Abstract

We propose an analytical approach to the study of graded photonic crystals operating in the metamaterial regime. Relationships are given to predict the optical index map and hole-drilling distribution required to make light follow a prescribed path. The method is applied to proof-of-concept structures based on silicon-on-insulator technology. Light propagation is studied using FDTD simulation to verify the light trajectory, study the influence of extended light beams, and evaluate the robustness of the semiclassical approach based on the equations of Hamiltonian optics. The overall approach can be used for the straightforward design of new optical functionalities within the photonic metamaterial regime.

© 2011 Optical Society of America

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References

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  1. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17, 14872–14879 (2009).
    [CrossRef] [PubMed]
  2. A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: recent advances and outlook,” Metamaterials 2, 1–17 (2009).
  3. 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]
  4. Z. L. Mei, J. Bai, and T. J. Cui, “Gradient index metamaterials realized by drilling hole arrays,” J. Phys. D 43, 055404(2010).
    [CrossRef]
  5. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
    [CrossRef] [PubMed]
  6. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. Lett. 3, 461–463 (2009).
    [CrossRef]
  7. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
    [CrossRef] [PubMed]
  8. W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. 96, 041102 (2010).
    [CrossRef]
  9. F. Borghero and G. Bozis, “A two-dimensional inverse problem of geometrical optics,” J. Phys. A: Math. Gen. 38, 175–184(2005).
    [CrossRef]
  10. P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
    [CrossRef]
  11. P. St. J. Russell and T. A. Birks, “Hamiltonian optics of non-uniform photonic crystals,” J. Lightwave Technol. 17, 1982–1988(1999).
    [CrossRef]
  12. Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic nanostructures: extended Hamiltonian method,” Phys. Rev. E 70, 036612 (2004).
    [CrossRef]
  13. P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E 72, 026615 (2005).
    [CrossRef]
  14. B. Vasic, G. Isic, R. Gajic, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18, 20321–20333 (2010).
    [CrossRef] [PubMed]
  15. A. F. Oskooi, D. Roundy, M. Ibanescu, P. BermelJ. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
    [CrossRef]
  16. S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
    [CrossRef]
  17. D. Bernier, E. Cassan, Le Roux, D. Marris-Morini, and L. Vivien, “Efficient band-edge light injection in two dimensional planar photonic crystals using a gradual interface,” Opt. Eng. Lett. 48, 070501 (2009).
    [CrossRef]

2010 (4)

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

W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. 96, 041102 (2010).
[CrossRef]

B. Vasic, G. Isic, R. Gajic, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18, 20321–20333 (2010).
[CrossRef] [PubMed]

A. F. Oskooi, D. Roundy, M. Ibanescu, P. BermelJ. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

2009 (5)

D. Bernier, E. Cassan, Le Roux, D. Marris-Morini, and L. Vivien, “Efficient band-edge light injection in two dimensional planar photonic crystals using a gradual interface,” Opt. Eng. Lett. 48, 070501 (2009).
[CrossRef]

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. Lett. 3, 461–463 (2009).
[CrossRef]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef] [PubMed]

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

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: recent advances and outlook,” Metamaterials 2, 1–17 (2009).

2008 (1)

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

2006 (1)

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 (2)

F. Borghero and G. Bozis, “A two-dimensional inverse problem of geometrical optics,” J. Phys. A: Math. Gen. 38, 175–184(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]

2004 (1)

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

2002 (1)

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

1999 (1)

1991 (1)

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

Bai, J.

Z. L. Mei, J. Bai, and T. J. Cui, “Gradient index metamaterials realized by drilling hole arrays,” J. Phys. D 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, 568–571(2009).
[CrossRef] [PubMed]

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]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. BermelJ. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Bernier, D.

D. Bernier, E. Cassan, Le Roux, D. Marris-Morini, and L. Vivien, “Efficient band-edge light injection in two dimensional planar photonic crystals using a gradual interface,” Opt. Eng. Lett. 48, 070501 (2009).
[CrossRef]

Bienstman, P.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Birks, T. A.

Boltasseva, A.

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: recent advances and outlook,” Metamaterials 2, 1–17 (2009).

Borghero, F.

F. Borghero and G. Bozis, “A two-dimensional inverse problem of geometrical optics,” J. Phys. A: Math. Gen. 38, 175–184(2005).
[CrossRef]

Bozis, G.

F. Borghero and G. Bozis, “A two-dimensional inverse problem of geometrical optics,” J. Phys. A: Math. Gen. 38, 175–184(2005).
[CrossRef]

Cardenas, J.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. Lett. 3, 461–463 (2009).
[CrossRef]

Cassan, E.

D. Bernier, E. Cassan, Le Roux, D. Marris-Morini, and L. Vivien, “Efficient band-edge light injection in two dimensional planar photonic crystals using a gradual interface,” Opt. Eng. Lett. 48, 070501 (2009).
[CrossRef]

Chen, L.

W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. 96, 041102 (2010).
[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]

Ding, W.

W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. 96, 041102 (2010).
[CrossRef]

Fan, S.

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

Gabrielli, L. H.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. Lett. 3, 461–463 (2009).
[CrossRef]

Gajic, R.

Hingerl, K.

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. BermelJ. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Isic, G.

J. Cui, T.

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

Jiao, Y.

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

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. BermelJ. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. BermelJ. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

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]

L. Mei, Z.

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

Landy, N. I.

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(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]

Lidorikis, E.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Lipson, M.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. Lett. 3, 461–463 (2009).
[CrossRef]

Liu, Y.

W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. 96, 041102 (2010).
[CrossRef]

Marris-Morini, D.

D. Bernier, E. Cassan, Le Roux, D. Marris-Morini, and L. Vivien, “Efficient band-edge light injection in two dimensional planar photonic crystals using a gradual interface,” Opt. Eng. Lett. 48, 070501 (2009).
[CrossRef]

Miller, D. A. B.

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

Mock, J. 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]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. BermelJ. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Padilla, W. J.

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]

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]

Poitras, C. B.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. Lett. 3, 461–463 (2009).
[CrossRef]

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. BermelJ. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Roux, Le

D. Bernier, E. Cassan, Le Roux, D. Marris-Morini, and L. Vivien, “Efficient band-edge light injection in two dimensional planar photonic crystals using a gradual interface,” Opt. Eng. Lett. 48, 070501 (2009).
[CrossRef]

Russell, P. St. J.

P. St. J. Russell and T. A. Birks, “Hamiltonian optics of non-uniform photonic crystals,” J. Lightwave Technol. 17, 1982–1988(1999).
[CrossRef]

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

Schurig, D.

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]

Shalaev, V. M.

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: recent advances and outlook,” Metamaterials 2, 1–17 (2009).

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]

Skorobogatiy, M. A.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Smith, D. R.

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]

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]

Sun, X.

W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. 96, 041102 (2010).
[CrossRef]

Tang, D.

W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. 96, 041102 (2010).
[CrossRef]

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef] [PubMed]

Vasic, B.

Vivien, L.

D. Bernier, E. Cassan, Le Roux, D. Marris-Morini, and L. Vivien, “Efficient band-edge light injection in two dimensional planar photonic crystals using a gradual interface,” Opt. Eng. Lett. 48, 070501 (2009).
[CrossRef]

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef] [PubMed]

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. 96, 041102 (2010).
[CrossRef]

Comput. Phys. Commun. (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. BermelJ. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

J. Phys. A: Math. Gen. (1)

F. Borghero and G. Bozis, “A two-dimensional inverse problem of geometrical optics,” J. Phys. A: Math. Gen. 38, 175–184(2005).
[CrossRef]

J. Phys. D (1)

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

Metamaterials (1)

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: recent advances and outlook,” Metamaterials 2, 1–17 (2009).

Nat. Mater. (1)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef] [PubMed]

Nat. Photon. Lett. (1)

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. Lett. 3, 461–463 (2009).
[CrossRef]

Opt. Eng. Lett. (1)

D. Bernier, E. Cassan, Le Roux, D. Marris-Morini, and L. Vivien, “Efficient band-edge light injection in two dimensional planar photonic crystals using a gradual interface,” Opt. Eng. Lett. 48, 070501 (2009).
[CrossRef]

Opt. Express (2)

Phys. Rev. E (3)

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic nanostructures: extended Hamiltonian method,” Phys. Rev. E 70, 036612 (2004).
[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. (1)

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

Science (1)

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]

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Figures (6)

Fig. 1
Fig. 1

Schematic picture of the problem under consideration: light path being known through { x ( σ ) , y ( σ ) } parametric curve, Hamiltonian optics propagation is considered to derive the required n ( x , y ) refractive index in-plane distribution.

Fig. 2
Fig. 2

FDTD simulation of the considered 90 ° circular turn: (a) structure permittivity overview and (b) field steady state at a / λ = 0.13 (the source used in FDTD simulation is responsible for field excitation in the two left and right horizontal directions).

Fig. 3
Fig. 3

Optical transmission (T) and reflection (R) power coefficients of the R = 40 a 90 ° bend based on the studied graded PhC ( loss = 1 T R ) .

Fig. 4
Fig. 4

Transmission properties of the 90 ° graded PhC turn for different values of the bending curvature radius R: (a)–(c) steady-state fields obtained for R = 10 a ( 2 μm ), R = 20 a ( 4 μm ), and R = 30 a ( 6 μm ), respectively, and (d) intrinsic optical transmission of 90 ° bend at a / λ = 0.13 , i.e., obtained by removing losses due to reflections at the input/output interfaces.

Fig. 5
Fig. 5

S-bend beam shifter: (a) permittivity in-plane distribution and (b) steady-state field profile obtained using the FDTD simulation.

Fig. 6
Fig. 6

Logarithmic spiral light path: (a) index map calculated analytically and considered after in FDTD simulation and (b)–(g) field snapshots at different increasing times.

Equations (15)

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

d r d σ = grad k ( H k ( k , r ) ) , d k d σ = grad r ( H k ( k , r ) ) ,
k = k x 2 + k y 2 = n ( x , y ) ω c ,
d x d σ = 2 k x / n 2 ( x , y ) , d y d σ = 2 k y / n 2 ( x , y ) ,
d k x d σ = 2 n ( x , y ) x ( k x 2 + k y 2 ) / n 3 ( x , y ) , d k y d σ = 2 n ( x , y ) y ( k x 2 + k y 2 ) / n 3 ( x , y ) ,
k x = n 2 ( x , y ) d x d σ 2 , k y = n 2 ( x , y ) d y d σ 2 .
ln ( n ( x , y ) ) x = d 2 x ( σ ) d σ 2 ( d x ( σ ) d σ ) 2 + ( d y ( σ ) d σ ) 2 , ln ( n ( x , y ) ) y = d 2 y ( σ ) d σ 2 ( d x ( σ ) d σ ) 2 + ( d y ( σ ) d σ ) 2 .
n ( x , y ) = ( 1 f ( x , y ) n 1 + f ( x , y ) n 2 ,
( r a ) ( x , y ) = 1 π n 1 n ( x , y ) n 1 n 2 .
ln ( n ( x , y ) ) x = x x 2 + y 2 , ln ( n ( x , y ) ) y = y x 2 + y 2 .
n ( x , y ) = C x 2 + y 2 = C ρ , with     ρ = x 2 + y 2 ,
ln ( n ( x , y ) ) x = x R 2 , ln ( n ( x , y ) ) y = y R 2 .
n ( x , y ) = n 0 exp ( x 2 + y 2 2 R 2 ) = n 0 exp ( ρ 2 2 R 2 ) ,
( r a ) ( x , y ) = 1 π n slab n 0 exp ( x 2 + y 2 2 R 2 ) n slab n hole .
ln ( n ( x , y ) ) x = b 2 1 b 2 + 1 x x 2 + y 2 2 b b 2 + 1 y x 2 + y 2 .
n ( x , y ) = D ρ 1 b 2 1 + b 2 , with     ρ = x 2 + y 2 .

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