1. Introduction

Plasmonic waveguides posses highly confined modes thus allowing to increase the level of circuit integration [1]. Different plasmonic waveguide structures like strips [2, 3], nano-particle chains [4], nanowires [5–7], grooves [8–11], wedges [12] and gaps [13–16] have been developed and investigated. Among various structures, the gap plasmon waveguide (GPW) [13–16] is particularly interesting, because its modes are extremely small and its fabrication process is comparably simple. However, most investigations of GPWs [13–16] have concentrated on the distribution, the dispersion and the losses of respective modes. Up to now only a few papers [4, 17–19] have dealt with the efficient excitation of GPW modes by external radiation. Prism, grating and end-fire coupling [20] have been used, but require a good matching of the spatial field distributions of the plasmonic waveguide with the exciting beam. This works well for weakly confined modes. Stegeman et al. [18] showed for that case that the coupling efficiency may exceed 90% when exciting plasmons at the interface between semi-infinite metallic and dielectric media by focusing the light onto the facet of the sample. Maier et al. [19] also used tapered optical fibers to bring light to the input facet of the waveguide. But for extremely confined modes as they are found in GPWs, the overlap between the exciting beam and the nanoscale waveguide mode becomes extremely small, because even in the case of extreme focusing the width of the excitation exceeds half a wavelength. Therefore the problem of an efficient excitation of GPWs with nanoscale cross-section is still under discussion. However, recently various investigators explored nanoantennas in the optical regime [21–27]. Nanoantennas can be seen as a miniaturized version of conventional antennas which may convert free-space radiation to guided waves in waveguides like transmission lines and vice versa. However, in the optical regime metals do not behave like ideal conductors. Nevertheless, some of the concepts developed in electrical engineering (see e.g.[34]) still work in the optical regime and others only have to be modified as it was shown recently [30, 31, 33]. The efficient delivery of electromagnetic radiations to nanoscale waveguides can be considered as a problem of either mode or impedance matching, where the first represents the optical point of view and the second that of electrical engineering. In what follows we simulate the excitation of GPWs by dipole nanoantennas using the commercial software COMSOL and compare the results with respective predictions obtained using standard antenna theory.

2. Field distributions of nanoantennas and gap waveguides

We first calculated the eigenmodes of plasmonic antennas and waveguides using the 2D mode analysis in COMSOL. We assumed a wavelength of λ = 1.55um and a silver layer of 100nm thickness (ε = -116.38+11.1i [28]) being deposited on a glass substrate (n = 1.444 [28]). The size of the computational domain was always 3um × 3um × 3.5um, which was large enough that fields were small enough at the boundaries to be efficiently absorbed by perfectly matched layers surrounding the computational domain. In order to get an accurate result, the minimum mesh size was set to 3nm. Figure 1(c) and (d) show typical distributions of the electric field in the gaps of a nanoantenna and a plasmonic waveguide. Obviously the fields are highly confined and almost coincide. The light of the mode inside the GPW leaks neither to the substrate nor to the air since the effective index of the fundamental modes are 1.65+0.023i, which has a real value larger than the refractive index of the glass substrate.

 

Fig. 1. The incident beam is focused from air to the center of the antenna, which feeds the GPW (see principal sketch of the structure as 3D (a) and top view (b)). Field distributions ∣E⃗∣ of fundamental modes are displayed for a nanoantenna (c) with a gap width G = 30nm and an arm length L = 675nm and for a GPW (d) with the same gap width.

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Besides the 2D mode analysis, the excitation of the GPW by an antenna (see Fig. 1(a) and (b)) was analyzed applying a 3D harmonic propagation analysis in COMSOL. The sample was assumed to be irradiated from air. The exciting beam is tightly focused onto the center of the nanoantenna reaching a diameter of 1um. Obviously the pure entrance facet of the GPW allows for a moderate excitation of the GPW mode. However, the presence of an antenna improves the coupling to external radiation considerably.

 

Fig. 2. The field distributions ∣E⃗∣ on the xz plane (y = 0) (a) when exciting with the antenna L = 675nm,G = 30nm; (b) when exciting without antennas G = 30nm. The antenna arm length L of (a) is optimized for a gap width of 30nm. Because the field enhancements change dramatically, different color bars are used in (a) and (b).

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As shown in Fig. 2(a), the surface plasmon polaritons are excited on the nanoantenna by the external incident field. At the same time, the confined mode with large intensity enhancement appears in the gap of the two nanorods and then propagates along the waveguide. We define the incident light power by integrating the time-averaged power flow over a circular area of 1um diameter. The coupled light power is determined by integrating the time-averaged power flow over the area of the gap at the terminal a - b shown in Fig. 1(b). For G = 30nm with the antenna in Fig. 2(a), the optimized coupling efficiency is 10.6%. And the coupling efficiency without antennas in Fig. 2(b) is 0.082%, which is nearly 129 times smaller. At the same time, the intensity enhancement inside the waveguide is around 400 relative to the incident beam when exciting with the resonant nanoantenna compared to less than 4 without nanoantennas. For a gap width of G = 100nm the coupling efficiency even for an optimized antenna does not reach more than 2.7%, while the coupling efficiency without antennas is 0.22%, which is 12 times smaller. The apparent differences of the coupling efficiency even for an optimized antenna length are surprising and will be discussed below.

3. Geometry optimization

At optical frequencies noble metals are no longer highly conductive materials. Hence, standard optimization schemes developed for the microwave domain cannot be applied in a straight forward way. Size and shape dependent plasmonic resonances influence the optical response considerably [29]. Still it would be highly desirable to reduce this increased set of free parameters and to characterize the individual elements by a few numbers only, e.g. a complex impedance. Following that approach Alù and Engheta have shown [32] that plasmonic waveguides can to a certain extend be regarded as optical transmission lines. Hence, our system can be optimized like an equivalent electric circuit (see Fig. 3), where impedance matching between source and load or antenna and GPW has to be obtained. To this end we determine the impedance of our elements first. The classical impedance is defined as the ratio between voltage and current at the termination port of an element. Here we set this port at the points a and b in Fig. 1(b) and Fig. 3.

 

Fig. 3. The nanoantenna in the receiving mode (a) can be described by an equivalent circuit model (b). V: optical voltage induced by the incident wave, R_a: resistance of the antenna, X_a: reactance of the antenna, R_g: resistance of the plasmonic waveguide, X_g: reactance of the plasmonic waveguide.

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Hence, we are now going to investigate how well voltage and current delivered by the antenna fit to the respective values required by the GPW. As long as the fields are quite homogenous in the gap between points a and b, the ratio between voltage and current can be replaced by the ratio of the appropriate components of the electric and the magnetic field [27, 33]. In what follows the ratio of E_x and H_y in the middle between the points a and b is chosen.

In general the impedance of the GPW Z_g can be easily deduced from the field structure of the respective waveguide mode. However, in our particular situation Z_g is the impedance of the GPW and of a correspondingly matched load because the GPW is terminated with the PML. Hence, we optimize the system for a slightly modified load impedance, which we determine by driving the GPW with an antenna and measuring the fields on port a-b. The impedance of the antenna Z_a is determined by reversing the experimental situation and operating the antenna as load in an emitting mode. We assume no incident field, but drive the antenna by a point like source placed in the GPW far away from the points a and b and close to the perfectly matched layers. Figure 4 (a) and (b) show the impedance of antennas of various lengths with gap widths of 30nm and 100nm. Actually the calculated impedance is not the real impedance of the dipole antenna but that of the antenna combined with a part of the GPW, because the impedance is calculated based on the terminal a–b. It should be further kept in mind that assigning a classical impedance to a plasmonic nanoantenna can also yield unphysical results. For vanishing antenna arms where the antenna consists of the GPW only the resistance approaches slightly negative values (see Fig. 4(a) for an antenna length smaller than 0.3λ). When looking to the respective field structures one notices that the field is no longer homogenous at the port between the points a and b. Strong back reflections from the residuals of the antenna and an excitation of fields propagating above the waveguide result in an inhomogenous field structure in the gap causing a local back flow of energy. However, for that parameter range, the impedance is dominated by its imaginary part. It will turn out that this value can still be used to optimize the structure.

Having determined the impedance of each of the elements we are now going to construct an equivalent circuit model (see Fig. 3(a)). The incident wave is assumed to create a voltage V. This voltage will induce a current defined by the impedance of respective circuit elements (see Fig. 3(b)) as

 

Fig. 4. Antenna resistance (a) and reactance (b) as well as power delivered to the waveguide (c) as a function of the antenna length for gap widths of 30nm and 100nm.

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Hence, the power delivered to the load or the GPW is given by

In what follows we look for the optimum antenna for a certain GPW. Hence we assume R_g and X_g to be fixed, but vary the impedance of the antenna. The maximum power will be delivered to the GPW, if

holds and R_a is as small as possible.

The above condition is the ideal case of conjugate matching resulting from an optimization of equation (4). Note, that it differs from the standard result, where the load is adapted to a given source. Still the above result must be interpreted carefully, because it is based on the assumption of a constant voltage V. But a vanishing R_a will finally also cause a drop of the voltage. Still we find the ratio between voltage and current delivered by the antenna to vary in a wide range and that the maximum power is in fact usually delivered to the GPW, if R_a is relatively small compared to R_g. However, it turns out that matching the reactance is by far more critical. It is indicated in Fig. 4(b) that only for the 30nm gap width, the reactance of the antenna and of the waveguide can match for certain antenna lengths. But only for some of these matching points the antenna resistance is small resulting in a maximum power transfer. In fact the two peaks of high power flow displayed in Fig. 4c correspond to extremely small values of the antenna resistance (i.e. Ra=14 and 39 Ohm), which are much smaller than the respective values of the waveguide (i.e.96 Ohm). This result is in good agreement with the power flows derived from FEM simulations of the complete system (see Fig. 4(c)).

For the 100nm gap width, antenna and waveguide reactance do never match. Therefore the power flow at the terminal a – b shown in Fig. 4(c) is considerably smaller and does not have pronounced peaks. The same behaviour is also found for other values of the gap width (see Fig. 5).

 

Fig. 5. The coupling efficiency to an incident beam obtained for an optimized antenna length is displayed as a function of the gap width.

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Although the cross section of the waveguide increases linearly, this cannot compensate the lack of impedance matching for larger waveguides. As shown in Fig. 5, the maximum coupling efficiency decreases dramatically, when the gap width increases to 40nm, and then increases slightly due to the increased gap area. However, the propagation length of the field inside the waveguide is smaller for narrower gaps due to the stronger confinement of the field inside the gap. Hence, a trade-off has to be found according to the respective application.

4. Conclusion

It has been found that the use of nanoantennas can increase the conversion of an incident beam into a waveguide mode dramatically. Although an excitation at optical frequencies is considered, classical arguments from the microwave domain can be applied. It turns out that impedance matching between the antenna and the waveguide plays an important role to achieve an optimum coupling efficiency. Only for a small gap width of 30nm the reactance of the waveguide and the antenna can be matched resulting in a considerably increased coupling efficiency.