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

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2008 (2)

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]

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]

2007 (1)

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]

2006 (5)

P. Berini, "Figures of merit for surface plasmon waveguides," Opt. Express 14, 13,030-13,042 (2006).
[CrossRef]

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]

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

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]

2005 (2)

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]

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]

2004 (5)

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]

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

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]

G. Tayeb and S. Enoch, "Combined fictitious-sources-scattering-matrix method," J. Opt. Soc. Am. A 21(8), 1417-1423 (2004).
[CrossRef]

2003 (1)

2002 (2)

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]

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]

1998 (2)

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]

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]

1994 (3)

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]

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

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]

1991 (1)

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]

1988 (2)

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 two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model," Rad. Sci. 23, 612-624 (1988).
[CrossRef]

1986 (1)

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]

1978 (1)

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

1967 (1)

V. D. Kupradze, "About approximate solutions of a mathematical physics problem," Success Math. Sci. 22(2), 59-107 (1967).

1953 (1)

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

Alu, A.

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]

Anastassiu, H. T.

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]

Aussenegg, F.

Berini, P.

P. Berini, "Figures of merit for surface plasmon waveguides," Opt. Express 14, 13,030-13,042 (2006).
[CrossRef]

Bienstman, P.

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]

Boag, A.

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. 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. 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]

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]

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]

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]

Bucci, O. M.

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]

Burke, J.

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]

Capolino, F.

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]

Chu, H.

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]

Citrin, D.

Crozier, K.

D’Elia, G.

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]

Durant, S.

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]

Engheta, N.

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]

Enoch, S.

Erni, D.

Ewe, W.

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]

Fairweather, G.

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]

Fan, S.

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]

Ford, G.

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

Hafner, C.

Harris, F. J.

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

Hochman, A.

Huang, K. C.

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]

Jackson, D. R.

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]

Jiang, X.

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]

Joannopoulos, J. D.

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]

Kaklamani, D. I.

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]

Karageorghis, A.

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]

Koenderink, A. F.

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]

Koh, W.

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]

Kotynski, R.

Krenn, J.

Kupradze, V. D.

V. D. Kupradze, "About approximate solutions of a mathematical physics problem," Success Math. Sci. 22(2), 59-107 (1967).

Leitner, A.

Leviatan, Y.

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]

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]

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]

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. 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. 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]

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]

Li, E.

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]

Lidorikis, E.

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]

Ludwig, A.

Moreno, E.

Nelson, K. A.

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]

Novotny, L.

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]

Panajotov, K.

Polman, A.

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]

Prade, B.

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

Quinten, M.

Santojanni, M.

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]

Schroeder, W.

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]

Serebryannikov, E.

Stegeman, G.

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]

Su, K. H.

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]

Szpulak, M.

Tamir, T.

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]

Tayeb, G.

Urbanczyk, W.

Vahldieck, R.

Vekua, I. N.

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

Vinet, J. Y.

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

Weber, W.

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

Wei, Q. H.

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]

Wilton, D. R.

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Figures (13)

Fig. 1.
Fig. 1.

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

Fig. 2.
Fig. 2.

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

Fig. 3.
Fig. 3.

Magnetic current filaments used to approximate the fields inside of the cylinder.

Fig. 4.
Fig. 4.

Boundary-value problem that defines the PPGF.

Fig. 5.
Fig. 5.

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

Fig. 6.
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.

Fig. 7.
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

Fig. 8.
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.
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.
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.
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.

Fig. 12.
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.
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.

Tables (1)

Tables Icon

Table 1. Convergence of neff with N

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

F(x)=F(xL)ejkxL,
F(L2)=F(L2)ejkxL,
[Z(kx,ω)]K=0,
[Z]=[[ZHin][ZHout][ZEin][ZEout]],
Gout=14jH0(2)[kc(xx0)2+(yy0)2],
Hz=jkcηcGout
Ex=Gouty
Ey=Goutx,
(2+k02)Gin=δ(xx0)δ(yy0),
subject to :
Ginx=L2=ejkxLGinx=L2,
Radiation condition for y ± .
Gin=12jL Σn= 1kyn exp {j[kxn(xx0)+kynyy0]}
kxn=kx+2πnL
kyn=k02kxn2.
kx=±k0+2πnL
Hz=jk0η0Gin
Ex=Giny
Ey=Ginx.
Jmz=f(xx0)δ(yy0)
Hzstrip=k02η0Σn=f̂nkyn ej[kxn(xx0)+kynyy0]
f̂n=1LL2L2f(x)ej2πnxLdx.
f(x)=0.35875+0.48829cos(2πxs)+0.14128cos(4πxs)+0.01168cos(6πxs).
minK[ΔE(K)]=minK[[Z]K2[Z˜]K2],
[Z˜]=12 [[ZHin][ZHout]] .
[Z][Z]=ξ [Z˜] [Z˜],
εc=1jτωp2ω(1+jτω)
x(ϕ)=Rb4v+1[4vcos(ϕ)cos(2ϕ)]
y(ϕ)=Rb4v+1[4vsin(ϕ)+sin(2ϕ)]

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