Abstract

We discuss the characteristics of surface plasmon modes guided on metallic nanowires of circular cross-section embedded in silica glass. Under certain conditions such wires allow low-loss guided modes, full account being taken of ohmic losses in the metal. We find that these modes can be bound to the wire even when the real part of their axial refractive index is less than that of the surrounding dielectric. We assess in detail the accuracy of a simple model in which SPs are viewed as spiralling around the nanowire in a helical path, forming modes at certain angles of pitch. The results are relevant for understanding the behavior of light in two-dimensional arrays of metallic nanowires in fiber form.

© 2008 Optical Society of America

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References

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  1. S. A. Maier, "Plasmonics: The promise of highly integrated optical devices," IEEE J. Sel. Top. Quantum Electron. 12, 1671-1677 (2006).
    [CrossRef]
  2. D. Hondros and P. Debye, "Elektromagnetische Wellen an dielektrischen Drähten," Annalen der Physik 337, 465 (1910).
    [CrossRef]
  3. C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, "Surface polaritons in a circularly cylindrical interface - surface plasmons," Phys. Rev. B 10, 3038-3051 (1974), http://link.aps.org/abstract/PRB/v10/p3038.
  4. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, "Guiding of a one-dimensional optical beam with nanometer diameter," Opt. Lett. 22, 475-477 (1997), http://www.opticsinfobase.org/abstract.cfm?URI=ol-22-7-475.
    [CrossRef] [PubMed]
  5. H. Khosravi, D. R. Tilley, and R. Loudon, "Surface-polaritons in cylindrical optical fibers," J. Opt. Soc. Am. A 8, 112-122 (1991), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-8-1-112.
    [CrossRef]
  6. B. Prade and J. Y. Vinet, "Guided optical waves in fibers with negative dielectric constant," IEEE J. Lightwave Technol. 12, 6-18 (1994), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=265728.
    [CrossRef]
  7. S. J. Al-Bader and M. Imtaar, "TM-polarized surface-plasma modes on metal-coated dielectric cylinders," IEEE J. Lightwave Technol. 10, 865-872 (1992), http://www.opticsinfobase.org/abstract.cfm?URI=josab-10-1-83.
    [CrossRef]
  8. S. S. Martinos and E. N. Economou, "Virtual surface-plasmons in cylinders," Phys. Rev. B 28, 3173-3181 (1983), http://prola.aps.org/abstract/PRB/v28/i6/p3173_1.
  9. C. G. Poulton, M. A. Schmidt, G. J. Pearce, G. Kakarantzas, and P. St.J. Russell, "Numerical study of guided modes in arrays of metallic nanowires," Opt. Lett. 32, 1647-1649 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ol-32-12-1647.
    [CrossRef] [PubMed]
  10. M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. St.J. Russell, "Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires," Phys. Rev. B 77, 33417 (2008), http://link.aps.org/abstract/PRB/v77/e033417.
  11. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, London, San Diego, 1985), pp. 350-357.
  12. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, London, San Diego, 2007), pp. 6.
  13. C. Miziumski, "Utilization of a cylindrical geometry to promote radiative interaction with slow surface excitations," Phys. Lett. A 40, 187-188 (1972).
  14. P. St.J. Russell, "Photonic-crystal fibers," IEEE J. Lightwave Technol. 24, 4729-4749 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=JLT-24-12-4729.
    [CrossRef]
  15. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, Weinheim, 2004) pp. 194-209.

2008

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. St.J. Russell, "Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires," Phys. Rev. B 77, 33417 (2008), http://link.aps.org/abstract/PRB/v77/e033417.

2007

2006

S. A. Maier, "Plasmonics: The promise of highly integrated optical devices," IEEE J. Sel. Top. Quantum Electron. 12, 1671-1677 (2006).
[CrossRef]

1997

1991

1983

S. S. Martinos and E. N. Economou, "Virtual surface-plasmons in cylinders," Phys. Rev. B 28, 3173-3181 (1983), http://prola.aps.org/abstract/PRB/v28/i6/p3173_1.

1974

C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, "Surface polaritons in a circularly cylindrical interface - surface plasmons," Phys. Rev. B 10, 3038-3051 (1974), http://link.aps.org/abstract/PRB/v10/p3038.

1972

C. Miziumski, "Utilization of a cylindrical geometry to promote radiative interaction with slow surface excitations," Phys. Lett. A 40, 187-188 (1972).

1910

D. Hondros and P. Debye, "Elektromagnetische Wellen an dielektrischen Drähten," Annalen der Physik 337, 465 (1910).
[CrossRef]

Debye, P.

D. Hondros and P. Debye, "Elektromagnetische Wellen an dielektrischen Drähten," Annalen der Physik 337, 465 (1910).
[CrossRef]

Economou, E. N.

S. S. Martinos and E. N. Economou, "Virtual surface-plasmons in cylinders," Phys. Rev. B 28, 3173-3181 (1983), http://prola.aps.org/abstract/PRB/v28/i6/p3173_1.

C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, "Surface polaritons in a circularly cylindrical interface - surface plasmons," Phys. Rev. B 10, 3038-3051 (1974), http://link.aps.org/abstract/PRB/v10/p3038.

Hondros, D.

D. Hondros and P. Debye, "Elektromagnetische Wellen an dielektrischen Drähten," Annalen der Physik 337, 465 (1910).
[CrossRef]

Kakarantzas, G.

Khosravi, H.

Kobayashi, T.

Loudon, R.

Maier, S. A.

S. A. Maier, "Plasmonics: The promise of highly integrated optical devices," IEEE J. Sel. Top. Quantum Electron. 12, 1671-1677 (2006).
[CrossRef]

Martinos, S. S.

S. S. Martinos and E. N. Economou, "Virtual surface-plasmons in cylinders," Phys. Rev. B 28, 3173-3181 (1983), http://prola.aps.org/abstract/PRB/v28/i6/p3173_1.

Miziumski, C.

C. Miziumski, "Utilization of a cylindrical geometry to promote radiative interaction with slow surface excitations," Phys. Lett. A 40, 187-188 (1972).

Morimoto, A.

Ngai, K. L.

C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, "Surface polaritons in a circularly cylindrical interface - surface plasmons," Phys. Rev. B 10, 3038-3051 (1974), http://link.aps.org/abstract/PRB/v10/p3038.

Pearce, G. J.

Pfeiffer, C. A.

C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, "Surface polaritons in a circularly cylindrical interface - surface plasmons," Phys. Rev. B 10, 3038-3051 (1974), http://link.aps.org/abstract/PRB/v10/p3038.

Poulton, C. G.

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. St.J. Russell, "Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires," Phys. Rev. B 77, 33417 (2008), http://link.aps.org/abstract/PRB/v77/e033417.

C. G. Poulton, M. A. Schmidt, G. J. Pearce, G. Kakarantzas, and P. St.J. Russell, "Numerical study of guided modes in arrays of metallic nanowires," Opt. Lett. 32, 1647-1649 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ol-32-12-1647.
[CrossRef] [PubMed]

Russell, P. St.J.

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. St.J. Russell, "Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires," Phys. Rev. B 77, 33417 (2008), http://link.aps.org/abstract/PRB/v77/e033417.

C. G. Poulton, M. A. Schmidt, G. J. Pearce, G. Kakarantzas, and P. St.J. Russell, "Numerical study of guided modes in arrays of metallic nanowires," Opt. Lett. 32, 1647-1649 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ol-32-12-1647.
[CrossRef] [PubMed]

Schmidt, M. A.

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. St.J. Russell, "Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires," Phys. Rev. B 77, 33417 (2008), http://link.aps.org/abstract/PRB/v77/e033417.

C. G. Poulton, M. A. Schmidt, G. J. Pearce, G. Kakarantzas, and P. St.J. Russell, "Numerical study of guided modes in arrays of metallic nanowires," Opt. Lett. 32, 1647-1649 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ol-32-12-1647.
[CrossRef] [PubMed]

Sempere, L. N. P.

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. St.J. Russell, "Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires," Phys. Rev. B 77, 33417 (2008), http://link.aps.org/abstract/PRB/v77/e033417.

Takahara, J.

Taki, H.

Tilley, D. R.

Tyagi, H. K.

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. St.J. Russell, "Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires," Phys. Rev. B 77, 33417 (2008), http://link.aps.org/abstract/PRB/v77/e033417.

Yamagishi, S.

Annalen der Physik

D. Hondros and P. Debye, "Elektromagnetische Wellen an dielektrischen Drähten," Annalen der Physik 337, 465 (1910).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

S. A. Maier, "Plasmonics: The promise of highly integrated optical devices," IEEE J. Sel. Top. Quantum Electron. 12, 1671-1677 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Lett. A

C. Miziumski, "Utilization of a cylindrical geometry to promote radiative interaction with slow surface excitations," Phys. Lett. A 40, 187-188 (1972).

Phys. Rev. B

C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, "Surface polaritons in a circularly cylindrical interface - surface plasmons," Phys. Rev. B 10, 3038-3051 (1974), http://link.aps.org/abstract/PRB/v10/p3038.

S. S. Martinos and E. N. Economou, "Virtual surface-plasmons in cylinders," Phys. Rev. B 28, 3173-3181 (1983), http://prola.aps.org/abstract/PRB/v28/i6/p3173_1.

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. St.J. Russell, "Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires," Phys. Rev. B 77, 33417 (2008), http://link.aps.org/abstract/PRB/v77/e033417.

Other

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, London, San Diego, 1985), pp. 350-357.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, London, San Diego, 2007), pp. 6.

B. Prade and J. Y. Vinet, "Guided optical waves in fibers with negative dielectric constant," IEEE J. Lightwave Technol. 12, 6-18 (1994), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=265728.
[CrossRef]

S. J. Al-Bader and M. Imtaar, "TM-polarized surface-plasma modes on metal-coated dielectric cylinders," IEEE J. Lightwave Technol. 10, 865-872 (1992), http://www.opticsinfobase.org/abstract.cfm?URI=josab-10-1-83.
[CrossRef]

P. St.J. Russell, "Photonic-crystal fibers," IEEE J. Lightwave Technol. 24, 4729-4749 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=JLT-24-12-4729.
[CrossRef]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, Weinheim, 2004) pp. 194-209.

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Figures (6)

Fig. 1.
Fig. 1.

(a)&(b): Real part of the modal refractive index for guided modes of a single silver nanowire embedded in silica for (a) a=500 nm and (b) a=100 nm. The dashed lines correspond to the bulk material refractive index of silica. (c)&(d): Corresponding loss of the guided modes for (c) a=500 nm and (d) a=100 nm. The integers in the figures indicate the mode order. The vertical dotted lines mark the cut-off wavelength at which guidance ceases and the loss goes to a minimum. The m=0 and m=1 modes do not cut-off.

Fig. 2.
Fig. 2.

Radial dependence of the z-components of the electric field for (a) the m=2 mode at 450 nm wavelength (n m =1.60+i 0.07) and (b) the m=4 mode at 390 nm wavelength (n m=0.16 +i 0.73) (a=100 nm). Note the oscillating real and imaginary parts in (b).

Fig. 3.
Fig. 3.

(a) Dispersion of the m=1 guided mode for two different wire radii. The inset shows the corresponding losses for both nanowire diameters. The dashed line is the index of bulk silica glass. (b) Mode field extension δ and (c) loss as function of wavelength of the m=1 mode for five different nanowire radii.

Fig. 4.
Fig. 4.

Schematic of the helical trajectory of a SP mode on a metallic nanowire. The green lines denote the local phase-fronts and the fields decay exponentially in the z-direction.

Fig. 5.
Fig. 5.

Comparison of the exact (full lines) and approximate (dashed lines) modal dispersion for nanowire radii (a): 500 nm and (b): 100 nm. The grey solid curve shows the material dispersion of silica and the integers indicate the mode order.

Fig. 6.
Fig. 6.

Comparison of the cut-off points for the approximate and exact solutions (the m=0 and m=1 modes do not cut-off). (a): Quasi-cut-off wavelengths of the modes of exact solution (solid) and analytic model (dashed) as functions of nanowire radius. (b) Percentage error of model versus nanowire radius. The integers indicate the mode order.

Equations (4)

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

( q D 2 ψ M + q M 2 ψ D ) ( ε M q D 2 ψ M + ε D q M 2 ψ D ) m 2 n m 2 ( ε D ε M ) 2 = 0
ψ D = m k 0 q D a K m + 1 ( k 0 q D a ) K m ( k 0 q D a ) , ψ M = m k 0 q M a J m + 1 ( k 0 q M a ) J m ( k 0 q M a )
q D = + n m 2 ε D ,   q M = ε M n m 2 .
n m = ε D ε M ε D + ε M ( ( m 1 ) k 0 a ) 2 , m 1

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