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Simultaneous measurement of group delay and transmission of a one-dimensional photonic crystal

Lisa J. Gamble, William M. Diffey, Spencer T. Cole, Richard L. Fork, Darryl K. Jones, Thomas R. Nelson, John P. Loehr, and James E. Ehret

Open Access Open Access

Abstract

We characterize both the group delay and the transmission of a layered semiconductor structure in a single easily interpreted plot. The data spans a 50 nm wide spectral range with 1.7 nanometer wavelength resolution, and a 1.3 picosecond wide temporal range with temporal resolution of tens of femtoseconds. Specific data for a 28 period GaAs/AlAs layered photonic band-gap structure that characterizes both group delay and transmission of multiple photonic resonances in a single display are presented and compared to theory.

©1999 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. Spectral variation of the theoretical transmission and delay of a 28 period GaAs/AlAs PBG structure.

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Fig. 2.
Fig. 2. Experimental setup.

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Fig. 3.
Fig. 3. The raw data sets. The leftmost data set represents the upconverted spectral region of the continuum, and the data set on the right is that of the continuum through the sample. For display purposes, here, zero delay has been arbitrarily set to coincide with the center of the continuum data set. The choice of where “zero” is located is not important as we are interested only in the difference between the delay of the two data sets.

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Fig. 4.
Fig. 4. The data sets after they have been fit to a Gaussian.

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Fig. 5.
Fig. 5. Transmission recovered from the raw data sets (red), predicted by theory (blue), and measured at WPAFB (green). The spectral width of the reference pulse (black), actually centered at 809 nm, is shown here for comparison with the size of the features.

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Fig. 6.
Fig. 6. The delay obtained from measurement (red) and from theory (red). The delay imparted by the substrate is shown here in black.

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Equations (6)

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Delay = d sample V g − d sample c = d ϕ d ω − d sample c .
E signal ( ω , τ ) = f Xtal ( ω ) ∫ − ∞ ∞ E p ( ω 1 ) e − i ω 1 τ E ref ( ω − ω 1 ) d ω 1 ,
E p ( ω 1 ) ≈ ∣ E p ( ω p ) ∣ e i [ ϕ p ( ω p ) + ( ω 1 − ω p ) ϕ ’ p ( ω p ) ]
I signal ( ω p , τ ) = ∣ E signal ( ω p , τ ) ∣ 2 ∝ e − [ τ − ϕ ’ p ( ω p ) ] 2 2 Δ ω r − 2
Delay ( ω ) = τ Sample max ( ω ) − τ NoSamp max ( ω ) = ϕ ’ Sample ( ω ) − d Sample c
T ( ω ) = I Sample [ ω , τ Sample max ( ω ) ] I NoSamp [ ω , τ NoSamp max ( ω ) ]
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