## Abstract

We present a broadband and efficient short-range plasmonic directional coupler design, for the delivery and collection of deeply sub-wavelength radiation to tapered plasmonic nanowires. We show a proof-of-concept design using a planar geometry operating at wavelengths between 1.2 −2.4 μm, showing that the propagation characteristics predicted by an Eigenmode analysis are in excellent agreement with finite element simulations. This analytical formulation is straightforward to implement and immediately provides the power-exchange properties of hybrid plasmonic waveguides. An investigation of both waveguide delivery and collection performance to and from a plasmonic nano-tip is performed. We show that this design strategy can be straightforwardly adapted to a realistic hybrid fiber geometry, containing wire diameters more than one order of magnitude larger than the planar geometries, with important applications in all-fiber plasmonic superfocussing, and nonlinear plasmonics.

© 2016 Optical Society of America

## 1. Introduction

Surface plasmon polaritons (SPPs) emerge when photons couple to the collective oscillations of electrons at metal/dielectric interfaces, and can give rise to a deeply sub-wavelength enhancement and confinement of the local electromagnetic field [1]. Over the past decades, numerous applications of SPPs have emerged where strong field enhancement and device miniaturization are advantageous, e.g. in field-enhanced microscopy [2], sensing [3], nonlinear optics [4, 5], photovoltaics [6], quantum optics [7], and near-field microscopy [8]. However, due to intrinsic metallic losses at optical and infrared frequencies, SPPs have attenuation lengths of the order of several tens of micrometers. Nevertheless, nanoscale hybrid plasmonic-photonic devices with superior functionalities with respect to their dielectric counterparts, especially in terms of compactness and field confinement, are routinely achieved. Since SPP-devices lengths are often much shorter than dielectric waveguides – typically no more than a few hundred micrometers – practical plasmonic devices require efficient coupling between a given low-loss dielectric mode and the lossy SPP modes.

When metallic nanofilm thicknesses approach the penetration depth of SPPs modes, the two SPPs on opposite sides of the film interact, leading to a splitting of the mode dispersions and the emergence of symmetric and antisymmetric modes (in terms of the magnetic field), commonly termed long-range (LR-) and short-range (SR-) SPPs, respectively [9, 10]. LR-SPPs possess smaller fractions of field in the metal and have lower loss, however they are only weakly bound to the metal, and simply behave like plane waves if the metal thickness is reduced to the nanoscale [11]. In contrast, SR-SPPs are tightly bound to the metal and show an *increasing* confinement to the metal surface for a reducing film thickness [11]. This makes SR-SPPs particularly advantageous for a number of applications, including subwavelength imaging via plasmonic superfocussing [12, 13], sensing [14], and all applications where nanoscale field enhancement is key [6, 15]; however, they are difficult to couple to due to their large effective mode, as discussed below. Note that SR- and LR-SPPs are supported not only in planar structures, but also in cylindrical metallic nano-wires: in this case, the SR-SPP is given by the radially polarized TM mode with azimuthal order zero (TM_{0}-mode), and the LR-SPP is the linearly polarized mode with azimuthal order of unity (HE_{1}-mode) [16]. The corresponding modes can be understood by a single interface SPP mode experiencing geometric momenta while spiralling around the nanowire circumference [16, 17].

Most generally, due to their comparably large wave vectors, SR-SPP modes cannot be excited via free space excitation, and more sophisticated coupling schemes are required, such as evanescent wave coupling [18], grating couplers [19], and tapers [20]. These techniques are widely used, but suffer from a narrow spectral bandwidth. The large momentum mismatch is a key problem, since SR-SPP modes commonly have index values much larger than those of typical dielectric systems. Another excitation method is end-fire coupling [21–23], relying on a direct excitation of the plasmonic mode by an incident beam, either from a dielectric waveguide or free space. This technique, commonly used in fiber optics, requires a matching of the fields of the modes involved, which is difficult when the plasmonic waveguides have sub-wavelength dimensions [13]. Moreover, end-fire coupling does not allow for the selective excitation of a desired mode, making this approach unfeasible for applications requiring SR-SPPs.

An alternate technique for the efficient excitation of SPPs is plasmonic directional coupling [24, 25], which relies on phase-matching of neighboring waveguides. Directional coupling is already widely used in integrated photonics [26] e.g., for integrated couplers, switches, and modulators. Previous plasmonics-based demonstrations have targeted SPPs on thin metal stripes, in the context of sensing [24], polarizers [25], and sub-wavelength field confinement in delivery mode [27, 28].

On the other hand, based on the idea of integrating increasingly sophisticated nanophotonic functionality into optical fibers (as is typically done in planar structures), recent years have shown a rapid development of the field of hybrid (multi-material) optical fibers [29, 30]. Vastly different materials such as metals (e.g., gold [31]) and semiconductors (e.g., chalcogenide [32] and silicon [33]) can be integrated into a single fiber using a variety of pre- and post- processing techniques, with the ultimate goal of interfacing nanoscale devices with common fiber optical circuitry. Gold wires with diameters of the order of a few hundred nanometers are now routinely integrated into fibers, either by direct drawing [34] or using pressure-assisted melt filling [31], where wire diameters as small as 30 nm can be achieved. It has been shown that such nanowire-enhanced fibers can act as near-field scattering nanoprobes using a localized surface plasmon resonance at the fiber end [35]. Indeed, due to the ductile nature of gold, typical cleaves of a gold-filled fiber naturally result in a gold nanotip which protrudes from of the fiber [35], see Fig. 1(a). It was shown that the apex size of such probes can be as small as ten nanometers, which is substantially smaller than the opening in currently used fiber-based SNOM probes. According to Bethe [36], the transmitted field amplitude scales with *b*^{3} (aperture size *b*) for apertures with *b* ≪ *λ*, and thus smaller apertures lead to a dramatic reduction in light throughput. However, fiber probes incorporating sharp tips are accompanied by a substantially improved spatial resolution [37]. An integrated and flexible fiber-based plasmonic nanoprobe would be essential for next generation near-field probes, allowing to efficiently deliver and collect radiation on the nanoscale on a convenient fiber platform [35, 38]. However the implementation of an integrated and flexible fiber-based nanoprobe device using SR-SPP plasmonic nanowires has not yet been achieved, mainly due to the lack of an effective and practical scheme to excite them.

Here, we propose an all-in-fiber nanoprobe design allowing for the efficient and broadband excitation of the strongly bounded cylindrical SPP mode for superfocussing via plasmonic directional mode coupling [Fig. 1(b)]. The design consists of a step-index fiber with a central dielectric core and a gold nanowire running parallel, resulting in a plasmonic coupler with a nanotip protruding from the fiber. Light launched into the dielectric core can directionally couple to the strongly bound cylindrically radially polarized SR-SPP on the gold nanowire, which then naturally tapers to a nanotip at the fiber output when cleaved. SR-SPPs generally possess effective indeces which dramatically increase as the metal width approaches small dimensions [both for planar [11] and cylindrical [17] geometries, see Fig. 1(c)], thus enabling plasmonic superfocussing. Note that the cylindrical nanowire and the planar film show comparable properties in terms of loss and effective index even though the nanowire diameter is ten times larger than the film thickness [Figs. 1(c)–1(d)], thus relaxing fabrication and material constrains. In turn, a comprehensive study of a planar (2D) plasmonic directional coupling structure will reveal the dependency of the modes on the various geometric parameters, thus acting as a pathway for establishing an efficient procedure for designing its cylindrical (3D) counterpart with substantially reduced computational efforts.

Inspired by previous demonstrations of SR-SPP directional coupling to plasmonic nanofilms [24, 25], we present a comprehensive Eigenmode (EM) and propagation analysis, which defines a design route to efficiently excite SR-SPPs on plasmonic waveguides using a directional coupling scheme. Our EM ansatz is rapid and precise, as validated by full-vectorial finite element (FE) calculations, and allows to obtain physical insights into the complex mode coupling behavior of such plasmonic-photonic structures, which cannot be obtained from FE simulations. As a proof-of-concept design, we first show that a planar multilayer geometry can be used to both deliver and collect light using a plasmonic nanotip at a deeply subwavelength scale, operating as an efficient and broadband near-field nanoprobe. Finally, we apply our calculation scheme to a nanowire-based fiber nanoprobe [Fig. 1(b)], showing that a plasmonic hybrid optical fiber can indeed be used for plasmonic superfocussing. Our simulation ansatz is straightforward to implement and can be adapted to suit a wide variety of materials and geometries.

## 2. Device concept, theory, design, performance

We illustrate our scheme by analyzing a planar 2D multilayer (Fig. 2) consisting of a gold nanofilm (thickness *w*) which is placed at a distance *t* from a dielectric waveguide with a refractive index *n _{core}* larger than the cladding (core thickness

*d*). This configuration is assumed to be embedded in silica, thus forming a hybrid plasmonic-dielectric waveguide. We only consider transverse-magnetic (TM) polarization, corresponding to the polarization state of SPPs. Isolated dielectric waveguides and gold nanofilms generally support a fundamental TM waveguide mode and a SR-SPP mode. We refer to these modes as the

*isolated*Eigenmodes (IEMs). Note that for the film thicknesses considered here, the LR-SPP has an effective index close to the background [11], and can thus be neglected. Due to the strong modal overlap, the modes hybridize [39] so that two new Eigenmodes are supported in the hybrid dielectric-plasmonic waveguide. We refer to these modes as the

*hybrid*Eigenmodes (HEMs). Within the context of SPPs, the formation of HEMs is sometimes referred as plasmonic hybridization [39, 40]. The geometry under investigation represents a strongly coupled system; preliminary calculations show that conventional coupled-mode theory for lossy systems [41], though certainly accurate for situations where perturbations are small (e.g. low losses, large separations between waveguides), is in fact inadequate for the particular case considered here. Instead, one can interpret the coupling process as a beating of the two HEMs within the coupling section. This reduces the problem to a simple calculation of the propagation constants of the EMs and the modal amplitudes [42], whereas the field at any point inside the coupling section is giving by a superposition of the HEMs. After a particular propagation length, most of the field will be located within the gold film, i.e. the SR-SPP mode [Fig. 2(i)]. Tapering this nanofilm at this length then allows to deliver/collect light on a deeply subwavelength scale [Fig. 2(ii)].

#### 2.1. Eigenmode propagation in hybrid waveguides

Generally, the total electric and magnetic field in a dual-waveguide structure are a superposition of the two supported HEMs,

*i*= 1, 2 labels the two HEMs.

**H**

*and*

_{tot}**E**

*are the transverse total magnetic and electric field vectors at any point in the waveguide, respectively,*

_{tot}**H**

*and*

_{i}**E**

*are the magnetic and electric field distributions of the HEMs, ${\beta}_{i}={\beta}_{i}^{R}+i{\beta}_{i}^{I}$ are their propagation constants, and*

_{i}*a*are the modal amplitudes which determine the contribution of the respective EMs to the total field. The above expansions are approximate since contributions from radiation modes have been neglected. Following the convention used in [42], we use modal fields that have unit normalization, so that

_{i}*A*

_{∞}. Modes can always be normalized to satisfy Eq. (3), which can then be used to calculate the complex modal amplitudes at input,

**H**

*(*

_{in}*x*,

*y*) =

**H**

*(*

_{core}*x*,

*y*) in this case is the magnetic field of the dielectric waveguide IEM at

*z*= 0 (see Fig. 2). Here we use the overlap integrals in Eqs. (3)–(4) in their unconjugated form, as appropriate for lossy (as well as for lossless) systems [42]. The modal amplitudes at output (i.e., at a certain position

*z*> 0) are given by:

*z*of the dual waveguide system [Eqs. (1)–(2)] onto the appropriate isolated modes using orthonormality, and take the modulus squared. After some algebra we obtain a closed-form expression for the fraction of power in the projected waveguide at the (output) position

*z*for each of the two waveguides:

*f*in the dielectric or plasmonic portion of the dual-waveguide system is obtained either by setting

_{P}**H**

*(*

_{out}*x*,

*y*) =

**H**

*(*

_{core}*x*,

*y*) (so that

*a*=

_{i}*b*) or

_{i}**H**

*(*

_{out}*x*,

*y*) =

**H**

*(*

_{SPP}*x*,

*y*) (so that

*a*≠

_{i}*b*) respectively, where

_{i}**H**

*(*

_{SPP}*x*,

*y*) is the the magnetic field of the SR-SPP IEM.

It is important to note that the modal amplitudes are complex quantities, indicating a phase delay between the two HEMs, which is to some extent similar to that occurring in circularly polarized light. This effect leads to an induced phase shift at the incoupling location (*z* = 0) and only occurs in unbalanced coupled mode systems (i.e. in situations where one mode has a significantly different loss or gain with respect to the other [31]). Note also that Eq. (6) clearly indicates that the energy exchange between the waveguides is only correlated to the real parts of the dispersions of the HEMs, i.e., to the dephasing of the EMs, whereas the imaginary parts of the propagation constant influence the overall loss.

#### 2.2. Hybrid waveguide design for SR-SPP excitation

The bounded solutions and corresponding field profiles are calculated by solving the transcendental dispersion equation, obtained after imposing continuity of the tangential electromagnetic field components at the interfaces and vanishing field amplitudes at infinity [43]. It is well known that directional couplers formed by adjacent non-identical waveguide optimally exchange energy at those wavelengths where the propagation constants of two modes match (i.e., possess the same effective mode index *n*_{eff}); this design rule, referred to as phase-matching, is well-known from lossless dielectric waveguide systems [44], but is less obvious in the case of hybrid plasmonic systems, where propagation constants are complex.

As a starting point for our design, we look for the structural parameters and material combinations that lead to phase-matching (in this complex case, we match the real part of *n*_{eff}) of the plasmonic and dielectric IEMs (corresponding to the case of an infinitely large waveguide separation *t* → ∞). As an example we choose an operation wavelength of *λ* = 1.55μm, TM-polarization (the only non-zero transverse components are *H _{y}* and

*E*), gold as the metal (Drude-Lorentz model [45]) and silica as the dielectric background (Sellmeier model [46]). We first calculate

_{x}*n*

_{eff}of the SR-SPP IEM as a function of the gold width

*w*(blue line in Fig. 3) and of the dielectric IEM as a function of core thickness

*d*for various core material indices

*n*(here we use

_{core}*n*= 1.67, 1.77 and 1.87), with the latter assumed to be lossless and dispersionless [colored dashed lines in Fig. 3(a)]. For a given dielectric waveguide thickness and core, we obtain the thickness of the gold film that is required for phase matching. Here we assume a dielectric core with a refractive index and thickness of

_{core}*n*= 1.87 and

_{core}*d*= 500nm, respectively, where a gold thickness of 15 nm is required (red arrows in Fig. 3).

#### 2.3. Eigenmode analysis

The next step in the design procedure relies on analyzing the propagation properties of the two resulting HEMs, which are excited at *z* = 0 with the modal amplitudes given by Eq. (4). These HEMs are calculated for a finite separation distance; we use *t* = 400nm to start, the dependence of the directional coupling properties on *t* is presented in Section 2.5. This corresponds to a highly coupled hybrid plasmonic system, resulting in HEM dispersions [solid lines in Fig. 4(a)] which strongly differ from those of the IEMs [dashed lines in Fig. 4(a)], as evident when inspecting the corresponding field profiles [Fig. 4(b)]. A strong splitting of the two HEMs is evident around the point where ℜ*e*(*n*_{eff}) of the IEMs cross. Equation (6) shows that a larger dispersion splitting leads to a maximum energy transfer on a short length scale due to a larger dephasing value Δ*β _{R}*. Weaker coupling results in shorter coupling lengths and excessive energy dissipation before energy is transferred to the plasmonic waveguide. The incident field is defined by the fundamental TM IEM of the dielectric waveguide [Fig. 4(b), core mode], so that the input modal amplitudes can be calculated via Eq. (4). The spectral distribution of such modal amplitudes are shown in Fig. 4(d).

As we will show, this device operates at a bandwidth of more than 800nm, and can be immediately understood by inspecting Eq. (6): firstly, the period of the exchange of energy between the two waveguides is only determined by the dephasing, and thus nearly parallel dispersions [solid lines in Fig. 4(a)] are beneficial for a broadband operation. Moreover, the modal amplitudes and losses of the HEMs are approximately constant for wavelengths longer than 1.4μm [solid lines in Fig. 4(c)], which makes our device comparably insensitive to wavelength.

#### 2.4. Numerical validation of Eigenmode approach

By determining the modal amplitudes *a _{i}*, the fields

**H**

*,*

_{i}**E**

*and the propagation constants*

_{i}*β*, the total fields [Eqs. (1) and (2)] and the time-averaged Poynting vector component in the propagation direction

_{i}*S*= ℜ

_{z}*e*(

**E**×

**H**

^{*}) ·

*ẑ*can be calculated at any position within the hybrid waveguide system. This allows for a direct comparison of the EM approach with FE simulations (COMSOL), which we show below for the example discussed here (

*λ*= 1.55μm,

*n*= 1.87,

_{core}*d*= 500nm,

*w*= 15nm,

*t*= 400nm, silica background). Port boundary conditions on the input (left) and output (right) ensure that only the fundamental TM mode of the waveguide is excited, and perfectly matched layers elsewhere suppress any reflections in the simulation volume.

The distributions of the axial component of the Poynting vector *S _{z}*(

*x*,

*z*) (normalized to the maximum in the window) using the simple EM analysis [Fig. 5(a)] show excellent agreement with the FE-simulations [Fig. 5(b)], confirming the feasibility of the EM approach in the context of plasmonics. The amplitudes of

*S*(

_{z}*x*,

*z*) in the center of the dielectric waveguide [ ${S}_{z}^{\mathit{core}}$, blue line in Fig. 5(c)] and on the upper edge (

*x*=

*d*+

*t*+

*w*= 915nm) of the gold strip [ ${S}_{z}^{\mathit{SPP}}$, red line in Fig. 5(c)] confirm the excellent agreement and show that 80% of the peak power of the input the mode can be transferred to the surface of the gold film over a length of 5.3 μm. To quantify the power transfer efficiency, we consider the total power in core- and film- region as a function of propagation distance. For this purpose, we choose to split the simulation space vertically into two regions [white arrows in Fig. 5(a)], with an artificial boundary half way between the two waveguides at

*x*=

_{b}*d*+

*t*/2, and define

*P*(

^{core}*z*) and

*P*(

^{SPP}*z*) are the total powers (per unit length) in core- and SPP- modes, respectively. This choice of

*x*then allows to estimate the fraction of power in the core and SPP modes [24],

_{b}*f*and

_{core}*f*respectively, as a function of propagation length:

_{SPP}*f*(

^{SPP}*z*) is defined as the plasmonic power transfer efficiency. Note that the sum of

*f*and

^{core}*f*is unity at

^{SPP}*z*= 0, and decreases below unity for

*z*> 0 due to absorption, i.e., power dissipation in the metal, however a significant amount of power can be transferred from the dielectric waveguide into the SR-SPP. The plasmonic coupling length

*L*is defined as the waveguide length after which a maximum power fraction

_{c}*η*= max(

*f*) is transferred to the plasmonic waveguide, where

^{SPP}*η*is the maximum transfer efficiency. A value of

*η*= 38% at 1.55 μm after a propagation distance of

*L*= 5.3μm is achieved for the example discussed here [Fig. 5(d)], where the EM analysis overlaps extremely well with the FE-simulations. We also find that the modal intensities calculated analytically from Eq. (6) provide a good estimate of the fraction of power in the dielectric and plasmonic mode [dashed-dotted lines in Fig. 5(d)]. This confirms that the superposition of the propagating (bounded) HEMs offers a straightforward pathway for analyzing the propagation characteristics of plasmonic/dielectric couplers with high degree of accuracy, greatly simplifying the design procedure with minimal simulation time, and providing additional physical insight. For example, the fact that the maximum SPP power transfer does not occur at the position where the power in the dielectric core is minimum [Fig. 5(b)] can be understood by inspecting Eq. (6), since in the presence of loss (non-zero imaginary field amplitudes), the power fractions in the dielectric (

_{c}*a*=

_{i}*b*) and plasmonic (

_{i}*a*≠

_{i}*b*) modes are phase-delayed.

_{i}#### 2.5. Bandwidth

We now investigate the wavelength dependence *S _{z}* in the dielectric and plasmonic portions of the hybrid waveguide system [Figs. 6(a)–6(d)]. We observe that the location of maximum of

*S*of the plasmonic mode ( ${S}_{z}^{\mathit{SPP}}$) is between ∼ 5 – 6μm [black dotted line in Figs. 6(b) and 6(d)] for the wavelengths considered here (1.2–2 μm), with the EM approach again overlapping with FE-simulations [Figs. 6(a) and 6(b)]. This insensitivity to wavelength is to some extent surprising, since directional mode coupling for dissimilar waveguides is often seen as a resonant phenomenon, occurring most efficiently at a specific resonance wavelength [31, 34, 39]. In the case presented here however, the insensitivity to wavelength is a result of the parallel HEM dispersions and the comparably small wavelength dependence of modal amplitudes and HEM losses. Therefore, our proposed device can transfer a significant amount of energy from the dielectric core mode into the SPP mode with a bandwidth of more than 800 nm, forming a broadband plasmonic mode coupler for the efficient excitation of SR-SPP modes.

_{z}We find that *η* increases towards longer wavelengths [Fig. 6(e) (blue)], approaching 50% at 2 μm. The slight discrepancy between FE-simulations (circles) and the EM approach (solid lines) for *λ* > 1.7μm is due to the partial dissipation of energy via radiation modes, which are not included in our simple EM approach. When analyzing the coupling length, we find that it only weakly depends on wavelength [Fig. 6(f), red], and is even constant between 1.44–1.75 μm. This broadband power transfer efficiency and wavelength-independent coupling length is particularly advantageous from a device design perspective, which we apply in the next section to produce sub-wavelength focussing on a plasmonic nanotip. As a final note on the directional coupler characteristics, Fig. 6(f) shows the dependence of *η* and *L _{c}* on the core/gold-strip separation

*t*at a fixed wavelength of

*λ*= 1.55μm: as expected, smaller separations lead to a decrease in the coupling length and an increase in the transfer efficiency.

## 3. Application: plasmonic tip

One particularly important property of the SR-SPP mode is its increasing electromagnetic field confinement with decreasing thickness of the supporting metal film, which can be employed for high-resolution near field scanning microscopy [12]. We now apply the proposed plasmonic mode coupler in the context of a superfocussing plasmonic nanotip, for applications in broadband efficient delivery and collection of sub-wavelength radiation. Henceforth, we consider a complete superfocussing device (see Fig. 2), consisting of an input section with only a dielectric waveguide (*z* < 0), a hybrid plasmonic/dielectric coupling section (0 < *z* < *L _{c}*) and a plasmonic nanotip (

*z*>

*L*). The geometric parameters of the coupling section correspond to those used in the previous Section, i.e.,

_{c}*L*= 5.3μm. The nanotip itself (which in this 2D analysis forms a nanowedge) is a tapered version of the 15 nm thick gold film, tapered down to 1 nm over a distance of 1 μm, with the apex of the tip possessing a 1 nm radius. Note that we have selected a coupling section corresponding to the length where the energy transfer to the plasmonic mode is maximum, as opposed to where the power in the dielectric core is minimum. This choice was motivated by the observation that the latter results in orders of magnitude lower power delivery to the nanotip, accompanied by a relative increase in the background noise when the nanotip operates in light delivery mode.

_{c}#### 3.1. Delivery operation

Figure 7 shows the power flow for this device at three example wavelengths (*λ* = 1.25μm, 1.55μm, 1.85μm), calculated using FE-simulations, assuming a fundamental TM core waveguide mode incident from the left, and perfectly matched layers elsewhere. By design, this device efficiently converts the electromagnetic power into a SR-SPP, which then travels down the tip. The delivery efficiency is quantified by calculating amount of power transferred from the input section to the apex of the tip [*η _{delivery}* =

*P*(

*z*=

*z*)/

_{apex}*P*(

*z*= 0)], where the latter has been calculated by integrating along the

*x*-direction over a 20 nm region (vertical dashed lines in Fig. 7). We observe up to 10% efficiency over a large spectral domain (Fig. 8), as a result of the wavelength insensitivity of the plasmonic mode coupler. Note that a resonant plasmonic cavity forms between the end of the coupling section (

*z*=

*L*) and the apex [35], leading to back reflections into the coupling section and thus to oscillations of

_{c}*η*over wavelength. This spectral dependence is an intrinsic feature of the tip and not of the plasmonic coupling section, which can be adjusted by choosing a different tip length (dotted line in Fig. 8). Finally, we characterize the intensity-enhancement properties of the tip by taking the ratio of the peak Poynting vector magnitude at the tip apex, with respect to that at input. Results for the 1 μm long tip are shown in Fig. 8 (red), showing up to 12-fold power enhancement. The fluctuations in delivery efficiency can be attributed to the fundamental plasmon mode being reflected at the apex and the base of the tip, forming Fabry-Perot type resonances [35].

_{delivery}#### 3.2. Collection operation

The collection performance of this device is evaluated by placing a *z*-oriented point dipole at a precise distance below the apex of the nanotip, with the objective of quantifying how much of the dipole power is collected via the excited SR-SPP and ultimately reaches the single waveguide section (Fig. 2). In first instance, we consider such a point dipole at an axial distance of 1 nm from the apex of the nanotip (*λ* = 1.55μm). In this case, a significant amount of power is collected and transferred to the output of the dielectric waveguide [Fig. 9(a)]. The spatial resolution of this device Δ*x* is then determined by placing the dipole source at different *x*-positions (direction perpendicular to the tip axis) and calculating the output power, showing that the device possesses a minimum resolvable feature size of Δ*x* ∼ 2.42nm (Full Width at Half Maximum FHWM), corresponding to *λ*/638 [Fig. 9(b)]. Remarkably, we find that Δ*x* is only weakly wavelength-dependent [blue line in Fig. 10(a)]; e.g., at *λ* = 2.2μm, Δ*x* = 2.54nm, corresponding to *λ*/787. The efficiency of this collection process is determined by calculating the ratio of the power at the output end of the device with respect to the dipole power, obtained by integrating the Poynting vector magnitude over a circle of radius 1 nm surrounding the dipole. Note in particular that for the dipole/tip distances considered, the dipole energy does not radiate isotropically; instead, calculations indicate that at least 94% of the point dipole energy absorbed by the tip, corresponding to a highly directional emission towards the tip and into the waveguide: note the small values of the Poynting vector magnitude in [Fig. 9(a)] for *z* > 6.3μm. We also find the collection efficiencies to be in the range of 6–12% between 1.2–2.4 μm [red line in Fig. 10(a)]. The collection efficiency reduces and the spatial resolution increases for a larger apex-dipole distance [Fig. 10(b)]. Nanoscale objects should be placed in general in close proximity to the nanotip. This is an intrinsic property of the plasmonic nanotip and not of the coupling section itself. Note that the collection performance is dependent on the nature of the excitation source (e.g., relative orientation of dipole with respect to tip symmetry axis). For example, an *x*-oriented dipole placed at 1 nm distance from the tip has a collection efficiency of only 0.03% at *λ* = 1.55μm. Furthermore, when a *z*-oriented dipole is placed at a distance of *x* = 250nm (i.e. aligned with the central axis of the dielectric core, with the same *z*-position discussed so far) the collection performance is 0.015%, suggesting that direct coupling of a point source to the dielectric core is negligible compared to that via the gold nanotip. Finally, we have observed that quenching of the point dipole as a result of Ohmic losses in proximity to the gold nanotip under consideration is significantly less than that typically observed e.g., in nanocylinders and nanospheres [47]; this, combined with the efficient excitation of the SRSPP mode which couples to the dielectric waveguide, yields the comparatively large collection efficiencies observed.

## 4. Future directions: hybrid optical fibers

Since the EM approach is of general origin and can be applied to any coupled waveguide situation, we now consider the case of a fiber-like superfocussing device including a plasmonic coupler for the excitation of the strongly-bounded plasmonic nanowire mode. The design [schematic in the inset of Fig. 11(a)] consists of two parallel running cylindrical wires embedded in silica, with one wire being made from a lossless dielectric and the other from gold [31]. This fiber structure is in fact a 3D version of the 2D multilayer situation discussed earlier. Effective indeces and losses for all relevant modes are respectively shown in Fig. 11(a) and 11(b). Generally, the metallic nanowire supports several modes, whereby the mode with the highest effective index (TM_{0}-mode) is radially polarized and corresponds to the tightly-bound SR-SPP IEM. Following our previous line of reasoning, we find that at *λ* = 1.55μm the effective index of the TM_{0} IEM of a 500 nm-diameter gold nanowire matches with the fundamental HE_{11} core IEM of a step index fiber waveguide of 1 μm diameter in the case the core has an index of *n _{core}* = 1.67 [Fig. 11(c)]. We also observe the formation of two new HEMs if a finite separation of

*t*= 500nm is considered [Fig. 11(c)]. Note that this refractive index value is comparable to that of many glasses that are compatible with filling and infiltration techniques (e.g., metaphosphate glasses [48]), which, together with gold, allows for the immediate implementation of such a device. This system possesses very similar dispersion properties to that of the planar case, such as mode splitting and parallel-running dispersions as discussed. The key difference is the modal loss, which is about ten times lower compared to the planar case [Fig. 11(b)] although the gold wire has a diameter more than one order of magnitude larger than the gold film thickness considered previously. These larger dimensions are favorable from the fabrication point of view, as they strongly relax fabrication constrains and prerequisites.

Applying the EM formalism outlined earlier [see Eqs. (1)–(6)], the power transfer properties of this hybrid device can be determined. The resulting *S _{z}* in the

*xz*plane (

*y*= 0) reveals that the plasmonic fiber coupler has a mode-coupling behavior being similar to that of the planar structure discussed earlier [Fig. 12(a)]. The Eigenmode approach reveals

*η*= 53% can be achieved for our example at

*z*=

*L*= 22μm, which is significantly shorter than the typical lengths of continuous 500 nm gold nanowires that can be integrated into fibers using pressure-assisted melt filling (approximately 50 μm), see [31], Fig. 4(b). Note that within the scope of FE-simulations such a length is extremely long and as a consequence, FE-calculations are impractical for the analysis of directional-coupling of fiber-based plasmonic couplers due to the prohibitively large mesh elements required, leading to unfeasibly long computation times. This puts the EM approach in favor of full numerical calculations, particularly for hybrid-fiber structures, since it only requires the Eigenmode fields and dispersions.

_{c}In a next step, we assume a conical nanotip at the end of the metallic nanowire, formed by tapering down the nanowire from 500 nm diameter to 5 nm over a length of 0.5 μm. We reveal the focussing properties of the TM_{0} mode on this nanotip using 3D FE-simulations [Fig. 12(b)–12(d)]. Since such a tip-like structure can be formed by cleaving a fiber incorporating a gold nanowire [Fig. 1(a) and [35]], we assume that the first part of the tip (with constant outer diameter) is embedded in silica, and that the remaining tapered part of the nanowire is in air. The resulting calculations [Figs. 12 (b)–12(d)] show a strong superfocussing behavior down to spot sizes of lateral dimensions of only 10 nm [Fig. 12(d)], thus providing a demonstration of the potential of this fiber-based approach. The properties of the plasmon propagating down the nanowire tip can be analyzed using an eikonal approach [12] and will be the focus of another study.

## 5. Conclusion

In conclusion, we have shown that an Eigenmode ansatz relying on a superposition of propagating bound modes can be used to design hybrid dielectric-plasmonic couplers for the excitation of the strongly bounded short-range surface plasmon polariton modes. In an example multilayer structure we showed that, when designed correctly, such a plasmonic-dielectric coupler can operate with high power transfer efficiency (more than 50%) over more than 800 nm spectral bandwidth, in excellent agreement with finite element simulations. Compared with full-vector calculations, this analytical formulation has the benefit of being straightforward to implement, immediately providing the most relevant parameters with regards to the power-exchange properties of hybrid plasmonic directional couplers, e.g. the characteristic directional coupling length and power transfer efficiency. We then applied this Eigenmode scheme to designing a broadband a superfocussing device showing subwavelength spatial resolution, which can either operate in collection and delivery mode with total efficiencies of 10% or more. We extended the Eigenmode scheme to fiber-based devices consisting of a metallic nanowire running parallel to a cylindrical dielectric core, to design a directional coupler exciting the radially polarized (short-range) plasmonic mode. This analysis reveals that such an all-in-fiber super-focussing device can be implemented using realistic materials and structural dimensions. This scheme can be adapted to other nanowire-enhanced optical fiber designs [49], and is applicable to other spectral domains. This approach will have important applications in all-fiber plasmonic superfocussing, nanoscale sensing and nonlinear plasmonics.

## Acknowledgments

A.T. acknowledges financial support from the Alexander Von Humboldt Foundation.

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