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Slow light in a semiconductor waveguide at gigahertz frequencies

Jesper Mørk, Rasmus Kjær, Mike van der Poel, and Kresten Yvind

Open Access Open Access

Abstract

We experimentally demonstrate slow-down of light by a factor of three in a 100 μm long semiconductor waveguide at room temperature and at a record-high frequency of 16.7 GHz. It is shown that the group velocity can be controlled all-optically as well as through an applied bias voltage. A semi-analytical model based on the effect of coherent population oscillations and taking into account propagation effects is derived and is shown to well account for the experimental results. It is shown that the carrier lifetime limits the maximum achievable delay. Based on the general model we analyze fundamental limitations in the application of light slowdown due to coherent population oscillations.

©2005 Optical Society of America

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Figures (7)
Fig. 1.
Fig. 1. Experimental set-up for measuring the propagation speed of a modulated optical signal in a semiconductor waveguide device (electro-absorption modulator – EAM).

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Fig. 2.
Fig. 2. Contour plot of measured absolute change in group refractive index versus input intensity and reverse voltage for a 100 μm long semiconductor waveguide. The measurements were carried out at a modulation frequency f0 = 15 GHz.

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Fig. 3.
Fig. 3. Measured transmission (a) and phase shift (time delay) (b) versus voltage for different input power levels. Modulation frequency (detuning): 16.7 GHz.

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Fig. 4.
Fig. 4. Imaginary (a) and real (b) parts of the third-order probe susceptibility due to coherent population oscillations and corresponding group refractive index (c) calculated for different values of α. Other parameters are: P = 10 mW, Psat , = 10 mW, τ s = 200 ps, and Γg0 = -5∙104 m-1.

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Fig. 5.
Fig. 5. Calculated transmission (a) and phase / time delay (b) versus reverse bias voltage for different input power levels. Frequency detuning: 16.7 GHz.

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Fig. 6.
Fig. 6. Calculated (a) transmission and (b) relative time delay ξ = Δtmod /τ (at zero detuning) versus input power for different levels of small-signal (unsaturated) transmission T0 .

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Fig. 7.
Fig. 7. Calculated variation of (a) delay relative to carrier lifetime and (b) length-averaged modulation refractive index versus waveguide length. The absorption is Γg0 =-5∙104 m-1.

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Equations (19)

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Δ n g = c L Δ t = c L Δ φ Ω
χ r ( ω 0 + Ω ) = c n b ω 0 g sat P P sat α + i − i Ω τ s + 1 + P P sat
g sat = Γ g 0 1 + P P sat ,
dP dz = ( g sat − α int ) P ,
n 2 = 1 + χ b ′ + χ r ′ ,
n g = n + ω 0 dn dω ≅ n gb + ω 0 2 n b d χ ′ r dω .
n g ( Ω = 0 ) = n gb − 1 2 c τ s Γ g 0 P P sat ( 1 + P P sat ) 3 .
n mod = n + ω 0 n ( ω 0 + Ω ) − n ( ω 0 − Ω ) 2 Ω ≅ n gb + 2 ω 0 2 n b χ r ( ω 0 + Ω ) − χ r ( ω 0 − Ω ) 2 Ω ,
n mod = n gb − c τ s Γ g 0 P P sat 1 + P P sat 1 ( Ω τ s ) 2 + ( 1 + P P sat ) 2 .
t mod = ∫ 0 L n mod ( z ) dz ≡ n ¯ mod c L
Δ φ mod = ΩΔ t mod = Ω L c ( n ¯ mod − n gb ) .
n ¯ mod = n gb − c L 1 Ω Arctan ⁡ { Ω τ s ( T sat − 1 ) P ( 0 ) / P sat ( Ω τ s ) 2 + 1 + ( T sat + 1 ) P ( 0 ) / P sat + T sat ( P ( 0 ) / P sat ) 2 }
T sat = P ( L ) P ( 0 ) .
ln ⁡ ( T sat ) + ( T sat − 1 ) P ( 0 ) P sat = Γ g 0 L ,
n ¯ mod = n gb − c L τ s ( T sat − 1 ) P ( 0 ) / P sat ( Ω τ s ) 2 + 1 + ( T sat + 1 ) P ( 0 ) / P sat + T sat ( P ( 0 ) / P sat ) 2 .
Δ P ( Ω , L ) Δ P ( Ω , 0 ) = G sat ( 1 − ( G sat − 1 ) P ( 0 ) / P sat − i Ω τ s + 1 + G sat P ( 0 ) / P sat ) .
n ¯ g = n gb + 1 2 c τ s 1 L ( 1 1 + T sat P ( 0 ) / P sat − 1 1 + P ( 0 ) / P sat ) .
Δ t g = L c Δ n ¯ g = 1 2 τ s ( 1 1 + T sat P ( 0 ) / P sat − 1 1 + P ( 0 ) / P sat ) .
g 0 = − a V ( V − V on ) 2 , τ s = τ s 0 exp ⁡ ( − V V ref ) ,
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