## Rigorous modal analysis of metallic nanowire chains

Optics Express, Vol. 17, Issue 16, pp. 13561-13575 (2009)

http://dx.doi.org/10.1364/OE.17.013561

Acrobat PDF (521 KB)

### 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

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]

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

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

3. F. Capolino, D. R. Jackson, and D. R. Wilton, “Fundamental properties of the field at the interface between air and a periodic artificial material excited by a line source,” IEEE Trans. Antennas Propag. **53**, 91–99 (2005). [CrossRef]

5. J. Burke, G. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B **33**, 5186–5201 (1986). [CrossRef]

6. L. Novotny and C. Hafner, “Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function,” Phys. Rev. E **50**, 4094–4106 (1994). [CrossRef]

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]

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]

3. F. Capolino, D. R. Jackson, and D. R. Wilton, “Fundamental properties of the field at the interface between air and a periodic artificial material excited by a line source,” IEEE Trans. Antennas Propag. **53**, 91–99 (2005). [CrossRef]

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]

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]

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

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. 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]

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. 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]

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]

## 2. Problem specification

*ε*and oriented parallel to the

_{c}*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*) is any of the field components and

*k*is a propagation constant. As the modes are proper, they must obey the radiation condition. This means that for

_{x}*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, p. 54]). For a given

*ω*, modes will exist only for certain values of

*k*. The determination of the (

_{x}*k*) pairs for which a mode exists is a central part of the mode determination process.

_{x},ω*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.

## 3. Source-Model Technique (SMT)

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]

*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]

*Z*] is an 2

*M*×2

*N*impedance matrix and

*K*⃗ 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 2

*M*equations. The number of current filaments is 2

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

*π/N*times the length of the boundary curves.

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]

*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.

*Z*

^{out}

_{H}] and [

*Z*

^{out}

_{E}] are easy to evaluate as they can be obtained from the 2D Green’s function

*H*

^{(2)}0(·) denotes the Hankel function of the second kind and zero order and

*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

*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)

*z*invariant geometry is defined as the solution to the following boundary-value problem

*k*, however, has not received much attention. The solution of the boundary-value problem is given by (see for example [28, p. 23])

_{x}*G*

_{in}has a denumerably infinite number of square-root-type branch points that occur at the propagation constants

*y*→±∞, we must have, for

*k*purely real, Re(

_{yn}*k*)≥0, and Im(

_{yn}*k*)<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

_{yn}*k*with a period of 2

_{x}*π/L*. Consequently, we can assume that

*k*is in the First Brillouin Zone (FBZ), i.e., between −

_{x}*π/L*and

*π/L*.

*x*=

*x*

_{0},

*y*=

*y*

_{0}, and when

*y*≠

*y*

_{0}the convergence is exponential. When

*y*≈

*y*

_{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]. 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]

*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

*H*field (from which

_{z}*E*and

_{x}*E*can be derived) due to such a current strip is given by

_{y}*f*̂

_{n}, are given by

*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]

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]

*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

*k*and

_{x}*k*

_{0}for which a non-trivial solution to (3) exists. As [

*Z*] is a non-square matrix, we cannot use its determinant to test whether a solution exists for a given pair of

*k*and

_{x}*k*

_{0}. Instead, the condition number of [

*Z*] could be used and it should be infinite for a solution to exist. In practice, as the solutions are always approximate, the condition number is never infinite. Rather, it is a local maximum of the condition number as a function of

*k*and

_{x}*k*

_{0}that indicates the existence of an approximate solution. So, a naive method for determining the modes would be to search for the maxima of the condition number. However, as explained 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]

*K*⃗=[1,-1,1,-1, …]

^{T}, where

*T*denotes transposition. These amplitudes are used to form approximations of the fields for each homogeneous region of the unit cell. When two such approximations corresponding to neighboring regions are evaluated at the testing points, which are some distance away from the sources, it is found that the fields of both approximations are very small. Hence, the absolute error in the continuity conditions associated with such a solution is small and this is manifested by a high condition number. Nevertheless, this solution is not a true solution because the absolute error in continuity conditions is not small compared with the values of the fields at the testing points.

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]

*E*(

*K*⃗), which is defined as the absolute error normalized to the norm of the fields at the testing points. The idea in [7

**15**, 14,431–14,453 (2007). [CrossRef]

*E*(

*K*⃗) 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

*K*⃗ for which Δ

*E*(

*K*⃗) is small. We are therefore interested in finding the smallest Δ

*E*(

*K*⃗) that can be obtained for a given matrix [

*Z*]. Once this smallest Δ

*E*(

*K*⃗) is obtained, we use its reciprocal as a measure of the singularity of [

*Z*]. Fortunately, finding the smallest Δ

*E*(

*K*⃗) is not significantly more difficult than evaluating the condition number of [

*Z*]. The smallest Δ

*E*(

*K*⃗) can be written as

*Z*̃] is a matrix that maps the current amplitude vector

*K*⃗ to tangential field values at the testing points. The matrix [

*Z*̃] can be readily obtained from the matrix [

*Z*], which maps

*K*⃗ 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 [

*Z*̃] can be written as

**15**, 14,431–14,453 (2007). [CrossRef]

*Z*̃]

*K*⃗ 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 Δ

*E*(

*K*⃗), this denominator would not be small for a spurious solution, and consequently, the rejection of the spurious solutions would be less reliable.

*E*(

*K*⃗) that can be obtained for a given [

*Z*]. It can be shown [33, Ch. 7] that the smallest Δ

*E*(

*K*⃗) is the square-root of the smallest generalized eigenvalue of the following generalized eigenvalue problem

_{min}, the smallest Δ

*E*(

*K*⃗) is simply

*k*

_{0}and

*k*for which a mode exists, we search for the local maxima of

_{x}**15**, 14,431–14,453 (2007). [CrossRef]

## 6. Numerical results

*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}=

*k*

_{x}/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]

*H*), 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

_{z}*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]

*L*=120 nm. The dispersion curves for the NC pair are shown in Fig. 9. The

*H*of the fundamental mode, shown in Fig. 10(a), is an even function of y, while the next mode, shown in Fig. 10(b), is an odd function of

_{z}*y*. In the case of the even mode, the power flows mostly in the air region between the NCs, whereas in the odd mode, the power flows mostly in the exterior air region.

*ϕ*is a parametric variable ranging from 0 to 2

*π*, and

*R*is the radius of a circle that tightly bounds the curve, which is a hypotrochoid. For

_{b}*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] 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]

*ν*=2 to obtain a shape which is triangular but not too sharp at the corners.

*n*

_{eff}≜

*n*∘

_{eff}−

*n*

^{▷}

_{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.

### 6.1. Accuracy and computational resources

*N*. The calculation involves searching for the complex effective index that yields the smallest error in the continuity conditions, where each evaluation of this error function requires the formation of the matrix [

*Z*] and the solution of the generalized eigenvalue problem (23). The search begins with an adaptive sampling of the error function at real values of

*n*

_{eff}ranging from 1 to

*λ*/(2

*L*). Once a minimum is found, a small region of the complex plane near this minimum is searched by a standard optimization algorithm (see [7

**15**, 14,431–14,453 (2007). [CrossRef]

*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

*E*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 Δ

*E*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 10

^{5}times greater (in absolute value) than the difference between consecutive terms.

## 7. Summary

## Acknowledgements

## References and links

1. | M. Quinten, A. Leitner, J. Krenn, and F. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. |

2. | P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express |

3. | F. Capolino, D. R. Jackson, and D. R. Wilton, “Fundamental properties of the field at the interface between air and a periodic artificial material excited by a line source,” IEEE Trans. Antennas Propag. |

4. | R. E. Collin and F. J. Zucker, |

5. | J. Burke, G. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B |

6. | L. Novotny and C. Hafner, “Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function,” Phys. Rev. E |

7. | A. Hochman and Y. Leviatan, “Efficient and spurious-free integral-equation-based optical waveguide mode solver,” Opt. Express |

8. | W. Weber and G. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B |

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. |

10. | D. Citrin, “Plasmon-polariton transport in metal-nanoparticle chains embedded in a gain medium,” Opt. Lett. |

11. | A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B |

12. | A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B |

13. | Q. H. Wei, K. H. Su, S. Durant, and X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. |

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 |

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. |

16. | Y. Leviatan, A. Boag, and A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies-theory and numerical solution,” IEEE Trans. Antennas Propag. |

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. |

18. | A. Ludwig and Y. Leviatan, “Analysis of bandgap characteristics of two-dimensional periodic structures by using the source-model technique,” J. Opt. Soc. Am. A |

19. | A. Hochman and Y. Leviatan, “Analysis of strictly bound modes in photonic crystal fibers by use of a source-model technique,” J. Opt. Soc. Am. A |

20. | C. Hafner, |

21. | O. M. Bucci, G. D’Elia, and M. Santojanni, “Non-redundant implementation of method of auxiliary sources for smooth 2D geometries,” Electronics Letters |

22. | G. Tayeb and S. Enoch, “Combined fictitious-sources-scattering-matrix method,” J. Opt. Soc. Am. A |

23. | D. Maystre, M. Saillard, and G. Tayeb, “Special methods of wave diffraction” in |

24. | G. Fairweather and A. Karageorghis, “The method of fundamental solutions for elliptic boundary value problems,” Advances in Computational Mathematics |

25. | V. D. Kupradze, “About approximate solutions of a mathematical physics problem,” Success of Mathematical Sciences |

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. |

28. | R. Petit, |

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 |

30. | W. Schroeder and I. Wolff, “The origin of spurious modes in numerical solutions of electromagnetic field eigenvalue problems,” IEEE Trans. Microwave Theory Tech. |

31. | A. Peterson, L. Scott, and R. Mittra, |

32. | F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE |

33. | R. Bellman, |

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 |

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 |

36. | B. Prade and J. Y. Vinet, “Guided optical waves in fibers with negative dielectric constant,” J. Lightwave Technol. |

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 |

**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

- 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]
- P. Berini, "Figures of merit for surface plasmon waveguides," Opt. Express 14, 13,030-13,042 (2006). [CrossRef]
- F. Capolino, D. R. Jackson, and D. R. Wilton, "Fundamental properties of the field at the interface between air and a periodic artificial material excited by a line source," IEEE Trans. Antennas Propag. 53, 91-99 (2005). [CrossRef]
- R. E. Collin and F. J. Zucker, Antenna Theory, Part II, (McGraw-Hill, 1969).
- J. Burke, G. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986). [CrossRef]
- L. Novotny and C. Hafner, "Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function," Phys. Rev. E 50, 4094-4106 (1994). [CrossRef]
- 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]
- W. Weber and G. Ford, "Propagation of optical excitations by dipolar interactions in metal nanoparticle chains," Phys. Rev. B 70, 125,429 (2004). [CrossRef]
- 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]
- D. Citrin, "Plasmon-polariton transport in metal-nanoparticle chains embedded in a gain medium," Opt. Lett. 31, 98-100 (2006). [CrossRef] [PubMed]
- 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]
- A. Alu and N. Engheta, "Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines," Phys. Rev. B 74, 205,436 (2006). [CrossRef]
- 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]
- 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]
- 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]
- Y. Leviatan, A. Boag, and A. Boag, "Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies-theory and numerical solution," IEEE Trans. Antennas Propag. 36, 1722-1734 (1988). [CrossRef]
- 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]
- A. Ludwig and Y. Leviatan, "Analysis of bandgap characteristics of two-dimensional periodic structures by using the source-model technique," J. Opt. Soc. Am. A 20, 1553-1562 (2003). [CrossRef]
- A. Hochman and Y. Leviatan, "Analysis of strictly bound modes in photonic crystal fibers by use of a sourcemodel technique," J. Opt. Soc. Am. A 21, 1073-1081 (2004). [CrossRef]
- C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics, (Artech House, 1990).
- O. M. Bucci, G. D’Elia, and M. Santojanni, "Non-redundant implementation of method of auxiliary sources for smooth 2D geometries," Electronics Letters 41(22), 1203-1205 (2005). [CrossRef]
- G. Tayeb and S. Enoch, "Combined fictitious-sources-scattering-matrix method," J. Opt. Soc. Am. A 21(8), 1417-1423 (2004). [CrossRef]
- D. Maystre, M. Saillard, and G. Tayeb, "Special methods of wave diffraction" in Scattering, P. Sabatier and E. R. Pike, eds., (Academic Press, 2001).
- 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]
- V. D. Kupradze, "About approximate solutions of a mathematical physics problem," Success Math. Sci. 22(2), 59-107 (1967).
- I. N. Vekua, Reports of the Academy of Science of the USSR 44(6), 901-909 (1953).
- 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]
- R. Petit, Electromagnetic Theory of Gratings, (Springer-Verlag, 1980).
- 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]
- 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]
- A. Peterson, L. Scott, and R. Mittra, Computational Methods for Electromagnetics, (IEEE Press, 1998).
- F. J. Harris, "On the use of windows for harmonic analysis with the discrete Fourier transform," Proc. IEEE 66, 51-83 (1978). [CrossRef]
- R. Bellman, Introduction to matrix analysis, (McGraw-Hill, 1970).
- 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]
- 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]
- B. Prade and J. Y. Vinet, "Guided optical waves in fibers with negative dielectric constant," J. Lightwave Technol. 12, 6-18 (1994). [CrossRef]
- 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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