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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 5, Iss. 11 — Nov. 22, 1999
  • pp: 267–272
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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, Jr., John P. Loehr, and James E. Ehret  »View Author Affiliations


Optics Express, Vol. 5, Issue 11, pp. 267-272 (1999)
http://dx.doi.org/10.1364/OE.5.000267


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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.

© 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

1. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996) [CrossRef]

3

3. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53,4101–4121 (1996) [CrossRef]

]. 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

3. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53,4101–4121 (1996) [CrossRef]

]. 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.

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

1. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996) [CrossRef]

]. 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

Delay=dsampleVgdsamplec=dϕdωdsamplec.
(2.1)

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

6. R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997) [CrossRef]

] as described by Foing et al.[7

7. J. P. Foing, J. P. Likforman, M. Joffre, and A. Migus, “Femtosecond pulse phase measurement by spectrally resolved up-conversion: Application to continuum compression,” IEEE J. Quantum Electron 28, 2285–2290 (1992) [CrossRef]

], 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

7. J. P. Foing, J. P. Likforman, M. Joffre, and A. Migus, “Femtosecond pulse phase measurement by spectrally resolved up-conversion: Application to continuum compression,” IEEE J. Quantum Electron 28, 2285–2290 (1992) [CrossRef]

]:

Esignal(ω,τ)=fXtal(ω)Ep(ω1)eiω1τEref(ωω1)dω1,
(2.2)

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

Ep(ω1)Ep(ωp)ei[ϕp(ωp)+(ω1ωp)ϕp(ωp)]
(2.3)

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:

Isignal(ωp,τ)=Esignal(ωp,τ)2e[τϕp(ωp)]22Δωr2
(2.4)

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):

Delay(ω)=τSamplemax(ω)τNoSampmax(ω)=ϕSample(ω)dSamplec
(2.5)

and the transmission can be obtained as:

T(ω)=ISample[ω,τSamplemax(ω)]INoSamp[ω,τNoSampmax(ω)]
(2.6)

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

8. R. L. Fork “Optical frequency filter for ultrashort pulses,” Opt. Let. 11, 629–631 (1986) [CrossRef]

,9

9. K. L. Schehrer, R. L. Fork, H. Avramopoulos, and E. S. Fry “Derivation and measurement of the reversible temporal lengthening of femtosecond pulses for the case of a four-prism sequence,” Opt. Let. 15, 550–552 (1990) [CrossRef]

] 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.

Fig. 2. Experimental setup.

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

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

1.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996) [CrossRef]

2.

T. R. Nelson, J. P. Loehr, Q. Xie, J. E. Ehret, J. E. VanNostrand, L. J. Gamble, D. K. Jones, S. T. Cole, R. A. Trimm, W. M. Diffey, R. L. Fork, and A. S. Keys, “Electrically tunable group delays using quantum wells in a distributed bragg reflector,” in Enabling Photonic Technologies for Aerospace Applications, Proc. SPIE 3714, 12–23 (1999)

3.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53,4101–4121 (1996) [CrossRef]

4.

Y. A. Vlasov, S. Petit, G. Klein, B. Honerlange, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E 60, 1030–1035 (1999) [CrossRef]

5.

S. Wang, H. Erlig, H. R. Fetterman, E. Yablonovitch, V. Grubsky, D. S. Starodubov, and J. Feinberg, “Measurement of the temporal delay of a light pulse through a one-dimensional photonic crystal,” Micro. Opt. Technol. Let. 20, 17–21 (1999) [CrossRef]

6.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997) [CrossRef]

7.

J. P. Foing, J. P. Likforman, M. Joffre, and A. Migus, “Femtosecond pulse phase measurement by spectrally resolved up-conversion: Application to continuum compression,” IEEE J. Quantum Electron 28, 2285–2290 (1992) [CrossRef]

8.

R. L. Fork “Optical frequency filter for ultrashort pulses,” Opt. Let. 11, 629–631 (1986) [CrossRef]

9.

K. L. Schehrer, R. L. Fork, H. Avramopoulos, and E. S. Fry “Derivation and measurement of the reversible temporal lengthening of femtosecond pulses for the case of a four-prism sequence,” Opt. Let. 15, 550–552 (1990) [CrossRef]

OCIS Codes
(300.6500) Spectroscopy : Spectroscopy, time-resolved
(320.7100) Ultrafast optics : Ultrafast measurements

ToC Category:
Research Papers

History
Original Manuscript: October 25, 1999
Published: November 22, 1999

Citation
Lisa Gamble, William Diffey, Spencer Cole, Richard L. Fork, and Darryl Jones, "Simultaneous measurement of group delay and transmission of a one-dimensional photonic crystal," Opt. Express 5, 267-272 (1999)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-5-11-267


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References

  1. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, R. P. Leavitt, "Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss," Phys. Rev. E 54, R1078-R1081 (1996) [CrossRef]
  2. T. R. Nelson, J. P. Loehr, Q. Xie, J. E. Ehret, J. E. VanNostrand, L. J. Gamble, D. K. Jones, S. T. Cole, R. A. Trimm, W. M. Diffey, R. L. Fork, A. S. Keys, "Electrically tunable group delays using quantum wells in a distributed bragg reflector," in Enabling Photonic Technologies for Aerospace Applications, Proc. SPIE 3714, 12-23 (1999).
  3. J. M. Bendickson, J. P. Dowling, M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53,4101-4121 (1996). [CrossRef]
  4. Y. A. Vlasov, S. Petit, G. Klein, B. Honerlange, C. Hirlimann, "Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal," Phys. Rev. E 60, 1030-1035 (1999). [CrossRef]
  5. S. Wang, H. Erlig, H. R. Fetterman, E. Yablonovitch, V. Grubsky, D. S. Starodubov, J. Feinberg, "Measurement of the temporal delay of a light pulse through a one-dimensional photonic crystal," Micro. Opt. Technol. Let. 20, 17-21 (1999). [CrossRef]
  6. R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, D. J. Kane, "Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating," Rev. Sci. Instrum. 68, 3277-3295 (1997). [CrossRef]
  7. J. P. Foing, J. P. Likforman, M. Joffre, A. Migus, "Femtosecond pulse phase measurement by spectrally resolved up-conversion: Application to continuum compression," IEEE J. Quantum Electron 28, 2285-2290 (1992). [CrossRef]
  8. R. L. Fork "Optical frequency filter for ultrashort pulses," Opt. Let. 11, 629-631 (1986). [CrossRef]
  9. K. L. Schehrer, R. L. Fork, H. Avramopoulos, E. S. Fry "Derivation and measurement of the reversible temporal lengthening of femtosecond pulses for the case of a four-prism sequence," Opt. Let. 15, 550-552 (1990). [CrossRef]

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