Abstract

Engineering of a refractive index profile is a powerful method for controlling electromagnetic fields. In this paper, we investigate possible realization of isotropic gradient refractive index media at optical frequencies using two-dimensional graded photonic crystals. They consist of dielectric rods with spatially varying radii and can be homogenized in broad frequency range within the lowest band. Here they operate in metamaterial regime, that is, the graded photonic crystals are described with spatially varying effective refractive index so they can be regarded as low-loss and broadband graded dielectric metamaterials. Homogenization of graded photonic crystals is done with Maxwell-Garnett effective medium theory. Based on this theory, the analytical formulas are given for calculations of the rods radii which makes the implementation straightforward. The frequency range where homogenization is valid and where graded photonic crystal based devices work properly is discussed in detail. Numerical simulations of the graded photonic crystal based Luneburg lens and electromagnetic beam bend show that the homogenization based on Maxwell-Garnett theory gives very good results for implementation of devices intended to steer and focus electromagnetic fields.

© 2010 Optical Society of America

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2010 (1)

Z. L. Mei, J. Bai, and T. J. Cui, "Gradient index metamaterials realized by drilling hole arrays," J. Phys. D Appl. Phys. 43, 055404 (2010).
[CrossRef]

2009 (11)

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, "Broadband Ground-Plane Cloak," Science 323, 366-369 (2009).
[CrossRef] [PubMed]

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, "Design and experimental realization of a broadband transformation media field rotator at microwave frequencies," Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 569-571 (2009).
[CrossRef]

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, "Silicon nanostructures cloak operating at optical frequencies," Nat. Photonics 3, 461-463 (2009).
[CrossRef]

Q. Cheng, H. F. Ma, and T. J. Cui, "Broadband planar Luneburg lens based on complementary metamaterials," Appl. Phys. Lett. 95, 181901 (2009).
[CrossRef]

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, "Coordinate transformation based design of confined metamaterial structures," Phys. Rev. B 79, 085103 (2009).
[CrossRef]

Z. L. Mei and T. J. Cui, "Arbitrary bending of electromagnetic waves using isotropic materials," J. Appl. Phys. 105, 104913 (2009).
[CrossRef]

B. K. Juluri, S. Chin, S. Lin, T. R. Walker, L. Jensen, and T. J. Huang, "Propagation of designer surface plasmons in structured conductor surfaces with parabolic gradient index," Opt. Express 17, 2997-3006 (2009).
[CrossRef] [PubMed]

J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, "Direct visualization of optical frequency invisibility cloak based on silicon nanorod array," Opt. Express 17, 12922-12928 (2009).
[CrossRef] [PubMed]

N. I. Landy and W. J. Padilla, "Guiding light with conformal transformations," Opt. Express 17, 14872-14879 (2009).
[CrossRef] [PubMed]

Z. L. Mei and T. J. Cui, "Experimental realization of a broadband bend structure using gradient index metamaterials," Opt. Express 17, 18354-18363 (2009).
[CrossRef] [PubMed]

2008 (3)

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, "Ray optics at a deep-subwavelength scale: a transformation optics approach," Nano Lett. 8, 4243-4247 (2008).
[CrossRef]

W. Smigaj, and B. Gralak, "Validity of the effective-medium approximation of photonic crystals," Phys. Rev. B 77, 235445 (2008).
[CrossRef]

J. Li and J. B. Pendry, "Hiding under the carpet: A new strategy for cloaking," Phys. Rev. Lett. 101, 203901 (2008).
[CrossRef] [PubMed]

2007 (5)

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, "Inhomogeneous dielectric metamaterials with space-variant polarizability," Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

G. Zouganelis and D. Budimir, "Effective dielectric constant and design of sliced Lüneburg lens," Microwave Opt. Technol. Lett. 49, 2332-2337 (2007).
[CrossRef]

H. Kurt and D. S. Citrin, "Graded index photonic crystals," Opt. Express 15, 1240-1253 (2007).
[CrossRef] [PubMed]

A. O. Pinchuk and G. C. Schatz, "Metamaterials with gradient negative index of refraction," J. Opt. Soc. Am. A 24, A39-A44 (2007).
[CrossRef]

2006 (6)

H.-T. Chien and C.-C. Chen, "Focusing of electromagnetic waves by periodic arrays of air holes with gradually varying radii," Opt. Express 14, 10759-10764 (2006).
[CrossRef] [PubMed]

U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

E. Centeno, D. Cassagne, and J.-P. Albert, "Mirage and superbending effect in two-dimensional graded photonic crystals," Phys. Rev. B 73, 235119 (2006).
[CrossRef]

F. S. Roux and I. De Leon, "Planar photonic crystal gradient index lens, simulated with a finite difference time domain method," Phys. Rev. B 74, 113103 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

2005 (5)

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

U. Levy, M. Nezhad, H.-C. Kim, C.-H. Tsai, L. Pang, and Y. Fainman, "Implementation of a graded-index medium by use of subwavelength structures with graded fill factor," J. Opt. Soc. Am. A 22, 724-733 (2005).
[CrossRef]

E. Centeno and D. Cassagne, "Graded photonic crystals," Opt. Lett. 30, 2278-2280 (2005).
[CrossRef] [PubMed]

P. A. Belov and C. R. Simovski, "Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers," Phys. Rev. E 72, 026615 (2005).
[CrossRef]

2004 (2)

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking Surface Plasmons with Structured Surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

Y. Jiao, S. Fan, and D. A. B. Miller, "Designing for beam propagation in periodic and nonperiodic photonic nanostructures: extended Hamiltonian method," Phys. Rev. 70, 036612 (2004).
[CrossRef]

2002 (1)

M. J. A. De Dood, E. Snoeks, A. Moroz, and A. Polman, "Design and optimization of 2D photonic crystal waveguides based on silicon," Opt. Quantum Electron. 34, 145-159 (2002).

2000 (1)

R. Ruppin, "Evaluation of extended Maxwell-Garnett theories," Opt. Commun. 182, 273-279 (2000).
[CrossRef]

1999 (3)

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

P. S. J. Russell and T. A. Birks, "Hamiltonian optics of nonuniform photonic crystals," J. Lightwave Technol. 17, 1982 (1999).
[CrossRef]

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic Crystal Optics and Homogenization of 2D Periodic Composites," Phys. Rev. Lett. 82, 719-722 (1999).
[CrossRef]

1998 (2)

1996 (1)

1993 (1)

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Phys. Rev. B 48, 14936-14943 (1993).
[CrossRef]

1989 (1)

W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
[CrossRef]

1982 (1)

W. G. Egan and D. E. Aspnes, "Finite-wavelength effects in composite media," Phys. Rev. B 26, 5313-5320 (1982).
[CrossRef]

1947 (1)

L. Lewin, "The electrical constants of a material loaded with spherical particles," Proc. Inst. Electr. Eng. 94, 65-68 (1947).

Abashin, M.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, "Inhomogeneous dielectric metamaterials with space-variant polarizability," Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

Albert, J.-P.

E. Centeno, D. Cassagne, and J.-P. Albert, "Mirage and superbending effect in two-dimensional graded photonic crystals," Phys. Rev. B 73, 235119 (2006).
[CrossRef]

Ao, X.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, "Design and experimental realization of a broadband transformation media field rotator at microwave frequencies," Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

Arriaga, J.

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic Crystal Optics and Homogenization of 2D Periodic Composites," Phys. Rev. Lett. 82, 719-722 (1999).
[CrossRef]

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

Aspnes, D. E.

W. G. Egan and D. E. Aspnes, "Finite-wavelength effects in composite media," Phys. Rev. B 26, 5313-5320 (1982).
[CrossRef]

Astilean, S.

Bai, J.

Z. L. Mei, J. Bai, and T. J. Cui, "Gradient index metamaterials realized by drilling hole arrays," J. Phys. D Appl. Phys. 43, 055404 (2010).
[CrossRef]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 569-571 (2009).
[CrossRef]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, "Ray optics at a deep-subwavelength scale: a transformation optics approach," Nano Lett. 8, 4243-4247 (2008).
[CrossRef]

Belov, P. A.

P. A. Belov and C. R. Simovski, "Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers," Phys. Rev. E 72, 026615 (2005).
[CrossRef]

Birks, T. A.

Blair, J.

Budimir, D.

G. Zouganelis and D. Budimir, "Effective dielectric constant and design of sliced Lüneburg lens," Microwave Opt. Technol. Lett. 49, 2332-2337 (2007).
[CrossRef]

Busch, K.

A. Kirchner, K. Busch, and C. M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998).
[CrossRef]

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Cambril, E.

Cardenas, J.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, "Silicon nanostructures cloak operating at optical frequencies," Nat. Photonics 3, 461-463 (2009).
[CrossRef]

Cassagne, D.

E. Centeno, D. Cassagne, and J.-P. Albert, "Mirage and superbending effect in two-dimensional graded photonic crystals," Phys. Rev. B 73, 235119 (2006).
[CrossRef]

E. Centeno and D. Cassagne, "Graded photonic crystals," Opt. Lett. 30, 2278-2280 (2005).
[CrossRef] [PubMed]

Centeno, E.

E. Centeno, D. Cassagne, and J.-P. Albert, "Mirage and superbending effect in two-dimensional graded photonic crystals," Phys. Rev. B 73, 235119 (2006).
[CrossRef]

E. Centeno and D. Cassagne, "Graded photonic crystals," Opt. Lett. 30, 2278-2280 (2005).
[CrossRef] [PubMed]

Chan, C. T.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, "Design and experimental realization of a broadband transformation media field rotator at microwave frequencies," Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Phys. Rev. B 48, 14936-14943 (1993).
[CrossRef]

Chavel, P.

Chen, C.-C.

Chen, H.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, "Design and experimental realization of a broadband transformation media field rotator at microwave frequencies," Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

Chen, S.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, "Design and experimental realization of a broadband transformation media field rotator at microwave frequencies," Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

Cheng, Q.

Q. Cheng, H. F. Ma, and T. J. Cui, "Broadband planar Luneburg lens based on complementary metamaterials," Appl. Phys. Lett. 95, 181901 (2009).
[CrossRef]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Chien, H.-T.

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, "Broadband Ground-Plane Cloak," Science 323, 366-369 (2009).
[CrossRef] [PubMed]

Chin, S.

Citrin, D. S.

Cui, T. J.

Z. L. Mei, J. Bai, and T. J. Cui, "Gradient index metamaterials realized by drilling hole arrays," J. Phys. D Appl. Phys. 43, 055404 (2010).
[CrossRef]

Z. L. Mei and T. J. Cui, "Arbitrary bending of electromagnetic waves using isotropic materials," J. Appl. Phys. 105, 104913 (2009).
[CrossRef]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, "Broadband Ground-Plane Cloak," Science 323, 366-369 (2009).
[CrossRef] [PubMed]

Z. L. Mei and T. J. Cui, "Experimental realization of a broadband bend structure using gradient index metamaterials," Opt. Express 17, 18354-18363 (2009).
[CrossRef] [PubMed]

Q. Cheng, H. F. Ma, and T. J. Cui, "Broadband planar Luneburg lens based on complementary metamaterials," Appl. Phys. Lett. 95, 181901 (2009).
[CrossRef]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Cunningham, J.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, "Inhomogeneous dielectric metamaterials with space-variant polarizability," Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

Datta, S.

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Phys. Rev. B 48, 14936-14943 (1993).
[CrossRef]

De Dood, M. J. A.

M. J. A. De Dood, E. Snoeks, A. Moroz, and A. Polman, "Design and optimization of 2D photonic crystal waveguides based on silicon," Opt. Quantum Electron. 34, 145-159 (2002).

De Leon, I.

F. S. Roux and I. De Leon, "Planar photonic crystal gradient index lens, simulated with a finite difference time domain method," Phys. Rev. B 74, 113103 (2006).
[CrossRef]

Doyle, W. T.

W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
[CrossRef]

Egan, W. G.

W. G. Egan and D. E. Aspnes, "Finite-wavelength effects in composite media," Phys. Rev. B 26, 5313-5320 (1982).
[CrossRef]

Fainman, Y.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, "Inhomogeneous dielectric metamaterials with space-variant polarizability," Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

U. Levy, M. Nezhad, H.-C. Kim, C.-H. Tsai, L. Pang, and Y. Fainman, "Implementation of a graded-index medium by use of subwavelength structures with graded fill factor," J. Opt. Soc. Am. A 22, 724-733 (2005).
[CrossRef]

Fan, S.

Y. Jiao, S. Fan, and D. A. B. Miller, "Designing for beam propagation in periodic and nonperiodic photonic nanostructures: extended Hamiltonian method," Phys. Rev. 70, 036612 (2004).
[CrossRef]

Gabrielli, L. H.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, "Silicon nanostructures cloak operating at optical frequencies," Nat. Photonics 3, 461-463 (2009).
[CrossRef]

Gajic, R.

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, "Coordinate transformation based design of confined metamaterial structures," Phys. Rev. B 79, 085103 (2009).
[CrossRef]

Garcia-Vidal, F. J.

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking Surface Plasmons with Structured Surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

Genov, D.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, "Ray optics at a deep-subwavelength scale: a transformation optics approach," Nano Lett. 8, 4243-4247 (2008).
[CrossRef]

Gralak, B.

W. Smigaj, and B. Gralak, "Validity of the effective-medium approximation of photonic crystals," Phys. Rev. B 77, 235445 (2008).
[CrossRef]

Halevi, P.

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic Crystal Optics and Homogenization of 2D Periodic Composites," Phys. Rev. Lett. 82, 719-722 (1999).
[CrossRef]

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

Han, S.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, "Ray optics at a deep-subwavelength scale: a transformation optics approach," Nano Lett. 8, 4243-4247 (2008).
[CrossRef]

Hingerl, K.

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, "Coordinate transformation based design of confined metamaterial structures," Phys. Rev. B 79, 085103 (2009).
[CrossRef]

Ho, K. M.

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Phys. Rev. B 48, 14936-14943 (1993).
[CrossRef]

Hou, B.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, "Design and experimental realization of a broadband transformation media field rotator at microwave frequencies," Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

Huang, T. J.

Ikeda, K.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, "Inhomogeneous dielectric metamaterials with space-variant polarizability," Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

Isic, G.

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, "Coordinate transformation based design of confined metamaterial structures," Phys. Rev. B 79, 085103 (2009).
[CrossRef]

Jensen, L.

Ji, C.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, "Broadband Ground-Plane Cloak," Science 323, 366-369 (2009).
[CrossRef] [PubMed]

Jiao, Y.

Y. Jiao, S. Fan, and D. A. B. Miller, "Designing for beam propagation in periodic and nonperiodic photonic nanostructures: extended Hamiltonian method," Phys. Rev. 70, 036612 (2004).
[CrossRef]

Juluri, B. K.

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Kildishev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Kim, H.-C.

Kirchner, A.

A. Kirchner, K. Busch, and C. M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998).
[CrossRef]

Koschny, T.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Krishnamoorthy, A.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, "Inhomogeneous dielectric metamaterials with space-variant polarizability," Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

Krokhin, A. A.

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic Crystal Optics and Homogenization of 2D Periodic Composites," Phys. Rev. Lett. 82, 719-722 (1999).
[CrossRef]

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

Kurt, H.

Lalanne, P.

Landy, N. I.

Launois, H.

Lee, J. H.

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U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

Levy, U.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, "Inhomogeneous dielectric metamaterials with space-variant polarizability," Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

U. Levy, M. Nezhad, H.-C. Kim, C.-H. Tsai, L. Pang, and Y. Fainman, "Implementation of a graded-index medium by use of subwavelength structures with graded fill factor," J. Opt. Soc. Am. A 22, 724-733 (2005).
[CrossRef]

Lewin, L.

L. Lewin, "The electrical constants of a material loaded with spherical particles," Proc. Inst. Electr. Eng. 94, 65-68 (1947).

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 569-571 (2009).
[CrossRef]

J. Li and J. B. Pendry, "Hiding under the carpet: A new strategy for cloaking," Phys. Rev. Lett. 101, 203901 (2008).
[CrossRef] [PubMed]

Lin, S.

Lipson, M.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, "Silicon nanostructures cloak operating at optical frequencies," Nat. Photonics 3, 461-463 (2009).
[CrossRef]

Liu, R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, "Broadband Ground-Plane Cloak," Science 323, 366-369 (2009).
[CrossRef] [PubMed]

Liu, Z.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, "Ray optics at a deep-subwavelength scale: a transformation optics approach," Nano Lett. 8, 4243-4247 (2008).
[CrossRef]

Ma, H. F.

Q. Cheng, H. F. Ma, and T. J. Cui, "Broadband planar Luneburg lens based on complementary metamaterials," Appl. Phys. Lett. 95, 181901 (2009).
[CrossRef]

Martin-Moreno, L.

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking Surface Plasmons with Structured Surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

Mei, Z. L.

Z. L. Mei, J. Bai, and T. J. Cui, "Gradient index metamaterials realized by drilling hole arrays," J. Phys. D Appl. Phys. 43, 055404 (2010).
[CrossRef]

Z. L. Mei and T. J. Cui, "Arbitrary bending of electromagnetic waves using isotropic materials," J. Appl. Phys. 105, 104913 (2009).
[CrossRef]

Z. L. Mei and T. J. Cui, "Experimental realization of a broadband bend structure using gradient index metamaterials," Opt. Express 17, 18354-18363 (2009).
[CrossRef] [PubMed]

Miller, D. A. B.

Y. Jiao, S. Fan, and D. A. B. Miller, "Designing for beam propagation in periodic and nonperiodic photonic nanostructures: extended Hamiltonian method," Phys. Rev. 70, 036612 (2004).
[CrossRef]

Mock, J. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, "Broadband Ground-Plane Cloak," Science 323, 366-369 (2009).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

Moroz, A.

M. J. A. De Dood, E. Snoeks, A. Moroz, and A. Polman, "Design and optimization of 2D photonic crystal waveguides based on silicon," Opt. Quantum Electron. 34, 145-159 (2002).

Nezhad, M.

Padilla, W. J.

Pang, L.

Park, W.

Pendry, J. B.

J. Li and J. B. Pendry, "Hiding under the carpet: A new strategy for cloaking," Phys. Rev. Lett. 101, 203901 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking Surface Plasmons with Structured Surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

Pinchuk, A. O.

Poitras, C. B.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, "Silicon nanostructures cloak operating at optical frequencies," Nat. Photonics 3, 461-463 (2009).
[CrossRef]

Polman, A.

M. J. A. De Dood, E. Snoeks, A. Moroz, and A. Polman, "Design and optimization of 2D photonic crystal waveguides based on silicon," Opt. Quantum Electron. 34, 145-159 (2002).

Rhee, S. J.

Roux, F. S.

F. S. Roux and I. De Leon, "Planar photonic crystal gradient index lens, simulated with a finite difference time domain method," Phys. Rev. B 74, 113103 (2006).
[CrossRef]

Ruppin, R.

R. Ruppin, "Evaluation of extended Maxwell-Garnett theories," Opt. Commun. 182, 273-279 (2000).
[CrossRef]

Russell, P. S. J.

Schatz, G. C.

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Simovski, C. R.

P. A. Belov and C. R. Simovski, "Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers," Phys. Rev. E 72, 026615 (2005).
[CrossRef]

Smigaj, W.

W. Smigaj, and B. Gralak, "Validity of the effective-medium approximation of photonic crystals," Phys. Rev. B 77, 235445 (2008).
[CrossRef]

Smith, D. R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, "Broadband Ground-Plane Cloak," Science 323, 366-369 (2009).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Snoeks, E.

M. J. A. De Dood, E. Snoeks, A. Moroz, and A. Polman, "Design and optimization of 2D photonic crystal waveguides based on silicon," Opt. Quantum Electron. 34, 145-159 (2002).

Soukoulis, C. M.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

A. Kirchner, K. Busch, and C. M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998).
[CrossRef]

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Phys. Rev. B 48, 14936-14943 (1993).
[CrossRef]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

Summers, C. J.

Tamma, V. A.

Tsai, C.-H.

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 569-571 (2009).
[CrossRef]

Vasic, B.

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, "Coordinate transformation based design of confined metamaterial structures," Phys. Rev. B 79, 085103 (2009).
[CrossRef]

Vier, D. C.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Walker, T. R.

Wen, W.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, "Design and experimental realization of a broadband transformation media field rotator at microwave frequencies," Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

Wu, Q.

Xiong, Y.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, "Ray optics at a deep-subwavelength scale: a transformation optics approach," Nano Lett. 8, 4243-4247 (2008).
[CrossRef]

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 569-571 (2009).
[CrossRef]

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 569-571 (2009).
[CrossRef]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, "Ray optics at a deep-subwavelength scale: a transformation optics approach," Nano Lett. 8, 4243-4247 (2008).
[CrossRef]

Zouganelis, G.

G. Zouganelis and D. Budimir, "Effective dielectric constant and design of sliced Lüneburg lens," Microwave Opt. Technol. Lett. 49, 2332-2337 (2007).
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Appl. Opt. (1)

Appl. Phys. Lett. (2)

Q. Cheng, H. F. Ma, and T. J. Cui, "Broadband planar Luneburg lens based on complementary metamaterials," Appl. Phys. Lett. 95, 181901 (2009).
[CrossRef]

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

J. Appl. Phys. (1)

Z. L. Mei and T. J. Cui, "Arbitrary bending of electromagnetic waves using isotropic materials," J. Appl. Phys. 105, 104913 (2009).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (2)

J. Phys. D Appl. Phys. (1)

Z. L. Mei, J. Bai, and T. J. Cui, "Gradient index metamaterials realized by drilling hole arrays," J. Phys. D Appl. Phys. 43, 055404 (2010).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

G. Zouganelis and D. Budimir, "Effective dielectric constant and design of sliced Lüneburg lens," Microwave Opt. Technol. Lett. 49, 2332-2337 (2007).
[CrossRef]

Nano Lett. (1)

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, "Ray optics at a deep-subwavelength scale: a transformation optics approach," Nano Lett. 8, 4243-4247 (2008).
[CrossRef]

Nat. Mater. (1)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 569-571 (2009).
[CrossRef]

Nat. Photonics (2)

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, "Silicon nanostructures cloak operating at optical frequencies," Nat. Photonics 3, 461-463 (2009).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Opt. Commun. (1)

R. Ruppin, "Evaluation of extended Maxwell-Garnett theories," Opt. Commun. 182, 273-279 (2000).
[CrossRef]

Opt. Express (6)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

M. J. A. De Dood, E. Snoeks, A. Moroz, and A. Polman, "Design and optimization of 2D photonic crystal waveguides based on silicon," Opt. Quantum Electron. 34, 145-159 (2002).

Phys. Rev. (1)

Y. Jiao, S. Fan, and D. A. B. Miller, "Designing for beam propagation in periodic and nonperiodic photonic nanostructures: extended Hamiltonian method," Phys. Rev. 70, 036612 (2004).
[CrossRef]

Phys. Rev. B (8)

E. Centeno, D. Cassagne, and J.-P. Albert, "Mirage and superbending effect in two-dimensional graded photonic crystals," Phys. Rev. B 73, 235119 (2006).
[CrossRef]

F. S. Roux and I. De Leon, "Planar photonic crystal gradient index lens, simulated with a finite difference time domain method," Phys. Rev. B 74, 113103 (2006).
[CrossRef]

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, "Coordinate transformation based design of confined metamaterial structures," Phys. Rev. B 79, 085103 (2009).
[CrossRef]

W. Smigaj, and B. Gralak, "Validity of the effective-medium approximation of photonic crystals," Phys. Rev. B 77, 235445 (2008).
[CrossRef]

A. Kirchner, K. Busch, and C. M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998).
[CrossRef]

W. G. Egan and D. E. Aspnes, "Finite-wavelength effects in composite media," Phys. Rev. B 26, 5313-5320 (1982).
[CrossRef]

W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
[CrossRef]

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Phys. Rev. B 48, 14936-14943 (1993).
[CrossRef]

Phys. Rev. E (3)

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

P. A. Belov and C. R. Simovski, "Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers," Phys. Rev. E 72, 026615 (2005).
[CrossRef]

Phys. Rev. Lett. (4)

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, "Inhomogeneous dielectric metamaterials with space-variant polarizability," Phys. Rev. Lett. 98, 243901 (2007).
[CrossRef] [PubMed]

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, "Design and experimental realization of a broadband transformation media field rotator at microwave frequencies," Phys. Rev. Lett. 102, 183903 (2009).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, "Hiding under the carpet: A new strategy for cloaking," Phys. Rev. Lett. 101, 203901 (2008).
[CrossRef] [PubMed]

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic Crystal Optics and Homogenization of 2D Periodic Composites," Phys. Rev. Lett. 82, 719-722 (1999).
[CrossRef]

Proc. Inst. Electr. Eng. (1)

L. Lewin, "The electrical constants of a material loaded with spherical particles," Proc. Inst. Electr. Eng. 94, 65-68 (1947).

Science (5)

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking Surface Plasmons with Structured Surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, "Broadband Ground-Plane Cloak," Science 323, 366-369 (2009).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

(a) Refractive index profile with 2D gradient and (b) its discrete approximation. (c) Implementation of the profile with the GPC.

Fig. 2.
Fig. 2.

Refractive index calculated from dispersion curves of a PC as a function of normalized frequency Ω = a/λ for the PC with (a) SiO2 and (b) Si rods. On the left side, (x1), the values for TM mode are shown and on the right side, (x2), the values for TE mode. Straight red lines represent values of refractive index calculated with EMT. Ratios r/a are changed from 0.1 to 0.5: □(r/a = 0.1), ○(r/a = 0.2), ♢(r/a = 0.3), △(r/a = 0.4), ∇(r/a = 0.5).

Fig. 3.
Fig. 3.

FEM simulation results of TE mode for (a) the original lens and (b) the GPC based Luneburg lens, Ω0 = 0.14 whereas Ω ∈ (0.14,0.2). On the left side, (x1), the z-component of magnetic field is shown and on the right side, (x2), the magnetic field intensity.

Fig. 4.
Fig. 4.

(Reflection (S 11 parameter) from the original (blue) and GPC (red) Luneburg lens.

Fig. 5.
Fig. 5.

FEM simulation results of the magnetic field intensity for TE mode in the GPC based Luneburg lens: (a) Ω0 = 0.08, (b) Ω0 = 0.3, (c) Ω0 = 0.33 and (d) Ω0 = 0.38.

Fig. 6.
Fig. 6.

The mapping z = exp ( w C ) , C = 1, for a design of an electromagnetic beam bend: (a) the square domain in Cartesian coordinates of complex plane w = u + iv and (b) its image, the annular segment in curvilinear coordinates of complex plane z = x + iy. The inner and outer bend radii are exp(u 1) and exp(u 2), respectively, whereas π/2 is the bend angle. Red arrows show propagation directions of an electromagnetic wave.

Fig. 7.
Fig. 7.

FEM simulation results of TM mode for (a) the original bend and (b) the GPC based bend, Ω0 = 0.11. On the left side, (x1), the z-component of electric field is shown and on the right side, (x2), the electric field intensity.

Fig. 8.
Fig. 8.

Transmission (S 21 parameter) through the original (blue) and GPC (red) electromagnetic beam bend. The small oscillations in transmissions are Fabry-Perot resonances within the bends due to impedance mismatching at the bend entrance and exit.

Fig. 9.
Fig. 9.

FEM simulation results of the z-component of electric field for TM mode in the GPC based bend: (a) Ω0 = 0.068 and (b) Ω0 = 0.15.

Equations (11)

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k = n ij ω c ,
k = ( n eff α ( k ) ) ij ω ( k ) c ,
( n eff α ( k ) ) ij = n ij .
( n eff α ) ij = n ij .
ε plane = ε host + f ε host ( ε rod ε host ) ε host + L plane ( 1 f ) ( ε rod ε host ) ,
ε z = f ε rod + ( 1 f ) ε host .
r ij = a ( ε host n ij 2 ) ( ε host + ε rod ) π ( ε host + n ij 2 ) ( ε host ε rod ) ,
r ij = a n ij 2 ε host π ( ε rod ε host ) .
n = 2 ( ρ R ) 2 , ρ < R ,
z = exp ( w C ) ,
n = dw dz = C x 2 + y 2 .

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