Light scattering, field localization and local density of states in co-axial plasmonic nanowires

Nate Lawrence; Luca Dal Negro

doi:10.1364/OE.18.016120

1. Introduction

Nanoscale structures consisting of layered metal and dielectric interfaces in different geometries have received much attention in the last few years due to the possibility of engineering plasmonic mode coupling, sub-wavelength localization and transport phenomena with unprecedented flexibility across wide spectral regions [1–6]. In particular, it was shown that planar Metal-Insulator-Metal (MIM) slot structures support highly localized electromagnetic fields confined in nanoscale insulator regions due to the excitation of coupled plasmon modes [1–3], and optical devices based on this concept have been fabricated in relation to sub-wavelength guiding of polariton fields [7]. Moreover, circularly symmetric MIM plasmonic nanoshells and ring-disk structures have also been largely investigated in the context of hybridization plasmon theory due to their ability to strongly localize polariton fields across the visible and near-IR spectrum by accurate tuning of their geometrical parameters [8–10]. The sub-wavelength field confinement achieved by these structures make them particularly attractive for a variety of nanoscale sensing and light emitting device applications based on the well-established nanowires platform [11–14]. It is also well known that the local density of photon states (LDOS) can be drastically altered by dielectric as well as metal-dielectric (plasmonic) structures which can either suppress or enhance radiation from dipoles (atoms, molecules) and semiconductor materials. To this regard, both weak and strong photon coupling regimes have been demonstrated using photonic nanostructures [15–17]. More recently, radiative rate enhancement through LDOS engineering in nanoscale slot (gap) regions [18–20] and quasi-periodic nanoplasmonic arrays [21] have also been demonstrated and show great promise for improving the light emission efficiency of indirect bandgap semiconductors, such as Si, which leverage on inexpensive processing technology for high density photonic-plasmonic integration.

In this paper, by using an efficient multipole expansion method for the rigorous calculation of fields originated by arbitrarily located sources in cylindrically-symmetric multilayered media, we systematically investigate the frequency dependent scattering spectra, near-field and LDOS enhancement spectra in concentric MIM cylindrical nanostructures, or coaxial plasmonic nanowires (CPNs). In particular, by carefully exploring the geometrical parameter space of the proposed CPNs, we show that it is possible to create sub-wavelength radially-polarized plasmon modes with large electromagnetic field enhancement distributed over multiple spectral bands and vastly tunable scattering resonances spanning across the visible and near-infrared (IR) spectral range. Interestingly, we also discover that the far-field scattering efficiency in these systems can be resonantly suppressed in proximity of distinctive localized modes that can be flexibly engineered across a wide spectral region depending on the geometrical parameters and dielectric index. Finally, we demonstrate that CPNs can be optimized to yield significant LDOS enhancement (almost 10³ increase over free space) confined in sub-wavelength dielectric regions, making them very attractive for a variety of nanoscale device applications including light emitters, photodetectors, optical sensors, and nanowires-based solar cells [13,14].

2. Computational method

In this section, we present the theoretical background of the method, based on the work in Refs [22,23], that we have modified to efficiently calculate the scattering, near-field intensity and LDOS properties of two-dimensional (2D) systems of concentrically layered cylinders with arbitrary materials properties. The geometry of the problem is shown in Fig. 1 , and consists of m concentric layers specified by m values of radii and m + 1 permittivities.

Fig. 1 The scattering geometry of a multilayered cylinder of m layers each having outer radius r_M, forming m + 1 zones each specified by a complex permittivity ε_l. The three coefficients in each zone, A,B and C will give the value of the fields in that zone. The vector r_s indicates a possible location for the source.

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The coefficients A^l, B^l, C^l, are uniquely associated to each layer and will be used to specify the fields within the l ^th layer. Since the problem is 2D, it simplifies to a scalar problem specified by the z-component of the field, from which all the other field components can be derived using Maxwell’s equations. The field solution U^l in the in the l ^th layer, must satisfy the scalar Helmholtz equation in 2D, and can be expressed as a linear combination of Bessel functions in the radial variable and complex exponential functions in the azimuthal variable. Therefore, the field amplitude in the l ^th layer can be expressed as:

U_{}^{l} (r, φ) = U_{s}^{l} (r, φ) + \sum_{n = - \infty}^{\infty} [A_{n}^{l} J_{n} (k_{l} r) + B_{n}^{l} H_{n}^{(1)} (k_{l} r)] e^{- i n φ}

where U_s^l is the field contribution due to the source within the l ^th layer, which is nonzero only within the layer containing the source (in all other layers, there is no distinction between fields originating from the source and fields due to response of the structure). To determine the field solution in the rest of the structure we need to apply the boundary conditions by enforcing the continuity of the tangential field components across the layer interfaces. Since the field radiated by the source must also satisfy the Helmholtz equation, it will be represented as Bessel functions centered at the location of the source. However, in order to take advantage of the cylindrical symmetry of the problem, we first need to express the source term as an expansion of functions centered at the origin of our reference system (see Fig. 1), using the Graf’s addition theorem [24]. As a result, we obtain the following representation for the fields everywhere in the structure:

U_{s}^{l} (r, φ) = \sum_{n = - \infty}^{\infty} [A_{n}^{l} J_{n} (k_{l} r) + B_{n}^{l} H_{n}^{(1)} (k_{l} r) + C_{n}^{l} X_{n} (k_{l} r)] e^{- i n φ}

where,

X_{n}^{l} (k_{l} r) = {\begin{matrix} J_{n}^{} (k_{l} r), & r < r_{s} \\ H_{n}^{(1)} (k_{l} r), & r > r_{s} \end{matrix}

and C_n^l are the expansion coefficients of a source with an arbitrary field profile and r_s is the amplitude of the position vector of the source. The function X_n^l is a Bessel function in regions including the origin and becomes a Hankel function in unbounded regions in order to correctly handle singularities and to allow for power conservation in the system [24]. We can now use the general field expression in Eq. (2) to write the boundary condition equations at each interface resulting in a matrix equation which, for a three layer structure, is shown in the Appendix (Eq. (8)). It is worth noting that the contribution of the source has been separated from the matrix, which only depends on the geometry, materials and wavelength of the problem, but not on the source location or its intensity profile. Since the source intensity profile is yet unspecified, this matrix equation can be used to calculate the system’s response to a plane wave or, more generally, to localized point sources positioned at arbitrary locations specified by the source vector coefficients (right-hand side of Eq. (8)). The local density of states (LDOS),

ρ ({\vec{r}}_{s}; ω)

, can be obtained by the calculation of the Green’s tensor according to [25]:

ρ (\vec{r}; ω) = - \frac{2 ω n^{2}}{π c^{2}} Im [T r [G (\vec{r}, \vec{r}; ω)]]

where n is the refractive index of the embedding medium (background index) and

G (\vec{r}, {\vec{r}}_{s}; ω)

represents the response field to an infinite line source (localized source in 2D) located at position

{\vec{r}}_{s}

. To find the three diagonal components of the Green’s tensor we will have to solve the problem for the three different orientations of the source, considering a dipole oriented along x, y and z axis of the Cartesian coordinate system. As derived in [23], the z-component of the field (electric or magnetic, depending on the dipole orientation) for an electric dipole can be expressed as:

\begin{array}{l} U_{s, z} = H_{0}^{(1)} (k | \vec{r} - {\vec{r}}_{s} |) / (4 i) \\ U_{s, x} = - H_{1}^{(1)} (k | \vec{r} - {\vec{r}}_{s} |) \sin (θ) / (4) \\ U_{s, y} = H_{1}^{(1)} (k | \vec{r} - {\vec{r}}_{s} |) \cos (θ) / (4) \end{array}

where U_s,a (a = x,y,z) is the source z-component for an a-oriented dipole in Eq. (1) and

θ = \arg (\vec{r} - {\vec{r}}_{s})

. Using Graf’s addition theorem and the well-known recursive properties of Bessel functions [24], we derived analytical expressions for the source term coefficients. The source coefficients for an x-oriented dipole with φ = 0 (positioned on the x-axis) are:

C_{n}^{s} = {\begin{matrix} \frac{- 1}{8 i} [H_{n - 1}^{(1)} (k_{s} r_{s}) + H_{n + 1}^{(1)} (k_{s} r_{s})], & r < r_{s} \\ \frac{- 1}{8 i} [J_{n - 1}^{} (k_{s} r_{s}) + J_{n + 1}^{} (k_{s} r_{s})], & r > r_{s} \end{matrix}

The

C_{n}^{s}

coefficients are zero outside of the layer containing the source. To solve for a plane wave excitation, we will simply set the source location to infinity and express a plane wave via its Bessel function expansion [26], which yields

C_{n} = i^{n}

. Once the coefficients for the fields have all been calculated, the field profiles in each region can be reconstructed using Eq. (1). Additionally, the LDOS at each point in space can be calculated by displacing a point source across the structure and solving, iteratively, the matrix equation (Eq. (8)) with proper redefinition of the source vector coefficients. Once the field coefficients have been determined, the scattering and extinction efficiencies can be readily calculated using the following well-known expressions [26]:

\begin{array}{l} Q_{s c a t} = \frac{λ}{π r_{M}} [{| B_{0}^{M + 1} |}^{2} + 2 \sum_{n = 1}^{\infty} {| B_{n}^{M + 1} |}^{2}] \\ Q_{e x t} = \frac{λ}{π r_{M}} Re [B_{0}^{M + 1} + 2 \sum_{n = 1}^{\infty} B_{n}^{M + 1}] \end{array}

As usual, the absorption efficiency is determined by the difference of the extinction and the scattering efficiency. In what follows, we will consider MIM systems assuming Ag metal layers described by the experimental dispersion data given in Ref [27].

3. Scattering and localized dark resonances in CPNs

In this paper, we will investigate the plasmonic mode localization and scattering properties of CPN structures in a large range of materials and geometrical parameters, as will be detailed in the analysis presented in section 4. In this section, we select two representative structures in order to discuss the most general characteristics of their near-field and scattering spectra as well as the salient features of their distinctive plasmonic resonances. The two structures under consideration consist of a metallic (Ag) core radius of 60 nm and an outer metallic shell thickness of 20nm surrounded by air, n_air = 1. The thickness of their internal dielectric regions is 50nm (Figs. 2(a) , and 2(c)) and 12nm (Figs. 2(b) and 2(d)), while its refractive index is n = 1.5. These structures simultaneously support strong plasmonic resonances in the visible and in the near-IR spectral regions, (a series of weak resonances are also observed at shorter wavelengths corresponding to higher order multipolar modes, and will not be discussed hereafter). The plots in Figs. 2(a) and 2(b) show the calculated wavelength spectra of the scattering (blue color lines) and absorption efficiencies (green color lines) of the structures. In Figs. 2(c) and 2(d) we plot the maxima of the electric field magnitudes inside the dielectric region. The structures are excited by a unit plane wave with a magnetic field vector H polarized along the wire axis (z-axis), corresponding to a transverse electric (TE) wave.

Fig. 2 (a) and (b) show the scattering efficiency (blue) and absorption efficiency (green) for two different MIM structures in air. (c) and (d) show the maximum of the electric field in the insulating layer for the same structures. The insulating layer for (a) and (c) has a thickness of 50nm and the insulating layer for (b) and (d) has a thickness of 12nm. The radius of the inner metallic core for both structures is 60nm and the outer shell thickness is 20nm. The blue dotted line in (b) is the scattering efficiency for a solid metallic wire 92nm radius.

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We observe in Fig. 2 the presence of multiple resonances in the absorption, scattering and near-field spectra of the CPNs structures, corresponding to the excitation of dipolar (labeled by the vertical dashed lines as peaks 2 and 4), quadrupolar (labeled as peaks 1 and 3) and higher order plasmonic resonances. In particular, for the structure with a dielectric gap of 50nm, we can distinguish in Figs. 2(a) and 2(c) very pronounced absorption and scattering peaks accompanied by significant enhancement of the electric field inside the dielectric region. On the other hand, when reducing the dielectric gap thickness to 12nm (Figs. 2(b) and 2(d)), the number of resonances decreases, their spectral positions shift to longer wavelengths, and the intensities of their scattering peaks significantly reduce. This behavior is consistent with plasmon mode hybridization theory [8–10], as discussed in more details in section 4. Moreover, close to the dipolar resonance (peaks 2 and 4), we notice that the far-field scattering spectra show resonant dips that are more pronounced for the smaller gap structure. For comparison, the dotted blue line in Fig. 2(b) shows the calculated scattering of a solid Ag cylinder with the same size as the CPN and it demonstrates that the presence of a 12nm-thick dielectric layer does not strongly perturb the scattering efficiency of a single cylinder but results in the formation of strongly localized plasmonic resonances. Moreover, Figs. 2(c) and 2(d) show a distinctive dip in the electric field spectra close to modes 2 and 4. Interestingly, at these spectral positions, the far-field scattering efficiency is very high while the internal field is almost zero. We also found that the scattered far-field can be almost entirely suppressed at the wavelengths of the scattering dips in Figs. 2(a) and 2(b), while the internal field in the dielectric gap remains appreciably high (Figs. 2(c) and 2(d)). We believe that such characteristic behavior originates from the resonant excitation of dark plasmon modes supported by layered CPN systems, analogously to the recently demonstrated anomalous localized resonances proposed for metamaterials cloaks [28,29]. This picture is confirmed by the simulations of the electric field and energy flow lines (Poynting vector streamlines) shown in Figs. 3 and 4 . In particular, Fig. 3(a) shows that the Poynting vector streamlines calculated at the wavelength of the resonant scattering dip (in this case, for a 50nm gap structure) can be rerouted around the thin insulator layer, effectively avoiding wave penetration into the inner metallic object and strongly reducing far-field scattering giving rise to dark plasmon modes confined in the dielectric gaps. At this wavelength, the dipolar field radiated by the inner metallic core appears completely screened by the outer metallic layer and the scattering is resonantly suppressed (Fig. 3(a)). On the other hand, at the peak of the dipolar scattering response (Fig. 3(b)), which is about 30nm red-shifted with respect to the scattering dip (Figs. 2(a) and 2(b)), the field lines can be significantly perturbed outside the structure resulting in strong far-field scattering.

Fig. 3 Shown here are field profiles with streamlines drawn tangential to the Poynting vectors for Dark (a) and highly scattering (b) modes. The MIM structures in both (a) and (b) are made of silver and a dielectric of refractive index n = 1.5 with core radius of 60nm, dielectric gap thickness 50nm and outer shell thickness 20nm, illuminated with a TE plane wave from the left side of the frame with a wavelength of 1152nm (a) and 1183nm (b).

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Fig. 4 Plots (a)-(c) are of absolute value of the total electric field with streamlines drawn tangential to the Poynting vector, (d)-(f) are relative polarization charge amplitude, all for an MIM structure made of Silver and a dielectric with n = 1.5, illuminated by a TE plane wave from the left side of the plot. Plots (a), (b), (d) and (e) have a core radius of 60 nm, dielectric thickness of 12 nm, an outer shell thickness of 20 nm. Plots (a) and (d) are illuminated at 1492 nm, (b) and (e) are illuminated at 803 nm. Plots (c) and (f) have core radius 60 nm, dielectric thickness 50 nm, outer shell thickness 20 nm by radiation at a wavelength of 1181.7 nm.(Media 1 and Media 2)

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In order to further understand the physical nature of these plasmonic resonances we plot in Fig. 4 the electric field profiles and energy flow lines of strongly scattering resonances (dipolar and quadrupolar) corresponding to structures with 12nm (Figs. 4(a) and 4(b)) and 50nm (Fig. 4(c)) dielectric gaps, respectively. The corresponding polarization charge densities of the modes are computed by the divergence of the electric field and are plotted in Figs. 4(d)-4(f). They show a build up of charge at the interfaces between the two materials indicating the formation of surface plasmons at both interfaces, which efficiently hybridize to form the plasmon modes confined in the dielectric gap regions [8–10]. On the other hand, we notice that the internal field corresponding to the strongly scattering dipolar resonance shown in Fig. 4(c) is almost zero. This significant drop in the intensity of the electric field inside the particle can be understood based on the corresponding polarization charge density of the mode (shown in Fig. 4(f)), that shows charge accumulation only on the outside of the structure. Additionally, Fig. 4(b) demonstrates the formation of a characteristic optical whirlpool in the Poynting vector streamlines. Optical whirlpools (or vortices) formed close to the dipolar plasmon resonance of metallic nanoparticles have been recently studied based on analytical Mie theory in Refs [30,31], and are interesting phenomena due to their potential to induce optical bistability and hysteresis effects on a nanoscale. Recently, it has been shown [30] that two different types of whirlpools exist in plasmonic nanoparticles. In the so-called “outward vortex”, the energy flow streamlines first penetrate into the particle near its center then, on exiting the particle, the flow lines turn away from the centerline and begin a spiral trajectory. These vortices are formed at wavelengths shorter than the dipolar plasmon resonance. For wavelengths longer than the plasmon dipolar resonance, “inward vortices” can form, and their power streamlines pass around the sides of the particle before entering the particle to enter a spiral trajectory. To our knowledge, the formation of optical whirlpools in more complex plasmonic systems such as CPNs, has not yet been reported. Figures 4(a) and 4(b) demonstrate that more complex vortex topologies can indeed be obtained in CPNs around their dipolar and quadrupolar resonances. The complex behavior of these optical whirlpools can be fully appreciated in the attached movies that show an animation of the energy flow around the CPN structure of Figs. 4(a) and 4(b) when the excitation wavelength is scanned across its quadrupolar (Media 1) and dipolar (Media 2) resonances. This complex whirlpool behavior, similar to the formation of “inward vortices”, in agreement with [31], can additionally be supported by the quadrupolar resonances of CPNs.

To characterize further the properties of the CPN modes we plot in Fig. 5 their radial and azimuthal components. We note that since these modes are excited by a plane TE wave there is no electric field component out of the plane. We can see from Figs. 5(a) and 5(b) that the electric field amplitude corresponding to the strongly confined plasmon modes is radially polarized within the sub-wavelength dielectric gap regions. This is consistent with the significant charge accumulation at the surface of the metal interfaces (see Fig. 4) which results in an electric field gradient between the layers. On the other hand, the azimuthal components of the localized fields (Figs. 5(d) and 5(e)) are found to be much weaker and spread spatially throughout the structures. For comparison, Figs. 5(c) and 5(f) show the radial and azimuthal electric field components of the dipolar bright mode of Fig. 4(c). Based on our analysis, we believe that CPNs structures are ideally suited for the engineering of radially polarized sub-wavelength modes, which are of particular interest for near-field plasmonic fibers and quantum optics applications [32].

Fig. 5 (a), (b) and (c) respectively show the radial component of the electric field of the modes in Figs. 2(a), 2(b) and 2(c), of the same structure and wavelength. Plots (d), (e) and (f) show the azimuthal component of the electric field for these same modes.

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4. Engineering resonances in CPNs

In order to engineer the resonant behavior of CPNs for a particular device application it is necessary to investigate systematically the respective roles of geometry and dielectric index on the optical response of these complex structures. In this section, we will study these effects in relation to the spectral position and intensity of the resonant peaks. To study the effects of geometry we have considered CPN structures with Ag inner/outer layer and a dielectric gap with index n = 1.5 illuminated by a TE polarized plane wave. The sketch in Fig. 6(a) shows the geometry of reference and labels the parameters we have varied, namely the radius of the core region (referred to as r₁), the thickness of the dielectric layer (referred to as d₂), and the thickness of the outer metal shell (referred to as d₃). The results of our analysis, summarized in Fig. 6, show the formation of two main resonances observed in the scattering efficiency and in the internal field, corresponding to the dipolar and the quadrupolar modes of the structures. The intensities of these resonances are maximized within separate wavelength regions, depending solely on the geometrical parameters and the dielectric index of the structures.

Fig. 6 (a) shows the geometry used for examining resonance of structures where the gray layers are silver and the gap is a dielectric with refractive index n = 1.5 which will be illuminated by a TE plane wave of unit amplitude. (b) and (e) are, respectively, the maximum of the total electric field in the dielectric layer and the scattering efficiency vs. wavelength and r₁ with d₂ = 20nm and d₃ = 20nm. Plots (c) and (f) are electric field and scattering efficiency vs. wavelength and d₂ with an r₁ = 60nm and d₃ = 20nm. Plots (d) and (g) are the electric field and scattering vs. wavelength and d₃ with an r₁ = 60 nm and a d₂ = 20 nm for (d) and d₂ = 50nm for (g).

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The plots in Figs. 6(b) and 6(e) show respectively the electric field magnitude in the dielectric layer and the scattering efficiency as a function of r₁ (i.e. core size) and wavelength. The other two geometrical parameters of the structure are kept constant equal to d₂ = 20nm and d₃ = 20nm, while the dielectric index is fixed to n = 1.5. We notice that strong dipolar and quadrupolar resonant peaks can be supported for a wide range of r₁. As the size of the core increases in radius from 10nm to 100nm, both these resonances red-shift almost linearly, the dipolar resonance shifting at a higher rate across the entire visible and near-IR spectrum (up to 1650nm). This can be explained, in the weak plasmon mode coupling limit, by a simple azimuthal standing wave model where resonances are formed when half integer numbers of wavelengths close around the core of the structure. We found that the electric field amplitude of the dipolar mode in the dielectric gap is optimized for r₁ around 50nm while the scattering efficiency of this mode is optimized for a smaller core size around r₁ = 30nm. On the other hand, the scattering efficiency of the quadrupolar mode (Fig. 6(e)) remains very small across the entire parameter space while the intensity of the field inside the dielectric gap becomes significant for larger values of r₁, as expected for higher order modes.

In Figs. 6(c) and 6(f) we study the effect thickness variations of the dielectric layer d₂ on the resonances of the structure, while the other parameters are fixed to r₁ = 60nm and d₃ = 20nm, n = 1.5. Both the scattering efficiency and field amplitude show two largely tunable resonance bands. However, when the thickness of the dielectric layer is increased, the frequency of the resonances blue-shifts in a non-linear way, varying very quickly at first and later approaching a plateau for thickness of about 40nm. This behavior is understood based on the established plasmon hybridization model [8,9] where the resonances of the CPN systems originate from the hybridization of coupled modes in the core and outer shell layers. Therefore, by increasing the thickness of d₂ we decrease the coupling between these modes and shift the resonance of the coupled system to shorter wavelengths (larger energy). Moreover, we also notice that, as already discussed in section 3, the scattering efficiency of CPN systems (Fig. 6(f)) dramatically reduces by decreasing the dielectric gap thickness, since dark plasmon modes form inside the dielectric region. In Figs. 6(d) and 6(g) we show the tunability of the resonances with respect to the thickness of the outer metallic layer, while fixing r₁ = 60nm, n = 1.5, d₂ = 20nm in 6(d) and d₂ = 50nm in 6(g). Two distinct modes (dipolar and quadrupolar) can be clearly observed and their spectral positions blue-shift non linearly by increasing d₃. This effect is due to the reduced coupling of the hybridized resonances [8–10]. Finally, we found (Fig. 6(g)) that within the range of interest (5nm-40nm) the scattering efficiency drops significantly by increasing the metallic outer layer thickness, since external plane waves cannot penetrate into the structure.

Next, we will explore the effects of varying the refractive index of the dielectric gap layer on the scattering and field amplitude of the CPN structures. In order to keep our analysis simple, we consider a CPN structure with fixed geometry r₁ = 60nm, d₂ = 20nm, d₃ = 20nm and a real refractive index in the gap region varying in the n = 1.5-3 range. (The main conclusions of our analysis are robust with respect to the introduction of dielectric losses in the sub-wavelength layer). Due to the very small scattering values observed at small gaps (Fig. 6(e)), the scattering efficiency has been plotted for a structure with d₂ = 50nm. We can see from the results in Fig. 7(a) that the dipolar and quadrupolar resonances can be engineered across a wide range of refractive index values. However, we observe a significant decrease in the scattering efficiency and in the amplitude of the internal electric field values as the refractive index of the dielectric layer is increased from its minimum value. This general behavior has been observed in planar metal-insulator-metal slot waveguides [1–3] and is explained by reduced plasmon hybridization. At large refractive index values, the plasmon fields at the two metal-dielectric interfaces decay very rapidly into the dielectric region thus reducing their coupling and preventing the formation of hybridized resonances within the sub-wavelength gap region. However, we notice in Fig. 7(b) that higher order plasmonic resonances become very significant, at short wavelengths, for high values of the dielectric refractive index.

Fig. 7 The maximum of the electric field in the dielectric layer and scattering efficiency are plotted respectively in (a) and (b) as the wavelength and refractive index of the dielectric layer, n, are varied. The structures are again illuminated by a TE polarized plane wave. For (a) and (b) r₁ = 60nm, d₃ = 20nm, for (a) d₂ = 20nm, (b) d₂ = 50nm

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5. LDOS calculation

In this section, we demonstrate that the strongly localized resonances of CPN structures are ideally suited to provide strong light-matter coupling by significantly enhancing the LDOS at multiple frequencies. Using the method described in section 2, we have calculated the LDOS of CPN structures by averaging the diagonal elements of the Green tensor. The LDOS data presented in Fig. 8 have been normalized with respect to the free space values, thus providing direct quantification of radiation rate enhancement of localized dipole sources.

Fig. 8 Here the maximum of the normalized LDOS in the dielectric layer is plotted in log₁₀ scale. In (a) the spectrum of LDOS is shown for a structure made of silver and insulator refractive index n_i = 1.5 with r₁ = 60nm, d₂ = 20nm and d₃ = 20nm. (b) Shows the LDOS as the wavelength and refractive index of the dielectric layer, n, are varied with r₁ = 60nm, d₂ = 20nm and d₃ = 20nm. (c) Shows the LDOS as the wavelength and radius of the core, r₁, are varied n = 1.5, d₂ = 20nm and d₃ = 20nm. (d) Shows the LDOS as the wavelength and the thickness of the dielectric layer, d₂, are varied with r₁ = 60nm, d₃ = 20nm and n_i = 1.5.

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In Fig. 8(a) we show, on a logarithmic scale, the maximum value of the LDOS calculated in the dielectric gap region from a randomly oriented dipole for a CPN structure with a 20nm dielectric gap (other parameters detailed in the caption). Our results demonstrate that over two orders of magnitude LDOS enhancement can be achieved at the dipolar plasmon resonance in CPN structures. We notice that this enhancement is spatially localized within the CPN dielectric gap region, corresponding to the maximum of the electric fields shown in Fig. 4. Moreover, differently from the previous results obtained under plane wave excitation, the LDOS features a series of higher order resonant peaks, with sizable intensity at shorter wavelengths, which can be efficiently excited by internal localized sources (line emitting dipole). This scenario offers the potential to enhance the radiative emission of randomly oriented dipoles (atoms, molecules, semiconductor nanostructures, etc) in the dielectric gap region simultaneously at multiple wavelengths. In order to better engineer this effect we studied systematically the LDOS enhancement for varying refractive indices and geometrical parameters. Figure 8(c) shows the calculated LDOS as a function of wavelength and r₁, the radius of the metallic core. Multiple resonances occur in Fig. 8(c). As the core size increases, all the resonances in the LDOS red-shift and their intensity increase. This increase appears to contrast the trend observed for the local electric field in Fig. 6(b), which shows a decreasing near-field intensity. We attribute this discrepancy to the difference in the source symmetry, which now consists of a localized line dipole. Despite that the LDOS is affected by the intensity of local fields, the results obtained under plane wave excitation cannot be directly used to explain the detailed LDOS trends. In Fig. 8(d) we show the effect of varying the thickness of the dielectric layer d₂, (which follows more directly the field amplitude plot shown in Fig. 6(c)) while in Fig. 8(b) we explore the effect of varying the refractive index of the dielectric layer. The results in Fig. 8 demonstrate that it is possible to obtain intense LDOS enhancement within deep sub-wavelength gaps (20nm) at both the dipolar and quadrupolar resonances across the entire range of explored refractive index values, potentially enabling significant light emission enhancement effects at multiple wavelengths in a large range of dielectric materials. Based on our analysis, CPN structures can be flexibly optimized, for instance, to strongly confine and to resonantly increase the pump intensity and the radiation rate of dipoles embedded within nanoscale dielectric layers over well-separated spectral regions, paving the way to the engineering of novel plasmonic nanowires devices for multi-spectral light emission, energy harvesting, optical sensing and near-field spectroscopy.

6. Conclusions

In conclusion, we have developed an efficient semi-analytical multipolar expansion method to investigate systematically the near-field enhancement, scattering/absorption spectra and LDOS in concentric co-axial plasmonic nanowires. We demonstrated that these structures support radially polarized plasmon modes with strongly enhanced local fields confined within deep sub-wavelength dielectric regions as well as distinctive scattering resonances widely tunable across the visible and near-infrared spectral regions. Finally, we calculated the LDOS of these structures and we demonstrated that more than two orders of magnitude enhancement could be obtained at multiple, controllable frequencies. These results enable the engineering of CPNs-based light emitters and optical sensors with simultaneously enhanced excitation and emission rates for nanowires plasmonics.

Appendix

The matrix equation for a three-layered structure is shown below, where each set of two rows are the two field continuity equations that exist at each boundary, starting from the innermost boundary. The zeros arise for coefficients associated with fields that are not present at that boundary.

\begin{array}{l} (\begin{matrix} J_{n} (k_{1} r_{1}) & - J_{n} (k_{2} r_{1}) & - H_{n}^{(1)} (k_{2} r_{1}) & 0 & 0 & 0 \\ \frac{1}{ε_{1}} J_{n}^{'} (k_{1} r_{1}) & - \frac{1}{ε_{2}} J_{n}^{'} (k_{2} r_{1}) & - \frac{1}{ε_{2}} H_{n}^{(1)'} (k_{2} r_{1}) & 0 & 0 & 0 \\ 0 & J_{n} (k_{2} r_{2}) & H_{n}^{(1)} (k_{2} r_{2}) & - J_{n} (k_{3} r_{2}) & - H_{n}^{(1)} (k_{3} r_{2}) & 0 \\ 0 & \frac{1}{ε_{2}} J_{n}^{'} (k_{2} r_{2}) & \frac{1}{ε_{2}} H_{n}^{(1)'} (k_{2} r_{2}) & - \frac{1}{ε_{3}} J_{n}^{'} (k_{3} r_{2}) & - \frac{1}{ε_{3}} H_{n}^{(1)'} (k_{3} r_{2}) & 0 \\ 0 & 0 & 0 & J_{n} (k_{3} r_{3}) & H_{n}^{(1)} (k_{3} r_{3}) & - H_{n}^{(1)} (k_{4} r_{3}) \\ 0 & 0 & 0 & \frac{1}{ε_{3}} J_{n}^{'} (k_{3} r_{3}) & \frac{1}{ε_{3}} H_{n}^{(1)'} (k_{3} r_{3}) & - \frac{1}{ε_{4}} H_{n}^{(1)'} (k_{4} r_{3}) \end{matrix}) (\begin{matrix} A_{n}^{1} \\ A_{n}^{2} \\ B_{n}^{2} \\ A_{n}^{3} \\ B_{n}^{3} \\ B_{n}^{4} \end{matrix}) = \\ (\begin{matrix} C_{n}^{2} X_{n} (k_{2} r_{1}) - C_{n}^{1} X_{n} (k_{1} r_{1}) \\ C_{n}^{2} \frac{1}{ε_{2}} X_{n}^{'} (k_{2} r_{1}) - C_{n}^{1} \frac{1}{ε_{1}} X_{n}^{'} (k_{1} r_{1}) \\ C_{n}^{3} X_{n}^{(1)} (k_{3} r_{2}) - C_{n}^{2} X_{n}^{(1)} (k_{2} r_{2}) \\ C_{n}^{3} \frac{1}{ε_{3}} X_{n}^{(1)'} (k_{3} r_{2}) - C_{n}^{2} \frac{1}{ε_{2}} X_{n}^{(1)'} (k_{2} r_{2}) \\ C_{n}^{4} X_{n}^{(1)} (k_{4} r_{3}) - C_{n}^{3} X_{n}^{(1)} (k_{3} r_{3}) \\ C_{n}^{4} \frac{1}{ε_{4}} X_{n}^{(1)'} (k_{4} r_{3}) - C_{n}^{3} \frac{1}{ε_{3}} X_{n}^{(1)'} (k_{3} r_{3}) \end{matrix}) (8) \end{array}

The A_n^{m + 1} coefficients (outermost layer) and B_n¹ coefficients (innermost layer) have been set to zero and do not appear in the matrix equation for reasons detailed in [22,26]. The contribution of the source has been moved to the right hand side of the equation so the matrix is only dependent on the geometry, materials and wavelength of the problem, but not on the source location or profile. As mentioned in Section 2, the C_n^l coefficients are non-zero only in the source layer; therefore most of the terms on the right hand side will vanish once the position of the source is determined. Since the source intensity profile is not yet specified, this matrix equation can be used to calculate the system’s response to a plane wave or, more generally, to localized point sources positioned at arbitrary locations as specified by source vector coefficients (right-hand side of Eq. (8)).

Acknowledgments

The authors thank Dr. Svetlana Boriskina helpful discussions on multipolar scattering theory, Dr. Markus Schmidt, Dr. Fabio Biancalana, and Prof. Siddharth Ramachandran for insightful discussions on plasmonic fibers applications. This work was supported by the NSF Career Award No. ECCS-0846651.

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Light scattering, field localization and local density of states in co-axial plasmonic nanowires

Abstract

1. Introduction

2. Computational method

3. Scattering and localized dark resonances in CPNs

4. Engineering resonances in CPNs

5. LDOS calculation

6. Conclusions

Appendix

Acknowledgments

References and links

Supplementary Material (2)

Cited By

Figures (8)

Equations (8)

Optics Express