## 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

Surface plasmon-polaritons (SPs) form at planar metal-dielectric interfaces because the free electrons in the metal introduce an additional phase change upon reflection that permits a bound state to exist despite the absence of any intervening waveguiding layer. The consequence is that light can be transversely confined in a space that is sub-wavelength in scale relative to the wavelength in the surrounding dielectric, leading to field enhancements that are useful in sensing and nonlinear optics. In recent years there has been growing interest in using SPs to construct nanoscale optical devices that can be tightly packed into very small volumes, the aim being to shrink the size of optical systems by orders of magnitude [1]. One drawback of this proposal is that metals have very high ohmic losses at optical frequencies, causing the SP attenuation to scale inversely with the modal volume. It has been known for many years that a metal film a few 10s of nm thick, sandwiched between two identical dielectrics so as to produce a symmetric structure, supports surface plasmons with greatly reduced attenuation. These “long-range” modes consist of anti-phased surface plasmons on opposite sides of the metal layer, with a field zero in the centre.

In this paper we discuss the characteristics of SP modes guided on circular metallic nanowires embedded in silica glass. Since such nanowires are intrinsically symmetric about their axis, it might be expected that they can also support long-range surface plasmons. Although plasmon modes on single wires have been studied since at least 1910 [2–4], more recent papers either treat absorption-free metals [5, 6] or theoretically discuss modes close to or above the plasma frequency (for silver this occurs at a vacuum wavelength of ~230 nm) [7, 8]. No comprehensive study exists that discusses very narrow wires or the behavior of the SP modes at cut-off (when their axial modal index equals that of the surrounding dielectric) – a region essential in the study of guidance and scattering by two-dimensional metallic nanowire arrays in photonic crystal fibers [9, 10]. In this paper we aim to fill this gap in the literature.

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], 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*

*(complex-valued in general). The nanowire (radius*

_{m}*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, 12]. Starting with Maxwell’s equations in cylindrical coordinates (

*r*,

*ϕ*,

*z*) and matching tangential field components (

*E*

*,*

_{z}*H*

*,*

_{z}*E*

*and*

_{ϕ}*H*

*) at the surface of the nanowire yields, after some manipulation, the following dispersion relation:*

_{ϕ}where *m* is the azimuthal mode order,

where *J*
* _{m}* is a Bessel function of the first kind, and

*K*

*a modified Bessel function. The radial wavevectors*

_{m}*q*

*and*

_{D}*q*

*are defined as follows:*

_{M}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]. 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*

*. 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*

_{D}*n*

*=*

_{m}*n*

*+*

_{mR}*in*

*mI*.

## 3. Results

We begin by discussing the dispersion and the loss of the first five azimuthal mode orders. The real part of the modal index and the attenuation is plotted in Fig. 1 for wires of radius of 500 nm and 100 nm. As expected the dispersion curves shift to shorter wavelengths for smaller wire diameters. There is also a clear overall reduction in loss with increasing wavelength; this is because the modes, as they approach cut-off, spread out more and more into the dielectric, reducing the overlap with the metal.

An intriguing feature of the plots is that bound modes with *m*=2 and higher do not cut-off exactly at the silica line, but rather extend into the range *n*
* _{mR}*<

*n*

*, where in a lossless structure total internal reflection cannot operate. This is a consequence of the large imaginary part of the modal refractive index, which causes the transverse (radial) index*

_{D}*q*

*to become complex-valued, creating the conditions for exponential decay into the dielectric and thus field confinement; if Im(*

_{D}*ε*

*) is set to zero, the cut-off points move exactly to the silica line. A further striking feature is that the attenuation becomes arbitrarily small (*

_{M}*n*

*→0) as cut-off is approached – a result of the modal field spreading out strongly into the dielectric. For example, the*

_{mI}*m*=3 mode shows a remarkably small effective index

*n*

_{3}

*=1.18 at cut-off (*

_{R}*a*= 100 nm). For the

*m*=4 mode the influence of Im(

*ε*

*) is so strong that*

_{M}*n*

_{4}

*reaches zero before cut-off, resulting in a loss curve that ends at a non-zero value. Below the silica line, the modal field amplitude*

_{R}*E*

*has out-of-phase oscillating real and imaginary parts, its field modulus |*

_{z}*E*

*| decaying evanescently away from the wire (Fig. 2(a)&(b)). In all cases, the transverse Poynting vector in the dielectric points towards the wire, as expected of a bound mode that loses energy to ohmic losses in the metal. These results show that Im(*

_{z}*ε*

*) has a critical influence on the properties of the SP dispersion and cannot be ignored as in many previous studies [5, 6].*

_{M}The *m*=0 and *m*=1 modes behave quite differently from the higher order modes. Their dispersion curves remain above the silica line for all nanowire radii and wavelengths, approaching it more and more closely as the wavelength increases (Fig. 3(a)). This effect is more pronounced for smaller nanowires. Interestingly, the corresponding losses (inset of Fig. 3(a)) decrease significantly for increasing wavelength in both cases. To our knowledge, this remarkable effect has not been reported before. The radial width of the *m*=1 field in the dielectric can be estimated as *δ*=*λ*[*π*
*R*e(*q*
* _{D}*)]

^{-1}. This quantity is plotted in Fig. 3(b), together with the associated loss (Fig. 3(c)), showing, e.g., that a nanowire with a radius of 200 nm has loss 9 dB/mm and modal width 2.4 µm at 1550 nm wavelength. This shows that long propagation lengths can be reached at reasonably small mode volumes. For wavelengths greater than 1500 nm the losses of this mode are competitive with those of long-range surface plasmons on thin metal films. The loss of the

*m*=0 mode increases for smaller wire diameters, whereas that of the

*m*=1 mode increases, allowing us to identify these modes as the nanowire equivalents of the short-range (even) and long-range (odd) SPs modes that exist on thin metal films sandwiched between two regions of identical dielectric material.

## 4. Approximate model for the modes

A simple ray model for the SP modes on a nanowire was recently reported [10], similar to one introduced in 1972, without verification of its validity, for very high mode orders (*m*>40) [13]. In this model, surface plasmon rays are viewed as travelling on a helical trajectory around the nanowire as indicated in Fig. 4.

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:

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}

*/2*

_{m}*ε*

^{2}

*>0 from Eq. (4). The dipolar*

_{M}*m*=1 mode, on the other hand, will not be so constrained, because the Coulomb effects are much weaker.

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*

*) of the guided modes are now examined as a function of nanowire radius.*

_{D}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-λ^{app}
_{co}/λ^{ex}
_{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

The SP modes guided on metallic nanowires display many interesting features. For example, higher order bound modes (*m*>1) can exist for which the real part of *n*
* _{m}* is less than that of the surrounding dielectric; this occurs when

*n*

*is non-zero. As these modes approach cut-off, beyond which point they are no longer bound to the wire, their losses go to zero in most cases. Since these modes are located below the silica line, they could phase-match to the guided mode of a solid-core air-silica photonic crystal fiber [14]. They could even be used to access guided modes above the silica line if a tapered nanowire with gradually increasing diameter were used. The*

_{mI}*m*=1 mode displays remarkably low loss and small modal volume over a broad range of wavelengths, with no cut-off point. This mode could have applications in plasmonic circuitry as a cylindrical replacement for long-range surface plasmons. Finally, a simple ray-model, in which the SP modes on the wires are viewed as being built up from spiralling surface plasmons, is shown to be better than 5% accurate down to wire radii of 100 nm, the error improving for increasing mode order.

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