Abstract

Transmission through two-dimensional photonic crystals (PhCs) with several (non)periodic line defects, each being created by removing a single row of rods, are studied with the emphasis put on angular selectivity. Most of the observed features appear due to a hybrid mechanism, which is realized as a common effect of the splitting of a transmission peak being the result of peculiar coupling of individual defect-mode resonators, and the angle-dependent guided-wave cavity effect, which depends on the chosen dispersion. In the case of zero-order propagation, the role of periodic location of line defects is demonstrated. A rich variety of effects can be obtained in the angle domain within a rather narrow frequency range, which contains eigenfrequencies of defect modes. Peculiarities of the transmission peaks arising in the case of first-order propagation are considered in both angle and frequency domains. It is shown that the defect-mode related peaks can be close by to the peaks, which appear due to resonances within the pieces of PhC separated by line defects and their coupling. For the effects observed while two beams are propagating, the presence of multiple defects is rather critical than the periodicity of their location.

© 2008 Optical Society of America

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References

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2006 (4)

A. E. Serebryannikov, T. Magath, and K. Schuenemann, "Bragg transmittance of s-waves through finite-thickness photonic crystals with periodically corrugated interface," Phys. Rev. E 74, 066607 (2006).
[CrossRef]

T. Magath, "Coupled integral equations for diffraction by profiled, anisotropic, periodic structures," IEEE Trans. Antennas Propag. 54, 681-686 (2006).
[CrossRef]

X. Hu, Q. Gong, Y. Li, B. Cheng, and D. Zhang, "Ultrafast tunable filter in two-dimensional organic photonic crystal," Opt. Lett. 31, 371-373 (2006).
[CrossRef] [PubMed]

C. Ciminelli, F. Peluso, M. N. Armenise, and R. De La Rue, "Variable oblique incidence for tunability in a two-dimensional photonic-crystal guided-wave filter," J. Lightwave Technol. 24, 470-476 (2006).
[CrossRef]

2005 (9)

S. Foteinopoulou and C. M. Soukoulis, "Electromagnetic wave propagation in two-dimensional photonic crystals: a study of anomalous refractive effects," Phys. Rev. B 72, 165112 (2005).
[CrossRef]

S. Kim, I. Park, and H. Lim, "Proposal for ideal 3-dB splitters-combiners in photonic crystals," Opt. Lett. 30, 257-259 (2005).
[CrossRef] [PubMed]

M. Yan, P. Shum, and J. Hu, "Design of air-guiding honeycomb photonic bandgap fiber," Opt. Lett. 30, 465-467 (2005).
[CrossRef] [PubMed]

A. Sentenac and A.-L. Fehrembach, "Angular tolerant resonant grating filters under oblique incidence," J. Opt. Soc. Am. A 22, 475-480 (2005).
[CrossRef]

I. Moreno, J. J. Araiza, and M. Avedano-Alejo, "Thin-film spatial splitters," Opt. Lett. 30, 914-917 (2005).
[CrossRef] [PubMed]

J. H. Wu, L. K. Ang, A. Q. Liu, H. G. Teo, and C. Lu, "Tunable high-Q photonic-bandgap Fabry-Perot resonator," J. Opt. Soc. Am. B 22, 1770-1777 (2005).
[CrossRef]

T. Magath and A. E. Serebryannikov, "Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs," J. Opt. Soc. Am. A 22, 2405-2418 (2005).
[CrossRef]

N. Burani and J. Laegsgaard, "Perturbative modeling of Bragg-grating-based biosensors in photonic-crystal fibers," J. Opt. Soc. Am. B 22, 2487-2492 (2005).
[CrossRef]

Y. J. Lee, J. Yeo, R. Mittra, and W. S. Park, "Applications of electromagnetic bandgap (EBG) superstrates with controllable defects for a class of patch antennas as spatial angular filters," IEEE Trans. Antennas Propag. 53, 224-235 (2005).
[CrossRef]

2004 (4)

A. Sugitatsu, T. Asano, and S. Noda, "Characterization of line-defect-waveguide lasers in two-dimensional photonic-crystal slabs," Appl. Phys. Lett. 84, 5395-5397 (2004).
[CrossRef]

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

J. Zimmermann, M. Kamp, A. Forschel, and R. Maerz, "Photonic crystal waveguide directional couplers as wavelength selective optical filters," Opt. Commun. 230, 387-392 (2004).
[CrossRef]

H. Altug and J. Vuckovic, "Two-dimensional coupled photonic crystal resonator arrays," Appl. Phys. Lett. 84, 161-163 (2004).
[CrossRef]

2003 (5)

2002 (2)

2000 (2)

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[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]

1999 (3)

1998 (2)

K. Othaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
[CrossRef]

T. Ueda, K. Othaka, N. Kawai, and K. Sakoda, "Limits on quality factors of localized defect modes in photonic crystals due to dielectric loss," J. Appl. Phys. 84, 6299-6304 (1998).
[CrossRef]

1997 (2)

K. Sakoda, T. Ueta, and K. Othaka, "Numerical analysis of eigenmodes localized at line defects in photonic lattices," Phys. Rev. B 56, 14905-14908 (1997).
[CrossRef]

G. Tayeb and D. Maystre, "Rigorous theoretical study of finite-size two-dimensional photonic crystals doped with microcavities," J. Opt. Soc. Am. A 14, 3323-3332 (1997).
[CrossRef]

1996 (1)

H. Benisty, "Modal analysis of optical guides with two-dimensional photonic band-gap boundaries," J. Appl. Phys. 79, 7483-7492 (1996).
[CrossRef]

1993 (1)

1991 (2)

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, "Microwave propagation in two-dimensional dielectric lattices," Phys. Rev. Lett. 67, 2017-2020 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, "Donor and acceptor modes in photonic band structure," Phys. Rev. Lett. 67, 3380-3383 (1991).
[CrossRef] [PubMed]

Appl. Phys. Lett. (4)

A. Martinez, F. Cuesta, A. Griol, D. Mira, J. Garcia, P. Sanchis, R. Llorente, and J. Marti, "Photonic-crystal 180° power splitter based on coupled-cavity waveguides," Appl. Phys. Lett. 83, 3033-3035 (2003).
[CrossRef]

A. Sugitatsu, T. Asano, and S. Noda, "Characterization of line-defect-waveguide lasers in two-dimensional photonic-crystal slabs," Appl. Phys. Lett. 84, 5395-5397 (2004).
[CrossRef]

D. Schurig and D. R. Smith, "Spatial filtering using media with indefinite permittivity and permeability tensors," Appl. Phys. Lett. 82, 2215-2217 (2003).
[CrossRef]

H. Altug and J. Vuckovic, "Two-dimensional coupled photonic crystal resonator arrays," Appl. Phys. Lett. 84, 161-163 (2004).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

T. Magath, "Coupled integral equations for diffraction by profiled, anisotropic, periodic structures," IEEE Trans. Antennas Propag. 54, 681-686 (2006).
[CrossRef]

Y. J. Lee, J. Yeo, R. Mittra, and W. S. Park, "Applications of electromagnetic bandgap (EBG) superstrates with controllable defects for a class of patch antennas as spatial angular filters," IEEE Trans. Antennas Propag. 53, 224-235 (2005).
[CrossRef]

J. Appl. Phys. (3)

F. Gadot, A. de Lustrac, J.-M. Lourtioz, T. Brillat, A. Ammouche, and E. Akmansoy, "High-transmission defect modes in two-dimensional metallic photonic crystals," J. Appl. Phys. 85, 8499-8501 (1999).
[CrossRef]

H. Benisty, "Modal analysis of optical guides with two-dimensional photonic band-gap boundaries," J. Appl. Phys. 79, 7483-7492 (1996).
[CrossRef]

T. Ueda, K. Othaka, N. Kawai, and K. Sakoda, "Limits on quality factors of localized defect modes in photonic crystals due to dielectric loss," J. Appl. Phys. 84, 6299-6304 (1998).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (5)

Opt. Commun. (1)

J. Zimmermann, M. Kamp, A. Forschel, and R. Maerz, "Photonic crystal waveguide directional couplers as wavelength selective optical filters," Opt. Commun. 230, 387-392 (2004).
[CrossRef]

Opt. Lett. (8)

Phys. Rev. B (3)

K. Sakoda, T. Ueta, and K. Othaka, "Numerical analysis of eigenmodes localized at line defects in photonic lattices," Phys. Rev. B 56, 14905-14908 (1997).
[CrossRef]

K. Othaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
[CrossRef]

S. Foteinopoulou and C. M. Soukoulis, "Electromagnetic wave propagation in two-dimensional photonic crystals: a study of anomalous refractive effects," Phys. Rev. B 72, 165112 (2005).
[CrossRef]

Phys. Rev. E (1)

A. E. Serebryannikov, T. Magath, and K. Schuenemann, "Bragg transmittance of s-waves through finite-thickness photonic crystals with periodically corrugated interface," Phys. Rev. E 74, 066607 (2006).
[CrossRef]

Phys. Rev. Lett. (4)

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

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, "Microwave propagation in two-dimensional dielectric lattices," Phys. Rev. Lett. 67, 2017-2020 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, "Donor and acceptor modes in photonic band structure," Phys. Rev. Lett. 67, 3380-3383 (1991).
[CrossRef] [PubMed]

Other (4)

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2005).

K. Inoue and K. Othaka, eds., Photonic Crystals: Physics, Fabrication and Applications (Springer, 2004).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975), Chap. 7.6.

See www.cst.com.

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Figures (11)

Fig. 1
Fig. 1

Geometry of the problem for one of the PhCs considered.

Fig. 2
Fig. 2

Transmission in defect-mode regime within the first opaque range of T at simultaneous variation of angle and frequency, when (a) m 1 = 3 and m 2 = 6 , (b) m 1 = 3 and m 2 = 5 , and (c) m 1 = 3 , m 2 = 6 , and m 3 = 9 , d a = 0.4 , ε r = 11.4 , N = 8 in cases (a) and (b), and N = 11 in case (c); ζ = ω a 2 π c . Compositions of rods within a single longitudinal period are shown in insets of Figs. 4a, 6b, 3a, respectively.

Fig. 3
Fig. 3

Transmission through an 11-layer PhC at (a) m 1 = 3 , m 2 = 6 and m 3 = 9 , and (b) m 1 = 3 , m 2 = 7 , and m 3 = 9 , and oblique incidence; ω a 2 π c = 0.315 (solid curve), ω a 2 π c = 0.304 (dashed curve), ω a 2 π c = 0.35 (dashed–dotted curve), d a = 0.4 , ε r = 11.4 . In case (a) the gray dotted curve corresponds to Eq. (6) with R = 0.91 and k b = 18 π 0.028 .

Fig. 4
Fig. 4

Dependence of transmittance on θ for an eight-layer PhC at (a) m 1 = 3 and m 2 = 6 , and (b) m 1 = 2 , m 2 = 5 , and m 3 = 7 , ω a 2 π c = 0.312 (solid curve), ω a 2 π c = 0.35 (dashed–dotted curve), ω a 2 π c = (a) 0.294 and (b) 0.277 (dashed curve), ω a 2 π c = (a) 0.304 and (b) 0.288 (thick solid curve); d a = 0.4 , ε r = 11.4 .

Fig. 5
Fig. 5

Electric field pattern at the triple peak shown by the dashed–dotted curve in Fig. 3a at θ = (a) 32 ° , (b) 33 ° , (c) 33.6 ° , (d) 34 ° , (e) 35.8 ° , (f) 36.6 ° , (g) 37.2 ° , (h) 38.5 ° , (i) 39.1 ° , and (j) 40 ° .

Fig. 6
Fig. 6

Peaks of transmission spectrum at the first-order propagation at (a) m 1 = 3 and m 2 = 6 , and (b) m 1 = 3 and m 2 = 5 , and the same other parameters as in Figs. 2a, 2b, at θ = π 3 , ζ = ω a 2 π c , solid and dashed curves correspond to n = 0 and n = 1 , respectively.

Fig. 7
Fig. 7

Maxima of different origin arising in transmission at θ = π 3 , d a = 0.4 , and ε r = 11.4 : (a) N = 8 , m 1 = 3 , and m 2 = 6 , (b) N = 8 , m 1 = 3 , and m 2 = 5 , (c) N = 8 without defects, (d) N = 11 , m 1 = 4 , and, m 2 = 8 , ζ = ω a 2 π c . Solid and dashed curves correspond to n = 0 and n = 1 , respectively.

Fig. 8
Fig. 8

Two enlarged fragments of Fig. 7d.

Fig. 9
Fig. 9

Near-field patterns for transmission peaks of eight-layer PhC. Cases (a) and (b) correspond to ω a 2 π c = 0.6815 and 0.6842 from Fig. 7a, cases (c)–(f) correspond to ω a 2 π c = 0.6823 , 0.686 , 0.6973 , and 0.6974 from Fig. 7b. Cases (e) and (f) demonstrate the possibility of field localization within the pieces of PhC, which are separated by line defects.

Fig. 10
Fig. 10

Near-field patterns corresponding to transmission peaks of the 11-layer PhC in Fig. 7d. Cases (a)–(f) correspond to ω a 2 π c = 0.6823 , 0.6826 , 0.6947 , 0.6953 , 0.6987 , and 0.6989 , respectively. Cases (c)–(f) correspond to NDMPs.

Fig. 11
Fig. 11

Transmission versus θ for PhCs with various locations of line defects from Fig. 7. Case (a) ω a 2 π c = 0.6852 , N = 8 , m 1 = 3 , and m 2 = 6 ; case (b) ω a 2 π c = 0.6852 , N = 8 , m 1 = 3 , and m 2 = 5 , case (c) ω a 2 π c = 0.6973 , N = 8 , m 1 = 3 , and m 2 = 5 . Solid and dashed curves correspond to n = 0 and n = 1 , respectively.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

E i ( x , y ) = E 0 i exp ( i α 0 x i η 0 y ) ,
E ( x , y ) = E i ( x , y ) + n = ρ n exp ( i α n x + i η n y ) ,
E ( x , y ) = n = τ n exp ( i α n x i η n y ) ,
m 2 m 1 = m 3 m 2 = = m M m M 1 = const.
m 1 = m 2 m 1 = N + 1 m M ,
T = t 0 + t 1 ,
T FP = ( 1 R ) 2 ( 1 R ) 2 + 4 R sin 2 ( k b cos θ ) ,
sin 2 ( k b cos θ i ) = 0 ,
k b = π cos θ 1 cos θ 2 cos θ 1 cos θ 2 .

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