Hybrid-mode assisted long-distance excitation of short-range surface plasmons in a nanotipenhanced

Transcription

1 Hybrid-mode assisted long-distance excitation of short-range surface plasmons in a nanotipenhanced step-index fiber Supporting Information Alessandro Tuniz 1*, Mario Chemnitz 1,2, Jan Dellith 1, Stefan Weidlich 1,3 and Markus A. Schmidt 1,2,4** 1. Leibniz Institute of Photonic Technology (IPHT Jena), Albert-Einstein-Str. 9, Jena, Germany 2. Abbe School of Photonics and Faculty of Physics, Max-Wien-Platz 1, Jena, Germany 3. Heraeus Quarzglas GmbH & Co. KG, Quarzstr. 8, Hanau, Germany 4. Otto Schott Institute of Materials Research, Fraunhoferstr. 6, Jena, Germany *alessandro.tuniz@leibniz-ipht.de **markus.schmidt@leibniz-ipht.de

2 Supporting Figures Figure S1. Typical gold nanotips protruding from nanobore step index fibers with bore radii of (a) 550nm (b) 600 nm, (c)-(d) 50nm.

3 Figure S2. Real part of the effective index (solid lines) and loss (dashed lines) for (a) the hybrid SPP 0 mode and (b) the HE 11, TM 01, TE 01 and HE 21 modes of the gold-filled nanobore fiber (GeO 2 -doped core with 2 µm radius, doping level 9 wt% and 550 nm gold bore radius and silica cladding). Note that the hybrid SPP 0 and dielectric-type TM 01 modes have vastly different real effective indices and losses. Figure S3. Detailed schematic of typical 3D simulation space used for the analysis in Fig. 2 of the manuscript. We consider a slice of the full structure, with a TM 01 input mode (empty bore step-index fiber), and perfectly matched layers at the edges of the simulation space to absorb the scattered radiation. Perfect magnetic boundary conditions (PMC) elsewhere impose radially polarized electric fields.

4 Figure S4. (a) Schematic of the simulations space considered here. We consider a quarter-slice of the full structure, with hybrid input mode (gold-filled bore step-index fiber), and perfectly matched layers at the edges of the simulation space to absorb the scattered radiation (geometric parameters of the wire, tip and fiber are the same of those of in Fig. 2 of the manuscript.) Perfect electric or magnetic boundary conditions elsewhere impose azimuthally or radially polarized excitation modes. (b) Spatial distribution of S in the vicinity of the nanotip, with a TE 01 azimuthally polarized (AP) excitation, and (c) with a TM 01 radially polarized (RP) excitation. The red arrows show the direction of the corresponding electric field vector at a fixed point of time. In both cases, a doughnut-shape mode is transmitted, but only the RP mode induces scattering at the apex of the nanotip, in agreement with our experimental observations (Figs. 3-4 of the main manuscript).

5 Figure S5. 3D simulations of the scattering properties of a gold tip excited by a short-range surface plasmon at λ = 650 nm. (a) Simplified schematic of the simulation space under consideration. The boundary conditions and geometry are analogous to those shown in Fig. S3. A radially polarized shortrange plasmonic mode is excited at the input of the gold wire embedded in silica, which then travels down the nanocone to the apex of 10nm radius in air. (b) Left: E 2, showing strong field concentration at the tip apex in the yz-plane at x=0, where E = (E x, E y, E z ) is the electric field. Right: saturated image of S with S = (S x, S y, S z ), showing the power scattered by SR-SPP on the tip. (c) Left: longitudinal component S z, right: transversal component S y, showing the scattering the in the forward- and upward- directions (z- and x-directions) respectively. (d) Far-field pattern in the xz plane at y=3µm for (d) E z 2 and E x 2, showing the expected longitudinally polarized intensity (z-direction) measured in the experiment (see Fig. 4g-i in the manuscript). Colorbar values are with respect to each window maximum.

6 Figure S6. Example simulation of the normalized Poynting vector magnitude for a dipole located at an axial distance of 20 nm from the apex of the nanotip considered in the main text (λ =650 nm). The calculated collection efficiency is ~2% this value is obtained from the ratio of the surface integral of the Poynting vector magnitude over the doped silica core surface at z = -1μm, with respect to the same integral calculated over the surface of a 10 nm sphere surrounding the point dipole.

7 Figure S7. (a) Conceptual schematic the propagation characteristics inside the air and nanowire sections of the nanobore fiber (all calculations refer to a wavelength of 650 nm). A radially polarized input mode in the air-bore nanobore fiber (Sec. i) excites the two Eigenmodes of the gold-filled section (Sec. ii), namely the hybrid dielectric TM 01 mode and the hybrid plasmonic SPP 0 mode. The beating of the two modes results in periodic energy transfer between wire and dielectric core. (b) Poynting vector profile of the three modes. Spatial distribution of the Poynting vector magnitude calculated using EM analysis (c) and (d) 3D FE-simulations.

8 Supporting Methods Eigenmode Theory The system under consideration is illustrated in Fig. S7a, cfr. Fig. 1 of the manuscript. For the hybrid nanowire-enhanced fiber considered here, the total electric and magnetic fields in the gold-filled section can be expressed a superposition of the two supported dielectric- and plasmonic-eigenmodes (EMs), H x, y, z) a H ( x, y)exp( iβ z) + a H ( x, y)exp( iβ ), (1) tot( z E x, y, z) a E ( x, y) exp( iβ z) + a E ( x, y) exp( iβ ), (2) tot( z where the subscript i = 1,2 labels the two EMs. Here, the label 1 and 2 refer to the hybrid TM 01 and SPP 0 plasmonic modes, respectively [see Fig. S7b]. H tot and E tot are the transverse total magnetic and electric field vectors at any point in the fiber, respectively, H i and E i are the magnetic and electric field distributions of the Eigenmode, β i = β i R + iβ i I are the EM propagation constants, and a i are the corresponding modal amplitudes determining the contribution of the respective EM to the total field. The above expansions are only approximate since contributions from radiation modes have been neglected. Following the convention used in [1], we use modal fields that have unit normalization, 1 2 A [ E ˆ, (3) i ( x, y) H j ( x, y)] zda=δi, j integrated over the entire lateral cross section A. Modes can always be normalized to satisfy Eq. (3), which can then be used to calculate the complex modal amplitudes at input, a i 1 = [ E i( x, y) Hinput ( x, y)] zda ˆ, (4) 2 A where H input (x,y) is the magnetic field of the radially polarized Eigenmode incoming from the unfilled (air-bore) fiber at z=0 (labelled as input in see Fig. S7a,b). Here we use the overlap integrals in Eqs. 3-4 in their unconjugated form, as appropriate for lossy (as well as for lossless) systems [1]. Figure 5c shows the spatial distribution of the Poynting vector magnitude in the gold-filled portion of the fiber (Sec. ii in Fig. S7a and Fig. 2a in the manuscript) predicted by EM theory, which is in agreement with finite element calculations of the full 3D structure [Fig. S7d]. Supporting References [1] Snyder A. W.; Love, J. D. Optical Waveguide Theory. Chapman and Hall: 1983.