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“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]

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]

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]

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]

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]

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]

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]

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

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[CrossRef]

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[CrossRef]

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[CrossRef]

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

[CrossRef]

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

[CrossRef]
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[PubMed]

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1839–1856 (2001).

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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L. C. Botten, M. S. Craig, R. C. McPhedran, J. L.
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(1981).

[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]

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]

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]

K. B. Dossou, L. C. Botten, R. C. McPhedran, C.
G. Poulton, A. A. Asatryan, and C. Martijn de Sterke, “Shallow
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K. B. Dossou, L. C. Botten, R. C. McPhedran, C.
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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]

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

[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]

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Y. Capdeboscq and M. Briane,
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anisotropic checkerboards,” Proc. R. Soc. London A 462,
2259–2279 (2006).

[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]

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]

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

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]

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]

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]

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

[CrossRef]

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]

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]

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]

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]

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

[CrossRef]
[PubMed]

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

[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, 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, 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]

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

[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]

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]

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

[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]

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]

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]

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]

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]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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]

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

[CrossRef]

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

[CrossRef]

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

[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]

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]

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

[CrossRef]

C. Kittel, Introduction to Solid State Physics,
7th ed. (Wiley, 1996).

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

[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]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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]

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]

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]

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]

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]

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

[CrossRef]

G. W. Milton, The Theory of Composites (Cambridge
University Press, 2002).

[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, 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, 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]

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]

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]

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]

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]

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]

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

[CrossRef]

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

[CrossRef]

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]

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

[CrossRef]

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]

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

[CrossRef]

G. Panasenko, Multi-Scale Modelling for
Structures and Composites (Springer, 2005).

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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

[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]

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]

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

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]

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]

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

[CrossRef]

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. 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]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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]

E. Sanchez-Palencia, Non-Homogeneous Media and
Vibration Theory (Springer-Verlag, 1980).

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]

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]

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]

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

[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]

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]

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]

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]

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]

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

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]

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]

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]

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

[CrossRef]
[PubMed]

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

[CrossRef]

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

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

[CrossRef]

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]

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

[CrossRef]
[PubMed]

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

[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]

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spontaneous emission in solid-state physics and electronics,”
Phys. Rev. Lett. 58, 2059–2062(1987).

[CrossRef]
[PubMed]

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

[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]

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

[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]

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

[CrossRef]

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

[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]

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]

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

[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.
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[CrossRef]

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on the conductivity of checkerboards,” J. Math. Phys. 42,
4873–4882 (2001).

[CrossRef]

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conductivity of a composite medium,” J. Math. Phys. 5,
548–549 (1964).

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[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]

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

[CrossRef]

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

[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]

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

[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]

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]

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]

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, “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]

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]

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

[CrossRef]

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

[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]

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

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

[CrossRef]
[PubMed]

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]

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]

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]

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

[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]

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]

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

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

[CrossRef]

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

[CrossRef]

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

[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]

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

[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]

C. Kittel, Introduction to Solid State Physics,
7th ed. (Wiley, 1996).

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

E. Sanchez-Palencia, Non-Homogeneous Media and
Vibration Theory (Springer-Verlag, 1980).

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

[CrossRef]

G. Panasenko, Multi-Scale Modelling for
Structures and Composites (Springer, 2005).

G. W. Milton, The Theory of Composites (Cambridge
University Press, 2002).

[CrossRef]

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

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