Abstract

We design an all–dielectric Lüneburg lens as an adiabatic space–variant lattice explicitly accounting for finite film thickness. We describe an all–analytical approach to compensate for the finite height of subwavelength dielectric structures in the pass–band regime. This method calculates the effective refractive index of the infinite–height lattice from effective medium theory, then embeds a medium of the same effective index into a slab waveguide of finite height and uses the waveguide dispersion diagram to calculate a new effective index. The results are compared with the conventional numerical treatment – a direct band diagram calculation, using a modified three–dimensional lattice with the superstrate and substrate included in the cell geometry. We show that the analytical results are in good agreement with the numerical ones, and the performance of the thin–film Lüneburg lens is quite different than the estimates obtained assuming infinite height.

© 2012 OSA

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  1. C. A. Swainson (alias J. C. Maxwell), “Problems,” Cambridge Dublin Math. J. 8, 188–189 (1854).
  2. R. K. Lüneburg, Mathematical Theory of Optics (Brown U. P., 1944).
  3. J. E. Eaton, An Extension of the Luneburg–Type Lenses (Rep. No. 4110, Naval Res. Lab., 1953).
  4. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [CrossRef] [PubMed]
  5. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef] [PubMed]
  6. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568 (2009).
    [CrossRef] [PubMed]
  7. D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express 18, 3181–3186 (2010).
    [CrossRef] [PubMed]
  8. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18, 20321–20333 (2010).
    [CrossRef]
  9. E. Hecht, Optics, (4th ed., Section 6.4) (Addison-Wesley, 2002).
  10. G. R. Schmidt, Compound Optical Arrays and Polymer Tapered Gradient Index Lenses (PhD Thesis) (University of Rochester, 2009).
  11. Manufacturable Gradient Index Optics (M–GRIN) (Broad Agency Announcement, Defense Advanced Research Projects Agency, 2010).
  12. Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method,” Phys. Rev. E 70, 036612 (2004).
    [CrossRef]
  13. P. S. J. Russel and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol. 17, 1982–1988 (1999).
    [CrossRef]
  14. S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, “Design and fabrication of dielectric nanostructured Luneburg lens in optical frequencies,” in Optical MEMS and Nanophotonics, (IEEE Photonics Society, 2010), Paper Th1–1.
    [CrossRef]
  15. S. Takahashi, Design and Fabrication of micro- and nano-dielectric structures for imaging and focusing at optical frequencies (PhD Thesis) (Massachusetts Institute of Technology, 2011).
    [PubMed]
  16. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
    [CrossRef] [PubMed]
  17. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
    [CrossRef]
  18. T. Zentgraf, J. Valentine, N. Tapia, J. Li, and X. Zhang, “An optical “Janus” device for integrated photonics,” Adv. Mater. 22, 2561–2564 (2010).
    [CrossRef] [PubMed]
  19. K. K. Y. Lee, Y. Avniel, and S. G. Johnson, “Design strategies and rigorous conditions for single-polarization single-mode waveguides,” Opt. Express 16, 15170–15184 (2008).
    [CrossRef] [PubMed]
  20. R. Ulrich and M. Tacke, “Submillimeter waveguiding on periodic metal structure,” Appl. Phys. Lett. 22, 251–253 (1973).
    [CrossRef]
  21. R. D. Meade, A. Devenyi, J. Joannopoulos, O. Alerhand, D. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
    [CrossRef]
  22. S. Fan, P. R. Villeneuve, J. Joannopoulos, and E. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997).
    [CrossRef]
  23. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751 (1999).
    [CrossRef]
  24. M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163 (2002).
    [CrossRef]
  25. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, (2nd ed.) (Princeton U. P., 2008).
  26. J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 8, 1246–1257 (2002).
    [CrossRef]
  27. M. Ahmadlou, M. Kamarei, and M. H. Sheikhi, “Negative refraction and focusing analysis in a left-handed material slab and realization with a 3D photonic crystal structure,” J. Opt. A, Pure Appl. Opt. 8, 199 (2006).
    [CrossRef]
  28. J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with uniform layered structures,” Opt. Express 19, 8625–8631 (2011).
    [CrossRef] [PubMed]
  29. X. Chen, T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
    [CrossRef]
  30. M. Hammer and O. V. Ivanova, “Effective index approximations of photonic crystal slabs: a 2-to-1-D assessment,” Opt. Quantum Electron. 41, 267–283 (2009).
    [CrossRef]
  31. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).
  32. R. Bräuer and O. Bryngdahl, “Design of antireflection gratings with approximate and rigorous methods,” Appl. Opt. 33, 7875–7882 (1994).
    [CrossRef] [PubMed]
  33. W. Yu, T. Konishi, T. Hamamoto, H. Toyota, T. Yotsuya, and Y. Ichioka, “Polarization-multiplexed diffractive optical elements fabricated by subwavelength structures,” Appl. Opt. 41, 96–100 (2002).
    [CrossRef] [PubMed]
  34. J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2008).
  35. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
    [CrossRef] [PubMed]
  36. H. Gao, S. Takahashi, L. Tian, and G. Barbastathis, “Aperiodic subwavelength Lüneburg lens with nonlinear Kerr effect compensation,” Opt. Express 19, 2257–2265 (2011).
    [CrossRef] [PubMed]
  37. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. 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]

2011

2010

D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express 18, 3181–3186 (2010).
[CrossRef] [PubMed]

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

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. 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]

T. Zentgraf, J. Valentine, N. Tapia, J. Li, and X. Zhang, “An optical “Janus” device for integrated photonics,” Adv. Mater. 22, 2561–2564 (2010).
[CrossRef] [PubMed]

2009

H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 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 (2009).
[CrossRef] [PubMed]

M. Hammer and O. V. Ivanova, “Effective index approximations of photonic crystal slabs: a 2-to-1-D assessment,” Opt. Quantum Electron. 41, 267–283 (2009).
[CrossRef]

2008

2006

M. Ahmadlou, M. Kamarei, and M. H. Sheikhi, “Negative refraction and focusing analysis in a left-handed material slab and realization with a 3D photonic crystal structure,” J. Opt. A, Pure Appl. Opt. 8, 199 (2006).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

2004

X. Chen, T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[CrossRef]

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

2002

W. Yu, T. Konishi, T. Hamamoto, H. Toyota, T. Yotsuya, and Y. Ichioka, “Polarization-multiplexed diffractive optical elements fabricated by subwavelength structures,” Appl. Opt. 41, 96–100 (2002).
[CrossRef] [PubMed]

M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163 (2002).
[CrossRef]

J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 8, 1246–1257 (2002).
[CrossRef]

2001

1999

P. S. J. Russel and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol. 17, 1982–1988 (1999).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751 (1999).
[CrossRef]

1997

S. Fan, P. R. Villeneuve, J. Joannopoulos, and E. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997).
[CrossRef]

1994

R. D. Meade, A. Devenyi, J. Joannopoulos, O. Alerhand, D. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

R. Bräuer and O. Bryngdahl, “Design of antireflection gratings with approximate and rigorous methods,” Appl. Opt. 33, 7875–7882 (1994).
[CrossRef] [PubMed]

1973

R. Ulrich and M. Tacke, “Submillimeter waveguiding on periodic metal structure,” Appl. Phys. Lett. 22, 251–253 (1973).
[CrossRef]

1956

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

1854

C. A. Swainson (alias J. C. Maxwell), “Problems,” Cambridge Dublin Math. J. 8, 188–189 (1854).

Ahmadlou, M.

M. Ahmadlou, M. Kamarei, and M. H. Sheikhi, “Negative refraction and focusing analysis in a left-handed material slab and realization with a 3D photonic crystal structure,” J. Opt. A, Pure Appl. Opt. 8, 199 (2006).
[CrossRef]

Alerhand, O.

R. D. Meade, A. Devenyi, J. Joannopoulos, O. Alerhand, D. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Avniel, Y.

Barbastathis, G.

H. Gao, S. Takahashi, L. Tian, and G. Barbastathis, “Aperiodic subwavelength Lüneburg lens with nonlinear Kerr effect compensation,” Opt. Express 19, 2257–2265 (2011).
[CrossRef] [PubMed]

S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, “Design and fabrication of dielectric nanostructured Luneburg lens in optical frequencies,” in Optical MEMS and Nanophotonics, (IEEE Photonics Society, 2010), Paper Th1–1.
[CrossRef]

Bartal, G.

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

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. 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]

Birks, T. A.

Bräuer, R.

Bryngdahl, O.

Cardenas, J.

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

Chang, C.

S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, “Design and fabrication of dielectric nanostructured Luneburg lens in optical frequencies,” in Optical MEMS and Nanophotonics, (IEEE Photonics Society, 2010), Paper Th1–1.
[CrossRef]

Chen, X.

X. Chen, T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Devenyi, A.

R. D. Meade, A. Devenyi, J. Joannopoulos, O. Alerhand, D. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Eaton, J. E.

J. E. Eaton, An Extension of the Luneburg–Type Lenses (Rep. No. 4110, Naval Res. Lab., 1953).

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[CrossRef] [PubMed]

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. E 70, 036612 (2004).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. Joannopoulos, and E. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997).
[CrossRef]

Gabrielli, H.

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

Gabrielli, L. H.

Gajic, R.

Gao, H.

Grzegorczyk, T. M.

X. Chen, T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Hamamoto, T.

Hammer, M.

M. Hammer and O. V. Ivanova, “Effective index approximations of photonic crystal slabs: a 2-to-1-D assessment,” Opt. Quantum Electron. 41, 267–283 (2009).
[CrossRef]

Hecht, E.

E. Hecht, Optics, (4th ed., Section 6.4) (Addison-Wesley, 2002).

Hingerl, K.

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. 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]

Ichioka, Y.

Isic, G.

Ivanova, O. V.

M. Hammer and O. V. Ivanova, “Effective index approximations of photonic crystal slabs: a 2-to-1-D assessment,” Opt. Quantum Electron. 41, 267–283 (2009).
[CrossRef]

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. E 70, 036612 (2004).
[CrossRef]

Joannopoulos, J.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. 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. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

S. Fan, P. R. Villeneuve, J. Joannopoulos, and E. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997).
[CrossRef]

R. D. Meade, A. Devenyi, J. Joannopoulos, O. Alerhand, D. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751 (1999).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, (2nd ed.) (Princeton U. P., 2008).

Johnson, S.

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. 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]

K. K. Y. Lee, Y. Avniel, and S. G. Johnson, “Design strategies and rigorous conditions for single-polarization single-mode waveguides,” Opt. Express 16, 15170–15184 (2008).
[CrossRef] [PubMed]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751 (1999).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, (2nd ed.) (Princeton U. P., 2008).

Kamarei, M.

M. Ahmadlou, M. Kamarei, and M. H. Sheikhi, “Negative refraction and focusing analysis in a left-handed material slab and realization with a 3D photonic crystal structure,” J. Opt. A, Pure Appl. Opt. 8, 199 (2006).
[CrossRef]

Kash, K.

R. D. Meade, A. Devenyi, J. Joannopoulos, O. Alerhand, D. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Kolodziejski, L. A.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751 (1999).
[CrossRef]

Kong, J. A.

X. Chen, T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[CrossRef]

J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2008).

Konishi, T.

Lee, K. K. Y.

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

Li, J.

T. Zentgraf, J. Valentine, N. Tapia, J. Li, and X. Zhang, “An optical “Janus” device for integrated photonics,” Adv. Mater. 22, 2561–2564 (2010).
[CrossRef] [PubMed]

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

Lipson, M.

D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express 18, 3181–3186 (2010).
[CrossRef] [PubMed]

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

Liu, L.

Loncar, M.

J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 8, 1246–1257 (2002).
[CrossRef]

Lüneburg, R. K.

R. K. Lüneburg, Mathematical Theory of Optics (Brown U. P., 1944).

Luo, Y.

Meade, R. D.

R. D. Meade, A. Devenyi, J. Joannopoulos, O. Alerhand, D. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, (2nd ed.) (Princeton U. P., 2008).

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. E 70, 036612 (2004).
[CrossRef]

Mortensen, N. A.

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. 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]

Pacheco, J.

X. Chen, T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Poitras, C. B.

D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express 18, 3181–3186 (2010).
[CrossRef] [PubMed]

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

Qiu, M.

M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163 (2002).
[CrossRef]

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. 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]

Russel, P. S. J.

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Scherer, A.

J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 8, 1246–1257 (2002).
[CrossRef]

Schmidt, G. R.

G. R. Schmidt, Compound Optical Arrays and Polymer Tapered Gradient Index Lenses (PhD Thesis) (University of Rochester, 2009).

Schubert, E.

S. Fan, P. R. Villeneuve, J. Joannopoulos, and E. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997).
[CrossRef]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Sheikhi, M. H.

M. Ahmadlou, M. Kamarei, and M. H. Sheikhi, “Negative refraction and focusing analysis in a left-handed material slab and realization with a 3D photonic crystal structure,” J. Opt. A, Pure Appl. Opt. 8, 199 (2006).
[CrossRef]

Smith, D.

R. D. Meade, A. Devenyi, J. Joannopoulos, O. Alerhand, D. Smith, and K. Kash, “Novel applications of photonic band gap materials: Low-loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Spadoti, D. H.

Swainson, C. A.

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Figures (9)

Fig. 1
Fig. 1

(a) Finite height rod lattice structure investigated in this paper. (b) 2D rod lattice structure assuming infinite height.

Fig. 2
Fig. 2

Effective guiding medium (EGM) approximation of 2D finite height rod lattice structure.

Fig. 3
Fig. 3

Graphical solutions of wave guidance condition [Eq. (6)&(7)] for TE (a) and TM (b) polarizations. Blue and red lines are the left and right hand sides of these equations, respectively. Operating frequencies ω1 = 0.11 × 2πc/a, ω2 = 0.16 × 2πc/a, ω3 = 0.14 × 2πc/a and ω4 = 0.18 × 2πc/a. Rod radius r = 0.50a.

Fig. 4
Fig. 4

(a) The supercell used in the DBD method for the finite height rod lattice structure. (b) Isofrequency contour of the supercell with r = 0.50a where the first band only is shown. Labels on the lines denote the corresponding normalized frequency ωa/2πc. The bold blue line corresponds to the wavelength λ = 6a used in this paper. (c) Field distribution of the waveguide slab at a particular x slice. Color shading denotes magnetic field (Hy) distribution and black contours illustrate silicon rods.

Fig. 5
Fig. 5

(a) Comparison between the dispersion relation for finite–height silicon rod lattice [Fig. 1(a)] calculated from the EGM and DBD method, and the dispersion relation for infinite–height 2D rod lattice [Fig. 1(b)]. For each case, the two lowest bands representing the TM and TE modes are shown. (b) Relationship between effective refractive index and rod radius calculated from both methods, compared with the relationship for infinite–height 2D rod lattice. Free space wavelength of light is λ = 6a = 1550 nm.

Fig. 6
Fig. 6

(a) Top view and side view of the thin–film subwavelength Lüneburg lens designed by EGM method for TE mode and (b) the corresponding 3D FDTD and Hamiltonian ray tracing results. (c) Top view and side view for TM mode and (d) the corresponding 3D FDTD and ray tracing results. Red circles outline the edge of Lüneburg lens, where radius R = 30a. Blue lines are the ray tracing results and color shading denotes the field [Hy for (b) and Ey for (d)] distribution, where red is positive and blue is negative.

Fig. 7
Fig. 7

Structure and the corresponding 3D FDTD and Hamiltonian ray tracing for the thin–film subwavelength Lüneburg lens shown in Fig. 6, but designed by the DBD method instead.

Fig. 8
Fig. 8

Structure and the corresponding 3D FDTD and Hamiltonian ray tracing for the thin–film subwavelength Lüneburg lens shown in Fig. 6, but designed using the EGM method without second–order terms when estimating the effective refractive indices.

Fig. 9
Fig. 9

FDTD and Hamiltonian ray–tracing results of the subwavelength Lüneburg lens made of finite height silicon rods, but designed assuming infinite height. The color conventions are the same as in Figs. 6(b) and 6(d).

Equations (8)

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n TE 2 = n 0 TE 2 + π 2 3 ( T λ ) f 2 ( 1 f ) 2 ( n 2 1 ) 2 ,
n TM 2 = n 0 TM 2 + π 2 3 ( T λ ) f 2 ( 1 f ) 2 n 0 TM 6 n 0 TE 2 ( 1 n 2 1 ) 2 ,
n 0 TE 2 = f n 2 + ( 1 f ) , n 0 TM 2 = 1 / ( f n 2 + ( 1 f ) )
n 2 D TE = 1 f + f n TE 2 ,
n 2 D TM = ( ( 1 f ) + f n TM 2 + n TE 2 n TE 2 ( 1 f ) + f ) / 2
( TE : ) tan ( k II y h ) = ɛ II k II y ( ɛ III k z 2 ɛ I ω 2 / c 2 + ɛ I k z 2 ɛ III ω 2 / c 2 ) ɛ I ɛ III k II y 2 ɛ II 2 k z 2 ɛ I ω 2 / c 2 k z 2 ɛ III ω 2 / c 2 F TE ( k II y h ) ,
( TM : ) ( tan k II y h ) = k II y ( k z 2 ɛ I ω 2 / c 2 + k z 2 ɛ III ω 2 / c 2 ) k II y 2 k z 2 ɛ I ω 2 / c 2 k z 2 ɛ III ω 2 / c 2 F TM ( k II y h ) ,
d q d σ = H p , d p d σ = H q ,

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