## Absolute three-dimensional coordinate measurement by the two-point diffraction interferometry

Optics Express, Vol. 15, Issue 8, pp. 4435-4444 (2007)

http://dx.doi.org/10.1364/OE.15.004435

Acrobat PDF (443 KB)

### Abstract

We describe a method of absolute xyz-coordinates measurement based on the two-point diffraction interferometer. In this paper we use a new optimization algorithm to the interferometer. Experimental results show that the systematic error of the interferometer is less than 1 μm (peak-to-valley value) within a 60 mm by 60 mm by 20 mm working volume. To extract the systematic error and verify the absolute performance of the interferometer we applied the Fourier self-calibration concept.

© 2007 Optical Society of America

## 1. Introduction

1. P. de Groot, “Grating interferometer for flatness
testing,” Opt. Lett. **21**, 228–230
(1996). [CrossRef] [PubMed]

2. R. Dandliker, R. Thalmann, and D. Prongue, “Two-wavelength laser interferometry
using superheterodyne detection,” Opt.
Lett. **13**, 339–341
(1988). [CrossRef] [PubMed]

3. Z. Sodnik, E. Fischer, T. Ittner, and H. J. Tiziani, “Two-wavelength double heterodyne
interferometry using a matched grating
technique,” Appl. Opt. **30**,
3139–3144(1991). [CrossRef] [PubMed]

4. H. Kikuta, K. Iwata, and R. Nagata, “Distance measurement by the
wavelength shift of laser diode light,”
Appl. Opt. **25**, 2976–2980
(1986). [CrossRef] [PubMed]

5. H. Kikuta, K. Iwata, and R. Nagata, “Absolute distance measurement by
wavelength shift interferometry with a laser diode light: some systematic
error sources,” Appl. Opt. **26**, 1654–1660
(1987). [CrossRef] [PubMed]

6. T. Li, A. Wang, K. Merphy, and R. Claus, “White-light scanning fiber Michelson
interferometer for absolute position-distance
measurement,” Opt. Lett. **20**, 785–787
(1995). [CrossRef] [PubMed]

7. U. Schnell and R. Dandliker, “Dispersive white-light
interferometry for absolute distance measurement with dielectric multilayer
systems on the target,” Opt. Lett. **21**, 528–530
(1996). [CrossRef] [PubMed]

8. 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]

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

10. 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]

11. K. Lau, R. J. Hocken, and W. C. Haight, “Automatic laser tracking
interferometer system for robot metrology,”
Prec. Eng. **8**, 3–8
(1986). [CrossRef]

14. H. Jiang, S. Osawa, T. Takatsuji, H. Noguchi, and T. Kurosawa, “High-performance laser tracker using
an articulation mirror for the calibration of coordinate measuring
machine,” Opt. Eng. **41**, 632–637
(2002). [CrossRef]

15. H.G. Rhee and S.W. Kim, “Absolute distance measurement by
two-point diffraction interferometry,”
Appl. Opt. **41**, 5921–5928
(2002). [CrossRef] [PubMed]

16. J. Chu and S. W. Kim, “Absolute distance measurement by
lateral shearing interferometry of point-diffracted spherical
waves,” Opt. Express **14**, 5961–5967
(2006). [CrossRef] [PubMed]

*x*

_{1},

*y*

_{1},

*z*

_{1}) and (

*x*

_{2},

*y*

_{2},

*z*

_{2}), respectively, which are to be determined to find out the xyz-location of the target. In this paper we describe the basic theory, the recently upgraded coordinate estimation algorithm, and error budget of the interferometer. To check the performance we compare the readings of our interferometer with the results of conventional coordinate measuring instruments. In addition, the systematic errors of the interferometer are identified adopting the concept of self-calibration [17

17. J. Ye, M. Takac, C. N. Berglund, G. Owen, and R. F. Pease, “An exact algorithm for
self-calibration of two-dimensional precision metrology
stages,” Prec. Eng. **20**, 16–32
(1997). [CrossRef]

## 2. Basic theory and the system configuration

_{1}and u

_{2}, can be derived as [15

15. H.G. Rhee and S.W. Kim, “Absolute distance measurement by
two-point diffraction interferometry,”
Appl. Opt. **41**, 5921–5928
(2002). [CrossRef] [PubMed]

*U*is the source strength, λ is the wavelength, ϕ is the initial phase, and

*r*is the diagonal distance from the source. The subscripts 1 and 2 affixed to the variables introduced in the above derivation correspond to

*u*and

_{1}*u*, respectively. The mean intensity ∏, the visibility Γ, and the phase Φ of the intensity field vary with the diagonal distances, while the initial phase difference Δϕ remains constant. Among the variables, the phase Φ relates to the diagonal distances most simply with the relationship of

_{2}*x*,

_{1}*y*,

_{1}*z*) and (

_{1}*x*,

_{2}*y*,

_{2}*z*) can be determined by solving inverse kinematics if the absolute value of Φ is provided from more than six different locations. For that, a two-dimensional array of photo-detectors is employed to capture the interferometric intensity I at multiple locations. From the measured intensity, the phase Φ+Δϕ of Eq. (1) is computed by applying the well-established phase measuring technique with phase shifting. For description, let us introduce the superscript k so that Φ

_{2}^{k}refers to the computed value of Φ at the location of (

*x*,

^{k}*y*,

^{k}*z*). All the measured values of Φ

^{k}^{k}are processed to be unwrapped, starting from a particular reference principal phase value that is for convenience designated as Φ

^{0}at location (

*x*,

^{0}*y*,

^{0}*z*). Then, we define a new geometric model Λ

^{0}^{k}as

_{1}, y

_{1}, z

_{1}) and (x

_{2}, y

_{2}, z

_{2}) are determined so as to minimize the cost function that is defined as

^{k}. The cost function E is so highly nonlinear in terms of the unknowns (

*x*,

_{1}*y*,

_{1}*z*) and (

_{1}*x*,

_{2}*y*,

_{2}*z*) that no explicit solutions exist. But the function E is convex with six unknowns, because ∇

_{2}^{2}E(

*x*,

_{1}*y*,

_{1}*z*,

_{1}*x*,

_{2}*y*,

_{2}*z*) is positive definite throughout the domain of

_{2}*x*,

*y*, and

*z*. Note that this condition theoretically means that the function E has a global minimum and there exists one true solution set. Thus, numerical technique is used to search for the global minimum of the cost function. In this paper we used a new optimization algorithm based on the simulated annealing technique [18] instead of the previous BFGS (suggested by Broyden

*et al*.) method [15

15. H.G. Rhee and S.W. Kim, “Absolute distance measurement by
two-point diffraction interferometry,”
Appl. Opt. **41**, 5921–5928
(2002). [CrossRef] [PubMed]

*E*, that decide whether the cost function is successfully converged or not. (When the cost function

_{c}*E*in Eq. (4) is smaller than

*E*, the optimization process is stopped.) A tight

_{c}*E*makes a poor probability but the estimated coordinates will be very close to the real solutions. On the other hand a loose

_{c}*E*increases the probability of convergence but we cannot avoid sacrificing the uncertainty. For a good balance between the probability of convergence and the uncertainty, we used 0.006 μm

_{c}^{2}

*E*, which allows the error budget listed in Table 2.

_{c}*k*= 1). In addition the genetic algorithm may be another applicable solution, but we suppose that the genetic algorithm takes longer than 60 sec for one point calculation.

*x*,

_{1}*y*,

_{1}*z*) and (

_{1}*x*,

_{2}*y*,

_{2}*z*) have the physical relationship such as

_{2}## 3. Performance test

### 3.1 One dimensional test

**41**, 5921–5928
(2002). [CrossRef] [PubMed]

### 3.2 Two-dimensional test

### 3.3 Fourier self-calibration

21. D. G. Cameron, J. K. Kauppinen, D. J. Moffatt, and H. H. Mantsch, “Precision in condensed phase
vibrational spectroscopy,” Appl.
Spectrosc. **36**, 245–250
(1982). [CrossRef]

17. J. Ye, M. Takac, C. N. Berglund, G. Owen, and R. F. Pease, “An exact algorithm for
self-calibration of two-dimensional precision metrology
stages,” Prec. Eng. **20**, 16–32
(1997). [CrossRef]

## 4. Conclusion

**41**, 5921–5928
(2002). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | P. de Groot, “Grating interferometer for flatness
testing,” Opt. Lett. |

2. | R. Dandliker, R. Thalmann, and D. Prongue, “Two-wavelength laser interferometry
using superheterodyne detection,” Opt.
Lett. |

3. | Z. Sodnik, E. Fischer, T. Ittner, and H. J. Tiziani, “Two-wavelength double heterodyne
interferometry using a matched grating
technique,” Appl. Opt. |

4. | H. Kikuta, K. Iwata, and R. Nagata, “Distance measurement by the
wavelength shift of laser diode light,”
Appl. Opt. |

5. | H. Kikuta, K. Iwata, and R. Nagata, “Absolute distance measurement by
wavelength shift interferometry with a laser diode light: some systematic
error sources,” Appl. Opt. |

6. | T. Li, A. Wang, K. Merphy, and R. Claus, “White-light scanning fiber Michelson
interferometer for absolute position-distance
measurement,” Opt. Lett. |

7. | U. Schnell and R. Dandliker, “Dispersive white-light
interferometry for absolute distance measurement with dielectric multilayer
systems on the target,” Opt. Lett. |

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

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

10. | 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. |

11. | K. Lau, R. J. Hocken, and W. C. Haight, “Automatic laser tracking
interferometer system for robot metrology,”
Prec. Eng. |

12. | O. Nakamura, M. Goto, K. Toyoda, Y. Tanimura, and T. Kurosawa, “Development of a coordinate
measuring system with tracking laser
interferometer,” Annals of CIRP |

13. | E. B. Hughes, A Wilson, and G. N. Peggs, “Design of a high-accuracy CMM based
on multi-lateration techniques,” Annals
of CIRP |

14. | H. Jiang, S. Osawa, T. Takatsuji, H. Noguchi, and T. Kurosawa, “High-performance laser tracker using
an articulation mirror for the calibration of coordinate measuring
machine,” Opt. Eng. |

15. | H.G. Rhee and S.W. Kim, “Absolute distance measurement by
two-point diffraction interferometry,”
Appl. Opt. |

16. | J. Chu and S. W. Kim, “Absolute distance measurement by
lateral shearing interferometry of point-diffracted spherical
waves,” Opt. Express |

17. | J. Ye, M. Takac, C. N. Berglund, G. Owen, and R. F. Pease, “An exact algorithm for
self-calibration of two-dimensional precision metrology
stages,” Prec. Eng. |

18. | A. D. Belegundu and T.R. Chandrupatla, “Simulated annealing
(SA),” |

19. | H. Kihm and S. W. Kim, “Nonparaxial free-space diffraction
from oblique end faces of single-mode optical
fibers,” Opt. Lett. |

20. | ISO, “Guide to the expression
of uncertainty in measurement,” |

21. | D. G. Cameron, J. K. Kauppinen, D. J. Moffatt, and H. H. Mantsch, “Precision in condensed phase
vibrational spectroscopy,” Appl.
Spectrosc. |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: January 23, 2007

Revised Manuscript: March 28, 2007

Manuscript Accepted: March 28, 2007

Published: April 3, 2007

**Citation**

Hyug-Gyo Rhee, Jiyoung Chu, and Yun-Woo Lee, "Absolute three-dimensional coordinate measurement by the two-point diffraction interferometry," Opt. Express **15**, 4435-4444 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4435

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### References

- P. de Groot, "Grating interferometer for flatness testing," Opt. Lett. 21, 228-230 (1996). [CrossRef] [PubMed]
- R. Dändliker, R. Thalmann, and D. Prongué, "Two-wavelength laser interferometry using superheterodyne detection," Opt. Lett. 13, 339-341 (1988). [CrossRef] [PubMed]
- Z. Sodnik, E. Fischer, T. Ittner, and H. J. Tiziani, "Two-wavelength double heterodyne interferometry using a matched grating technique," Appl. Opt. 30, 3139-3144(1991). [CrossRef] [PubMed]
- H. Kikuta, K. Iwata, and R. Nagata, "Distance measurement by the wavelength shift of laser diode light," Appl. Opt. 25, 2976-2980 (1986). [CrossRef] [PubMed]
- H. Kikuta, K. Iwata, and R. Nagata, "Absolute distance measurement by wavelength shift interferometry with a laser diode light: some systematic error sources," Appl. Opt. 26, 1654-1660 (1987). [CrossRef] [PubMed]
- T. Li, A. Wang, K. Merphy, and R. Claus, "White-light scanning fiber Michelson interferometer for absolute position-distance measurement," Opt. Lett. 20, 785-787 (1995). [CrossRef] [PubMed]
- U. Schnell and R. Dändliker, "Dispersive white-light interferometry for absolute distance measurement with dielectric multilayer systems on the target," Opt. Lett. 21, 528-530 (1996). [CrossRef] [PubMed]
- 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]
- J. Ye, "Absolute measurement of long, arbitrary distance to less than an optical fringe," Opt. Lett. 29, 1153-1155 (2004). [CrossRef] [PubMed]
- 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]
- K. Lau, R. J. Hocken, and W. C. Haight, "Automatic laser tracking interferometer system for robot metrology," Prec. Eng. 8, 3-8 (1986). [CrossRef]
- O. Nakamura, M. Goto, K. Toyoda, Y. Tanimura, and T. Kurosawa, "Development of a coordinate measuring system with tracking laser interferometer," Annals of CIRP 40, 523-526 (1991). [CrossRef]
- E. B. Hughes, A Wilson, and G. N. Peggs, "Design of a high-accuracy CMM based on multi-lateration techniques," Annals of CIRP 49, 391-394 (2000). [CrossRef]
- H. Jiang, S. Osawa, T. Takatsuji, H. Noguchi, and T. Kurosawa, "High-performance laser tracker using an articulation mirror for the calibration of coordinate measuring machine," Opt. Eng. 41, 632-637 (2002). [CrossRef]
- H.G. Rhee, and S.W. Kim, "Absolute distance measurement by two-point diffraction interferometry," Appl. Opt. 41, 5921-5928 (2002). [CrossRef] [PubMed]
- J. Chu, and S. W. Kim, "Absolute distance measurement by lateral shearing interferometry of point-diffracted spherical waves," Opt. Express 14, 5961-5967 (2006). [CrossRef] [PubMed]
- J. Ye, M. Takac, C. N. Berglund, G. Owen, and R. F. Pease, "An exact algorithm for self-calibration of two-dimensional precision metrology stages," Prec. Eng. 20, 16-32 (1997). [CrossRef]
- A. D. Belegundu, and T.R. Chandrupatla, "Simulated annealing (SA)," in Optimization concepts and applications in engineering, M. Horton, ed. (Prentice-Hall, Inc., New Jersey, 1999).
- H. Kihm, and S. W. Kim, "Nonparaxial free-space diffraction from oblique end faces of single-mode optical fibers," Opt. Lett. 29, 2366-2368 (2004). [CrossRef] [PubMed]
- ISO, "Guide to the expression of uncertainty in measurement," in International vocabulary of basic and general terms in metrology, International Organization for Standardization ed. (International Organization for Standardization, Switzerland, 1993).
- D. G. Cameron, J. K. Kauppinen, D. J. Moffatt, and H. H. Mantsch, "Precision in condensed phase vibrational spectroscopy," Appl. Spectrosc. 36, 245-250 (1982). [CrossRef]

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