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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 16 — Aug. 3, 2009
  • pp: 13561–13575
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Rigorous modal analysis of metallic nanowire chains

Amit Hochman and Yehuda Leviatan  »View Author Affiliations


Optics Express, Vol. 17, Issue 16, pp. 13561-13575 (2009)
http://dx.doi.org/10.1364/OE.17.013561


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Abstract

Nanowire chains (NCs) are analyzed by use of a rigorous, full-wave, Source-Model Technique (SMT). The technique employs a proper periodic Green’s function which converges regardless of whether the structure is lossless or lossy. By use of this Green’s function, it is possible to determine the complex propagation constants of the NC modes directly and accurately, as solutions of a dispersion equation. To demonstrate the method, dispersion curves and mode profiles for a few NCs are calculated.

© 2009 Optical Society of America

1. Introduction

Linear chains of metallic nano-particles and nanowires have been the subject of intensive research recently. This effort has been motivated, primarily, by the possible use of these structures as integrated optical waveguides that can confine light transversally to sub-wavelength dimensions [1

1. M. Quinten, A. Leitner, J. Krenn, and F. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23, 1331–1333 (1998). [CrossRef]

]. However, metallic structures are inherently lossy at optical frequencies, and there appears to be a correlation between losses and confinement [2

2. P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14, 13,030–13,042 (2006). [CrossRef]

]. Hence, reliable methods for determining these parameters are essential for waveguide design.

A number of ways around this problem have been proposed. In [8

8. W. Weber and G. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125,429 (2004). [CrossRef]

], the infinite array is truncated to a finite length array. While this solves the problem, it has the drawback that a large number of particles must be modeled. In [10

10. D. Citrin, “Plasmon-polariton transport in metal-nanoparticle chains embedded in a gain medium,” Opt. Lett. 31, 98–100 (2006). [CrossRef] [PubMed]

], the poly-logarithm function is used to continue analytically the values of an approximate dispersion equation from the real-line into the complex propagation-constant plane, where it would have otherwise diverged. This method is also used in [11

11. A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006). [CrossRef]

, 12

12. A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74, 205,436 (2006). [CrossRef]

]. A different approach is to solve a scattering problem and to infer the dispersion curve from the variation of the reflection with the angle of incident light, as in [13

13. Q. H. Wei, K. H. Su, S. Durant, and X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. 4, 1067–1072 (2004). [CrossRef]

, 14

14. T. Yang and K. Crozier, “Dispersion and extinction of surface plasmons in an array of gold nanoparticle chains: influence of the air/glass interface,” Opt. Express 16, 8570–8580 (2008). [CrossRef] [PubMed]

]. It is also possible to approximately determine the propagation length from the attenuation of the total fields (scattered+incident) along a finite chain as in [15

15. H. Chu, W. Ewe, W. Koh, and E. Li, “Remarkable influence of the number of nanowires on plasmonic behaviors of the coupled metallic nanowire chain,” Appl. Phys. Lett. 92, 103,103 (2008). [CrossRef]

]. In a scattering problem, the divergence does not arise, even if the structure is lossy and infinite, because the incident wave impinges on all the scatterers with the same amplitude.

In this paper, we focus on nanowire chains (NCs), i.e., the periodic elements are assumed to be periodic cylinders invariant in some direction transverse to the propagation direction. As this problem is 2D, it is somewhat simpler than the nano-particle chain problem. But the mentioned divergence occurs in the 2D case as well, and the techniques we propose could be applied to the 3D case.

2. Problem specification

We consider a linear chain of cylinders of arbitrary cross-section, all of them characterized by a relative permittivity εc and oriented parallel to the z direction, as shown in Fig. 1. The linear chain is periodic in the x direction and the period is denoted by L. The material surrounding the cylinders is characterized by free-space permittivity, ε 0, and free-space permeability, µ 0. For all fields and sources, exp(jωt) harmonic time variation is assumed and suppressed. Our aim is to determine proper modes of this structure, i.e., source-free electromagnetic fields which obey the periodicity condition

F(x)=F(xL)ejkxL,
(1)

where F(x) is any of the field components and kx is a propagation constant. As the modes are proper, they must obey the radiation condition. This means that for y→±∞, the fields must be either zero or composed entirely of outgoing waves. A more formal mathematical statement of the radiation condition for periodic structures is somewhat involved (see [28

28. R. Petit, Electromagnetic Theory of Gratings, Springer-Verlag, (1980).

, p. 54]). For a given ω, modes will exist only for certain values of kx. The determination of the (kx) pairs for which a mode exists is a central part of the mode determination process.

Fig. 1. Linear chain of nanowires, of arbitrary cross-section and relative permittivity εc, surrounded by free-space.

In this paper, we will consider metallic cylinders characterized by a plasma-type permittivity. Consequently, we will only be interested in TE to z modes, which have a z-directed magnetic field, as these are generally not attenuated as much as TM to z modes, which have a z-directed electric field. The electric field of the latter modes is tangential to the metallic interface and is therefore shorted-out.

Lastly, the mode determination problem may be reduced, by the Floquet theorem, to consideration of a single unit cell with periodic boundary conditions

F(L2)=F(L2)ejkxL,
(2)

at the unit cell boundaries.

3. Source-Model Technique (SMT)

The following formulation of the SMT for periodic structures follows [9

9. A. Boag, Y. Leviatan, and A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Rad. Sci. 23, 612–624 (1988). [CrossRef]

], in which scattering from a linear chain of cylinders is analyzed. In the SMT, the fields in each homogenous region are approximated by the fields due to a set of elementary sources, of yet to be determined amplitudes, that is placed outside of each homogeneous region and is used to approximate the fields in that region. To approximate modes that are TE to z, all the sources are magnetic current filaments oriented parallel to the z axis. The fields within the unit cell and outside of the cylinder are approximated by the fields due to magnetic current filaments, of yet to be determined current amplitudes, operating in the unit cell with the cylinder removed and the periodic conditions and radiation conditions enforced. This is shown in Fig. 2. Generally, the filaments are distributed uniformly on curves which are slightly contracted and dilated versions of the media boundary curves. A simple method for generating such curves for arbitrary smooth boundary curves is described in [29

29. M. Szpulak, W. Urbanczyk, E. Serebryannikov, A. Zheltikov, A. Hochman, Y. Leviatan, R. Kotynski, and K. Panajotov, “Comparison of different methods for rigorous modeling of photonic crystal fibers,” Opt. Express 14, 5699–5714 (2006). [CrossRef] [PubMed]

]. The fields inside the cylinder are approximated by the fields due to magnetic current filaments operating in a homogeneous, unbounded, obstacle-free medium that has the same material parameters as the cylinder, as shown in Fig. 3.

Fig. 2. Magnetic current filaments used to approximate the fields within the unit cell and outside of the cylinder.
Fig. 3. Magnetic current filaments used to approximate the fields inside of the cylinder.

Enforcing the continuity of the field components tangential to the media boundaries at a discrete set of testing points leads to a homogeneous matrix equation

[Z(kx,ω)]K=0,
(3)

where [Z] is an 2M×2N impedance matrix and K⃗ is a column vector of unknown current amplitudes. Here, M is the number of testing points, which are uniformly distributed on the media boundaries, and enforcing the continuity of the tangential electric and magnetic fields leads to 2M equations. The number of current filaments is 2N, of which there is an equal number inside and outside of the cylinder. To avoid linear dependence of the columns of [Z], as N is increased, the sources should approach the boundary. A simple rule of thumb used in this paper (adapted from [20

20. C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics, Artech House, (1990).

, pp. 169–170]) is to make the length of the contracted and dilated curves on which the sources are placed equal to 1±2π/N times the length of the boundary curves.

Using more testing points than sources spreads the error in the continuity conditions more uniformly along the boundary. It is also useful for avoiding certain spurious solutions [30

30. W. Schroeder and I. Wolff, “The origin of spurious modes in numerical solutions of electromagnetic field eigenvalue problems,” IEEE Trans. Microwave Theory Tech. 42, 644–653 (1994). [CrossRef]

]. The structure of the impedance matrix is as follows

[Z]=[[ZHin][ZHout][ZEin][ZEout]],
(4)

where the entries of [Z in H] are the magnetic fields at the testing points due to current filaments of unit amplitude that lie inside the cylinder and those of [Z out H] are the magnetic fields at the testing points due to current filaments that lie outside the cylinder. The sub-matrices [Z in E] and [Z out E] are analogous to [Z in H] and [Z out H], except that their entries are the tangential electric fields (divided by η 0) at the testing points.

The entries of the sub-matrices [Z out H] and [Z out E] are easy to evaluate as they can be obtained from the 2D Green’s function

Gout=14jH0(2)[kc(xx0)2+(yy0)2],
(5)

where H (2)0(·) denotes the Hankel function of the second kind and zero order and kc=εck0 is the wave number inside the cylinder, with k 0 being the free-space wave number. In (5), the filament coordinates are denoted by (x 0,y 0) and the testing point coordinates are denoted by (x,y). As a function of G out, the fields are given by

Hz=jkcηcGout
(6)
Ex=Gouty
(7)
Ey=Goutx,
(8)

where ηc=μ0εc is the characteristic impedance inside the cylinder. The entries of the sub-matrices [Z in H] and [Z in E] are given in terms of the PPGF, as detailed in Section 4.

4. Proper Periodic Green’s Function (PPGF)

The PPGF for a z invariant geometry is defined as the solution to the following boundary-value problem

(2+k02)Gin=δ(xx0)δ(yy0),
subjectto:
Ginx=L2=ejkxLGinx=L2,
Radiationconditionfory±.
(9)

This boundary-value problem, depicted in Fig. 4, is a standard Sturm-Liouville problem, which leads to a well-known spectral representation of the periodic Green’s function. The case of complex kx, however, has not received much attention. The solution of the boundary-value problem is given by (see for example [28

28. R. Petit, Electromagnetic Theory of Gratings, Springer-Verlag, (1980).

, p. 23])

Gin=12jLΣn=1kynexp{j[kxn(xx0)+kynyy0]}
(10)

where

kxn=kx+2πnL
(11)
kyn=k02kxn2.
(12)

The function G in has a denumerably infinite number of square-root-type branch points that occur at the propagation constants

kx=±k0+2πnL
(13)

corresponding to Wood’s anomalies. For each of the square-roots in (12), a choice of branch and branch-cut must be made. To comply with the radiation conditions at y→±∞, we must have, for kyn purely real, Re(kyn)≥0, and Im(kyn)<0 otherwise. This implies using the proper Riemann sheet with branch-cuts as shown in Fig. 5. On this Riemann sheet, Gin is periodic in kx with a period of 2π/L. Consequently, we can assume that kx is in the First Brillouin Zone (FBZ), i.e., between −π/L and π/L.

Fig. 4. Boundary-value problem that defines the PPGF.

The fields of a unit-amplitude magnetic current filament, operating in the presence of the periodic boundary and radiation conditions, can be readily obtained from the PPGF. We have

Hz=jk0η0Gin
(14)
Ex=Giny
(15)
Ey=Ginx.
(16)

where η0=μ0ε0 denotes the characteristic impedance of free-space. This current may source or sink power depending on the parameters of the boundary-value problem (9).

Fig. 5. Branch-cuts for Im(kyn)<0. In this figure, k 0<π/L.

The series (10) converges everywhere except at x=x 0,y=y 0, and when yy 0 the convergence is exponential. When yy 0, however, the rate of convergence is slow. This is a well-known problem of periodic Green’s functions, for which a number of solutions have been proposed [31

31. A. Peterson, L. Scott, and R. Mittra, Computational Methods for Electromagnetics, IEEE Press, (1998).

]. When the SMT is used, this problem has a simple solution which was proposed in [9

9. A. Boag, Y. Leviatan, and A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Rad. Sci. 23, 612–624 (1988). [CrossRef]

]. The idea is to replace the current filaments with current strips of magnetic current density

Jmz=f(xx0)δ(yy0)
(17)

where f(x) is a smooth, real-valued window function that is zero outside the interval [−s/2, s/2], with s being the width of the strip. The Hz field (from which Ex and Ey can be derived) due to such a current strip is given by

Hzstrip=k02η0Σn=f̂nkynej[kxn(xx0)+kynyy0]
(18)

where the Fourier coefficients, f̂n, are given by

f̂n=1LL2L2f(x)ej2πnxLdx.
(19)

By choosing f(x) to have rapidly decaying Fourier coefficients the evaluation of the fields can be accelerated. In this paper, we used the Blackman-Harris [32

32. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978). [CrossRef]

] window, which was used in [17

17. A. Boag, Y. Leviatan, and A. Boag, “Analysis of electromagnetic scattering from linear periodic arrays of perfectly conducting bodies using a cylindrical-current model,” IEEE Trans. Antennas Propag. 39, 1332–1337 (1991). [CrossRef]

] and is given by

f(x)=0.35875+0.48829cos(2πxs)+0.14128cos(4πxs)+0.01168cos(6πxs).
(20)

The width of the strip s should be as large as possible, since a larger width increases the rate of decay. Of course, the strip should be small enough so as not to intersect or even approach the boundary. We took the strip width to be a quarter of the distance from the center of the strip to the boundary.

5. Determination of modal solutions

To filter out the spurious solutions, we use the measure of singularity proposed in [7

7. A. Hochman and Y. Leviatan, “Efficient and spurious-free integral-equation-based optical waveguide mode solver,” Opt. Express 15, 14,431–14,453 (2007). [CrossRef]

]. This measure of singularity is based on the normalized error ΔE(K⃗), which is defined as the absolute error normalized to the norm of the fields at the testing points. The idea in [7

7. A. Hochman and Y. Leviatan, “Efficient and spurious-free integral-equation-based optical waveguide mode solver,” Opt. Express 15, 14,431–14,453 (2007). [CrossRef]

] is to exploit the fact that ΔE(K⃗) is small for a true solution and large for a spurious solution, and to devise a measure of singularity which, when applied to [Z], is large if and only if there exists some solution vector K⃗ for which ΔE(K⃗) is small. We are therefore interested in finding the smallest ΔE(K⃗) that can be obtained for a given matrix [Z]. Once this smallest ΔE(K⃗) is obtained, we use its reciprocal as a measure of the singularity of [Z]. Fortunately, finding the smallest ΔE(K⃗) is not significantly more difficult than evaluating the condition number of [Z]. The smallest ΔE(K⃗) can be written as

minK[ΔE(K)]=minK[[Z]K2[Z˜]K2],
(21)

where [Z̃] is a matrix that maps the current amplitude vector K⃗ to tangential field values at the testing points. The matrix [Z̃] can be readily obtained from the matrix [Z], which maps K⃗ to the difference of the tangential fields on both sides of the media boundaries, by reversing the signs of half of its entries, and dividing by two. For TE fields, such as the ones considered in this paper, the matrix [Z̃] can be written as

[Z˜]=12[[ZHin][ZHout]].
(22)

This choice differs from that used in [7

7. A. Hochman and Y. Leviatan, “Efficient and spurious-free integral-equation-based optical waveguide mode solver,” Opt. Express 15, 14,431–14,453 (2007). [CrossRef]

], where the fields were hybrid (neither TE nor TM). The significance of this choice is that [Z̃]K⃗ is a vector of the magnetic field values at the testing points, and it is the norm of this vector that is used to normalize the absolute error. Only the magnetic field is used because, for TE fields, the electric field due to a spurious solution is significantly larger than the magnetic field multiplied by the characteristic impedance of the medium in which the sources operate. Hence, if the electric field were included in the denominator of the expression for ΔE(K⃗), this denominator would not be small for a spurious solution, and consequently, the rejection of the spurious solutions would be less reliable.

As mentioned above, we are interested in finding the solution to (21), namely, the smallest ΔE(K⃗) that can be obtained for a given [Z]. It can be shown [33

33. R. Bellman, Introduction to matrix analysis, McGraw-Hill, (1970).

, Ch. 7] that the smallest ΔE(K⃗) is the square-root of the smallest generalized eigenvalue of the following generalized eigenvalue problem

[Z][Z]=ξ[Z˜][Z˜],
(23)

where the dagger sign indicates conjugate-transpose. Denoting the smallest generalized eigenvalue by ξmin, the smallest ΔE(K⃗) is simply ξmin,, and the measure of singularity is 1ξmin..

6. Numerical results

To demonstrate the capabilities of the proposed modeling scheme, a few numerical results are given. Dispersion curves for a NC of circular cylinders, of radius R=50 nm and period L=120 nm, are shown in Fig. 6. The parameter varied in these curves is λ, and the values shown are the real and imaginary parts of the effective refractive index n eff=kx/k 0. In this and all numerical examples, the cylinders are assumed to be made of silver, which is characterized by a Drude-model permittivity function given by [34

34. E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, “Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures,” J. Opt. Soc. Am. A 19, 101–111 (2002). [CrossRef]

]

εc=1jτωp2ω(1+jτω)
(24)

Fig. 6. Real, (a), and imaginary, (b), parts of neff as function of L/λ (only λ is varied, while L is kept fixed). Here, R=50 nm and L=120 nm.

A few mode profiles, corresponding to four points on the dispersion curves of Fig. 6, are shown in Fig. 7. These points are marked A-D in Fig. 6; to avoid clutter, the points are marked only in Fig. 6(a). In the SMT, the solutions obey Maxwell’s equations exactly and the continuity conditions approximately. Therefore, the continuity of Re(Hz), which graphically appears perfect in Fig. 7, attests to the validity of the solutions. As could be anticipated from the symmetry of the NC, the modes are either odd or even with respect to y. The attenuation is not apparent in these short segments of the NC, but it can be observed in a longer section shown in Fig. 8. Also evident in Fig. 8, is that the x-directed time-average power flow density assumes negative values inside the cylinders. This is a well-known phenomenon in plasmonic waveguides [36

36. B. Prade and J. Y. Vinet, “Guided optical waves in fibers with negative dielectric constant,” J. Lightwave Technol. 12, 6–18 (1994). [CrossRef]

].

Fig. 7. Mode profiles for points A-D in Fig. 6 are shown in (a)–(d), respectively. The value shown is Re(Hz) and the arrows represent the time average power flow density. The modes have been normalized to have max|Hz|=1 A/m

The last structure considered is a NC shown in the inset in Fig. 11. The cylinders in this structure are of a more general cross-section defined by hypotrochoid curves, which have the following parametric representation

x(ϕ)=Rb4v+1[4vcos(ϕ)cos(2ϕ)]
(25)
y(ϕ)=Rb4v+1[4vsin(ϕ)+sin(2ϕ)]
(26)

where ϕ is a parametric variable ranging from 0 to 2π, and Rb is the radius of a circle that tightly bounds the curve, which is a hypotrochoid. For ν≥1, the hypotrochoid resembles a triangle with rounded corners, and the radius of curvature of the rounded corners increases with increasing ν. Modeling a sharp triangular shape would require careful placement of the sources near the corners, such as was done in [37

37. S. Eisler and Y. Leviatan, “Analysis of electromagnetic scattering from metallic and penetrable cylinders with edges using a multifilament current model,” IEE-Proc. H 136, 431–438 (1989).

] or [34

34. E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, “Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures,” J. Opt. Soc. Am. A 19, 101–111 (2002). [CrossRef]

]. Hence, we used ν=2 to obtain a shape which is triangular but not too sharp at the corners.

Fig. 8. Attenuation of x-directed time-average power flow density. The effective index is n eff=1.1228-0.0054 j, corresponding to point B in Fig. 6.
Fig. 9. Real, (a), and imaginary, (b), parts of n eff as function of L/λ (only λ is varied, while L is kept fixed). Here, R=40 nm and L=120 nm.
Fig. 10. Mode profiles for points A and B in Fig. 9 are shown in (a) and (b), respectively. The value shown is Re(Hz) and the arrows represent the time average power flow density. The modes have been normalized to have max|Hz|=1 A/m
Fig. 11. Real, (a), and imaginary, (b), parts of n eff as function of L/λ (only λ is varied, while L is kept fixed). Here, Rb=50 nm, ν=2, and L=120 nm.

The dispersion curves of this NC are quite similar to those of the circular cylinder NC (shown in Fig. 6). The mode profiles at points A and B, shown in Fig. 12, are also quite similar to their counterparts in Fig. 7. It may be interesting to look at the influence of the shape of the NC on some parameters of interest, such as the real and imaginary parts of the effective indices. In Fig. 13, we show the difference, Δn effneffn eff, between the effective indices of the fundamental modes of the hypotrochoidal-cylinder NC of Fig. 11 and a circular-cylinder NC. To try and isolate the effect of the shape from the effect of the size of the cylinders, we choose them to have the same circumference. As can be inferred from Fig. 13, the attenuation rates can differ quite significantly, while the phase velocities differ very little. At longer wavelengths, the fundamental mode of the circular-cylinder NC is attenuated slightly more rapidly than the fundamental mode of the hypotrochoidal-cylinder NC, while at shorter wavelengths it is the other way around. The difference becomes quite significant as the edge of the FBZ is approached.

Fig. 12. Mode profiles for points A and B in Fig. 11 are shown in (a) and (b), respectively. The value shown is Re(Hz) and the arrows represent the time average power flow density. The modes have been normalized to have max|Hz|=1 A/m
Fig. 13. Percent of difference in the real and imaginary parts of the propagation constants of the fundamental modes of two structures. One is the NC with hypotrochoidal cylinders of Fig. 11, and the other is a NC with circular cylinders, of radius R≈45 nm, that have the same circumference as the hypotrochoids. For both structures, L=120 nm.

6.1. Accuracy and computational resources

The results of the calculation with increasing N are shown in Table. 6.1, where T is the computation time for a MATLAB implementation running on a 3GHz PC. The number of sources

Table 1. Convergence of neff with N

table-icon
View This Table

is doubled from one line in the table to the next, and ΔE is seen to decrease by about an order of magnitude. The computation time increases at first linearly, and for N=40, a bit more sharply. Lastly, the error in continuity conditions appears to be a reliable indicator of the accuracy of the solution. Throughout this work, it was ensured that the error ΔE did not exceed 2%. This required N=20 for the analysis of the circular-cylinder NC, N=30 for the NC pair, and N=40 for the hypotrochoidal-cylinder NC. The Green’s function summations in (10) were truncated when the partial sum was at least 105 times greater (in absolute value) than the difference between consecutive terms.

7. Summary

Acknowledgements

We would like to thank the anonymous reviewers for their valuable comments and suggestions. This work was supported in part by the Israel Science Foundation and the Russell Berrie Nanotechnology Institute (NTU-Technion grant). AH gratefully acknowledges the financial support of the Andrew and Erna Finci Viterbi Foundation.

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23.

D. Maystre, M. Saillard, and G. Tayeb, “Special methods of wave diffraction” in Scattering, P. Sabatier and E.R. Pike, Academic Press, (2001).

24.

G. Fairweather and A. Karageorghis, “The method of fundamental solutions for elliptic boundary value problems,” Advances in Computational Mathematics 9(1), 69–95 (1998). [CrossRef]

25.

V. D. Kupradze, “About approximate solutions of a mathematical physics problem,” Success of Mathematical Sciences 22(2), 59–107 (1967).

26.

I. N. Vekua, Reports of the Academy of Science of the USSR 44(6), 901–909 (1953).

27.

D. I. Kaklamani and H. T. Anastassiu, “Aspects of the Method of Auxiliary Sources (MAS) in computational electromagnetics,” IEEE Antennas and Propag. Mag. 44, 48–64 (2002). [CrossRef]

28.

R. Petit, Electromagnetic Theory of Gratings, Springer-Verlag, (1980).

29.

M. Szpulak, W. Urbanczyk, E. Serebryannikov, A. Zheltikov, A. Hochman, Y. Leviatan, R. Kotynski, and K. Panajotov, “Comparison of different methods for rigorous modeling of photonic crystal fibers,” Opt. Express 14, 5699–5714 (2006). [CrossRef] [PubMed]

30.

W. Schroeder and I. Wolff, “The origin of spurious modes in numerical solutions of electromagnetic field eigenvalue problems,” IEEE Trans. Microwave Theory Tech. 42, 644–653 (1994). [CrossRef]

31.

A. Peterson, L. Scott, and R. Mittra, Computational Methods for Electromagnetics, IEEE Press, (1998).

32.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978). [CrossRef]

33.

R. Bellman, Introduction to matrix analysis, McGraw-Hill, (1970).

34.

E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, “Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures,” J. Opt. Soc. Am. A 19, 101–111 (2002). [CrossRef]

35.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B 69, 195,111 (2004). [CrossRef]

36.

B. Prade and J. Y. Vinet, “Guided optical waves in fibers with negative dielectric constant,” J. Lightwave Technol. 12, 6–18 (1994). [CrossRef]

37.

S. Eisler and Y. Leviatan, “Analysis of electromagnetic scattering from metallic and penetrable cylinders with edges using a multifilament current model,” IEE-Proc. H 136, 431–438 (1989).

OCIS Codes
(230.7370) Optical devices : Waveguides
(240.6680) Optics at surfaces : Surface plasmons
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: April 2, 2009
Revised Manuscript: May 26, 2009
Manuscript Accepted: June 7, 2009
Published: July 23, 2009

Citation
Amit Hochman and Yehuda Leviatan, "Rigorous modal analysis of metallic nanowire chains," Opt. Express 17, 13561-13575 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13561


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References

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  24. G. Fairweather and A. Karageorghis, "The method of fundamental solutions for elliptic boundary value problems," Advances in Computational Mathematics 9(1), 69-95 (1998). [CrossRef]
  25. V. D. Kupradze, "About approximate solutions of a mathematical physics problem," Success Math. Sci. 22(2), 59-107 (1967).
  26. I. N. Vekua, Reports of the Academy of Science of the USSR 44(6), 901-909 (1953).
  27. D. I. Kaklamani and H. T. Anastassiu, "Aspects of the Method of Auxiliary Sources (MAS) in computational electromagnetics," IEEE Antennas Propag. Mag. 44, 48-64 (2002). [CrossRef]
  28. R. Petit, Electromagnetic Theory of Gratings, (Springer-Verlag, 1980).
  29. M. Szpulak, W. Urbanczyk, E. Serebryannikov, A. Zheltikov, A. Hochman, Y. Leviatan, R. Kotynski, and K. Panajotov, "Comparison of different methods for rigorous modeling of photonic crystal fibers," Opt. Express 14, 5699-5714 (2006). [CrossRef] [PubMed]
  30. W. Schroeder and I. Wolff, "The origin of spurious modes in numerical solutions of electromagnetic field eigenvalue problems," IEEE Trans. Microwave Theory Tech. 42, 644-653 (1994). [CrossRef]
  31. A. Peterson, L. Scott, and R. Mittra, Computational Methods for Electromagnetics, (IEEE Press, 1998).
  32. F. J. Harris, "On the use of windows for harmonic analysis with the discrete Fourier transform," Proc. IEEE 66, 51-83 (1978). [CrossRef]
  33. R. Bellman, Introduction to matrix analysis, (McGraw-Hill, 1970).
  34. E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, "Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures," J. Opt. Soc. Am. A 19, 101-111 (2002). [CrossRef]
  35. K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195,111 (2004). [CrossRef]
  36. B. Prade and J. Y. Vinet, "Guided optical waves in fibers with negative dielectric constant," J. Lightwave Technol. 12, 6-18 (1994). [CrossRef]
  37. S. Eisler and Y. Leviatan, "Analysis of electromagnetic scattering from metallic and penetrable cylinders with edges using a multifilament current model," IEE-Proc. H 136, 431-438 (1989)

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