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Slow-light, band-edge waveguides for tunable time delays

M. L. Povinelli, Steven G. Johnson, and J. D. Joannopoulos

Open Access Open Access

Abstract

We propose the use of slow-light, band-edge waveguides for compact, integrated, tunable optical time delays. We show that slow group velocities at the photonic band edge give rise to large changes in time delay for small changes in refractive index, thereby shrinking device size. Figures of merit are introduced to quantify the sensitivity, as well as the accompanying signal degradation due to dispersion. It is shown that exact calculations of the figures of merit for a realistic, three-dimensional grating structure are well predicted by a simple quadratic-band model, simplifying device design. We present adiabatic taper designs that attain <0.1% reflection in short lengths of 10 to 20 times the grating period. We show further that cascading two gratings compensates for signal dispersion and gives rise to a constant tunable time delay across bandwidths greater than 100 GHz. Given typical loss values for silicon-on-insulator waveguides, we estimate that gratings can be designed to exhibit tunable delays in the picosecond range using current fabrication technology.

©2005 Optical Society of America

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Figures (7)
Fig. 1.
Fig. 1. (a) Types of 1-D periodic gratings. (b) Typical band structure of a 1-D periodic grating. The band gap is shaded yellow. (c) Magnified view of the band structure near the band edge, illustrating how a small shift in refractive index can lead to a large shift in group velocity. (The case shown corresponds to n′ < n, or Δn < 0.)

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Fig. 2.
Fig. 2. (a) 3-D perspective view of a slow-light grating structure. (b) Top view. (c) Band structure.

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Fig. 3.
Fig. 3. (a) Sensitivity figure of merit for the structure of Fig. 2 as a function of fractional frequency from the band edge. (b) Required length for different amounts of tunable time delay. Symbols/solid lines are exact/quadratic-approximation calculations for a fractional index shift Δn/n = -0.01.

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Fig. 4.
Fig. 4. Dispersion figures of merit for the structure of Fig. 2 as a function of fractional frequency from the band edge for (a) fixed bandwidth Δω/2n=10 GHz and (b) fixed time delay Δτ=1ns. Symbols/solid lines are exact/quadratic-approximation calculations for a fractional index shift Δn/n = -0.01.

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Fig. 5.
Fig. 5. (a) Reflection as a function of the length of a taper (in units of a, the waveguide period) connecting a uniform waveguide to the grated waveguide of Fig. 2 for n Si = 3.45 and ω= 0.22491[2πc/a] (< 2% from the band edge). Results were obtained from 3d numerical integration of the coupling integral in Eq. 8. (b) Normalized taper profile s(z) for linear taper (dashed line) and optimized, variable-rate taper (solid line).

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Fig. 6.
Fig. 6. Device design including adiabatically-tapered waveguide segments and dispersion compensation. Insets show the band diagram corresponding to each of the two grating regions.

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Fig. 7.
Fig. 7. Effect of dispersion compensation on the tunable time delay. Using grating 1 or grating 2 alone causes the tunable delay to vary strongly across the bandwidth. Cascading the two gratings as in Fig. 7 (labeled “average”) gives a flat tunable delay in the center of the bandwidth.

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Equations (22)

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s ≡ Δ τ τ Δ n n = Δ ( 1 v g ) Δ n 1 v g 1 Δ n n ,
L = ∣ Δ τ Δ ( 1 v g ) Δ n ∣ = ∣ Δ τ Δ n n v g s ∣ .
d ≡ τ ∣ ω − τ ∣ ω − Δ ω 1 Δ ω = L Δ ( 1 v g ) Δ ω 1 Δ ω
d = ∣ Δ τ ∣ Δ ω Δ ( 1 v g ) Δ ω ∣ Δ ( 1 v g ) Δ n ∣ .
ω ( k ) = ω be − α ( π a − k ) 2 ,
ω ′ ( k ) = ω be + δω − ( α + δα ) ( π a − k ) 2 .
s ≈ 1 Δ n n ( 1 − 1 ( 1 + δα Δ ω be ) ( 1 + δα α ) ) .
L ≈ ∣ 2 Δτ αΔ ω be 1 − [ ( 1 + δω Δ ω be ) ( 1 + δα α ) ] − 1 2 ∣ .
d ≈ L Δ ω 2 α [ 1 Δ ω be − 1 Δ ω be + Δ ω ] .
c r = − ∫ 0 L dz ∑ k 〈 r ∣ e 2 πik a ∂ C ̂ ∂ z ∣ i 〉 Δ β k ( z ) × exp [ i ∫ 0 z Δ β k ( z ′ ) dz ′ ] ,
Δ ω FP ω ≈ v g Δ L 2 L Δ β ω .
L ~ 1 ( αΔ ω be ) 3 2 .
β ˜ 3 = d dω ( d 2 k 1 d ω 2 + d 2 k 2 d ω 2 ) ∣ ω o ≈ 3 4 α ( Δ 2 ) − 5 2
ω 1 ( k 1 ) = ω o + Δ 2 − α ( π a 1 − k 1 ) 2
ω 1 ′ ( k 1 ) = ω o + Δ 2 + δω − α ( π a 1 − k 1 ) 2
ω 2 ( k 2 ) = ω o − Δ 2 + α ( k 2 − π a 2 ) 2
ω 2 ′ ( k 2 ) = ω o − Δ 2 − δω + α ( k 2 − π a 2 ) 2
Δ τ = L 2 [ Δ ( 1 v g , 1 ) + Δ ( 1 v g , 2 ) ]
Δ ( 1 v g , 1 ) = 1 2 α [ 1 Δ 2 + δω − ( ω − ω o ) − 1 Δ 2 − ( ω − ω o ) ] ,
Δ ( 1 v g , 2 ) = 1 2 α [ 1 Δ 2 + δω − ( ω o − ω ) − 1 Δ 2 − ( ω o − ω ) ] ,
Δ τ L = 1 2 α [ 1 Δ 2 + δω − 1 Δ 2 ] ≈ − 0 . 8503 c − 1 .
L 1 e = − 10 log 10 ( 1 e ) loss in dB mm = 4.3429 dB 2.1 dB mm = 2.1 mm .
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