## Abstract

We theoretically investigate composite cylindrical nanowires for the waveguiding of the lowest-order surface plasmon-polariton (SPP) mode. We find that the confinement of the SPP fields in a metallic nanowire can be significantly improved by a dielectric cladding and show that by adjusting the thickness of the optically-pumped cladding, the gain required to compensate for the losses can be minimized. If this structure is coated with an additional metal layer to form a metal–dielectric–metal (MDM) nanowire, we show that the field can be predominantly confined within the dielectric layer, to have amplitudes of three orders of magnitude higher than those in the metallic regions. We also show that the propagation lengths of SPPs can be maximized by the proper selection of the geometrical parameters. We further demonstrate that the mode is strongly confined in subwavelength scale, e.g.,
$\sim {\lambda}_{0}^{2}/1220$ for a 60-nm-thick nanowire, where *λ*
_{0} is the wavelength in vacuum. We also find that regardless of the size of nanowire, it is possible to carry over 98.5% of the mode energy within the nanowire. In addition, we demonstrate that by appropriate choice of the material thicknesses, the losses of an MDM nanowire can be compensated by a considerably low level of optical gain in the dielectric region. For example, the losses of a 260-nm-thick Ag–ZnO–Ag nanowire can be entirely compensated by a gain of ∼ 400 cm^{−1}. Our results will be useful for the optimum design of nanowires as interconnects for high-density nanophotonic circuit integration.

© 2011 OSA

## 1. Introduction

Manipulating light in nanoscale is challenging, as light cannot be guided within structures having dimensions smaller than approximately half of its wavelength, commonly known as the diffraction limit [1]. However, light can be used to excite surface plasmon-polaritons (SPPs)—a form of surface waves which have much smaller effective wavelength at the same frequency—that can propagate in nanoscale dimensions [2, 3]. These waves represent collective oscillations of photons and electrons, and travel along metal-dielectric interfaces [4]. The ability to approach the nanoscale with optical signals opens up a wide range of promising applications including bio-sensing [5], scanning near-field optical microscopy (SNOM) [6, 7], surface enhanced raman spectroscopy (SERS) [8], and the realization of nanophotonic circuit elements such as interconnects, modulators, couplers, and switches [9–11]. For the efficient operation of these devices, subwavelength guiding of SPPs is essential, which can be achieved using plasmonic waveguides [12]. Several geometries have been proposed for plasmonic waveguiding such as metal films [13], planar metal-dielectric-metal waveguides [14], cylinders of circular [15] and square [16] cross-section, triangular metal wedges [17], grooves [18], dielectric gaps [19, 20], and coupled metal nanoparticles [21]. The circular cylindrical structure is particularly known for guiding electromagnetic energy, e.g., optical fibers in telecommunication applications [22] and coaxial cables in power transmission [23]. The availability of precise fabrication techniques at nanoscale [24] has made the cylindrical nanowaveguide (nanowire) a promising candidate for guiding optical signals in subwavelength dimensions [25], and has allowed it to be considered as a model for probes in high resolution optical microscopy applications [26] and as optical interconnects for nanophotonic circuit integration [15, 27].

The propagation losses play a crucial role when determining the performance of a plasmonic waveguide. These losses occur mainly due to the absorption of metal, which may limit the propagation length of SPPs to a few or tens of micrometers [28]. Therefore, a significant portion of the mode must be kept outside the metallic parts of the waveguide in order to reduce the effects of losses [29, 30]. However, because the metals in such a waveguide are essential to sustain and guide SPPs, losses due to interaction with metallic parts of the waveguide cannot be completely eliminated. The remedy is to provide optical gain to adjacent dielectrics of these metallic regions as shown in Ref. [31]. Many studies have been done where dielectrics with gain have been investigated to reduce the losses in various waveguide geometries; e.g., planar metallic waveguides [32], gap waveguides [19], and cylindrical nanowires [25]. It is always possible to overcome the losses by intensifying the pump-power, but in certain cases this could be highly challenging from a practical point of view, due to various limitations imposed by factors such as the material properties of the gain medium and the temperature of operation. Thus, it is essential to explore means of reducing the effects of losses by optimal design of the waveguide, so that the losses could be compensated with lower gain levels. In Ref. [25], we demonstrated that the gain required for a metal–dielectric nanowire can be minimized by adjusting its geometrical parameters.

In this paper, we analyze composite nanowires for the propagation of SPPs, and develop important design guidelines that are critical for the realization of high-performance optical interconnects. To start with, in Section 2, we present the general theory of SPPs in a composite cylindrical nanowire and define the parameters that are used throughout the paper. We consider the simple case of a metallic nanowire in Section 3, where we discuss its drawbacks as an optical waveguide and the need for more complex nanowire structures. In Section 4, we find that the performance of a metallic nanowire can be enhanced by a dielectric cladding. In addition, we consider the compensation of propagation losses by optical gain in the cladding, and show that the losses of the nanowire can be overcome with minimal gain by adjusting the thickness of the cladding to an optimal value. By considering a metal–dielectric–metal (MDM) structure in Section 5, we reveal that the field confinement can be significantly improved and the majority of the mode energy can be carried within the nanowire. We further show that by appropriately choosing the material thicknesses, the propagation length of SPPs can be maximized. We find that the amplitudes of the lowest-order SPP fields in the dielectric layer can be as high as three orders of magnitude larger than those in the metallic regions. By considering optical gain in the dielectric layer, we show that optimum geometrical parameters exist that ensure the compensation of losses with minimal pump-power. Finally, we devise important design guidelines that ensure high-performance of the composite nanowires as optical waveguides.

## 2. Theory of SPPs in Composite Cylindrical Nanowires

With decreasing diameter of the cylindrical waveguide, the number of supported modes decrease, and only the fundamental (azimuthally symmetric) transverse magnetic (TM) SPP mode exists where the diameter is extremely small [33]—the case which is assumed in this paper. In cylindrical coordinates *ρ* and *z* (see Fig. 1), the electric field of this mode, associated with SPPs propagating in the *z* direction with propagation constant *β*, is of the form [34, 35]

**e**

*and*

_{ρ}**e**

*are the unit vectors along*

_{z}*ρ*and

*z*directions. Assuming that the function exp(−

*iωt*) governs the time dependency of the fields, the

*φ*component of the magnetic field can be represented as

*k*=

*ω*/

*c*,

*ω*is the SPP frequency, and

*c*is the speed of light in vacuum.

The electric and magnetic fields in the *j*th medium of the composite nanowire shown in Fig. 1 are given by the solution of Maxwell’s equations as [34]

*α*= (

_{j}*β*

^{2}–

*ɛ*

_{j}k^{2})

^{1/2},

*χ*(

_{j}*x*) = d[

*ψ*(

_{j}*x*)]/d

*x*, and the function

*ψ*(

_{j}*x*) is given by the expression

*A*and

_{j}*B*are the complex constants;

_{j}*I*

_{0}(

*x*) and

*K*

_{0}(

*x*) are the modified Bessel functions of the first and second kinds, respectively. It should be noted that

*B*

_{1}= 0 and

*A*= 0, since the fields must be finite at

_{n}*ρ*= 0 and vanish at

*ρ*→ ∞.

The real and imaginary parts of *β* determine the wavelength and the attenuation coefficient of the SPPs, respectively. The value of *β* can be found by the solution of the dispersion equation that is derived by ensuring the continuity of the tangential components of the fields at material interfaces [35]. At an arbitrary interface *ρ* = *R _{j}*, the following relations can be obtained by considering the field components

*E*and

_{z}*H*

_{φ}To simplify the analysis, we assume the nanowires to be smooth, and neglect the scattering losses arising from surface roughness. We note that the dielectric losses can be easily neglected at the telecommunication wavelengths, and it is the metallic losses that contribute to the attenuation of SPPs. These losses are characterized by the collision frequency (*δ*) of the free electrons, in the Drude formula
${\varepsilon}_{\text{m}}\hspace{0.17em}(\omega )\hspace{0.17em}\approx \hspace{0.17em}{\varepsilon}_{\infty}\hspace{0.17em}[1\hspace{0.17em}-\hspace{0.17em}{\omega}_{\text{p}}^{2}/({\omega}^{2}\hspace{0.17em}+\hspace{0.17em}i\delta \omega )]$, where *ɛ*
_{∞} is the high-frequency dielectric constant and *ω*
_{p} is the plasma frequency [25]. The propagation length of SPPs *L*
_{SPP} is defined as the distance at which the amplitudes of the SPP fields attenuate by a factor of 1/e, i.e., *L*
_{SPP} = 1/(2Im*β*) [4]. A reasonable mearsure of the losses in a nanowire can be obtained by the quantity *L*
_{SPP}. Typically, high propagation lengths of SPPs are associated with poor mode confinement, and one should also consider the degree of mode confinement to assess the performance of a nanowire [36]. An estimate of the relative degree of confinement can be obtained by the parameter *η*, which is calculated by normalizing the mode area by the diffraction limited area in vacuum (*λ*
_{0}/2)^{2} = (*π*
*/k*)^{2} [29], i.e.,

*R*

_{0}= 0,

*R*= ∞, and

_{n}*w*(

_{j}*ρ*) is the electromagnetic energy density [37] given by

*w*(

_{j}*ρ*) corresponding to metallic regions, the losses in metal were readily neglected, since

*δ*≪

*ω*at the telecommunication wavelengths that are of interest. However, one can even use a weakly lossy model of permittivity but then the permittivity used in the energy density formula needs to be replaced by the real value of the permittivity. Because none of these derivations rely on a specific feature/form of a permittivity model that we adopted for metals of interest, it is even possible to replace them completely by a curve-fitted model matching exact experimental data. For example, seven pole Drude-Lorentz model [38] can be used here if desired.

The applicability of the nanowire as an interconnect in nanophotonic circuitry largely relies on the extent of electromagnetic interference it may have with the nearby components. We obtain an estimate for such interference via the parameter power containment factor *ζ*, which gives the fraction of modal power carried within the nanowire. The power flow inside the nanowire is calculated by integrating the component of the Poynting vector (time-averaged) in the *z* direction over the cross-sectional area of the nanowire. Thus *ζ* can be written as

*ζ*close to 100%.

In what follows, we consider different nanowire structures and investigate their suitability as efficient SPP waveguides. We first analyze the pure metallic nanowire and then examine the effects of (1) a dielectric cladding, and (2) a dielectric cladding and an outer metal cladding, on its performance.

## 3. Pure Metallic Nanowire

The pure metallic nanowire is the simplest cylindrical structure that enables SPP guiding. It is useful to explore the limitations of this simple structure as a nanowaveguide, which raises the need for more complex nanowire geometries.

We consider an infinitely long metallic nanowire of radius *R*
_{1} having permittivity *ɛ*
_{1}, embedded in an infinitely extending medium of dielectric permittivity *ɛ*
_{2}. According to Eq. (1)
, the amplitudes of the electric field and the magnetic field are:

*A*

_{1}and

*A*

_{2}are the complex constants. The dispersion of the SPPs can be obtained by ensuring the continuity of the tangential components (

*z*and

*φ*) of the fields at the nanowire surface (

*ρ*=

*R*

_{1}). This leads to the dispersion equation

Assuming real SPP frequencies, we solve Eq. (3) to calculate the dispersion of SPPs. For a Ag nanowire surrounded by air, the dispersion is shown by the solid blue curve in Fig. 2(a). We focus our attention to low SPP frequencies, for which the absorption losses of the dielectrics and metallic losses are comparatively less. In this case, the wave vectors of SPPs are approximately the same as that of light, and the wavelengths of the SPPs (*λ*
_{SPP}) are close to the wavelength of light in air (*λ*
_{0}) (shown by the solid red curve). As a result, to a great extent, the SPPs show light-like behavior and are not well confined to the interface. This is evidenced by the variation of the electric field along the nanowire shown in Fig. 2(b), calculated at the wavelength *λ*
_{0} = 1.55 *μ*m (*ω* ≈ 0.21*ω*
_{p}).

For application of the pure metallic nanowire as an interconnect in nanophotonic circuitry, poor field confinement is a serious disadvantage, considering the high electromagnetic interference it may cause to the nearby components. For example, for the case of 200-nm-thick metallic nanowire shown in Fig. 2(b), *ζ* is as low as ∼ 0.02%. It is clear that *ζ* will remain in similar order of magnitude regardless of the nanowire diameter, because the penetration of the fields into the metal is significantly less than that into the surrounding dielectric [28]. While such high interference remains unavoidable, we note that by reducing the diameter of the nanowire, the mode area can be reduced. However in this case, the associated losses become significantly high and the propagation length of SPPs decreases drastically. For example, when the diameter is decreased from 100 nm to 10 nm, the mode area reduces from ∼ 0.01 to ∼ 0.002, but at the same time, *L*
_{SPP} decreases from 260 nm to 10 nm. To reduce such losses, one may consider embedding the metallic nanowire in a dielectric medium that can be pumped to provide optical gain. Yet, this will increase the size of the circuit, and thus will limit the applicability of the device in nanoscale.

## 4. Metallic Nanowire with a Single Dielectric Cladding

In order to overcome the limitations of the pure metallic nanowire, we investigate the effect of a dielectric cladding. We would like to note that this is the inverse geometrical structure of the metal–dielectric nanowire we considered in Ref. [25], which consisted of a dielectric core and a metallic cladding. For the metal–dielectric nanowire, we showed that an optimal cladding thickness exists, for which the metallic losses can be fully compensated with a minimum amount of optical gain inside the dielectric. Here, we show that similar optimization is possible for the metallic nanowire with dielectric cladding, and find that for the same outer radius, it is possible to overcome the losses with comparatively low levels of gain.

From Eq. (1) , the amplitudes of the electric and magnetic fields in a metallic nanowire with a dielectric cladding (see Fig. 3) are

*A*and

_{j}*B*are the complex constants corresponding to

_{j}*j*th medium and

*ɛ*

_{1}=

*ɛ*

_{m}.

By ensuring the continuity of the tangential components of the fields at the interfaces *ρ* = *R*
_{1} and *ρ* = *R*
_{2}, the dispersion equation can be derived as

The dispersion of the SPPs in a Ag–ZnO nanowire surrounded by air is shown by the dashed blue curve in Fig. 2(a). With comparison to the case of pure metallic nanowire, the wavevectors of SPPs are considerably higher than that of light, and the wavelengths of SPPs are much shorter (see the dashed red curve). The improved confinement of SPP fields in this case can be readily observed from Fig. 2(c). In this case, the value of *ζ* is ∼ 67%, which marks a significant boost from the value *ζ* ≈ 0.02% reported for the pure metallic nanowire of the same size.

We note that for fixed *R*
_{2}, the maximum propagation length of SPPs is achieved in the limit *R*
_{1} → *R*
_{2}. This implies that for any finite dielectric thickness, *L*
_{SPP} is lower in the metallic nanowire with dielectric cladding, compared to the pure metallic nanowire. The only possible way of overcoming this problem is the compensation for losses in metal at optical frequencies by supplying gain to the cladding [40]. The gain is generally enabled by pumping the dielectric either optically or electrically [41, 42]. For the Ag–ZnO nanowire, the desired gain may be readily achieved via optically pumping the nanowire in the radial direction. Even though several experimental work exist that demonstrate optical gain in ZnO at wavelengths in the vicinity of its bandgap (∼ 3.3 eV [43]) [44, 45], realizing gain at *λ*
_{0} = 1.55 *μ*m (≈ 0.8 eV) could be challenging. A potential way to overcome this difficulty would be to dope ZnO with Er ions—the same method that was employed to achieve optical emission in the 1.55-*μ*m band [46, 47]. To model a specific gain *γ*, we take dielectric permittivity of the form [25]

Gain counteracts to compensate for the metallic losses and slows the attenuation of SPPs. Figure 4(a) shows the effect of gain on *L*
_{SPP} in a 200-nm-thick nanowire, for several values of the relative cladding thickness *q* = 1 – *R*
_{1}/*R*
_{2}. For comparison, the propagation length of SPPs in a similar size metallic nanowire surrounded by air is shown by the dashed line. The *L*
_{SPP} for the latter case is a constant value since this setup has no provision to add gain. One may observe that for certain values of *q*, propagation lengths higher than that of the pure metallic nanowire surrounded by air, can be achieved by gain levels below 1000 cm^{−1}. We note that losses can be entirely compensated and lossless propagation of SPPs (*L*
_{SPP} → ∞) can be achieved at a certain gain value *γ _{c}*, known as the critical gain. For

*γ*>

*γ*, we find that Im

_{c}*k*< 0, and thus the SPPs are amplified due to optical pumping. It is significant that the value of

*γ*corresponding to

_{c}*q*= 60% is less than of those observed at

*q*= 40% and

*q*= 80%, implying that

*γ*becomes minimal at a specific relative cladding thickness of the nanowire. We refer to this value as the optimum relative cladding thickness (

_{c}*q*

_{0}) of the nanowire, and the minimum value of

*γ*is denoted by

_{c}*γ*

_{0}. This variation is clearly understood from Fig. 4(b) where the critical gain is plotted as a function of

*q*, for different nanowire sizes. It can be observed that the pump-power required to overcome the losses becomes significantly less, when the thickness of the cladding is appropriately chosen. For example, the losses in a nanowire of

*R*

_{2}= 100 nm can be entirely compensated by a gain of ∼ 882 cm

^{−1}at

*q*= 67%, whereas for the same nanowire size, a pump power of more than five times this value (∼ 4610 cm

^{−1}) is needed at

*q*= 10%.

Henceforth, we analyze the performance of this nanowire at the optimal relative cladding thickness. We note that when the nanowire diameter is relatively small, the thickness of the metal which correspond to *q*
_{0} is in the order of a few nanometers. In this case, the bulk material value of *δ* is inadequate to accurately describe the collision frequency of the electrons, as one needs to take into account the increase of *δ* due to the larger surface-to-volume ratio. Thus, we understand that the accuracy of our results may diminish when *R*
_{2} ≲ 50 nm, where the metal thickness is less than 6 nm [48]. Throughout this paper, we denote the corresponding regions of inapplicability by shading the relevant areas in the figures (e.g., Fig. 5).

From the blue curve in Fig. 5(a), it can be seen that for nanowires with smaller diameters, more intense pumping is required to overcome the losses. This variation is explained by considering the confinement of the SPP fields, which increases with decreasing *λ*
_{SPP}, and vice versa. As shown by the red curve, small nanowires have lower *λ*
_{SPP}, which means that the mode is strongly confined to the metal and suffers from high losses. In contrast, *λ*
_{SPP} is relatively high in larger nanowires, and owing to the poor mode confinement, the losses are less. Another contributing factor for *γ*
_{0} is the thickness of the dielectric layer which provides gain. With decreasing nanowire size, the optimum dielectric thickness also decreases [see the red curve in Fig. 5(b)], and the pump-power needed to reach *γ*
_{0} becomes higher. However, it should be noted that, although the gain levels needed for small nanowires are high, quantitatively the values are comparatively lower than those reported for similar size metal–dielectric nanowires [25]. For example, the gain required for a 100-nm-thick metallic nanowire with dielectric cladding is ∼ 1320 cm^{−1}, whereas a gain of ∼ 2750 cm^{−1} is needed for a metal–dielectric nanowire. The variation of the optimum relative cladding thickness with nanowire size is shown in Fig. 5(b). It is worth noting that *q*
_{0} remains almost constant around ∼ 65% within the range 100 nm ≲ *R*
_{2} < 300 nm, which can be used as a simple design guideline for the optimum design of nanowires.

Figure 5(c) shows the variation of *ζ* with nanowire size. It is significant that over 90% of the mode power can be carried within the nanowire for 50 nm < *R*
_{2} < 75 nm, but the value of *ζ* drops below 80% for nanowires with 80 nm < *R*
_{2} < 260 nm. Thus, we find that the operation of this nanowire as an optical interconnect is highly efficient only within a narrow range of sizes. Hence, it is important to investigate means of improving the field confinement.

## 5. MDM Nanowire

It is reasonable to expect strong confinement of the SPP fields in the metallic nanowire with dielectric cladding, when it is coated with an additional metal cladding. This would ensure that the majority of the mode energy is concentrated within the nanowire, and will minimize the interference to the nearby components.

For a composite nanowire having two shells (e.g., Fig. 6), the field amplitudes can be written as

*A*,

_{j}*B*are the complex constants associated with

_{j}*j*th medium and

*R*

_{2}=

*R*

_{1}+

*d*. After matching the tangential components of the fields at

*ρ*=

*R*,

_{j}*j*= 1, 2, 3, we obtain the dispersion relation for the SPPs

*K*(

_{p}*α*) and

_{a}R_{b}*I*(

_{p}*α*), respectively.

_{a}R_{b}For the MDM nanowire with *ɛ*
_{1} = *ɛ*
_{3} = *ɛ*
_{m}, the solution of Eq. (5) gives three dispersion branches, as shown by Fig. 7(a). From an application point of view, only the two dispersion branches passing through the origin are important, as the other branch is cutoff at the telecommunication frequencies that are of interest. The dispersion branches can be characterized according to the concentration of SPP energy at different interfaces of the nanowire. For example, at *λ*
_{0} = 1.55*μ*m, the solution of the lower dispersion branch [point B in Fig. 7(a)] corresponds to the case where majority of the SPP energy is localized at the inner metal-dielectric interface (*ρ* = *R*
_{1}) [see the blue curve in Fig. 7(b)]. In contrast, the solution of the upper branch [point A in Fig. 7(a)] leads to the case where SPP energy is predominantly localized at the nanowire surface (*ρ* = *R*
_{3}) [shown by the red curve in Fig. 7(b)]. The values of *ζ* corresponding to the above two cases are 99.8% and 0.1%, respectively. This suggests that the solutions of the lower dispersion branch allows extremely high mode confinement, resulting in negligible interference to the neighborhood. Hence, in our analysis, we focus only on the solutions of the lower dispersion branch. We note that when *R*
_{2} ≫ *R*
_{1}, this dispersion branch is similar to that of a pure metallic nanowire of radius *R*
_{1} embedded in a medium of dielectric permittivity *ɛ*
_{2} [49].

To reduce the effect of metallic losses on the propagation of SPPs, it is vital to minimize the field concentration inside the metallic regions and confine most of the mode energy within the dielectric. Using this approach, significantly high propagation lengths of SPPs was demonstrated for hybrid plasmonic waveguides [29], where the mode energy was confined within a nanoscale dielectric gap between a metal plane and a dielectric cylinder. Similarly, strong field confinement within a dielectric layer was reported for the fundamental mode of a cylindrical hybrid plasmonic waveguide, a structure with a metal core and two dielectric cladding layers [30]. Using a Ag–SiO_{2}–Si waveguide, the authors demonstrated the possibility of achieving significantly high field concentration in the SiO_{2} layer, considering nanowires with diameters larger than 400 nm. However, since the lateral size of such a nanowire is relatively large, its applicability in nanoscale circuit integration is limited.

In order to examine the field confinement of smaller cylindrical hybrid plasmonic waveguides, we solve Eq. (5) for the same parameters given in Ref. [30], when the diameter is 200 nm. The electric field distribution along a longitudinal cross-section of the Ag–SiO_{2}–Si cylindrical hybrid plasmonic waveguide is shown in Fig. 8(a). It can be observed that for this nanowire size, the electric field is not predominantly confined to the SiO_{2} layer, and the confinement is much weaker than that for a 700 nm-thick waveguide shown in Fig. 3 of Ref. [30]. From the blue curve in Fig. 8(c), we note that the amplitude of the electric field in the surrounding medium is comparable to that in the SiO_{2} layer, which leads to *ζ* ≈ 43%. Hence we find that when the diameter is relatively small, cylindrical hybrid plasmonic waveguide causes considerable interference, and therefore is less favorable for nanophotonic applications. We show that this limitation can be easily overcome by replacing the outer dielectric cladding with metal. As evidenced by Fig. 8(b), the metal cladding significantly reduces the extension of the fields to the surrounding medium, and ensures extremely high field confinement within the nanowire. From Fig. 8(c), it can be observed that the amplitude of the electric field in the dielectric region is one order of magnitude higher than that in the surrounding medium, and almost three orders of magnitude higher than that in the metal region. The value of *ζ* corresponding to this case is ∼ 99.8%, which indicates negligible interference to the neighborhood.

Propagation of SPPs along the interface *ρ* = *R*
_{1} is not supported if *R*
_{1} = 0 or *R*
_{1} = *R*
_{2}, due to the absence of either the metal or the dielectric region. Hence, the propagation length of SPPs should tend to zero when *R*
_{1} → 0 or *R*
_{1} → *R*
_{2}. One may reasonably argue that at a certain value of *R*
_{1} between these two extremities, *L*
_{SPP} should be maximum, implying an optimum propagation condition for the SPPs in the nanowire. From Figs. 9(a) and 9(b), it can be observed that *L*
_{SPP} becomes maximum in this range, and the above argument is true. In Fig. 9(c), *L*
_{SPP} is plotted as a function of both the parameters *R*
_{1} and *d* for a fixed nanowire size, and it is seen that by appropriate choice of these parameters, higher *L*
_{SPP} can be realized. The maximum achievable propagation length of SPPs is shown in Fig. 9(d), which indicates that longer SPP propagation lengths can be realized at the expense of larger nanowire size.

In order to comprehensively evaluate the performance of the MDM nanowire, it is useful to examine the mode confinement of the nanowire at the parameters for which the propagation length of SPPs becomes maximum. From the blue curve in Fig. 9(e), it can be seen that the SPP mode is confined well below the diffraction limit. We find that the mode confinement is extremely strong when the diameter is small (
$\sim \hspace{0.17em}{\lambda}_{0}^{2}/1220$ for *R*
_{3} = 30 nm) and remains satisfactorily strong even for large diameters (
$\sim \hspace{0.17em}{\lambda}_{0}^{2}/80$ for *R*
_{3} = 200 nm). The red curve shows the parameter *A*
_{d}, which is defined as the cross-sectional area of the dielectric cladding normalized to the diffraction limited area in vacuum, i.e.,
${A}_{\text{d}}\hspace{0.17em}=\hspace{0.17em}\pi ({R}_{2}^{2}\hspace{0.17em}-\hspace{0.17em}{R}_{1}^{2})/{({\lambda}_{0}/2)}^{2}$. This quantity is the hypothetical limit of 100% mode confinement within the dielectric region. It is observed that *η* is approximately the same as *A*
_{d} when the nanowire size is relatively small, which reflects the fact that the mode is predominantly confined within the dielectric cladding. The difference between *A*
_{d} and *η* becomes more prominent for large nanowire sizes. This can be explained by considering the variation of electromagnetic energy density within the dielectric region [*w*
_{2}(*ρ*)], while noting that its maximum value occurs at *ρ* = *R*
_{1}. According to Eq. (2), if *η* is calculated by assuming that *w*
_{2}(*ρ*) remains constant at its maximum value throughout *R*
_{1} < *ρ* < *R*
_{2}, it is equal to *A*
_{d}. However in reality, *w*
_{2}(*ρ*) decays away from the interface *ρ* = *R*
_{1}, and this accounts for the difference between *A*
_{d} and *η*. With increasing nanowire size, the optimum value of the dielectric thickness increases, and hence the decay in *w*
_{2}(*ρ*) within the dielectric region becomes more noticeable. As a result, the difference between *A*
_{d} and *η* becomes more pronounced for larger nanowires. Nevertheless, we note that *η* remains in the same order of magnitude as *A*
_{d}, and hence the mode confinement within the dielectric cladding is considerably high even in this case. The extremely high field confinement ensures that more than 98.5% of the mode power is carried within the nanowire, regardless of its size [see Fig. 9(f)].

Although the geometrical parameters of the nanowire can be chosen to maximize the propagation length of SPPs, even this maximum value can be inadequate for certain applications, particularly when the diameter is small. To overcome this limitation, one can pump the dielectric cladding to provide gain. The required gain can be realized by electrical pumping, using the two metal regions as electrodes. Alternatively, the dielectric region can be optically pumped from the radial direction, if the metal cladding is sufficiently thin [25]. Assuming that the permittivity of the dielectric is given by Eq. (4), we calculate the gain coefficient that corresponds to the case where the metallic losses are entirely compensated by optical gain (*γ* = *γ _{c}*). The variation of

*γ*for a 200-nm-thick Ag–ZnO–Ag nanowire is shown in Fig. 10(a), as a function of

_{c}*R*

_{1}and

*d*. It is noted that lossless propagation of SPPs can be achieved with significantly less pump-power, by properly choosing the nanowire parameters. It is interesting to note that

*γ*remains considerably low over a wide range of geometrical parameters (e.g.,

_{c}*γ*≲ 700 cm

^{−1}for 10 nm <

*d*< 70 nm and 20 nm <

*R*

_{1}< 70 nm). In large nanowires, the losses can be compensated with less gain, as evidenced by the blue curve in Fig. 10(b). For example, the red curve shows the amount of gain required for metallic nanowires with dielectric cladding, having the same outer radius. We note that the gain needed for the MDM nanowire is always lesser, and for certain sizes it can even be approximately half of that required by the metallic nanowire with dielectric cladding (e.g., for a nanowire with a diameter of 160 nm, 658 cm

^{−1}vs 1195 cm

^{−1}).

Based on our numerical results, several general conclusions can be drawn, which are important for the design of efficient composite nanowires. For relatively large metallic nanowires with dielectric cladding, the optimum relative cladding thickness remains almost constant over a wide range of nanowire sizes (*q*
_{0} ≈ 65% for Ag–ZnO nanowires with 100 nm ≲ *R*
_{2} < 300 nm). Thus, for optimum design of the nanowire, the dielectric thickness can simply be determined via scaling the nanowire radius by the above *q*
_{0} value. If the focus is on containing majority of the mode energy within the nanowire to minimize the interference—as desired for optical interconnects—the diameter can be increased, or alternatively, the diameter can be decreased with larger pump-power. For any nanowire size, the extension of the fields to the surrounding medium can be almost entirely eliminated by coating the nanowire with a metal layer. This also helps to considerably lower the gain required to overcome the losses in the nanowires, particularly when the nanowire size is relatively small.

It is important to examine the feasibility of fabricating the aforementioned composite nanowire structures with existing technology. Recent experimental results reveal evidences of dielectric coated metal nanowires of subwavelength dimensions. For example, in the work of Ref. [50], Ag nanorods coated with 15-20 nm thick ZnO layers were demonstrated, while maintaining a smooth interface between the two materials. The diameters of the reported composite nanorods were ∼ 100 nm, which is of the same order of diameter of the proposed metallic nanowires with dielectric cladding. The realization of MDM nanowires with material thicknesses in the nanometer range could be challenging, but may be achieved with the advancements in materials engineering technology.

## 6. Conclusions

In this paper, we have analyzed cylindrical composite nanowires and developed key design guidelines that ensure efficient propagation of SPPs. We showed that the confinement of SPP fields can be significantly improved if the metallic nanowire is coated with a dielectric layer. In addition, we considered reducing the propagation losses by pumping the doped cladding to achieve optical gain, and showed that the losses of the nanowire can be entirely compensated with minimal gain by adjusting the thickness of the dielectric cladding. If the nanowire is fabricated with an additional metal cladding, we demonstrated that excellent field confinement can be achieved inside the sandwiched dielectric layer, with amplitudes of the lowest-order SPP fields as high as three orders of magnitude larger than those in the metallic regions. We further showed that the thicknesses of the metallic core and the dielectric layer can be chosen to maximize the propagation length of SPPs, and the mode is strongly confined in subwavelength dimensions. We found that regardless of the nanowire size, the fraction of mode energy carried within the nanowire is close to 100%. By considering optical gain in the dielectric layer, we showed that for a specific choice of geometrical parameters, the losses can be overcome with minimal pump-power. Our results may prove useful in realizing high performance optical interconnects for nanowire-based high-density nanophotonic circuit integration.

## Acknowledgments

The work of M. Premaratne and I. D. Rukhlenko is sponsored by the Australian Research Council (ARC) through its Discovery Grant scheme under grant DP110100713. C. Jagadish also gratefully acknowledges the Australian Research Council for financial support.

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