## Abstract

Modal properties of a plasmon guiding structure consisting of two parallel metallic nanowires embedded in a dielectric material have been analyzed employing the finite element method. The coupling characteristics of the surface plasmon supermode reveal extreme yet linearly changing confinement for a moderate range of nanowire separation. The potentialities of this structure in different nanophotonic applications have been explored in this study based on modal characteristics variation with the alteration of both the operating wavelength and the cross-sectional geometry.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Metallic nanostructures supporting surface plasmon excitation are being investigated extensively because of its high promise in different research areas especially in optical waveguiding, laser development, biosensors and molecular engineering [1–4]. Plasmon waveguides are being thought of as the replacement of the traditional electronic interconnects because of their subwavelength light confinement ability and highspeed data propagation capability [5–7]. Although their diffraction limited mode confinement leads to high integration density in photonic chips, they suffer from large modal attenuation [3]. Therefore, the waveguides are in extensive research to circumvent the problem of high modal attenuation. Previously, different multilayer structures like metal-insulator-metal (MIM), insulator-metal-insulator (IMI) and dielectric-metal hybrid waveguides have been proposed in this regard [7–9].

Another important aspect of plasmonics is surface plasmon resonance, whose spectral location is highly sensitive on the refractive index variation of surrounding dielectric medium of the metal waveguide. This property makes metallic nanodevices remarkable for sensing applications [3,4]. Thus, different types of biosensors, gas sensors, and nanogap sensors are attracting the attention of researchers for their ever-growing demands in medical science and nanoscience [10–12]. Moreover, a plasmonic guiding structure can be suggested for sensing applications if any modal property shows linear variation with high responsivity with respect to the sensing parameter.

Metallic nanodevices also play their parts in surface-enhanced Raman spectroscopy (SERS), where a specific material or molecule is characterized by a red or blue shift in the scattered light field spectra. This spectral shift results from the coupling of the incident optical field with the vibrational frequency of molecule. Though the shift is unique for a certain material, it appears in spectra with a very small peak due to an insignificant amount of inelastic scattering [3]. It demands highly localized electric field in the near field of SERS sample to strengthen and detect the shift of incoming frequency [3,13]. As mentioned earlier, metallic nano-particles support surface plasmon polariton, which can obtain the required field enhancement accumulating surface charge into the deep subwavelength dimensions. This field localization is known as plasmonic hotspot [14]. Plasmonic hotspots also demonstrate their potential in many other growing applications like controlling SPP launching with hotspot cylindrical waves which have a good prospect in future high performance photonic integrated circuit design [15]. In the literature, a wide variety of structures are already proposed as plasmonic hotspot sources [16–18]. However, there is still need for research to understand the coupling characteristics of SPP modes.

In this work, plasmonic waveguide supermode characteristics in coupled elliptical nanowires have been examined. This guiding structure containing two elliptical gold nanowires in ZnO cladding has been analyzed for different nanowire spacing. The modal properties of some lower order non-localized surface plasmon modes have been derived and compared with those of some previously proposed geometries. Although a similar type of analysis is done in [19], the feasibilities of this structure as plasmonic hotspot source, optical interconnect, optical nonlinear effect study as well as nano-pressure sensor have been analyzed here. Although, ZnO has been considered as the cladding for its outstanding optical and electrical properties with well-established nanoscale fabrication compatibility [20,21], it is possible to replace ZnO with other dielectric materials to achieve SPP modes. Importantly, the modal analysis reveals that the fundamental gap mode excited here is the only supermode that offers novelties in single mode plasmonic applications. In addition to the gap mode, this structure supports higher order modes. The propagation length of these modes can be tuned by manipulating the cross-sectional geometry.

## 2. Design and methodology

The guiding structure consists of two identical elliptical gold nanowires embedded in ZnO dielectric medium as shown in Fig. 1. Each of the nanowires has a major axis length of 2a along the x-direction and minor axis length of 2b along the y-direction. The z-axis has been considered as its propagation axis. For small d the surface plasmon modes of two nanowires couple to give rise to coupled surface plasmon modes. In this type of nanowire structure, FEM can precisely represent curved dielectric interfaces as it uses many triangular meshes of different shapes and sizes [10,22–24]. Apart from this, in an elliptical metal/dielectric waveguide, the optical modes appear with all six components of the E and H fields, which consequently invalidates scalar-FEM analysis [10]. In this work, an E-field based full-vectorial FEM with a cylindrical perfectly matched layer (PML) has been employed for eigen mode analysis using COMSOL Multiphysics, a commercially available software. Here, triangular mesh with a minimum size of 0.48 nm was chosen in this study, which is geometry controlled.

Since gold is a plasmonic material, the dispersion of the complex refractive index should reflect the Drude model and interband transitions [3]. Also, the gold nanowire dimensions are comparable with the mean free path of free electrons in bulk gold, hence the size-dependent damping frequency has been included in the design [25]. Figure 2 shows the dispersion of the complex refractive index of the gold. It has been obtained by an analytical formula given by Etchegoin et al. [26,27]. The real part of refractive index of the ZnO cladding region changes from 2.2 to 2 in the operating wavelength range as shown in Fig. 3. The complex index of ZnO has been formulated by Holden et al. [28] and Jellison [29]. In the modal analysis of thestructure, the waveguide parameters a = 120 nm and b = 100 nm have been considered. Firstly, d = 270 nm has been regarded to obtain the modal solutions, then d is varied to determine the effect of gap size on the plasmon modal properties. The tunability of mode propagation is also obtained by taking b = 30 nm.

Regarding the fabrication of this structure, the Sol-Gel template synthesis technique reported in [30] can be employed. It suggests that cylindrical nanowires of metals, polymers or semiconductors can be fabricated in cylindrical pores of nanoporous membrane or other solids. Accordingly, a single gold nanowire or even its bundle can also be fabricated in ZnO template [30]. Hence, the proposed structure can be easily fabricated using the above mentioned technique.

## 3. Results and discussion

#### 3.1 Polarizabilities and effective index

In a single elliptical waveguide, the electromagnetic interactions between the surface charges along the x and the y-axes are not identical, unlike a circular optical waveguide. Therefore, the horizontally and vertically polarized dipole modes excited in this kind of waveguide are non-degenerate in nature with different effective indices. As mentioned earlier, for small inter-nanowire separation, mode coupling takes place in this structure. The effect of mode coupling has been presented by comparing the modal properties of some lower order plasmon modes in the coupled structure with that of the fundamental dipole mode along the major axis of a single elliptical nanowire. It has been reported that in a circular gold nanowire with ZnO cladding, the effective indices of electromagnetic modes exceed the refractive index of cladding at λ = 0.52 μm and propagates as SPP mode for longer wavelengths which is also applicable for elliptical nanowires [10]. The location of the resonant peak of the effective index dispersion depends on the location of the pole of the polarizability factor [10,31].

The polarizability factor is defined as the ability to form instantaneous dipole in response to a harmonic electric field. The plasmon oscillations in nanostructures can be attributed to the electrical polarizations according to the optical theorem of anisotropic Rayleigh particles[31]. Thus, plasmon resonance corresponds to the maxima of transverse polarizabilities to attain the maximum electric displacement. The electrical polarizabilities have been expressed as a function of the cross-sectional area of the cylindrical nanowire and the respective dielectric functions of the constituent materials and is given by [10],

*α*,

_{x}*α*, and

_{y}*α*are the electrical polarizabilities along the

_{z}*x*,

*y*and

*z*direction, respectively. Also,

*ε*and

_{m}*ε*are the dielectric functions of gold and ZnO, respectively. To attribute the location of SPP resonance, the transverse and longitudinal polarizability factors of a cylindrical nanowire of a unit cross section have been regarded in Fig. 4. Here, evidently the transverse polarizability attains the resonant peak at λ = 0.55 μm. So, the plasmon modes excited here should find their resonant peaks near λ = 0.55 μm. More importantly, the location of SPP resonance does not depend on the cross-sectional geometry as implied by the Eqs. (1) and (2). Therefore, infinitely long cylindrical nanowires of different shapes support non-localized surface plasmon modes.

_{d}The real parts of effective indices of three lower order SPP modes excited in the coupled structure for d = 270 nm and the fundamental mode of single nanowire structure have been depicted in Fig. 5. Each of the SPP modes finds its resonant peak at λ = 0.56 µm and then the effective index monotonically reduces following the trend of the real part of dielectric index as in circular waveguide [10]. Therefore, the location of SPP resonance is independent of the cross-sectional geometry. This dispersion is closely related to the trend of other modal properties which will be shown in the following sections.

#### 3.2 Field distribution and power confinement

The SPP modes excited in this structure evolve due to the near field coupling between the nanowires [16]. Different higher order modes are resulted due to monopole-monopole, dipole-dipole and quadrupole-quadrupole hybridizations between the two elliptical nanowires [16]. However, only three lowest order modes are studied here which are evolved due to monopole-monopole and dipole-dipole mode hybridizations as shown in Fig. 6. From the normalized electric field distribution at SPP resonance, both mode 1 and mode 2 evolve from the hybridization of the monopole modes. While, the fundamental gap mode (mode 1) is resulted from the hybridization between the monopole modes of opposite polarities, and mode 2 is resulted from the hybridization between the monopole modes of same polarities. On the contrary, mode 3 is emerged from the coupling between the parallel dipole modes polarized along the minor axes the elliptical nanowires. In addition to the modes of the coupled structure, the electric field distribution of the fundamental dipole SPP mode (m = 1) has been demonstrated in Fig. 6.

The evolution of the fundamental supermode could also be attributed to MIM like mode coupling between the two elliptical nanowires at proximity. However, the coupled structure does not support the leaky SPP modes like planar MIM structures [32]. From the electric field profile, it is also clear that around SPP resonant frequency, each mode finds field maxima at the inner interfaces inside the metallic region and then evanescently decays from the maxima into the metal and dielectric regions. The location of field maxima shifts towards the dielectric region at longer wavelengths and the evanescent field decay length inside the dielectric region increases. So, at longer wavelengths, the field confinement to the interface decreases as theoretically pointed out in [3].

The Poynting vectors along the propagation axis of different SPP modes at resonance are demonstrated in Fig. 7, where the respective color bars are in W/m^{2}. As the fundamental supermode excited here, has electromagnetic energy confined in the inter nanowire gap region as well as in the metallic nanowires, the power confinement in the dielectric region signifies the gap confinement. The confinement factor is defined as the fraction of total propagating mode power that resides inside the metallic region. The gap confinement on the other hand is defined as the fraction of total mode power residing inside the cladding for the fundamental supermode (i.e., gap confinement = 1− confinement factor). At shorter wavelengths near resonance, a larger fraction of mode power resides inside the metallic region as shown in Fig. 8. This is due to much larger mode effective index than the dielectric refractive index. Adversely, as wavelength increases the effective index decreases monotonically resulting in less amount of power confinement in the metal. For the fundamental supermode in the coupled structure, high field amplitude at the boundary is resulted from the phase matching of electron oscillation. In addition to that the continuity of tangential field components along the interface increases the field decay length inside metal. Hence, mode 1 exhibits higher confinement factor in metal than other SPP modes even at longer wavelengths which has been depicted in Fig. 8. However, the supermode exhibits appreciable energy localization in the dielectric gap region instead of getting dispersed at longer wavelengths. Specifically, in theinfrared regime less than 25% confinement factor leads to a remarkably high gap confinement (75%-90%). Evidently, the highly intense electric field localization in the dielectric region (75%-90% gap confinement) associated with this mode can be exploited to study the nonlinear effects of the dielectric cladding material. Moreover, a liquid solution instead of ZnO can be explored for its single molecule florescence emission enhancement of some micro-molar concentration solution [33].

#### 3.3 Attenuation characteristics

The attenuation characteristics of a waveguide are determined by the wavelength dependence of mode propagation length. The propagation length is defined as the distance traveled by an electromagnetic mode along the z-axis to fall its mode power amplitude by 1/e [4]. The propagation length has been obtained by,

where,*K*is the magnitude of the imaginary part of effective index and

_{eff}*k*being the free space wavenumber. Figure 9 demonstrates propagation lengths of different plasmon modes in the operating wavelength range. The dispersion of propagation length can be explained by virtue of the modal confinement to the metal-dielectric interface. In the shorter wavelength regime, a higher attenuation is observed since the larger portion of total mode power propagates through the metallic region and causes ohmic heating [3]. As the operating wavelength increases, the confinement in metal reduces resulting in longer propagation length. Evidently, the florescence enhancement and nonlinear effect study would have the optimum performance at longer wavelengths. The orthodox tradeoff between confinement and attenuation has been clearly demonstrated by the plasmon modes excited here. As the mode 1 of the coupled structure has the highest confinement factor, it has the smallest propagation length at a given wavelength. On the contrary, the mode 3 encounters the lowest confinement hence the highest propagation length as shown in Fig. 9.

_{0}Another point to note that although both mode 2 and mode 3 have nearly equal confinement in metal, the latter has surprisingly longer propagation length in the longer wavelength regime. This fact can be elucidated by looking at the longitudinal component of the electric field of the modes of interest. Generally, the long-ranging SPP modes supported by planar IMI structures have antisymmetric E_{z} field distribution [4]. For vanishing metallic layer thickness, the modes evolve into TEM modes. Hence propagation length increases with reducing intermediate layer thickness at a given wavelength. In the coupled structure, both mode 2 and mode 3 have antisymmetric E_{z} field as shown in Fig. 10. But the separation between the accumulated opposite surface charge is smaller for mode 3. So, mode 3 is more like TEM mode than mode 2 and results in smaller attenuation. Importantly, the propagation length of mode 3 can be further increased by decreasing the minor axis length (2b) of each elliptical nanowire. Specifically, the reduction of b decreases the separation between the antisymmetric peaks of E_{z} field profile resulting in more resemblance with the TEM mode supported by ZnO cladding. So, the propagation length at 1000 nm wavelength can be tuned from 6.8 μm to 22 μm by changing b from 100 nm to 30 nm as indicated in Fig. 11. Therefore, mode 3 has a better promise in waveguiding compared to that of traditional nanowire waveguide modes as the modal attenuation can be tuned by engineering its cross-sectional geometry.

#### 3.4 Effective mode area

The effective mode area is defined as the equivalent cross sectional area through which the electromagnetic power propagates and is formulated using the following equation [10],

*E*is the normalized electric field intensity and the ∫∫(.)

_{norm}*ds*indicates surface integration over the cross-section of the waveguide. Around resonant frequency, the plasmon modes are tightly bound to the interface and hence rendering smaller effective area. Conversely, at longer wavelengths, the confinement in metal falls significantly thus having higher effective area. Therefore, each of the SPP modes has resonant minima around λ = 0.56 μm followed by monotonically increasing effective mode size as demonstrated in Fig. 12. Since the longer ranging mode 3 in coupled structure has the lowest confinement in metal, the mode attains the highest effective area at a given wavelength. The fundamental mode with the highest attenuation has the minimum effective mode area consequently.

Although the effective area of the mode increases at longer wavelengths as shown in the inset of Fig. 12, it renders very small $d{A}_{eff}/d\lambda ,$ leading to a very small effective area for large bandwidth compared to other plasmon modes. The nearly zero gradient of effective area of mode 1 compared to the other modes in this structure can be attributed to the larger fraction of mode power propagating through the metallic region and the MIM like mode coupling in the gap region. The MIM structures excite plasmon modes with a monotonically decreasing effective area dispersion with increasing wavelength [4,34]. Therefore, the MIM like mode coupling localize the mode power of mode 1 inside the gap region and the gradient of effective area dispersion curve is reduced. Again, the extremely low effective area for a large bandwidth can be attributed to the dominance of term in the denominator of Eq. (4). This is a strong evidence of electric field localization in the inter-nanowire gap region as the field enhancement factor is proportional to the local electric field raised to the power of four ($\propto {E}_{loc}^{4}$) [3,4]. Therefore, this structure can be proposed as a wideband plasmonic hotspot source. Also, a very high integration density is achievable due to the extremely low effective mode area of mode 1. Furthermore, the field enhancement factor can be quantified by the Purcell factor, which relates the quality factor and mode volume [17]. As the effective mode area is analogue to the effective mode volume, the field enhancement factor increaseswith decreasing effective area. Hence, the proposed geometry may serve as a wideband plasmonic hotspot source for rendering very small effective area for a wide range of the operating wavelengths [17].

#### 3.5 Effect of the gap between nanowires and figures of merit

In the previous sections, all the modal properties have been obtained for fixed inter nanowire spacing d = 270 nm. Now let us see the effect of this gap width. Figure 13 shows the variation of effective index, confinement in metal, modal attenuation and effective mode area of mode 1 in coupled structure for varying gap width at λ = 0.6 μm. With the decrease in inter nanowire spacing, the effective mode area decreases. As the field enhancement factor is proportional to the quality factor and inversely proportional to the mode volume [17], reduced effective area significantly increases the electric field enhancement factor, and thereby increasing the resolution of SERS based characterization, fluorescence enhancement and nonlinear dielectric effect observation.

Here all the parameters except the effective area mostly resemble those of an uncoupled single nanowire mode for d > 300 nm. So, with respect to these parameters the structure is highly sensitive on the gap width for d < 300 nm. However, the effective area demonstrates almost linear variation with respect to the gap size for a longer range (up to d = 400 nm). Therefore, it is possible to obtain even higher field localization for smaller gap size. It has been also found that the sensitivity of modal attenuation is more prominent for higher order modes than that of the fundamental mode. Hence, this structure may be suggested to sense small pressure variations for the reported high sensitivity of modal parameters. It can be seen from Fig. 13 that for smaller gap size the loss is highly sensitive (the slope of the dispersion has a remarkably high absolute value) with respect to the variation of gap size. Therefore, very small pressure variation may possibly be characterized by this waveguide. It can also be seen that the effective index is almost constant after 400 nm gap. We have also seen that at about 1200 nm, the effective index remains almost same indicating its identical interaction. Next, we have simulated the design for d = 650 nm at λ = 600 nm (d > λ). In this case, as isseen in [35], we have also found strong field localization for higher order modes. Furthermore, we have found that the modes remain almost same when an offset of one waveguide along y axis is introduced and the properties also remain almost identical. As for example, if an offset of 20 nm along y axis is introduced, the confinement factor, propagation length and effective area change about 2.2%, 2.3% and 5%, respectively, from zero offset values. These small deviations render flexibility in designing the waveguide structure.

In the preceding sections, it has been found that plasmon waveguides manifest a tradeoff between surface confinement and modal attenuation. The less the propagation the more the confinement. So, to compare the performances of different waveguides, the figures of merit based on the tradeoff parameters (confinement and loss) should be considered. The mode 1 in coupled structure shows high promise in achieving high integration density of optical interconnects in a photonic integrated circuit, which can be confirmed looking at the figures of merit based on confinement and propagation length as well as effective area and propagation length presented in Fig. 14 and Fig. 15, respectively. Although the mode 1 in coupled structure shows smaller propagation lengths than that of the dipole (m = 1) mode in a single nanowire, it exhibits larger propagation length at the same confinement in metal than the latter as shown in Fig. 14. In case of effective area, the mode 1 is noticeable, offering remarkably low effective area with the same modal attenuation compared to the dipole (m = 1) mode of single nanowire as depicted in Fig. 15.

Furthermore, the coupled structure offers a higher order mode (mode 3) which has larger propagation length than the fundamental mode as has been stated previously. Also, the mode coupling facilitates the engineering of modal attenuation of this mode. Since the long propagating surface plasmon modes have lower surface confinement, a figure of merit based on propagation length as well as confinement factor should be considered. The mode 3 in the coupled structure has higher surface confinement with moderately lower modal attenuation than the dipole (m = 1) mode in single elliptical nanowire as shown in Fig. 16. To use a plasmonic waveguide as an optical interconnect, the effective mode size of long-ranging mode should be considered to obtain high integration density of photonic devices. Therefore, a figure of merit based on effective area and propagation length has been presented in Fig. 17. Although, mode 3 is longer ranging, the effective mode size is moderately lower than that of the dipole (m = 1) mode in single-nanowire as exhibited in Fig. 17. So, the figures of merit suggest the coupled structure as a moderate waveguiding geometry for supporting mode 3. This mode can guide light for comparatively longer range with an appreciable subwavelength confinement.

## 4. Conclusion

The modal properties of the gold elliptical nanowire based coupled structure reveal that the fundamental mode has a very small effective area for a wideband of operating wavelength. With increasing wavelength, the drastic fall of metal confinement offers highly intense electric field in gap with small effective area for a large bandwidth. Interestingly, the field maxima shift from metallic region to dielectric region as the wavelength increases. Specifically, in the infra-red region more than 90% of total SPP mode power propagates through the gap region. This highly confined electromagnetic mode can be employed effectively in applications where large field enhancement, hence smaller effective area is essential. Also, the only one supermode tenders the opportunity to use it as a single mode plasmonic waveguide structure. Furthermore, the mode 3 excited in this structure shows a high promise in waveguiding in the light of a good figure of merit and tunable propagation length. By proper geometrical modification like increasing the ellipticity of each nanowire and forming metal-dielectric hybrid waveguides, the propagation length can further be increased. This may yield improved performance in highly confined long ranging SPP propagation.

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