We theoretically show that a weakly-focused radially polarized beam can excite surface-plasmon-polaritons in metal nanowires and nanocones with efficiencies of the order of 90% and large bandwidths. The coupling mechanism relies on the formation of a standing wave on the nanowire facet, which imposes a relationship between the operating wavelength and the nanowire radius. An immediate application of this finding is nanofocusing of optical energy for implementations of ultra-fast and high-throughput linear and nonlinear nanoscopies, optical nanolithographies, quantum nano-optics and photochemistry at the nanoscale.
© 2010 Optical Society of America
Since its birth in the mid-1980s, scanning near-field optical microscopy (SNOM) has suffered from the small fraction of optical energy that can be concentrated near the tip apex. For high-resolution probes, this factor is at most of the order of 10-3 but usually much less, depending on the probe parameters . Optical antennas can perform better, but their implementation in a scanning device is still restricted by difficulties associated with the high-throughput fabrication or attachment of a well-defined metal nanoparticle to the end of a tapered fiber [2, 3]. Furthermore, these probes exhibit a nearly-dipolar radiation pattern, which requires high numerical-aperture (NA) optics to obtain large coupling efficiencies .
An emerging approach to concentrate light into a subwavelength spot size relies on the so-called nanofocusing of surface plasmon-polaritons (SPPs) [5–7]. Nanofocusing could indeed revolutionize SNOM by largely improving fluorescence, Raman and other nonlinear nanoscopies [8–10]. Furthermore, the possibility of feeding optical energy into a nanoscale volume has also immediate implications for photochemistry , quantum optics  and nanolithography . However, practical exploitations of this concept require a rapid and effective conversion of SPPs into photons, especially in the visible and UV spectral range, where absorption losses by real metals lead to very small propagation lengths.
We recently demonstrated that guided photons of a dielectric nanofiber are converted into SPPs in metal nanowires (NWs) and vice versa with close to 100% efficiency. Based on these findings, we suggested that a high-throughput SNOM could be realized by butt-coupling a metal cone with a tapered fiber . Since SNOMs based on cantilevers are gaining interest due to their reliability and performances , it is relevant to know whether SPPs in metal nanocones attached to cantilevers can be efficiently excited by focused beams.
Several designs of cantilever-based SNOMs are found in the literature. For example, a Gaussian beam focused into an aperture SNOM  or a fully metal coated dielectric tip [17, 18], and an aperture probe combined with a microsphere . Other schemes used a grating etched on the side of the nanocone  or a photonic-crystal cavity  to improve the coupling efficiency. On the other hand, the polarization and profile of TM0 SPPs suggest that a promising candidate for their efficient excitation in a nanofocusing device could be a focused radially-polarized beam (FRB) . Indeed, radially-polarized light has already been applied to nanocones [23–26], but the conversion efficiency of photons into SPPs did not exhibit a large improvement in comparison to the other arrangements. Furthermore, when light is focused on to the tip apex it gives rise to a strong background illumination.
Here, we show that the conversion mechanism of our fiber-based high-throughput SNOM, namely the molding of SPPs at the cone base , holds also for free-space coupling. When a weakly-focused radially-polarized beam is incident on the nanocone base, the conversion of photons into SPPs can reach 90% efficiency. Moreover, since the tip apex is out of focus, background noise due to direct illumination is further suppressed .
2. Results and Discussion
The primary processes that we need to consider in a high-throughput SNOM are nanofocusing of SPPs and conversion of SPPs into photons. Since there exists much literature on nanofocusing [5–7, 28–32], we direct our attention only to the coupling between photons and SPPs in the collection and illumination modes. To this end, in Sec. 2.1 we first analyze the reflection and scattering of SPPs at the end of a metal NW [33,34] in the presence of a semi-infinite dielectric that accounts for the cantilever. In Sec. 2.2, we consider the far-field pattern and compare it to the field profile of FRBs. In Sec. 2.3, we study the conversion of photons into SPPs by illuminating the NW end with FRBs. Finally, in Sec. 2.4 we combine our findings with nanofocusing to assess field enhancement and spatial resolution.
2.1. Reflection and Directional Emission
We first considered the reflection and radiation properties of TM0 SPPs when they reach the end of a semi-infinite metal NW. In contrary to butt-coupling with a dielectric nanofiber  and free-space coupling [33, 34], here we placed a semi-infinite dielectric at the NW termination. Besides holding the NW, the substrate changes reflection and radiation of SPPs. Figure 1(a) sketches the situation for a gold  NW on a glass substrate (refractive index n=1.5) together with the simulation layout. Our calculations were carried out using the body-of-revolution (BOR) finite-difference time-domain (FDTD) method, whose details and advantages are briefly explained in the Appendix. Throughout this work we chose a working wavelength of λ = 633 nm, keeping in mind that these results are generally valid over a broad spectral range if the NW radius (r) is properly scaled [14, 34].
A TM0 SPP is launched on the gold NW and when it reaches the NW end it can be reflected into the same SPP mode, into free space, and scattered in the forward direction. Figure 1(b) displays the amount of reflection back into the SPP as a function of the NW radius for different metals and substrates, showing that it is minimized for some values of r. Figures 1(c) and 1(d) plot the time-averaged magnetic field at two reflection minima corresponding to a gold NW on glass with respectively r = 160 nm and r = 340 nm. It turns out that the NW facet supports a standing wave , which leads to a directional radiation pattern with a profile determined by the field near the NW facet. There is no radiated power along the z-axis, a result that simply stems from the spatial symmetry and polarization of TM0 SPPs. Figure 1(b) also shows that when gold NWs are replaced by silver  NWs reflection and emission are almost the same.
We then investigated the effect of changing the dielectric substrate. For example, when the refractive index is set to n = 2, reflections increase and the minima shift towards shorter NW radii. This is a further indication that the standing wave on the NW facet plays an important role in lowering reflection, as we found for the case of butt-coupling with a dielectric nanofiber . We have also considered the amount of reflection that is not channeled into SPPs and found that it can be negligible.
In summary, when SPPs reach the NW end they radiate in the forward direction with a very high efficiency if, for a given wavelength, the NW radius is appropriately chosen. Furthermore, the radius can be reduced by increasing the refractive index of the supporting substrate, but at the cost of increasing reflection.
2.2. Optimizing the Beam Parameters
Here we are interested in the conversion of focused beams into SPPs of metal NWs. Reciprocity tells us that if the out-coupling efficiency is high, the same holds for the opposite direction. However, one has to clarify what beam profile should be used to perform this task. We thus considered the near field obtained from the BOR-FDTD calculations and transformed it to the far field using an algorithm described in the Appendix. In the far region the field is a spherical transverse wave polarized along θ since the φ component must be zero by symmetry considerations.
In the spherical coordinates (r, θ, φ) we define a Gaussian reference sphere (GRS) to interface the NW with the optical focusing system, as sketched in Fig. 2(a). The polarization and spatial properties of the electric field on the GRS suggest that a good candidate for coupling optical energy in the NW would be a radially-polarized beam . Its electric-field profile at the beam waist reads
where Eo is the field amplitude, w the beam waist, ρ the radial coordinate, and its unit vector. Fig. 2(a) depicts how this enters the optical system to reach the GRS with a transformed field
where a = f/w, is the unit vector, and √cosθ is the apodization function for an aplanatic system ·f is the lens focal length, which corresponds to the radius of the GRS, and w is the beam waist. The idea is to optimize the beam parameter a such that the FRB matches the SPPs radiation pattern, whose far field is given by Eq. (4) in the Appendix.
Figures 2(b)–2(e) illustrate the calculated far field for different NWs and substrates at λ = 633 nm. In Fig. 2(b) the field of a gold NW on glass is maximum at about θ = 18° when r = 160 nm, showing that the FRB does not need to be tightly focused. Indeed, good overlap between the two fields is empirically obtained by setting a = 3.1, a value that leads to moderate focusing even with high-NA lenses . Interestingly, the radiation pattern does not depend very much on the refractive index of the substrate if one tunes the NW radius to minimize reflections, as evident in the comparison of Fig. 2(b) with Fig. 2(c). An intuitive explanation is found if the NW is considered as an aperture of radius r. A larger n increases the wavevector in the forward direction kz = √(2π/λ)2n2 − kρ2, but a smaller aperture increases the span of the transverse wavevector kρ ∈ [0,2π/r]. These effects compensate each others and lead to a small change in the radiation pattern.
As shown in Fig. 2(d), more directionality can be obtained by working with higher-order standing waves [see Fig. 1(d)]. The peak of the radiation pattern is now close to θ = 12°, and its width is significantly narrower. However, besides the existence of wide secondary lobe, the main drawback is the larger NW radius, which for the same tapering angle implies a longer path for nanofocusing. Thus, higher-order patterns are interesting only for applications in a spectral range where propagation lengths are much greater than the nanocone dimensions .
As one last representative case, Fig. 2(e) displays the SPPs radiation pattern for a silver NW on glass for r = 160 nm. The curve is very close to that of Fig. 2(b), as expected if one notes that the reflection minima in Fig. 1(b) occur for nearly the same NW radii. Therefore, the advantage of using silver in place of gold NWs is only in the longer propagation length of SPPs due to lower absorption losses . While in Fig. 2(b) the beam parameter a is set to overlap the NW and the FRB electric fields for both small and large angles, in Fig. 2(e) the optimization targets only small angles. In the next section we will investigate how this affects the excitation of SPP in the NW.
2.3. Efficient Excitation of SPP in Nanowires
Having found that a FRB can match the radiation pattern of SPPs in semi-infinite metal NWs, we assessed the coupling efficiency in a more quantitative manner. To this end, we computed the electromagnetic field of a FRB in the focal region  and used it as a source for the BOR-FDTD simulations.
Figure 3(a) shows the time-averaged magnetic field of a FRB in an infinite glass background for a = 3.1 and full NA=n sin α, with α = 90°. Next, we performed BOR-FDTD simulations for gold and silver NWs on glass, varying a and the position of the focal spot with respect to the NW facet. The conversion efficiency was calculated by taking the ratio of the power coupled in the TM0 SPP and the power in the incident FRB. Selected data are shown in Fig. 3(b). For both gold and silver NWs the efficiency is about 90% if the NW end is close to the focal spot and it remains larger than 60% even 400 nm away from the optimal position. The maximum does not occur exactly when the NW facet is in focus because the glass-air interface changes the properties of a focused beam . Figure 3(c) confirms that NW radius and operating wavelength are not independent from each others since the coupling efficiency drops when the r departs from the value that minimizes reflections [see Fig. 1(b)].
Figure 3(b) shows that choosing a = 3.6 yields similar performances to the case for a = 3.1, suggesting that the FRB should be foremost optimized in the peak region. Therefore, lenses with a lower NA should not affect these results. For example, in Fig. 2(d) the field amplitude of the FRB at θ= 45° is about 10% of the maximum (1% for the intensity), meaning that an NA of 0.7 in air would be enough to couple most of the beam energy into the SPP.
To give more insight on the conversion process, Fig. 3(d) displays the time-averaged magnetic field for the case of a FRB impinging on a gold NW on glass when r = 160 nm, a = 3.1, and the focal spot is 100 nm before the NW end facet. The beam is partially reflected, but the color scale shows that most of the energy is coupled into the SPP mode. Moreover, the field pattern confirms that the excitation of a standing wave on the NW facet plays a very important role in the conversion of photons into SPPs.
2.4. Nanofocusing and Spatial Resolution
The efficient excitation of SPPs in NWs can be immediately transferred to nanocones, provided that the tapering angle of the nanocone is not large (adiabatic focusing) [5–7]. Figure 4(a) sketches a possible implementation of a cantilever-based high-throughput SNOM. The radially-polarized beam is focused by a lens onto the nanocone base. The dielectric interface between the lens and the cantilever has a small effect on the beam profile, which if necessary can be compensated by placing a solid-immersion lens. The energy is then converted into TM0 SPPs and nanofocused. Since the tip apex is out of focus, the direct light of the FRB is almost negligible in the scanning region . The device can also be operated in the collection mode, where SPPs generated by a local source near the nanocone tip propagate along the nanocone and radiate with a directional pattern towards the collection optics. The weak resonant character of the standing wave adds the important advantage of large operation bandwidths, which were found also for the case of butt-coupling with a nanofiber . For example, the minima in Fig. 1(b) and, likewise, the peak in Fig. 3(c) have a width that is sufficient for collecting and launching fs pulses in the device.
Since the propagation properties of SPPs on metal nanocones have been thoroughly discussed in the literature [5–7, 28–32], here we only emphasize the field enhancement and the spatial resolution of the SNOM device. To this purpose we consider the normalized energy density W = 0.5(ε|E|2 + μ|H|2)/WBL, where WBL = Pinck2/(3πc) is the maximum achievable by far-field focusing for a given incident power Pinc and wavevector k . Since in our model the FRB is propagating from a glass substrate, we set WBL for a homogeneous medium with a refractive index equal to 1.5.
We then chose a gold nanocone with a base radius of 160 nm and a tapering angle of 8° illuminated by a FRB with a = 3.1. The tip apex was a paraboloid (z = ρ2/(3.2nm)) and the cantilever was modeled as a semi-infinite glass substrate. Figure 4(b) plots W at a distance z = 1115 nm from the cantilever, which corresponds to a plane 5 nm away from the cone tip. A zoom of W is shown in Fig. 4(c), where the contributions associated with the longitudinal (Ez) and transverse (Eρ) electric field components are also indicated. The FWHM for W is of the order of 10 nm and it is primarily due to Ez. The maximum value of W reveals that for the same Pinc the nanocone allows energy concentrations that are nearly 1000 times larger than what can be achieved by far-field focusing. Furthermore, the total energy at the observation plane is about 65% of that near the focus of the FRB, proving that a large fraction of optical energy can be transported to the nanoscale. Recent experiments on SPP excitation in NWs by adiabatic compressors have indeed found similar efficiencies in the near-infrared spectral range .
At last it is interesting to note how the features of a FRB can be exploited to minimize background illumination. To this aim, Fig. 4(b) displays the W obtained without gold cone at different distances from the cantilever and for the same incident FRB. We found that the W of the FRB near the cone tip (z = 1100 nm) is more than two orders of magnitude smaller than the W in the focal region (z = 20 nm). This corresponds to a strong background suppression compared to illuminations where the incident beam is focused on the cone tip.
We demonstrated an efficient scheme for converting free-space photons into SPPs in NWs, and combined it with nanofocusing to concentrate optical energy below the diffraction limit with a high throughput. Our approach relies on the directional radiation and low reflection of SPPs at the NW end, which occur if the NW radius is chosen according to the operation wavelength and the supporting substrate. These properties are associated with the formation of a standing wave at the NW facet, as previously found for the case of butt-coupling with a nanofiber . Furthermore, by analyzing the radiation pattern and polarization in the far region we identified weakly-focused radially-polarized beams as the best way to excite SPPs from the NW facet. We showed indeed that conversion efficiencies of 90% can be reached by optimizing the beam parameters and the position of the NW in the focal region.
In contrast to previous works on metal nanocones and radially-polarized light, which do not focus the beam on the nanocone base [23–26], our scheme yields a better conversion efficiency and lower background noise caused by direct illumination of the sample. These results were presented for λ = 633 nm, but any wavelength from the UV to the near-IR range would work by adjusting the NW radius and composition . Moreover, in comparison to a fiber-based high-throughput SNOM , the conversion efficiency is only slightly lower and the device presented here is easier to implement with existing scanning-probe technology .
For the huge intensity that can be achieved at the nanocone tip and the large operation bandwidth, we envision not only better implementations of fluorescence, Raman and other nonlinear (time-resolved) nanoscopies [8–10], but also applications in all areas that would benefit from high-throughput concentration of optical energy in nanoscale spots and fs time scales .
A. Body-of-Revolution FDTD
The electromagnetic properties of a BOR are conveniently studied in cylindrical coordinates (ρ, ϕ, z). The general solution of Maxwell’s equations can be expanded into even and odd cylindrical modes with azimuthal dependence cos(mϕ) and sin(mϕ), respectively. Since the TM0 SPP mode is even with m = 0, the field takes the simple form
The same holds for the field radiated from the NW facet, because coupling to other modes is avoided by symmetry. These equations clearly show that the full electromagnetic problem can be solved by considering only two dimensions. We performed this task using the BOR-FDTD algorithm, where the Maxwell curl equations in cylindrical coordinates are discretized in the ρz-plane . By this method we could use very fine meshes without compromising computational speed and memory usage. Furthermore, the implementation of cylindrical symmetry increases the accuracy in comparison with a full three-dimensional FDTD approach that has the same mesh pitch.
The simulation domain was truncated using perfectly matched layers (PML). Either the SPP or the FRB fields were launched using a line source with amplitude and phase given by the incident field at that location. This is indicated in Figs. 1(a)–1(d), 3(a) and 3(d) by a red line. The dispersive dielectric function of silver or gold was included by fitting the tabulated values  with a Drude dispersion model around the working wavelength. Reflection and coupling efficiency were obtained by projecting the field on the SPP mode, as described in Ref. . The FDTD mesh was set to 1 nm for the NW and 0.5 nm for the nanocone studies.
B. Near-to-Far-Field Transformation
The near-to-far field transformation was performed starting from the electromagnetic field obtained by BOR-FDTD calculations. On the reference plane shown in Fig. 2(a), one defines equivalent electric and magnetic surface current densities, which respectively are Js = −n̂ × E and Ms = n̂ × H, where n̂ is the unit vector normal to the surface. Each current element radiates to the far field as a dipolar source. By integrating the contribution of these elements over that plane, one obtains the electromagnetic field in the far region . Symmetry considerations imply that on the GRS the electric field has only the θ component in spherical coordinates (r, θ, φ), which reads
J1 is the Bessel function of the first kind , zo is where the reference plane intercepts the z-axis, Z is the medium impedance and k is the wavevector. ρmax should be large enough to make the contribution of the excluded field negligible.
We thank F. De Angelis, E. Di Fabrizio, M. Celebrano, K.-G. Lee and S. Götzinger for helpful conversations. This work was supported by ETH Zurich grant TH-49/06-1.
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