## Abstract

Optical connects will become a key point in the next generation of integrated circuits, namely the upcoming nanoscale optical chips. In this context, nano-optical wireless links using nanoantennas have been presented as a promising alternative to regular plasmonic waveguide links, whose main limitation is the range propagation due to the metal absorption losses. In this paper we present the complete design of a high-capability wireless nanolink using matched directive nanoantennas. It will be shown how the use of directive nanoantennas clearly enhances the capability of the link, improving its behavior with respect to non-directive nanoantennas and largely outperforming regular plasmonic waveguide connects.

© 2013 OSA

## 1. Introduction

With the great advances in nanotechnology, plasmonic integrated circuits (ICs) using sub-diffraction propagation plasmonic waveguides have been proposed to achieve nanoscale integration in upcoming optical ICs. Plasmonic guiding provides highly confined propagation compared with the usual dielectric photonic guides, thereby enabling miniaturization of future optical chips [1–5]. Nevertheless, it is also well known that plasmonic waveguides suffer from metal absorption and, consequently, they do not provide long propagation distances, thus being limited to communications in reduced ranges in terms of wavelength. One alternative could be to revert to the use of dielectric photonic guides, but this would be at the expense of greatly increasing the footprint of transmission lines due to the diffraction optics.

To circumvent the previous trade-off between miniaturization and losses in nanoscale chips, wireless connectivity based on optical nanoantennas has been suggested as a promising alternative [6]. Nanoantennas enable optical wireless links, which would present much less absorption losses, largely outperforming conventional plasmonic waveguides. Replacing waveguide networks by nanoantenna chip-to-chip and intra-chip links also provides more on-chip space that can be used to house other circuitry, hence enabling further miniaturization.

When designing plasmonic nanolinks, special care must be taken in order to account for the different behavior of the plasmonic elements (waveguides, antennas, impedance matching units, etc.) compared with their low-frequency counterpart. Although many of the properties remain similar, important differences may arise in the optical domain, because material properties do not scale with frequency [7]. Consequently, the classical low frequency antenna design principles and rules cannot be straightforwardly downscaled. Despite these differences, many efforts are being done to take advantage of the wide experience in radiofrequency (RF) link designs, applying it to optics with great success. At first, nano-optical equivalents of simple antennas were designed using the classical RF principles, and they were studied as isolated elements [8–10]. The advances in the fabrication of nano-optical antennas were also applied to achieve directive antennas in the optical regime [11–14]. Recently, optical antennas have been properly designed for connection to plasmonic waveguides, as an integral part of nanoscale circuits [6, 15]. This has been a major step towards the application of the extensive RF know-how to the optical domain. The fundamental concepts of input impedance, radiation resistance, and impedance matching between antennas and transmission lines, so extensively used in RF, have been adapted and extended to the optical regime. The use of optical circuits made of lumped nanoloads has also been theoretically proposed for the impedance matching between antennas and plasmonic waveguides, greatly improving the power transfer between the antenna and the waveguide [6].

In this paper, we propose the complete design of an optical wireless nanolink using specifically designed directive nanoantennas and impedance matching elements. As in RF and microwave regimes, the use of directive antennas can significantly enhance the performance of port-to-port communications compared with omnidirectional antenna broadcast links. Directive links greatly increase the power balance (ratio of received to transmitted power) and the link efficiency while minimizing interference with other parts of the circuitry. This reduces undesired coupling, allowing the level of circuit integration to be increased. Thereby, directive nanolinks may soon become a pervasive building block in the design of upcoming nanoscale optical ICs.

## 2. Numerical results

The proposed layout is shown in the lower inset of Fig. 1. Two directive Yagi-Uda nanoantennas facing each other are considered. On the left side, the transmitting (Tx) nanoantenna is connected to a feeding metal-insulator-metal (MIM) plasmonic waveguide. The waveguide is excited by a delta-gap source at its shorted end to mainly excite the fundamental TM plasmon mode (other excitations such as a quantum-dot inside the guide gap have also been tested, providing similar results). The transmitter details are shown in Fig. 1(a). On the right side, the receiving (Rx) antenna is connected to a MIM waveguide terminated with a matched load, as shown in Fig. 1(b). Antennas and waveguides are coplanar and single layer, and the entire link is fully embedded in glass to prevent most of the power from being refracted into the supporting substrate, as observed in [13]. As it would be done in an equivalent RF or microwave link, the design of such an optical link includes the following steps: i) appropriate selection of the plasmonic waveguide for the targeted frequency; ii) design and optimization of the directive Yagi-Uda antennas to provide maximum gain at the selected frequency; and iii) design and optimization of the proper lumped-element impedance matching network for the purpose of maximizing the power transfer between the nanoantennas and the plasmonic waveguides.

As a first step we characterize the MIM plasmonic waveguide, consisting of two silver parallel wires with a 20×20 nm^{2} section separated by a 10 nm glass gap. The link frequency is set to 461.2185 THz, corresponding to a wavelength of *λ*_{0} = 650 nm in vacuum and *λ*_{g} = 447.97 nm in glass. We simulate a long enough piece of MIM waveguide (6 *μ*m, about 13.4 *λ*_{g}) using the surface integral equation-method of moments (SIE-MoM) technique. The waveguide is terminated by a matched load to further avoid any end reflection. A detailed image of the matched load is shown in the upper-right inset of Fig. 1. It consists of a pyramid-shaped lossy material whose shape and constitutive parameters were selected to match the vacuum gap impedance; the values of *ε _{r}* = (1 − 2

*j*)

*ε*

_{r}_{g}and

*μ*= 1 − 2

_{r}*j*are considered, with

*ε*

_{r}_{g}the relative permittivity of glass, providing a very small reflection coefficient amplitude of 0.0256 (below 0.066% in power reflection.) The surfaces of the waveguide and the matched load were modeled using 47898 Rao-Wilton-Glisson (RWG) [16] basis functions, leading to 95796 unknowns for the equivalent electric and magnetic currents. Once these currents are calculated, the electric and magnetic fields can be straightforwardly obtained from them. Using these fields we calculate the waveguide characteristic impedance defined as

*Z*

_{0}=

*V/I*, where

*V*is the voltage, obtained as the line integral of the electric field from the center of one arm to the other, and where

*I*is the displacement current, obtained according to Ampère’s law as the circulation of the magnetic field along a closed-loop enclosing one arm of the waveguide. The characteristic impedance obtained in this way is

*Z*

_{0}= 199.13 − 1.28

*j*Ω. We also calculate the propagation constant

*γ*=

*α*+

*jβ*, consisting of the attenuation constant

*α*and the wave vector

*β*. Figure 2(a) shows a line cut of the amplitude of the electric field along the center of the gap for the waveguide terminated by the matched load. The field profile obtained for the waveguide terminated by an open end is also shown for comparison. In the latter case, a standing wave pattern is observed close to the open end due to the wave reflection. Exploiting this pattern we determine the effective wavelength of the guided plasmon mode and the respective wave vector, leading to

*β*= 56.61 rad/

*μ*m. Owing to the virtual absence of power reflection, the standing wave pattern is almost negligible in the case of the matched terminated waveguide. From the exponential decay with distance for this profile we determine the attenuation constant as

*α*= 0.757 Np/

*μ*m, a strong attenuation which drastically limits the interconnect distance when using the MIM waveguide.

Next, appropriate directive Yagi-Uda antennas are designed. The optimization of a plasmonic Yagi-Uda antenna is a challenging task, since performance strongly depends on the lengths of the elements, the inter-element distances and the nearfield mutual couplings, which must be accurately handled [17]. The optimization process was performed using the standard genetic algorithm (GA) explained in [18]. The number of antenna elements was set to four, consisting of a feed element and three parasitic directors. The lengths of the elements and the inter-element distances were simultaneously optimized to maximize the antenna directivity *D*_{0}[19]. The range of possible values was set from 0.1 to 0.5 *λ*_{g} both for lengths and distances. The SIE-MoM analysis technique was applied for the accurate evaluation of *D*_{0} (which constitutes the GA fitness function) for each of the individuals in each generation of the GA. The final design achieved after 82 generations (starting with a random population of 64 individuals and considering mutation and crossover probabilities of 0.01 and 0.6 respectively) is shown in Fig. 1(a) and Fig. 1(b) for the transmitting and receiving sides. Through reciprocity, the same design is used for the transmit and the receive antenna. The length of the feed element is 96.4 nm, and the lengths of the successive directors are 51.2 nm, 50.9 nm, and 51.9 nm (0.215 *λ*_{g}, 0.114 *λ*_{g}, 0.114 *λ*_{g}, and 0.116 *λ*_{g} respectively). The distance between the feed element and the first director is 64.3 nm and the successive distances between directors are 81.4 nm and 121.4 nm (0.144 *λ*_{g}, 0.182 *λ*_{g}, and 0.271 *λ*_{g} respectively.) The attained directivity with this design is *D*_{0} = 3.471 n.u. or 5.4 dBi (dB with respect to an isotropic theoretical antenna).

So far, the designed Yagi-Uda nanoantennas were excited either by nearfield coupling with a quantum dot in [11, 13] or by lasing radiation [12]. In the framework of nano-optical ICs, however, Yagi-Uda antennas must be coupled to MIM waveguides. In this case, looking for a realistic design we opted for a coplanar configuration, as shown in Fig. 1. This prevents the inclusion of the usual reflector element of the Yagi-Uda design, which indeed means some sacrifice of maximum attainable directivity, but in return simplifies fabrication. Otherwise, much like in RF or microwave regimes, in order to maximize the power transfer between Tx and Rx, we must match the nanoantenna input impedance to the MIM waveguide impedance *Z*_{0}, which can be achieved by means of a properly designed matching network. For this, we first calculate the reflection coefficient for fields at the antenna connection point, Γ = |Γ|*e ^{jϕ}*. Figure 2(b) shows the standing wave pattern of the electric field on a linear path along the gap of the transmitting waveguide. The reflection coefficient can be determined by curve fitting of this standing wave pattern. We only consider the fundamental mode for the fitting procedure, since other higher order modes are negligible in comparison with the fundamental one. The field amplitude for this mode at position

*x*in the waveguide can be described as

*E*=

*E*

_{0}

*e*

^{−γx}(1 + Γ

*e*

^{2γx}), with the reflection plane (antenna connection point) located at

*x*= 0.

*E*

_{0}is the field amplitude of the wave in the forward direction (direction from the source to the antenna) and

*γ*is the propagation constant. The reflection coefficient so determined for the nanoantenna alone, without matching network, is Γ = 0.6362

*e*

^{j1.4343}, meaning that 40.5% of the power available on the waveguide is reflected and only 59.5% is accepted by the antenna. Similarly as done in [6], the power transmission is improved by including a dielectric nanoparticle filling the gap of the feed element (see Fig. 1). This nanoparticle acts as a lumped element constituting the actual matching network nanocircuit. Changing the value of the relative dielectric permittivity

*ε*of this nanoparticle the impedance matching can be achieved, obtaining the point of minimum reflection (maximum transmission) for

_{r}*ε*= 3.1, with Γ = 0.1919

_{r}*e*

^{−j0.3473}. This means a reflection below 3.7 % of the available power and 96.3 % of power accepted by the nanoantenna. The input impedance of the nanoantenna with the matching network can be obtained from the reflection coefficient and the waveguide characteristic impedance as

*Z*=

_{in}*Z*

_{0}(1 + Γ)/(1 − Γ), leading to

*Z*= 283.5 − 40.306

_{in}*j*Ω. The standing wave pattern and the best fit for this case are also collected in Fig. 2(b). Now, by computing a line integral over the electric field from one arm to the other at each point along the waveguide, we obtain a standing wave pattern for the voltage (not shown since it is analogous to the electric field pattern of Fig. 2(b)). Applying the fitting procedure described above to this voltage pattern we determine the amplitude of the voltage wave flowing in the forward direction. Taking the value of this voltage at the antenna connection point,

*V*, and the characteristic impedance,

*Z*

_{0}, we can calculate the power available at the antenna feeding point as ${P}_{tx}=\frac{1}{2}{\left|V\right|}^{2}/\text{Re}\left({Z}_{0}\right)$, so the power accepted by the nanoantenna (or input power) can be obtained as

*P*= (1 − |Γ|

_{in}^{2})

*P*. On the other hand, the power being effectively radiated by the nanoantenna,

_{tx}*P*, can be calculated by computing the flux of the Poynting vector across a closed surface containing both the antenna and the waveguide [19]. The efficiency of the transmit antenna can then be obtained as the ratio

_{rad}*η*=

*P*/

_{rad}*P*, leading to

_{in}*η*= 0.412, which, by reciprocity, will be also the efficiency of the receiving antenna.

We then simulated a complete wireless link, consisting of the previously designed Tx and Rx nanoantennas, matched to their respective MIM waveguides, looking toward one another and separated by a distance (defined as the distance between the feed elements) of *d* = 17.92 *μ*m (40 *λ*_{g}); the sketch is depicted in the lower inset of Fig. 1. Figures 3(a) and 3(b) illustrate the electric near field amplitude on transverse planes to the transmit and receive nanoantennas respectively. Looking at Fig. 3(a) and the respective line cut in Fig. 2(b), we can observe that the amplitude of the field is almost constant on the gap of the transmitting waveguide, showing a smooth standing wave pattern. This is due to the good impedance match between the antenna and the waveguide. Similarly, we observe in Fig. 3(b) that the amplitude of the field is almost constant in the gap of the receiving waveguide. In this case, this is due to the effect of the matched load termination, absorbing almost all the energy. The power balance (ratio of the received power at the output of the Rx nanoantenna, *P _{rx}*, to the available power at the feeding point of the Tx nanoantenna,

*P*) obtained from this full-wave simulation is

_{tx}*P*/

_{rx}*P*= 6.9948 · 10

_{tx}^{−6}(−51.55 dB).

This power balance could also be obtained from circuit line theory in terms of the above calculated parameters by means of the Friis equation [19], as *P _{rx}*/

*P*= [

_{tx}*λ*

_{g}/(4

*πd*)]

^{2}(1−|Γ

*|*

_{tx}^{2})(1 − |Γ

*|*

_{rx}^{2})

*η*. The first term on the second hand of Friis equation accounts for free space path losses in the external (glass) region with the link distance

_{tx}η_{rx}D_{tx}D_{rx}*d*, second and third account for impedance mismatch between nanoantennas and their respective optical waveguides,

*η*and

_{tx}*η*are the radiation efficiencies of the nanoantennas, and

_{rx}*D*and

_{tx}*D*are the directivities of the two nanoantennas facing each other. We are assuming that the transmit and receive nanoantennas are perfectly aligned, so there are no polarization losses. The result provided by the Friis formula with Γ

_{rx}*= Γ*

_{tx}*= Γ,*

_{rx}*η*=

_{tx}*η*=

_{rx}*η*, and

*D*=

_{tx}*D*=

_{rx}*D*

_{0}, where Γ,

*η*and

*D*

_{0}have been calculated previously, is

*P*/

_{rx}*P*= 7.5024 · 10

_{tx}^{−6}(−51.25 dB), which is in perfect agreement with the full wave simulation.

To illustrate the benefit of the directive wireless connects in the context of optical communications, Fig. 4 shows a comparison of the power balance versus distance *d* for three different connects: (i) MIM plasmonic waveguide, (ii) broadcast wireless connect using matched dipole nanoantennas, and (iii) the proposed wireless connect using matched directive Yagi-Uda nanoantennas. The curves have been obtained using the Friis equation for the wireless links, and using the exponential decay with the attenuation constant *α* for the plasmonic waveguide. Full wave simulations of the complete Tx/Rx directive link have also been carried out for a number of distances (between 1.5 *μ*m and 32.5 *μ*m) and represented with marks. Looking at Fig. 4, it can be seen that for distances larger than about 6 to 8 *μ*m, the wireless connects clearly outperform the plasmonic waveguide connect. The reason is that the power density decays as 1/*d*^{2} for wireless links, while it drops exponentially as exp(−2*αd*) for the waveguide connect. This slope change means that the wireless link will always outperform the waveguide connect after a certain distance. In the particular case of plasmonic waveguides, this distance is moderately short due to high metal absorption. Otherwise, the directive link always provides a better power balance than the broadcast link. The received power is almost 7 dB higher due to the higher directivity of the Yagi-Uda nanoantennas. An additional advantage of this higher directivity is a better field confinement compared with the broadcast link, reducing interference and increasing the density of connections with minimal footprint, which indeed is a key point for miniaturization.

Finally, some studies about the experimental realization of the proposed directive wireless link have been addressed. Although fabrication techniques in the field of nanoantennas are progressing very rapidly, nowadays it is still difficult to obtain resolution within a few nanometers. Current conventional nanofabrication techniques such as electron beam (e-beam) lithography, focused ion beam milling or nanoimprint lithography, can achieve sub-10 nm resolutions (albeit feature sizes beyond this limit can be obtained by indirect fabrication procedures [20]). Other less extended (and more expensive) techniques, such as scanning probe lithography, can pattern features on a surface with atomic resolution (< 1 nm) [21], although with very low throughput. To give some insight on how the directivity of the antenna, and hence the power balance of the whole nanolink, are affected by the fabrication tolerances, a Monte Carlo simulation was carried out. The directivities of 100 Yagi-Uda antennas whose element lengths were randomly varied up to ±1 nm (with 0.5 nm steps) have been calculated. It has been concluded that a maximum degradation of about 0.5 dB was obtained for the directivity, which implies a reduction of about 1 dB in the power balance for the worst case.

All the designs in this paper have been carried out using a very efficient and accurate frequency-domain surface integral equation-method of moments software [22–24]. SIE formulations solved by MoM have demonstrated to be very accurate, robust and versatile for the analysis of conductors and dielectrics in RF and microwave domains [25–27]. Although not yet widespread in optics, they bring important advantages for the rigorous analysis of penetrable plasmonic bodies compared with volumetric approaches [22–24, 28]. Otherwise, the latest advances in fast integral equation solvers [29–33], together with the vast computational capabilities of modern high performance computing (HPC) computers, have allowed us to use this precise SIE-MoM software as the basic electromagnetic analysis tool underlying the hard optimizations used for our design purposes. For the optimization procedures, we developed a C++ routine based on the standard GA explained in [18] with non-overlapping populations. This routine was integrated with the SIE-MoM software and parallelized with message passing interface (MPI) to take advantage of massively parallel computers.

## 3. Conclusions

In conclusion, in this paper we have presented the complete design of a high-capability wireless link between two points at nano scale (including both the antennas and the matching networks between them and the corresponding transmitter and receiver). This idea had been already presented in [6], but in this case we introduce the use of specifically designed directive nanoantennas and matching elements. Although the radiated fields between the nanoantennas are not confined, the use of such a wireless link has demonstrated to be an efficient alternative, improving the capability of conventional plasmonic waveguides which suffer from metal absorption. The use of directive nanoantennas greatly enhances the capability of the link, improving its behavior with respect to conventional nanoantennas and largely outperforming regular plasmonic waveguide connects.

## Acknowledgments

This work was supported by the Spanish Government and European Regional Development Fund (ERDF), under projects TEC2011-28784-C02-01, TEC2011-28784-C02-02, CONSOLIDER-INGENIO 2010 CSD2008-00068, and by ERDF and the Galician Regional Government under project CN 2012/260. The authors thank CénitS and CESGA Spanish supercomputing centers for their support to run the simulations. The authors also thank Kathryn Williams at Northeastern University (Boston, USA) for carefully reviewing the English in this paper.

## References and links

**1. **L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express **13**, 6645–6650 (2005). [CrossRef]

**2. **G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. **30**, 3359–3361 (2005). [CrossRef]

**3. **G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. **87**, 131102 (2005). [CrossRef]

**4. **J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B **73**, 035407 (2006). [CrossRef]

**5. **G. Veronis, Z. Yu, S. E. Kocabas, D. A. B. Miller, M. L. Brongersma, and S. Fan, “Metal-dielectric-metal plasmonic waveguide devices for manipulating light at the nanoscale,” Chin. Opt. Lett. **7**, 302–308 (2009). [CrossRef]

**6. **A. Alù and N. Engheta, “Wireless at the nanoscale: optical interconnects using matched nanoantennas,” Phys. Rev. Lett. **104**, 213902 (2010). [CrossRef]

**7. **S. A. Maier, *Plasmonics: Fundamentals and Applications* (Springer, New York, 2007).

**8. **D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single bowtie nanoantennas resonant in the visible,” Nano Lett. **4**, 957–961 (2004). [CrossRef]

**9. **P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science **308**, 1607–1608 (2005). [CrossRef]

**10. **L. Novotny and N. F. van Hulst, “Antennas for light,” Nat. Photon. **5**, 83–90 (2011). [CrossRef]

**11. **H. F. Hofmann, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical yagi-uda antenna,” New J. Phys. **9**, 207 (2007). [CrossRef]

**12. **T. Kosako, Y. Kadoya, and H. F. Hofmann, “Directional control of light by a nano-optical yagi-uda antenna,” Nat. Photon. **4**, 312–315 (2010). [CrossRef]

**13. **A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science **329**, 930–933 (2010). [CrossRef]

**14. **M. Klemm, “Directional plasmonic nanoantennas for wireless links at the nanoscale,” in *Proceedings of Antennas and Propagation Conference*, (Loughborough, 2011).

**15. **J.-S. Huang, T. Feichtner, P. Biagioni, and B. Hecht, “Impedance matching and emission properties of nanoantennas in an optical nanocircuit,” Nano Lett. **9**, 1897–1902 (2009). [CrossRef]

**16. **S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. **30**, 409–418 (1982). [CrossRef]

**17. **M. G. Araújo, D. M. Solís, J. Rivero, J. M. Taboada, F. Obelleiro, and L. Landesa, “Design of optical nanoantennas with the surface integral equation method of moments,” in *Proceedings of the International Conference on Electromagnetics in Advanced Applications*, (Cape Town, 2012).

**18. **D. Goldberg, *Genetic Algorithms in Search, Optimization and Machine Learning* (Addison-Wesley, Reading, MA, 1989).

**19. **C. A. Balanis, *Antenna Theory: Analysis and Design* (Wiley & Sons, New York, 1982).

**20. **Z. Cui, *Nanofabrication: Principles, Capabilities and Limits* (Springer, New York, 2008).

**21. **B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Willson, and G. M. Whitesides, “New approaches to nanofabrication: Molding, printing, and other techniques,” Chem. Rev. **105**, 1171–1196 (2005). [CrossRef]

**22. **J. M. Taboada, J. Rivero, F. Obelleiro, M. G. Araújo, and L. Landesa, “Method of moments formulation for the analysis of plasmonic nano-optical antennas,” J. Opt. Soc. Am. A **28**, 1341–1348 (2011). [CrossRef]

**23. **M. G. Araújo, J. M. Taboada, D. M. Solís, J. Rivero, L. Landesa, and F. Obelleiro, “Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers,” Opt. Express **20**, 9161–9171 (2012). [CrossRef]

**24. **L. Landesa, M. G. Araújo, J. M. Taboada, L. Bote, and F. Obelleiro, “Improving condition number and convergence of the surface integral-equation method of moments for penetrable bodies,” Opt. Express **20**, 17237–17249 (2012). [CrossRef]

**25. **S. M. Rao and D. R. Wilton, “E-field, h-field, and combined field solution for arbitrarily shaped three-dimensional dielectric bodies,” Electromagnetics **10**, 407–421 (1990). [CrossRef]

**26. **P. Yla-Oijala, M. Taskinen, and S. Jarvenpaa, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. **40**, RS6002 (2005). [CrossRef]

**27. **P. Yla-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. **53**, 1168–1173 (2005). [CrossRef]

**28. **A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3d simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A **26**, 732–740 (2009). [CrossRef]

**29. **J. Song, C.-C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag. **45**, 1488–1493 (1997). [CrossRef]

**30. **O. Ergul and L. Gurel, “A hierarchical partitioning strategy for an efficient parallelization of the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. **57**, 1740–1750 (2009). [CrossRef]

**31. **J. Taboada, M. Araújo, J. Bértolo, L. Landesa, F. Obelleiro, and J. Rodríguez, “MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics,” Prog. Electromagn. Res. **105**, 15–20 (2010). [CrossRef]

**32. **J. Taboada, M. Araújo, F. Obelleiro, J. Rodríguez, and L. Landesa, “MLFMA-FFT parallel algorithm for the solution of extremely large problems in electromagnetics,” Proceedings of the IEEE **PP**(99), 1–14 (2013).

**33. **M. G. Araújo, J. M. Taboada, J. Rivero, D. M. Solís, and F. Obelleiro, “Solution of large-scale plasmonic problems with the multilevel fast multipole algorithm,” Opt. Lett. **37**, 416–418 (2012). [CrossRef]