Abstract

The counterintuitive properties of photonic crystals, such as all-angle negative refraction (AANR) [J. Mod. Opt. 34, 1589 (1987)] and high-directivity via ultrarefraction [Phys. Rev. Lett. 89, 213902 (2002)], as well as localized defect modes, are known to be associated with anomalous dispersion near the edge of stop bands. We explore the implications of an asymptotic approach to uncover the underlying structure behind these phenomena. Conventional homogenization is widely assumed to be ineffective for modeling photonic crystals as it is limited to low frequencies when the wavelength is long relative to the microstructural length scales. Here a recently developed high-frequency homogenization (HFH) theory [Proc. R. Soc. Lond. A 466, 2341 (2010)] is used to generate effective partial differential equations on a macroscale, which have the microscale embedded within them through averaged quantities, for checkerboard media. For physical applications, ultrarefraction is well described by an equivalent homogeneous medium with an effective refractive index given by the HFH procedure, the decay behavior of localized defect modes is characterized completely, and frequencies at which AANR occurs are all determined analytically. We illustrate our findings numerically with a finite-size checkerboard using finite elements, and we emphasize that conventional effective medium theory cannot handle such high frequencies. Finally, we look at light confinement effects in finite-size checkerboards behaving as open resonators when the condition for AANR is met [J. Phys. Condens. Matter 15, 6345 (2003)].

© 2011 Optical Society of America

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  49. S. Guenneau, S. A. Ramakrishna, S. Enoch, S. Chakrabarti, G. Tayeb, and B. Gralak, “Cloaking and imaging effects in plasmonic checkerboards of negative epsilon and mu and dielectric photonic crystal checkerboards,” Photon. Nanostruct. Fund. Appl. 5, 63–72 (2007).
    [CrossRef]
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    [CrossRef]
  51. M. Farhat, S. Guenneau, S. Enoch, G. Tayeb, A. Movchan, and N. Movchan, “Analytical and numerical analysis of lensing effect for linear surface water waves through a square array of close to touching rigid square cylinders,” Phys. Rev. E 77, 046308(2008).
    [CrossRef]
  52. M. Farhat, S. Guenneau, S. Enoch, A. Movchan, and G. Petursson, “Focussing bending waves via negative refraction in perforated thin plates,” Appl. Phys. Lett. 96, 081909 (2010).
    [CrossRef]
  53. M. Farhat, S. Guenneau, and S. Enoch, “High-directivity and confinement of flexural waves through ultrarefraction in thin perforated plates,” Euro. Phys. Lett. 91, 54003 (2010).
    [CrossRef]
  54. X. Hu, Y. Shu, X. Liu, R. Fu, and J. Zi, “Superlensing effect in liquid surface waves,” Phys. Rev. E 69, 030201 (2004).
    [CrossRef]
  55. M. Farhat, S. Guenneau, S. Enoch, and A. Movchan, “All-angle negative refraction and ultra-refraction for liquid surface waves in 2-D phononic crystals,” J. Comput. Appl. Math. 234, 2011–2019 (2010).
    [CrossRef]

2011 (1)

E. Nolde, R. V. Craster, and J. Kaplunov, “High frequency homogenization for structural mechanics,” J. Mech. Phys. Solids 59, 651–671 (2011).
[CrossRef]

2010 (5)

R. V. Craster, J. Kaplunov, and A. V. Pichugin, “High frequency homogenization for periodic media,” Proc. R. Soc. London A 466, 2341–2362 (2010).
[CrossRef]

R. V. Craster, J. Kaplunov, and J. Postnova, “High frequency asymptotics, homogenization and localization for lattices,” Q. J. Mech. Appl. Math. 63, 497–519 (2010).
[CrossRef]

M. Farhat, S. Guenneau, S. Enoch, A. Movchan, and G. Petursson, “Focussing bending waves via negative refraction in perforated thin plates,” Appl. Phys. Lett. 96, 081909 (2010).
[CrossRef]

M. Farhat, S. Guenneau, and S. Enoch, “High-directivity and confinement of flexural waves through ultrarefraction in thin perforated plates,” Euro. Phys. Lett. 91, 54003 (2010).
[CrossRef]

M. Farhat, S. Guenneau, S. Enoch, and A. Movchan, “All-angle negative refraction and ultra-refraction for liquid surface waves in 2-D phononic crystals,” J. Comput. Appl. Math. 234, 2011–2019 (2010).
[CrossRef]

2009 (2)

S. Guenneau and S. A. Ramakrishna, “Negative refractive index, perfect lenses and checkerboards: trapping and imaging effects in folded optical spaces,” C. R. Phys. 10, 352–378 (2009).
[CrossRef]

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Guided and standing Bloch waves in periodic elastic strips,” Waves Random Complex Media 19, 321–346 (2009).
[CrossRef]

2008 (2)

K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. Martijn de Sterke, “Shallow defect states in two-dimensional photonic crystals,” Phys. Rev. A 77, 063839 (2008).
[CrossRef]

M. Farhat, S. Guenneau, S. Enoch, G. Tayeb, A. Movchan, and N. Movchan, “Analytical and numerical analysis of lensing effect for linear surface water waves through a square array of close to touching rigid square cylinders,” Phys. Rev. E 77, 046308(2008).
[CrossRef]

2007 (5)

A. Alù, M. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[CrossRef]

S. Guenneau, S. A. Ramakrishna, S. Enoch, S. Chakrabarti, G. Tayeb, and B. Gralak, “Cloaking and imaging effects in plasmonic checkerboards of negative epsilon and mu and dielectric photonic crystal checkerboards,” Photon. Nanostruct. Fund. Appl. 5, 63–72 (2007).
[CrossRef]

A. B. Movchan, N. V. Movchan, and R. C. McPhedran, “Bloch–Floquet bending waves in perforated thin plates,” Proc. R. Soc. London A 463, 2505–2518 (2007).
[CrossRef]

J. M. Harrison, P. Kuchment, A. Sobolev, and B. Winn, “On occurrence of spectral edges for periodic operators inside the Brillouin zone,” J. Phys. A: Math. Theor. 40, 7597–7618 (2007).
[CrossRef]

K. D. Cherednichenko and S. Guenneau, “Bloch wave homogenisation for spectral asymptotic analysis of the periodic Maxwell operator,” Waves Random Complex Media 17, 627–651 (2007).
[CrossRef]

2006 (4)

A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media 16, 293–392 (2006).
[CrossRef]

K. D. Cherednichenko, V. P. Smyshlyaev, and V. V. Zhikov, “Non-local homogenised limits for composite media with highly anisotropic periodic fibres,” Proc. R. Soc. Edinburgh 136 A, 87–114 (2006).
[CrossRef]

G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London A 462, 3027–3059 (2006).
[CrossRef]

Y. Capdeboscq and M. Briane, “Expansion formulae for the homogenized determinant of anisotropic checkerboards,” Proc. R. Soc. London A 462, 2259–2279 (2006).
[CrossRef]

2005 (1)

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005).
[CrossRef]

2004 (3)

J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191–202 (2004).
[CrossRef]

R. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E 70, 046608 (2004).
[CrossRef]

X. Hu, Y. Shu, X. Liu, R. Fu, and J. Zi, “Superlensing effect in liquid surface waves,” Phys. Rev. E 69, 030201 (2004).
[CrossRef]

2003 (2)

J. Pendry and A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. Condens. Matter 15, 6345–6364 (2003).
[CrossRef]

F. Zolla and S. Guenneau, “Duality relation for the Maxwell system,” Phys. Rev. E 67, 026610 (2003).
[CrossRef]

2002 (4)

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

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

M. Notomi, “Negative refraction in photonic crystals,” Opt. Quantum Electron. 34, 133–143 (2002).
[CrossRef]

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684(2002).
[PubMed]

2001 (2)

R. V. Craster and Y. V. Obnosov, “Four phase checkerboard composites,” SIAM J. Appl. Math. 61, 1839–1856 (2001).
[CrossRef]

G. W. Milton, “Proof of a conjecture on the conductivity of checkerboards,” J. Math. Phys. 42, 4873–4882 (2001).
[CrossRef]

2000 (4)

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behaviour in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[CrossRef]

B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, 1012–1020(2000).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, V. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187(2000).
[CrossRef] [PubMed]

1997 (1)

T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic-crystal model,” Phys. Rev. E 55, 6024–6038 (1997).
[CrossRef]

1994 (1)

J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351(1994).
[CrossRef]

1987 (3)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062(1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

R. Zengerle, “Light propagation in singly and doubly periodic waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

1986 (1)

V. V. Zhikov, “Estimates for the trace of the averaged matrix,” Math. Notes Acad. Sci. USSR 40, 628–634 (1986).
[CrossRef]

1982 (1)

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
[CrossRef]

1981 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

1979 (1)

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

1964 (1)

J. B. Keller, “A theorem on the conductivity of a composite medium,” J. Math. Phys. 5, 548–549 (1964).
[CrossRef]

1931 (1)

R. L. Kronig and W. G. Penney, “Quantum mechanics in crystals lattices,” Proc. R. Soc. London A 130, 499–531 (1931).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Adams, S. D. M.

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Guided and standing Bloch waves in periodic elastic strips,” Waves Random Complex Media 19, 321–346 (2009).
[CrossRef]

Alù, A.

A. Alù, M. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Asatryan, A. A.

K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. Martijn de Sterke, “Shallow defect states in two-dimensional photonic crystals,” Phys. Rev. A 77, 063839 (2008).
[CrossRef]

Bakhvalov, N.

N. Bakhvalov and G. Panasenko, Homogenization: Averaging Processes in Periodic Media (Kluwer, Amsterdam, 1989).
[CrossRef]

Botten, L. C.

K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. Martijn de Sterke, “Shallow defect states in two-dimensional photonic crystals,” Phys. Rev. A 77, 063839 (2008).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Bowden, C. M.

J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351(1994).
[CrossRef]

Briane, M.

Y. Capdeboscq and M. Briane, “Expansion formulae for the homogenized determinant of anisotropic checkerboards,” Proc. R. Soc. London A 462, 2259–2279 (2006).
[CrossRef]

Brillouin, L.

L. Brillouin, Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, 2nd ed. (Dover, 1953).

Capdeboscq, Y.

Y. Capdeboscq and M. Briane, “Expansion formulae for the homogenized determinant of anisotropic checkerboards,” Proc. R. Soc. London A 462, 2259–2279 (2006).
[CrossRef]

Chakrabarti, S.

S. Guenneau, S. A. Ramakrishna, S. Enoch, S. Chakrabarti, G. Tayeb, and B. Gralak, “Cloaking and imaging effects in plasmonic checkerboards of negative epsilon and mu and dielectric photonic crystal checkerboards,” Photon. Nanostruct. Fund. Appl. 5, 63–72 (2007).
[CrossRef]

Cherednichenko, K. D.

K. D. Cherednichenko and S. Guenneau, “Bloch wave homogenisation for spectral asymptotic analysis of the periodic Maxwell operator,” Waves Random Complex Media 17, 627–651 (2007).
[CrossRef]

K. D. Cherednichenko, V. P. Smyshlyaev, and V. V. Zhikov, “Non-local homogenised limits for composite media with highly anisotropic periodic fibres,” Proc. R. Soc. Edinburgh 136 A, 87–114 (2006).
[CrossRef]

Conca, C.

C. Conca, J. Planchard, and M. Vanninathan, Fluids and Periodic Structures (Masson, 1995).

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Craster, R. V.

E. Nolde, R. V. Craster, and J. Kaplunov, “High frequency homogenization for structural mechanics,” J. Mech. Phys. Solids 59, 651–671 (2011).
[CrossRef]

R. V. Craster, J. Kaplunov, and A. V. Pichugin, “High frequency homogenization for periodic media,” Proc. R. Soc. London A 466, 2341–2362 (2010).
[CrossRef]

R. V. Craster, J. Kaplunov, and J. Postnova, “High frequency asymptotics, homogenization and localization for lattices,” Q. J. Mech. Appl. Math. 63, 497–519 (2010).
[CrossRef]

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Guided and standing Bloch waves in periodic elastic strips,” Waves Random Complex Media 19, 321–346 (2009).
[CrossRef]

R. V. Craster and Y. V. Obnosov, “Four phase checkerboard composites,” SIAM J. Appl. Math. 61, 1839–1856 (2001).
[CrossRef]

Dossou, K. B.

K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. Martijn de Sterke, “Shallow defect states in two-dimensional photonic crystals,” Phys. Rev. A 77, 063839 (2008).
[CrossRef]

Dowling, J. P.

J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351(1994).
[CrossRef]

Engheta, N.

A. Alù, M. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[CrossRef]

Enoch, S.

M. Farhat, S. Guenneau, S. Enoch, and A. Movchan, “All-angle negative refraction and ultra-refraction for liquid surface waves in 2-D phononic crystals,” J. Comput. Appl. Math. 234, 2011–2019 (2010).
[CrossRef]

M. Farhat, S. Guenneau, and S. Enoch, “High-directivity and confinement of flexural waves through ultrarefraction in thin perforated plates,” Euro. Phys. Lett. 91, 54003 (2010).
[CrossRef]

M. Farhat, S. Guenneau, S. Enoch, A. Movchan, and G. Petursson, “Focussing bending waves via negative refraction in perforated thin plates,” Appl. Phys. Lett. 96, 081909 (2010).
[CrossRef]

M. Farhat, S. Guenneau, S. Enoch, G. Tayeb, A. Movchan, and N. Movchan, “Analytical and numerical analysis of lensing effect for linear surface water waves through a square array of close to touching rigid square cylinders,” Phys. Rev. E 77, 046308(2008).
[CrossRef]

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M. Farhat, S. Guenneau, and S. Enoch, “High-directivity and confinement of flexural waves through ultrarefraction in thin perforated plates,” Euro. Phys. Lett. 91, 54003 (2010).
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M. Farhat, S. Guenneau, S. Enoch, A. Movchan, and G. Petursson, “Focussing bending waves via negative refraction in perforated thin plates,” Appl. Phys. Lett. 96, 081909 (2010).
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M. Farhat, S. Guenneau, S. Enoch, and A. Movchan, “All-angle negative refraction and ultra-refraction for liquid surface waves in 2-D phononic crystals,” J. Comput. Appl. Math. 234, 2011–2019 (2010).
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M. Farhat, S. Guenneau, S. Enoch, G. Tayeb, A. Movchan, and N. Movchan, “Analytical and numerical analysis of lensing effect for linear surface water waves through a square array of close to touching rigid square cylinders,” Phys. Rev. E 77, 046308(2008).
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M. Farhat, S. Guenneau, and S. Enoch, “High-directivity and confinement of flexural waves through ultrarefraction in thin perforated plates,” Euro. Phys. Lett. 91, 54003 (2010).
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M. Farhat, S. Guenneau, S. Enoch, A. Movchan, and G. Petursson, “Focussing bending waves via negative refraction in perforated thin plates,” Appl. Phys. Lett. 96, 081909 (2010).
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M. Farhat, S. Guenneau, S. Enoch, and A. Movchan, “All-angle negative refraction and ultra-refraction for liquid surface waves in 2-D phononic crystals,” J. Comput. Appl. Math. 234, 2011–2019 (2010).
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Appl. Phys. Lett. (1)

M. Farhat, S. Guenneau, S. Enoch, A. Movchan, and G. Petursson, “Focussing bending waves via negative refraction in perforated thin plates,” Appl. Phys. Lett. 96, 081909 (2010).
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C. R. Phys. (1)

S. Guenneau and S. A. Ramakrishna, “Negative refractive index, perfect lenses and checkerboards: trapping and imaging effects in folded optical spaces,” C. R. Phys. 10, 352–378 (2009).
[CrossRef]

Contemp. Phys. (1)

J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191–202 (2004).
[CrossRef]

Euro. Phys. Lett. (1)

M. Farhat, S. Guenneau, and S. Enoch, “High-directivity and confinement of flexural waves through ultrarefraction in thin perforated plates,” Euro. Phys. Lett. 91, 54003 (2010).
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J. Comput. Appl. Math. (1)

M. Farhat, S. Guenneau, S. Enoch, and A. Movchan, “All-angle negative refraction and ultra-refraction for liquid surface waves in 2-D phononic crystals,” J. Comput. Appl. Math. 234, 2011–2019 (2010).
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J. Mech. Phys. Solids (1)

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J. Opt. Soc. Am. A (1)

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

J. M. Harrison, P. Kuchment, A. Sobolev, and B. Winn, “On occurrence of spectral edges for periodic operators inside the Brillouin zone,” J. Phys. A: Math. Theor. 40, 7597–7618 (2007).
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J. Pendry and A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. Condens. Matter 15, 6345–6364 (2003).
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Opt. Acta (1)

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Opt. Express (1)

Opt. Quantum Electron. (1)

M. Notomi, “Negative refraction in photonic crystals,” Opt. Quantum Electron. 34, 133–143 (2002).
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Photon. Nanostruct. Fund. Appl. (1)

S. Guenneau, S. A. Ramakrishna, S. Enoch, S. Chakrabarti, G. Tayeb, and B. Gralak, “Cloaking and imaging effects in plasmonic checkerboards of negative epsilon and mu and dielectric photonic crystal checkerboards,” Photon. Nanostruct. Fund. Appl. 5, 63–72 (2007).
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Phys. Rev. A (1)

K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. Martijn de Sterke, “Shallow defect states in two-dimensional photonic crystals,” Phys. Rev. A 77, 063839 (2008).
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Phys. Rev. B (4)

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Figures (8)

Fig. 1
Fig. 1

AANR and ultrarefraction through a three-phase finite dielectric checkerboard oriented along the line y = x . (a) A line source located on the left side of the checkerboard displays an image on the other side in a symmetric fashion (plot of u) for an excitation frequency of Ω = 1.136 . (b) A Gaussian beam incident from the left at an angle π / 4 (parallel to the x axis) is reflected at an angle π / 4 (plot of u) and transmitted at an angle close to π / 4 (nearly parallel to the x axis). (c) A line source at frequency Ω = 1.3 located within the checkerboard emits a concentric wave that emerges as a plane wave in the surrounding medium. (d) Geometry of the checkerboard alternating unit squares of relative permittivity 1 (i.e., vacuum), 1 + r 2 , 1 + 2 r 2 , and 1 + r 2 (clockwise) (see Fig. 2) with r = 1 . Each cell is rotated by an angle π / 4 , and the checkerboard is oriented along the line y = x .

Fig. 2
Fig. 2

(a) Single cell for the checkerboard geometry, with a side length of two, for the piecewise constant media. (b) Reciprocal Brillouin lattice in the wavenumber, κ = ( κ 1 , κ 2 ) , space.

Fig. 3
Fig. 3

Dispersion curves for high-contrast r = 10 : exact solutions (dashed curves), CH limit (dotted curves), and HFH asymptotics (solid curves) from Eq. (13). Stop bands, whose edges are accurately predicted by HFH, are shown shaded.

Fig. 4
Fig. 4

Response to localized line source excitation with frequency Ω = 0.172 for the checkerboard structure with r = 10 . (a)  | u | as a globally decaying solution modulating an oscillatory behavior. (b) u along the line y = 1 / 2 as the solid line lying within a dashed envelope given by the asymptotic relation [Eq. (14)]. (c) Further detail of the solution along this line with the points marking the cell edges demonstrating the local out-of-phase oscillations from cell to cell. (d) Detailed view of the dispersion curves and stop band near the excitation frequency: the dashed lines are from the full numerics, and the asymptotics are the solid lines.

Fig. 5
Fig. 5

Ultrarefraction shown for r = 1 and Ω = 1.3 (close to the standing wave frequency Ω 0 = 1.3627 at wavenumber C) where the tensor T of the homogenized PDE [Eq. (7)], as given by the asymptotic formulas (8, 9), is isotropic, with T 11 = T 22 = 2.7697 . The checkerboard lies within the square | x | , | y | 21 . (a)  | u | with the detail of the fine oscillations within the checkerboard in panel (c) (this shows u). (b)  | u | for a source within a square of effective material with refractive index n eff ( Ω , Ω 0 ) 1 according to asymptotic formula (17). (d) Dispersion curves close to the operating frequency Ω, which is the horizontal dotted line at Ω = 1.3 . The dispersion curves have the numerics as the dashed curves and asymptotics as solid. Ultrarefraction at Ω 0 = 1.3627 is less pronounced, as T is no longer isotropic.

Fig. 6
Fig. 6

(a) Dispersion diagram for the acoustic mode pertinent for AANR for r = 10 with the exact dispersion relation for the checkerboards as the dashed line, the CH approximation Ω | κ | / ( 2 1 + r 2 ) is the heavy dotted line, and the asymptotics from HFH as the solid line. A stop band is also evident and is shown shaded. AANR occurs when the dispersion line for the incoming waves, shown as dotted, intersects the exact checkerboard dispersion curve. (b) Isofrequencies for the lowest dispersion surface (the acoustic mode). The isofrequency contour for Ω = 1.136 (dotted line) has the opposite curvature to equifrequency contours for Ω 1.1152 which is the hallmark of AANR. (c), (d) Dispersion and isofrequency curves for r = 1 .

Fig. 7
Fig. 7

Plots of | u | for AANR for (a) an harmonic line source at frequency 1.136 on the upper edge of a finite-size dielectric checkerboard with eight rows of rotated cells and parameters as in Fig. 1, i.e., r = 1 , generates an image about a distance 0.15 below along the direction orthogonal to the checkerboard; (b) idem when the source is located a distance 0.15 above the checkerboard, in which case the image appears on the lower edge of the checkerboard. Panels (c) and (d) are for r = 10 . (c) Same as (b) but now for a frequency 0.211 given by CH [Eq. (19)]. (d) Same as (b) for a frequency 0.171 given by HFH [Eq. (18)]. All panels show plots of | u | in a color scale ranging from dark blue (vanishing field) to red (larger values of field). White regions are for values of | u | outside the color scale.

Fig. 8
Fig. 8

Open resonators via AANR: A ray picture of the cloaking effect (a) unveils the underlying physics of light confinement via negative refraction. A point source at frequency Ω = 1.136 located within a square checkerboard of alternating cells with r = 1 , emits light that is confined, as shown by the plot of | u | (b). Note that Ω is a frequency at which AANR occurs (i.e., in a passband); see Fig. 5d and asymptotic formula (18). The color scale ranges from dark blue (vanishing field) to red (larger values of field). White regions are for values of | u | outside the color scale.

Tables (1)

Tables Icon

Table 1 First Five Standing Wave Frequencies, Ω 0 , and the Corresponding T 11 and T 22 for the Standing Wave Cases, κ = ( 0 , 0 ) (Wavenumber A), κ = ( π , 0 ) (Wavenumber B), and κ = ( π , π ) (Wavenumber C) a

Equations (19)

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2 u x 1 2 + 2 u x 2 2 + Ω 2 [ 1 + g 1 ( x 1 ) + g 2 ( x 2 ) ] u = 0 ,
g i ( x i ) = { r 2 for     0 x i < 1 , 0 for     1 x i < 0.
u ( 1 , x 2 ) = e i κ 1 u ( 1 , x 2 ) , u x 1 ( 1 , x 2 ) = e i κ 1 u x 1 ( 1 , x 2 ) ,
u ( x 1 , 1 ) = e i κ 2 u ( x 1 , 1 ) , u x 2 ( x 1 , 1 ) = e i κ 2 u x 2 ( x 1 , 1 ) ,
2 γ i β i ( cos κ i cos γ i cos β i ) + ( β i 2 + γ i 2 ) sin γ i sin β i = 0 ,
β i 2 = Ω 2 / 2 μ 2 , γ i 2 = Ω 2 ( 1 / 2 + r 2 ) μ 2 ,
T i j 2 f X i X j + ( Ω 2 Ω 0 2 ) ϵ 2 f = 0 ,
T 11 = 8 γ 1 2 β 1 2 K 2 K 1 K 2 + r 2 K 1 ( 2 ) K 2 + r 2 K 1 K 2 ( 2 ) ( sin γ 1 p 1 + cos γ 1 ) ,
T 22 = 8 γ 2 2 β 2 2 K 1 K 1 K 2 + r 2 K 1 ( 2 ) K 2 + r 2 K 1 K 2 ( 2 ) ( sin γ 2 p 2 + cos γ 2 ) ,
K i ( 1 ) = γ i [ 2 γ i ( cos γ i cos β i ) + ( γ i 2 β i 2 ) sin γ i ] ,
K i ( 2 ) = β i [ 2 β i ( cos γ i cos β i ) ( γ i 2 β i 2 ) sin β i ] ,
p i = β i sin γ i + γ i sin β i β i ( cos β i cos γ i ) .
Ω Ω 0 ( 1 + T i j κ j 2 8 Ω 0 2 ) ;
u ( x , y ) 1 2 π K 0 ( Ω 2 Ω 0 2 | T | x 2 + y 2 ) ,
[ 2 + Ω 2 n eff 2 ] u = 0 ,
[ 2 + Ω 2 ] u = 0 ,
n eff ( Ω , Ω 0 ) = 2 | T | ( Ω 2 Ω 0 2 ) Ω ,
Ω HFH = Ω 0 1 T / 4 ,
Ω CH = 2 π 1 + 2 1 + r 2 ,

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