Abstract

Two-dimensional periodic arrays of noble metal nanospheres support a variety of optical phenomena, including bound and leaky modes of several types. The scope of this paper is the characterization of the modal dispersion diagrams of planar arrays of silver nanospheres, with the ability to follow individual modal evolutions. The metal spherical nanoparticles are described using the single dipole approximation technique by including all the retarded dynamic field terms. Polarizability of the nanospheres is provided by the Mie theory. Dispersion diagrams for both physical and nonphysical modes are shown for a square lattice of Ag nanospheres for the case of lossless and lossy metal particles, with dipole moments polarized along the x, y, and z directions. Though an array with one set of parameters has been studied, the analysis method and classification are general. The evolution of modes through different Riemann sheets and analysis of guidance and radiation are studied in detail.

© 2011 Optical Society of America

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    [Crossref]

2009 (4)

Y. Saito and P. Verma, “Imaging and spectroscopy through plasmonic nano-probe,” Eur. Phys. J. Appl. Phys. 46, 20101 (2009).
[Crossref]

C. Mateo-Segura, C. R. Simovski, G. Goussetis, and S. Tretyakov, “Subwavelength resolution for horizontal and vertical polarization by coupled arrays of oblate nanoellipsoids,” Opt. Lett. 34, 2333–2335 (2009).
[Crossref] [PubMed]

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

R. Rodríguez-Berral, F. Mesa, P. Baccarelli, and P. Burghignoli, “Excitation of a periodic microstrip line by an aperiodic delta-gap source,” IEEE Antennas Wirel. Propag. Lett. 8, 641–644 (2009).
[Crossref]

2008 (5)

F. T. Celepcikay, D. R. Wilton, D. R. Jackson, and F. Capolino, “Choosing splitting parameters and summation limits in the numerical evaluation of 1-D and 2-D periodic Green’s functions using the Ewald method,” Radio Sci. 43, RS6S01(2008).
[Crossref]

Y. R. Zhen, K. H. Fung, and C. T. Chan, “Collective plasmonic modes in two-dimensional periodic arrays of metal nanoparticles,” Phys. Rev. B 78, 035419 (2008).
[Crossref]

C. Li, Y. Liu, L. Li, Z. Du, S. Xu, M. Zhang, X. Yin, and T. Wang, “A novel amperometric biosensor based on NiO hollow nanospheres for biosensing glucose,” Talanta 77, 455–459 (2008).
[Crossref] [PubMed]

R. Gordon, D. Sinton, K. L. Kavanagh, and A. G. Brolo, “A new generation of sensors based on extraordinary optical transmission,” Acc. Chem. Res. 41, 1049–1057 (2008).
[Crossref] [PubMed]

A. Alú and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B 78, 085112 (2008).
[Crossref]

2007 (10)

C. R. Simovski, S. A. Tretyakov, and A. J. Viitanen, “Subwavelength imaging in a superlens of plasmon nanospheres,” Tech. Phys. Lett. 33, 264–266 (2007).
[Crossref]

E. V. Ponizovskaya and A. M. Bratkovsky, “Ensembles of plasmonic nanospheres at optical frequencies and a problem of negative index behavior,” Appl. Phys. A 87, 175–179(2007).
[Crossref]

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
[Crossref]

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Sub-wavelength resolution in linear arrays of plasmonic particles,” J. Appl. Phys. 101, 123102 (2007).
[Crossref]

C. Hsu and H. H. Liu, “Optical behaviours of two dimensional Au nanoparticle arrays within porous anodic alumina,” J. Phys. Conf. Ser. 61, 440–444 (2007).
[Crossref]

C. M. Linton, R. Porter, and I. Thompson, “Scattering by a semi-infinite periodic array and the excitation of surface waves,” SIAM J. Appl. Math. 67, 1233–1258 (2007).
[Crossref]

P. Baccarelli, S. Paulotto, and C. D. Nallo, “Full-wave analysis of bound and leaky modes propagating along 2D periodic printed structures with arbitrary metallisation in the unit cell,” IET Microw. Antennas Propag. 1, 217–225 (2007).
[Crossref]

F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3d Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Computat. Phys. 223, 250–261 (2007).
[Crossref]

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76, 201403 (2007).
[Crossref]

2006 (7)

A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74, 205436 (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]

P. Baccarelli, C. D. Nallo, S. Paulotto, and D. R. Jackson, “A full-wave numerical approach for modal analysis of 1-D periodic microstrip structures,” IEEE Trans. Microwave Theory Tech. 54, 1350–1362 (2006).
[Crossref]

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

Y. Guo and R. Xu, “Planar metamaterials supporting multiple left-handed modes,” PIER 66, 239–251 (2006).
[Crossref]

P. Alitalo, C. Simovski, A. Viitanen, and S. Tretyakov, “Near-field enhancement and subwavelength imaging in the optical region using a pair of two-dimensional arrays of metal nanospheres,” Phys. Rev. B 74, 235425 (2006).
[Crossref]

A. Alú, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14, 1557–1567 (2006).
[Crossref] [PubMed]

2005 (5)

G. Gantzounis, N. Stefanou, and V. Yannopapas, “Optical properties of a periodic monolayer of metallic nanospheres on a dielectric waveguide,” J. Phys. Condens. Matter 17, 1791–1802 (2005).
[Crossref]

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

L. A. Sweatlock, S. A. Maier, and H. A. Atwater, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B 71, 235408 (2005).
[Crossref]

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

R. A. Shore and A. D. Yaghjian, “Travelling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett. 41, 578–580 (2005).
[Crossref]

2004 (2)

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
[Crossref]

S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B 69, 125418 (2004).
[Crossref]

2003 (2)

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[Crossref]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[Crossref]

2002 (2)

A. D. Yaghjian, “Scattering-matrix analysis of linear periodic arrays,” IEEE Trans. Antennas Propag. 50, 1050–1064(2002).
[Crossref]

N. Félidj, J. Aubard, and G. Lévi, “Enhanced substrate-induced coupling in two-dimensional gold nanoparticle arrays,” Phys. Rev. B 66, 245407 (2002).
[Crossref]

2000 (1)

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[Crossref]

1986 (1)

K. E. Jordan, G. E. Richter, and P. Sheng, “An efficient numerical evaluation of the Green’s function for the Helmholtz operator in periodic structures,” J. Comp. Phys. 63, 222–235(1986).
[Crossref]

1983 (1)

P. K. Aravind and H. Metiu, “The effects of the interaction between resonances in the electromagnetic response of a sphere-plane structure; applications to surface enhanced spectroscopy,” Surf. Sci. 506–528 (1983).
[Crossref]

Alitalo, P.

P. Alitalo, C. Simovski, A. Viitanen, and S. Tretyakov, “Near-field enhancement and subwavelength imaging in the optical region using a pair of two-dimensional arrays of metal nanospheres,” Phys. Rev. B 74, 235425 (2006).
[Crossref]

S. Steshenko, F. Capolino, P. Alitalo, and S. Tretyakov, “Effective model and investigation of the near-field enhancement and subwavelength imaging properties of multilayer arrays of plasmonic nanospheres,” Phys. Rev. E (to be published).

Alú, A.

A. Alú and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B 78, 085112 (2008).
[Crossref]

A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74, 205436 (2006).
[Crossref]

A. Alú, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14, 1557–1567 (2006).
[Crossref] [PubMed]

A. Alú and N. Engheta, “Negative refraction in infrared and visible domains,” in Theory and Phenomena of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 23.

Aravind, P. K.

P. K. Aravind and H. Metiu, “The effects of the interaction between resonances in the electromagnetic response of a sphere-plane structure; applications to surface enhanced spectroscopy,” Surf. Sci. 506–528 (1983).
[Crossref]

Atwater, H. A.

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

L. A. Sweatlock, S. A. Maier, and H. A. Atwater, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B 71, 235408 (2005).
[Crossref]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[Crossref]

Aubard, J.

N. Félidj, J. Aubard, and G. Lévi, “Enhanced substrate-induced coupling in two-dimensional gold nanoparticle arrays,” Phys. Rev. B 66, 245407 (2002).
[Crossref]

Baccarelli, P.

R. Rodríguez-Berral, F. Mesa, P. Baccarelli, and P. Burghignoli, “Excitation of a periodic microstrip line by an aperiodic delta-gap source,” IEEE Antennas Wirel. Propag. Lett. 8, 641–644 (2009).
[Crossref]

P. Baccarelli, S. Paulotto, and C. D. Nallo, “Full-wave analysis of bound and leaky modes propagating along 2D periodic printed structures with arbitrary metallisation in the unit cell,” IET Microw. Antennas Propag. 1, 217–225 (2007).
[Crossref]

P. Baccarelli, C. D. Nallo, S. Paulotto, and D. R. Jackson, “A full-wave numerical approach for modal analysis of 1-D periodic microstrip structures,” IEEE Trans. Microwave Theory Tech. 54, 1350–1362 (2006).
[Crossref]

Baker-Jarvis, J.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[Crossref]

Biswas, R.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absortion and Scattering of Light by Small Particles (Wiley, 1983).

Bratkovsky, A. M.

E. V. Ponizovskaya and A. M. Bratkovsky, “Ensembles of plasmonic nanospheres at optical frequencies and a problem of negative index behavior,” Appl. Phys. A 87, 175–179(2007).
[Crossref]

Brolo, A. G.

R. Gordon, D. Sinton, K. L. Kavanagh, and A. G. Brolo, “A new generation of sensors based on extraordinary optical transmission,” Acc. Chem. Res. 41, 1049–1057 (2008).
[Crossref] [PubMed]

Burghignoli, P.

R. Rodríguez-Berral, F. Mesa, P. Baccarelli, and P. Burghignoli, “Excitation of a periodic microstrip line by an aperiodic delta-gap source,” IEEE Antennas Wirel. Propag. Lett. 8, 641–644 (2009).
[Crossref]

Campione, S.

S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: fabrication, optical properties and applications,” Selected Topics in Metamaterials and Photonic Crystals, A.Andreone, A.Cusano, A.Cutolo, and V.Galdi, eds. (World Scientific, in press, 2011), Chap. 5.
[Crossref]

Capolino, F.

F. T. Celepcikay, D. R. Wilton, D. R. Jackson, and F. Capolino, “Choosing splitting parameters and summation limits in the numerical evaluation of 1-D and 2-D periodic Green’s functions using the Ewald method,” Radio Sci. 43, RS6S01(2008).
[Crossref]

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3d Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Computat. Phys. 223, 250–261 (2007).
[Crossref]

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

F. Capolino, D. R. Jackson, and D. R. Wilton, “Field representations in periodic artificial materials excited by a source,” Theory and Phenomena of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 12.
[Crossref]

S. Steshenko, F. Capolino, P. Alitalo, and S. Tretyakov, “Effective model and investigation of the near-field enhancement and subwavelength imaging properties of multilayer arrays of plasmonic nanospheres,” Phys. Rev. E (to be published).

S. Steshenko, F. Capolino, S. Tretyakov, and C. R. Simovski, “Super-resolution and near-field enhancement with layers of resonant arrays of nanoparticles,” in Applications of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 4.

A. Vallecchi and F. Capolino, “Metamaterials based on pairs of tightly coupled scatterers,” in Theory and Phenomena of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 19.
[Crossref]

S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 8.

S. Steshenko, A. Vallecchi, and F. Capolino, “Electric and magnetic resonances in arrays with elements made of tightly coupled silver nanospheres,” presented at Metamaterials 2008, Pamplona, Spain, 21–26 Sept. 2008.

A. Vallecchi, S. Steshenko, and F. Capolino, “Artificial magnetism at optical frequencies in composite materials made of particles with pairs of tightly coupled metallic nanospheres,” presented at the 2008 URSI General Assembly, Chicago, Illinois, 11–16 Aug. 2008.

S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: fabrication, optical properties and applications,” Selected Topics in Metamaterials and Photonic Crystals, A.Andreone, A.Cusano, A.Cutolo, and V.Galdi, eds. (World Scientific, in press, 2011), Chap. 5.
[Crossref]

Celepcikay, F. T.

F. T. Celepcikay, D. R. Wilton, D. R. Jackson, and F. Capolino, “Choosing splitting parameters and summation limits in the numerical evaluation of 1-D and 2-D periodic Green’s functions using the Ewald method,” Radio Sci. 43, RS6S01(2008).
[Crossref]

Chan, C. T.

Y. R. Zhen, K. H. Fung, and C. T. Chan, “Collective plasmonic modes in two-dimensional periodic arrays of metal nanoparticles,” Phys. Rev. B 78, 035419 (2008).
[Crossref]

Citrin, D. S.

de Waele, R.

A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76, 201403 (2007).
[Crossref]

Du, Z.

C. Li, Y. Liu, L. Li, Z. Du, S. Xu, M. Zhang, X. Yin, and T. Wang, “A novel amperometric biosensor based on NiO hollow nanospheres for biosensing glucose,” Talanta 77, 455–459 (2008).
[Crossref] [PubMed]

El-Kady, I.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[Crossref]

Engheta, N.

A. Alú and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B 78, 085112 (2008).
[Crossref]

A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74, 205436 (2006).
[Crossref]

A. Alú, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14, 1557–1567 (2006).
[Crossref] [PubMed]

A. Alú and N. Engheta, “Negative refraction in infrared and visible domains,” in Theory and Phenomena of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 23.

Félidj, N.

N. Félidj, J. Aubard, and G. Lévi, “Enhanced substrate-induced coupling in two-dimensional gold nanoparticle arrays,” Phys. Rev. B 66, 245407 (2002).
[Crossref]

Felsen, L. B.

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

L. B. Felsen and N. Marckuvitz, Radiation and Scattering of Waves (IEEE Press, 1984).

Ford, G. W.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
[Crossref]

Fung, K. H.

Y. R. Zhen, K. H. Fung, and C. T. Chan, “Collective plasmonic modes in two-dimensional periodic arrays of metal nanoparticles,” Phys. Rev. B 78, 035419 (2008).
[Crossref]

Gantzounis, G.

G. Gantzounis, N. Stefanou, and V. Yannopapas, “Optical properties of a periodic monolayer of metallic nanospheres on a dielectric waveguide,” J. Phys. Condens. Matter 17, 1791–1802 (2005).
[Crossref]

Gordon, R.

R. Gordon, D. Sinton, K. L. Kavanagh, and A. G. Brolo, “A new generation of sensors based on extraordinary optical transmission,” Acc. Chem. Res. 41, 1049–1057 (2008).
[Crossref] [PubMed]

Goussetis, G.

Guo, Y.

Y. Guo and R. Xu, “Planar metamaterials supporting multiple left-handed modes,” PIER 66, 239–251 (2006).
[Crossref]

Ho, K. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[Crossref]

Holloway, C. L.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[Crossref]

Hsu, C.

C. Hsu and H. H. Liu, “Optical behaviours of two dimensional Au nanoparticle arrays within porous anodic alumina,” J. Phys. Conf. Ser. 61, 440–444 (2007).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absortion and Scattering of Light by Small Particles (Wiley, 1983).

Jackson, D. R.

F. T. Celepcikay, D. R. Wilton, D. R. Jackson, and F. Capolino, “Choosing splitting parameters and summation limits in the numerical evaluation of 1-D and 2-D periodic Green’s functions using the Ewald method,” Radio Sci. 43, RS6S01(2008).
[Crossref]

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

P. Baccarelli, C. D. Nallo, S. Paulotto, and D. R. Jackson, “A full-wave numerical approach for modal analysis of 1-D periodic microstrip structures,” IEEE Trans. Microwave Theory Tech. 54, 1350–1362 (2006).
[Crossref]

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

F. Capolino, D. R. Jackson, and D. R. Wilton, “Field representations in periodic artificial materials excited by a source,” Theory and Phenomena of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 12.
[Crossref]

Johnson, W. A.

F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3d Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Computat. Phys. 223, 250–261 (2007).
[Crossref]

Jordan, K. E.

K. E. Jordan, G. E. Richter, and P. Sheng, “An efficient numerical evaluation of the Green’s function for the Helmholtz operator in periodic structures,” J. Comp. Phys. 63, 222–235(1986).
[Crossref]

Kabos, P.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[Crossref]

Kavanagh, K. L.

R. Gordon, D. Sinton, K. L. Kavanagh, and A. G. Brolo, “A new generation of sensors based on extraordinary optical transmission,” Acc. Chem. Res. 41, 1049–1057 (2008).
[Crossref] [PubMed]

Kik, P. G.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[Crossref]

Koenderink, A. F.

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

A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76, 201403 (2007).
[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]

Kuester, E. F.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[Crossref]

Lévi, G.

N. Félidj, J. Aubard, and G. Lévi, “Enhanced substrate-induced coupling in two-dimensional gold nanoparticle arrays,” Phys. Rev. B 66, 245407 (2002).
[Crossref]

Li, C.

C. Li, Y. Liu, L. Li, Z. Du, S. Xu, M. Zhang, X. Yin, and T. Wang, “A novel amperometric biosensor based on NiO hollow nanospheres for biosensing glucose,” Talanta 77, 455–459 (2008).
[Crossref] [PubMed]

Li, L.

C. Li, Y. Liu, L. Li, Z. Du, S. Xu, M. Zhang, X. Yin, and T. Wang, “A novel amperometric biosensor based on NiO hollow nanospheres for biosensing glucose,” Talanta 77, 455–459 (2008).
[Crossref] [PubMed]

Linton, C. M.

C. M. Linton, R. Porter, and I. Thompson, “Scattering by a semi-infinite periodic array and the excitation of surface waves,” SIAM J. Appl. Math. 67, 1233–1258 (2007).
[Crossref]

Liu, H. H.

C. Hsu and H. H. Liu, “Optical behaviours of two dimensional Au nanoparticle arrays within porous anodic alumina,” J. Phys. Conf. Ser. 61, 440–444 (2007).
[Crossref]

Liu, Y.

C. Li, Y. Liu, L. Li, Z. Du, S. Xu, M. Zhang, X. Yin, and T. Wang, “A novel amperometric biosensor based on NiO hollow nanospheres for biosensing glucose,” Talanta 77, 455–459 (2008).
[Crossref] [PubMed]

Maier, S. A.

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

L. A. Sweatlock, S. A. Maier, and H. A. Atwater, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B 71, 235408 (2005).
[Crossref]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[Crossref]

Marckuvitz, N.

L. B. Felsen and N. Marckuvitz, Radiation and Scattering of Waves (IEEE Press, 1984).

Mateo-Segura, C.

Mesa, F.

R. Rodríguez-Berral, F. Mesa, P. Baccarelli, and P. Burghignoli, “Excitation of a periodic microstrip line by an aperiodic delta-gap source,” IEEE Antennas Wirel. Propag. Lett. 8, 641–644 (2009).
[Crossref]

Metiu, H.

P. K. Aravind and H. Metiu, “The effects of the interaction between resonances in the electromagnetic response of a sphere-plane structure; applications to surface enhanced spectroscopy,” Surf. Sci. 506–528 (1983).
[Crossref]

Nallo, C. D.

P. Baccarelli, S. Paulotto, and C. D. Nallo, “Full-wave analysis of bound and leaky modes propagating along 2D periodic printed structures with arbitrary metallisation in the unit cell,” IET Microw. Antennas Propag. 1, 217–225 (2007).
[Crossref]

P. Baccarelli, C. D. Nallo, S. Paulotto, and D. R. Jackson, “A full-wave numerical approach for modal analysis of 1-D periodic microstrip structures,” IEEE Trans. Microwave Theory Tech. 54, 1350–1362 (2006).
[Crossref]

Park, S. Y.

S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B 69, 125418 (2004).
[Crossref]

Paulotto, S.

P. Baccarelli, S. Paulotto, and C. D. Nallo, “Full-wave analysis of bound and leaky modes propagating along 2D periodic printed structures with arbitrary metallisation in the unit cell,” IET Microw. Antennas Propag. 1, 217–225 (2007).
[Crossref]

P. Baccarelli, C. D. Nallo, S. Paulotto, and D. R. Jackson, “A full-wave numerical approach for modal analysis of 1-D periodic microstrip structures,” IEEE Trans. Microwave Theory Tech. 54, 1350–1362 (2006).
[Crossref]

Polman, A.

A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76, 201403 (2007).
[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]

Ponizovskaya, E. V.

E. V. Ponizovskaya and A. M. Bratkovsky, “Ensembles of plasmonic nanospheres at optical frequencies and a problem of negative index behavior,” Appl. Phys. A 87, 175–179(2007).
[Crossref]

Porter, R.

C. M. Linton, R. Porter, and I. Thompson, “Scattering by a semi-infinite periodic array and the excitation of surface waves,” SIAM J. Appl. Math. 67, 1233–1258 (2007).
[Crossref]

Prangsma, J. C.

A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76, 201403 (2007).
[Crossref]

Richter, G. E.

K. E. Jordan, G. E. Richter, and P. Sheng, “An efficient numerical evaluation of the Green’s function for the Helmholtz operator in periodic structures,” J. Comp. Phys. 63, 222–235(1986).
[Crossref]

Rodríguez-Berral, R.

R. Rodríguez-Berral, F. Mesa, P. Baccarelli, and P. Burghignoli, “Excitation of a periodic microstrip line by an aperiodic delta-gap source,” IEEE Antennas Wirel. Propag. Lett. 8, 641–644 (2009).
[Crossref]

Saito, Y.

Y. Saito and P. Verma, “Imaging and spectroscopy through plasmonic nano-probe,” Eur. Phys. J. Appl. Phys. 46, 20101 (2009).
[Crossref]

Salandrino, A.

Sheng, P.

K. E. Jordan, G. E. Richter, and P. Sheng, “An efficient numerical evaluation of the Green’s function for the Helmholtz operator in periodic structures,” J. Comp. Phys. 63, 222–235(1986).
[Crossref]

Shore, R.

R. Shore and A. Yaghjian, “Complex waves on 1D, 2D, and 3D periodic arrays of lossy and lossless magnetodielectric spheres,” In-house report, AFRL-RY-HS-TR-2010-0019 (Air Force Research Laboratory, 2010).

Shore, R. A.

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
[Crossref]

R. A. Shore and A. D. Yaghjian, “Travelling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett. 41, 578–580 (2005).
[Crossref]

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless magnetodielectric spheres,” presented at EuCAP 2006, Nice, France, 6–10 Nov. 2006.

Sigalas, M. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[Crossref]

Simovski, C.

P. Alitalo, C. Simovski, A. Viitanen, and S. Tretyakov, “Near-field enhancement and subwavelength imaging in the optical region using a pair of two-dimensional arrays of metal nanospheres,” Phys. Rev. B 74, 235425 (2006).
[Crossref]

Simovski, C. R.

C. Mateo-Segura, C. R. Simovski, G. Goussetis, and S. Tretyakov, “Subwavelength resolution for horizontal and vertical polarization by coupled arrays of oblate nanoellipsoids,” Opt. Lett. 34, 2333–2335 (2009).
[Crossref] [PubMed]

C. R. Simovski, S. A. Tretyakov, and A. J. Viitanen, “Subwavelength imaging in a superlens of plasmon nanospheres,” Tech. Phys. Lett. 33, 264–266 (2007).
[Crossref]

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Sub-wavelength resolution in linear arrays of plasmonic particles,” J. Appl. Phys. 101, 123102 (2007).
[Crossref]

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Subwavelength imaging and resolution by two linear chains of plasmonic particles,” in Proceedings of the 2007 IEEE Antennas and Propagation Society International Symposium (IEEE, 2007), pp. 4344–4347.

S. Steshenko, F. Capolino, S. Tretyakov, and C. R. Simovski, “Super-resolution and near-field enhancement with layers of resonant arrays of nanoparticles,” in Applications of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 4.

Sinton, D.

R. Gordon, D. Sinton, K. L. Kavanagh, and A. G. Brolo, “A new generation of sensors based on extraordinary optical transmission,” Acc. Chem. Res. 41, 1049–1057 (2008).
[Crossref] [PubMed]

Soukoulis, C. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[Crossref]

Stefanou, N.

G. Gantzounis, N. Stefanou, and V. Yannopapas, “Optical properties of a periodic monolayer of metallic nanospheres on a dielectric waveguide,” J. Phys. Condens. Matter 17, 1791–1802 (2005).
[Crossref]

Steshenko, S.

A. Vallecchi, S. Steshenko, and F. Capolino, “Artificial magnetism at optical frequencies in composite materials made of particles with pairs of tightly coupled metallic nanospheres,” presented at the 2008 URSI General Assembly, Chicago, Illinois, 11–16 Aug. 2008.

S. Steshenko, F. Capolino, P. Alitalo, and S. Tretyakov, “Effective model and investigation of the near-field enhancement and subwavelength imaging properties of multilayer arrays of plasmonic nanospheres,” Phys. Rev. E (to be published).

S. Steshenko, F. Capolino, S. Tretyakov, and C. R. Simovski, “Super-resolution and near-field enhancement with layers of resonant arrays of nanoparticles,” in Applications of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 4.

S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 8.

S. Steshenko, A. Vallecchi, and F. Capolino, “Electric and magnetic resonances in arrays with elements made of tightly coupled silver nanospheres,” presented at Metamaterials 2008, Pamplona, Spain, 21–26 Sept. 2008.

Stroud, D.

S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B 69, 125418 (2004).
[Crossref]

Sweatlock, L. A.

L. A. Sweatlock, S. A. Maier, and H. A. Atwater, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B 71, 235408 (2005).
[Crossref]

Thompson, I.

C. M. Linton, R. Porter, and I. Thompson, “Scattering by a semi-infinite periodic array and the excitation of surface waves,” SIAM J. Appl. Math. 67, 1233–1258 (2007).
[Crossref]

Tretyakov, S.

C. Mateo-Segura, C. R. Simovski, G. Goussetis, and S. Tretyakov, “Subwavelength resolution for horizontal and vertical polarization by coupled arrays of oblate nanoellipsoids,” Opt. Lett. 34, 2333–2335 (2009).
[Crossref] [PubMed]

P. Alitalo, C. Simovski, A. Viitanen, and S. Tretyakov, “Near-field enhancement and subwavelength imaging in the optical region using a pair of two-dimensional arrays of metal nanospheres,” Phys. Rev. B 74, 235425 (2006).
[Crossref]

S. Steshenko, F. Capolino, P. Alitalo, and S. Tretyakov, “Effective model and investigation of the near-field enhancement and subwavelength imaging properties of multilayer arrays of plasmonic nanospheres,” Phys. Rev. E (to be published).

S. Steshenko, F. Capolino, S. Tretyakov, and C. R. Simovski, “Super-resolution and near-field enhancement with layers of resonant arrays of nanoparticles,” in Applications of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 4.

Tretyakov, S. A.

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Sub-wavelength resolution in linear arrays of plasmonic particles,” J. Appl. Phys. 101, 123102 (2007).
[Crossref]

C. R. Simovski, S. A. Tretyakov, and A. J. Viitanen, “Subwavelength imaging in a superlens of plasmon nanospheres,” Tech. Phys. Lett. 33, 264–266 (2007).
[Crossref]

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Subwavelength imaging and resolution by two linear chains of plasmonic particles,” in Proceedings of the 2007 IEEE Antennas and Propagation Society International Symposium (IEEE, 2007), pp. 4344–4347.

Vallecchi, A.

S. Steshenko, A. Vallecchi, and F. Capolino, “Electric and magnetic resonances in arrays with elements made of tightly coupled silver nanospheres,” presented at Metamaterials 2008, Pamplona, Spain, 21–26 Sept. 2008.

A. Vallecchi and F. Capolino, “Metamaterials based on pairs of tightly coupled scatterers,” in Theory and Phenomena of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 19.
[Crossref]

A. Vallecchi, S. Steshenko, and F. Capolino, “Artificial magnetism at optical frequencies in composite materials made of particles with pairs of tightly coupled metallic nanospheres,” presented at the 2008 URSI General Assembly, Chicago, Illinois, 11–16 Aug. 2008.

Verma, P.

Y. Saito and P. Verma, “Imaging and spectroscopy through plasmonic nano-probe,” Eur. Phys. J. Appl. Phys. 46, 20101 (2009).
[Crossref]

Viitanen, A.

P. Alitalo, C. Simovski, A. Viitanen, and S. Tretyakov, “Near-field enhancement and subwavelength imaging in the optical region using a pair of two-dimensional arrays of metal nanospheres,” Phys. Rev. B 74, 235425 (2006).
[Crossref]

Viitanen, A. J.

C. R. Simovski, S. A. Tretyakov, and A. J. Viitanen, “Subwavelength imaging in a superlens of plasmon nanospheres,” Tech. Phys. Lett. 33, 264–266 (2007).
[Crossref]

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Sub-wavelength resolution in linear arrays of plasmonic particles,” J. Appl. Phys. 101, 123102 (2007).
[Crossref]

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Subwavelength imaging and resolution by two linear chains of plasmonic particles,” in Proceedings of the 2007 IEEE Antennas and Propagation Society International Symposium (IEEE, 2007), pp. 4344–4347.

Wang, T.

C. Li, Y. Liu, L. Li, Z. Du, S. Xu, M. Zhang, X. Yin, and T. Wang, “A novel amperometric biosensor based on NiO hollow nanospheres for biosensing glucose,” Talanta 77, 455–459 (2008).
[Crossref] [PubMed]

Weber, W. H.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
[Crossref]

Wilton, D. R.

F. T. Celepcikay, D. R. Wilton, D. R. Jackson, and F. Capolino, “Choosing splitting parameters and summation limits in the numerical evaluation of 1-D and 2-D periodic Green’s functions using the Ewald method,” Radio Sci. 43, RS6S01(2008).
[Crossref]

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3d Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Computat. Phys. 223, 250–261 (2007).
[Crossref]

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

F. Capolino, D. R. Jackson, and D. R. Wilton, “Field representations in periodic artificial materials excited by a source,” Theory and Phenomena of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 12.
[Crossref]

Xu, R.

Y. Guo and R. Xu, “Planar metamaterials supporting multiple left-handed modes,” PIER 66, 239–251 (2006).
[Crossref]

Xu, S.

C. Li, Y. Liu, L. Li, Z. Du, S. Xu, M. Zhang, X. Yin, and T. Wang, “A novel amperometric biosensor based on NiO hollow nanospheres for biosensing glucose,” Talanta 77, 455–459 (2008).
[Crossref] [PubMed]

Yaghjian, A.

R. Shore and A. Yaghjian, “Complex waves on 1D, 2D, and 3D periodic arrays of lossy and lossless magnetodielectric spheres,” In-house report, AFRL-RY-HS-TR-2010-0019 (Air Force Research Laboratory, 2010).

Yaghjian, A. D.

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
[Crossref]

R. A. Shore and A. D. Yaghjian, “Travelling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett. 41, 578–580 (2005).
[Crossref]

A. D. Yaghjian, “Scattering-matrix analysis of linear periodic arrays,” IEEE Trans. Antennas Propag. 50, 1050–1064(2002).
[Crossref]

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Zhang, M.

C. Li, Y. Liu, L. Li, Z. Du, S. Xu, M. Zhang, X. Yin, and T. Wang, “A novel amperometric biosensor based on NiO hollow nanospheres for biosensing glucose,” Talanta 77, 455–459 (2008).
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F. Capolino, D. R. Jackson, and D. R. Wilton, “Field representations in periodic artificial materials excited by a source,” Theory and Phenomena of Metamaterials, F.Capolino, ed. (CRC Press, 2009), Chap. 12.
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Figures (16)

Fig. 1
Fig. 1

Spatial array of metal nanospheres with 2D periodicity. r m n = m a x ^ + n b y ^ is the 2D vector pointing at the nanospheres (m, n = 0 , ± 1 , ± 2 , ).

Fig. 2
Fig. 2

Path deformation in the complex k x plane for a lossless host medium assuming an observer along the positive x. The meaningful physical modes are those captured in the deformation of the original path (detouring around the branch-point singularities). Physical proper poles (top Riemann sheet) are shown with a solid contour, and physical improper poles (bottom Riemann sheet) are shown with a dashed contour. The poles and branch points are periodically repeated in the k x plane with period 2 π / a . k p q = k p 2 π / a q 2 π / b .

Fig. 3
Fig. 3

Evolution of the real part of proper (solid curve) and improper (dotted curve) modes in the first BZ for (a) x, (b) y, and (c) z polarization in a lossless structure assuming k B = k x x ^ . The physical branches for an observer along positive x are tagged by circular markers (○), and the nonphysical branches by crossed circular markers (⊗).

Fig. 4
Fig. 4

Evolution of the imaginary part of proper (solid curve) and improper (dotted curve) modes for (a) x, (b) y, and (c) z polarization in a lossless structure assuming k B = k x x ^ .

Fig. 5
Fig. 5

Loci of the wavenumbers of the modes in Figs. 3a, 4a (x polarization). Arrows show the increasing frequency.

Fig. 6
Fig. 6

Detail of the transition of the real improper mode to the real proper mode and then to the complex proper mode shown in Figs. 3a, 4a (x polarization). At the normalized frequency k a / π = 0.414 , two real forward and backward proper modes (solid red line) meet at the boundary of the BZ, generating a pair of complex conjugate proper modes (solid blue line).

Fig. 7
Fig. 7

Loci of the wavenumbers of the modes shown in Figs. 3b, 4b (y polarization). Arrows show increasing frequency.

Fig. 8
Fig. 8

Detail of the evolution of the modes for y polarization in Figs. 3b, 4b. At the normalized frequency k a / π = 0.37 , two real forward and backward proper modes [solid red curve in Fig. 3b] meet at the boundary of the BZ, generating at A a pair of complex conjugate proper modes (purple curve). These modes evolve up and down the BZ limit line until they meet other complex proper modes [solid green curve in Fig. 3b] at points B and B . Starting from these points, two new pairs of complex modes [solid blue curve in Fig. 3b] move horizontally to the right- and left-hand sides.

Fig. 9
Fig. 9

Loci of the wavenumbers of the modes shown in Figs. 3c, 4c (z polarization). Arrows show the increasing frequency.

Fig. 10
Fig. 10

Details of the evolution of the modes shown in Fig. 9. Proper sheet: at the normalized frequency k a / π = 0.39 , two real forward and backward proper modes [solid red and cyan curves in Figs. 3c, 9] meet at Re ( k x ) a / π = 0.67 to give rise to a pair of complex conjugate proper modes [solid blue curve in Figs. 3c, 9]. These modes evolve toward the origin, reached at k a / π = 0.46 . Improper sheet: at k a / π = 0.46 , a pair of complex conjugate improper modes [blue dotted curve in Figs. 3c, 9] transition into two real improper modes at Re ( k x ) a / π = 0.6 [dotted green and pink curves in Figs. 3c, 9].

Fig. 11
Fig. 11

Evolution of the real part of the proper (solid curve) and improper (dotted curve) modes in the first BZ for (a) x, (b) y, and (c) z polarization in a lossy structure assuming k B = k x x ^ . The physical branches for an observer along positive x are tagged by circular markers (○) and the nonphysical branches by crossed circular markers (⊗).

Fig. 12
Fig. 12

Evolution of the imaginary part of the proper (solid curve) and improper (dotted curve) modes for (a) x, (b) y, and (c) z polarization in a lossy structure assuming k B = k x x ^ .

Fig. 13
Fig. 13

Loci of the wavenumbers of the modes in Figs. 11a, 12a (x polarization). Arrows show the increasing frequency.

Fig. 14
Fig. 14

Loci of the wavenumbers of the modes in Figs. 11b, 12b (y polarization). Arrows show the increasing frequency.

Fig. 15
Fig. 15

Details of the evolution of the modes in the complex k x plane for y polarization (solid blue and red curves in Fig. 14).

Fig. 16
Fig. 16

Loci of the wavenumbers of the modes in Figs. 11c, 12c (z polarization). Arrows show the increasing frequency.

Equations (23)

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p m n = p 00 exp ( i k B · r m n ) ,
k B = k x x ^ + k y y ^ = β + i α
E mode = p , q = E p q ( z ) exp [ i ( k x , p x + k y , q y ) ] .
k B , p q = k x , p x ^ + k y , q y ^ = β p q + i α ,
k z , p q = k 2 ( k B , p q · k B , p q ) = β z , p q + i α z , p q
p m n = α ee E m n loc ,
α ee = 6 i π ε 0 ε r , h k 3 n r ψ 1 ( s 1 ) ψ 1 ( s 2 ) ψ 1 ( s 2 ) ψ 1 ( s 1 ) n r ψ 1 ( s 1 ) ξ 1 ( s 2 ) ξ 1 ( s 2 ) ψ 1 ( s 1 ) ,
ε r , m = ε r , ω p 2 ω ( ω + i γ ) ,
E loc ( r 00 , k B ) = Ğ ̲ ( r 00 , r 00 , k B ) · p 00 + E inc ( r 00 ) ,
p 00 = α ee [ Ğ ̲ ( r 00 , r 00 , k B ) · p 00 ] .
A ̲ ( k B ) · p 00 = 0 ,
A ̲ ( k B ) = 1 α ee I ̲ Ğ ̲ ( r 00 , r 00 , k B ) ,
det [ A ̲ ( k B ) ] = 0 .
A ̲ ( k B ) = A ̲ t ( k B ) + A z z ( k B ) z ^ z ^ ,
A ̲ t ( k B ) = ( 1 α ee x x x y y x 1 α ee y y ) ,
A z z ( k B ) = 1 α ee z z ,
det [ A ̲ t ( k B ) ] = 0 ,
A z z ( k B ) = 0 .
k z , p q = k 2 k x , p 2 k y , q 2 = 0 .
Ğ ̲ ( r , r 00 , k B ) = G ̲ ( r , r 00 , k B ) G ̲ ( r , r 00 ) ,
G ̲ ( r , r 00 , k B ) = m n G ̲ ( r , r m n ) e i k B · r m n ,
G ̲ ( r , r m n ) = 1 ε 0 ε r , h [ k 2 G ( r , r m n ) I ̲ + G ( r , r m n ) ] ,
G ( r , r 00 ) = e i k | r r 00 | 4 π | r r 00 | .

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