Abstract

The band structure of periodic multilayers that include left- and right-handed materials is studied under the formalism of equivalent layers. Although the band structure of these systems looks quite complicated, it is shown that bandgaps and bulk bands follow a simple behavior that can be explained in terms of the phase thickness of one period and the Brewster condition that can be satisfied in both polarizations. The optical tunneling is explained in terms of optical holes that are points where the gaps narrow and close owing to the zero optical thickness of the period under given conditions. It is shown that optical tunneling can be present also under nonnormal incidence.

© 2006 Optical Society of America

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  1. G. Dewar, "Candidates for ϵ<0,µ<0 nanostructures," Int. J. Mod. Phys. B 15, 3528-3265 (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]
  3. 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]
  4. D. R. Smith, P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, "Design and measurement of anisotropic metamaterials that exhibit negative refraction," IEICE Trans. Electron. E87-C, 359-370 (2004).
  5. J. B. Pendry and D. R. Smith, "Reversing light: negative refraction," Phys. Today 57, 37-43 (2004).
    [CrossRef]
  6. K. Sakoda, Optical Properties of Photonic Crystals (Elsevier, 2001).
  7. F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "Surface waves in finite one-dimensional photonic crystals: mode coupling," Opt. Commun. 216, 361-367 (2003).
    [CrossRef]
  8. J. A. Gaspar-Armenta and F. Villa, "Photonic surface-wave excitation: photonic crystal-metal interface," J. Opt. Soc. Am. B 21, 405-412 (2004).
    [CrossRef]
  9. L. Wu, S. He, and L. Chen, "On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials," Opt. Express 11, 1283-1290 (2003).
    [CrossRef] [PubMed]
  10. L. Wu, S. He, and L. Shen, "Band structure for a one-dimensional photonic crystal containing left-handed materials," Phys. Rev. B 67, 235103 (2003).
    [CrossRef]
  11. J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901 (2003).
    [CrossRef] [PubMed]
  12. D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
    [CrossRef]
  13. J. A. Gaspar-Armenta and F. Villa, "Band-structure properties of one-dimensional photonic crystals under the formalism of equivalent systems," J. Opt. Soc. Am. B 21, 405-412 (2004).
    [CrossRef]

2004 (5)

D. R. Smith, P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, "Design and measurement of anisotropic metamaterials that exhibit negative refraction," IEICE Trans. Electron. E87-C, 359-370 (2004).

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

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

J. A. Gaspar-Armenta and F. Villa, "Band-structure properties of one-dimensional photonic crystals under the formalism of equivalent systems," J. Opt. Soc. Am. B 21, 405-412 (2004).
[CrossRef]

J. A. Gaspar-Armenta and F. Villa, "Photonic surface-wave excitation: photonic crystal-metal interface," J. Opt. Soc. Am. B 21, 405-412 (2004).
[CrossRef]

2003 (5)

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]

L. Wu, S. He, and L. Chen, "On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials," Opt. Express 11, 1283-1290 (2003).
[CrossRef] [PubMed]

F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "Surface waves in finite one-dimensional photonic crystals: mode coupling," Opt. Commun. 216, 361-367 (2003).
[CrossRef]

L. Wu, S. He, and L. Shen, "Band structure for a one-dimensional photonic crystal containing left-handed materials," Phys. Rev. B 67, 235103 (2003).
[CrossRef]

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

2001 (2)

G. Dewar, "Candidates for ϵ<0,µ<0 nanostructures," Int. J. Mod. Phys. B 15, 3528-3265 (2001).

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

Akjouj, A.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Bria, D.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

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]

Chan, C. T.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Chen, L.

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]

Dewar, G.

G. Dewar, "Candidates for ϵ<0,µ<0 nanostructures," Int. J. Mod. Phys. B 15, 3528-3265 (2001).

Djafari-Rouhani, B.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Dobrzynski, L.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

El Boudouti, E. H.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Gaspar-Armenta, J. A.

He, S.

L. Wu, S. He, and L. Chen, "On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials," Opt. Express 11, 1283-1290 (2003).
[CrossRef] [PubMed]

L. Wu, S. He, and L. Shen, "Band structure for a one-dimensional photonic crystal containing left-handed materials," Phys. Rev. B 67, 235103 (2003).
[CrossRef]

Houck, A. A.

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]

Li, J.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Mock, J. J.

D. R. Smith, P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, "Design and measurement of anisotropic metamaterials that exhibit negative refraction," IEICE Trans. Electron. E87-C, 359-370 (2004).

Nougaoui, A.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Pendry, J. B.

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

Perram, T.

D. R. Smith, P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, "Design and measurement of anisotropic metamaterials that exhibit negative refraction," IEICE Trans. Electron. E87-C, 359-370 (2004).

Ramos-Mendieta, F.

F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "Surface waves in finite one-dimensional photonic crystals: mode coupling," Opt. Commun. 216, 361-367 (2003).
[CrossRef]

Rye, P.

D. R. Smith, P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, "Design and measurement of anisotropic metamaterials that exhibit negative refraction," IEICE Trans. Electron. E87-C, 359-370 (2004).

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Elsevier, 2001).

Schultz, S.

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

Shelby, R. A.

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

Shen, L.

L. Wu, S. He, and L. Shen, "Band structure for a one-dimensional photonic crystal containing left-handed materials," Phys. Rev. B 67, 235103 (2003).
[CrossRef]

Sheng, P.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith, P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, "Design and measurement of anisotropic metamaterials that exhibit negative refraction," IEICE Trans. Electron. E87-C, 359-370 (2004).

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

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

Starr, A. F.

D. R. Smith, P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, "Design and measurement of anisotropic metamaterials that exhibit negative refraction," IEICE Trans. Electron. E87-C, 359-370 (2004).

Vier, D. C.

D. R. Smith, P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, "Design and measurement of anisotropic metamaterials that exhibit negative refraction," IEICE Trans. Electron. E87-C, 359-370 (2004).

Vigneron, J. P.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Villa, F.

Wu, L.

L. Wu, S. He, and L. Shen, "Band structure for a one-dimensional photonic crystal containing left-handed materials," Phys. Rev. B 67, 235103 (2003).
[CrossRef]

L. Wu, S. He, and L. Chen, "On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials," Opt. Express 11, 1283-1290 (2003).
[CrossRef] [PubMed]

Zhou, L.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

IEICE Trans. Electron. (1)

D. R. Smith, P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, "Design and measurement of anisotropic metamaterials that exhibit negative refraction," IEICE Trans. Electron. E87-C, 359-370 (2004).

Int. J. Mod. Phys. B (1)

G. Dewar, "Candidates for ϵ<0,µ<0 nanostructures," Int. J. Mod. Phys. B 15, 3528-3265 (2001).

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "Surface waves in finite one-dimensional photonic crystals: mode coupling," Opt. Commun. 216, 361-367 (2003).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (1)

L. Wu, S. He, and L. Shen, "Band structure for a one-dimensional photonic crystal containing left-handed materials," Phys. Rev. B 67, 235103 (2003).
[CrossRef]

Phys. Rev. E (1)

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudouti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Phys. Rev. Lett. (2)

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]

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Phys. Today (1)

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

Science (1)

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

Other (1)

K. Sakoda, Optical Properties of Photonic Crystals (Elsevier, 2001).

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Figures (9)

Fig. 1
Fig. 1

(Color online) Equivalent functions as a function of reduced frequency under normal incidence ( β ¯ = 0 ) . The system considered has a LHM with n p = 1 , μ p = 1 , d p = 320 and a RHM with n q = 1 , μ q = 2 , d q = 120 . The real part of δ e is given in units of π radians. Equivalent admittance is normalized by dividing by the optical admittance of vacuum y.

Fig. 2
Fig. 2

(Color online) Band structure of the system with n p = 2 , μ p = 3 , d p = 0.5 Λ , n q = 1 , μ q = 1 , d q = 0.5 Λ for the TE case. The double thin line represents the light line for vacuum, the dotted–dotted–dashed line represents the light line of the LHM, and the dotted–dashed line represents the Brewster line.

Fig. 3
Fig. 3

(Color online) Band structure of the system given in Fig. 2 for the TM case.

Fig. 4
Fig. 4

(Color online) Band structure of the system with n p = 2 , μ p = 1.5 , d p = 0.5 Λ , n q = 1 , μ q = 1 , d q = 0.5 Λ for the TM case. The double thin line represents the light line for vacuum, the dotted–dotted–dashed line represents the light line of the LHM, and the dotted–dashed line represents the Brewster line.

Fig. 5
Fig. 5

(Color online) Band structure of the system given in Fig. 4 for the TE case.

Fig. 6
Fig. 6

(Color online) Band structure for TE polarization when optical tunneling points are present under normal incidence. Center curves are indicated by dotted–dotted–dashed curves. In this case the parameters of our system were n p = 2 , μ p = 1 , d p = 0.25 Λ , n q = 1 , μ q = 1 , d q = 0.5 Λ .

Fig. 7
Fig. 7

(Color online) Same system of Fig. 6 is considered but for the TM case.

Fig. 8
Fig. 8

(Color online) Band structure for TE polarization when optical tunneling points reside in a line that is oblique. Light lines of both component materials are given in double thin lines. The curves corresponding to conditions δ p = m p π , and δ q = m q π are indicated by solid and dashed curves. respectively, as in previous figures.

Fig. 9
Fig. 9

(Color online) Same system of Fig. 8 is considered for the TM case.

Equations (38)

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cos ( δ e ) = cos ( δ p ) cos ( δ q ) ρ + sin ( δ p ) sin ( δ q ) ,
η e = η p sin ( δ e ) [ sin ( δ p ) cos ( δ q ) + ρ + cos ( δ p ) sin ( δ q ) ρ sin ( δ q ) ] .
η j = { y k ¯ z j μ j ω ¯ TE polarization y n j 2 ω ¯ μ j k ¯ z j TM polarization } ,
δ j = 2 π Λ k ¯ z j d j ,
Re ( δ p + δ q ) = m π
η p = η q ,
n p μ p n p 2 ω ¯ 2 β ¯ 2 = n q μ q n q 2 ω ¯ 2 β ¯ 2 ,
ω ¯ B = a β ¯ B ,
a = ( n p 4 μ q 2 n q 4 μ p 2 ) [ ( n p 2 n q 2 ) ( n p 2 μ q 2 n q 2 μ p 2 ) ] .
a = ( μ q 2 μ p 2 ) ( n p 2 μ q 2 n q 2 μ p 2 ) .
Re ( n p 2 ω ¯ 2 β ¯ 2 d p + n q 2 ω ¯ 2 β ¯ 2 d q ) = m Λ 2
β ¯ B = m Λ 2 1 n p 2 a 1 d p + n q 2 a 1 d q .
β ¯ B = m Λ 2 1 ( n p μ q n q d p + n q μ p n p d q ) ( n p 2 n q 2 n p 2 μ q 2 n q 2 μ p 2 ) 1 2
β ¯ B = m Λ 2 1 ( μ p d p + μ q d q ) ( n p 2 + n q 2 n p 2 μ q 2 n q 2 μ p 2 ) 1 2
n p 2 μ p > n q 2 μ q , n p μ p > n q μ q , n p > n q
n p 2 μ p < n q 2 μ q , n p μ p < n q μ q , n p < n q ,
μ p > μ q , n p μ p < n q μ q , n p > n q
μ p < μ q , n p μ p > n q μ q , n p < n q ,
if n p μ p n q μ q , then a for both polarizations ,
if μ p μ q , then a n p 2 + n q 2 ( n p n q ) in the TM case ,
if n p n q , then a 1 n p 2 n q 2 in the TE case .
Re ( δ p ) = m p π ,
Re ( δ q ) = m q π ,
m p + m q = m .
β ¯ m p , m q = Λ 2 1 d p d q ( m q 2 n p 2 d p 2 m p 2 n q 2 d q 2 n q 2 n p 2 ) 1 2 ,
ω ¯ m p , m q = Λ 2 1 d p d q ( m q 2 d p 2 m p 2 d q 2 n q 2 n p 2 ) 1 2 ,
m q n q d q n p d p m p .
m q n q d q n p d p m p .
ω ¯ ( n p d p + n q d q ) = m Λ 2 .
δ p + δ q = 0 .
ω ¯ = ( d p 2 d q 2 n p 2 d p 2 n q 2 d q 2 ) 1 2 β ¯ .
cos ( δ e ) = cos 2 ( δ ) ρ + sin 2 ( δ ) ;
δ q = m q π ,
[ cos ( δ e ) i η e sin ( δ e ) i η e sin ( δ e ) cos ( δ e ) ] = [ 1 0 0 1 ] .
β ¯ m q = Λ 2 m q d p d q ( n p 2 d p 2 n q 2 d q 2 n q 2 n p 2 ) 1 2 ,
ω ¯ m q = Λ 2 m q d p d q ( d p 2 d q 2 n q 2 n p 2 ) 1 2 .
d p > d q , n p < n q , n p d p > n q d q ,
d p < d q , n p > n q , n p d p < n q d q .

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