Abstract

A new formulation of the plane-wave method to study the characteristics (dispersion curves and field patterns) of photonic crystal structures is proposed. The expression of the dielectric constant is written using the superposition of two regular lattices, the former for the perfect structure and the latter for the defects. This turns out to be simpler to implement than the classical one, based on the supercell concept. Results on mode coupling effects in two-dimensional photonic crystal waveguides are studied and successfully compared with those provided by a Finite Difference Time Domain method. In particular the approach is shown able to determine the existence of “mini-stop bands” and the field patterns of the various interfering modes.

© 2003 Optical Society of America

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References

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Appl. Phys. Lett.

C. J. M. Smith, R. M. De La Rue, M. Rattier ,S. Olivier, H. Benisty, C. Weisbuch, T. F . Krauss, R. Houdrè, U. Osterle �??Coupled guide and cavity in a two-dimensional photonic crystal,�?? Appl. Phys. Lett. 78, 1487-1489 (2001).
[CrossRef]

J. Appl. Phys.

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

J. Opt. Soc. Am. B

Opt. Express

Phy. Rev. B

P. R. Villeneuve, S. Fan, J. D. Joannopoulos, �??Microcavity in photonic crystals: Mode symmetry, tenability and coupling efficiency,�?? Phy. Rev. B 54, 7837-7842 (1996).
[CrossRef]

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, B. Jaskorzynska, �??Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,�?? Phy. Rev. B 64, 155113 (2001).
[CrossRef]

Phys. Rev. B.

S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Osterle, R. Houdrè, �??Mini-stopbands of one-dimensional system: The channel waveguide in a two-dimensional photonic crystal,�?? Phys. Rev. B. 63, 11311 (2001).
[CrossRef]

Phys. Rev. E

M. Agio, C. M. Soukoulis, �??Ministop bands in single defect photonic crystal waveguides,�?? Phys. Rev. E 64, 055603 (2001).
[CrossRef]

Proc. of IEEE-WFOPC 2000

F. Fogli, J. Pagazaurtundua Alberte, G. Bellanca, P. Bassi, �??Analysis of Finite 2-D Photonic Bandgap Lightwave Devices using the FD-TD Method,�?? Proc. of IEEE-WFOPC 2000, Pavia, June 8-9, 236-241 (2000).

Other

A.Taflove, Computational electrodynamics �?? The Finite Difference Time-Domain Method, (Artech House, Norwood, MA, 1995).

K. Sakoda, Optical properties of Photonic Crystals, Springer (2001).

S. G. Johnson, J. D. Joannopoulos, Photonic crystals The Road from Theory to Practice, Kluwer Academic Publishers, (2001).

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Figures (3)

Fig. 1.
Fig. 1.

Modeling of 2D triangular lattice PC waveguide W1: (a) direct lattice of the W1 and (b) reciprocal lattices of the perfect lattice (larger gray circles) and the defect perfect lattice (smaller green dots).

Fig. 2.
Fig. 2.

(a) Dispersion curves computed by the PW method and (b) transmission coefficient computed by FDTD of a W1 waveguide. (c) Local picture of the “mini-stop band” associated to the coupling among the fundamental mode and higher order modes. (d)-(e)-(f) Electric displacement pattern associated to the fundamental mode. (g)-(h) Electric displacement pattern associated to the high order modes.

Fig. 3.
Fig. 3.

(1.7 MB) Movies of Electric field at (a) 0.47 a/λ and (b) 0.49 a/λ. [Media 2]

Equations (13)

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ε g ( r ) = G ε g ( G ) e j G · r
ε g ( G ) = ε a f g + ε b ( 1 f g )
ε g ( G ) = ( ε a ε b ) f g 2 J 1 ( G · R ) G · R
ε s ( r ) = S ε s ( S ) e j S · r
ε s ( S ) = ε d f s
ε s ( S ) = ε d f s 2 J 1 ( S · R ) S · R
ε ( r ) = ε g ( r ) + ε s ( r ) = S ε c ( S ) e j S · r
ε c ( S ) = f s ( ε a ε b ) ( n 1 n 2 1 ) + ε b
ε c ( S ) = f s ( ε a ε b ) ( n 1 n 2 ) · 2 J 1 ( S · R ) S · R
ε c ( S ) = f s ( ε b ε a ) · 2 J 1 ( S · R ) S · R .
S ε 1 ( S S ) ( k + S ) · ( k + S ) H z ( S ) = ω 2 c 2 H z ( S )
S ε 1 ( S S ) k + S 2 E z ( S ) = ω 2 c 2 E z ( S )
E ( r ) = S E ( S ) e j ( k + S ) · r , H ( r ) = S H ( S ) e j ( k + S ) · r .

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