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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 28, Iss. 7 — Jul. 1, 2011
  • pp: 1341–1348
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Method-of-moments formulation for the analysis of plasmonic nano-optical antennas

José M. Taboada, Javier Rivero, Fernando Obelleiro, Marta G. Araújo, and Luis Landesa  »View Author Affiliations


JOSA A, Vol. 28, Issue 7, pp. 1341-1348 (2011)
http://dx.doi.org/10.1364/JOSAA.28.001341


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Abstract

We present a surface integral equation (SIE) to model the electromagnetic behavior of metallic objects at optical frequencies. The electric and magnetic current combined field integral equation considering both tangential and normal equations is applied. The SIE is solved by using a method-of-moments (MoM) formulation. The SIE-MoM approach is applied only on the material boundary surfaces and interfaces, avoiding the cumbersome volumetric discretization of the objects and the surrounding space required in differential-equation formulations. Some canonical examples have been analyzed, and the results have been compared with analytical reference solutions in order to prove the accuracy of the proposed method. Finally, two plasmonic Yagi–Uda nanoantennas have been analyzed, illustrating the applicability of the method to the solution of real plasmonic problems.

© 2011 Optical Society of America

1. INTRODUCTION

In recent years, with the advent of nanoscience and nanotechnology, the optical plasmonic properties of metallic nanoparticles have generated increasing research interest due to their ability to control light surpassing the diffraction limit. A review of the basic principles of nanoplasmonics in the context of several photonic applications can be found in [1

1. S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005). [CrossRef]

, 2

2. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010). [CrossRef] [PubMed]

]. On the other hand, the optical response of metallic nanoparticles is well described by classical electrodynamics [3

3. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

]. It is therefore possible to apply ideas from electrical engineering in the radio frequency (RF) and microwave regimes directly to the design of nano-optical devices, thus enabling miniaturization in terms of wavelength for their electronic counterparts [4

4. J.-J. Greffet, “Nanoantennas for light emission,” Science 308, 1561–1563 (2005). [CrossRef] [PubMed]

, 5

5. M. L. Brongersma, “Plasmonics: engineering optical nanoantennas,” Nat. Photon. 2, 270–273 (2008). [CrossRef]

, 6

6. P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1, 438–483 (2009). [CrossRef]

]. Nevertheless, the optical response of metals is quite different from the metallic conductivity observed at low frequencies. At RF, metals are usually considered as perfect electrical conductors supporting only surface electric currents. At optical frequencies, however, the penetration of fields can no longer be neglected. Here it is necessary to take into account the precise plasmonic electromagnetic response of metals as given by their complex dielectric constant. Concretely, the optical response of metals is characterized by a negative permittivity real part that originates from a complex conductivity, which in turn is given by the retarded coherent collective electron oscillations (plasmons) distinctive of strongly coupled plasmas. For small particles, this results in a size-independent localized surface plasmon resonance (LSPR) determined by the shape and the dielectric constant of the particle. The ability of metal nanoparticles to support such LSPRs is the responsibility of the nanoantenna properties at optical frequencies.

One well-researched consequence of the above plasmon resonances is that they strongly couple to the optical radiation field, enhancing and directing the spontaneous emission of light by single molecules or other pointlike light sources in much the same way that properly designed RF antennas can enhance and direct radiofrequency emissions from electrical circuits [7

7. T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “λ/4 Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33 (2007). [CrossRef] [PubMed]

, 8

8. P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15, 14266–14274 (2007). [CrossRef] [PubMed]

, 9

9. F. Tam, G. P. Goodrich, B. R. Johnson, and N. J. Halas, “Plasmonic enhancement of molecular fluorescence,” Nano Lett. 7, 496–501 (2007). [CrossRef] [PubMed]

]. Recently, this idea was combined with the principles of antenna design known from radio and microwave technology, resulting in the experimental realizations of nano-optical equivalents of half-wave, bow-tie, and other simple antennas [10

10. K. Sendur and W. Challener, “Near-field radiation of bow-tie antennas and apertures at optical frequencies,” J. Microsc. 210, 279–283 (2003). [CrossRef] [PubMed]

, 11

11. 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]

, 12

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

, 13

13. J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. García de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71, 235420 (2005). [CrossRef]

, 14

14. A. Sundaramurthy, P. J. Schuck, N. R. Conley, D. P. Fromm, G. S. Kino, and W. E. Moerner, “Toward nanometer-scale optical photolithography: utilizing the near-field of bowtie optical nanoantennas,” Nano Lett. 6, 355–360 (2006). [CrossRef] [PubMed]

, 15

15. O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Express 15, 17736–17746 (2007). [CrossRef] [PubMed]

] (a further review and additional references can be found in [6

6. P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1, 438–483 (2009). [CrossRef]

]). The advances in the fabrication of nano-optical antennas were also applied to achieve the highly directional emission of radio antennas in the optical regime [16

16. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Single emitters coupled to plasmonic nano-antennas: angular emission and collection efficiency,” New J. Phys. 10, 105005 (2008). [CrossRef]

]. An example is the well-known Yagi–Uda antenna, which is a typical design for high directivity in the RF regime. By fabricating a corresponding array of nanoparticles, similar radiation patterns are being obtained in the optical regime [17

17. J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: a Yagi–Uda nanoantenna in the optical domain,” Phys. Rev. B 76, 245403 (2007). [CrossRef]

, 18

18. H. F. Hofmann, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi–Uda antenna,” New J. Phys. 9, 217 (2007). [CrossRef]

, 19

19. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi–Uda antenna,” Opt. Express 16, 10858–10866 (2008). [CrossRef] [PubMed]

, 20

20. 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]

, 21

21. 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] [PubMed]

].

Despite the great advances achieved to date, and even though several fabricated nanoantennas demonstrated their applicability and feasibility [20

20. 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]

, 21

21. 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] [PubMed]

], the design of plasmonic nanosystems is still a challenging task. The plasmonic optical properties of metals make it impossible to directly downscale the microwave traditional antenna designs to the optical regime [22

22. J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic nanowire antennas: experiment, simulation, and theory,” Nano Lett. 10, 3596–3603 (2010). [CrossRef] [PubMed]

]. There is therefore an increasing demand for rigorous and efficient tools to aid the accurate engineering of the optical response of arbitrarily shaped plasmonic antenna systems. Nowadays, the design and optimization of antennas at RF or microwave regime are largely done with the aid of rigorous numeric Maxwell’s solvers. Nevertheless, at optical frequencies, it is necessary to carefully consider the precise electromagnetic response of the penetrable nanoparticles due to their high-contrast plasmonic dielectric constants, with strong LSPRs leading to extremely localized field variations. To date, the most common way to rigorously accomplish the resolution of such problems has been the use of differential-equation formulations, such as the finite difference time domain (FDTD) [23

23. A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time- dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975). [CrossRef]

], the finite integration technique (FIT) [24

24. T. Weiland, “A discretization method for the solution of Maxwell’s equations for six-component fields,” AEU Arch. Elektron. Übertragungstech. 31, 116–120 (1977).

] or the finite element method (FEM) [25

25. P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, 2003). [CrossRef]

]. The main advantage of the differential-equation techniques is that they can be easily implemented from differential Maxwell’s equations. However, it is well known that they are volumetric formulations strongly burdened with the discretization of the object and the surrounding space, and that they need to impose artificial termination boundary conditions. As a consequence, the precise analysis of extended three-dimensional (3D) plasmonic systems often exceeds current computational limits. Besides, in some practical cases involving sharp wedges or very small gaps and dims, the rapid field variations due to the strong plasmon resonances are also difficult to handle for these field-based techniques. Thereby, numerical dispersion or instability may occur.

Recently, the authors have successfully applied a JMCFIE-MoM formulation to the solution of homogenized left-handed metamaterials [39

39. J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18, 15876–15886 (2010). [CrossRef] [PubMed]

]. In the present work, the formulation of [39

39. J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18, 15876–15886 (2010). [CrossRef] [PubMed]

] is generalized for the simulation of arbitrary configurations of composite plasmonic nanoparticles. The implemented JMCFIE-MoM approach yields a very efficient and highly accurate representation of the electromagnetic problem, taking into account all the physical constraints as determined by the shape and the complex dielectric constants of the particles. It is worth pointing out that with this approach, the rapidly changing plasmonic near fields or the presence of hot spots does not compromise at all the accuracy of the solution or impose new restrictions on the surface meshing. This enables the accurate and straightforward analysis of extremely shaped geometries involving sharp wedges or arbitrarily small gaps and dims. On the other hand, the application of the MoM also brings the possibility of applying the latest breakthrough developments in fast integral equation methods, such as those presented in [40

40. J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, J. M. Bertolo, M. G. Araujo, J. C. Mouriño, and A. Gomez, “High scalability FMM-FFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propag. Mag. 51 (6), 20–28 (2009). [CrossRef]

, 41

41. M. G. Araújo, J. M. Taboada, F. Obelleiro, J. M. Bértolo, L. Landesa, J. Rivero, and J. L. Rodríguez, “Supercomputer aware approach for the solution of challenging electromagnetic problems,” PIER 101, 241–256 (2010). [CrossRef]

, 42

42. J. M. Taboada, M. G. Araújo, J. M. Bértolo, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics,” PIER 105, 15–30 (2010), invited paper. [CrossRef]

], to the solution of large-scale problems in plasmonics.

The paper is structured as follows: in Section 2 the JMCFIE-MoM formulation for multiple homogeneous penetrable bodies is detailed. In Section 3, numerical examples are presented to demonstrate the accuracy and the applicability of the method. First, a plasmonic gold nanosphere is analyzed and compared to the analytical reference provided by the Mie series. Second, the radiation pattern of a Yagi–Uda antenna made of plasmonic spherical particles that has been subject of interesting analyses in recent works is obtained. Finally, a plasmonic Yagi–Uda antenna is analyzed both for directional light emission and for field enhancement. The conclusions of the work are summarized in Section 4.

2. GENERAL SURFACE INTEGRAL EQUATION FORMULATION FOR THE ANALYSIS OF PLASMONIC MATERIALS

Let us consider several homogeneous penetrable objects in a homogeneous (unbounded) medium. Each homogeneous region is denoted by Ri, with i=1M the number of regions. The material properties of each region Ri corresponding to the penetrable objects or the background is defined by the complex permittivity εi=εriε0 and the complex permeability μi=μriμ0. εri and μri are the complex valued relative permittivity and permeability constants of region Ri, respectively, and ε0, μ0 are the constitutive parameters of vacuum. A time harmonic variation exp(jωt) is assumed and suppressed from the formulation. Using a similar notation as in [35

35. K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003). [CrossRef]

], let us denote with Sij (or Sji) the interface surface between regions Ri and Rj. n^ij denotes the unit vector normal to Sij and pointing toward Ri, hence n^ij=n^ji.

In region Ri, the electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) can be formulated in two alternative ways, depending on the method applied to project the fields onto the surfaces surrounding this region. The tangential (T) equations in Ri are given by [33

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

]
T-EFIEi:j(ηiLi(Jij)Ki(Mij))tan+12n^ij×Mij=(Eiinc)tan,
(1)
T-MFIEi:j(Ki(Jij)+1ηiLi(Mij))tan12n^ij×Jij=(Hiinc)tan,
(2)
respectively. In a similar way, the normal (N) equations in Ri can be written as
N-EFIEi:n^ij×j(ηiLi(Jij)Ki(Mij))12Mij=n^ij×Eiinc,
(3)
N-MFIEi:n^ij×j(Ki(Jij)+1ηiLi(Mij))+12Jij=n^ij×Hiinc.
(4)
Here ηi is the intrinsic impedance in medium Ri, and the following integro-differential operators have been defined:
Li(Xij)=jki[SXij(r)Gi(r,r)dS+1ki2S·XijGi(r,r)dS],
(5)
Ki(Xij)=S,PVXij(r)×Gi(r,r)dS,
(6)
with r the observation points approaching to Sij from the inner of region Ri and with rSij the source points. · denotes the divergence in the primed (source) coordinates, PV denotes the principal value of the integral in Eq. (6), ki is the wavenumber in Ri, and we define
Gi(r,r)=ejki|rr|4π|rr|,
(7)
which is the homogeneous Green’s function in Ri. Jij(r) and Mij(r) are the induced electric and magnetic currents on the surface Sij. These currents are given by
Jij(r)=n^ij×Hi(r),
(8)
Mij(r)=n^ij×Ei(r),
(9)
where Ei(r) and Hi(r) are the total electric and magnetic fields in Ri. Eiinc(r) and Hiinc(r) are the incident fields due to the sources located inside region Ri. Finally, the summations in Eqs. (1, 2, 3, 4) account for all the interfaces Sij surrounding Ri (including Sij).

In the same way, we combine the MCFIE equations of regions Ri and Rj to obtain another integral equation on Sij:
MCFIEij:cin^ij×j(ηiLi(Jij)Ki(Mij))cjn^ji×j(ηjLj(Jjj)Kj(Mjj))+dij(ηiKi(Jij)+Li(Mij))tan+djj(ηjKj(Jjj)+Lj(Mjj))tan+12(ciMij+cjMjidiηin^ij×Jijdjηjn^ji×Jji)=cin^ij×Eiinc+cjn^ji×Ejincdi(ηiHiinc)tandj(ηjHjinc)tan,rSij.
(13)

Each block of the impedance matrix in Eq. (29) can be written as
Z¯ij,ij=[Z¯ij,ij1JZ¯ij,ij1MZ¯ij,ij2JZ¯ij,ij2M],
(30)
with
Z¯ij,ij1J[m,n]=aiAmni+ajAmnj+biBmnibjBmnj+12(bi+bj)Imn,
(31)
Z¯ij,ij1M[m,n]=aiηiBmniajηjBmnj+biηiAmnibjηjAmnj+12(aiηiajηj)Imn,
(32)
Z¯ij,ij2J[m,n]=diηiBmni+djηjBmnjciηiAmni+cjηjAmnj12(diηidjηj)Imn,
(33)
Z¯ij,ij2M[m,n]=diAmni+djAmnj+ciBmnicjBmnj+ci+cj2Imn,
(34)
for the self-coupling of surface Sij (that is, case i=i, j=j). For the mutual coupling between different surfaces or interfaces, off-diagonal blocks in Eq. (29), we obtain
Z¯ij,ij1J[m,n]=sijaiAmni+sijbiBmni,
(35)
Z¯ij,ij1M[m,n]=sijaiηiBmni+sijbiηiAmni,
(36)
Z¯ij,ij2J[m,n]=sijdiηiBmnisijciηiAmni,
(37)
Z¯ij,ij2M[m,n]=sijdiAmni+sijciBmni,
(38)
if both surfaces share region Ri, and
Z¯ij,jj1J[m,n]=sjjajAmnj+sjjbjBmnj,
(39)
Z¯ij,jj1M[m,n]=sjjajηjBmnj+sjjbjηjAmnj,
(40)
Z¯ij,jj2J[m,n]=sjjdjηjBmnjsijcjηjAmnj,
(41)
Z¯ij,jj2M[m,n]=sjjdjAmnj+sjjcjBmnj,
(42)
if both surfaces are adjacent to region Rj. In other case, the off-diagonal block vanishes. Similarly, we define
Iij=[JijMij],
(43)
with Jij[n]=Jn, nSij and Mij[n]=Mn, nSij, and
Vij=[Vij1Vij2],
(44)
where
Vij1[m]=Emi,j+Hmi,j,
(45)
Vij2[m]=Emi,j+Hmi,j.
(46)

Different known formulations can be obtained depending on the selection of the complex combination parameters al, bl, cl, and dl, with l=i, j [33

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

, 34

34. 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]

, 35

35. K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003). [CrossRef]

, 36

36. P. Ylä-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]

, 37

37. P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general integral equation method for general composite metallic and dielectric structures with junctions,” PIER 52, 81–108 (2005). [CrossRef]

, 38

38. Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009). [CrossRef]

]. For this work, we have chosen al=bl=cl=dl=1, which leads to the JMCFIE combined formulation [36

36. P. Ylä-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]

]. The JMCFIE is free of internal resonances, and it simultaneously provides faster convergence and better accuracy than other approaches [38

38. Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009). [CrossRef]

]. Nevertheless, other formulations could be straightforwardly applied simply by choosing different combination parameters. The comparison between different formulations for plasmonic applications is left for a future contribution.

For the discretization of the problem, we have employed the extensively used Rao–Wilton–Glisson (RWG) basis functions [43

43. 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]

]. Accurate calculation of the integrals involved in the operators of Eqs. (5, 6) for the evaluation of Eqs. (18, 19, 20, 21) using the RWG functions is crucial for the precision of the method. We have applied Gaussian quadrature rules for the numerical integration of smooth varying integrands, together with the analytical extraction procedures of [44

44. D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984). [CrossRef]

, 45

45. R. E. Hodges and Y. Rahmat-Samii, “The evaluation of MFIE integrals with the use of vector triangle basis functions,” Microw. Opt. Technol. Lett. 14, 9–14 (1997). [CrossRef]

, 46

46. R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag. 41, 1448–1455 (1993). [CrossRef]

, 47

47. P. Ylä-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n^×RWG functions,” IEEE Trans. Antennas Propag. 51, 1837–1846 (2003). [CrossRef]

] for the accurate evaluation of singular integrals. On the other hand, for the application of the above-described formulation to the analysis of plasmonic nanoparticles, special care must be taken with the definition of the electromagnetic parameters and properties of the material. Concretely, if region Ri defines a penetrable plasmonic particle, the following definitions should be applied for the wavenumber and the intrinsic impedance:
ki=ωμiεi;ηi=μiεi.
(47)

3. SIMULATION RESULTS

In this section we present numerical examples to demonstrate the validity, accuracy and versatility of the JMCFIE-MoM formulation for the efficient analysis and design of plasmonic nanosystems. First, we examine the accuracy of the method by comparing our simulation results for a plasmonic gold nanosphere with the analytical solution provided by the Mie series. This verification is important in order to guarantee that both the formulation and the implementation are consistent with the constitutive parameters of metals at optical frequencies, as well as to test the ability of the method to accurately handle the surface plasmon resonances. We have considered the gold nanosphere analyzed in [10

10. K. Sendur and W. Challener, “Near-field radiation of bow-tie antennas and apertures at optical frequencies,” J. Microsc. 210, 279–283 (2003). [CrossRef] [PubMed]

], with a radius of 200nm. An incident plane wave at a wavelength of 550nm is impinging onto the sphere. At this wavelength, the relative dielectric constant of gold is εr=8.01.66j [48

48. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

] [note the negative imaginary part due to the exp(jωt) convention]. The problem was solved using 2430 unknowns for the equivalent electric and magnetic currents on the surface. Once these currents have been solved, all the other parameters of interest, such as the near or far fields, can be straightforwardly obtained with almost negligible cost. In this case, the bistatic radar cross section (RCS) [49

49. C. A. Balanis, Antenna Theory: Analysis and Design (Wiley & Sons, 1982).

] patterns of the θθ and ϕϕ polarizations have been computed. The results are plotted in Fig. 1, compared with the Mie series solutions. A close agreement can be observed between both results, which confirms the validity and accuracy of the method.

Next, analysis of a Yagi–Uda antenna composed of several spherical particles is undertaken. Arrays of nanospheres constitute very interesting plasmonic problems that have been tackled in recent research works [50

50. A. Koenderink, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Lett. 9, 4228–4233 (2009). [CrossRef] [PubMed]

, 51

51. B. Stout, A. Devilez, B. Rolly, and N. Bonod, “Multipole methods for nano-antennas design: applications to Yagi–Uda configurations,” J. Opt. Soc. Am. B 28, 1213–1223 (2011). [CrossRef]

]. These examples can further support the reliability and consistency of the method proposed here. As in the analysis corresponding to Fig. 1 of [50

50. A. Koenderink, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Lett. 9, 4228–4233 (2009). [CrossRef] [PubMed]

], we have considered a Yagi–Uda antenna consisting of one reflector sphere with radius of 60nm and an array of eight director spheres with radius of 55nm and spaced 150nm. There is a distance of 100nm between the reflector sphere and the first director sphere. The antenna is made of silver (εr=20.090.45j [48

48. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

] at an operating wavelength of 650nm), and it is embedded in glass (εr=2.25). The array arrangement and the surface mesh details are shown in Fig. 2. See Fig. 1 of [50

50. A. Koenderink, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Lett. 9, 4228–4233 (2009). [CrossRef] [PubMed]

] for further details of the antenna geometry. In Fig. 3 we show the calculated Yagi–Uda antenna directivity [49

49. C. A. Balanis, Antenna Theory: Analysis and Design (Wiley & Sons, 1982).

] in the H and E planes (which correspond to the xy and xz planes, respectively) for a near-field coupled Hertzian dipole emitter. The infinitesimal emitter is placed between the reflector and the feed spheres, at a distance of 10nm from the reflector, and it is oriented in the z^ direction. The antenna was analyzed using a total of 4860 basis functions to represent the equivalent electric and magnetic currents on the surfaces of the nanospheres. Looking at Fig. 3, a perfect agreement is observed with the results of [50

50. A. Koenderink, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Lett. 9, 4228–4233 (2009). [CrossRef] [PubMed]

].

In order to show the applicability of the method for general 3D plasmonic nanostructures, the Yagi–Uda optical antenna designed in [19

19. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi–Uda antenna,” Opt. Express 16, 10858–10866 (2008). [CrossRef] [PubMed]

] has been analyzed with the JMCFIE-MoM formulation. The antenna consists of five cylindrical elements made of aluminum, with radius 20nm and terminated with hemispherical ends. The elements are one reflector, one feed dipole, and three directors. They are placed along the x^ axis and oriented in the z^ direction, as depicted in Fig. 4. The antenna was optimized for an operating wavelength of λ=570nm (in which the relative permittivity constant of aluminum is εr=38.010.9j). See Fig. 1 of [19

19. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi–Uda antenna,” Opt. Express 16, 10858–10866 (2008). [CrossRef] [PubMed]

] for more details.

The antenna directivity in the H and E planes for a near-field coupled Hertzian dipole emitter is depicted in Fig. 5. The infinitesimal dipole is placed at a distance of 4nm from the lower extreme of the antenna feed element, and it is oriented in the z^ direction. The antenna was analyzed using a total of 9036 basis functions to represent the equivalent electric and magnetic currents on the surfaces of the nanodipoles. Looking at Fig. 5, an excellent agreement with the results of [19

19. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi–Uda antenna,” Opt. Express 16, 10858–10866 (2008). [CrossRef] [PubMed]

] is observed.

Finally, the capabilities of the Yagi–Uda antenna to provide field enhancement have been investigated. A very thin flat gap 4nm wide (about λ/142) was introduced to the feed dipole, as shown in the inset of Fig. 4. An incident plane wave with θ^ polarization was considered impinging onto the directors’ side of the antenna. In Fig. 6, the predicted near electric field is shown for an incidence in the maximum directivity direction, θinc=90° and ϕinc=0°. The magnitude of the electric field is shown in the xz and xy planes, both of them crossing the antenna gap. Looking at this figure, it can be seen how the light is strongly confined inside the gap region, providing a high field enhancement with regard to the incident field in absence of the antenna. On the other hand, the movie (Media 1) of the electric field versus the impinging direction for an elevation angular sweep from θinc=0° to 180° (with ϕinc=0°) clearly shows that the local field enhancement can be further augmented and become directive due to the directive guiding of light given by the Yagi–Uda antenna. As expected, the strongest enhancement is obtained for an incidence in the maximum directivity direction (θinc=90°). This enhancement, in combination with the spatial confinement of light in the feed gap, demonstrates the applicability of the high-directivity optical antenna designs to field enhancement purposes (such as spontaneous emission enhancement of light emitters), as well as for directional emission of light. It is worth mentioning that the precise analytic treatment of all the singularities enables the accurate calculation of fields at every point inside or outside the nanoparticles without imposing any additional constraints or surface mesh refinement. This is observed even for this extreme example, with a very thin gap and sharp wedges around it, which demonstrates the robustness and versatility of the MoM formulation for the resolution of plasmonic problems.

Finally, it must be pointed out that the proposed method is general and valid for any arbitrary configuration of composite plasmonic and dielectric particles. The interest of this ability becomes evident in the consideration of some aspects not addressed in depth in this paper, such as the crucial influence of the dielectric substrate [21

21. 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] [PubMed]

], or the analysis of metallo- dielectric antennas, such as those proposed in [52

52. A. Devilez, N. Bonod, and B. Stout, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano 4, 3390–3396 (2010). [CrossRef] [PubMed]

].

4. CONCLUSION

The JMCFIE SIE has been presented for electromagnetic anal ysis of 3D metallic objects at optical frequencies solved by using the MoM. Because the SIE-MoM approach is stated only on the surfaces of the objects, it avoids the cumbersome discretization of the objects and the surrounding space required in differential-equation formulations such as FDTD, FIT, or FEM. Besides, the SIE-MoM formulation inherently satisfies Sommerfeld radiation at infinity without termination condition requirements.

Some numerical examples have been presented confirming the validity and versatility of the proposed JMCFIE-MoM approach for the solution of practical plasmonic problems. The simulations were carried out using much fewer unknowns than would be required when using differential-equation or volumetric formulations. The method has shown an excellent agreement with reference results obtained from theory and with interesting results of recent research works. This accuracy has been observed even for extreme cases with very thin gaps (resulting in very rapidly changing fields and hot spots), demonstrating the robustness of the method.

Finally, by introducing a small gap at the feed element of a Yagi–Uda nanoantenna, we have observed that the local field enhancement associated with the plasmon resonances on an individual gap nanowire antenna can be further augmented and become directive by considering this element as the feed element of a Yagi–Uda antenna. This demonstrates the applicability of the directive Yagi–Uda design to field enhancement in optical applications.

ACKNOWLEDGMENTS

This work was supported by the Spanish Government (through projects TEC2008-06714-C02-01, TEC2008-06714-C02-02, and CONSOLIDER-INGENIO2010 CSD2008-00068), and by Xunta de Galicia (through project IN-CITE08PXIB322250PR).

Fig. 1 Bistatic RCS of a gold nanosphere with a radius of 200nm at a wavelength of 550nm (εr=8.01.66j).
Fig. 2 Surface mesh composed of 1620 flat triangular facets of the Yagi–Uda antenna embedded in glass (εr=2.25) of [50]. The antenna is made of spherical silver nanoparticles with εr=20.090.45j [48] at an operating wavelength of 650nm. Note that only the metal– dielectric interfaces must be modeled.
Fig. 3 Directivity of the Yagi–Uda antenna of [50] in dBi for a near-field coupled Hertzian dipole emitter: (a) H plane and (b) E plane.
Fig. 4 Surface mesh composed of 3012 flat triangular facets for the analysis of the Yagi–Uda antenna of [19] with the proposed JMCFIE-MoM formulation. The antenna is made of aluminum, and it is optimized for an operating wavelength of 570nm. The relative permittivity constant of aluminum at this wavelength is εr=38.010.9j. Note that only the metal–dielectric interfaces must be modeled. Inset, detail of the 4nm gap introduced to the feed element to analyze the usability of the antenna for field enhancement.
Fig. 5 Directivity of the Yagi–Uda antenna of [19] in dBi for a near-field coupled Hertzian dipole emitter: (a) H plane and (b) E plane.
Fig. 6 We show (Media 1, 1744K) the total near-electric-field distribution in the vertical and horizontal planes crossing the Yagi–Uda antenna of [19] versus the impinging direction, for an elevation angular sweep from θinc=0° to 180° (ϕinc=0°). Media 1 illustrates the gap confinement of light and the directional field enhancement provided by the Yagi–Uda antenna. The electric field distribution is shown both inside and outside the metallic nanoelements. Dimensions are in nanometers.
1.

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005). [CrossRef]

2.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010). [CrossRef] [PubMed]

3.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

4.

J.-J. Greffet, “Nanoantennas for light emission,” Science 308, 1561–1563 (2005). [CrossRef] [PubMed]

5.

M. L. Brongersma, “Plasmonics: engineering optical nanoantennas,” Nat. Photon. 2, 270–273 (2008). [CrossRef]

6.

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1, 438–483 (2009). [CrossRef]

7.

T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “λ/4 Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33 (2007). [CrossRef] [PubMed]

8.

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15, 14266–14274 (2007). [CrossRef] [PubMed]

9.

F. Tam, G. P. Goodrich, B. R. Johnson, and N. J. Halas, “Plasmonic enhancement of molecular fluorescence,” Nano Lett. 7, 496–501 (2007). [CrossRef] [PubMed]

10.

K. Sendur and W. Challener, “Near-field radiation of bow-tie antennas and apertures at optical frequencies,” J. Microsc. 210, 279–283 (2003). [CrossRef] [PubMed]

11.

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]

12.

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

13.

J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. García de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71, 235420 (2005). [CrossRef]

14.

A. Sundaramurthy, P. J. Schuck, N. R. Conley, D. P. Fromm, G. S. Kino, and W. E. Moerner, “Toward nanometer-scale optical photolithography: utilizing the near-field of bowtie optical nanoantennas,” Nano Lett. 6, 355–360 (2006). [CrossRef] [PubMed]

15.

O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Express 15, 17736–17746 (2007). [CrossRef] [PubMed]

16.

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Single emitters coupled to plasmonic nano-antennas: angular emission and collection efficiency,” New J. Phys. 10, 105005 (2008). [CrossRef]

17.

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: a Yagi–Uda nanoantenna in the optical domain,” Phys. Rev. B 76, 245403 (2007). [CrossRef]

18.

H. F. Hofmann, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi–Uda antenna,” New J. Phys. 9, 217 (2007). [CrossRef]

19.

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi–Uda antenna,” Opt. Express 16, 10858–10866 (2008). [CrossRef] [PubMed]

20.

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]

21.

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] [PubMed]

22.

J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic nanowire antennas: experiment, simulation, and theory,” Nano Lett. 10, 3596–3603 (2010). [CrossRef] [PubMed]

23.

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time- dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975). [CrossRef]

24.

T. Weiland, “A discretization method for the solution of Maxwell’s equations for six-component fields,” AEU Arch. Elektron. Übertragungstech. 31, 116–120 (1977).

25.

P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, 2003). [CrossRef]

26.

R. F. Harrington, Field Computation by Moment Methods, IEEE Series on Electromagnetic Wave Theory (IEEE, 1993). [CrossRef]

27.

A. J. Poggio and E. K. Miller, Computer Techniques for Electromagnetics (Pergamon, 1973).

28.

Y. Chang and R. F. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag. 25, 789–795 (1977). [CrossRef]

29.

T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977). [CrossRef]

30.

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]

31.

B. Gallinet, A. M. Kern, and O. J. F. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010). [CrossRef]

32.

M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional dielectric objects,” IEEE Trans. Antennas Propag. 47, 1615–1622 (1999). [CrossRef]

33.

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

34.

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]

35.

K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003). [CrossRef]

36.

P. Ylä-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]

37.

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general integral equation method for general composite metallic and dielectric structures with junctions,” PIER 52, 81–108 (2005). [CrossRef]

38.

Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009). [CrossRef]

39.

J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18, 15876–15886 (2010). [CrossRef] [PubMed]

40.

J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, J. M. Bertolo, M. G. Araujo, J. C. Mouriño, and A. Gomez, “High scalability FMM-FFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propag. Mag. 51 (6), 20–28 (2009). [CrossRef]

41.

M. G. Araújo, J. M. Taboada, F. Obelleiro, J. M. Bértolo, L. Landesa, J. Rivero, and J. L. Rodríguez, “Supercomputer aware approach for the solution of challenging electromagnetic problems,” PIER 101, 241–256 (2010). [CrossRef]

42.

J. M. Taboada, M. G. Araújo, J. M. Bértolo, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics,” PIER 105, 15–30 (2010), invited paper. [CrossRef]

43.

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]

44.

D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984). [CrossRef]

45.

R. E. Hodges and Y. Rahmat-Samii, “The evaluation of MFIE integrals with the use of vector triangle basis functions,” Microw. Opt. Technol. Lett. 14, 9–14 (1997). [CrossRef]

46.

R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag. 41, 1448–1455 (1993). [CrossRef]

47.

P. Ylä-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n^×RWG functions,” IEEE Trans. Antennas Propag. 51, 1837–1846 (2003). [CrossRef]

48.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

49.

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

50.

A. Koenderink, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Lett. 9, 4228–4233 (2009). [CrossRef] [PubMed]

51.

B. Stout, A. Devilez, B. Rolly, and N. Bonod, “Multipole methods for nano-antennas design: applications to Yagi–Uda configurations,” J. Opt. Soc. Am. B 28, 1213–1223 (2011). [CrossRef]

52.

A. Devilez, N. Bonod, and B. Stout, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano 4, 3390–3396 (2010). [CrossRef] [PubMed]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(260.2110) Physical optics : Electromagnetic optics
(260.3910) Physical optics : Metal optics
(350.4238) Other areas of optics : Nanophotonics and photonic crystals

ToC Category:
Optics at Surfaces

History
Original Manuscript: March 18, 2011
Revised Manuscript: May 9, 2011
Manuscript Accepted: May 15, 2011
Published: June 3, 2011

Citation
José M. Taboada, Javier Rivero, Fernando Obelleiro, Marta G. Araújo, and Luis Landesa, "Method-of-moments formulation for the analysis of plasmonic nano-optical antennas," J. Opt. Soc. Am. A 28, 1341-1348 (2011)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-28-7-1341


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References

  1. S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005). [CrossRef]
  2. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010). [CrossRef] [PubMed]
  3. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  4. J.-J. Greffet, “Nanoantennas for light emission,” Science 308, 1561–1563 (2005). [CrossRef] [PubMed]
  5. M. L. Brongersma, “Plasmonics: engineering optical nanoantennas,” Nat. Photon. 2, 270–273 (2008). [CrossRef]
  6. P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1, 438–483 (2009). [CrossRef]
  7. T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “λ/4 Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33(2007). [CrossRef] [PubMed]
  8. P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15, 14266–14274 (2007). [CrossRef] [PubMed]
  9. F. Tam, G. P. Goodrich, B. R. Johnson, and N. J. Halas, “Plasmonic enhancement of molecular fluorescence,” Nano Lett. 7, 496–501 (2007). [CrossRef] [PubMed]
  10. K. Sendur and W. Challener, “Near-field radiation of bow-tie antennas and apertures at optical frequencies,” J. Microsc. 210, 279–283 (2003). [CrossRef] [PubMed]
  11. 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]
  12. P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1608(2005). [CrossRef] [PubMed]
  13. J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. García de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71, 235420 (2005). [CrossRef]
  14. A. Sundaramurthy, P. J. Schuck, N. R. Conley, D. P. Fromm, G. S. Kino, and W. E. Moerner, “Toward nanometer-scale optical photolithography: utilizing the near-field of bowtie optical nanoantennas,” Nano Lett. 6, 355–360 (2006). [CrossRef] [PubMed]
  15. O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Express 15, 17736–17746(2007). [CrossRef] [PubMed]
  16. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Single emitters coupled to plasmonic nano-antennas: angular emission and collection efficiency,” New J. Phys. 10, 105005 (2008). [CrossRef]
  17. J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: a Yagi–Uda nanoantenna in the optical domain,” Phys. Rev. B 76, 245403 (2007). [CrossRef]
  18. H. F. Hofmann, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi–Uda antenna,” New J. Phys. 9, 217(2007). [CrossRef]
  19. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi–Uda antenna,” Opt. Express 16, 10858–10866(2008). [CrossRef] [PubMed]
  20. 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]
  21. 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] [PubMed]
  22. J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic nanowire antennas: experiment, simulation, and theory,” Nano Lett. 10, 3596–3603 (2010). [CrossRef] [PubMed]
  23. A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975). [CrossRef]
  24. T. Weiland, “A discretization method for the solution of Maxwell’s equations for six-component fields,” AEU Arch. Elektron. Übertragungstech. 31, 116–120 (1977).
  25. P. Monk, Finite Element Methods for Maxwell’s Equations(Oxford University Press, 2003). [CrossRef]
  26. R. F. Harrington, Field Computation by Moment Methods, IEEE Series on Electromagnetic Wave Theory (IEEE, 1993). [CrossRef]
  27. A. J. Poggio and E. K. Miller, Computer Techniques for Electromagnetics (Pergamon, 1973).
  28. Y. Chang and R. F. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag. 25, 789–795 (1977). [CrossRef]
  29. T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977). [CrossRef]
  30. 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]
  31. B. Gallinet, A. M. Kern, and O. J. F. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010). [CrossRef]
  32. M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional dielectric objects,” IEEE Trans. Antennas Propag. 47, 1615–1622 (1999). [CrossRef]
  33. P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40, RS6002 (2005). [CrossRef]
  34. 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]
  35. K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003). [CrossRef]
  36. P. Ylä-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]
  37. P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general integral equation method for general composite metallic and dielectric structures with junctions,” PIER 52, 81–108 (2005). [CrossRef]
  38. Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009). [CrossRef]
  39. J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18, 15876–15886(2010). [CrossRef] [PubMed]
  40. J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, J. M. Bertolo, M. G. Araujo, J. C. Mouriño, and A. Gomez, “High scalability FMM-FFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propag. Mag. 51 (6), 20–28 (2009). [CrossRef]
  41. M. G. Araújo, J. M. Taboada, F. Obelleiro, J. M. Bértolo, L. Landesa, J. Rivero, and J. L. Rodríguez, “Supercomputer aware approach for the solution of challenging electromagnetic problems,” PIER 101, 241–256 (2010). [CrossRef]
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