## Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser

Optics Express, Vol. 14, Issue 13, pp. 5954-5960 (2006)

http://dx.doi.org/10.1364/OE.14.005954

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### Abstract

We describe an interferometric method that enables to measure the optical path delay between two consecutive femtosecond laser pulses by way of dispersive interferometry. This method allows a femtosecond laser to be utilized as a source of performing absolute distance measurements to unprecedented precision over extensive ranges. Our test result demonstrates a non-ambiguity range of ~1.46 mm with a resolution of 7 nm over a maximum distance reaching ~0.89 m.

© 2006 Optical Society of America

## 1. Introduction

1. M. R. Hee, J. A. Izatt, J. M. Jacobson, J. G. Fujimoto, and E.A. Swanson, “Femtosecond transillumination optical coherence tomography,” Opt. Lett. **18**, 950–951 (1993). [CrossRef] [PubMed]

2. M. Wojtkowski and A. Kowalczyk, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. **27**, 1415–1417 (2002). [CrossRef]

3. 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]

4. C. E. Towers, D. P. Towers, D. T. Reid, W. N. MacPherson, R. R. J. Maier, and J. D. C. Jones, “Fiber interferometer for simultaneous multiwavelength phase measurement with a broadband femtosecond laser,” Opt. Lett. **29**, 2722–2724 (2004). [CrossRef] [PubMed]

5. J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. **29**, 1153–1155 (2004). [CrossRef] [PubMed]

6. J. Ye, H. Schnatz, and L. W. Hollberg, “Optical frequency combs: from precision frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. **9**, 1041–1058 (2003). [CrossRef]

7. S. T. Cundiff and J. Ye, “Femtosecond optical frequency combs,” Rev. Mod. Phys. **75**, 325–342 (2003). [CrossRef]

8. J. Schwider and L. Zhou, “Dispersive interferometric profiler,” Opt. Lett. **19**, 995–997 (1994). [CrossRef] [PubMed]

9. K. Sakai, “Michelson-type Fourier Spectrometer for the far infrared,” Appl. Opt. **11**, 2894–2901(1972). [CrossRef] [PubMed]

10. L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B **12**, 2467–2474(1995). [CrossRef]

## 2. Principles

_{R}is fixed stationary while the measurement mirror M

_{M}is movable along the optical axis of the measurement beam. The interference intensity between the reference and measurement beams is observed by use of a spectrometer that consists of a line grating and a line array of 3648 photodetectors. A FPE(Fabry-Perot Etalon) made from fused silica with 2.0 mm thickness is put before the spectrometer, of which the resonance filtering function trims down the mode density of the comb so that only ~3 filtered consecutive modes are selectively picked out to fall on each photodetector of the spectrometer.

_{r}(ν) and r

_{m}(ν) denote the reflection coefficients of the reference and measurement mirrors, respectively. In general, the reflection coefficients tend to vary slowly with ν and for simplification, they may be assumed unity within the spectral range of s(ν). Consequently the spectral power density of Eq. (1) can be rewritten as

11. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**, 156–160(1982). [CrossRef]

_{NAR}=c/(4Np), which is referred to as the non-ambiguity range (NAR). If all the modes of the optical comb are possibly be sampled, the frequency resolution p would equal the pulse repetition rate, resulting in a L

_{NAR}of 2.0 m that is identical to the cavity length of the femtosecond laser. However, due to the practical limitation in the total number of photodetectors available for the spectrometer in use, the modes need to be filtered using a FPE (Fig. 1). In this circumstance, p is decided by the free spectral range (F.S.R.) of the FPE, accompanying a reduction of L

_{NAR}, which is measured 1.458 mm in our current configuration.

_{NAR}. When the distance to be measured is larger than L

_{NAR}, its interference signal is sampled in a folded state as illustrated in Fig. 3(a). This aliasing phenomenon results in the repeated triangle-shaped variation of the phase value as the distance increases beyond L

_{NAR}, as illustrated in Fig. 3(b). The maximum measurable distance L

_{MAX}is therefore extendable up to the temporal coherence length defined by the line width (Δ) of the optical comb modes, which is worked out to be more than ~150 m. In practice, as several FPE-filtered modes are sampled by each photodetector of the spectrometer as in our current configuration, the temporal coherence length is determined by the mode spacing of the filtered optical comb. Table 1 summarized the performance of our current configuration with reference to the case of ideal sampling in which FPE filtering is excluded.

## 3. Experimental results

_{NAR}was measured 1458.5 µm with a measuring resolution of 7 nm. The measuring resolution is mainly affected by two factors; the hardware resolution of analog-to-digital conversion of the spectrometer and the computational resolution of the Fourier-transform software adopted for determining L. To improve the measuring resolution, the spectrometer resolution needs to be as fine as possible with a high level of S/N ratio. In addition, a large size of zero padding needs to be incorporated in the Fourier-transform of measured g(ν) of Eq. (5) together with an appropriate Hanning window to enhance the computational resolution in the ν-domain.

_{NAR}, an important issue is to determine the integer multiple of L

_{NAR}to obtain absolute distances, i.e., L=mL

_{NAR}±f where m is the integer multiple and f is the fraction being measured directly from the dispersive interferometer. To determine the integer multiple m, the scheme of synthetic wavelength interferometry presented in Ref.[3

3. 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]

_{NAR}. This allows the integer m to be accurately determined so that large distances more than L

_{NAR}can be measured. Fig. 3(c) shows a test result in which a step motion of 500 µm was repeatedly induced over a distance range of 100 mm.

## 4. Conclusion

^{5}. The exploitation of a femtosecond laser for the dispersive interferometry as attempted in this investigation permits producing an abundance of interference signals of monochromatic frequencies simultaneously. This advantage results in a significant extension of the measurable range far beyond the low-coherence limit of short pulses with no need of time-delay line of mechanical scanning, which may also finds applications in the field of Fourier interferometry for spectroscopy and spectral interferometry in ultrafast technology.

## References and links

1. | M. R. Hee, J. A. Izatt, J. M. Jacobson, J. G. Fujimoto, and E.A. Swanson, “Femtosecond transillumination optical coherence tomography,” Opt. Lett. |

2. | M. Wojtkowski and A. Kowalczyk, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. |

3. | 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. |

4. | C. E. Towers, D. P. Towers, D. T. Reid, W. N. MacPherson, R. R. J. Maier, and J. D. C. Jones, “Fiber interferometer for simultaneous multiwavelength phase measurement with a broadband femtosecond laser,” Opt. Lett. |

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

6. | J. Ye, H. Schnatz, and L. W. Hollberg, “Optical frequency combs: from precision frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. |

7. | S. T. Cundiff and J. Ye, “Femtosecond optical frequency combs,” Rev. Mod. Phys. |

8. | J. Schwider and L. Zhou, “Dispersive interferometric profiler,” Opt. Lett. |

9. | K. Sakai, “Michelson-type Fourier Spectrometer for the far infrared,” Appl. Opt. |

10. | L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B |

11. | M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. |

12. | R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3940) Instrumentation, measurement, and metrology : Metrology

(140.7090) Lasers and laser optics : Ultrafast lasers

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: April 26, 2006

Revised Manuscript: June 12, 2006

Manuscript Accepted: June 15, 2006

Published: June 26, 2006

**Citation**

Ki-Nam Joo and Seung-Woo Kim, "Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser," Opt. Express **14**, 5954-5960 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-5954

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### References

- M. R. Hee, J. A. Izatt, J. M. Jacobson, J. G. Fujimoto, and E.A. Swanson, "Femtosecond transillumination optical coherence tomography," Opt. Lett. 18, 950-951 (1993). [CrossRef] [PubMed]
- M. Wojtkowski and A. Kowalczyk, "Full range complex spectral optical coherence tomography technique in eye imaging," Opt. Lett. 27, 1415-1417 (2002). [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]
- C. E. Towers, D. P. Towers, D. T. Reid, W. N. MacPherson, R. R. J. Maier, and J. D. C. Jones, "Fiber interferometer for simultaneous multiwavelength phase measurement with a broadband femtosecond laser," Opt. Lett. 29, 2722-2724 (2004). [CrossRef] [PubMed]
- J. Ye, "Absolute measurement of a long, arbitrary distance to less than an optical fringe," Opt. Lett. 29, 1153-1155 (2004). [CrossRef] [PubMed]
- J. Ye, H. Schnatz and L. W. Hollberg, "Optical frequency combs: from precision frequency metrology to optical phase control," IEEE J. Sel. Top. Quantum Electron. 9, 1041-1058 (2003). [CrossRef]
- S. T. Cundiff and J. Ye, "Femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003). [CrossRef]
- J. Schwider and L. Zhou, "Dispersive interferometric profiler," Opt. Lett. 19, 995-997 (1994). [CrossRef] [PubMed]
- K. Sakai, "Michelson-type Fourier Spectrometer for the far infrared," Appl. Opt. 11, 2894-2901(1972). [CrossRef] [PubMed]
- L. Lepetit, G. Chériaux and M. Joffre, "Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy," J. Opt. Soc. Am. B 12, 2467-2474(1995). [CrossRef]
- M. Takeda, H. Ina and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, 156-160(1982). [CrossRef]
- R. Holzwarth, T. Udem, and T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, "Optical frequency synthesizer for precision spectroscopy," Phys. Rev. Lett. 85, 2264-2267(2000). [CrossRef] [PubMed]

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