## Analysis of the temporal coherence function of a femtosecond optical frequency comb

Optics Express, Vol. 17, Issue 9, pp. 7011-7018 (2009)

http://dx.doi.org/10.1364/OE.17.007011

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### Abstract

The temporal coherence function of the femtosecond pulse train from femtosecond optical frequency comb (FOFC) has been studied. The theoretical derivation, which is based on the electric field equations of a pulse train, has been used to model the temporal coherence function of the FOFC and shows good agreement with experimental measurements which are taken with a modified Michelson interferometer. The theoretical and experimental points of view provide useful information for applications of FOFC in imaging and metrology.

© 2009 Optical Society of America

## 1. Introduction

2. L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hansch, ”Route to phase control of ultrashort light pulses,” Opt. Lett. **21**, 2008–2010 (1996). [CrossRef] [PubMed]

3. A. M. Chekhovsky, A. N. Golubev, and M. V. Gorbunkov, ”Optical Pulse Distance-Multiplying Interferometry,” Appl. Opt. **37**, 3480–3483 (1998). [CrossRef]

4. K. Minoshima and H. Matsumoto, ”High-Accuracy Measurement of 240-m Distance in an Optical Tunnel by Use of a Compact Femtosecond Laser,” Appl. Opt. **39**, 5512–5517 (2000). [CrossRef]

5. Y. Yamaoka, K. Minoshima, and H. Matsumoto, ”Direct Measurement of the Group Refractive Index of Air with Interferometry between Adjacent Femtosecond Pulses,” Appl. Opt. **41**, 4318∣*γ*(*τ*)∣4324 (2002). [CrossRef] [PubMed]

6. T. Yasui, K. Minoshima, and H. Matsumoto, ”Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum Electron. , **37**, 12–19 (2001). [CrossRef]

9. J. S. Oh and S.-W. Kim, “Femtosecond laser pulses for surface-profile metrology,” Opt. Lett. **30**, 2650–2652 (2005). [CrossRef] [PubMed]

## 2. Principles

*E*

_{train}(

*t*) and

*E*̃

_{train}(

*f*) are the electric fields of a pulse train in the time domain and the frequency domain as shown in Fig. 1(a), (b), respectively, and

*A*(

*t*) is the pulse envelope,

*φ*

_{0}is an arbitrary initial phase of the “carrier” pulse. In the time domain, the “carrier” pulse moves with the (angular) carrier frequency

*ω*. When the electric field packet repeats at the pulse repetition period

_{c}*T*, the “carrier” phase slips by Δ

_{R}*φ*

_{ce}to the carrier-envelope phase because of the difference between the group and phase velocities. In the frequency domain, a mode-locked laser generates equidistant frequency comb lines with the pulse repetition frequency

*f*

_{rep}∝1/

*T*, and due to phase slip Δ

_{R}*φ*

_{ce}, the whole equidistant frequency comb is shifted by

*f*

_{CEO}.

*S*(

*f*), which is proportional to the squared modulus of the Fourier spectrum ∣

*E*̃

_{train}(

*f*)∣

^{2}. The power spectrum of an FOFC light source can be expressed as

*τ*) is given by the inverse Fourier transform of Eq. (3):

*E*

_{train 1}(

*t*) and

*E*

_{train 2}(

*t*) at the beam splitter BS,

*E*

_{train 2}(

*t*) delays relatively to

*E*

_{train 1}(

*t*) and they finally are recombined at the BS. When the pulse train

*E*

_{train 1}(

*t*) and the relatively delayed pulse train

*E*

_{train 2}(

*t*) overlap in space, one would expect that interference fringes can be observed.

*E*

_{train 1}(

*t*) and

*E*

_{train 2}(

*t*) with parallel polarization is just

*h*× Δ

*φ*

_{ce}, 2

*π*) returns

*h*× Δ

*φ*

_{ce}-

*n*× 2

*π*where

*n*= floor(

*h*× Δ

*φ*

_{ce}/2

*π*) ( floor(

*h*× Δ

*φ*

_{ce}/2

*π*) rounds the elements of

*h*× Δ

*φ*

_{ce}/2

*π*to the nearest integers less than or equal to

*h*× Δ

*φ*

_{ce}/2

*π*),

*hT*is the relative delay between

_{R}*E*

_{train 1}(

*t*) and

*E*

_{train 2}(

*t*), and Am is the “carrier” phase slip defined by Eq. (1). After performing the time integration we obtain

*hT*an integer multiple

_{R}*h*of the pulse spacing

*T*. And the information about the time-averaged value first provides

_{R}2. L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hansch, ”Route to phase control of ultrashort light pulses,” Opt. Lett. **21**, 2008–2010 (1996). [CrossRef] [PubMed]

*I*(

*t*) actually measured experimentally. First, Fourier-transforming Eq. (11) leads to the result of

*δ*(

*f*) is the Dirac delta function and

*f*is the frequency.

*G*(

*f*) is the Fourier transform of

*I*(

*t*). The interference fringe

*I*(

*t*) is a real function, so its Fourier spectra become symmetrical at about

*f*

_{CEO}= 0 and are separated by the frequency

*f*

_{COE}, as shown in Fig. 3(b). The unwanted noise has been filtered out by a band pass filter, and the peak at

*f*=+

*f*

_{COE}is inverse Fourier-transformed into the time domain, whose result is derived as

*γ*(

*t*)∣ can be obtained as,

## 3. Experiment

_{1}, and an object mirror (half-reflection mirror) HM

_{1}. One unbalanced optical-path Michelson interferometer is composed of the common BS and M

_{1}, and a different object mirror (half-reflection mirror) HM

_{2}. The other unbalanced optical-path Michelson interferometer is composed of the common BS and M

_{1}, and a different object mirror M

_{2}. The mirrors HM

_{2}and M

_{2}are arranged at space position far away from HM

_{1}about 1.5 m and 3 m in space, respectively.

_{1}. The other part of the pulse train goes into the other arm with lengths Lo, Lo+cT

_{R}+cΔ

_{P1}, and Lo+2cT

_{R}+cΔ

_{P1}+cΔ

_{P2}(c is the light velocity in air.) and are sequentially reflected by mirrors HM

_{1}, HM

_{2}, and M

_{2}, respectively. The displacement cΔ

_{P1}and cΔ

_{P2}are introduced to avoid overlap with each other between interference fringes in space. During the measurement, by moving the common reference arm of the interferometers by means of a computer-controlled and calibrated ultrasonic stepping motor (TULA-OP-03, Technohands, Inc), we could vary the relative delay between the two output pulse trains of the three pairs.

_{1}and S

_{2}, respectively, are opened. As predicted, the interference fringe signals exhibit a high contrast between the two pairs of pulse trains by the relative displacements cT

_{R}+cΔ

_{P1}(about 1.5 m) and 2cT

_{R}+cΔ

_{P1}+cΔ

_{P2}(about 3 m). To see more clearly, we analyzed the interference fringe signals in Fig. 5(c) by the Fourier transform method that is described in the principles. Because two half-reflection mirrors were used, the intensities of the interference signals were adjusted, respectively. In consequence, Fig. 6 shows the reconstructed TCF with the relative different delay times.

## 4. Summary and future work

## Acknowledgments

## References and links

1. | J. Ye and S. T. Cundiff, |

2. | L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hansch, ”Route to phase control of ultrashort light pulses,” Opt. Lett. |

3. | A. M. Chekhovsky, A. N. Golubev, and M. V. Gorbunkov, ”Optical Pulse Distance-Multiplying Interferometry,” Appl. Opt. |

4. | K. Minoshima and H. Matsumoto, ”High-Accuracy Measurement of 240-m Distance in an Optical Tunnel by Use of a Compact Femtosecond Laser,” Appl. Opt. |

5. | Y. Yamaoka, K. Minoshima, and H. Matsumoto, ”Direct Measurement of the Group Refractive Index of Air with Interferometry between Adjacent Femtosecond Pulses,” Appl. Opt. |

6. | T. Yasui, K. Minoshima, and H. Matsumoto, ”Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum Electron. , |

7. | J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. |

8. | M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Europ. Opt. Soc. Rap. Public. 08003 |

9. | J. S. Oh and S.-W. Kim, “Femtosecond laser pulses for surface-profile metrology,” Opt. Lett. |

10. | P. A. Atanasov, 14th International School on Quantum Electronics : laser physics and applications : 18–22 September, 2006, Sunny Beach, Bulgaria (SPIE, Bellingham, Wash., 2007), Chap. 2. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(120.2830) Instrumentation, measurement, and metrology : Height measurements

(140.4050) Lasers and laser optics : Mode-locked lasers

(320.7120) Ultrafast optics : Ultrafast phenomena

**ToC Category:**

Ultrafast Fiber Lasers

**History**

Original Manuscript: February 20, 2009

Revised Manuscript: March 28, 2009

Manuscript Accepted: April 3, 2009

Published: April 13, 2009

**Citation**

Dong Wei, Satoru Takahashi, Kiyoshi Takamasu, and Hirokazu Matsumoto, "Analysis of the temporal coherence function of a femtosecond optical frequency comb," Opt. Express **17**, 7011-7018 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7011

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### References

- J. Ye and S. T. Cundiff, Femtosecond optical frequency comb : principle, operation, and applications (Springer, New York, NY, 2005).
- L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hansch, "Route to phase control of ultrashort light pulses," Opt. Lett. 21, 2008-2010 (1996). [CrossRef] [PubMed]
- A. M. Chekhovsky, A. N. Golubev, and M. V. Gorbunkov, "Optical Pulse Distance-Multiplying Interferometry," Appl. Opt. 37, 3480-3483 (1998). [CrossRef]
- K. Minoshima and H. Matsumoto, "High-Accuracy Measurement of 240-m Distance in an Optical Tunnel by Use of a Compact Femtosecond Laser," Appl. Opt. 39, 5512-5517 (2000). [CrossRef]
- Y. Yamaoka, K. Minoshima, and H. Matsumoto, "Direct Measurement of the Group Refractive Index of Air with Interferometry between Adjacent Femtosecond Pulses," Appl. Opt. 41, 4318-4324 (2002). [CrossRef] [PubMed]
- T. Yasui, K. Minoshima, and H. Matsumoto, "Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry," IEEE J. Quantum Electron. 37, 12-19 (2001). [CrossRef]
- J. Ye, "Absolute measurement of a long, arbitrary distance to less than an optical fringe," Opt. Lett. 29, 1153-1155 (2004). [CrossRef] [PubMed]
- M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, "Experimental demonstration of distance measurement with a femtosecond frequency comb laser," J. Europ. Opt. Soc. Rap. Public.08003 Vol 3 (2008).
- J. S. Oh and S.-W. Kim, "Femtosecond laser pulses for surface-profile metrology," Opt. Lett. 30, 2650-2652 (2005). [CrossRef] [PubMed]
- P. A. Atanasov, 14th International School on Quantum Electronics : laser physics and applications : 18-22 September, 2006, Sunny Beach, Bulgaria (SPIE, Bellingham, Wash., 2007), Chap. 2.

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