We propose and analyze a new type of mechanically robust optical nanofocusing probe with minimized external environmental interference. The probe consists of a dielectric optical fiber terminated by a dielectric hemisphere – both covered in thin gold film whose thickness is reduced (tapered) along the surface of the hemisphere toward its tip. Thus the proposed probe combines the advantages of the diffraction-limited focusing due to annular propagation of the plasmon with its nanofocusing by a tapered metal wedge (i.e. a metal film with reducing local thickness). The numerical finite-element analysis demonstrates strongly subwavelength resolution of the described structure with the achievable size of the focal spot of ~20 nm with up to ~150 times enhancement of the local electric field intensity. Detailed physical interpretations of the obtained results are presented and possible application as a new type of SNOM probe for subwavelength imaging, spectroscopy and sensing are also discussed.
©2012 Optical Society of America
It is well known that diffraction presents a fundamental limit to how tightly propagating electromagnetic waves can be localized or focused in dielectric media, restricting the minimum size of a guided mode or spatial resolution of any conventional far-field imaging device to about half the wavelength . Focusing of light far beyond the diffraction limit into nanoscale spatial regions as small as a few nanometers (and thus achieving the effective delivery of electromagnetic energy to nanoscale objects, such as quantum dots, nanostructures or separate molecules) is called nanofocusing . One of the simplest ways for achieving nanofocusing is based on using tapered metallic nanostructures guiding strongly localized surface plasmon-polariton (SPP) modes . Examples of metallic nanofocusing structures include tapered metal rods [3–8], sharp metal wedges [9–11], a tapered section of a metallic film on a dielectric substrate [12–14], a tapered nanogap between two metallic media [15–19], metallic V-grooves [20,21], a tapered two-wire transmission line , etc. In many cases, nano-focal spots from a few tens down to just several nanometers have been predicted, as well as strong enhancement of the local plasmonic electric field intensity by factors of hundreds or even thousands . This opens unique opportunities for the development of a new generation of nano-optical sensors, near-field imaging techniques with the resolution as good as a few nanometers, near-field super-resolution spectroscopy (including surface-enhanced Raman spectroscopy of single molecules), interfacing between optical communication systems and nanoscale electronic devices, nanobiophotonics applications, and precise nanomanipulation of nanoscale objects and separate molecules, etc.
A somewhat alternative way of efficient coupling of light energy to nanoscale structures is based on the use of plasmonic nanoantennas [2,23–34]. Unlike nanofocusing structures, nanoantennas work as plasmonic resonators that accumulate the light energy from the incident radiation, leading to significant local field enhancement near the antenna elements. Thus plasmonic nanoantennas act as efficient coupling intermediaries between the relatively long-wavelength bulk optical radiation and nanoscale objects [25,30]. It is currently still an open question which of the coupling structures for light, plasmonic nanofocusing structures or nanoantennas, are capable of producing the strongest local field enhancement. At the same time, their significant structural differences open vast opportunities and a range of important options for the design and fabrication of nano-optical sensors, detection and near-field imaging techniques with a range of operational capabilities.
Attempts to combine the resonance antenna effect with plasmonic nanofocusing in tapered metallic structures were undertaken by several recent papers [35–39], where plasmonic modes were considered in coupled metal nanoparticles, including the so-called ‘kissing’ metal cylinders [37–39]. Such structures are capable of simultaneously working as a nanoantenna for coupling electromagnetic energy from a bulk incident wave into a resonance mode of the nanoparticles, and nanofocusing due to either changing dimensions of the nanoparticles [35,36] or tapered gaps between closely spaced nano-cylinders [37–39]. Further, the conducted theoretical analysis of plasmonic modes on circular paraboloidal surfaces  may also be helpful for the analysis of surface plasmons in such structures.
One of the disadvantages of these structures combining nanofocusing with the resonant (nanoantenna) effects in nanoparticles is related to difficulties with their fabrication and/or use as a tip of a near-field sensor or a scanning microscopic probe. At the same time, the idea of plasmon nanofocusing by a spherical or otherwise curved metal surface with tapered elements may seem attractive for the design of new types of near-field sensing/imaging probes with nanoscale resolution.
Therefore, in this paper, we propose and describe a nanofocusing structure that consists of a microscopic dielectric hemisphere covered in a tapered metal film with the smallest thickness at the tip of the hemisphere. This structure is shown to combine the benefits of diffraction-limited focusing due to plasmon annular propagation towards the tip of the hemisphere, and nanofocusing in the tapered metal film . As a result, the full-width half-maximum (FWHM) of the nano-focal spot at the tip of the hemisphere is predicted to approach ~20 nm (i.e. ~λvac/30; λvac is the wavelength in vacuum) with the electric field intensity enhancement factor of ~150. We will also demonstrate that this structure attached to an optical fiber represents a new type of efficient and robust probe for a near-field scanning microscope or nano-optical sensor with increased mechanical strength (due to the absence of free-standing sharply tapered metallic structures) and significantly reduced interference from external factors and contaminants.
2. Structure and methods
The proposed structure is presented in Fig. 1(a) . A dielectric (e.g. glass) fiber of radius R1 and refractive index n1 = (ε1)1/2 is terminated by a dielectric hemisphere with the same refractive index n1 and radius R1. The fiber and the hemisphere are coated in metal film of complex refractive index n2 = (ε2)1/2. The initial (at the point of attachment of the hemisphere to the fiber) thickness of the metal film is t0 = R2 – R1, where R2 is the radius of the external surface of the metal film covering the hemisphere. The smallest thickness of the metal film t = tmin is at the tip of the hemisphere (i.e. at z = 0, r = 0 – Fig. 1(a)).
The analysis of this nanofocusing structure was conducted as a two-dimensional problem in the cylindrical co-ordinates, using the finite-element analysis software package COMSOL Multiphysics®. The incident plasmon in the structure was generated at either the gold-fiber interface (inner plasmon), or at the gold-air interface (outer plasmon) at the distance z ~– 1.5 μm from the tip of the hemisphere (Fig. 1(a)). To achieve this, a port boundary condition was used, which involved the generating magnetic field with the radial distribution corresponding to either the inner or outer plasmon, and the perfectly matched layer boundary condition to eliminate reflections of plasmons propagating back toward the point of excitation.
3. Numerical results and discussion
Figures 1(b)-1€ show the typical distributions of the magnetic and electric fields in the probe with tmin = 5 nm for the inner (Figs. 1(b), 1(d), 1(e)) and outer (Fig. 1(c)) plasmon excitation at λvac = 632.8 nm. An excited plasmon propagates towards the hemisphere along the gold-coated fiber and is then focused by the hemisphere. The standing wave patterns of the field along the surface of the probe are formed by the plasmons passing through the focal point at the tip of the hemisphere and propagating backwards along the hemisphere and the fiber towards the point of excitation. Note that the nanofocusing mechanism here differs significantly to the conventional tapered structures terminated by a sharp tip [3–14]. Indeed, due to the large radius of curvature of the probe termination (~1 μm), the plasmons may propagate continuously along the tip surface and then back in the opposite direction with negligible radiative loss. This is in contrast with the previously considered nanofocusing tapered structures where the very small radius of curvature at the tip (~5 – 10 nm) may result in significant radiative loss into the bulk .
The inner plasmon corresponds to the quasi-symmetric (with respect to charge distribution) plasmon in a thin metal film on a dielectric substrate [12,13], while the outer plasmon corresponds to the quasi-anti-symmetric film plasmon [12,13]. Therefore, the inner plasmon experiences additional nanofocusing by the tapered metal film on the hemisphere, while the outer plasmon has a cut-off film thickness and thus cannot experience nanofocusing [12,13].
The considered cylindrical symmetry of the problem (i.e. the independence of all the field components of the φ-coordinate) means that the magnetic field vectors having only φ-components point in opposite directions at the opposite sides of the optical fiber. Such a transverse magnetic (TM) plasmon will converge towards the tip of the hemisphere with the magnetic field cancelled at the tip, which results in a node of the standing wave patterns (zero magnetic field) at the tip of the hemisphere for both the inner and outer plasmon excitations (Figs. 1(b), 1(c)). The radial components of the electric field Er also point in the opposite directions at the opposite sides of the fiber. However, as the plasmon propagates along the surface of the hemisphere, the electric field components normal to the metal-fiber interface (i.e. Er at the point where it is attached to the fiber) gradually rotate and transform into Ez at the tip of the hemisphere. Because of the rotational symmetry of the structure, this transformation occurs in such a way that the normal to the metal-hemisphere component of the electric field experiences constructive interference at the tip of the structure with an anti-node of the standing wave pattern (Fig. 1(e)). The z-components of the electric field at the surface of the fiber are similarly rotated and transformed into the r-components of the electric field at the tip, but pointing in the opposite directions and thus cancelling each other, resulting in the node of the standing wave pattern at the tip (Fig. 1(d)).
For the inner plasmon excitation (Figs. 1(b), 1(d), 1(e)), the field of the plasmon is effectively screened by the thick (up to 200 nm) gold film, and the field at the metal-air interface of the film is negligible, except only for the region near the tip of the hemisphere where the thickness of the tapered gold film is smaller than the skin depth (Fig. 1(b)). This is very different from the case of the outer plasmon excitation, where the field is significant at the air-metal interface everywhere along the probe (Fig. 1(c)). This is seen as a significant practical advantage of the probe excitation by an inner plasmon – the screening effect eliminates the possibility of interference of the plasmon with the environment and contaminants on the metal-air interface. For example, the described near-field optical probe with the inner plasmon excitation could be equally used in the air or water, as the inner plasmon interacts with the environment only within the nanoscale region near the tip of the hemisphere, where its field emerges at the external metal interface. Thus, only the nanoscale region near the tip may have a noticeable impact on the propagating inner plasmon, which will enable highly targeted testing of nanostructures and/or materials within this region irrespective of the general environment surrounding the probe.
Further differences in the obtained field patterns for the inner and outer plasmons (and, in particular, significant leakage near the hemisphere for the outer plasmon excitation Fig. 1(c)) can be explained by the fact that the outer plasmon propagates along the convex (spherical) gold-air interface. On the contrary, the inner plasmon propagates along a concave gold-glass interface, and its bend losses are effectively eliminated by the screening gold film. This is another advantage of the inner plasmon excitation, which should lead to significantly larger local field enhancement at the tip (see below).
The spatial resolution of the proposed near-field optical probe is given by the size of the hotspot (plasmon waist) at the focal point at the tip of the hemisphere. For the inner plasmon, it should be about half of the minimum wavelength λp = λpmin of the quasi-symmetric film plasmon achieved at the tip with the minimum film thickness t = tmin. Reducing tmin results in a reduction of λpmin [12,13], and this means reduced nanoscale size of the focal region and increased subwavelength resolution of the probe.
Quantitatively, the spatial resolution of the probe can be evaluated as the FWHM of the electric field intensity |E|2 in the plane z = 1 nm (i.e. ~1 nm above the tip of the hemisphere covered in tapered metal film – Fig. 1(a)). The maximum value of |E|2 at the tip determines the local enhancement of the field intensity in the structure. Figures 2(a) -2(c) present the typical dependencies of the electric field intensity |E|2 on r in the plane z = 1 nm (positioned at 1 nm above the tip of the probe) for the three different film thicknesses at the tip: (a) tmin = 2 nm, (b) tmin = 5 nm, and (c) tmin = 8 nm. These figures confirm the expectation that the FWHM (and thus the spatial resolution of the probe) for the inner plasmon excitation (solid curves) is significantly reduced with decreasing tmin.
Furthermore, the field intensity distributions corresponding to the inner plasmon excitation in the considered nanofocusing structures are practically indistinguishable near the focal point from the respective distributions produced by the annularly converging quasi-symmetric plasmon in a uniform flat gold film of thickness t = tmin on a dielectric (glass) substrate – compare the solid and dotted curves in Figs. 2(a)-2(c). This clearly confirms that the focal spot, FWHM and the resolution of the probe are physically determined by the quasi-symmetric plasmon and its wavelength at the gold film thickness t = tmin. In this regard, the limiting factor to the resolution of this structure is the minimum film thickness. This is in contrast to, for example, the conical rod, where the limiting factor for the resolution is the radius of curvature of the tip [3–8].
Figure 2(b) also demonstrates that there can be major differences between the intensity distributions near the tip for the inner plasmon and outer plasmon excitations of the probe. In particular, the pronounced side-lobe structure in the case of the outer plasmon excitation (Fig. 2(b) – dashed curve) results in a significant reduction in the resolution of the probe. Interestingly, the same figure shows that in addition to the two major maximums in the side-lobe structure for the outer plasmon excitation, there is also a maximum of the field at exactly the tip of the structure, which has the same width as the maximum for the inner plasmon excitation (dotted curve). This can be explained as follows. The radius of the coated hemisphere R2 ~700 nm is close to the wavelength of the outer plasmon, which is close to λvac = 632.8 nm. As a result, the outer plasmon should experience significant scattering effects from the curvature of the guiding surface. These scattering effects naturally include radiation losses from the outer (quasi-anti-symmetric) plasmon into the bulk radiation (Fig. 1(c)) and scattering into the inner (quasi-symmetric) plasmon through the metal film. Scattering into the inner plasmon may only be efficient in the regions where the thickness of the metal film is close to or smaller than the skin depth δ ~30 nm, i.e. relatively close to the tip. As a result, even if we initially have pure outer plasmon excitation, the inner plasmon is still excited in the structure due to the scattering of the outer plasmon on the curved hemispherical surface. This scattered inner plasmon experiences nanofocusing in the probe and results in the middle maximum on the dashed curve in Fig. 2(b). This is also the explanation for the fact that the resolution of the probe is often independent of whether the initial inner plasmon excitation or outer plasmon excitation was used (Figs. 2(a), 2(c), 2(d)). It is only in special cases where the side lobe structure caused by the outer plasmon appears to be dominant, that the resolution is significantly reduced during the outer plasmon excitation (Fig. 2(b)).
Though the resolution of the probe is often independent of whether the inner or outer plasmon excitation is used (Figs. 2(a), 2(c)), the local field enhancement at the tip is strongly dependent on the excitation type. For example, the dashed and solid curves in Fig. 2(d) demonstrate that the local electric field enhancement at the tip is substantially higher for the inner plasmon excitation. This is easy to understand, as the efficiency of scattering of the outer plasmon into the inner plasmon on the curved surface of the hemisphere is rather low, which results only in a fraction of the initial outer plasmon energy being scattered into the inner plasmon. Therefore, the local field at the tip is significantly lower in the case of outer plasmon excitation (the dashed curve in Fig. 2(d)). The rest of the energy of the outer plasmon is lost to the radiation leakage from the probe mainly due to bend losses for the outer plasmon at the curved surface of the hemisphere (Fig. 1(c)).
On the contrary, if the inner plasmon is excited in the probe, it experiences only insignificant radiation bend losses on the glass hemisphere, mainly because of the screening effect of the gold film covering the hemisphere. In addition, the wavelength of the inner plasmon significantly decreases towards the tip of the hemisphere. This makes the radius of curvature of the hemisphere significantly larger than the inner plasmon wavelength, which is another reason for the low bend losses of the inner plasmon. Therefore, practically all the energy of the inner plasmon (apart from the fraction of its energy lost to dissipation in the metal) reaches the tip of the probe, resulting in significantly larger local field enhancements (the solid curve in Fig. 2(d)). This is another significant advantage of using the inner plasmon excitation in the proposed probe.
Significant subwavelength resolution of the proposed probe is demonstrated by Figs. 2(a)-2(c). As discussed above, the resolution of the probe is increased with decreasing thickness of the metal film at the tip of the probe and may reach ~20 nm for the minimum gold film thickness of tmin = 2 nm (Fig. 2(a)). For more practical larger gold film thicknesses with tmin = 5 nm, the resolution of the probe is reduced to allow (for the inner plasmon excitation) resolution of objects as small as ~45 nm (i.e., ~λvac/14) – see the solid curve in Fig. 2(b).
The dotted curve in Fig. 2(d) shows that approximately the same resolution could be achieved if we use a near-field optical probe in the form of a uniform gold rod terminated by a gold hemisphere – both with the diameter of ~60 nm. Nevertheless, using such a probe may be associated with significant technical difficulties including fabrication complexity, mechanical weakness of the metal nano-rod, difficulties with efficient coupling of electromagnetic energy into the rod, and low levels of local field enhancement (Fig. 2(d)), unless nanofocusing tapered rods are used [3–8] etc. The proposed new probe eliminates or significantly eases these difficulties.
Figure 3(a) shows the comparison between the resolution of the probe for the inner and outer plasmon excitation. In particular, it can again be seen that, apart from the values of tmin around 4.5 nm or 5 nm, the resolution defined as the FWHM is independent of the type of plasmon excitation, because of scattering of the outer plasmon into the inner plasmon on the curved surface of the hemisphere (see above). Figure 3(b) shows not only the substantial difference in the levels of the local field enhancement in the probe corresponding to the inner and outer plasmon excitation, but it also shows that the local field enhancement tends to have a minimum for the outer plasmon excitation at the values of tmin ~4.5 – 5 nm, corresponding to the reduced resolution of the probe. This observation may offer an explanation for the appearance of the dominant side-lobe structure on the dashed curve in Fig. 2(b) for the outer plasmon excitation, which significantly reduces the resolution of the probe (filled circles in Fig. 3(a)). This effect is caused not by an enhancement of the side-lobe structure in the electric field intensity distribution in the z = 1 nm plane, but rather by a reduction of the strength of the central maximum, which makes the side-lobe structure relatively more pronounced and causes the reduction of the resolution of the probe (see the dashed curve in Fig. 2(b) and outlying filled circles in Fig. 3(a)). This also naturally results in a reduction of the local field enhancement in the probe (circles in Fig. 3(b)), because the enhancement is typically determined by the field strength in the central maximum.
The discussed decrease in the height of the central maximum should be caused by a reduced efficiency of generation of the inner plasmon in the probe. This occurs due to destructive interference effects during scattering of the outer plasmon into the inner plasmon. As indicated above, scattering of the outer plasmon into the inner plasmon is only efficient where the thickness of the gold film is close to or smaller than the skin depth δ ~30 nm. For example, consider the two points A and B on the surface of the hemisphere (Fig. 4 ). Let us assume that the thickness of the metal film at the position of point A is ≤ δ. The wavelength of the outer plasmon is relatively large and increases with reducing thickness of the metal film. At the same time, the wavelength of the inner plasmon is significantly smaller and rapidly decreases with decreasing thickness of the metal film. Therefore, if the metal thickness at the tip is reduced, then the metal thickness at both points A and B also decreases, and the optical path (the number of wavelengths) for the inner plasmon between points A and B may significantly increase (Figs. 4(a) and 4(b)), which means significantly changing phase of the plasmon arriving at point B from point A. On the other hand, because the wavelength of the outer plasmon is large and does not change substantially with decreasing thickness of the metal film, the conditions for excitation of the inner plasmon by the outer plasmon at point B are not expected to change significantly either (except for the film thickness). As a result, the phase difference between the inner plasmon arriving at point B from point A and the inner plasmon excited by the outer plasmon at point B may significantly change with changing tmin. Depending on this phase shift, we obtain either constructive or destructive interference between these inner plasmons at point B. This is one of the physical mechanisms that explains the significant reduction of the central maximum (due to the destructive interference of the inner plasmons) on the dashed curve in Fig. 2(b) for tmin ~5 nm, compared to Figs. 2(a) and 2(c) for tmin = 2 nm and tmin = 8 nm, respectively.
The curve in Fig. 3(a) is not plotted as a fit to the data points, but is rather a dependence calculated using the determined ~18 nm for the probe with the minimum gold film thickness tmin = 2 nm (Fig. 2(a) and the leftmost cross in Fig. 3(a)) and the following simple proportion for other values of the minimum thickness of the gold film in the probe:Fig. 3(a) clearly confirms the physical nature of the focal spot and its expected size determined by the local wavelength of the annularly converging quasi-symmetric film plasmon in the tapered gold film with the minimum thickness tmin at the tip of the probe. Note that the resolution of the probe will also depend on the refractive index of the fiber, as the wavelength of the quasi-symmetric plasmon mode decreases with an increase in ε1.
While we have investigated the proposed structure for nanofocusing in the visible spectral region at λvac = 632.8 nm, it is expected to be suitable also for operation in the near IR. However, as the resolution is determined by the wavelength λpmin of the quasi-symmetric plasmon mode in the metal film of thickness t = tmin, the resolution does not scale linearly with the vacuum wavelength. For example, at tmin = 5 nm and λvac = 632.8 nm we have λpmin ~0.13λvac, while at λvac = 1550 nm we have λpmin ~0.46λvac. In this regard, at fixed tmin, the resolution of the probe is reduced as the wavelength increases (due to the dispersive properties of gold and the quasi-symmetric plasmon mode).
In this paper, we have proposed and analyzed in detail a new nanofocusing structure/probe having rotational symmetry and using a microscopic hemisphere attached to a micro-scale optical fiber – both covered in thin metal film whose thickness is gradually reduced along the surface of the hemisphere to a minimum thickness at the tip of the hemisphere. The proposed nanofocusing probe combines the advantages of diffraction-limited focusing on the surface of the sphere (due to the annular propagation of the excited plasmon) and nanofocusing due to reducing (tapered) thickness of the metal film covering the glass hemisphere. Thus, the proposed new probe represents a combination of efficient nanofocusing in wedges [12–14] with the annular propagation of the plasmon (for further field enhancement) for the design of a robust and efficient near-field optical probe.
As has been indicated, the described probe has significant practical benefits including its mechanical robustness and strength (due to the absence of any free-standing sharply tapered metal nanostructures), and substantially reduced interference with the environment and external contaminations (due to the discussed screening effect of the gold film covering the glass fiber and hemisphere). It also combines these structural advantages with strong local field enhancement at the tip of the hemisphere, and strong localization of the plasmonic field ensuring subwavelength spatial resolution of the probe. For example, using the rigorous numerical analysis, it was predicted that the focal spot size at the tip of the hemisphere can be as small as ~20 nm (i.e. ~λvac/30) for reasonable structural parameters allowing the approximation of macroscopic electrodynamics in the absence of noticeable spatial dispersion . Simultaneously, the achievable local electric field intensity enhancement was shown to approach and even exceed ~150 times. Further optimization of the geometrical and material parameters is expected to further enhance the performance of the device.
In addition, the proposed structural design represents a possible new type of SNOM tip for near-field microscopy, spectroscopy and nano-sensing with subwavelength resolution. The considered structures are not limited to the exact spherical shape of the dielectric hemisphere. The shape of the terminating dielectric section can in principle be any smooth surface, such as a hemisphere, paraboloid, etc. Furthermore, the spherical profile of the tapered section of the metal film is not crucial for the proposed probe either. The probe will work with any sufficiently smooth taper profile of the covering metal film (as long as the minimum film thickness is achieved at the tip of the structure), though structural optimization in this regard may be worthwhile in future. The metal film with a required taper profile (with the minimum metal thickness at the tip) can be fabricated by means of the standard evaporative deposition of the metal onto the optical fiber (with a hemisphere) rotating around its axis, and with the deposition occurring at a significant (e.g., normal) angle with respect to this axis. This demonstrates the potential simplicity of fabrication of the proposed probe for its subsequent experimental investigation and optimization.
Efficient excitation of the plasmons in the probe can be realized in practice by means of a grating coupler transforming a mode of the optical fiber with non-zero Hφ component into the inner or outer surface plasmon mode. Broadband nanofocusing can be achieved in the probe by using chirped gratings  to couple the incident optical fiber mode with varying frequency into the respective inner or outer plasmons. At the same time, the question about the possibility of nanofocusing femtosecond pulses  still remains open. Though the proposed probe does not rely upon single-mode optical fibers, the coupling grating should typically be adjusted to each fiber mode (with the respective polarization characteristics) to ensure generation of the surface plasmons with the required radial symmetry. In addition, the coupling efficiency may be significantly different for different fiber modes. These factors may result in significant distortion of ultra-short pulses, which will require additional analysis that is beyond the scope of this paper.
It is also worth noting that the proposed structure/probe will be particularly useful for the optical analysis, local spectroscopy and imaging with nanoscale resolution of surface nanostructures with small-scale surface topography, e.g., thin non-uniform films including mono-layers and sub-mono-layers of substances, or as a sensor tip in liquid or gaseous environments (including biological specimens). It may be of a particular interest to use the proposed probe for the analysis of small nanoparticles and large molecules trapped by the strongly localized near-field of the inner plasmon near the tip of the hemisphere (for the analysis of different schemes of such near-field trapping near sharp metal tips see [43–45]).
This work was supported by NRF (National Honor Scientist Program: 2010-0020414, WCU: R32-2008-000-10180-0), KISTI (KSC-2011-G3-02), and BK21 (Dept. of EECS, Seoul National University). Daniel R. Mason and Dmitri K. Gramotnev contributed equally to this work.
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