A stable nonlinear optical point light source is investigated, based on field enhancement at individual, pointed gold nanocones with sub-wavelength dimensions. Exciting these cones with near-infrared, focused radially polarized femtosecond beams allows for tip-emission at the second harmonic wavelength (second harmonic generation, SHG) in the visible range. In fact, gold nanocones with ultra-sharp tips possess interesting nonlinear optical (NLO) properties for SHG and two-photon photoluminescence (TPPL) emission, due to the enhanced electric field confinement at the tip apex combined with centrosymmetry breaking. Using two complementary optical setups for bottom or top illumination a sharp tip SHG emission is discriminated from the broad TPPL background continuum. Moreover, comparing the experiments with theoretical calculations manifests that these NLO signatures originate either from the tip apex or the base edge of the nanocones, clearly depending on the cone size, the surrounding medium, and illumination conditions. Finally, it is demonstrated that the tip-emitted signal vanishes when switching from radial to azimuthal polarization.
© 2014 Optical Society of America
The search for time-stable and tunable point light sources with ultra-small dimensions constitutes one of the most challenging tasks since the early days of nano-optics [1–3]. Clearly quantum dots, single molecules, or N-V-centers in nano-diamonds may manifest the ideal and ultimately small point light source when attached to an appropriate handle [4–7]. Nevertheless, many of such emitters suffer in practice from e.g. blinking or photo-bleaching effects . Plasmonic nanostructures, in contrast, such as gold spheres, bow-tie antennas, or pointed metal tips, are highly stable even when being illuminated with focused laser beams for strong field enhancement or for three-dimensional steering of a focus that is formed by their emitted radiation [3,8–14].
Hence, by making use of the lightning rod effect, well-defined and spatially confined light sources can be realized with such metallic tips [15–21]. In particular nonlinear effects such as second-harmonic generation (SHG) should benefit greatly from the strong and highly localized near-fields leading to intense and small light sources. SHG presents the advantages of enabling extremely localized light sources with an extremely narrow emission spectrum, thus offering very well defined nano-light sources. However the following challenges are to be met:
- Firstly, for nanospheres with sizes much smaller than the wavelength λ, the polarizability scales with the third power of the radius, and hence only a tiny scattering cross-section for small nanoparticles results. This also holds for SHG, which scales with a higher power of the particle radius as well .
- Secondly, background suppression is mandatory in order to achieve a reasonable signal-to-noise-ratio (SNR) for any such strategy [23–25]. One effective way has been realized with adiabatic plasmon transport along the tip shaft where the areas of optical excitation and emission are spatially well separated [26,27], as necessary for instance in tip-enhanced Raman scattering . Another strategy is to replace elongated metal tips by nanoscale particles at the tip apex .
To address these challenges we developed a dedicated fabrication process allowing the preparation of sharp-tipped gold nanocones in different sizes on glass/indium tin oxide (ITO) substrates [Fig. 1(a)] [30–32]. Other methods to prepare nanocones by e.g. nanosphere lithography, nanoimprint or laser interference also exist [33–37]. Only recently cones were grown in hole masks for SHG studies . In comparison, our technique allows to place these cones also at the tips of silicon cantilevers for scanning probe applications, which was demonstrated in initial measurements . Metallic nanocones provide an elegant solution in efficiently absorbing impinging electromagnetic waves due to their large volume, breaking the centrosymmetry inherently through their geometry (thus providing stronger second-harmonic polarizability along the axis of symmetry-breaking)  and concentrating the field energy via both plasmon oscillations and the lightning-rod effect into the cone’s tip [Figs. 1(b) and 1(c)] [26, 40,41]. The importance of field enhancement and symmetry breaking has already motivated recent investigations on SHG at metallic or metal-coated tips with elongated shafts [42–44] or at cones of a single size . Also micron-sized LiNbO3 cones (good second-order polarizability) combined with Au coating (plasmonic field enhancement) were investigated .
In this work, the sharp tips of cones of approximately 100 nm in height that end in tip radii on the order of 10 nm act as well confined nonlinear optical (NLO) point light sources. Other than elongated tips, due to their truly three-dimensional nanoscale size the cone excitation can no longer be modeled as a single dipole. Our investigations aim at finding ways how to excite the tip dipole most efficiently at the second-harmonic frequency in these nanocones, and how to discriminate the tip dipole radiation from spurious background signal arising from the cones’ base edge emission [Figs. 1(c) and 1(d)]. For this purpose, a series of cones with different cone sizes in the range up to 200 nm as the varying geometrical parameter were investigated, while keeping the aspect ratio and tip radius roughly constant in order to investigate the influence of the cone size on their NLO properties. The role of the linear plasmon resonance with respect to this size variation is taken into account. Additionally, the SHG was measured in different media. Moreover, the SHG is separated from and compared with the broadband two-photon photoluminescence (TPPL) emission from these tips that is common to metallic nanostructures [Fig. 1(e)] [46–50].
The measurements were performed with two different experimental setups designed either for bottom or top illumination of the nanocones, both operating in back-reflection (see Fig. 1(b)). We demonstrate that NLO emission from the cone tip can be efficiently collected with both setups. Notably, the emission dramatically depends on the cone size, environment, and focus conditions. The experiments are in excellent agreement with theoretical calculations of SHG emission using the Discontinuous Galerkin (DG) method [51,52].
Gold nanocones were fabricated on cover glass substrates with a 50 nm thick ITO layer in regular arrays with 2-4 µm spacing (see Fig. 2 and Methods Section). They all possess full opening angles of ~50 to 60° at heights of 80, 130, 160 and 200 nm (with deviations of up to ± 10 nm, see Fig. 1(a)). The cone tip radius was evaluated by scanning electron microscopy (SEM) and is on the order of 10 nm. The radii of curvature at the sharp base edge are hard to determine precisely, but are typically smaller than the tip radii . Owing to the fabrication process, the nominally circular cone bases often exhibit some anisotropy (ellipticity). In the following the different cones are labeled 80NC, 130NC, 160NC and 200NC, respectively.
For experimentally investigating the NLO properties of these cones, two complementing setups were used, equipped for bottom or top illumination of individual cones as shown in Fig. 3. The first setup employs an oil objective of numerical aperture N.A. = 1.49 and uses bottom illumination [Figs. 3(a) and 3(c)], while the second setup uses a parabolic mirror for top illumination and signal collection [Figs. 3(b) and 3(d)]. Both setups have the advantage of large detection angles. All measurements were performed by scanning single cones through the laser focus of radially or azimuthally polarized beams (in the following denoted as “radial focus” or “azimuthal focus”) with the help of 3-D piezoelectric scanners, thus actually probing the different foci with nanocones of a variable size.
In the oil objective setup the cones are illuminated with 100 fs laser pulses at a wavelength of 790 nm through the glass substrate that carries the nanocones. This setup allows embedding cones in media of a variable refractive index; here we used immersion oil or air. Both SHG and TPPL emission signals are collected with the same objective in backscattering configuration over a detection angle of up to 80°. The SHG signal is detected by an avalanche photo diode (APD) after passing a filter transmitting between 390 and 400 nm. The TPPL signal is recorded separately by a spectrometer in the range between 420 and 640 nm.
A complementary investigation was carried out using top illumination of nanocones with a parabolic mirror setup . Here, cones are measured in air, and illuminated with femtosecond laser light (120 fs) at 774 nm. SHG and TPPL signals are collected over an opening angle from 28 to 85° by the parabolic mirror. However, the SHG signal alone is too weak to be well measured in this setup. For this reason the nonlinear signals of SHG and TPPL are collected together (filter range 350 to 680 nm) and sent to a single photon counting detector (for imaging) or a spectrometer CCD camera (for recording spectra). The images thus show a superposition of SHG and TPPL information. In comparison with the oil-objective setup, the axial extension of the laser focus is much larger (see Figs. 4 and 5).
Figure 1(e) shows a typical spectrum of the nonlinear emission by a single cone excited with a radial and an azimuthal laser focus in the parabolic mirror setup, respectively. A sharp SHG peak appears at a wavelength of 387 nm together with a broad TPPL background at longer wavelengths. The radial spectrum was taken with the cone in the center of the focus. The azimuthal spectrum was taken at four points on the boundary of the focus where the NLO signal was maximal and then averaged over these points.
For efficient tip SHG and TPPL excitation, the electric field polarization needs to be aligned parallel to the cone (symmetry) axis, which can be achieved by using a radial focus. In fact, the Ez electric field component parallel to the cone symmetry axis (z-axis) is clearly dominant in the center of such a focus and thus favorable for achieving our goal . Applying such a field to the cone would excite a tip mode in the cone with strong near fields near the tip [Fig. 1(c)]. However, when zooming into the boundary of such a laser focus (i.e. at a distance of λ/2 away from the focal axis), the electric field vector is oriented perpendicular to the z-axis and hence lies in the xy-plane, manifesting a dominant in-plane component. This can be seen in the foci from both objective and parabolic mirror setup, shown in Figs. 4 and 5. Such field components would excite the base edge mode of the cone [Fig. 1(d)].
In contrast an azimuthal focus exclusively exhibits in-plane field components [53–55]. These in-plane fields can excite in-plane dipoles that yield SHG and TPPL from the base edges of the cones. The numerically calculated electric field distributions in the foci for both setups are shown in Figs. 4 and 5.
Scanning a single cone through the radial focus in the oil-objective setup shows SHG and TPPL emission from both the cone base and the cone tip. Clearly, their relative strengths depend on the cone size (height and base diameter) as shown in Figs. 6 and 7. Figure 6(a) illustrates the SHG (left column) and TPPL (right column) for 80NC, 130NC, 160NC and 200NC, respectively, measured in air in a radial focus. Figure 6(b) shows the SHG and TPPL images for the same four cone sizes but measured in oil. Note that the scans of cones in air and cones in oil were taken from different individual cones, while the corresponding SHG and TPPL image always stem from the same cone. For all of the image patterns shown, whole arrays of nominally identical nanocones were scanned. Comparable image patterns were reproducibly observed from the individual cones within the array. Each image shows a pattern representative for the corresponding array. Variations between the images in each array typically result from nanoscale differences in the cone geometry and are not further discussed in this study.
SHG and TPPL show very similar emission patterns. For 80NC, 130NC, and 160NC in air as well as for 80NC in oil, we observe a ring-shaped emission pattern. This ring-shape indicates dominant base edge signal radiation excited by the in-plane field components at the rim of the focus. These rings have a typical diameter of 0.7 to 0.8 µm, in agreement with the shape of the focus as shown in Fig. 4, with the in-plane field reaching its maximum at a distance of ~400 nm from the center of the focus.
In contrast, 200NCs exhibit a single SHG and TPPL maximum that solely originates from the center of the focus. The ring pattern vanishes in air, and is reduced to a weak halo in oil. Furthermore, medium-sized cones (i.e. 130NC and 160NC) exhibit both tip and base edge emission in oil, yielding a pattern with no clear minimum in the center. The single-cone patterns shown are typical for the cones of each kind of sample. To conclude, all 80NC show ring-shaped patterns, while the tip-like patterns are typical for 200NC. 130NC and 160NC show ring-shaped patterns in air and patch-like patterns in oil.
Figure 7 shows SHG scans of 200NC and 130NC excited by a radial and azimuthal focus, respectively. Since no tip emission is expected for azimuthal illumination, these measurements may directly be used as a reference in order to discriminate tip from base edge emission . As seen for 200NCs in air, the SHG tip emission completely disappears when switching from radial to azimuthal polarization. For 130NC in air [Fig. 7(b)] the ring shaped pattern remains unchanged between radial and azimuthal polarization. However, the lobes of highest intensity within the ring pattern are rotated by ~90°, in agreement with the rotation angle between the exciting in-plane fields in the radial and the azimuthal focus. This indicates that the lobes result from anisotropies (ellipticity) of the cone bases. In oil, the superposition of the ring-shaped pattern from the base mode and the spot-like feature from the tip mode changes to pure base edge emission that shows up as a ring pattern for the 130NC and as a halo for the 200NC. As in Fig. 6, the corresponding TPPL images display very similar patterns (data not shown).
Figure 8 shows the cones’ performance in the parabolic mirror setup, using radial or azimuthal foci as well. In contrast to the oil-objective setup, the radial focus excites a clear and dominant tip signal for all four cone sizes. Consistently with the oil-objective setup, the tip emission signal increases with cone size and reaches its strongest value for 200NC. Moreover, the central maximum is accompanied by an additional concentric ring feature that stems from the ring-shaped second maximum of the Ez field component in the radial focus (cf. Figure 5(b)). The weak second-maximum signal (besides the main tip emission) is also found for 80NC and 130NC. Changing to an azimuthal focus [Fig. 8, bottom row] reveals a ring-shaped pattern of weak intensity. The intensity rises gradually from 80NC to 130NC to 160NC. The ring radii are smaller than the second-maximum rings as seen in the case of the radial focus, as the first maximum of the azimuthal focus appears at a smaller radius. No ring-shaped pattern can be observed for 200NC.
In addition to these measurements, we performed simulations of the cones’ SHG behavior using the Discontinuous Galerkin (DG) method [51,52], as shown in Fig. 9. The electrodynamic properties of the gold cones were modeled by a Drude-Lorentz-model, with the Drude-term treated as a hydrodynamic electron gas. This model can describe second order nonlinearities of the electron gas polarization and thus provides SHG as well. The ITO substrate was assumed to be mainly dielectric (in consistence with the very low losses measured experimentally) having a refractive index of 1.9.
To model the radial focus for the DG simulation in the objective setup, we superposed eight plane-waves incident at an angle of 30° to the optical axis, resulting in a Bessel-like beam (see Methods) which has a Gaussian temporal profile centered at 790 nm. The cone heights were modeled as given experimentally, while tip and base radii of 20 nm were assumed. As was the case in the experiment, the tip dipole was excited by placing the nanocone in the center of the focus, denoted as position 1 in Fig. 9(a), while the base dipole was excited by laterally shifting the cone 400 nm off center, where the out-of-plane field of the simulated focus has its minimum intensity (denoted as position 2). The SHG signal was extracted via filtering the scattered light at 395 nm and integrating the result over a full 4π solid angle. Figures 9(c) and 9(d) show the obtained SHG intensity values as a function of the cone size, the cone position with respect to the focus (and hence the position of the excited dipole), and with respect to the surrounding medium. In air, the tip signal gradually increases with cone size, whereas the base signal has its maximum for ~160NCs. This results in a base-dominated SHG signal for the smaller and a tip-dominated SHG signal for the lager cones. This behavior compares well with the experimental data [Fig. 6(a)], where for smaller cones only the base signal is visible, whereas the larger 200NC manifest clear tip emission only. Changing the surrounding medium to oil allows for a clear tip-signal to develop already for smaller cone sizes. Only the 80NCs still show a dominant base signal in agreement with our experimental findings [Fig. 6(b)].
When the cone symmetry axis coincides with the center of the radial focus, excitation of the tip dipole by Ez dominates. Scanning the cone through the focus, the tip signal thus reflects the spot-like out-of-plane field distribution . Scanning the cone through the boundary of the focus where in-plane polarization dominates forces the sharp cone base edges to emit light reflecting the ring-like in-plane field distribution.
The behavior of the measured tip and base edge signals with the increasing cone sizes in Fig. 6 is in good agreement with the tip:base SHG ratio of the simulations in Fig. 9. In the experiment, the base edge signal is typically dominant for scans of smaller cones, i.e. 80NC to 160NC at lower refractive index (air, n = 1.0) and 80NC at higher refractive index (oil, n = 1.5). For larger cones (200NC in air, 130NC to 200NC in oil), the tip apex emits increasingly strong SHG and TPPL signal whilst the base NLO response drops. Especially in air, the 200NC exhibit usually a sharp spot from tip emission. In the simulations shown in Fig. 9(c) the tip:base SHG ratio increases strongly with the cone size, as the tip signal rises and the base edge signal drops drastically for the 200NC. 130NC and 160NC in oil [Fig. 6(b)] typically show an intermediate behavior with a superposition of base and tip signal. Here the ratio of tip to base SHG is larger in oil than in air, which is in further agreement with the simulations.
Please note again that the SHG and TPPL images show the same patterns. This indicates that strong TPPL signal as well as strong SHG signal originates from the excitation of either tip or base edge.
In the parabolic mirror setup both signals SHG + TPPL were detected together. Considering the similarity between the SHG and TPPL patterns in the oil objective setup, both signals are expected to yield similar contributions to the overall pattern, while the TPPL contribution is dominant. The tip signal is dominant for all cone sizes in Fig. 8 when excited with the radial focus, with the intensity increasing with cone size. Switching to the azimuthal focus results in weak ring-shaped halos from 80NC to 160NC, however no signal for 200NC. These observations again reflect the trends underlined by the simulations shown in Fig. 9(c), where the tip signal increases with cone size, while the base edge signal goes through a maximum for intermediate cone sizes. The dominant tip signal measured by the parabolic mirror setup is most likely related to the large extension of the radial focus in the z-direction (see Fig. 5). In contrast, for the measurements with the objective setup in air, the Ez field decays strongly over the length scale of the cone heights (see Fig. 4), which may lead to a weaker excitation of the cone tip compared to the base edge. This strong drop can be explained by the evanescent modes that occur when the focused light hits the interface from below and large parts of it are totally reflected.
For the measurements with the objective setup in oil the z-extension of the Ez field is again larger than in air. This may be a further reason for the clear presence of tip signal for 130NC and 160NC in oil.
The nonlinear emission properties of the nanocones are closely connected with their linear optical properties via the local near-fields excited by plasmon resonances. Therefore white light dark-field scattering spectra were recorded from all cone samples in air and in oil under illumination angles of 60° as additional information. In these spectra plasmon resonances are observed in the far-field . The detection scheme is more sensitive to the in-plane mode, for which clear maxima are observed. The out-of-plane mode (tip signal) can only in some spectra be discerned as a weak shoulder. In reference  also examples of the near-field distributions around a cone are shown for the in-plane and out-of-plane modes.
The measurements are shown in Fig. 10 and compared with simulated (DG method) scattering spectra of cones with similar dimensions, which show a superposition of scattering from excitation with in-plane and out-of-plane polarized beams (see Methods). Instead of spectrally narrow Gaussian pulses the plane waves are designed as Differentiated Gaussian pulses that cover the whole visible and NIR spectrum. A comparison with calculations independently illuminating the cones with a radial focus (exciting the tip) and a linear focus (exciting the base edge) allows to identify the positions of the tip and the base edge resonances of the cones (marked by arrows).
The measured scattering spectra show clear resonances around 640 nm (in air) and 680 nm (in oil) for the 80NC, which are shifted towards the infrared spectral range for larger cones. Both measurements and simulations show comparable tendencies. In air, the simulated tip resonance maximum moves gradually from ~550 nm towards ~650 nm for increasing cone sizes. The simulated and measured peaks in Figs. 10(a)-10(c) become broader, which together with the approach of the tip mode towards the excitation wavelength at ~800 nm, allows for a more effective excitation of the tip. This is in good agreement with the simulations shown in Fig. 9, where the tip signal increases strongly with cone size. In contrast to the tip mode, the simulated base edge resonance, which starts out at ~600 nm, almost matches the excitation laser wavelength for the 160NC. For the 200NC no resonances could be assigned (data not shown). Given the increasing red-shift, the base edge becomes off-resonant with respect to the excitation wavelength, which may explain the strong drop of the base edge SHG signal.
Similar to the “air” case, the measured resonances as well as the simulated tip and base edge resonances in oil also become broader and red shifted. In contrast to the simulations in air, both the tip and the base resonance are shifted across the excitation wavelength (from ~650 nm to beyond 900 nm), resulting in the occurrence of tip excitation also for smaller cones than 200NC (130NC and 160NC, see Figs. 6(b) and 9). For the 200NC in oil no dipole resonances could be observed in the spectrum (data not shown). The dipolar base edge resonance can be assumed to be far away from the excitation wavelength (and outside the detected range), which would explain the weak base edge SHG.
In conclusion, gold nanocones are found to constitute non-blinking individual NLO sources for localized SHG and TPPL emission. These optical nanoantennas efficiently focus optical energy into a tip with sub-wavelength dimensions via plasmon excitation and the lightning rod effect. Hence, well-defined spatially confined point light sources are realized. In the objective setup both tip signal and strong base edge signals are observed for cones of four different sizes from 80 to 200 nm. The 200 nm high gold cones show the brightest signal from the tip while the base edge signal is strongly suppressed in oil as well as in air. In the parabolic mirror setup, all cones show nonlinear signal originating from the tip, with the highest intensity measured for 200 nm high cones. Note that SHG and TPPL show comparable behavior, indicating that both equally depend on the excitation of the tip and base edge modes. By size variation and choice of the illumination configuration (oil-objective vs. parabolic mirror) the tip-to-base-edge ratio of SHG/TPPL emission can be tuned and optimized. Moreover, the SHG/TPPL tip emission was shown to be enhanced or suppressed by using either different polarization modes or a cone size variation. These experimental findings are supported by numerical investigations, calculating the tip and base emissions as functions of cone size. The role of the plasmon resonance in the observed behavior is underlined by measurements and simulations of linear scattering spectra of the cones. As shown, the tip SHG emission is favorable for larger cone sizes, i.e. for cone heights above 130 nm. Hence, such nanocones can be used as advantageous nonlinear single point light sources, which potentially can be applied for modern-type scanning near-field probing or lab-on-chip sensor applications.
5.1 Cone fabrication
The nanocones are fabricated according to a variation of a process previously described by the authors [30–32, 56,57]. Here, Al2O3 is used instead of hydrogen silsesquioxane (HSQ) as the etch mask material for improved stability and better reproducibility of the mask diameter. First ITO and gold are sputtered on a clean cover glass slide (type #1, thickness ~150 µm). The thickness of the gold film defines the height of the cones. In the next step positive resist PMMA is structured by electron beam lithography. After evaporation of aluminum oxide and lift-off, the resulting aluminum oxide disks act as etching masks for an argon ion etch step. In the etch step the gold nanocones are formed. The etching time is calculated from the evaporated mask thickness and the known etch rates, and verified by intermittently inspecting the cone formation in the SEM. The etching is stopped when the mask is just completely removed to obtain cones with sharp tips (tip radii ~10 nm). Cones with four different heights (80NC, 130NC, 160NC and 200NC) and tip opening angles of ~50 to 60° were fabricated according to this process, which is shown in Fig. 2. Note that this process can be equally applied in order to fabricate nanocones on cantilevers .
5.2 Bottom-illumination / objective measurement setup
We use a Ti-sapphire femtosecond laser to generate linearly polarized 100 fs pulses at 790 nm. The average power measured in front of the objective was about 4 mW. The pulses are focused on the sample by a 100x oil immersion objective with an N.A. of 1.49. The measurement principle and setup are shown in Figs. 3(a) and 3(c). The focused beam has an effective aperture angle of around 60°. In front of the objective the beam is widened and its polarization is changed to radial polarization by a four-segment wave plate . Conducting the beam through a pinhole after the wave plate creates a proper radial mode by mode cleaning. Rotating the beam polarization before the wave plate by 90° changes the mode to azimuthal polarization. A scanning piezo stage moves the sample with the nanocones relative to the focus, thus scanning individual cones through the focus. When scanning the cones through the focus, the integration time over the signal was 50 ms for each pixel. The SHG and TPPL light is collected by the objective and returned along the beam path back to a beam splitter which couples half of the light out and directs it towards a spectrometer (for TPPL detection) and an avalanche photodiode (APD) (for SHG detection). In the detection path, an interference filter of OD 8 at 790 nm blocks the infrared stray light. A further band-pass filter in front of the APD selects the spectral range from 390 to 400 nm. The beam is focused by a lens with f = 30 mm on the 100 µm x 100 µm large APD area.
5.3 Top-illumination / parabolic mirror setup
In the experiment a home-built confocal optical microscope with a high numerical aperture (N. A. 0.998) parabolic mirror as the laser focusing and signal collection element was used, as shown in Figs. 3(b) and 3(d). Cylindrical vector beams (radial and azimuthal polarizations) were employed to study the optical properties of the sample. A femtosecond erbium fiber laser (774 nm) beam for sample excitation was converted from a linearly polarized Gaussian beam to either radial or azimuthal polarization by a home-built mode converter. The average applied power was on the order of 1.5 mW. When a radially polarized beam is focused by a parabolic mirror, this instrument yields a focal spot with an area of 0.134 λ2, which is small compared to that of aplanatic lenses . The laser beam was focused on the sample in reflection mode, and the emitted or scattered light from the sample was collected by the parabolic mirror under angles from 28° to 85°. By applying short pass filters (Semrock FF01-680/SP.), only the SHG and TPPL (combined SHG and TPPL signal over a range of 350 to 680 nm) from the sample were collected by either an APD for photon counting imaging or by a thermoelectrically cooled charge-coupled device (CCD) camera coupled to a spectrometer for spectral measurements. All spectra were obtained using a spectrometer grating with 150 grooves per mm. As in the objective setup the sample was moved with a piezo stage. The integration time per pixel during the scans was 20 ms.
5.4 Calculation of the field distributions generated by the oil objective and the parabolic mirror
The electric field in the region of the focus as shown in Figs. 4 and 5 can be calculated using the angular spectrum representation introduced by Richards and Wolf . For the case of a radially polarized beam, the electric field in the focus of an aplanatic lens was calculated by Youngworth and Brown  and the electric field in the focus of a parabolic mirror objective was calculated by Lieb and Meixner [54, 62]. The presence of an interface is accounted for using the Fresnel reflection and transmission coefficients in combination with Snell's law. The incident radially or azimuthally polarized laser beam being focused by the optics is modeled as a Bessel-Gaussian beam with parameters according to the experimental setups shown in Fig. 3.
5.5 Discontinuous Galerkin (DG) method
The DG method is a numerical method for the solution of electromagnetic scattering problems in the time domain [51,52, 63,64]. An important advantage of the DG method compared to FDTD is the possibility to implement a flexible volume discretization that especially allows to model curved surfaces - without sharp edges that in general yield singularities. On the other hand, nonlinear-optical calculations require operating in the time-domain and not the frequency-domain. However, DG as well as FDTD uses an explicit time-step algorithm. This is a big advantage especially for nonlinear-optical calculations based on fundamental models such as the hydrodynamic model.
As in the Finite Element Method (FEM) the space is divided into mostly tetrahedrical cells. In each cell the electric and magnetic field components are expressed by a finite set of polynomial basis functions. To obtain the electromagnetic field that solves a scattering problem the expansion coefficients have to be determined. In FEM the expansion coefficients are determined by matching the electromagnetic fields to the continuity conditions at the cell boundaries. The DG method instead uses the fact that the Maxwell equations can be written in the form of a continuity equation. As there is a conserved current, DG does not demand continuity of the field components at the cell boundaries but just current conservation; i.e. at each boundary of two adjacent cells 1 and 2 the defined currents F(1) and F(2) defined in each cell at that boundary must be equal: F(1) = F(2). In practice, the difference F(1) - F(2) has to be minimized.
For the simulations shown in Fig. 9 the nanocones are excited by a superposition of eight plane waves to simulate the radial focus as generated in the objective setup. In contrast to the focus calculations described in the former section we use a simplified model superposing only eight plane waves with an incident angle of 30° to the optical axis. When we assume the radial laser beam as a bundle of parallel rays, the rays at the intensity maximum of the beam profile are focused under that angle of 30°.
No harmonic waves were assumed but Gaussian pulses at a central wavelength of 790 nm and with a pulse duration of 8.3 fs. To simulate the dielectric response of gold, a Drude-Lorentz-model for the polarization of the electron gas with a Drude term and a Lorentz term was used . The values of these terms for gold were taken from Ref . To simulate nonlinear effects of gold, the Drude term of the polarization was replaced by a complete hydrodynamic model with the same basic parameters . For ITO a constant refractive index of 1.9 was assumed, referring to .
The simulations for Fig. 10 were carried out similarly, but with the simple Drude-Lorentz model as only the linear scattering spectra were looked for. The cone was excited with a single plane wave with a transmission angle of 45°, since in the experiment the dark field illumination takes place under oblique angles. As unpolarized light was used in the experiment, both p- and s-polarization were employed, and the results were superposed. As the exciting wave a differentiated Gaussian pulse with a central wavelength of 790 nm was introduced that provides a broad spectrum covering the whole visible and NIR range within one calculation. Similar calculations were done respectively exciting the cone with a radial focus (as above, to excite the tip) and with a plane wave under perpendicular incidence (to excite the base edge). With these calculations the tip and the base edge resonances could be identified as marked by the arrows in Fig. 10.
5.6. Dark field scattering spectra
Far-field dark field scattering spectra of 80NC and 130NC were measured in an inverted microscope (Nikon Eclipse TE2000-U) using an unpolarized white light source and a dark field condenser (N.A. 0.80-0.95) in transmission mode. Only the light scattered by the sample was collected by an objective (60x, N.A. 0.7). The collected signal was passed through a pinhole for spatial filtering such that only signal from a sample area with a diameter of ~2.5 µm was collected (single particle measurements). Spectra were recorded by a spectrometer (Princeton Instruments Acton Spectra Pro 2300i). The displayed curves represent background corrected single particle spectra after normalization with the lamp spectrum. For measurements of nanocones embedded in oil an oil dark field condenser (NA 1.2-1.43) was used.
Scattering spectra of 160NC and 200NC were measured with a similar setup. An inverted microscope (Nikon Eclipse Ti-U) with the same illumination technique (dark field condenser, NA 0.80-0.95) and an oil immersion objective (100x, N.A. 0.5) was used. For single structure measurements the spatial filtering was achieved in the same way as above and spectra were recorded by a spectrometer (Ocean Optics QE 6500). For measurements in oil the sample was turned upside down, such that the nanocones were embedded in oil. The spectra were again background corrected and normalized by the lamp spectrum.
P. Reichenbach, A. Horneber and D.A. Gollmer contributed equally to this work. The authors thank Stefan Grafström and Phillip Olk for helpful discussions. P.R. acknowledges financial support by the Deutsche Forschungsgemeinschaft under the Priority Program SPP1327. M.F. is grateful for support by the Ministry of Science, Research and the Arts Baden-Württemberg and the European Social Fund. A.M. and D.Z. acknowledge financial support from the Deutsche Forschungsgemeinschaft (ME1600/5-2) and SPP1391 ultrafast nanooptics priority program. A.H. and L.M.E. acknowledge support by the Dresden Center for Computational Materials Science within the DFG Cluster of Excellence “Center of Advancing Electronics Dresden”. The authors acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of Tuebingen University. Part of the scattering measurements was performed at The Molecular Foundry under User Proposal No. 917 and supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Support by P.J. Schuck, A. Weber-Bargioni and D. Gargas is gratefully acknowledged.
References and links
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