Abstract

Artificial periodic structures (metamaterials and photonic crystals) with feature sizes smaller than the wavelength can be capable of supporting backward waves and producing negative refraction. However, backward waves may only exist if the lattice cell size is a sufficiently large fraction of the vacuum wavelength and/or the Bloch wavelength. Explicit lower bounds for the cell size are established and imply, in particular, a limit on the optical resolution of negative-index lenses. A key tool in the analysis is Fourier decomposition of Bloch waves and related eigenfrequency estimates. Numerical examples and a detailed exposition of the mechanism of backward waves are included.

© 2008 Optical Society of America

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2007 (6)

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41-48 (2007).
[CrossRef]

S. Tretyakov, “On geometrical scaling of split-ring and double-bar resonators at optical frequencies,” Metamaterials 1, 40-43 (2007).
[CrossRef]

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780nm wavelength,” Opt. Lett. 32, 53-55 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, H.-K. Yuan, V. C. de Silva, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Metamagnetics with rainbow colors,” Opt. Express 15, 3333-3341 (2007).
[CrossRef] [PubMed]

M. Davanço, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15, 9681-9691 (2007).
[CrossRef] [PubMed]

2006 (8)

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic metamaterials: magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097-1105 (2006).
[CrossRef]

A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14, 1557-1567 (2006).
[CrossRef] [PubMed]

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23, 391-403 (2006).
[CrossRef]

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Demonstration of metal-dielectric negative-index metamaterials with improved performance at optical frequencies,” J. Opt. Soc. Am. B 23, 434-438 (2006).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

R. Meisels, R. Gajic, F. Kuchar, and K. Hingerl, “Negative refraction and flat-lens focusing in a 2d square-lattice photonic crystal at microwave and millimeter wave frequencies,” Opt. Express 14, 6766-6777 (2006).
[CrossRef] [PubMed]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[CrossRef]

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

2005 (8)

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. R. Soc. London, Ser. A 461, 3999-4034 (2005).
[CrossRef]

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

R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal,” Phys. Rev. B 71, 085106 (2005).
[CrossRef]

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matter 17, 3717-3734 (2005).
[CrossRef] [PubMed]

I. A. Larkin and M. I. Stockman, “Imperfect perfect lens,” Nano Lett. 5, 339-343 (2005).
[CrossRef] [PubMed]

B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B 22, 1179-1190 (2005).
[CrossRef]

R. Gajic, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express 13, 8596-8605 (2005).
[CrossRef] [PubMed]

D. Sjöberg, C. Engstrom, G. Kristensson, D. J. N. Wall, and N. Wellander, “A Floquet-Bloch decomposition of Maxwell's equations applied to homogenization,” Multiscale Model. Simul. 4, 149-171 (2005).
[CrossRef]

2004 (7)

G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. 93, 243902 (2004).
[CrossRef]

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microwave Opt. Technol. Lett. 41, 315-316 (2004).
[CrossRef]

P. A. Belov, C. R. Simovski, and S. A. Tretyakov, “Backward waves and negative refraction in photonic (electromagnetic) crystals,” J. Commun. Technol. Electron. 49, 1199-1207 (2004).

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84, 1290-1292 (2004).
[CrossRef]

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57, 37-43 (2004).
[CrossRef]

D. R. Smith and D. C. Vier, “Design of metamaterials with negative refractive index,” Proc. SPIE 5359, 52-63 (2004).
[CrossRef]

2003 (5)

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell's law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
[CrossRef]

2002 (3)

M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, “The negative index of refraction demystified,” Eur. J. Phys. 23, 353-359 (2002).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, “Sign of refractive index and group velocity in lefthanded media,” Solid State Commun. 124, 283-287 (2002).
[CrossRef]

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

2001 (2)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

2000 (3)

D. R. Smith, W. J. Padilla, D. 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]

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

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

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

1994 (1)

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479-8482 (1994).
[CrossRef]

1987 (1)

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

1979 (1)

1968 (1)

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

1945 (1)

L. I. Mandelshtam, “Group velocity in crystalline arrays,” Zh. Eksp. Teor. Fiz. 15, 475-478 (1945).

Aitchison, J. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[CrossRef]

Alù, A.

Aydin, K.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Balmain, K. G.

G. V. Eleftheriades and K. G. Balmain, Negative Refraction Metamaterials: Fundamental Principles and Applications (Wiley-IEEE Press, 2005).
[CrossRef]

Belov, P. A.

P. A. Belov, C. R. Simovski, and S. A. Tretyakov, “Backward waves and negative refraction in photonic (electromagnetic) crystals,” J. Commun. Technol. Electron. 49, 1199-1207 (2004).

Brock, J. B.

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell's law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Brueck, S. R. J.

Burger, S.

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic metamaterials: magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097-1105 (2006).
[CrossRef]

Cai, W.

Chettiar, U. K.

Chuang, I. L.

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell's law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Cubukcu, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Davanço, M.

de Silva, V. C.

Depine, R. A.

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microwave Opt. Technol. Lett. 41, 315-316 (2004).
[CrossRef]

Derov, J. S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Dolling, G.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780nm wavelength,” Opt. Lett. 32, 53-55 (2007).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic metamaterials: magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097-1105 (2006).
[CrossRef]

Drachev, V. P.

Dunbar, L. A.

Efros, A. L.

A. L. Pokrovsky and A. L. Efros, “Sign of refractive index and group velocity in lefthanded media,” Solid State Commun. 124, 283-287 (2002).
[CrossRef]

Eleftheriades, G. V.

G. V. Eleftheriades and K. G. Balmain, Negative Refraction Metamaterials: Fundamental Principles and Applications (Wiley-IEEE Press, 2005).
[CrossRef]

Engheta, N.

Engstrom, C.

D. Sjöberg, C. Engstrom, G. Kristensson, D. J. N. Wall, and N. Wellander, “A Floquet-Bloch decomposition of Maxwell's equations applied to homogenization,” Multiscale Model. Simul. 4, 149-171 (2005).
[CrossRef]

Enkrich, C.

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic metamaterials: magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097-1105 (2006).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

Fan, W.

Ferrini, R.

Foteinopolou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Foteinopoulou, S.

R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal,” Phys. Rev. B 71, 085106 (2005).
[CrossRef]

Gajic, R.

Greegor, R. B.

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Appl. Phys. Lett. (2)

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84, 1290-1292 (2004).
[CrossRef]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
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Figures (4)

Fig. 1
Fig. 1

Photonic band diagram of the Gajic et al. crystal. TE modes (p-polarization, one-component H-field) (squares, bold curves). TM modes (s-polarization, one-component E-field) (circles, dashed curves).

Fig. 2
Fig. 2

Plane-wave Poynting components P n for the Gajic et al. crystal (arb. units). The TE2 mode near the Γ point on the Γ X line (From [40], p. 471, © 2008 Springer Science and Business Media, with kind permission of Springer Science and Business Media).

Fig. 3
Fig. 3

Amplitudes h m of the plane-wave harmonics for the Gajic et al. crystal (arb. units). Second H-mode (TE2) near the Γ point on the Γ X line (From [40], p. 469, ©2008 Springer Science and Business Media, with kind permission of Springer Science and Business Media).

Fig. 4
Fig. 4

Bounds on the normalized cell size and a few representative data points from the literature.

Equations (61)

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E = E y = E 0 exp ( i k x ) ,
H = H z = H 0 exp ( i k x ) ,
H 0 = k ω μ E 0 ,
k = ω μ ϵ ( which branch of the square root ? ) .
k > 0 .
exp ( i k x ) = exp ( i ( k + i k ) x ) = exp ( k x ) exp ( i k x )
P = P x = 1 2 Re E 0 H 0 * = 1 2 Re k ω μ E 0 2 .
ϵ = ϵ exp ( i ϕ ϵ ) ; μ = μ exp ( i ϕ μ ) ; 0 < ϕ ϵ , ϕ μ < π .
k = ω μ ϵ exp ( i ϕ ϵ + ϕ μ 2 ) .
sign P = sign Re k μ = sign cos ϕ ϵ ϕ μ 2 .
sign k = sign cos ϕ ϵ + ϕ μ 2
ϕ ϵ + ϕ μ > π .
cos ϕ ϵ < cos ( π ϕ μ ) ,
cos ϕ ϵ + cos ϕ μ < 0 .
ϵ ϵ + μ μ < 0 .
E ( x ) + k 2 E ( x ) = 0 ,
k 2 = ω 2 μ 0 ϵ ( x ) .
E ( x ) = E PER ( x ) exp ( i K B x ) ,
2 E + k 2 E = 0 ,
k 2 = ω 2 μ 0 ϵ ( x , y ) .
E ( r ) = E PER ( r ) exp ( i K B r ) ; r ( x , y ) .
ϵ 1 H + ω 2 μ 0 H = 0 ,
× × E ω 2 μ 0 ϵ E = 0 ,
E ( r ) = E PER ( r ) exp ( i K r ) ,
r ( x , y , z ) , K B = ( K B x , K B y , K B z ) .
× ϵ 1 × H ω 2 μ 0 H = 0 .
E PER ( x ) = m = e m exp ( i m κ 0 x ) , κ 0 = 2 π a 1 .
e m = a 1 a E PER ( x ) exp ( i m κ 0 x ) d x ,
E ( r ) = E PER ( r ) exp ( i K B r ) , r ( x , y , z ) ,
E PER ( r ) = m Z 3 e m exp ( i k m r ) ,
E ( r ) = m Z 3 E m m Z 3 e m exp ( i k m r ) exp ( i K B r ) .
H ( r ) = m Z 3 H m m Z 3 h m exp ( i k m r ) exp ( i K B r )
P = m Z 3 P m ; P m = k m 2 ω μ 0 e m 2 .
K B x + κ 0 m = k x air .
ϵ ( r ) = m Z 2 ϵ ̃ m exp ( i k m r ) .
k n 2 e n = ω 2 μ 0 m Z 2 ϵ ̃ n m e m .
̃ 2 E + ω ̃ 2 ϵ E = 0 ,
̃ × ̃ × E = ω ̃ 2 ϵ ( r ) E ,
̃ × ϵ 1 ( r ) ̃ × H = ω ̃ 2 H ,
K ̃ B + 2 π n 2 e n = ω ̃ 2 m Z 2 ϵ ̃ n m e m .
E ( x ̃ ) = m Z e m exp ( i 2 π m x ̃ ) .
P ̃ 2 ω μ 0 a P = K ̃ B e 0 ( K ̃ B ) 2 + m = 1 ( K ̃ B + 2 π m ) e m ( K ̃ B ) 2 + ( K ̃ B 2 π m ) e m ( K ̃ B ) 2 .
P ̃ = K ̃ B e 0 ( K ̃ B ) 2 + m = 1 ( K ̃ B + 2 π m ) e m ( K ̃ B ) 2 + ( K ̃ B 2 π m ) e m ( K ̃ B ) 2 = K ̃ B e 0 ( K ̃ B ) 2 + K ̃ B m = 1 ( e m ( K ̃ B ) 2 + e m ( K ̃ B ) 2 ) + 2 π m = 1 m ( e m ( K ̃ B ) 2 e m ( K ̃ B ) 2 ) .
P ̃ = K ̃ B [ e 0 2 + m = 1 ( e m ( K ̃ B ) 2 + e m ( K ̃ B ) 2 ) + 2 π m = 1 m e m 2 K ̃ B ] .
e n = ω ̃ 2 ( K ̃ B + 2 π n ) 2 m Z ϵ n m e m ω ̃ 2 ( K ̃ B + 2 π n ) 2 ϵ l 2 e l 2 , n 0 ,
ϵ E = η ̃ 2 E .
E 0 = ϵ ̃ 0 1 Ω ϵ E d Ω , ϵ ̃ 0 0 .
ϵ [ E ϵ ̃ 0 1 Ω ϵ E d Ω ] = η ̃ 2 E .
̃ 2 { ϵ [ E ϵ ̃ 0 1 ( ϵ , E ) ] } = η E ,
η ( 4 π 2 ) 1 ϵ max ( 1 + ϵ max ϵ ̃ 0 ) .
( a λ 0 ) 2 = ω ̃ 2 4 π 2 ϵ max 1 ( 1 + ϵ ̃ 0 1 ϵ max ) 1 .
ϵ E = η ̃ × ̃ × E .
̃ 2 u 2 2 = ( ̃ × ̃ × u , ̃ × ̃ × u ) = ( ̃ 2 u , ̃ 2 u ) ( 4 π 2 ) 2 ( u , u ) = ( 4 π 2 ) 2 u 2 2 .
( ̃ × ) 2 ( 4 π 2 ) 1 .
ϵ ( E L ϵ 1 ̃ ( ϵ E ) ) = η ̃ × ̃ × E ,
( ̃ × ) 2 { ϵ [ E L ϵ 1 ( ϵ E ) ] } = η E .
η ( 4 π 2 ) 1 ϵ max ( 1 + λ max ( L ϵ 1 ) ϵ max ) .
λ max ( L ϵ 1 ) ϵ min 1 λ max ( 2 ) ( 4 π 2 ϵ min ) 1
( a λ 0 ) 2 = ( ω ̃ 2 π ) 2 = 1 4 π 2 η 1 ϵ max ( 1 + ϵ max ( 4 π 2 ϵ min ) ) .
( ϵ ϕ , ϕ ) = ϵ h W h + ( ϵ i + i ϵ i ) W i ,
λ max ( L ϵ 1 ) ( ϵ h ϵ i ) 2 + ϵ i 2 4 π 2 ϵ h ϵ i .

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