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Reformulation of the plane wave method to model photonic crystals

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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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Supplementary Material (2)

Media 1: AVI (1743 KB)     
Media 2: AVI (1718 KB)     

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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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