We investigate numerically a Scanning Near field Optical Microscope (SNOM) that uses a Parabolic Mirror (PM) to focus a radially polarized beam on a metallic tip. In order to overcome problems - like overestimated near fields or resonances - that arise when only considering finite tips, we have introduced a semi-infinite continuation of the tip, which incorporates the analytic solution of surface waves. For a realistic modeling the right description of the incident field is essential and we have complied with this requirement by a Bessel expansion of the focal fields, which is also applicable to an aplanatic objective. The established numerical model is used for an extensive study of model parameters like tip geometry, illumination directions and tip materials (Ag, Au, Al and Cu). Compared with a simplified inverted microscope configuration, the PM setup shows an increased field enhancement (factor of 2–2.5), which can be ascribed to the efficient coupling of the exciting field to tip surface plasmons.
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Over the last decades scanning near field microscopy has proven to be a useful technique for different research areas like nano optics, material science and Raman spectroscopy. The core of this technique is a sharp probe which allows for simultaneous measurement of topographic and optical information far beyond the diffraction limit [1–3]. Typically a sharp metal tip is used to interact with the local near-field, which is then coupled to the far-field and can be collected as signal. With this technique a spatial resolution down to ∼ 10nm is achievable [4–6], and thereby also the total signal is increased.
Aperture SNOMs, which consist of tapered metal-coated fibers, stood at the beginning of the evolution of different SNOM techniques. However, these probes suffer from very low throughput, since the propagating modes become very lossy with decreasing core diameter .
Apertureless probes, which are mostly sharp metal tips illuminated externally by a laser beam, can serve as localized near-field source and also serve for the collection of the near-field by efficiently releasing it to the far-field similar to an antenna. This type of operating mode is attractive, since opaque samples can be investigated. However, often the sample is also illuminated with a laser spot having a diameter on the order of one micrometer or larger. Hence the sample generates a far-field background signal , which leads to a loss of signal to noise ratio. To overcome this drawback indirect excitation schemes were proposed [7–11] which use guided surface plasmon polaritons (SPPs) for focusing energy below the tip apex. In this category also fall the so-called back illumination schemes, which consist of fully coated dielectric probes [12–14].
In the present study we considered an apertureless SNOM which uses a parabolic mirror (PM) with high numerical aperture (NA ≈ 1) to focus a radially polarized beam on a metal tip (see Fig. 1(a)). This setup fulfills various requirements for achieving super resolution combined with a high signal to noise ratio. One requirement, which is crucial for obtaining a high field enhancement is a high field component along the tip-axis . This requirement is met by illuminating the PM with a radially polarized laser beam with a diameter matching the NA of the mirror. This gives a diffraction limited field distribution in the focal volume, where the electric field component in the focus is polarized along the tip-axis . Furthermore, the application of a PM for focusing this type of light results in a tighter focus [17, 18] and therefore reduces the far-field background signal. Finally, the high NA allows for a high collection efficiency for the field emerging from the focus into the half-space above the sample and at the same time to an efficient suppression of the far-field background signal generated outside the focal volume. Numerical modelling is often the key for optimizing the free parameters in physical systems and gaining a deeper insight into the complex interrelationships of that system. The numerical modeling of SNOMs is demanding since it is necessary to model large systems, extending over several micrometres and to capture also small details in the nanometre regime. Truncated tips in the range of 200nm to a few microns [19–24] were often considered, due to the computational cost. However, the tip length has shown to have essential influence on the field enhancement. In particular, too small metal tips show an overestimated field enhancement, which in addition is overlaid by resonances and therefore gives rise to misleading results not corresponding to the experimental situation . If the size of a modelled tip is sufficient, is determined by the the propagation length of bulk and surface waves, which should be well below the size of the tip. The mentioned resonant behavior is based on delocalized SPPs which, due to reflection of the tip ends, form cavity modes . This resonant behavior is evitable by using absorbing boundary conditions mimicking semi-infinite structures, which then can guide away the SPPs [25,26]. For large scattering objects -like SNOM tips- a plane wave excitation is inappropriate for obtaining experimentally meaningful results [22, 25], because retardation effects cannot be neglected.
We note that in the considered system there are converging rays with large angles of incidence and consequently a paraxial approximation is clearly inadequate. Therefore, the use of a Gaussian beam is inappropriate for approximating the incident field and it is important to provide an efficient method for a realistic representation of this field.
In this study we aim to establish a realistic model for the PM SNOM configuration that overcomes the mentioned issues, by (i) providing a method for modeling the incident vector fields, which can be included in the used simulation software, (ii) by introducing a semi-infinite extension of the tip shaft by which the otherwise reflected SPPs are guided away and dissipated. Our model is used for an extensive study of the field enhancement for the available parameters, and also the application of different tip materials (Ag, Au, Cu, Al) is considered.
This paper is organized as follows: after the general introduction to the established electromagnetic model we lay out the mathematical basis for describing the vector field in the focal volume of a radially polarized beam focused by a PM (section 2.1), followed by a discussion of the surface wave solutions of a plasmonic wire (section 2.2). Finally, the results of the parameter studies are presented and discussed, with a key aspect on the excitation of focused SPPs in a PM SNOM that lead to strongly enhanced near fields (section 3).
2. The model
For solving Maxwell’s equations we have applied the multiple multipole program (MMP) [27, 28] contained in OpenMaX, which is available under open source . This method uses analytic solutions of the Maxwell equations for a finite series expansion of the fields in the different domains of a given problem. The parameters of the series are evaluated by imposing the field’s continuity conditions in matching points along the boundaries separating the different domains and then minimize the weighted error in a least squares sense. Consequently, the magnitude of the error integral along the boundaries can serve as quality control allowing for an accurate solution of the considered problem.
The domains of the present problem are sketched (with different length scale) in Fig. 1(b), where we have introduced a virtual boundary (Γ2,4 and Γ1,3) that separates the lower tip-end from the tip-shaft. The permittivity was set to the measured values for Ag, Au, Cu  and Al , respectively. The tip-end is located at the geometrical focus of the incident beam. Due to the centre-hole in the PM the infinitely extended tip shaft stays in the shadow region (see Fig. 1(a)). The incident beam is expressed in terms of a Bessel expansion (section 2.1) valid in in both domains defined as vacuum (D1 and D3 in Fig. 1(b)).
The impinging radiation is being scattered by the tip and also waves propagating along the wire can be excited. We have placed the virtual boundary sufficiently far away from the tip end (l = 11μm) and so we can represent the field in the upper region by a scattered space wave and a wave propagating along the wire.
It is well known that various modes can propagate along a circular wire . In the low frequency range, under the assumption of a purely imaginary permittivity, Sommerfeld  has shown that there exists a transverse magnetic surface wave with a low attenuation, which is also called the principal wave. Under the same conditions Hondros has shown  that beside the principal wave there are higher order waves (so-called complementary waves), which however are strongly attenuated. In the optical regime these surface waves are known as surface plasmons and they were independently studied by Ashley and Emerson  and Pfeiffer et. al. . We will show in section 2.2.1 that Hondros’ formalism is applicable for the considered range λ = 300 – 1000nm where the permittivity does not fulfill the assumption of being purely imaginary. In this context it turns out that only the principal wave will give a significant contribution to the field near the wire in the upper region because complementary waves are strongly attenuated.
The scattered space wave is expanded by a series of multipoles located at the tip end a so-called complex origin multipole  located near the edge of the wire (at R2, the transition of the conical to the cylindrical section) mimicking the field scattered from the edge of the wire. The fields in the lower region outside the wire are modeled by complex origin multipoles and for the fields inside the wire so-called ring multipoles  are used. The whole system including the illumination preserves azimuthal symmetry and consequently it is sufficient to distribute the matching points along a tiny segment in azimuthal direction, reducing the problem essentially to two dimensions.
In the following we consider time-harmonic fields, with a time dependence ∝ e−iωt, which was omitted in the field equations for the sake of clarity and brevity.
2.1. The focal fields in a parabolic mirror
A method for calculating the field in the vicinity of the focus of an optical system is the so-called Richards-Wolf formalism, which is based on the numerical evaluation of Debye diffraction integrals . In a previous work Lieb and Meixner  have used this method for a radially polarized beam focused with the PM objective. As stated by Sheppard and Török , the focal distributions via multipole expansion can be calculated much more efficiently than by using direct quadrature. Bessel expansions and multipoles constitute one of the basic expansions for the field representation in the MMP method and so the incident field can be modeled easily if the coefficients for the incident field are known. A derivation of these coefficients is given in the following lines.
Assuming the generating sources are located infinitely far away from a considered spherical volume, the field in that volume can be expanded in terms of Bessel vector spherical harmonics, which then represent a complete orthogonal set41]. However, here we will use a different approach where we take only one sphere, the so-called Gaussian reference sphere (GRS), which expands into the far-field (kR ≫ 1) and consider the field in the limit of geometrical optics at two distinct instants of time. At the first instant of time the amplitude is purely real (t0 = 0) and at the second the amplitude is purely imaginary (t1 = −π/(2ω)).
For the evaluation of the integrals, the knowledge of the incident field on the entire GRS is necessary. The amplitude of the incident field on the upper hemisphere can be calculated in paraxial approximation using geometrical optics . The reflection of the incident beam at the PM transforms the incoming cylindrical beam (with cylindrical coordinates ρ, ϕ, z) into a spherically converging beam, where the unit vector êρ gets mapped to êθ (see Fig. 2). For the aperture of the PM the radial coordinate is related to the elevation angle according to , where f is the focal length of the PM objective. The incident field on the back aperture is assumed to be a purely radial polarized mode, which is a superposition of two Hermite-Gaussian modes (TEM10 and TEM01) with a beam waist of w0 and a field amplitude of E0, which in paraxial approximation can be written as:16, 42]) and so the field in spherical coordinates on the upper hemisphere of the GRS is given by:
The field on the lower hemisphere can be derived from the diffraction integrals  from which it follows that
The formula for the coefficients bn now reduces to:
In the far-field, where kr is large, jn−1(kr) is the dominant term in (13) providing the appropriate normalization of the coefficients. In order to avoid numerical problems which may arise when the radius of the GRS is located accidentally on the zeros of jn−1(kr) we will readjust the radius so that
Using the asymptotic expansion for the spherical Bessel function 6,16,23,44]. The range of the wavelength was 300 – 1000nm and the order of the Bessel expansion was set to 30, where the asymptotic condition in (15) is still valid. For the calculation of the integrals the standard quadpack routine were used. The field on the GRS and the corresponding Bessel expansion is shown in Fig. 3(a) and 3(b), and the field intensity near the focus is shown in Fig. 3(b). Although the expanded field on the GRS deviates from the field specified by geometrical optics, we note that this still has no big influence in the proximity of the focal region. As stated by Brock  a good match within a sphere of radius R can be obtained by choosing the maximum order by . For N = 30 this corresponds to a sphere radius of approximately 5λ for which the field is precisely described and the inclusion of higher order terms brings negligible improvement.
2.2. Wave propagation along a cylindric wire with circular cross section
In this section we are considering the upper half space with an infinitely extended cylindrical wire of radius ϱ in vacuo. A summary of the formalism to the problem of waves propagating along a circular cylinder and the main results are given in .
In our model the electromagnetic field is TM, as the exciting radiation, and can be written in cylindrical coordinates (ρ, ϕ, z) as(20) the determinant of these equations must vanish. Using the relations (20), where we still have a free parameter β that defines the strength of the surface wave (23)
2.2.1. Complementary waves
As shown by Hondros  there is a series of other solutions to (23) distinguished from the principal wave by a high imaginary part in the propagation constant which he denoted as complementary waves. In the following we will show by adopting Hondros’ approach that in the considered range of our stated problem (λ = 300 – 1000nm, ϱ = 500 – 1500nm) the complementary waves will also have high imaginary parts for γ0 and that our assumption of considering just the principal wave is justified.
Considering (23) for values of γ0 in the complex plane, one realizes that beside the principal solution there is a series of roots that appear above k2/k1 beginning close to that point (e.g. in Fig. 4(a)). Assuming that there will be a solution where γ0 is close to k2/k1, we see that x2 defined by (22) will be small or in a moderate range (∼ 1). Furthermore, for the considered range mostly ϱ ∼ 1μm and k2/k1 > 1 holds (except for Ag with λ < 340nm), which means that the argument of the Hankel-functions in (23) becomes large and we can replace the Hankel-functions in (22) by their asymptotic representation (25). After some simple algebraic manipulation we obtain(22) we then obtain the corresponding propagation constant Fig. 4(b)). Since the cylindrical section extends over several wavelengths before it reaches the virtual boundary, we can assume that higher order modes are so strongly damped that they can be neglected in the region above the virtual boundary and consequently only the principal wave has to be taken into account.
2.2.2. Root-finding for the principal wave
For finding a solution of (23) that corresponds to the principal wave, we have applied a stepping scheme, which decreases stepwise the wire radius and then uses the Minpack  implementation of Powell’s Method for finding the roots of the determinantal equation. To ensure that the found solution is the principal wave, a starting value close to this solution is needed. For large radii we can use the asymptotic dispersion relation (26) as initial starting value for the stepping scheme which then starts with a large wire radius and solves (23) using Powell’s method. In the next step the wire radius is reduced by a specified factor cϱ according to ϱ(i+1) = ϱ(i)/cϱ and the solution of the previous step ( ) is used as starting value. As shown before, the complementary wave solutions are located sufficiently far away and due to the good search conditions the solution is found reliably.
For a fast start value evaluation we have set up a linear interpolation grid with the solutions of a predefined parameter set γ0(ρ(i), λk) where λk is the wavelength corresponding to the measured data of the dielectric function. The results for the considered materials and wavelengths are shown in Fig. 5 where also the asymptotic solution (indicated by ϱ∞) is shown. These results may also serve for a choice between materials for plasmonic nano tips and wires. The imaginary part of the propagation constant determines the propagation length (distance at which the field energy decays to exp(−1) measured in free space wavelengths), which is given by 1/(2Im(γ0)). Furthermore, a high value of Re(γ0) results in a transversal wavenumber with high imaginary part, which means a high transversal field confinement and gives a hint in which spectral region a high field confinement and enhancement can be expected. The frequency of the peak value of Re(γ0) will hereafter be referred to as surface plasmon frequency ωsp or, similarly, as surface plasmon wavelength λsp. Note that the surface plasmon frequency of Al lies above the considered range and is not visible in Fig. 5(a).
3. Results and discussion
In the following we will mainly concentrate on the field enhancement, which represents an essential quantity for SNOMs and Raman spectroscopy. We defined the field enhancement as the ratio of the time averaged total field 1nm below the tip to the maximum of the time averaged incident field.
We have considered the illumination from top and bottom. Although the bottom illumination does not correspond to a realistic operating mode for the parabolic mirror SNOM, it is beneficial to study this type of illumination since it is similar to a setup with inverse illumination and can therefore help to understand the conceptual differences between these experimental setups. Comparing the results for top and bottom illumination for the set of several parameters (see Figs. 6, 8 and 9) there are fundamental differences manifested in the field enhancement. Top illumination results in a field enhancement, which over a broad spectral region is much higher than for bottom illumination (factor of 2–2.5). For top illumination there is a strong dependence for the region close above λsp (see Fig. 6(b)), which expresses itself, for example, in the appearance of several peaks for silver. In particular for silver there is a peak in the field enhancement for top illumination, which is not visible for the bottom illuminated tip.
The reason for this substantial difference in field enhancement can be explained by the coupling of the incident field to SPPs. For top illumination the interaction of the incident field with the conduction electrons in the tip results in SPPs travelling towards the tip end and leading there to a high field enhancement. Considering the time averaged field along the tip (see Fig. 7(a)) this effect is manifested in a wave pattern that is similar to a standing wave, indicating the presence of upward and downward propagating surface waves. This feature is missing for the bottom illumination (see Fig. 7(b)). From the scattered field of the tip - see Fig. 7 (c) and 7(d)-it is evident that the excited SPPs can again couple to the far field through the tapered region and by the tip edge (R2 in Fig. 1(b)). It can be assumed, due to reciprocity, that this are also the channels through which the SPPs can be excited. The excited SPPs can add up constructively, leading to a resonant behavior in the field enhancement below the tip if the incident field matches to the geometrical shape of the tip. Consequently, for top illumination the tip can act as a nano antenna which by excitation of SPPs can concentrate the incident field energy to a small volume below the tip apex. In this context, special care has to be taken since the tip edge provides a possible channel for SPP excitation by the incident field, which for an unrealistic geometric model of the tip might lead to misleading results. It is evident that the coupling is strongly dependent on the shape of the tip, the material function and also the angular spectrum of the incident field. The angular spectrum of the incident field is in principle accessible, for example, through a spatial light modulator, and could be tailored in order to maximize the field enhancement for a given tip geometry. However, this is well beyond the scope of the present study.
We note that the inverse illumination mode is usually used for transparent substrates, where due to total internal reflection evanescent field components are available and which can also have an effect on the field enhancement. Furthermore, the presence of the substrate changes the angular distribution of the emission . Consequently, the results will certainly differ from the presented results if the substrate is taken into account, but nonetheless we expect that the fundamental findings are transferable to this system. We note that the inclusion of a substrate is rather demanding and can be modeled by a layered media Green’s function .
3.2. Wire radius and tip angle
Considering the influenence of the wire radius on the field enhancement, the most prominent feature is that bottom illumination results in a field enhancement almost independent of the wire radius (see Fig. 6(a)). This effect can be understood by the previously mentioned coupling to SPPs, where we have stated that bottom illumination couples to the upward propagating SPPs. The excited SPPs propagate along the tip with only little reflection at the tip edge and so the influence on the field below the tip is low. For calculating the field enhancement in a setup with bottom illumination this also suggests that it should be sufficient to consider thin wires, which can be simulated with less computational effort.
For top illumination the most noticeable change of field enhancement takes place for silver where several peaks are visible (Fig. 6(b) at 380, 420, 440nm). For smaller tip angles these peaks are less pronounced and become very small. It is evident that this behavior stems from the conical part of the tip which determines the coupling of the incident field with the tip, and is related to the previous mentioned antenna effect.
For small tip angles a change of the wire radius changes mainly the boundary segment that is located in the geometrical shadow of the incoming field. Consequently, the coupling of the incident field with the tapered part of the tip is less dependant of the wire radius for smaller tip angles (not shown), because the incident field extends over a small focal region and falls off rapidly along the central axis. This reduced coupling also explains why the peaks are less pronounced for smaller angles (see Fig. 8(b)). The efficient coupling to SPPs is of particular interest for optical spectroscopy, where the antenna effect on one hand can be exploited for exciting focused SPPs for achieving a strongly confined field below the tip apex; on the other hand, due to reciprocity, the emission of an emitter below the tip will have an angular distribution predominantly towards the PM leading to a high collection signal.
In the study of the tip angle, it is evident that a decreasing angle leads to a red shift of the field enhancement in the region close above λsp. This behaviour is visible for both types of illumination in (see Fig. 8). This observation is in agreement with the results of Zhang et. al.  who considered silver tips with the same set of angles. The rippled shape in the field enhancement - particularly pronounced for α = 15° and λ > 700nm in Fig. 8 - could be traced back to reflections and scattering of the SPPs at the tip edge and is a manifestation of the strong geometry dependence of SPPs.
For the spectral region (λ > 750nm) where Ag, Au and Cu have similar material parameters (see Fig. 4(b)) the field enhancement is similar in shape and level for those materials. In this spectral region, the field enhancement is higher for smaller angles for the shown values in Fig. 8. We note however, that for a tip radius of R1 = 10nm this increase in field enhancement with decreasing tip angle is less, and for tip radii R1 ≥ 20nm this behaviour changes and the field enhancement is stronger for the larger angles. A higher field enhancement for larger angles was also observed by Zhang et. al. . However, these authors have not considered smaller radii than 20nm for their study of the influence of the tip angle.
The red shift in the region close above λsp and the increase of field enhancement for decreasing angles can also be found in a simplified model of a tip, represented by a small prolate ellipsoid with a major (a) and minor (b) axis such that a, b ≪ λ. For this system (in the quasi static limit) the field at the tip of the ellipsoid reads as :Fig. 3(c) the lateral field Eρ drops to zero in the focus and the field is polarized along the major axis, which corresponds to the exciting field in the simplified model.
The depolarization factor determines the position of the plasmon resonance, which is determined by the wavelength where the denominator of (29) approaches zero. A deeper analysis shows that the resonance position depends approximately linearly on the aspect ratio such that an increased aspect ratio leads to a red shift of the resonance . For the SNOM tip this increased aspect ratio corresponds to a smaller tip angle, for which we also see the previously mentioned red shift in the region above λsp (see Figs. 8(a) and 8(b)).
At the tip of the ellipsoid the field is concentrated by the lightning rod factor γ, which reads as :Figs. 8(a) and 8(b). For the larger radii (20, 30, 50nm), however, the field enhancement is actually higher for larger tip angles, which already shows the limitations of this simplified model.
Finally, we note that the choice of an optimal tip angle is governed by the considered wavelength, since this determines the dominant enhancement effect. Close above λsp a larger tip angle seems more favourable since it offers better coupling to the incident field and allows for high field enhancements reaching down to lower wavelengths. For lower frequencies the materials approach the case of good conductors and it is the lightning rod factor that substantially influences the field enhancement by the shape of the boundary and in this context a smaller tip angle seems to be more favourable, at least for small apex radii.
3.3. Tip radius
As can be seen in Fig. 9 the tip radius is a crucial parameter and compared to the other geometrical parameters it turns out that it offers the highest possibility for influencing the field enhancement and consequently the resolution. For the SNOM tip the small radius of the spherical cap represents a singularity which gives rise to the lightning rod effect and leads to a high field enhancement. Note that aluminium has an interband absorption maximum at around λ ∼ 800nm, which renders this material less attractive for the near infrared region.
As already mentioned before, for the region close above λsp efficient coupling to SPPs can occur and lead to an increased enhancement, which is particularly pronounced for silver (λ ≃ 440nm).
A general comparison of the field enhancement between the present study and other studies is difficult, since the field below the tip is strongly evanescent and consequently the field enhancement strongly depends on the position below the tip, which is mostly different for the respective studies. Furhtermore, the incident field, which is undoubtedly crucial for the comparability, differs in most studies. A similar model by Zhang et. al. , which also considered an infinitely extended tip with a focused beam shows lower enhancement values, which might be ascribed to the lower numerical aperture for their incident beam.
In the present paper we established a numerical model of a SNOM where a radially polarized beam is focused by a PM on a metal tip. The inclusion of a semi-infinite wire in the model successfully prevented the appearance of artificial resonances and therefore corresponds better to the real system. Furthermore, the presented solutions of surface waves on the wire for the considered materials, wire radii and wavelengths provide information about propagation characteristics and can serve for a choice when electromagnetic energy is guided along nano-wires.
Since the proper description of the incident illumination is crucial for obtaining meaningful results , we have derived the necessary formulas, which allow for calculating the focal fields in terms of a Bessel expansion for a PM objective and also for an aplanatic lens.
We studied top illumination and an inverse illumination configuration and found mostly an increased enhancement (factor ∼ 2 – 2.5) for top illumination. The increased enhancement for top illumination could be traced back to an antenna effect where the incident field can efficiently couple to SPPs, which propagate towards the tip apex and concentrate the energy below the tip. In this regard also resonances appear which are strongly dependent on the tip geometry. As for antennas these resonances should be accessible by optimizing the incident illumination. For this purpose the PM with a high NA is well suited since it offers a broad angular spectrum for the incident field.
The influence of the tip geometry on the field enhancement was studied for a large set of different tip geometries, and as stated already by other studies, the radius of the tip apex turned out as the most influential parameter. The field enhancement for top illumination is strongly dependent on the geometry, which could be explained by the coupling of the incident field to SPPs. For bottom illumination, where mainly upward propagating SPPs are excited, the field enhancement showed to be almost independent of the wire radius, which allows for a simplification of the numerical model by only considering small wire radii.
The study of the tip angle showed a red shift in the plasmonic band of the materials, which we found to be similar to the red shift in a simplified model of a prolate spheroid.
In the study of gold, silver, copper and aluminum as tip materials, the choice is strongly governed by the intended working wavelength. In this context the presence of plasmon resonances gives rise to high field enhancements and distinct dependence on the tip geometry.
Josip Mihaljevic gratefully acknowledges funding from the Carl Zeiss Foundation. A. J. Meixner thanks for financial support from the DFG through Grant No. ME 1600/5-3 and ME 1600/12-2. We would also like to thank the reviewers for comments and suggestions, which helped to improve the quality and clarity of this work.
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