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

Optics Express, Vol. 5, Issue 11, pp. 267-272 (1999)

http://dx.doi.org/10.1364/OE.5.000267

Acrobat PDF (718 KB)

### 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

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. 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]

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]

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. 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]

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]

*et al*.[4

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]

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]

*et al*.[5

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]

## 2. Theory

### 2.1 Group delay in photonic band-gap structures

**54**, R1078–R1081 (1996) [CrossRef]

^{iϕ(ω)}, so that the effective group velocity for the structure, v

_{g}=

*d*ω/

*d*k=d

_{sample}(

*d*ϕ/

*d*ω)

^{-1}, where d

_{sample}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

### 2.2 Measurement technique

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]

*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]

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]

_{ref}is the reference field, E

_{p}is the probe field,

*τ*is the relative time delay between the reference and the probe, and f

_{Xtal}(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.

_{r}, with a small width, Δω

_{r}. Letting ω

_{p}=ω-ω

_{r}and assuming that Δω

_{r}is small, we can evaluate the magnitude of the probe field at ω

_{p}and keep only the first term in the Taylor series expansion of the phase about ω

_{p}, such that

_{p}, the τ dependence of the measured intensity is:

_{p}). By curve fitting in time the measured intensity to a Gaussian for each frequency, the location of the peak, τ

^{max}(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):

## 3. Experiment

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]

## 4. Results

## 5. Conclusions

## 6. Acknowledgements

## 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 |

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 |

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 |

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 |

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. |

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. |

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 |

8. | R. L. Fork “Optical frequency filter for ultrashort pulses,” Opt. Let. |

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. |

**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

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