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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
1. Introduction
Recent theoretical and experimental work suggests that one-dimensional photonic crystals fabricated from layered semiconductors are an attractive candidate for constructing an optical delay line (ODL) for use in agile beam steering or optical phased arrays [1–3]. It has been shown that for a one-dimensional, N-period, photonic crystal there exists a series of N transmission resonances. A unique feature of these resonances is the combination of large group index and peak transmittance near unity, with the largest group index values occurring in the band-edge transmission resonance [3]. By controlling the frequency location of the transmission resonances relative to an optical pulse, one could theoretically control and correct the group delay and the phase of the pulse. It is important to be able to experimentally characterize the spectral and temporal behavior of photonic crystals to better understand and utilize these effects.
Several groups have examined the propagation of ultrashort optical pulses in photonic crystals both experimentally and theoretically.[1, 3–5] The previous experimental work done by Vlasov et al.[4] consisted of measuring via auto-correlation methods the delay imparted by a three-dimensional synthetic opal photonic crystal as a function of wavelength. Scalora and co-workers [1] used cross correlation techniques to examine the propagation of a picosecond pulse centered at one particular wavelength through a one-dimensional layered photonic crystal as a function of tilt of the sample. Wang et al.[5] used auto-correlation methods to measure group delay in a fiber Bragg grating. In all previous work, however, a separate experimental measurement was required to obtain the transmission spectrum of the sample, and a separate cycle of delay scan was required for each spectral (or tilt) location. An ideal experiment would be one that would give the sample-induced delay of the optical pulse concomitantly with the pulse transmission through the sample in one measurement. In this paper we present the results of group delay and transmission measurements made with just such a method.
2. Theory
2.1 Group delay in photonic band-gap structures
In order to calculate the theoretical group delay caused by a layered semiconductor photonic crystal, the propagation matrix method can be used to first calculate the complex transmission, t(ω), [1]. The transmission can be expressed as |t(ω)|eiϕ(ω), so that the effective group velocity for the structure, vg=dω/dk=dsample(dϕ/dω)-1, where dsample is the physical thickness of the sample. For an optical pulse passing through the device, the change in optical path length relative to free space will result in a time delay of
Figure 1 shows a plot of the theoretical transmission and delay of a 28 period GaAs/AlAs photonic crystal near the band-edge resonance at 1038 nanometers.
Fig. 1. Spectral variation of the theoretical transmission and delay of a 28 period GaAs/AlAs PBG structure.
2.2 Measurement technique
In order to measure the group delay and the transmission shown in Fig. 1 in one measurement sequence, we employ a form of SHG-FROG[6] as described by Foing et al.[7], using spectrally resolved upconversion of a probe pulse with a reference pulse that has a very narrow spectral width. The upconverted signal produced by mixing two non collinear pulses in a non-linear crystal can be expressed by the function [7]:
where Eref is the reference field, Ep is the probe field, τ is the relative time delay between the reference and the probe, and fXtal(ω) is a function describing the phase matching conditions in the crystal. This function depends on the thickness of the crystal, the angle between the two beams, the angle of the crystal axis, and the frequencies of the two beams.
We assume that the spectrum of the reference pulse is Gaussian, centered on the frequency ωr, with a small width, Δωr. Letting ωp=ω-ωr and assuming that Δωr is small, we can evaluate the magnitude of the probe field at ωp and keep only the first term in the Taylor series expansion of the phase about ωp, such that
where ϕ’ is the first derivative of the phase with respect to ω. It can be further shown that for a given ωp, the τ dependence of the measured intensity is:
which can be seen to be a maximum at τ=ϕ’(ωp). By curve fitting in time the measured intensity to a Gaussian for each frequency, the location of the peak, τmax(ω), will correspond to the first derivative of the phase of the probe beam. If two measurements are taken, one with the sample in the probe beam, and one without, the delay can be found according to Eq. (2.1):
and the transmission can be obtained as:
We should note, here, that the validity of Eq. (2.4) depends strongly upon the spectral width of the reference pulse. Also, the above analysis assumes an unchirped reference pulse, or equivalently a pulse spectrally narrow enough so that any residual chirp will be negligible.
3. Experiment
The sample upon which the measurements were made was a 28 period GaAs/AlAs one dimensional photonic crystal on a 670 µm GaAs substrate, with the long band edge resonance at 1.038 µm. The structure was grown by molecular beam epitaxy at Wright Patterson Air Force Base (WPAFB).
The output of a Coherent RegA 9000 regenerative amplifier (the center wavelength is 809 nm, the spectral bandwidth is ~10 nm, and the pulse duration is ~100 fs) is split into a probe and a reference path. As seen in Fig. 2, the probe field is produced by focussing 30% of the laser output into a sapphire flat to generate a continuum, and either allowing it to pass through the sample or not. The reference pulse passes through a prism pair compressor/filter [8,9] with a variable slit width to narrow the spectrum. A variable delay stage in the path of the reference pulse allows control over the relative delay between the probe and reference pulses. The two pulses are recombined co-linearly with a separation of 6 mm, and focussed at f/12.5 into a BBO crystal (1 mm thick, 24.7° angle cut) for background free upconversion.
Because the probe field is a continuum generated by focussing into a sapphire flat, any group delay or amplitude variation imparted by the continuum generation process will be present in the measured results. This effect is removed by making two sets of measurements, one with the sample in the probe beam path, and one without. The spectral dependence of the transmission and relative delay imparted by the sample can then be obtained according to equations (2.5) and (2.6).
For each data set, the variable delay stage was scanned in 6.6 fs (2 µm) steps through the region of interest alternately with and without the sample in the probe path until two sets of measurements were made with the sample present, and three were made with the sample absent. A reference pulse having a spectral width of 1.7 nm, and a duration of 560 fs was used. All of the like data sets were averaged together, and then for every frequency, were fit to a Gaussian in time to determine the location and magnitude of its maximum.
4. Results
Figure 3 shows the raw data sets. The horizontal axis is wavelength and the vertical axis is relative delay. Both data sets span 1.3 ps time intervals and 50 nm wavelength intervals.
Figure 4 shows the data sets after they have been fit to a Gaussian in time for each frequency. The fit was done by varying the amplitude, width, and center of a Guassian to give the best fit. For the no sample data set, the average width of the Gaussian fit was 645 fs. The accuracy of the center of the fit is estimated to be +/- 10 fs.
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.
Figure 5 shows the transmission recovered from the raw data sets compared to the theoretical transmission spectrum and the transmission spectrum measured at WPAFB using a Hewlett-Packard 70951B Optical Spectrum Analyzer with a built in light source, with 1 nm resolution. The deviation both of the measured transmission spectrums from theory in the first resonance peak could be attributed to small variations (<0.5%) of the thickness of the individual semiconductor layers.
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.
Figure 6 shows the delay recovered from the data sets compared to the delay predicted by theory. The group velocity dispersion of the substrate can be clearly seen in the overall slope of the delay, and the theoretical delay due to just the substrate is also shown as a reference. The measured delay, corrected for the effect of the substrate, varies from 200 fs at the peak of the band edge resonance to 27 fs in the first transmission valley.
Fig. 6. The delay obtained from measurement (red) and from theory (red). The delay imparted by the substrate is shown here in black.
5. Conclusions
We have demonstrated a measurement technique that provides an easily understood display of the influence of a structure having complex optical properties on a short optical pulse. We have applied this technique to the simultaneous measurement of the group delay and transmission of a one-dimensional photonic crystal. We find strong agreement between theory and experiment over a spectral range that spans two photonic resonances. We expect this method to be valuable in studying the spectral and temporal influence of complex optical structures on short optical pulses.
6. Acknowledgements
This work was supported in part by AFOSR, contract number F49620-96-1-0206, and in part by NSF, grant number IFCA-UAS1.
References and Links
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