Abstract

A finite element method (FEM) for solving a complex valued k(ω) vs. ω dispersion curve of a 3D metamaterial/photonic crystal system is presented. This 3D method is a generalization of a previously reported 2D eigenvalue method [Opt. Express 15, 9681 (2007)]. This method is particularly convenient for analyzing periodic systems containing dispersive (e.g., plasmonic) materials, for computing isofrequency surfaces in the k-space, and for calculating the decay length of the evanescent waves. Two specific examples are considered: a photonic crystal comprised of dielectric spheres and a plasmonic fishnet structure. Hybridization and avoided crossings between Mie resonances and propagating modes are numerically demonstrated. Negative index propagation of four electromagnetic modes distinguished by their symmetry is predicted for the plasmonic fishnets. By calculating the isofrequency contours, we also demonstrate that the fishnet structure is a hyperbolic medium.

© 2011 OSA

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2010

2009

C. Engström, C. Hafner, and K. Schmidt, “Computations of lossy bloch waves in two-dimensional photonic crystals,” J. Comput. Theor. Nanosci. 6, 1–9 (2009).
[CrossRef] [PubMed]

X. Zhang, M. Davanco, Y. Urzhumov, and G. Shvets, “A subwavelength near-infrared negative index material,” Appl. Phys. Lett. 94, 131107 (2009).
[CrossRef]

2008

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, and G. Bartal, “Three-dimensional optical metamaterial with a negative refractive index,” Science 455, 376–380 (2008).

M. Navarro-Ca, M. Beruete, M. Sorolla, and I. Campillo, “Negative refraction in a prism made of stacked subwavelength hole arrays,” Opt. Express 16, 560–566 (2008).
[CrossRef] [PubMed]

Y. A. Urzhumov and G. Shvets, “Optical magnetism and negative refraction in plasmonic metamaterials,” Solid State Commun. 146, 208–220 (2008).
[CrossRef]

2007

M. Davanco, Y. Urzhumov, and G. Shvets, “The complex bloch bands of a 2d plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15, 9681–9691 (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, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials 1, 62–80 (2007).
[CrossRef]

2006

2005

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 613, 4922 (2005).

2004

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. 93, 243902 (2004).

2003

Z. Y. Li and L. L. Lin, “Photonic band structure solved by a plane-wave–based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

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]

2001

F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Review 43, 235–286 (2001).

2000

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696 (2000).
[CrossRef]

1996

K. J. Maschhoff and D. C. Soerensen, “PARPACK: An efficient portable large scale eigenvalue package for distributed memory parallel architectures,” Lect. Notes Comp. Sci. 1184, 478–486 (1996).
[PubMed]

1995

T. Suzuki and P. L. Yu, “Tunneling in photonic band strucures,” J. Opt. Soc. Am. B 12, 804 (1995).
[CrossRef]

A. J. Ward, J. B. Pendry, and W. J. Stewart, “Photonic dispersion surfaces,” J. Phys.: Condens. Matter 7, 2217–2224 (1995).

1988

Alú, A.

A. Alú, “First-principle homogenization theory for periodic metamaterial arrays,” arXiv:1012.1351v2(2011).
[CrossRef]

Asatryan, A. A.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[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]

Balay, S.

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc users manual,” Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010).
[CrossRef]

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, and G. Bartal, “Three-dimensional optical metamaterial with a negative refractive index,” Science 455, 376–380 (2008).

Beruete, M.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).

Botten, L. C.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Bozhevolnyi, S. I.

A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: a rigorous approach,” arXiv:1104.2972v1(2011).
[CrossRef]

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).

Brown, J.

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc users manual,” Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010).
[CrossRef]

Brueck, S. R. J.

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 613, 4922 (2005).

Buschelman, K.

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc users manual,” Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010).
[CrossRef]

Campillo, I.

Conforti, M.

Davanco, M.

de Sterke, C. M.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Eijkhout, V.

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc users manual,” Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010).
[CrossRef]

Engström, C.

C. Engström, C. Hafner, and K. Schmidt, “Computations of lossy bloch waves in two-dimensional photonic crystals,” J. Comput. Theor. Nanosci. 6, 1–9 (2009).
[CrossRef] [PubMed]

Fan, W.

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 613, 4922 (2005).

Ferrari, R. L.

P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers3rd ed. (Cambridge University Press, 1996).

Fietz, C.

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

X. Zhang, M. Davanco, K. Miller, T. W. J. C. Wu, C. Fietz, D. Korobkin, X. Li, G. Shvets, and S. R. Forrest, “Interferometric characterization of a subwavelength near-infrared negative index metamaterial,” Opt. Express 18, 17788–17795 (2010).
[CrossRef]

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Phys. B 405, 2930–2934 (2010).
[CrossRef] [PubMed]

Forrest, S. R.

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, and G. Bartal, “Three-dimensional optical metamaterial with a negative refractive index,” Science 455, 376–380 (2008).

Green, A. A.

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005).
[CrossRef]

Gropp, W. D.

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc users manual,” Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010).
[CrossRef]

Guasoni, M.

Hafner, C.

C. Engström, C. Hafner, and K. Schmidt, “Computations of lossy bloch waves in two-dimensional photonic crystals,” J. Comput. Theor. Nanosci. 6, 1–9 (2009).
[CrossRef] [PubMed]

Halevi, P.

P. Halevi, Spatial Dispersion in Solids and Plasmas (Elsevier Science Publishers, 1992).
[CrossRef] [PubMed]

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]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).

Istrate, E.

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005).
[CrossRef]

Jelinek, L.

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (John Wiley & Sons, Inc., 2002).

Joannopoulos, J. D.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[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]

Kaushik, D.

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc users manual,” Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010).
[CrossRef]

Knepley, M. G.

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc users manual,” Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010).
[CrossRef]

Korobkin, D.

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]

Lehoucq, R. B.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide, Solution of Large-Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods (SIAM, Philadelphia).

Li, X.

Li, Z. Y.

Z. Y. Li and L. L. Lin, “Photonic band structure solved by a plane-wave–based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Lin, L. L.

Z. Y. Li and L. L. Lin, “Photonic band structure solved by a plane-wave–based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Machac, J.

Machorro, R.

Malloy, K. J.

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 613, 4922 (2005).

Marques, R.

Maschhoff, K. J.

K. J. Maschhoff and D. C. Soerensen, “PARPACK: An efficient portable large scale eigenvalue package for distributed memory parallel architectures,” Lect. Notes Comp. Sci. 1184, 478–486 (1996).
[PubMed]

McInnes, L. C.

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc users manual,” Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010).
[CrossRef]

McOrist, J.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

Meerbergen, K.

F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Review 43, 235–286 (2001).

Miller, K.

Navarro-Ca, M.

Nicorovici, N. A.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Notomi, M.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696 (2000).
[CrossRef]

Osgood, R. M.

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 613, 4922 (2005).

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Acedemic Press, 1985), vol. 1.
[CrossRef]

Panoiu, N. C.

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 613, 4922 (2005).

Pendry, J. B.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

A. J. Ward, J. B. Pendry, and W. J. Stewart, “Photonic dispersion surfaces,” J. Phys.: Condens. Matter 7, 2217–2224 (1995).

Pors, A.

A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: a rigorous approach,” arXiv:1104.2972v1(2011).
[CrossRef]

Regalado, L. E.

Roman, J. E.

J. E. Roman, E. Romero, and A. Tomas, “SLEPc users manual,” Tech. Rep. DSIC-II/24/02 - Revision 3.1, D. Sistemas Informáticos y Computación, Universidad Politécnica de Valencia (2010).

Romero, E.

J. E. Roman, E. Romero, and A. Tomas, “SLEPc users manual,” Tech. Rep. DSIC-II/24/02 - Revision 3.1, D. Sistemas Informáticos y Computación, Universidad Politécnica de Valencia (2010).

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2004).

Sargent, E. H.

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005).
[CrossRef]

Schmidt, K.

C. Engström, C. Hafner, and K. Schmidt, “Computations of lossy bloch waves in two-dimensional photonic crystals,” J. Comput. Theor. Nanosci. 6, 1–9 (2009).
[CrossRef] [PubMed]

Shapiro, M. A.

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]

Shvets, G.

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Phys. B 405, 2930–2934 (2010).
[CrossRef] [PubMed]

X. Zhang, M. Davanco, K. Miller, T. W. J. C. Wu, C. Fietz, D. Korobkin, X. Li, G. Shvets, and S. R. Forrest, “Interferometric characterization of a subwavelength near-infrared negative index metamaterial,” Opt. Express 18, 17788–17795 (2010).
[CrossRef]

X. Zhang, M. Davanco, Y. Urzhumov, and G. Shvets, “A subwavelength near-infrared negative index material,” Appl. Phys. Lett. 94, 131107 (2009).
[CrossRef]

Y. A. Urzhumov and G. Shvets, “Optical magnetism and negative refraction in plasmonic metamaterials,” Solid State Commun. 146, 208–220 (2008).
[CrossRef]

M. Davanco, Y. Urzhumov, and G. Shvets, “The complex bloch bands of a 2d plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15, 9681–9691 (2007).
[CrossRef]

M. A. Shapiro, G. Shvets, J. R. Sirigiri, and R. J. Temkin, “Spatial dispersion in metamaterials with negative dielectric permittivity and its effect on surface waves,” Opt. Lett. 31, 2051 (2006).

G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. 93, 243902 (2004).

Silvester, P. P.

P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers3rd ed. (Cambridge University Press, 1996).

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C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials 1, 62–80 (2007).
[CrossRef]

Siqueiros, J. M.

Sirigiri, J. R.

Smith, B. F.

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc users manual,” Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010).
[CrossRef]

Soerensen, D. C.

K. J. Maschhoff and D. C. Soerensen, “PARPACK: An efficient portable large scale eigenvalue package for distributed memory parallel architectures,” Lect. Notes Comp. Sci. 1184, 478–486 (1996).
[PubMed]

Sorensen, D. C.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide, Solution of Large-Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods (SIAM, Philadelphia).

Sorolla, M.

Stewart, W. J.

A. J. Ward, J. B. Pendry, and W. J. Stewart, “Photonic dispersion surfaces,” J. Phys.: Condens. Matter 7, 2217–2224 (1995).

Suzuki, T.

Temkin, R. J.

Tisseur, F.

F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Review 43, 235–286 (2001).

Tomas, A.

J. E. Roman, E. Romero, and A. Tomas, “SLEPc users manual,” Tech. Rep. DSIC-II/24/02 - Revision 3.1, D. Sistemas Informáticos y Computación, Universidad Politécnica de Valencia (2010).

Tsukerman, I.

A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: a rigorous approach,” arXiv:1104.2972v1(2011).
[CrossRef]

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, and G. Bartal, “Three-dimensional optical metamaterial with a negative refractive index,” Science 455, 376–380 (2008).

Urzhumov, Y.

X. Zhang, M. Davanco, Y. Urzhumov, and G. Shvets, “A subwavelength near-infrared negative index material,” Appl. Phys. Lett. 94, 131107 (2009).
[CrossRef]

M. Davanco, Y. Urzhumov, and G. Shvets, “The complex bloch bands of a 2d plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15, 9681–9691 (2007).
[CrossRef]

Urzhumov, Y. A.

Y. A. Urzhumov and G. Shvets, “Optical magnetism and negative refraction in plasmonic metamaterials,” Solid State Commun. 146, 208–220 (2008).
[CrossRef]

G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. 93, 243902 (2004).

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, and G. Bartal, “Three-dimensional optical metamaterial with a negative refractive index,” Science 455, 376–380 (2008).

Ward, A. J.

A. J. Ward, J. B. Pendry, and W. J. Stewart, “Photonic dispersion surfaces,” J. Phys.: Condens. Matter 7, 2217–2224 (1995).

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

Wu, T. W. J. C.

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]

Yang, C.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide, Solution of Large-Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods (SIAM, Philadelphia).

Yu, P. L.

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, and G. Bartal, “Three-dimensional optical metamaterial with a negative refractive index,” Science 455, 376–380 (2008).

Zhang, H.

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc users manual,” Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010).
[CrossRef]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, and G. Bartal, “Three-dimensional optical metamaterial with a negative refractive index,” Science 455, 376–380 (2008).

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 613, 4922 (2005).

Zhang, X.

Zimmerman, W. B. J.

W. B. J. Zimmerman, Process Modelling and Simulation with Finite Element Methods (World Scientific, 2004).

Appl. Opt.

Appl. Phys. Lett.

X. Zhang, M. Davanco, Y. Urzhumov, and G. Shvets, “A subwavelength near-infrared negative index material,” Appl. Phys. Lett. 94, 131107 (2009).
[CrossRef]

IEEE Trans. Antennas Propag.

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]

J. Comput. Theor. Nanosci.

C. Engström, C. Hafner, and K. Schmidt, “Computations of lossy bloch waves in two-dimensional photonic crystals,” J. Comput. Theor. Nanosci. 6, 1–9 (2009).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B

J. Phys.: Condens. Matter

A. J. Ward, J. B. Pendry, and W. J. Stewart, “Photonic dispersion surfaces,” J. Phys.: Condens. Matter 7, 2217–2224 (1995).

Lect. Notes Comp. Sci.

K. J. Maschhoff and D. C. Soerensen, “PARPACK: An efficient portable large scale eigenvalue package for distributed memory parallel architectures,” Lect. Notes Comp. Sci. 1184, 478–486 (1996).
[PubMed]

Metamaterials

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials 1, 62–80 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. B

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Phys. B 405, 2930–2934 (2010).
[CrossRef] [PubMed]

Phys. Rev. B

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696 (2000).
[CrossRef]

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71, 195122 (2005).
[CrossRef]

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Phys. Rev. E

Z. Y. Li and L. L. Lin, “Photonic band structure solved by a plane-wave–based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Phys. Rev. Lett.

G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. 93, 243902 (2004).

Radio Sci.

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]

Science

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, and G. Bartal, “Three-dimensional optical metamaterial with a negative refractive index,” Science 455, 376–380 (2008).

SIAM Review

F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Review 43, 235–286 (2001).

Solid State Commun.

Y. A. Urzhumov and G. Shvets, “Optical magnetism and negative refraction in plasmonic metamaterials,” Solid State Commun. 146, 208–220 (2008).
[CrossRef]

Other

A. Alú, “First-principle homogenization theory for periodic metamaterial arrays,” arXiv:1012.1351v2(2011).
[CrossRef]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).

P. Halevi, Spatial Dispersion in Solids and Plasmas (Elsevier Science Publishers, 1992).
[CrossRef] [PubMed]

W. B. J. Zimmerman, Process Modelling and Simulation with Finite Element Methods (World Scientific, 2004).

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (John Wiley & Sons, Inc., 2002).

P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers3rd ed. (Cambridge University Press, 1996).

E. D. Palik, Handbook of Optical Constants of Solids (Acedemic Press, 1985), vol. 1.
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2004).

A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: a rigorous approach,” arXiv:1104.2972v1(2011).
[CrossRef]

Both .m and .mph files for the sphere photonic crystal model can be acquired by contacting Chris Fietz at fietz.chris@gmail.com.

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).

MUltifrontal Massively Parallel Solver (MUMPS 4.9.2) User’s guide (Lyon, 2009).

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide, Solution of Large-Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods (SIAM, Philadelphia).

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc users manual,” Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010).
[CrossRef]

J. E. Roman, E. Romero, and A. Tomas, “SLEPc users manual,” Tech. Rep. DSIC-II/24/02 - Revision 3.1, D. Sistemas Informáticos y Computación, Universidad Politécnica de Valencia (2010).

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Figures (6)

Fig. 1
Fig. 1

Complex k dispersion curves and field profiles for eigenmodes of the photonic crystal pictured in the inset assuming k 0 = 0 and n = . (a) Real part of kx (ω) for a transversely polarized mode and a diagram of the crystal unit cell. (b) Imaginary part of kx (ω) for a transversely polarized mode and a field profile for the polarized transverse mode. There are two transverse modes, ŷ and electrically polarized, which are degenerate. (c) Real part of kx (ω) for two longitudinally polarized modes and a field profile for the magnetic longitudinal mode. (d) Imaginary part of kx (ω) for two longitudinally polarized modes and a field profile for the electric longitudinal mode. The longitudinal mode with the passband near ω = 4.5c/a is magnetically polarized in the direction and the longitudinal mode with the passband near ω = 5c/a is electrically polarized in the direction. The longitudinal modes correspond to Mie’s dipole resonances. For all dispersion curves the dotted lines are the result of a conventional ω(k) eigenvalue simulation. For all field profiles the frequency is ω = 2c/a with arrows representing D y and D z and color representing D x .

Fig. 2
Fig. 2

Complex wavenumber dispersion curves and field profiles for eigenmodes of the photonic crystal pictured in Fig. 1(a) assuming k 0 = ω/csin(π/6)ŷ and n = . Modes excited by p or s polarized incident light are plotted with solid or dashed lines respectively. (a) Real part of kx (ω) for two transverse hybrid modes and a expanded view of the avoided crossing in Re(kx ) space. (b) Imaginary part of kx (ω) for two transverse hybrid modes, an expanded view of the avoided crossing in Im(kx ) space (plotting the same modes as the expanded view in Re(kx ) space), and a field profile for the E z polarized transverse hybrid mode. (c) Real part of kx (ω) for two longitudinal hybrid modes and a field profile for the magnetic longitudinal hybrid mode. (d) Imaginary part of kx (ω) for two longitudinal hybrid modes and a field profile for the electric longitudinal hybrid mode. The magnetic longitudinal hybrid mode is excited by s polarized incident light and the electric longitudinal hybrid mode is excited by p polarized incident light. For all field profiles the frequency is ω = 2c/a with arrows representing D y and D z and color representing D x .

Fig. 3
Fig. 3

The fishnet metamaterial from Ref. [22]. The lattice constants of the unit cell are ax = ay = 860nm and az = 80nm. The fishnet is made up of alternating layers of Ag with a thickness of 30nm and MgF2 with a thickness of 50nm. The widths of the the crisscrossing fishnet strips are b 1 = 265nm and b 2 = 565nm.

Fig. 4
Fig. 4

TOP: Real (a) and imaginary (b) parts of kz (ω) for several eigenmodes of the fishnet metamaterial shown in Fig. 3. In addition to the transverse mode electrically polarized in the direction labeled E x which was identified in Ref. [22] we see two other transverse modes electrically polarized in the ŷ direction labeled E y and a longitudinal mode electrically polarized in the direction labeled E z . MIDDLE: Field profiles for the transverse E x mode (c) and the longitudinal E z mode (d) on a cross-section laying on the x-y plane in the middle of the MgF2 layer. Arrows represent in-plane electric field and color represents the E z field. BOTTOM: (e) Field profile of the transverse E x mode on a cross-section laying on the x-z plane halfway between two thin Ag-MgF 2 strips. Arrows represent in-plane electric field and color represents the H y field. For all field profiles the frequency is 175THz and each region is labeled Ag or MgF 2 according the the material of the region. Unlabeled regions are vacuum.

Fig. 5
Fig. 5

The FOM (a) and propagation length (b) for the four fishnet eigenmodes identified in Fig. 4. Note that though the mode labeled Ex has the largest propagation length it also has the smallest FOM. The negative value of FOM for the Ex mode for frequencies above 220THz indicates that it is no longer a negative index mode at these higher frequencies.

Fig. 6
Fig. 6

(a) Isofrequency contours for vacuum plotted with respect to kz and ky (b) Isofrequency contours for the E x polarized eigenmode of the fishnet crystal plotted with respect to Re(kz ) and ky . (c) Isofrequency contours for the E x polarized eigenmode of the fishnet crystal plotted with respect to Im(kz ) and ky . The black arrows indicate the direction of the phase velocity and the red arrows indicate the direction of the group velocity at a frequency of 170THz and ky = ω/csin(π/6).

Equations (12)

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

× ( 1 ɛ × H ) μ ω 2 c 2 H = 0 .
H ( x ) = u ( x ) exp [ i ( ω t k x ) ] ,
k 2 ɛ u k ɛ ( k u ) i k × ( 1 ɛ × u ) i × ( 1 ɛ k × u ) + × ( 1 ɛ × u ) μ ω 2 c 2 u = 0 ,
E ( x ) = 1 i ɛ ω / c × H = 1 i ɛ ω / c ( i k × u + × u ) exp [ i ( ω t k x ) ]
F H ( v , u ) = k 2 ɛ v u 1 ɛ ( k v ) ( k u ) i 1 ɛ v [ k × ( × u ) ] i ( × v ) 1 ɛ ( k × u ) + ( × v ) 1 ɛ ( × u ) μ ω 2 c 2 v u ,
0 = Ω d 3 x F H ( v , u ) = Ω d 3 x v [ 1 ɛ k × ( k × u ) i 1 ɛ k × ( × u ) i × ( 1 ɛ k × u ) + × ( 1 ɛ × u ) μ ω 2 c 2 u ] + Ω d A v [ n ^ × 1 ɛ ( i k × u + × u ) ] ,
A u + λ B u + λ 2 C u = 0 ,
( A B 0 1 ) ( u λ u ) = λ ( 0 C 1 0 ) ( u λ u ) .
× ( 1 μ × E ) ɛ ω 2 c 2 E = 0 .
E ( x ) = u ( x ) exp [ i ( ω t k x ) ] ,
k 2 μ u k μ ( k u ) i k × ( 1 μ × u ) i × ( 1 μ k × u ) + × ( 1 μ × u ) ɛ ω 2 c 2 u = 0.
F E ( v ,  u ) = k 2 μ v u 1 μ ( k v ) ( k u ) i 1 μ v [ k × ( × u ) ] i ( × v ) 1 μ ( k × u ) + ( × v ) 1 μ ( × u ) ɛ ω 2 c 2 v u ,

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