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Supermodes in multiple coupled photonic crystal waveguides

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Abstract

We analyze the supermodes in multiple coupled photonic crystal waveguides for long-wavelengths. In the tight-binding limit we obtain analytic results that agree with fully numerical calculations. We find that when the field flips sign after a single photonic crystal period, and there is an odd number of periods between adjacent waveguides, the supermode order is reversed, compared to that in conventional coupled waveguides, generalizing earlier results obtained for two coupled waveguides.

©2006 Optical Society of America

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Figures (5)

Fig. 1.
Fig. 1. Schematic of the geometry that we are considering, consisting of m coupled PCWs, separated by m-1 barriers, all sandwiched between two semi-infinite PCs. Each barrier consists of l PC periods.
Fig. 2.
Fig. 2. Exact numerical results for β 2 s - β 0 2 with increasing layer thickness l for (a) m = 5, and (b) m = 6 identical waveguides. Parameter values are as given in the text, with the red and blue dots respectively showing values for even and odd symmetric supermodes.
Fig. 3.
Fig. 3. Comparison of exact numerical results for a set of (a) m = 5, and (b) m = 6 identical waveguides with parameters given in the text. The horizontal lines give the limiting values of F(βs ) in the tight binding approximation, cos[/(m + 1)], whereas the dots give exact results as discussed in the text. The red dots are for even modes while the blue dots indicate odd modes.
Fig. 4.
Fig. 4. Fields of the supermodes for m = 5 and (a) l = 6 and (b) l = 7, with the associated propagation constants βsd indicated above the figures. In each case, the fundamental mode is shown on the right, with consecutive higher order modes shown towards the left. The guides are oriented parallel to the x-axis.
Fig. 5.
Fig. 5. As Figure 4, but for m = 6 and (a) l = 6 and (b) l = 7.

Equations (28)

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V j ( r ) = p = χ P 1 2 [ g j , p e P ( y y i ) + g j , p + e P ( y y j ) ] e i β P x ,
R l = ( R Q l R Q l ) ( I R Q l R Q l ) 1 ,
T l = ( I R 2 ) Q l ( I R Q l R Q l ) 1 ,
f 1 = R ˜ f 1 + , f 1 + = R ˜ m 1 f 1 + T ˜ m 1 f m + ,
f m + = R ˜ f m , f m = T ˜ m 1 f 1 + R ˜ m 1 f m + ,
˜ n = ( T ˜ n R ˜ n T ˜ n 1 R ˜ n R ˜ n T ˜ n 1 T ˜ n 1 R ˜ n T ˜ n 1 ) A n B n C n D n = ˜ n ( A B C D ) n ,
g m g m + = ( A n B n C n D n ) g 1 g 1 + .
[ f 1 + ± f m f 1 ± f m + ] = ˜ [ ± ( f 1 ± f m + ) ± ( f 1 + ± f m ) ] ,
( A n R ˜ + B n ) g = σg , ( C n R ˜ + D n ) g = σ R ˜ g ,
M ( k , β o ) g [ I ( R ˜ n + σ T ˜ n ) R ˜ ] g = 0 ,
A n = A u n 1 ( t ) u n 2 ( t ) , B n = B u n 1 ( t ) C n = C u n 1 ( t ), D n = D u n 1 ( t ) u n 2 ( t ) ,
t = 1 2 ( A + D ) , A = ( T l 2 R l 2 ) P T l ,
B = C = R l T l , D = 1 T l P .
R l = ρ ( 1 μ 2 l ) 1 ρ 2 μ 2 l , T l = ( 1 ρ 2 ) μ l 1 ρ 2 μ 2 l ,
= ± exp [ iv ( ξ ) ] ,
( APρ + B ) u n 1 ( t ) u n 2 ( t ) σ = 0
t = α ( 2 ρ ) + O ( ξ ) , + B = 2 t + O ( ξ ) , = 1 + O ( ξ ) .
u n ( t ) σ = 0 .
k 2 β s 2 h + arg [ ρ ( β s ) ] = 2 [ ρ ( β s ) ] cos ( ϑ s ) μ ( β s ) j + .
β s 2 β 0 2 4 [ ρ ( β s ) ] h χ s + β s 1 ( arg [ ρ ( β s ) ] β ) μ l cos ϑ s .
β s 2 β 0 2 4 χ 0 [ ρ ( β 0 ) ] h eff μ l cos ϑ s .
g j = T ˜ j 1 g 1 + R ˜ j 1 g j + , g j + = T ˜ m j g m + + R ˜ m j g j
g j + g j + = T ˜ j 1 ( 1 + R ˜ m j ) 1 g j + T ˜ m j ( 1 + R ˜ j 1 ) 1 g m + 1 R ˜ j 1 R ˜ m j .
g j + g j + sin j ϑ s + O ( ξ ) ,
F ( β s ) = ( β s 2 β 0 2 ) h eff 4 χ 0 [ ρ ( β 0 ) ] μ l ,
v 1 ρP v 1 + ( 1 ρ 2 ) ξP g 2 + , v 2 ( 1 ρ 2 ) ξP g 1 + ρP v 2 ,
v 1 ρP v 1 + ( 1 ρ 2 ) ξ 2 ρ v 2 ,
v 2 ( 1 ρ 2 ) ξ 2 ρ v 1 ρP v 2 ,
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