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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 18 — Sep. 1, 2008
  • pp: 13617–13623
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Long-range spiralling surface plasmon modes on metallic nanowires

M. A. Schmidt and P. St.J. Russell  »View Author Affiliations


Optics Express, Vol. 16, Issue 18, pp. 13617-13623 (2008)
http://dx.doi.org/10.1364/OE.16.013617


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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

1. Introduction

We will show that under certain conditions such nanowires allow ultra-low loss guided modes, even when full account is taken of ohmic losses in the metal. We find that these modes are bound to the wire even when the real part of their axial refractive index (n m=n mR+i n mI) is less than that of the surrounding dielectric n mR<n D (the presence of a non-zero n mI allows this to occur). Finally we study in detail the accuracy of a simple model briefly presented in [10

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.

], where SPs are viewed as spiralling around the nanowire in a helical path, forming modes for certain trajectories.

2. Dispersion relation

The fields are taken to be monochromatic with vacuum wavevector k 0 and axial wavevector component β m=k 0 n m (complex-valued in general). The nanowire (radius a) is oriented along the z-axis, the complex-valued dielectric constant of the metal is ε M(k 0) and the surrounding dielectric has real-valued dielectric constant ε D(k 0). The frequency-dependent dielectric functions for silver (complex-valued) and silica (real-valued) are included in the simulations [11

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

, 12

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

]. Starting with Maxwell’s equations in cylindrical coordinates (r,ϕ,z) and matching tangential field components (E z, H z, E ϕ and H ϕ) at the surface of the nanowire yields, after some manipulation, the following dispersion relation:

(qD2ψM+qM2ψD)(εMqD2ψM+εDqM2ψD)m2nm2(εDεM)2=0
(1)

where m is the azimuthal mode order,

ψD=mk0qDaKm+1(k0qDa)Km(k0qDa),ψM=mk0qMaJm+1(k0qMa)Jm(k0qMa)
(2)

where J m is a Bessel function of the first kind, and K m a modified Bessel function. The radial wavevectors q D and q M are defined as follows:

qD=+nm2εD, qM=εMnm2.
(3)

It is interesting that the denominators of the scattering coefficients of Mie theory for an infinitely long cylindrical metal wire embedded in glass, illuminated by an infinite plane wave, are proportional to the LHS of Eq. (1) [15

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

]. The positive sign in q D selects modes whose amplitudes decay exponentially away from the wire, restricting the solutions to bound modes – the subject of this paper. In a lossless structure bound modes exist for n 2 m>ε D. When absorption in the metal is included, all quantities in the expressions become complexvalued and, as we shall show, the conditions for bound modes become more complicated. In the simulations, Eq. (1) is solved numerically for the bound modes, yielding complex values n m=n mR+in mI.

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.

3. Results

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. (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.

4. Approximate model for the modes

A simple ray model for the SP modes on a nanowire was recently reported [10

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.

], similar to one introduced in 1972, without verification of its validity, for very high mode orders (m>40) [13

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

]. In this model, surface plasmon rays are viewed as travelling on a helical trajectory around the nanowire as indicated in 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.

Such spiralling SPs are in fact inhomogeneous “plane waves” with phase-fronts that are oriented at an angle to the axis (the green lines in Fig. 4), while the direction of exponential decay is along the z-axis. A mode will form when an integral number of azimuthal wavelengths fits around the wire. The model is expected to be more accurate when a is much larger than the skin-depth.

This model leads to a discrete set of modes with axial refractive index:

nm=εDεMεD+εM((m1)k0a)2,m1
(4)

where m is as before the azimuthal mode order. The mode order in Eq. (4) is reduced by 1 to obtain agreement with the results of the exact model. It is interesting that the m=1 (dipolar) mode has an effective index that coincides quite accurately (for large enough wire radius) with that of a planar SP. The m=0 mode has a higher effective index, presumably because, in the absence of azimuthal variations, the formation of a space-charge layer near the surface is constrained by Coulomb effects. This will make Re(ε M) less negative and increase the refractive index above that of a planar SP, as may be shown by evaluating ∂n M/∂ε M=n 3 m/2ε 2 M>0 from Eq. (4). The dipolar m=1 mode, on the other hand, will not be so constrained, because the Coulomb effects are much weaker.

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. 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.

We now examine in detail the limits of this model, concentrating in particular on lower-order modes. Fig. 5 compares the exact and approximate values of n m for two different nanowire diameters. For a=500 nm (Fig. 5(a)) the index of the model clearly follows the dispersion of the exact solution with an increasingly good agreement for higher mode orders, even below the silica line (the curves are terminated at the cut-offs predicted by the exact solutions). The same overall behavior is observed for a=100 nm (Fig. 5(b)), though the agreement is less good. This can be explained by the larger curvature of the nanowire, which renders the local planar approximation increasingly invalid. To evaluate the accuracy of the model, the quasi-cut-offs (n mR=n D) of the guided modes are now examined as a function of nanowire radius.

As a falls, the solutions show the expected reduction in cut-off wavelength (Fig. 6(a)), while the error between the approximate and exact solutions increases. The percentage error 100(1-λapp coex co) increases for nanowire radii down to 100 nm (Fig. 6(b)). This we attribute to an increasing failure in the local planar approximation.

5. Conclusions

References and links

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. HuffmanAbsorption and Scattering of Light by Small Particles (Wiley-VCH, Weinheim, 2004) pp. 194–209.

OCIS Codes
(130.2790) Integrated optics : Guided waves
(240.6680) Optics at surfaces : Surface plasmons
(160.4236) Materials : Nanomaterials
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Optics at Surfaces

History
Original Manuscript: May 27, 2008
Revised Manuscript: August 13, 2008
Manuscript Accepted: August 17, 2008
Published: August 20, 2008

Citation
M. A. Schmidt and P. S. Russell, "Long-range spiralling surface plasmon modes on metallic nanowires," Opt. Express 16, 13617-13623 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13617


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References

  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.

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