## 3-dimensional eigenmodal analysis of plasmonic nanostructures |

Optics Express, Vol. 20, Issue 5, pp. 5481-5500 (2012)

http://dx.doi.org/10.1364/OE.20.005481

Acrobat PDF (1179 KB)

### Abstract

We introduce a 3-dimensional electromagnetic eigenmodal algorithm for the theoretical analysis of resonating nano-optical structures. The method, a variant of the Jacobi–Davidson algorithm, solves the electric field vector wave, or *curl-curl*, equation for the electromagnetic eigenmodes of resonant optical structures with a finite element method. In particular, the method includes transparent boundary conditions that enable the analysis of resonating structures in unbounded space. We demonstrate the performance of the method. First, we calculate the modes of several dielectric resonator antennas and compare them to theoretically determined results. Second, we calculate the modes of a nano-cuboid and compare them to theoretically determined results. Third, we numerically analyze spherical nanoparticles and compare the result to the theoretical Mie solution. Fourth, we analyze optical dipole antenna configurations in order to assess the method’s capability for solving technologically relevant problems.

© 2012 OSA

## 1. Introduction

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

4. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**, 824–830 (2003). [CrossRef] [PubMed]

5. H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, “Bright unidirectional fluorescence emission of molecules in a nanoaperture with plasmonic corrugations,” Nano Lett. **11**, 637–644 (2011). [CrossRef] [PubMed]

8. M. F. Garcia-Parajo, “Optical antennas focus in on biology,” Nat. Photon. **2**, 201–203, (2008). [CrossRef]

9. P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science **325**, 594–597 (2009). [CrossRef] [PubMed]

10. G. W. Hanson and A. B. Yakovlev, *Operator Theory for Electromagnetics – An Introduction* (Springer, 2002). [PubMed]

- The dielectric resonator antenna (DRA) [15,16
15. A. Okaya and L. F. Barash, “The dielectric microwave resonator,” Proc. IRE

**50**, 2081–2092 (1962). [CrossRef]]: the problem is solved in the microwave region. A theoretical model, applicable under specific conditions, is available to benchmark the method.16. R. K. Mongia and A. Ittipiboon, “Theoretical and experimental investigations on rectangular dielectric resonator antennas,” IEEE Trans. Antennas Propag.

**45**, 1348–1356 (1997). [CrossRef] - The single nano-cuboid: the problem is solved in the optical region. We compare the numerical solution with the theoretical model proposed in
**(i)**. - The single spherical nanoparticle: we compare the numerical solution with the Mie solution [17].
- The optical dipole antenna of finite lateral dimensions: we realistically model the dipole antenna with two smoothly rounded arms. This geometry has traditionally been studied via the
*thin wire*approximation [18]. The FDTD method [19] experiences intrinsic difficulties when modeling rounded objects. Other work [2019. A. Dhawan, S. J. Norton, M. D. Gerhold, and T. Vo-Dinh, “Comparison of FDTD numerical computations and analytical multipole expansion method for plasmonics-active nanosphere dimers,” Opt. Express

**17**, 9688–9703 (2009). [CrossRef] [PubMed]–2220. X. Cui and D. Erni, “The influence of particle shapes on the optical response of nearly touching plasmonic nanoparticle dimers,” J. Comput. Theor. Nanosci.

**47**, 1610–1615 (2010). [CrossRef]] is restricted to 2-dimensional geometries. In [2322. J. Smajic, C. Hafner, L. Raguin, K. Tavzarashvili, and M. Mishrikey, “Comparison of numerical methods for the analysis of plasmonic structures,” J. Comput. Theor. Nanosci.

**6**, 763–774 (2009). [CrossRef], 2423. J. Smajic, C. Hafner, K. Tavzarashvili, and R. Vahldieck, “Numerical analysis of channel plasmon polaritons enhanced optical antennas,” J. Comput. Theor. Nanosci.

**5**, 725–734 (2008). [CrossRef]], the 3-D FEM is restricted to relatively small-scale problems due to the huge memory consumption and excessive computation time, further suffering from relatively low accuracy. In [25, 2624. C. G. Khoury, S. J. Norton, and T. Vo-Dinh, “Plasmonics of 3-D nanoshell dimers using multipole expansion and finite element method,” ACS Nano

**3**, 2776–2788 (2009). [CrossRef] [PubMed]], the surface integral equation (SIE) method experiences difficulties when modeling the substrate. Here, we study a technologically relevant geometrical discretization of the optical dipole antenna, free of the aforementioned limitations. Both bright and dark modes [2726. A. M. Kern and O. J. F. Martin, “Excitation and reemission of molecules near realistic plasmonic nanostructures,” Nano Lett.

**11**, 482–487 (2011). [CrossRef] [PubMed]] are investigated.27. M. W. Chu, V. Myroshnychenko, C. H. Chen, J. P. Deng, C. Y. Mou, and F. J. García de Abajo, “Probing bright and dark surface-plasmon modes in individual and coupled noble metal nanoparticles using an electron beam,” Nano Lett.

**9**, 399–404 (2009). [CrossRef]

## 2. Formulation of the problem

**H**(

**x**), the electric field

**E**(

**x**) satisfies Here, Ω ⊂

^{3}is a bounded domain,

*complex*wavenumber in free space;

*ω*̃ (=

*ω*+

*iα*) is the

*complex*angular frequency,

*ω*is the angular frequency and

*α*is the exponential decay rate;

*μ*,

_{r}*ε*are relative magnetic permeability and relative electric permittivity, respectively;

_{r}*σ*is the ohmic loss of the material;

*ε*, normally with negative real part, instead of ohmic loss

_{r}*σ*. The dielectric function is not generally constant, but

*ε*depends on the frequency

_{r}*ω*, or, equivalently, the real part of

*k*

_{0}[28

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

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

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

29. G. M. Hale and M. R. Querry, “Optical constants of water in the 200 nm to 200 *μ*m wavelength region,” Appl. Opt. **12**, 555–563 (1973). [CrossRef] [PubMed]

_{1}∪ Ω

_{2}∪ Ω

_{3}. It encloses the nano-optical device Ω

_{1}, its substrate Ω

_{2}, and the environment Ω

_{3}, cf. Fig. 1. We solve the eigenproblem Eq. (1), where

*ε*

_{sub}and

*ε*

_{env}are the constant relative permittivity for the substrate and the environment, respectively.

*ε*depends on Re(

_{r}*k*

_{0}) in the subdomain Ω

_{1}.

_{13}the tangential component of

**E**(

**x**) must be continuous. On the artificial surface Γ we use the 1st order absorbing, aka. transparent, boundary condition (ABC) [12] where

*R*.

_{b}*R*should be as large as possible to minimize the artificial reflections on Γ introduced by the ABC, which arise in the case of non-orthogonal incidence of out-going wave. On the other hand,

_{b}*R*should be small to limit the size of the discretized system. By numerical experiment we fix

_{b}*R*such that a change of

_{b}*R*does not result in a significant deviation of the computed resonant wavelength, cf. the discussion in Sections 4.3. Following this rule, we observe that the minimum distance between Γ and the device Ω

_{b}_{1}is at least (and usually larger than) 1/3 of the resonance wavelength.

**E**(

**x**) and the wavenumber

*k*

_{0}. The corresponding magnetic field is obtained by The resonant frequency is

*f*=

*ω*/2

*π*(

*ω*= Re(

*ω*̃)). The decay rate

*α*(= Im(

*ω*̃)) is directly connected to the loss. In analogy to an oscillating system with damping, the quality factor is defined as

*U*, stored energy

*U*, dissipated energy

_{s}*U*, and the radiated power

_{d}*P*of the dispersive metallic nano-structure, respectively. Under the condition of no magnetic dispersion (

_{r}*μ*= 1), the total energy is [30

_{r}30. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A: Gen. Phys. **3**, 233245 (1970). [CrossRef]

31. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A **299**, 309–312 (2002). [CrossRef]

*γ*is the damping frequency of the dispersive metal with

_{e}*γ*=

_{e}*τ*

^{−1}where

*τ*is the relaxation time.

*τ*≈ 31fs for silver and

*τ*≈ 9.3fs for gold [28

**6**, 4370–4379 (1972). [CrossRef]

**(i)**the metal dielectric function can be described by the Drude model [17], and

**(ii)**

*ωτ*≫ 1, we obtain the stored energy [32

32. T. G. Philbin, “Electromagnetic energy momentum in dispersive media,” Phys. Rev. A **83**, 013823 (2011). [CrossRef]

33. F. D. Nunes, T. C. Vasconcelos, M. Bezerra, and J. Weiner, “Electromagnetic energy density in dispersive and dissipative media,” J. Opt. Soc. Am. B **28**, 1544–1552 (2011). [CrossRef]

32. T. G. Philbin, “Electromagnetic energy momentum in dispersive media,” Phys. Rev. A **83**, 013823 (2011). [CrossRef]

33. F. D. Nunes, T. C. Vasconcelos, M. Bezerra, and J. Weiner, “Electromagnetic energy density in dispersive and dissipative media,” J. Opt. Soc. Am. B **28**, 1544–1552 (2011). [CrossRef]

**(i)**and

**(ii)**are satisfied. The second assumption, i.e.,

*ωτ*≫ 1, holds for all eigenmodes computed in this paper, e.g.,

*ωτ*> 100 for silver spheres, Section 4.3, and

*ωτ*> 20 for gold dipole antennas, Section 4.4. While the Drude model is a good fit for dispersive silver, still, there is a difference between the analytical model and the experimental data from [28

**6**, 4370–4379 (1972). [CrossRef]

33. F. D. Nunes, T. C. Vasconcelos, M. Bezerra, and J. Weiner, “Electromagnetic energy density in dispersive and dissipative media,” J. Opt. Soc. Am. B **28**, 1544–1552 (2011). [CrossRef]

*ε*and the experimental data is considerable for the photon energy above 2 eV [35

_{r}35. A. Vial, A. S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B **71**, 085416 (2005). [CrossRef]

**28**, 1544–1552 (2011). [CrossRef]

*Q*

_{2}will appear to be a less reliable definition than

*Q*

_{1}, since

*U*in (7) may not be accurate enough for nano-metallic structures. Therefore, In our numerical examples, we list both

_{s}*Q*

_{1}and

*Q*

_{2}for comparison purposes. The radiative quantum yield

*η*is evaluated as the ratio of radiation loss and total loss, i.e.,

## 3. Numerical solution

### 3.1. The finite element method

**E**(

**x**) and

**f**(

**x**) that satisfy the boundary condition (3) we have [12, 18] Similarly, for any sufficiently smooth scalar function

*q*(

**x**) vanishing on Γ we get Therefore, a natural weak form of (1)–(3) is:

*Find k*

_{0}∈

*and*

**E**∈

*V*,

**E**≠

**0**,

*such that for all*

**f**∈

*V and all q*∈

*W*Here,

*V*denotes the functions in

*H*(

**curl**;Ω) that satisfy the boundary condition (3) and

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

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

*V*and

*W*. To that end we triangulate Ω by tetrahedra with Gmsh [36

36. C. Geuzaine and J.-F. Remacle, “Gmsh: A 3-D finite element mesh generator with built-in pre- and postprocessing facilities,” Int. J. Numer. Methods Eng. **79**, 1309–1331 (2009). [CrossRef]

_{13}and Γ

_{23}. The mesh must be fine in the vicinity of the surface of the optical device Ω

_{1}, but it can be coarse in the far-field zone. This strategy maintains efficiency and accuracy while, on the other hand, it reduces the computational cost.

*V*are then approximated by Nédélec (edge) elements, while the scalar functions in

*W*are approximated by Lagrange (nodal) finite elements [13

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

**N**

*, 1 ≤*

_{i}*i*≤

*n*, be the Nédélec basis functions, while the scalar functions

*N*, 1 ≤

_{ℓ}*ℓ*≤

*m*, denote the Lagrange basis functions. Then, Eq. (13) yield a constrained complex quadratic eigenvalue problem where

*λ*(=

*k*

_{0}) is the eigenvalue and

**x**is the eigenvector. The matrices

*A*,

*R*,

*M*, and

*C*in (14) have entries

*ε*and

_{r}*μ*are element-wise constant,

_{r}*A*,

*M*, and

*R*are composed of real symmetric element matrices multiplied by a complex factor. Thus, they are complex symmetric.

### 3.2. The eigensolver

37. F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev . **43**, 235–286 (2001). [CrossRef]

**x**

_{2}=

*λ*

**x**

_{1}, (14) is transformed into a constrained linear eigenproblem of the form The identity matrix

*I*in (16) could be replaced by any nonsingular matrix

*X*, i.e., we could write

*X*

**x**

_{2}=

*λX*

**x**

_{1}. Choosing

*X*=

*M*would make both 𝒜 and complex symmetric. Since complex symmetry does not necessarily increase numerical simplicity [38

38. P. Arbenz and M. E. Hochstenbach, “A Jacobi–Davidson method for solving complex symmetric eigenvalue problems,” SIAM J. Sci. Comput. **25**, 1655–1673 (2004). [CrossRef]

*X*=

*I*.

39. Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, *Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide* (SIAM, Philadelphia PA, 2000). [CrossRef]

40. D. R. Fokkema, G. L. G. Sleijpen, and H. A. Van der Vorst, “Jacobi–Davidson style QR and QZ algorithms for the partial reduction of matrix pencils,” SIAM J. Sci. Comput. **20**, 94–125 (1996). [CrossRef]

*Q*and

*Z*are 2

*n*×

*r*matrices, and

*T*and

_{A}*T*are upper-triangular

_{B}*r*×

*r*matrices. The quotients of corresponding diagonal elements of

*T*and

_{A}*T*provide the eigenvalues. We are looking for a few eigenvalues closest to a prescribed target value

_{B}*τ*. The initial target is chosen to be an estimate of the expected eigenvalue. More details on selecting the initial target are given in Section 4.1. The columns of

*Q*are the Schur vectors from which the desired eigenvectors are extracted.

*Z*is a so-called shadow space. For details on eigenvalue/eigenvector extraction and on restarting we refer to [39

39. Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, *Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide* (SIAM, Philadelphia PA, 2000). [CrossRef]

*correction equation*has to be solved, where the solution

**t**is used to span the search space.

*λ*̃ is the actual best eigenvalue approximation,

*Q*̃ is the matrix

*Q*expanded by the corresponding Schur vector

**u**.

*Z*̃ is the matrix

*Z*expanded by the new shadow vector

**p**. So,

*Q*̃ and

*Z*̃ are 2

*n*× (

*r*+ 1) matrices.

**t**of (18) is not needed to high accuracy. Therefore, we can approximately solve the correction equation by executing a few steps of a preconditioned Krylov space method. The Jacobi–Davidson method has the advantages of short execution time and low memory consumption, which is crucial for solving large scale eigenproblems. To impose the divergence-free condition in (14), we construct an appropriate projector to assert that each vector in the search space is in the null space of 𝒞

*[41–43*

^{T}43. P. Arbenz and R. Geus, “Multilevel preconditioned iterative eigensolvers for Maxwell eigenvalue problems,” Appl. Numer. Math. **54**, 107–121 (2005). [CrossRef]

### 3.3. Implementation

44. Trilinos Project Home Page, http://trilinos.sandia.gov/.

42. P. Arbenz, M. Bečka, R. Geus, U. Hetmaniuk, and T. Mengotti, “On a parallel multilevel preconditioned Maxwell eigensolver,” Parallel Comput. **32**, 157–165 (2006). [CrossRef]

43. P. Arbenz and R. Geus, “Multilevel preconditioned iterative eigensolvers for Maxwell eigenvalue problems,” Appl. Numer. Math. **54**, 107–121 (2005). [CrossRef]

44. Trilinos Project Home Page, http://trilinos.sandia.gov/.

45. Paraview Home Page, http://www.paraview.org/.

## 4. Validation and application of the algorithm

**(i)**the dielectric resonator antenna (DRA),

**(ii)**the nano-cuboid, and

**(iii)**a spherical nanoparticle with variable radius. Eventually, in addition to the benchmark cases, we apply the algorithm to the analysis of the optical dipole antenna.

46. Home Page of the Swiss National Supercomputing Centre (CSCS), http://www.cscs.ch/.

### 4.1. Dielectric resonator antenna

15. A. Okaya and L. F. Barash, “The dielectric microwave resonator,” Proc. IRE **50**, 2081–2092 (1962). [CrossRef]

16. R. K. Mongia and A. Ittipiboon, “Theoretical and experimental investigations on rectangular dielectric resonator antennas,” IEEE Trans. Antennas Propag. **45**, 1348–1356 (1997). [CrossRef]

*a*,

*b*,

*d*, and dielectric function

*ε*. The meshes usually contain ≈ 80′000 tetrahedra for each DRA, so that the discretization counts ≈ 600′000 degrees of freedom (dof).

_{r}*R*= 30 mm, see Fig. 2(a). Theoretical solutions from [16

_{b}**45**, 1348–1356 (1997). [CrossRef]

*τ*to a real value, the equivalent of 4.8 GHz. Each simulation completes in ≈ 3 minutes using 64 cores on the Cray XT5. The normalized residual (||

*𝒜*

**x**

*−*

*λ*

**x||**

_{2}/||

**x||**

_{2}) of the converged eigenpair is in the order of 10

^{−5}.

**45**, 1348–1356 (1997). [CrossRef]

_{1}, i.e. the DRA, assuming perfect magnetic boundary conditions (PMC)

**E**·

**n**= 0 on the surface Γ

_{13}of the DRA. In femaxx Γ

_{13}is treated as an interface where just the continuity of the tangential component of

**E**is enforced.

**E**·

**n**need not be zero.

*Q*

_{1}matches very well with

*Q*

_{2}. However, the different treatment of the DRA walls causes a noticeable difference between

*f*and

*f*

_{MI}and also between

*Q*

_{1}and

*Q*

_{MI}. We note that, interestingly, our frequencies

*f*are closer to the experimental data cited in [16

**45**, 1348–1356 (1997). [CrossRef]

*f*

_{MI}. Thus, the model implemented in femaxx appears to be more realistic.

*N*of tetrahedral mesh elements. We let

*N*be 8′137, 28′150, 87′380 and 645′868, respectively. The computed resonant frequencies are, respectively, 6.548 GHz, 6.545 GHz, 6.545 GHz and 6.545 GHz, while the corresponding quality factors

*Q*

_{1}are, respectively, 13.9, 13.4, 13.3 and 13.3. Thus, a tetrahedral mesh containing ≈ 80′000 elements for each DRA in Table 1 is good enough.

**E**| of the last DRA in Table 1 (

*a*=

*b*= 10 mm,

*d*= 4 mm) is shown in Fig. 2(b). Here, the number of tetrahedra of the mesh is 645′868. According to the theoretical model in [16

**45**, 1348–1356 (1997). [CrossRef]

*y*-axis satisfies

*E*=

_{y}*E*= 0 and

_{z}*E*=

_{x}*Ak*sin(

_{y}*k*) where

_{y}y*A*is a complex constant and

*k*=

_{y}*π/b*is the wavenumber. We plot the three components of the electric field along the

*y*-axis in Fig. 2(d). They are in good agreement with the theoretical model, except that

*E*varies a bit more than half a cycle inside the DRA. This, again, can be ascribed to the differing treatments of the DRA surface.

_{x}*ε*of the employed materials is generally constant. However, in the optical region of the electromagnetic spectrum the dielectric permittivity is in general not constant but depends on the real part of eigenvalue Re(

_{r}*k*

_{0}), cf. (2). The dispersive material property implies that the matrices

*M*and

*C*in (15) also depend on Re(

*k*

_{0}). So, in the nano-optical region, the eigenproblem (14) becomes a ‘truly’ non-linear, i.e. not linearizable, eigenproblem. At present, femaxx is capable of solving linear and quadratic eigenproblems. Nevertheless, we can address this non-linear eigenproblem via solving a

*sequence*of quadratic eigenproblems. We start with an estimate

*λ*

^{(0)}for the eigenvalue of the expected eigenmode. By means of Re(

*λ*

^{(0)}), we determine the dielectric permittivity

*ε*=

_{r}*ε*(Re(

_{r}*λ*

^{(0)})), based on [28

**6**, 4370–4379 (1972). [CrossRef]

*M*(Re(

*λ*

^{(0)})), and

*C*(Re(

*λ*

^{(0)})) can be constructed. Then, we solve the quadratic eigenvalue problem (14) and consequently obtain an improved eigenvalue estimate

*λ*

^{(1)}. This procedure is repeated to yield further estimates

*λ*

^{(2)},

*λ*

^{(3)}, etc. We terminate the iteration as soon as two consecutive estimates are close enough. In what follows, we use this iterative procedure. We first validate our method with the nano-cuboid and nano-sphere geometries. Then we use femaxx for the analysis of specific optical devices. In the individual quadratic eigenvalue problems we use

*τ*= Re (

*λ*

^{(}

^{j}^{)}) as the target.

### 4.2. Cuboid

*a*=

*b*= 100 nm,

*d*= 40 nm,

*R*= 300 nm and assume gold and silver, respectively, for the material of the nano-cuboid. We also reuse the theoretical DRA model [15

_{b}**50**, 2081–2092 (1962). [CrossRef]

**45**, 1348–1356 (1997). [CrossRef]

*f*

_{MI}and the electromagnetic field. The theoretically determined wavenumber

*k*

_{0}is complex. The quality factor

*Q*

_{MI}is also theoretically determined via Eq. (5). We comment that Mongia’s model was developed in the microregion of the electromagnetic spectrum. Thus, strictly speaking, it may not be used as a reference or benchmark in the optical region where the dielectric permittivity of metals is highly dispersive and exhibits considerable dielectric loss. Nevertheless, we believe that, at present, there is no better analytical model available than Mongia’s for cuboid geometries. Therefore, we use Mongia’s model as a signpost into the optical region. Fully aware that we stretch Mongia’s model beyond its original limits of applicability, we consider it useful to help acquire at least qualitative insight into the mode structure of the nano-cuboid. The results are shown in Table 2. The mesh contains 645′868 tetrahedra leading to 4′120′676 degrees of freedom. For all simulations, the initial value of

*τ*(= Re(

*λ*

^{(0)})) is equivalent of 1000 THz. The eigensolver calculation time for a single run is less than 10 minutes with 256 cores, while total computation time for one eigenmode is then below 40 minutes for 3–4 runs. The normalized residual of the converged eigenpair is around 10

^{−5}.

**E**| of the gold cuboid is shown in Fig. 2(c). The three components of

**E**, plotted along the

*y*-axis, are displayed in Fig. 2(e). Again they are in good agreement with the fields of the theoretical model. In Fig. 2(e),

*E*≈ 0 and

_{y}*E*≈ 0.

_{z}*E*varies by a bit more than half a cycle inside the cuboid which indicates that the PMC is not strictly satisfied on the surfaces of the cuboid. We also notice that the field is strong inside the cuboid and drops to almost zero in the far-field zone, see Fig. 2(e). Therefore, the mode experiences rather high dissipative loss, but radiates only scant electromagnetic power. Thus,

_{x}*η*is very small.

*ε*is not at all constant for gold and silver in the optical region of the spectrum, The dispersive material property, in combination with the different treatment of the DRA wall, largely explains the noticeable difference between

_{r}*f*and

*f*

_{MI}. Compared to dielectric resonator antennas, the quality factors (

*Q*

_{1},

*Q*

_{2}, or

*Q*

_{MI}) of the nano-cuboid are very small, significantly below 1. This is attributed to high dissipative loss. The difference between

*Q*

_{1}and

*Q*

_{2}is noticeable, cf. the discussion in Section 2.

### 4.3. Sphere

*R*= 25,30,35,40,60 and 70 nm. Thus,

*ε*

_{sub}=

*ε*

_{env}= 1.0. The radius of the computational domain is

*R*= 300 nm. For each sphere, the mesh contains about 250′000 tetrahedra. For all simulations, the initial value of

_{b}*τ*is equivalent of 3.00eV. In this example we had problems with the quadratic elements. The computation of

*E*

_{res1}for

*R*= 25 nm and of

*E*

_{res2}(see below) for

*R*= 60 nm failed to converge such that we had to resort to linear elements. In general, quadratic elements deliver more accurate solutions for fewer elements but they are more demanding w.r.t. iterative solver schemes. The eigensolver calculation time for a single run is less than 10 minutes with 256 cores (or 32 cores) if quadratic (or linear) elements are used. The normalized residual of the converged eigenpair is around 5 × 10

^{−4}. The total computation time for one eigenmode is thus below 1 hour for 5–6 runs. We study the energy range from 2 eV to 4 eV, in wavelength units, 310 nm to 620 nm. We use

*λ*for the numerically determined resonance wavelength and the resonance energy

*E*=

*hc/λ*, with

*h*the Plank constant and

*c*the speed of light in vacuum, respectively.

*R*= 25, 30, 35, or 40 nm, the peaks of scattering (

*Q*

_{sca}) and absorption efficiency (

*Q*

_{abs}) occur at almost the same position,

*R*= 30 nm. The full-linewidth-at-half-maximum (FWHM) energy width Δ

*E*

_{ext}of the extinction efficiency,

*Q*

_{ext}=

*Q*

_{sca}+

*Q*

_{abs}, can be evaluated. The quality factor is computed as

47. C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, and J. Feldmann, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. **88**, 077402 (2002) [CrossRef] [PubMed]

*R*= 60 or 70 nm, the peaks of

*Q*

_{sca}and

*Q*

_{abs}are separated from each other, such that there are two distinct resonances close together. Figure 3(b) shows the Mie solution for

*R*= 60 nm. We first study the resonance

*Q*

_{sca}reaches maximum. Let Δ

*E*

_{sca}be the FWHM of

*Q*

_{sca}, and the quality factor of this resonance

*Q*

_{sca}deviates from a Lorentzian shape [48

48. R. Fuchs and K. L. Kliewer, “Optical modes of vibration in an ionic crystal spheres,” J. Opt. Soc. Am. **58**, 319–330 (1968) [CrossRef]

*E*

_{sca}is not accurate.

*η*

^{mie}, computed with the Mie solution, is defined as the ratio between scattering and extinction efficiency, i.e.,

*η*

^{mie}=

*Q*

_{sca}

*/Q*

_{ext}.

*E*

_{res1}and

*η*and

*η*

^{mie}) show good agreements. The differences are mainly attributed to the 1st order ABC employed by femaxx, while the Mie model is unbounded. Quality factors (

*Q*

_{1}and

*R*= 25,30,35 or 40 nm. When

*R*= 60 or 70 nm, two peaks of resonances (

*Q*

_{1}is more reliable. As noted in Section 2, in the range of 2.5eV to 3.5eV, the difference of silver

*ε*between the Drude model and the experimental data is noticeable. Therefore

_{r}*Q*

_{2}is not accurate either. Note that

*E*

_{res1}approximates a 3-fold degenerate resonance. The corresponding computed mode splits in three simple modes since the mesh does not respect the spherical symmetry of the problem. For

*R*= 60 nm, the relative differences among the three computed resonances are below 1%.

*R*= 60nm as an example (see Fig. 3(b)), and study the resonance

*Q*

_{abs}reaches the maximum. (Our computations indicate that

*E*

_{res2}is 5-fold degenerate.) The numerical value of this mode is

*E*

_{res2}= 3.48eV and

*Q*

_{1}= 8.1. Let the angular frequency, dissipated energy and radiated powers of resonance

*E*

_{res2}be

*ω*

_{res2},

*E*

_{res1}be

*ω*

_{res1},

_{Ω}Re (

*ε*)|

_{r}**E**(

**x**)|

^{2}

*d*

**x**are equal. We define

*r*= 8.9 and

_{d}*r*= 0.21. Resonance

_{r}*E*

_{res2}has higher dissipative loss since

*Q*

_{abs}reaches maximum, while resonance

*E*

_{res1}has higher radiation loss since

*Q*

_{sca}reaches maximum. The Mie solution of this mode is

*E*

_{abs}is not accurate. The electric field plots of the two modes, when

*R*= 60nm, are shown in Figures 5(a) and 5(b).

*E*

_{res1}of the silver sphere (

*R*= 60nm) to study the influence of the 1st order ABC on the accuracy of the result. If we let the boundary radius

*R*vary from 250 to 300nm (keeping the number of tetrahedra around 250′000), the deviations of the computed resonant wavelength are as small as 3nm. For

_{b}*R*= 300nm, the computed resonant wavelength (408.9nm) is closest to the resonant wavelength of the Mie solution (≈ 410.5nm). For smaller

_{b}*R*, the error gets larger. For

_{b}*R*= 200nm, e.g., the computed resonant wavelength is 385.0nm.

_{b}### 4.4. An optical dipole antenna fabricated with gold dielectric permittivity

*a*=

*b*= 40 mm,

*l*= 100 nm. The corners of each arm are rounded, with a radius of curvature = 5 nm. The gap

*g*between the two arms is 20 nm wide but will be varied in order to study the electromagnetic field enhancement as a function of the gap width. The radius

*R*of the spherical computational domain is 400 nm.

_{b}**(1)**the antenna hovers in vacuum, thus

*ε*

_{sub}=

*ε*

_{env}= 1.0;

**(2)**the antenna resides on a silica substrate, thus

*ε*

_{sub}= 2.25

*,ε*

_{env}= 1.0;

**(3)**the antenna resides on a silica substrate and is covered with water, thus

*ε*

_{sub}= 2.25,

*ε*

_{env}= 1.77 [29

29. G. M. Hale and M. R. Querry, “Optical constants of water in the 200 nm to 200 *μ*m wavelength region,” Appl. Opt. **12**, 555–563 (1973). [CrossRef] [PubMed]

*bright*, aka. optically active, mode and the

*dark*mode split. The modes can then be well distinguished in the scattering spectra [27

27. M. W. Chu, V. Myroshnychenko, C. H. Chen, J. P. Deng, C. Y. Mou, and F. J. García de Abajo, “Probing bright and dark surface-plasmon modes in individual and coupled noble metal nanoparticles using an electron beam,” Nano Lett. **9**, 399–404 (2009). [CrossRef]

49. S. C. Yang, H. Kobori, C. L. He, M. H. Lin, H. Y. Chen, C. Li, M. Kanehara, T. Teranishi, and S. Gwo, “Plasmon hybridization in individual gold nanocrystal dimers: direct observation of bright and dark modes,” Nano Lett. **10**, 632–637 (2010). [CrossRef] [PubMed]

#### 4.4.1. Bright mode

*τ*is equivalent of 660

*nm*. The results are presented in Table 4. The fields of model (1), an antenna in vacuum, are evaluated in the sample domain Ω

_{sample}= 400nm×200nm×200 nm. Both magnetic and electric fields are shown in Fig. 7(a) and 7(b), respectively. A close-up view of the electric field in the gap region is shown in Fig. 7(c). The results in Table 4 confirm the influence of the substrate onto the antenna’s resonant wavelength. With increasing permittivity

*ε*

_{sub}the antenna resonance is red-shifted; with increasing

*ε*

_{env}the resonance wavelength is also red-shifted [50

50. H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express **16**, 9144–9154 (2008). [CrossRef] [PubMed]

^{−4}. The memory per core required by the eigensolver is 33.38 MB for the model (1), and 28.65 MB for models (2) and (3). Thus, the method is particularly attractive when compared to other 3-dimensional FEM approaches [22

22. J. Smajic, C. Hafner, L. Raguin, K. Tavzarashvili, and M. Mishrikey, “Comparison of numerical methods for the analysis of plasmonic structures,” J. Comput. Theor. Nanosci. **6**, 763–774 (2009). [CrossRef]

24. C. G. Khoury, S. J. Norton, and T. Vo-Dinh, “Plasmonics of 3-D nanoshell dimers using multipole expansion and finite element method,” ACS Nano **3**, 2776–2788 (2009). [CrossRef] [PubMed]

*g*affects the resonance wavelength

*λ*and the field intensity |

**E**|

^{2}in the gap. We use the dielectric arrangement of model (2). Let the gap width

*g*be 20 nm, 10 nm, 5 nm, and 1 nm, respectively. The corresponding resonance wavelengths are then 717.5 nm, 751.9 nm, 796.3 nm, and 913.5 nm, respectively. A red shift, i.e., increasing wavelength, of the antenna resonance occurs with decreasing gap widths. Since the gap widths are significantly smaller than the resonance wavelengths, we can approximate the gap region with an electrostatic regime. Thus, we understand the gap region, including those portions of the dipole antenna, that immediately border it, to form a plate capacitor whose capacitance is given as

*C*=

*ε*

_{0}

*ε*. Here,

_{r}A/d*A*,

*d*, and

*ε*correspond to the plate area, the distance between the plates and the dielectric property of the medium between the plates, respectively; in particular,

_{r}*ε*= 1 since the gap is in vacuum. Then, it becomes clear that, with decreasing gap width, capacitance increases. A capacitance increase on the other hand is roughly proportional to a higher value of the dielectric permittivity and thus, for constant length of the dipole arms, the resonance wavelength increases since the resonance frequency is reduced. We comment that adding a capacitor parallel to an antenna’s terminal contacts is a well-known technique in microwave electronics in order to geometrically shorten the antenna, thus mimicking higher dielectric permittivity in the vicinity of the antenna.

_{r}*g*= 10 nm, 5 nm, and 1 nm, the corresponding field intensities |

**E**|

^{2}increase and are 3.4, 9.6, and 56 times larger than when a gap width

*g*= 20 nm is employed, see Fig. 8(b). Decreasing the gap width thus dramatically increases the field intensity in the gap. An example field distribution, for a 5 nm gap width, is shown in Fig. 8(a). Due to the substrate effect the field is not symmetric with respect to the

*x*-axis.

#### 4.4.2. Dark mode

**9**, 399–404 (2009). [CrossRef]

*ω*

_{dark},

*ω*

_{bright},

_{Ω}Re(

*ε*)|

_{r}**E**(

**x**)|

^{2}

*d*

**x**are equal. We define

*η*is computed via Eq. (11). Associated results are listed in Table 5. The mesh contains about 350′000 tetrahedra which results in more than 2′000′000. For all simulations of the dark modes, the initial value of

*τ*is equivalent of 550

*nm*. The eigensolver calculation time for a single run is 10 to 15 minutes with 256 cores, and the normalized residual of the converged eigenpair is around 5 × 10

^{−4}for the bright mode and 10

^{−4}for the dark mode. The total computation time for one eigenmode is then below 90 minutes for 5–6 runs.

*g*of the gap between the two arms is reduced, the single arm’s degenerate fundamental mode with its resonance at 623.5 nm splits into a bright and a dark mode. With decreasing gap width, the bright mode red-shifts, due to the effect of increased capacity for smaller width, and the dark mode shifts to blue. The dark mode radiates less, i.e. (

*r*< 0.1), which implies that it cannot efficiently radiate energy into the far-field, thus earning its dark reputation. On the other hand, the dark mode experiences higher dissipative loss inside the two gold arms, i.e.

_{r}*r*> 10. Its dissipative loss approximates almost 100% of the total loss, namely

_{d}*η*< 0.005, and thus the quality factor

*Q*

_{1}of the dark mode is smaller than the bright mode.

*Q*

_{2}is in good agreement with

*Q*

_{1}for bright modes, but is significantly off for dark modes.

## 5. Discussion and conclusions

49. S. C. Yang, H. Kobori, C. L. He, M. H. Lin, H. Y. Chen, C. Li, M. Kanehara, T. Teranishi, and S. Gwo, “Plasmon hybridization in individual gold nanocrystal dimers: direct observation of bright and dark modes,” Nano Lett. **10**, 632–637 (2010). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photon. |

2. | L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. |

3. | P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. |

4. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

5. | H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, “Bright unidirectional fluorescence emission of molecules in a nanoaperture with plasmonic corrugations,” Nano Lett. |

6. | D. W. Pohl, S. G. Rodrigo, and L. Novotny, “Stacked optical antennas,” Appl. Phys. Lett. |

7. | A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photon. |

8. | M. F. Garcia-Parajo, “Optical antennas focus in on biology,” Nat. Photon. |

9. | P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science |

10. | G. W. Hanson and A. B. Yakovlev, |

11. | Ch. Hafner, |

12. | J. Jin, |

13. | P. Monk, |

14. | J. L. Volakis, A. Chatterjee, and L. C. Kempel, |

15. | A. Okaya and L. F. Barash, “The dielectric microwave resonator,” Proc. IRE |

16. | R. K. Mongia and A. Ittipiboon, “Theoretical and experimental investigations on rectangular dielectric resonator antennas,” IEEE Trans. Antennas Propag. |

17. | C. Bohren and D. Huffmann, |

18. | S. Ramo, J. R. Whinnery, and T. V. Duzer, |

19. | A. Dhawan, S. J. Norton, M. D. Gerhold, and T. Vo-Dinh, “Comparison of FDTD numerical computations and analytical multipole expansion method for plasmonics-active nanosphere dimers,” Opt. Express |

20. | X. Cui and D. Erni, “The influence of particle shapes on the optical response of nearly touching plasmonic nanoparticle dimers,” J. Comput. Theor. Nanosci. |

21. | J. M. McMahon, A. I. Henry, K. L. Wustholz, M. J. Natan, R. G. Freeman, R. P. Van Duyne, and G. C. Schatz, “Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy,” Anal. Bioanal. Chem. |

22. | J. Smajic, C. Hafner, L. Raguin, K. Tavzarashvili, and M. Mishrikey, “Comparison of numerical methods for the analysis of plasmonic structures,” J. Comput. Theor. Nanosci. |

23. | J. Smajic, C. Hafner, K. Tavzarashvili, and R. Vahldieck, “Numerical analysis of channel plasmon polaritons enhanced optical antennas,” J. Comput. Theor. Nanosci. |

24. | C. G. Khoury, S. J. Norton, and T. Vo-Dinh, “Plasmonics of 3-D nanoshell dimers using multipole expansion and finite element method,” ACS Nano |

25. | A. M. Kern and O. J. F. Martin, “Modeling near-field properties of plasmonic nanoparticles: a surface integral approach,” in Plasmonic: Nanoimaging, Nanofabrication, and their Applications V, V. M. Shalaev and D. P. Tsai, eds., |

26. | A. M. Kern and O. J. F. Martin, “Excitation and reemission of molecules near realistic plasmonic nanostructures,” Nano Lett. |

27. | M. W. Chu, V. Myroshnychenko, C. H. Chen, J. P. Deng, C. Y. Mou, and F. J. García de Abajo, “Probing bright and dark surface-plasmon modes in individual and coupled noble metal nanoparticles using an electron beam,” Nano Lett. |

28. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

29. | G. M. Hale and M. R. Querry, “Optical constants of water in the 200 nm to 200 |

30. | R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A: Gen. Phys. |

31. | R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A |

32. | T. G. Philbin, “Electromagnetic energy momentum in dispersive media,” Phys. Rev. A |

33. | F. D. Nunes, T. C. Vasconcelos, M. Bezerra, and J. Weiner, “Electromagnetic energy density in dispersive and dissipative media,” J. Opt. Soc. Am. B |

34. | L. Brillouin, |

35. | A. Vial, A. S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B |

36. | C. Geuzaine and J.-F. Remacle, “Gmsh: A 3-D finite element mesh generator with built-in pre- and postprocessing facilities,” Int. J. Numer. Methods Eng. |

37. | F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev . |

38. | P. Arbenz and M. E. Hochstenbach, “A Jacobi–Davidson method for solving complex symmetric eigenvalue problems,” SIAM J. Sci. Comput. |

39. | Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, |

40. | D. R. Fokkema, G. L. G. Sleijpen, and H. A. Van der Vorst, “Jacobi–Davidson style QR and QZ algorithms for the partial reduction of matrix pencils,” SIAM J. Sci. Comput. |

41. | R. Geus, “The Jacobi–Davidson algorithm for solving large sparse symmetric eigenvalue problems.” PhD Thesis No. 14734, ETH Zurich 2002. |

42. | P. Arbenz, M. Bečka, R. Geus, U. Hetmaniuk, and T. Mengotti, “On a parallel multilevel preconditioned Maxwell eigensolver,” Parallel Comput. |

43. | P. Arbenz and R. Geus, “Multilevel preconditioned iterative eigensolvers for Maxwell eigenvalue problems,” Appl. Numer. Math. |

44. | Trilinos Project Home Page, http://trilinos.sandia.gov/. |

45. | Paraview Home Page, http://www.paraview.org/. |

46. | Home Page of the Swiss National Supercomputing Centre (CSCS), http://www.cscs.ch/. |

47. | C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, and J. Feldmann, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. |

48. | R. Fuchs and K. L. Kliewer, “Optical modes of vibration in an ionic crystal spheres,” J. Opt. Soc. Am. |

49. | S. C. Yang, H. Kobori, C. L. He, M. H. Lin, H. Y. Chen, C. Li, M. Kanehara, T. Teranishi, and S. Gwo, “Plasmon hybridization in individual gold nanocrystal dimers: direct observation of bright and dark modes,” Nano Lett. |

50. | H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express |

**OCIS Codes**

(140.4780) Lasers and laser optics : Optical resonators

(240.6680) Optics at surfaces : Surface plasmons

(260.3910) Physical optics : Metal optics

(260.5740) Physical optics : Resonance

(220.4241) Optical design and fabrication : Nanostructure fabrication

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: January 18, 2012

Revised Manuscript: February 10, 2012

Manuscript Accepted: February 10, 2012

Published: February 21, 2012

**Virtual Issues**

Vol. 7, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Hua Guo, Benedikt Oswald, and Peter Arbenz, "3-dimensional eigenmodal analysis of plasmonic nanostructures," Opt. Express **20**, 5481-5500 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-5-5481

Sort: Year | Journal | Reset

### References

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- L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett.98, 266802 (2007). [CrossRef] [PubMed]
- P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon.1, 438–483 (2009). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–830 (2003). [CrossRef] [PubMed]
- H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, “Bright unidirectional fluorescence emission of molecules in a nanoaperture with plasmonic corrugations,” Nano Lett.11, 637–644 (2011). [CrossRef] [PubMed]
- D. W. Pohl, S. G. Rodrigo, and L. Novotny, “Stacked optical antennas,” Appl. Phys. Lett.98, 023111 (2011). [CrossRef]
- A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photon.3, 654–657 (2009). [CrossRef]
- M. F. Garcia-Parajo, “Optical antennas focus in on biology,” Nat. Photon.2, 201–203, (2008). [CrossRef]
- P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science325, 594–597 (2009). [CrossRef] [PubMed]
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- R. K. Mongia and A. Ittipiboon, “Theoretical and experimental investigations on rectangular dielectric resonator antennas,” IEEE Trans. Antennas Propag.45, 1348–1356 (1997). [CrossRef]
- C. Bohren and D. Huffmann, Absorption and Scattering of Light by Small Particles (John Wiley, New York, 1983).
- S. Ramo, J. R. Whinnery, and T. V. Duzer, Fields and Waves in Communication Electronics (John Wiley, New York, 1993).
- A. Dhawan, S. J. Norton, M. D. Gerhold, and T. Vo-Dinh, “Comparison of FDTD numerical computations and analytical multipole expansion method for plasmonics-active nanosphere dimers,” Opt. Express17, 9688–9703 (2009). [CrossRef] [PubMed]
- X. Cui and D. Erni, “The influence of particle shapes on the optical response of nearly touching plasmonic nanoparticle dimers,” J. Comput. Theor. Nanosci.47, 1610–1615 (2010). [CrossRef]
- J. M. McMahon, A. I. Henry, K. L. Wustholz, M. J. Natan, R. G. Freeman, R. P. Van Duyne, and G. C. Schatz, “Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy,” Anal. Bioanal. Chem.394, 1819–1825 (2009). [CrossRef] [PubMed]
- J. Smajic, C. Hafner, L. Raguin, K. Tavzarashvili, and M. Mishrikey, “Comparison of numerical methods for the analysis of plasmonic structures,” J. Comput. Theor. Nanosci.6, 763–774 (2009). [CrossRef]
- J. Smajic, C. Hafner, K. Tavzarashvili, and R. Vahldieck, “Numerical analysis of channel plasmon polaritons enhanced optical antennas,” J. Comput. Theor. Nanosci.5, 725–734 (2008). [CrossRef]
- C. G. Khoury, S. J. Norton, and T. Vo-Dinh, “Plasmonics of 3-D nanoshell dimers using multipole expansion and finite element method,” ACS Nano3, 2776–2788 (2009). [CrossRef] [PubMed]
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- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972). [CrossRef]
- G. M. Hale and M. R. Querry, “Optical constants of water in the 200 nm to 200 μm wavelength region,” Appl. Opt.12, 555–563 (1973). [CrossRef] [PubMed]
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- R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A299, 309–312 (2002). [CrossRef]
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- A. Vial, A. S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005). [CrossRef]
- C. Geuzaine and J.-F. Remacle, “Gmsh: A 3-D finite element mesh generator with built-in pre- and postprocessing facilities,” Int. J. Numer. Methods Eng.79, 1309–1331 (2009). [CrossRef]
- F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. 43, 235–286 (2001). [CrossRef]
- P. Arbenz and M. E. Hochstenbach, “A Jacobi–Davidson method for solving complex symmetric eigenvalue problems,” SIAM J. Sci. Comput.25, 1655–1673 (2004). [CrossRef]
- Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide (SIAM, Philadelphia PA, 2000). [CrossRef]
- D. R. Fokkema, G. L. G. Sleijpen, and H. A. Van der Vorst, “Jacobi–Davidson style QR and QZ algorithms for the partial reduction of matrix pencils,” SIAM J. Sci. Comput.20, 94–125 (1996). [CrossRef]
- R. Geus, “The Jacobi–Davidson algorithm for solving large sparse symmetric eigenvalue problems.” PhD Thesis No. 14734, ETH Zurich 2002.
- P. Arbenz, M. Bečka, R. Geus, U. Hetmaniuk, and T. Mengotti, “On a parallel multilevel preconditioned Maxwell eigensolver,” Parallel Comput.32, 157–165 (2006). [CrossRef]
- P. Arbenz and R. Geus, “Multilevel preconditioned iterative eigensolvers for Maxwell eigenvalue problems,” Appl. Numer. Math.54, 107–121 (2005). [CrossRef]
- Trilinos Project Home Page, http://trilinos.sandia.gov/ .
- Paraview Home Page, http://www.paraview.org/ .
- Home Page of the Swiss National Supercomputing Centre (CSCS), http://www.cscs.ch/ .
- C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, and J. Feldmann, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett.88, 077402 (2002) [CrossRef] [PubMed]
- R. Fuchs and K. L. Kliewer, “Optical modes of vibration in an ionic crystal spheres,” J. Opt. Soc. Am.58, 319–330 (1968) [CrossRef]
- S. C. Yang, H. Kobori, C. L. He, M. H. Lin, H. Y. Chen, C. Li, M. Kanehara, T. Teranishi, and S. Gwo, “Plasmon hybridization in individual gold nanocrystal dimers: direct observation of bright and dark modes,” Nano Lett.10, 632–637 (2010). [CrossRef] [PubMed]
- H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express16, 9144–9154 (2008). [CrossRef] [PubMed]

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