Abstract

We determine the band structure of two-dimensional photonic crystals that are composed of left-handed materials and dielectrics, based on the numerical solution of the Helmholtz equation by using integral equations. It is found that plasmonic resonances appear constituting a band that is independent of the filling fraction. Wide bandgaps are present where the penetration depth of the electromagnetic field inside the photonic crystal is quite short compared to purely dielectric photonic crystals.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  24. 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]
  25. R. Ruppin, "Surface polaritons of a left-handed medium," Phys. Lett. A 227, 61-64 (2000).
    [CrossRef]

2007

F. Villa, J. A. Gaspar, and A. Mendoza, "Surface modes in one dimensional photonic crystals that include left handed materials," J. Electromagn. Waves Appl. 21, 485-489 (2007).
[CrossRef]

2006

J. Li, W. Huang, and Y. Han, "Tunable photonic crystals by mixed liquids," Colloids Surf. 279, 213-217 (2006).
[CrossRef]

T. Stomeo, V. Errico, A. Salhi, A. Passaseo, R. Cingolani, A. D'Orazio, M. De Sario, V. Marrocco, V. Petruzzelli, F. Prudenzano, and M. De Vittorio, "Design and fabrication of active and passive photonic crystal resonators," Microelectron. Eng. 83, 1823-1825 (2006).
[CrossRef]

S. H. G. Teo, A. Q. Liu, M. B. Yu, and J. Singh, "Fabrication and demonstration of square lattice two-dimensional rod-type photonic band gap crystal optical intersections," Photonics Nanostruct. Fundam. Appl. 4, 103-115 (2006).
[CrossRef]

F. Yuntuan, S. Haijin, and S. Tinggen, "New evidences of negative refraction in photonic crystals," Opt. Mater. 28, 1156-1159 (2006).
[CrossRef]

F. Villa-Villa and J. A. Gaspar-Armenta, "Brewster angle and optical tunneling in one-dimensional photonic crystals composed of left and right handed materials," J. Opt. Soc. Am. B 23, 375-380 (2006).
[CrossRef]

A. Mendoza-Suárez, F. Villa-Villa, and J. A. Gaspar-Armenta, "Numerical method based on the solution of integral equations for the calculation of the band structure and reflectance of one and two-dimensional photonic crystals," J. Opt. Soc. Am. B 23, 2249-2256 (2006).
[CrossRef]

2005

2004

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudoti, 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]

2003

V. A. Podolsky, "Plasmon modes and negative refraction in metal nanowire composites," Opt. Express 11, 735-745 (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]

N. C. Panoiu, M. Bahl, and R. M. Osgood, Jr., "Optically tunable superprism effect in nonlinear photonic crystals," Opt. Lett. 28, 2503-2505 (2003).
[CrossRef] [PubMed]

A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observation of a left-handed material that obeys Snell's law," Phys. Rev. Lett. 90, 137401 (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]

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]

2002

2001

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
[CrossRef]

2000

R. Ruppin, "Surface polaritons of a left-handed medium," Phys. Lett. A 227, 61-64 (2000).
[CrossRef]

1999

1997

1990

A. A. Maradudin, E. R. Mendez, and T. Michel, "Enhanced backscattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

Ann. Phys. (N.Y.)

A. A. Maradudin, E. R. Mendez, and T. Michel, "Enhanced backscattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

Appl. Opt.

Colloids Surf.

J. Li, W. Huang, and Y. Han, "Tunable photonic crystals by mixed liquids," Colloids Surf. 279, 213-217 (2006).
[CrossRef]

IEEE J. Sel. Areas Commun.

B. Momeni and A. Adibi, "Systematic design of superprism-based photonic crystal demultiplexers," IEEE J. Sel. Areas Commun. 23, 1355-1364 (2005).
[CrossRef]

J. Electromagn. Waves Appl.

F. Villa, J. A. Gaspar, and A. Mendoza, "Surface modes in one dimensional photonic crystals that include left handed materials," J. Electromagn. Waves Appl. 21, 485-489 (2007).
[CrossRef]

J. Opt. Soc. Am. B

Microelectron. Eng.

T. Stomeo, V. Errico, A. Salhi, A. Passaseo, R. Cingolani, A. D'Orazio, M. De Sario, V. Marrocco, V. Petruzzelli, F. Prudenzano, and M. De Vittorio, "Design and fabrication of active and passive photonic crystal resonators," Microelectron. Eng. 83, 1823-1825 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Mater.

F. Yuntuan, S. Haijin, and S. Tinggen, "New evidences of negative refraction in photonic crystals," Opt. Mater. 28, 1156-1159 (2006).
[CrossRef]

Photonics Nanostruct. Fundam. Appl.

S. H. G. Teo, A. Q. Liu, M. B. Yu, and J. Singh, "Fabrication and demonstration of square lattice two-dimensional rod-type photonic band gap crystal optical intersections," Photonics Nanostruct. Fundam. Appl. 4, 103-115 (2006).
[CrossRef]

Phys. Lett. A

R. Ruppin, "Surface polaritons of a left-handed medium," Phys. Lett. A 227, 61-64 (2000).
[CrossRef]

Phys. Rev. B

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
[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]

Phys. Rev. E

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudoti, 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.

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]

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

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Figures (13)

Fig. 1
Fig. 1

Unit cell of a 2DPC composed of two different materials with refractive indices n 1 ( ω ) and n 2 ( ω ) . The two different regions are limited by the contours Γ a , Γ b , , Γ e , whose normal vectors are given in the figure. Dashed curves represent closed contours inside each medium.

Fig. 2
Fig. 2

Scheme of a finite LR2DPC. The integration contours are indicated in dashed curves. R 0 and R q represent the regions enclosing the incident and transmission media, respectively.

Fig. 3
Fig. 3

Dielectric function and magnetic permeability of a dispersive LHM as a function of frequency.

Fig. 4
Fig. 4

Band structure under TE polarization and normal incidence ( β ¯ = 0 ) for a 1DPC whose unit cell is composed of two materials, vacuum ( n 1 ) and a LHM ( n 2 ) , with d 1 = 0.3 D , d 2 = 0.7 D . Solid curves indicate the results obtained with the proposed method, while dashed curves correspond to results obtained by the characteristic matrix method.

Fig. 5
Fig. 5

Band structure under TM polarization. Same parameters as the previous figure.

Fig. 6
Fig. 6

Band structure of a LR2DPC with a square Bravais lattice and square inclusions under TE polarization. The inset on the left shows the unit cell in the real space; the right inset shows the first Brillouin zone in the k space.

Fig. 7
Fig. 7

Same system as previous figure under TM polarization.

Fig. 8
Fig. 8

Magnetic field intensity distribution at the reduced frequency of ω ¯ = 0.87 located within the first bandgap. The region outside the LR2DPC corresponds to a near field ( 10 D ) . The incident beam goes from the left to the right. In this case the amplitude of the field in the transmission region has been amplified by a factor of an order of 10 9 to get a visible pattern.

Fig. 9
Fig. 9

Magnitude of the magnetic field H at the line x = 0 and the position indicated by the dashed–dotted line. At this frequency ( ω ¯ 0 = 0.87 ) the field strongly decays within one period.

Fig. 10
Fig. 10

Scattered far field. The LR2DPC is in the plane y = 0 . The transmission intensity of the field (180°–360°) was amplified 10 9 in order to make it visible on the graphic.

Fig. 11
Fig. 11

Band structure of a LR2DPC with a square Bravais lattice and circular inclusions of LHM under TE polarization.

Fig. 12
Fig. 12

Band structure of LR2DPC under TM polarization. Same parameters as previous figure.

Fig. 13
Fig. 13

Comparison of determinants at the point M ¯ for both LR2DPCs with square and cylindrical inclusions, respectively. TE polarization.

Equations (31)

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2 Ψ j ( r ) + k j 2 Ψ j ( r ) = 0 .
k j = n j ( ω ) ω c ,
2 G j ( r , r ) + k j 2 G j ( r , r ) = 4 π δ ( r r ) ,
G j ( r , r ) = i π H 0 ( 1 ) ( k j r r ) ,
S j δ ( r r ) Ψ j ( r ) d A = 1 4 π C j [ G j ( r , r ) Ψ j ( r ) n j G j ( r , r ) n j Ψ j ( r ) ] d s .
R n ( q ) = ( X n ( q ) , Y n ( q ) ) = [ X ( s n ( q ) ) , Y ( s n ( q ) ) ] ,
1 4 π Γ q Ψ j ( r ) G j ( r , r ) n d s n = 1 N q N m n ( q ) ( j ) ψ n ( q ) ( j ) ,
1 4 π Γ q G j ( r , r ) Ψ j ( r ) n d s n = 1 N q L m n ( q ) ( j ) Φ n ( q ) ( j ) ,
Φ n ( q ) ( j ) = ψ j ( r ) n j r = R n ( q ) ,
L m n ( q ) ( j ) = i Δ s 4 H 0 ( 1 ) ( k j d m n ( q ) ) ( 1 δ m n ( q ) ) + i Δ s 4 H 0 ( 1 ) ( k j Δ s 2 e ) δ m n ( q ) ,
N m n ( q ) ( j ) = i Δ s 4 k j H 1 ( 1 ) ( k j d m n ( q ) ) D m n ( q ) d m n ( q ) ( 1 δ m n ( q ) ) + ( 1 2 + Δ s 4 π D n ( q ) ) δ m n ( q ) ,
d m n ( q ) = ( X m X n ( q ) ) 2 + ( Y m Y n ( q ) ) 2 ,
D m n ( q ) = Y n ( q ) ( X m X n ( q ) ) + X n ( q ) ( Y m Y n ( q ) ) ,
D n ( q ) = X n ( q ) Y n ( q ) X n ( q ) Y n ( q ) ,
Ψ n ( e ) ( 1 ) = Ψ n ( e ) ( 2 ) , Φ n ( e ) ( 2 ) = f 2 f 1 Φ n ( e ) ( 1 ) ,
f j = { μ j ( ω ) for TE polarization ε j ( ω ) for TM polarization } .
Ψ ( r + R ) = Ψ ( r ) e i K R ,
n = 1 N a ( N m n ( a ) ( 1 ) + e i K x D x N m n ( c ) ( 1 ) ) Ψ n ( a ) ( 1 ) + n = 1 N a ( L m n ( a ) ( 1 ) + e i K x D x L m n ( c ) ( 1 ) ) Φ n ( a ) ( 1 ) + n = 1 N b ( N m n ( b ) ( 1 ) + e i K y D y N m n ( d ) ( 1 ) ) Ψ n ( b ) ( 1 ) + n = 1 N b ( L m n ( b ) ( 1 ) + e i K y D y L m n ( d ) ( 1 ) ) Φ n ( b ) ( 1 ) + n = 1 N e N m n ( e ) ( 1 ) Ψ n ( e ) ( 1 ) n = 1 N e L m n ( e ) ( 1 ) Φ n ( e ) ( 1 ) = 0 ,
f 2 f 1 n = 1 N e L m n ( e ) ( 2 ) Φ n ( e ) ( 1 ) + n = 1 N e ( δ m n ( e ) N m n ( e ) ( 2 ) ) Ψ n ( e ) ( 1 ) = 0 ,
D t ( k , ω ) = ln ( Det ( M ) ) ,
Ψ ( r ) = Ψ inc ( r ) + 1 4 π Γ 1 [ G 0 ( r , r ) n 1 Ψ 0 ( r ) G 0 ( r , r ) Ψ 0 ( r ) n 1 ] d s ,
n = 1 N 1 [ δ m n ( 1 ) N m n ( 1 ) ( 0 ) ] ψ n ( 1 ) ( 1 ) + f 0 f 1 n = 1 N 1 L m n ( 1 ) ( 0 ) Φ n ( 1 ) ( 1 ) = ψ m inc ,
n = 1 N 1 N m n ( 1 ) ( 1 ) ψ n ( 1 ) ( 1 ) + n = 1 N 1 L m n ( 1 ) ( 1 ) Φ n ( 1 ) ( 1 ) n = 1 N 2 N m n ( 2 ) ( 1 ) ψ n ( 2 ) ( 1 ) + n = 1 N 2 L m n ( 2 ) ( 1 ) Φ n ( 2 ) ( 1 ) + n = 1 N q N m n ( q ) ( 1 ) ψ n ( q ) ( 1 ) + n = 1 N q L m n ( q ) ( 1 ) Φ n ( q ) ( 1 ) = 0 ,
n = 1 N 2 [ δ m n ( 2 ) N m n ( 2 ) ( 2 ) ] ψ n ( 2 ) ( 1 ) + f 2 f 1 n = 1 N 2 L m n ( 2 ) ( 2 ) Φ n ( 2 ) ( 1 ) = 0 ,
n = 1 N 3 [ δ m n ( 3 ) N m n ( 3 ) ( 2 ) ] ψ n ( 3 ) ( 1 ) + f 2 f 1 n = 1 N 3 L m n ( 3 ) ( 2 ) Φ n ( 3 ) ( 1 ) = 0 ,
n = 1 N q 1 [ δ m n ( q 1 ) N m n ( q 1 ) ( 2 ) ] ψ n ( q 1 ) ( 1 ) + f 2 f 1 n = 1 N q 1 L m n ( q 1 ) ( 2 ) Φ n ( q 1 ) ( 1 ) = 0 ,
n = 1 N q [ δ m n ( q ) N m n ( q ) ( 3 ) ] ψ n ( q ) ( 1 ) + f 3 f 1 n = 1 N q L m n ( q ) ( 3 ) Φ n ( q ) ( 1 ) = 0 .
ε p ( ω ) = 1 ω p 2 ω 2 ,
μ p ( ω ) = 1 F ω 2 ω 2 ω 0 2 .
k + G 2 = ω 2 c 2 μ 1 μ 2 ( ε 2 μ 1 ε 2 μ 1 μ 1 2 + μ 2 2 ) ,

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