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Analysis and design of arrayed waveguide gratings with MMI couplers

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Abstract

We present an extension of the AWG model and design procedure described in [1] to incorporate multimode interference, MMI, couplers. For the first time to our knowledge, a closed formula for the passing bands bandwidth and crosstalk estimation plots are derived.

©2001 Optical Society of America

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Figures (7)

Fig. 1.
Fig. 1. AWG physical layout. Insets, waveguide parameters (left) and FPR coupler layout (right)
Fig. 2.
Fig. 2. MMI coupler layout
Fig. 3.
Fig. 3. Cross talk level @ Δνc with MMI at the IW’s
Fig. 4.
Fig. 4. Cross talk level @ Δνc with MMI at the IW’s
Fig. 5.
Fig. 5. MMI-based 1×16 frequency cyclic AWG module response versus detunning from the design frequency
Fig. 6.
Fig. 6. MMI-based AWG module (blue) and delay (green) response versus detunning from the design frequency
Fig. 7.
Fig. 7. MMI-based (red) and ordinary (blue) AWG module responses

Tables (1)

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Table 1. High Level Requirements for the designed AWG’s

Equations (35)

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β i ( x 0 ) = 2 π ω i 2 4 e ( x 0 ω i ) 2
B i ( x 1 ) = 2 π ω i 2 α 2 4 e ( π ω i x 1 α ) 2
α = c L f n s ν 0
β g ( x 1 ) = 2 π ω g 2 4 e ( x 1 ω g ) 2
f 1 ( x 1 ) = [ Π ( x 1 N d w ) B i ( x 1 ) δ ω ( x 1 ) ] 2 π ω g 2 4 β g ( x 1 )
Π ( x 1 N d ω ) = { 1 x 1 N d ω 2 0 otherwise
δ w ( x 1 ) = r = + δ ( x 1 r d w )
Δ l = m λ 0 n c = m c n c ν 0
f 2 ( x 2 , ν ) = [ B i ( x 2 ) Π ( x 2 N d w ) δ w ( x 2 ) ϕ ( x 2 , ν ) ] 2 π ω g 2 4 β g ( x 2 )
ϕ ( x 2 , ν ) = ψ ( ν ) e j 2 π m ν ν 0 x 2 d w
ψ ( ν ) = e i 2 π ν ( n c l 0 c + m N ν 0 2 )
f 3 ( x 3 , ν ) = 2 π ω g 2 α 2 4 B g ( x 3 ) ψ ( ν ) r = f M ( x 3 r α d w + ν γ )
γ = d ω ν 0 α m
B g ( x 3 ) = F { β g ( x 2 ) } u = x 3 α = 2 π ω g 2 4 e ( π ω g x 3 α ) 2
f M ( x 3 ) = ( α 2 8 π ω i 2 ) 1 4 e ( x 3 ω i ) 2 [ er f ( π ω i N d w 2 α + i x 3 α ) + er f ( π ω i N d w 2 α i x 3 α ) ]
Δ x 3 , FSR = α d w
Δ ν FSR = ν m
f 3 ( x 3 , ν ) = 2 π ω g 2 α 2 4 B g ( x 3 ) ψ ( ν ) r = f M ( x 3 Δ x 3 , FSR [ r ν Δ ν FSR , 0 ] )
t 0 , q ( ν ) = + f 3 ( x 3 , ν ) β o ( x 3 q d o ) x 3
L m = 3 π 8 ( ζ 0 ζ 1 )
β i ( x 0 ) = [ 2 ω i π 2 ( 1 + e Δ x m 2 2 ω i 2 ) ] 1 2 [ e ( x 0 1 2 Δ x m ω i ) 2 + e ( x 0 + 1 2 Δ x m ω i ) 2 ]
B i ( x 1 ) = [ 2 α 2 π ω i ( 1 + e Δ x m 2 2 ω i 2 ) ] 1 4 [ e i π Δ x m x 1 α + e i π Δ x m x 1 α ] e ( π ω i x 1 α ) 2
f 3 ( x 3 , ν ) = π ω g 2 2 α 2 ( 1 + e Δ x m 2 2 ω i 2 ) 4 B g ( x 3 ) ψ ( ν ) r = [ f M ( x 3 r α d w + ν γ + Δ x m 2 ) ]
+ f M ( x 3 r α d w + ν γ Δ x m 2 ) ]
t 0,0 ( Δ ν ) + β i ( x 3 Δ ν γ ) β o ( x 3 ) x 3
t 0,0 , n ( Δ ν ) = t 0,0 ( Δ ν ) t 0,0 ( 0 ) = e 1 2 ( Δ ν ω o γ ) 2 cosh ( Δ x m Δ ν 2 ω o 2 γ )
Δ x m = 2 ω i
t 0,0 , n ( x ) = e 1 2 x 2 cosh ( x ) = 10 3 20
Δ ν b ω = 2 γ ω o 1.6173
Δ ν b ω = 2 γ ω o 0.8311
t 0,1 ( σ , σ o ) = + β i ( u n ) β o ( u n ) u n
σ = α π N d w ω o
σ o = d o ω o
B i ( x 1 n ) = e ( x 1 n σ ) 2 ( e i 2 x 1 n σ + e i 2 x 1 n σ )
β o ( u n ) = e ( π σ u n σ o ) 2
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