## Abstract

We examine the possibility of optimizing the emission and the near-field signal of apertureless silver and gold tips by using an optimized non-periodic grating. In this context, we consider the emission of a single quantum emitter in close proximity to optimized tips. Additionally, we study the far-field coupling efficiency of a tightly focused beam to the near-field of the tip. The gain in performance is compared with unstructured tips and the comparison with a pure plasmonic excitation of an unstructured tip is discussed. The optimized, structured tips show a significant enhancement of the total decay rate, as a result of standing plasmonic waves between the grating and the tip apex, leading to a resonant behavior. The resonances can be explained well with a Fabry-Pérot model. Furthermore, the total decay rate of an emitter near a structured tip can also be decreased as compared to an unstructured tip, when the grating is shifted from the optimal resonant position. The proposed scheme represents an interesting novel nano-antenna, for which the resonance as well as the directivity can be controlled by the grating.

© 2015 Optical Society of America

## 1. Introduction

Sharp metallic tips excited by optical fields have various applications in several scientific domains, such as Tip Enhanced Raman Spectroscopy (TERS), Near-field-Scanning-Optical-Microscopy (NSOM) [1], hot-electron nanoscopy [2], optical trapping [3, 4] or optical near-field mapping by utilizing the plasmon-mediated luminescence from the apex [5]. At the apex of such a tip, an externally applied optical field can give rise to charge oscillations, which lead to high field enhancements, due to the lightning rod effect. This effect enables the tip to act as near-field source. Besides the direct excitation of the tip apex by an external field, there is also the possibility of an indirect excitation by surface plasmon polaritons (SPPs) [6], which is also known as adiabatic focusing. This type of excitation can be realized by a grating coupling of radiation to SPPs that propagate to the tip apex [7–9].

Besides the use as near-field source, the tip also represents a nano-antenna, which can couple to an emitter nearby and efficiently release the energy of the emitter to the far-field [10]. In this regard it is worth to mention that the tip can alter the local mode density at the position of the emitter, and therefore it is possible to influence the emitter’s radiative decay rate [11, 12].

In a study by Issa and Guckenberger [13] for an infinitely extended SNOM-tip it was shown that the electromagnetic-mediated decay channels can be separated into radiative decay, SPP generation and local energy transfer (LET). The latter becomes important for distances ≲ 5 nm and is responsible for quenching. For distances ≳ 5 nm it was shown that the generation of SPPs can become very efficient.

The coupling of this SPP to free space radiation is strongly dependent on the propagation constant of the corresponding SPP. In particular, high propagation constants lead to low coupling to free space radiation, and so this decay channel is mainly non-radiative, leading to a loss of useful signal. In the light of this finding, we analyzed the possibility of transferring the otherwise non-radiative portion of SPPs via a grating coupler to free space radiation. Thereby, the loss of useful signal should be prevented and the antenna efficiency should be significantly increased.

Indeed, there are many studies which have considered the analogous reverse case, i.e. the excitation of SPPs by radiation impinging on a grating [7–9, 14–17]. Here, to the best of our knowledge, only the use of periodic gratings was considered. This circumstance can be ascribed to the fact that there is a rule of thumb for estimating the coupling efficiency of periodic gratings. By using the reciprocal vector of the grating (2*π/P*), the coupling condition can be written as [18]

*k*is the propagation constant of the SPP,

^{spp}*n*the refractive index of the dielectric medium around the tip,

*k*

_{0}the free space propagation constant,

*ϕ*the angle of incidence and

*P*the period of the grating. Consequently, the required grating period can be calculated by this formula for a given set of parameters. However, the shape of the grating itself is also a very critical property for the coupling efficiency. For instance, it is strongly dependent on the grating modulation, and so strongly modulated gratings do not necessarily provide a better coupling efficiency than weakly modulated ones [19]. For the application of Eq. (1) in the domain of circular wires or conical tips, an additional restriction has to be kept in mind, namely the dependence of

*k*on the wire radius.

^{spp}Compared to periodic gratings, non-periodic gratings can offer a higher degree of freedom, which can help to design devices with highly specialized functions. To cite a few examples, a non-periodic grating coupler can be used for the efficient suppression of an unwanted second diffraction order [20], or, it can be used for unidirectional launching of SPPs [21].

The lacking heuristics in the design of non-periodic structures makes a numerical optimization necessary, which can become computationally very demanding. The higher freedom in the optical design, therefore, comes at the cost of the necessity of a numerical optimization. However, it is clear that the application of a numerical optimization can also help to reduce the demands and potential costs of the manufacturing process by restricting the parameter space of the optimization to structures, which can be manufactured easily or at least with reasonable effort.

In the following we aim at optimizing the emission of SNOM tips by applying a non-periodic grating [see Fig. 1(a)], which then can be experimentally realized by focused ion beam milling. The guidelines in this optimization process were set as follows. Firstly, the grating should be geometrically as simple as possible because the structuring of such a near field tip will be quite challenging. Therefore, we have studied rounded rectangular grooves. In order to keep manufacturing demands low, only two grooves were considered. The grating position was restricted to a small range around 1 *μ*m from the tip apex, which is a compromise, because the distance of the grating to the apex is subject to two conflicting demands. On one hand, the coupling of a tightly focused beam is intended to reduce the background signal, which requires the grating to be located far from the tip apex. On the other hand, the grating should be located near the apex for reducing propagation losses of the excited SPPs. According to the results in the following sections, the grating position of ∼ 1 *μ*m turned out to be reasonable.

For the optimization we have used an evolutionary algorithm, namely the covariance matrix adaption evolutionary strategy [22], which was implemented in the framework of DEAP [23]. The solution of Maxwell’s equations for evaluating the cost function, was performed by the Multiple Multipole Programm (MMP) contained in OpenMaXwell [24, 25]. Further details about the optimization are given in section 1.2.

The following sections of the manuscript are organized as follows. First, we consider the emission of a single quantum emitter near an optimized tip. In this regard, we begin with a presentation of the theoretical background, then we continue with the process of tip optimization and finally we discuss the corresponding results. In the second part, we study the far-field coupling efficiency of a tightly focused beam and discuss achievable improvements in comparison with different excitation schemes.

#### 1.1. Emission enhancement and inhibition of a single quantum emitter

In this section we consider a single quantum emitter, such as an atom or molecule, located below a metallic tip. We restrict the analysis to a two level system with a high intrinsic quantum yield (i.e., *q _{i}* ≈ 1), which can be excited by an external field in the non-saturated regime. Note that the enhancement of fluorescence and Raman scattering near plasmonic structures can depend on factors like Stokes shift, quantum yield and the strength of the external field [26].

### 1.1.1. Theoretical background

For an emitter with transition dipole moment **p** and transition frequency *ω*_{0}, the spontaneous decay rate can be calculated by Fermi’s golden rule [11]:

*ρ*(

_{L}**r**

*,*

_{e}*ω*

_{0}) is the local density of states (LDOS) at the location of the emitter, i.e.

**r**

*. The LDOS can be expressed by the dyadic Green’s function [11]:*

_{e}**n**

*is a unit vector along the dipole axis. For free space the LDOS is given by: and so the corresponding spontaneous decay rate reads*

_{p}The virtue of the representation in Eq. (3) becomes evident, when comparing it with the energy dissipation of a classical dipole in an inhomogeneous environment. There one realizes that the expressions are the same. The latter follows from Poynting’s theorem and can be simply calculated by integrating the Poynting vector over a spherical surface, containing the dipole. Consequently, the relative change of transition rates can be described in the frame of classical electrodynamics. For the considered system, with *q _{i}* ≈ 1, this allows us to write the relative transition rate in terms of the totally emitted power

*P*and the power emitted by a dipole in free space

*P*

_{0}

*γ*is the non-radiative-SPP rate at which SPPs are thermally dissipated in the infinite tip shaft.

_{spp}*γ*represents the local energy transfer decay rate and is associated with short ranged dissipation due to ohmic losses. Furthermore,

_{LET}*γ*can also be separated into the different spatial components according to the power emission of the system. In antenna theory this corresponds to the directivity of an antenna. In NSOM the use of a substrate, which may also be opaque in the emission frequency range, often restricts the measurable signal to the upper halfspace. Therefore, we separate the radiative decay rate into contributions to the upper

_{rad}*γ*

^{+}and lower

*γ*

^{−}halfspace In principle, these values can be obtained by integrating the Poynting vector over the corresponding hemisphere [e.g. boundary

*P*

^{−}in Fig. 1(b)]. In the presence of an infinitely extended tip the situation is more complicated because of the SPPs which can propagate along the wire. However, by noting that the propagating SPP decays very quickly in the transverse direction, it is possible to use a circular ring [see

*P*in Fig. 1(b)] for the power integration. In this segment the scattered field from the tip is very low and can be neglected. This allows for separating the power decay into SPPs and radiation in the upper halfspace [

_{spp}*P*

^{+}in Fig. 1(b)].

For nano-spheres it is known that the emission rate is enhanced dominantly for the orientation of the transition dipole parallel to the central axis of the particle [27–29]. For a perpendicular orientation mainly quenching is observed and only small enhancements ∼ 2 can be obtained, if the emission frequency is close to the plasmon resonance [29].

It can be assumed that the situation is similar for a SNOM tip. Therefore, we restricted the following study to emitters with a transition dipole oriented along the central axis of the tip.

Furthermore, it is important to note that a field enhancement most efficient when the tip is excited along the central axis, which leads to a *z*-polarized electric field below the tip, and therefore we will only use emitters with a transition dipole moment parallel to the *z*-axis.

Issa and Guckenberger [13] have studied a similar configuration with a dipole below a SNOM tip. As already mentioned before, they have found for dipole-to-apex distances ≲ 5nm quenching becomes important due to a local energy transfer, being restricted to a very small volume at the tip apex. Furthermore, they have stated that the generation of SPPs becomes extraordinarily efficient. The character of the occurring SPPs was mainly non-radiative, whereas also coupling to the far-field by the tapered part of the tip was observed.

We want to operate in the regime where a strong SPP generation is present. Therefore the distance of the dipole to the apex was fixed to 10nm. The excitation by a vertically oriented dipole located on the central axis of a rotationally symmetric tip can only excite SPPs with the same azimuthal symmetry. On a circular wire the corresponding axisymmetric modes can be written in cylindrical coordinates (*ρ*, *ϕ*, *z*) as [30]:

*l*= 2 denotes the interior domain of the wire,

*l*= 1 the exterior domain,

*κ*is the transverse wavenumber and

_{l}*Z*represents either the Bessel function

_{n}*J*or the Hankel function of first kind ${H}_{n}^{(1)}$. The coefficients

_{n}*A*are obtained by the solution of the corresponding boundary value problem.

_{l}Here *k _{z}* fully determines the corresponding mode. Higher order modes have a bulk-wave character, associated with high propagation losses, and consequently, only the fundamental TM0 mode, which has a surface-wave behavior, is responsible for long distance energy transport. Hereafter, the propagation constant of the fundamental mode will be referred to as
${k}_{z}^{\mathit{spp}}$ or also
${k}_{z,0}^{\mathit{spp}}$. The latter is the relative propagation constant that is given by dividing
${k}_{z}^{\mathit{spp}}$ by the free space propagation constant

*k*

_{0}.

A dipole emitter below the tip with an dipole axis aligned along the central axis of this tip will excite radial symmetric SPPs propagating along the tip. The SPP propagation on a conical tip with a small opening angle can be approximated by the propagation of the TM0 mode on a circular wire, whereby the propagation constant of the fundamental TM0 mode strongly depends on the cylinder radius; see Fig. 2.

The fundamental TM0 mode has a purely evanescent transverse field and hence will not radiate to the far-field while it propagates along a cylindrical wire. However, due to the breaking of the translational invariance, a propagating SPP on the conical tip can couple to free space radiation. Therefore, the SPP is not purely non-radiative as in the case of a cylinder and it is necessary to mention that the approximation by the TM0 mode does not contain this coupling to free space radiation. Nevertheless, the approximation is still valuable, in the sense that the coupling to free space radiation can be estimated by the propagation constant. A SPP with a propagation constant much higher than the free space value will hence only inefficiently couple to free space radiation. For the considered materials and wavelengths silver at 405nm has the highest relative propagation constant (see Fig. 2) and so coupling to the far field can be expected to be the worst. For near-field microscopy this represents a problem, since SPPs with large propagation constants are associated with high spatial frequencies that are needed for super-resolution. Thus, the corresponding SPP-modes will be unnoticeably dissipated along the wire. The use of a grating coupler can compensate for this signal-loss by helping to bridge the mismatch between the propagation constants of free space and the SPP.

#### 1.2. Optimization of the tip

We have considered silver tips for excitation wavelengths of 405nm and 550 nm and gold tips for 633nm, whereby the surrounding medium was vacuum. In the present study we have assumed a harmonic time dependence, i.e. ∝ exp(−*iω*_{0}*t*). The geometry of the tip and the grating are depicted in Fig. 1(a). For the optimization, the constraints of the geometric parameters were the following: the interspacing (*b*_{2}) and the heights (*h*_{1}, *h*_{2}) of the grooves were in the range between 40nm and 150nm; the amplitudes (*d*_{1}, *d*_{2}) of the grating were in the range between 30nm and 150nm and the distance of the first groove to the apex (*b*_{1}) was limited to the range 1μm ± *λ*_{0}/3, where *λ*_{0} is the free space wavelength. The remaining geometric parameters were set to *θ _{c}* = 15°,

*R*

_{1}= 10nm,

*R*

_{2}= 10nm,

*R*

_{3}= 3μm and the wire diameter was 2μm. For modeling the SNOM tip, the MMP model of an infinitely extended metallic tip was used, which is presented in detail in [30].

We were interested in maximizing the radiative decay into the upper halfspace, i.e. *γ*^{+}, and so we used −*P*^{+} as cost function in the CMA-ES algorithm. The resulting parameters of this optimization are summarized in Table 1.

#### 1.3. Results and discussion for the optimized emission enhancement

For all considered systems the results in Table 1 show a significant enhancement of the radiative decay *F*^{+} compared to a tip without grating. The corresponding emission patterns are shown in Fig. 3 and it is clearly visible that the emission in the upper halfspace is dominant.

At first glance it might seem surprising that the total decay rate is also increased compared to a tip without a grating. As it was mentioned before, the grating is intended for transferring the generated SPPs to propagating radiation. If the grating would only convert the SPPs to radiation, we would expect the total decay rate to be unaffected by the introduced grating. However, one must bear in mind that the presence of a grating also gives rise to reflected SPPs, which can propagate back to the tip apex, and therefore will influence the scattered field at the position of the emitter. The effect of the scattered field on the total decay rate can be understood by using Poynting’s theorem and by using the normalized total rate of energy dissipation expressed in terms of the scattered field **E*** _{s}*, which can be written as [11]:

*γ*

_{tot}/γ_{0}. Therefore, it is evident that the change of the decay rate depends on the secondary field of the dipole at its position

**r**

*. The SPP reflected at the grating will contribute to the secondary field*

_{e}**E**

*. Depending on the phase of this contribution, the energy dissipation — and equivalently the decay rate — can either be increased or decreased. It is again worth noting that we would expect no change of the decay rate, if the grating would only transfer the SPP into radiation without any SPP reflection.*

_{s}If one pictures the grating as reflecting mirror, like inside an interferometer, it seems obvious that the phase of the scattered field can be manipulated by the position of the grating; see *b*_{1} in Fig. 1(a). Therefore, we studied the influence of a grating shift on the decay rates. The results are shown in Fig. 4. Note that the dashed line represents the values of the decay rates of a similar tip, however, without a grating. Furthermore, the fundamental mode for Ag @ 550nm (see Fig. 2) has the lowest losses and consequently the highest quality factor. This leads to higher resonance peaks and smaller FWHM values. However, in terms of an experimental realization this requires high precision in positioning of the grating.

The shape of the curves in Fig. 4 clearly reveal the presence of resonances, and similar to a resonator, the decay rate can either be enhanced or inhibited [31].

Indeed, the system has some characteristics in common with a Fabry-Pérot resonator. We will adapt this correspondence to a Fabry-Pérot resonator, as it was done for one dimensional single arm antennas, where the Fabry-Pérot model has proven to be appropriate for describing the occurring resonances [32–34].

In a Fabry-Pérot resonator a characteristic variable is the free spectral range Δ*f*, which is given by

*n*is the refractive index of the resonator medium,

*c*the speed of light and

*L*is the distance of the mirrors. For a constant frequency the appearance of resonances will depend on the distance between the mirrors, and the interspacing of resonances Δ

*L*can be computed by where we have used the propagation constant

*k*of the mode in the resonator. This can be used to estimate the spacing of the resonances in Fig. 4 by identifying the real part of the SPP as mode of the resonator, i.e. $k=\text{Re}\left({k}_{z}^{\mathit{spp}}\right)$, and by using an effective length Δ

*L*/cos

*θ*instead of Δ

_{c}*L*. This effective length is due to the tapering of the wire, on which the grating is located. For ${k}_{z}^{\mathit{spp}}$ we can use the solution of the fundamental mode of a cylindrical wire (see Fig. 2), where the diameter is chosen according to the corresponding diameter between two resonances. Then, the interspacing can be approximated by

However, there are also some differences to a Fabry-Pérot resonator. In contrast to a Fabry-Pérot resonator, where the interspacing Δ*L* between resonances is independent of the resonator length, we see in Fig. 4 that the height of the resonance peaks decreases with increasing distance of the grating to the apex, i.e. the cavity length. Considering the SPP mode — which forms a standing wave — as cavity mode, we must take into account that this mode suffers from propagation losses that become larger for an increasing cavity length (grating to apex distance). Furthermore, the coupling of the SPP to radiation can lead to an additional loss channel, which needs to be taken into account. Consequently the quality factor decreases with increasing grating distance.

Trying to capture the main effects of a frequency change, we have simulated the spectral response of the optimized tips. The results are shown in Fig. 5. The occurrence of resonances is clearly visible for all tips with a grating. It is worthwhile noting that the resonance peaks tend to decrease with increasing wavelength. This effect can be easily understood by the observation that the grating represents an ever smaller perturbation for an increasing wavelength. Therefore, in the limit of large wavelengths the decay rates of a tip with a grating converge asymptotically to the values of a tip without a grating.

## 2. Far-field excitation

In this section we consider the optimized tips of the previous section under far-field excitation. We are interested in coupling to SPPs through the grating on the tip in order to generate a high near-field at the tip apex. As stated by Stockman [6] a radially symmetric SPP mode (transverse magnetic TM) is necessary for adiabatic focusing. Consequently, a radially symmetric grating for coupling to TM modes also demands a TM excitation. Furthermore, the indirect excitation of SPPs at the grating should reduce the far-field background emanating from the sample surface located at the tip apex. These two requirements can be fulfilled by using a tightly focused radially polarized beam, which can be focused, for example, by a parabolic mirror (PM) [30, 35, 36]. In the following we will use this type of excitation for our simulations. The details of the implementation can be found in [30].

For reducing the background signal, the focus needs to be shifted towards the grating [as sketched in Fig. 6(b)]. In a PM this shift could be realized by using a weakly focusing lens in front of the PM or by simply shifting the tip with respect to the focus as shown in Fig. 6(b).

To quantify the achievable improvement by such a configuration, we introduced a Signal-to-Background Ratio (SBR) similar to the approach used by Novotny and Hecht in [11, p. 210]. As useful signal the integrated near-field signal below the tip was considered. The near-field strongly depends on the tip apex radius [30] and the FWHM value of the near-field is approximately given by the apex radius *R*_{1}. Therefore, we have defined the useful signal *S _{nf}* as

*ρ*is the radial coordinate and |

**E**(

*ρ*)|

^{2}is the squared field modulus at the normal plane of the tip 10nm below the apex. The far-field background signal

*S*is proportional to the illuminated confocal area of the sample and was approximated by

_{ff}Note that we have used the same wavelengths for the external excitation by the focused beam as for the simulation of the emission process, which was modeled by a dipole source below the tip. However, for the simulation of fluorescence or Raman scattering the excitation wavelength would in general differ from the emission wavelength, which would lead to different results [26].

#### 2.1. Results and discussion for far-field excitation

As mentioned before, the SPPs on the tip can couple to the far-field when they are propagating along a conical segment. The coupling efficiency depends strongly on the propagation constant of the SPP mode and, as described above, can be estimated by the fundamental mode of a circular wire. It is clear that the coupling efficiency is very low for a large mismatch to the free space propagation constant. In this situation, the use of the optimized gratings strongly increases the coupling efficiency between the far-field and plasmonic modes on the tip and leads to a high field amplitude at the tip apex, which can be seen in Fig. 7. In particular for the silver tip at a wavelength of 405nm (for an optimal focus position) the near-field amplitude increases by a factor of ∼ 90.

The optimal focus location, for which the amplitude is maximized, was found to be slightly (75–110nm) below the position of the lower edge of the grating [*b*_{1} in Fig. 1(a)]. This position is indicated by the dashed vertical line in the intensity plot in Fig. 7. According to the shape of the field amplitude plot, the sensitivity to a misalignment of the focus to the optimal position is not very critical. For example, for the optimized silver tip at 405nm a shift of the focus position of ±100nm leads to a maximum falloff of 10% in field intensity.

As can be seen from Fig. 7 the SBR can be strongly improved by shifting the focus towards the grating position. For the silver tip at *λ* = 550nm and the gold tip at *λ* = 633nm an improvement of a factor up to ∼ 3 can be achieved, whereas for the silver tip at *λ* = 405nm even a factor of ∼ 90 can be obtained. This can be explained by two effects. Firstly, a tightly focused beam diverges quickly [11], which leads to a fast intensity decrease with increasing distance from the focus, and so the background signal is reduced. Secondly, the better coupling via the grating to SPPs and the associated adiabatic focusing increase the useful near-field signal.

In addition to comparing the optimized tips to tips without a grating, we have also used the tips without a grating in combination with a pure SPP (TM0 mode) excitation; as depicted in Fig. 6(c). This corresponds well to the situation of adiabatic focusing on a conical wire, as shown in [6] by Stockman. Note that in contrast to ideal adiabatic focusing, there is still a background contribution when we use a pure SPP excitation, since there is emission to free space due to the symmetry breaking by the tapered wire. The results of this analysis are summarized in Table 2. For the considered optimized tips the resulting SBRs are similar to those obtained by a pure SPP excitation. Note that the SBR value of the optimized tip excited with a wavelength of 405nm is even slightly higher than for adiabatic focusing, i.e. 0.41 vs. 0.40. The reason for this is a lower *S _{ff}* value due to a lower field intensity in the corresponding area. This lower field intensity might be ascribed to interference effects with the exciting field. However, no clear proofs for this assumption are present at this stage and further investigations would be needed.

## 3. Conclusions

We have studied a concept for enhancing the useful signal in scanning near-field optical microscopy by adding an optimized grating coupler on an apertureless metallic tip.

The expected enhancement of the optimized SNOM tips can be estimated by the product of emission enhancement *F*^{+} and excitation enhancement, which can be represented by the maximum of the normalized field intensity in Fig. 7. For the silver tip and the gold tip at a wavelength of 550nm and 633nm, respectively, an enhancement of up to ∼ 15 can be obtained. For the optimized silver tip at *λ* = 405nm a much higher enhancement of ∼ 3 × 10^{3} can be obtained, due to the stronger confinement of the occurring SPPs. Consequently, the structuring of tips with optimized gratings could be of great interest for entering a new operating range of apertureless SNOMs, which is inaccessible for unstructured ones. It is important to mention that the high enhancement is also due to the use of a radially polarized beam in a radially symmetric setup, which however is not the most common way to excite apertureless tips. Furthermore, the oxidation of the silver tips can lead to a degradation of the near field enhancement.

For maximizing the emission to the far-field by a structured tip, we have found that the grating must have i) a good outcoupling and ii) the grating must also possess good reflection properties in order to exhibit localized plasmon resonances. These resonances can strongly influence both the emission rate of a nearby emitter and the field enhancement of the tip itself.

We believe that the proposed scheme represents an interesting nano-antenna, for which the resonance as well as the directivity can be manipulated by the grating. However, the price for this improved flexibility is the need for a numerical optimization and increased fabrication requirements.

Compared to other scannable nano-antennas, such as spheres on fiber tips [28] or a bowtie antenna on an AFM tip [37], the proposed scheme does not need a direct illumination, and therefore the background signal can be drastically reduced. Furthermore, the tip excitation via the grating leads to an enhanced near-field amplitude at the apex as compared to an excitation where the tip apex is simply placed into the focus. The reason for this enhancement is the larger interaction volume of the tip via the grating, from which more electromagnetic energy can be collected. For the situation where the tip apex is placed directly into the focus, the energy density at the focus is clearly much higher but also restricted to a very small volume. The transferable energy depends on both the energy density and the interacting volume, and so the grating coupling can efficiently enhance the near-field amplitude.

In this study we have used the CMA-ES algorithm for optimization, due to its robustness. However, for the considered problem also other derivative free methods, such as the Nelder-Mead algorithm, could be applied in order to speed up the optimization process when starting from a good guess.

In this context, it is also worth to note that it can be useful to define another cost function. For instance, the shown results reveal that not only the enhancement of the emission into the upper halfspace, but also the emission into the lower one is enhanced. This might lead to unwanted interference effects, for example, with the substrate. By changing the cost function to (*P*^{−} − *P*^{+}) it should be possible to reduce the emission in the lower halfspace.

## Acknowledgments

We acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of University of Tübingen. A.M. acknowledge financial support from the SPP1391 ultrafast nanooptics priority program. J. M. gratefully acknowledges funding from the Carl-Zeiss Foundation.

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