We analyze the properties of HE2 surface plasmon polaritons on fiber-integrated gold, silver and copper nanowires and find a systematic blue-shift of the measured resonance wavelengths which we attribute to the emergence of a narrow nanometer-sized gap between the nanowire surface and the surrounding silica cladding. Our analysis relies on the determination of the nanogap width from the experimentally measured phase-matching wavelength by comparison with numerical simulations, revealing that these gaps are much smaller than expected from the bulk material considerations. This implies a diminished coefficient of thermal expansion along the radial direction that we believe results from the domination of interfacial van der Waals forces at high temperatures. These results are important for future fiber designs involving nanowires fabricated by pressure-assisted melt filling.
© 2017 Optical Society of America
The excitation of surface plasmon polaritons (SPPs) on gold nanowires (NWs) has been observed in various types of hybrid optical fibers (HOFs) [1–3]. The guided NW-SPPs are closely related to their planar counterparts, traveling on helical trajectories with discrete helix angles along the wire surface and experiencing curvature-induced geometric momenta [4,5]. It was also shown that arrays of metal NWs impose plasmonic hybridization, leading to plasmonic supermodes and plasmonic band gaps [6–8]. So far, research has mostly targeted gold as the plasmonic metal to be integrated into optical fiber, with only minor effort being invested for investigating other metals such as silver , copper  or platinum . One issue that has been regularly observed in experiments but has not been rigorously addressed so far is the systematic blue-shift of the measured phase-matching wavelength of the SPP and the fiber core mode [1,2,6,11]. This spectral blue-shift appears in all systems fabricated by either direct drawing or pressure-assisted melt filling and is typically associated with a nanometer-sized air gap (nanogap) between the capillary wall and the surface of the wire, imposed by the mismatch of the thermal expansion coefficients of silica and metals. The blue-shift originates from the partial location of the evanescent field of the SPP mode inside the air nanogap surrounding the nanowire. Since the relative permittivity of air is much smaller than that of silica (air: 1.0, silica: 2.1), the SPP mode experiences a smaller effective refractive index, which overall blue-shifts the plasmon dispersion curve and thus the crossing point of SPP and HE11-like mode.
Here, we present a detailed analysis of these nanogaps and their influence on the plasmonic properties of the hybrid fiber, with a focus on the HE2 SPP mode. This study will play an important role for novel HOF materials  and designs including fiber-integrated polarizers and polarization converters  and directional couplers [14,15].
2. Fiber design and sample preparation
The HOF geometries considered here consist of a dielectric core and a parallel metal NW (see Fig. 1), which are prepared by fiber drawing and subsequent post-processing. The pristine silica fibers possess a GeO2-doped parabolic graded-index core (diameter D, maximum doping level: 9.5 mol %) with a parallel cylindrical hole (center-to-center distance Λ, diameter dhole). Two different types of such modified graded index fibers (MGIFs) were fabricated (MGIF1: D = 1.7 µm, Λ = 3.8 µm; MGIF2: D = 2.9 µm, Λ = 4.9 µm), exhibiting different light guiding characteristics (e.g., cutoff wavelength). The hole diameter was adjusted during fiber drawing in the range of 0.3 µm < dhole < 2.0 µm by changing the applied air pressure.
2.1. Nanowire fabrication
Pressure-assisted melt filling (PAMF)  was used to press liquid gold, silver or copper into the empty hole of the MGIFs at appropriate temperatures (see Table 1). The sample fabrication procedure involves (i) placing a few millimeter long bulk metal wire (diameter: 50 µm) into a capillary (inner diameter ≈ 80 µm), (ii) splicing this arrangement to an intermediate capillary (inner diameter ≈ 20 µm), (iii) splicing this combination to the MGIF and (iv) heating it in order to melt the metal and pressing it into the empty hole via Argon gas. The additional step (ii) yields mechanically stable splices through better matching of the hole diameters since a direct splice of the 80 µm capillary to the MGIF can yield weak connections. Oxidation or other chemical reactions were prevented by collapsing the hole of the MGIF at its open side and evacuating the system between step (iii) and (iv) for several hours at 600 °C.
The filling dynamics within PAMF are governed by the Washburn equation, allowing to determine the filling time Table 1. For typical hole diameters of 1 µm and a pressure of 200 bar, the time to fill a 30 cm sample is of the order of a few minutes.
2.2. Formation of nanogaps
The linear coefficient of thermal expansion (CTE) of fused silica is comparably small and can be assumed to be constant between room and filling temperature Tfill, with an average value of αSiO2 = 0.5 × 10−6 K−1 . However, metals like gold, silver and copper typically exhibit CTEs more than one order of magnitude larger than that of silica. This mismatch of the CTEs suggests the formation of a nanogap between the surfaces of the wire and the hole during cooling to ambient temperature (293 K) after PAMF. The width w of this nanogap can be calculated by Table 2) and Tm is the melting temperature (in Kelvin). The resulting relative gap width w/dhole is listed in Table 2, suggesting the emergence of gap widths in the magnitude of 10 nm per micrometer of hole diameter.
In order to investigate the existence and the spectral influence of a possible nanogap around the metal NW, the attenuation spectra of several gold-, silver- and copper-filled MGIFs with different hole diameters were measured. It is well-known from experiments and numerical simulation that the attenuation of the fundamental core mode drastically increases around the phase-matching wavelength. In this study we want to focus on the interaction of the HE11 core mode with the HE2 NW-SPP.
3.1. Experimental setup
The optical characterization of the MGIFs relies on measuring the phase-matching wavelength between the fundamental fiber core mode and the HE2 SPP using the cutback technique (setup shown in Fig. 2). The input side of the transmission setup is composed of an unpolarized supercontinuum source (NKT Photonics SuperK COMPACT) and a combination of mirrors (M1, M2), a linear polarizer (LP) and a half-wave plate (HWP) for beam and polarization control. Efficient light coupling into the sample was ensured by a 40× objective (O1, NA=0.65). Typical sample lengths were about 40 cm, with an unfilled part at the input side (length: ~20 cm) in order to strip off undesired cladding light by applying mode-stripping gel. The output light is collimated using a 60× microscope objective (O2, NA=0.90) and directed into a multimode fiber (MMF) via another objective (O3, NA=0.25). An aperture (I1) is inserted into the beam path to spatially block the light propagating in the cladding, which is excited at the sample input in addition to the dielectric core mode. The collected light is finally analyzed by an optical spectrum analyzer (OSA).
3.2. Experimental results
In this work we investigated eight samples filled with gold, silver and copper. Fig. 3(b) shows an example polarization-resolved attenuation spectrum of a silver-filled MGIF1 (hole diameter: 380 nm). A distinct increase of the attenuation at around 614 nm is observed for both polarization states. This attenuation peak originates from a coupling of the fundamental core mode and the HE2 SPP on the silver NW, as confirmed by finite-element method (FEM) simulations. As Fig. 3(a) shows, it is found that if no nanogap between the silica wall and the wire is considered, the simulated attenuation peak is located around 630 nm, being 16 nm longer than observed in the experiment. A narrow air gap of constant width w = 0.8 nm around the wire shifts the phase-matching position towards shorter wavelengths and matches well with the experimental results. It is important to note that the centered alignment of the NW inside the hole is an idealized approximation since the NW necessarily touches the wall. The position of the NW strongly affects the magnitude of the attenuation but, has only negligible influence on the phase-matching wavelength.
Using FEM simulations we identified two origins of the different loss magnitudes of the experimental data compared to calculations: First, discontinuities of the NW along the fiber reduce the total amount of metal the optical mode interacts with during propagation. Secondly, the transverse position of the nanowire inside the hole changes randomly along the sample length. To analyze the influence of the transverse NW position we performed example FEM simulations (geometry MGIF1, inset of Fig. 3) for the situation of a centered NW with a azimuthally uniform nanogap width (4.3 nm, case i), a NW touching the wall at the location facing the fiber core (case ii) and a NW touching the wall opposite to the core (case iii). All three cases have their resonance at around the same wavelength (resonance wavelength difference is less than 1 nm) but with different loss magnitudes. Our simulations also showed that the transverse alignment of the wire inside the hole has no influence on the spectral bandwidth of the resonance peak, as verified by calculating the average of the three above-mentioned cases of wire location inside the hole. The partial filling along the fiber axis (i.e., the junctions between the filled and unfilled parts) was not included into the simulations, as preliminary simulations revealed that the overlap of the dielectric-type modes in the filled and unfilled sections is >90 %, indicating that an excitation of the plasmonic-like mode in the wire section can be neglected .
4. Determination of the nanogap width
For all simulations the permittivities of pure silica and Germanium-doped silica were obtained from a Sellmeier equation [28,29], while the permittivity of the metals has been modeled using a Drude plus two critical points (D2CP) model :31, 32], the corresponding values for the permittivity of copper have been fitted to experimental data  in this work, yielding an well matching analytic expression for the dielectric function of copper for the first time. The D2CP parameters for all three metals are listed in Table 3.
In order to determine the actual width of the nanogap inside a fiber sample, a full-vectorial transfer-matrix method was utilized , treating the fiber core and the metal wire as two independent waveguides. In a first step, the effective index of the fundamental dielectric core mode nHE11 was calculated at the experimentally observed resonance wavelength λr by a discretization of the parabolic refractive index profile (Fig. 1(c)) into 15 concentric layers. Secondly, the coaxially wire/nanogap/silica system was solved at the phase-matching condition as a function of w and in order to find the gap width matching the measured blue-shift. Secondly, the coaxially wire/nanogap/silica system was solved at the phase-matching condition as a function of the gap width w and in order to find the gap width matching the measured blue-shift, where and is the effective refractive index of the SPP and the fundamental core mode, respectively. The resulting absolute and relative nanogap widths for all fiber samples are listed in Table 4. Because w naturally vanishes for dhole → 0, an offset-free linear function was fitted to the estimated gap width, yielding values w/dhole of 4.3(3.6, 4.8) nm/µm, 3.3(2.7, 3.8) nm/µm and 4.4(3.6, 5.2) nm/µm for gold, silver and copper, respectively (last column in Table 4).
The error of the determined gap width was estimated by repeating the calculations at the lower and upper limit of the error interval of dhole and λr. Based on these values the phase-matching wavelengths for all hole diameters between 300 nm and 1400 nm were calculated and are shown in Fig. 4. The minimum and maximum values for the phase-matching wavelength correspond to simulations assuming a gap width obtained from Eq. (2) (dahsed lines in Fig. 4) and the absence of a gap (dotted lines in Fig. 4), respectively. The experimentally observed phase-matching wavelengths consistently show excellent agreement match with the simulations when the appropriate nanogaps are considered. However, it is important to note that the experimentally observed phase-matching wavelengths are shorter than those calculated neglecting the air gap but longer than those assuming the CTE of the bulk metal. This indirectly proves the existence of a narrow nanogap around the NW but also implies that its width must be smaller than predicted by the bulk CTE. It needs to be pointed out that to our knowledge there is no approach to study nanogaps of these dimensions directly, as cleaving fibers with incorporated NWs leads to the formation of nanotips that typically start tapering at a position inside the capillary, so that the nanogap dimensions cannot be visually discerned [15,35].
A possible explanation for this phenomenon is the very small roughness of the inner surface of the hole which is caused by frozen-in surface capillary waves, typically of the order of a few hundred picometers . Since the capillary was filled with liquid metal at high pressure, it can be assumed that the average distance h between the surfaces of the hole and NW is of the order of the surface roughness (see Fig. 5). The attracting van der Waals force (per unit area) between the two surfaces can be estimated by FW = A/6πh3, with A being the material-dependent Hamaker constant (typically in the magnitude of 10−19 J for solids ). Assuming an average distance h ≈ 0.2 nm, the van der Waals force per unit area FW (which attracts the two surfaces) is ∼ 109 N/m2. However, a counteracting thermal-contraction-induced force Fc at a given temperature T also exists, and can be roughly estimated to be Fc = E(Tm − T)(α − αSiO2)  and is assumed to vanish at the melting temperature of the metal. Here, E is Young’s modulus and α is the linear CTE of the nanowire material. Literature values of E for gold, silver and copper are 78 GPa, 83 GPa and 130 GPa  respectively, yielding contraction forces of the same magnitude as the van der Waals force for a complete cooling to room temperature. Therefore, the cooling process can be divided into two phases as follows: (i) The NW cools from the melting temperature to a threshold temperature Ttr without detaching from the silica wall because Fc < FW. In this phase the contraction of the NW is constrained and it is shrinking solely along its axial direction (see Fig. 5(a)). (ii) During cooling the temperature T drops below Ttr where Fc > FW. Below this temperature the NW detaches from the silica wall and contracts in both the axial and radial direction (see Fig. 5(b)). As a result, the NW is not shrinking radially during the entire cooling period, but only below the threshold temperature. This leads to an effectively smaller gap width than predicted by Eq. (2) which may also be affected by other process conditions, e.g., cooling rate.
In conclusion, we have analyzed the properties of gold, silver and copper nanowires in MGIFs and determined the spectral attenuation of the fundamental core mode, showing a blue-shift of the resonance wavelength compared to finite-element simulations. We attribute this discrepancy to the emergence of an nano-sized air gap between wire surface and silica wall, which is caused by the mismatch of the CTE between metals and silica. Our simulations showed that the spectral blue-shift can indeed be explained by this nanogaps. However, using a detailed analysis procedure it was found that their width is smaller by approximately 50 % to what is expected from the literature values of the CTE in order to fit the experiments. We explained this effect by the interaction of van der Waals and contraction forces at the metal-silica interface, constraining the radial contraction of the metal during cooling and resulting in an effectively smaller gap width. Future studies will aim to analysis other filling metals using our nanogap analysis procedure with one goal being the integration of a plasmonically relevant metal or alloy with a negligible air gap width.
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