okamoto, J. Simonen, and S. Kawata, Phys. Rev. B 77, 115425 (2008).
[CrossRef]

A. Kocabas, G. Ertas, S. S. Senlik, and A. Aydinli, Opt. Express 16, 12469 (2008).
[CrossRef]

2006

M. Luisier, A. Schenk, W. Fichtner, and G. Klimeck, Phys. Rev. B 74, 205323 (2006).
[CrossRef]

R.-L. Chern, C. C. Chang, and C. C. Chang, Phys. Rev. E 73, 036605 (2006).
[CrossRef]

2004

T. Okamoto, F. H’Dhili, and S. Kawata, Appl. Phys. Lett. 85, 3968 (2004).
[CrossRef]

1998

1996

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, Phys. Rev. B 54, 6227 (1996).
[CrossRef]

1995

1994

W. C. Chew and W. H. Weedon, Microwave Opt. Technol. Lett. 7, 599 (1994).
[CrossRef]

Adam, P.

Akozbek, N.

D. de Ceglia, M. A. Vincenti, M. Scalora, N. Akozbek, and M. J. Bloemer, AIP Adv. 1, 032151 (2011).
[CrossRef]

Aydinli, A.

Barnes, W. L.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, Phys. Rev. B 54, 6227 (1996).
[CrossRef]

Bloemer, M. J.

Buncick, M. C.

Callahan, J. M.

Chang, C. C.

R.-L. Chern, C. C. Chang, and C. C. Chang, Phys. Rev. E 73, 036605 (2006).
[CrossRef]

R.-L. Chern, C. C. Chang, and C. C. Chang, Phys. Rev. E 73, 036605 (2006).
[CrossRef]

Chern, R.-L.

R.-L. Chern, C. C. Chang, and C. C. Chang, Phys. Rev. E 73, 036605 (2006).
[CrossRef]

Chew, W. C.

W. C. Chew and W. H. Weedon, Microwave Opt. Technol. Lett. 7, 599 (1994).
[CrossRef]

Choy, W. C. H.

X. H. Li, W. E. I. Sha, W. C. H. Choy, D. D. S. Fung, and F. X. Xie, J. Phys. Chem. C 116, 7200 (2012).
[CrossRef]

Co, D. T.

W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, Nat. Nanotechnol. 8, 506 (2013).
[CrossRef]

D’Aguanno, G.

de Ceglia, D.

D. de Ceglia, M. A. Vincenti, M. Scalora, N. Akozbek, and M. J. Bloemer, AIP Adv. 1, 032151 (2011).
[CrossRef]

Djurisic, A. B.

Dostalek, J.

Dridi, M.

W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, Nat. Nanotechnol. 8, 506 (2013).
[CrossRef]

Elazar, J. M.

Ertas, G.

Everitt, H. O.

Fan, S. H.

A. Raman and S. H. Fan, Phys. Rev. Lett. 104, 087401 (2010).
[CrossRef]

Fichtner, W.

M. Luisier, A. Schenk, W. Fichtner, and G. Klimeck, Phys. Rev. B 74, 205323 (2006).
[CrossRef]

Fietz, C.

Foreman, J. V.

Fung, D. D. S.

X. H. Li, W. E. I. Sha, W. C. H. Choy, D. D. S. Fung, and F. X. Xie, J. Phys. Chem. C 116, 7200 (2012).
[CrossRef]

Gaylord, T. K.

Grann, E. B.

H’Dhili, F.

T. Okamoto, F. H’Dhili, and S. Kawata, Appl. Phys. Lett. 85, 3968 (2004).
[CrossRef]

Homola, J.

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Kawata, S.

T. Okamoto and S. Kawata, Opt. Express 20, 5168 (2012).
[CrossRef]

T. Okamoto, J. Simonen, and S. Kawata, Phys. Rev. B 77, 115425 (2008).
[CrossRef]

T. Okamoto, F. H’Dhili, and S. Kawata, Appl. Phys. Lett. 85, 3968 (2004).
[CrossRef]

Kim, C. H.

W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, Nat. Nanotechnol. 8, 506 (2013).
[CrossRef]

Kitson, S. C.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, Phys. Rev. B 54, 6227 (1996).
[CrossRef]

Klimeck, G.

M. Luisier, A. Schenk, W. Fichtner, and G. Klimeck, Phys. Rev. B 74, 205323 (2006).
[CrossRef]

Knoll, W.

Kocabas, A.

Li, C.

Li, X. H.

X. H. Li, W. E. I. Sha, W. C. H. Choy, D. D. S. Fung, and F. X. Xie, J. Phys. Chem. C 116, 7200 (2012).
[CrossRef]

Liu, Q. H.

Luisier, M.

M. Luisier, A. Schenk, W. Fichtner, and G. Klimeck, Phys. Rev. B 74, 205323 (2006).
[CrossRef]

Luo, M.

Majewski, M. L.

Mattiucci, N.

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Moharam, M. G.

Odom, T. W.

W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, Nat. Nanotechnol. 8, 506 (2013).
[CrossRef]

Okamoto, T.

T. Okamoto and S. Kawata, Opt. Express 20, 5168 (2012).
[CrossRef]

T. Okamoto, J. Simonen, and S. Kawata, Phys. Rev. B 77, 115425 (2008).
[CrossRef]

T. Okamoto, F. H’Dhili, and S. Kawata, Appl. Phys. Lett. 85, 3968 (2004).
[CrossRef]

Pommet, D. A.

Preist, T. W.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, Phys. Rev. B 54, 6227 (1996).
[CrossRef]

Rakic, A. D.

Raman, A.

A. Raman and S. H. Fan, Phys. Rev. Lett. 104, 087401 (2010).
[CrossRef]

Sambles, J. R.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, Phys. Rev. B 54, 6227 (1996).
[CrossRef]

Scalora, M.

D. de Ceglia, M. A. Vincenti, M. Scalora, N. Akozbek, and M. J. Bloemer, AIP Adv. 1, 032151 (2011).
[CrossRef]

Schatz, G. C.

W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, Nat. Nanotechnol. 8, 506 (2013).
[CrossRef]

Schenk, A.

M. Luisier, A. Schenk, W. Fichtner, and G. Klimeck, Phys. Rev. B 74, 205323 (2006).
[CrossRef]

Senlik, S. S.

Sha, W. E. I.

X. H. Li, W. E. I. Sha, W. C. H. Choy, D. D. S. Fung, and F. X. Xie, J. Phys. Chem. C 116, 7200 (2012).
[CrossRef]

Shvets, G.

Simonen, J.

T. Okamoto, J. Simonen, and S. Kawata, Phys. Rev. B 77, 115425 (2008).
[CrossRef]

Suh, J. Y.

W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, Nat. Nanotechnol. 8, 506 (2013).
[CrossRef]

Toma, K.

Toma, M.

Urzhumov, Y.

Vincenti, M. A.

D. de Ceglia, M. A. Vincenti, M. Scalora, N. Akozbek, and M. J. Bloemer, AIP Adv. 1, 032151 (2011).
[CrossRef]

Wang, H.-Yu.

Wasielewski, M. R.

W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, Nat. Nanotechnol. 8, 506 (2013).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, Microwave Opt. Technol. Lett. 7, 599 (1994).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Xie, F. X.

X. H. Li, W. E. I. Sha, W. C. H. Choy, D. D. S. Fung, and F. X. Xie, J. Phys. Chem. C 116, 7200 (2012).
[CrossRef]

Zhou, W.

W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, Nat. Nanotechnol. 8, 506 (2013).
[CrossRef]

Zhou, Y.-S.

AIP Adv.

D. de Ceglia, M. A. Vincenti, M. Scalora, N. Akozbek, and M. J. Bloemer, AIP Adv. 1, 032151 (2011).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

T. Okamoto, F. H’Dhili, and S. Kawata, Appl. Phys. Lett. 85, 3968 (2004).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. Chem. C

X. H. Li, W. E. I. Sha, W. C. H. Choy, D. D. S. Fung, and F. X. Xie, J. Phys. Chem. C 116, 7200 (2012).
[CrossRef]

Microwave Opt. Technol. Lett.

W. C. Chew and W. H. Weedon, Microwave Opt. Technol. Lett. 7, 599 (1994).
[CrossRef]

Nat. Nanotechnol.

W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, Nat. Nanotechnol. 8, 506 (2013).
[CrossRef]

Opt. Express

Phys. Rev. B

M. Luisier, A. Schenk, W. Fichtner, and G. Klimeck, Phys. Rev. B 74, 205323 (2006).
[CrossRef]

T. Okamoto, J. Simonen, and S. Kawata, Phys. Rev. B 77, 115425 (2008).
[CrossRef]

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, Phys. Rev. B 54, 6227 (1996).
[CrossRef]

Phys. Rev. E

R.-L. Chern, C. C. Chang, and C. C. Chang, Phys. Rev. E 73, 036605 (2006).
[CrossRef]

Phys. Rev. Lett.

A. Raman and S. H. Fan, Phys. Rev. Lett. 104, 087401 (2010).
[CrossRef]

Other

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Schematic pattern of a squarely modulated metallic grating with the geometric parameters W=410nm, P=820nm, H=30nm, and T=50nm. The red ellipses and yellow arrows denote the excited SPs and incident light, respectively. (b), (c) The normalized spatial overlap integrals between the near-field profile of the grating and the mth Floquet mode at 410 nm [Peak 1 of Fig. 2(a)] and 480 nm [Peak 3 of Fig. 2(a)], respectively.

Fig. 2.
Fig. 2.

(a) Optical absorption of the metallic grating with a varying thickness T, (b)–(d) Hz field distributions of the metallic grating corresponding to the absorption peaks denoted by arrows (T=50nm).

Fig. 3.
Fig. 3.

(a) Reflectance of the metallic grating with a varying thickness T, (b) the phase distribution of Hz field at the PBG with respect to Peak 2 in Fig. 2(a).

Fig. 4.
Fig. 4.

Band structure of a periodic dielectric strip shielded with a PEC plate. The dielectric constant and the side length of each strip are 4 and P/2, respectively.

Fig. 5.
Fig. 5.

Band structure of the 2D metallic grating as a function of the real part of the Bloch wavenumber. λ is the incident wavelength, and the rightmost vertical line denotes Re(kB)=π/P. The imaginary part of (physically real) metal permittivity by the B-B model increases or decreases by a factor of 2 for simulating the high or low ohmic loss, respectively.

Fig. 6.
Fig. 6.

Band structure of the 2D metallic grating as a function of the imaginary part of the Bloch wavenumber. λ is the incident wavelength. The imaginary part of (physically real) metal permittivity by the B-B model increases or decreases by a factor of 2 for simulating the high or low ohmic loss, respectively.

Equations (7)

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

η=|0P0ycHz(x,y)exp(jkxx+jky(yyc))dxdy|,
kx=k0sinθ+2πn/P,kx2+ky2=k02,Im(ky)<0,
ksp+Δksp=k0sinθ+2πPn,
(D11T0TejkBPTD22T00TD33TTejkBP0TD44)(ϕ1ϕ2ϕ3ϕ4)=0,
H=(D11T00TD22T00TD33TT000),Q=(000T0000000000TD44)
Hϕ=ejkBPQϕ.
Hϕ=ejkBPQϕ+QϕQϕ,(HQ)ϕ=(ejkBP1)Qϕ,(HQ)1Qϕ=1ejkBP1ϕ,Mϕ=λϕ.

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