## Absolute distance measurement by lateral shearing interferometry of point-diffracted spherical waves

Optics Express, Vol. 14, Issue 13, pp. 5961-5967 (2006)

http://dx.doi.org/10.1364/OE.14.005961

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### Abstract

We describe a method of absolute distance measurement based on the lateral shearing interferometry of point-diffracted spherical waves. A unique feature is that the distance measurement is not confined only along a single line of the optical axis, but the target is allowed to take movement freely within a volumetric measurement space formed by the aperture angle of point-diffraction. Detailed measurement theory is explained along with experimental verification.

© 2006 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. Dändliker, R. Thalmann, and D. Prongué, “Two-wavelength laser interferometry using superheterodyne detection,” Opt. Lett. **13**, 339–341 (1988). [CrossRef] [PubMed]

3. F. Bien, M. Camac, H. J. Caulfield, and S. Ezekiel, “Absolute distance measurements by variable wavelength interferometry,” Appl. Opt. **20**, 400–403 (1981). [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. U. Schnell, R. Dändliker, and S. Gray, “Dispersive white-light interferometry for absolute distance measurement with dielectric multilayer systems on the target,” Opt. Lett. **21**, 528–530 (1996). [CrossRef] [PubMed]

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

7. 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]

8. 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]

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

## 2. System setup

10. 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]

## 2.1. Lateral shearing interferometry

_{c},y

_{c},z

_{c}) that represents the central point of the four diffraction sources formed by the four fiber emitters housed within the transmitter. To reconstruct the master wavefront W(x,y), a polynomial approximation of W(x, y)=

_{nm}x

^{m}y

^{n-m}is adopted with k being the degree of polynomials. Then the wavefronts of the two laterally-sheared interferograms are derived, respectively, as

_{x}and ΔW

_{y}are actually measured by applying the well-established Fourier-transform technique with subsequent phase-unwrapping [11

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

_{nm}and D

_{nm}are computed by fitting the measured data of ΔW

_{x}and ΔW

_{y}into the above polynomial expressions. Finally, the linear algebraic relations between B

_{nm}and C

_{nm}in Eq. (1) and B

_{nm}and D

_{nm}in Eq. (2) allow B

_{nm}to be precisely determined, which leads to the final step of reconstructing the master wavefront W(x,y) [12

12. M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a Lateral Shear Interferometer having variable shear,” Appl. Opt. **14**, 142–150 (1975). [PubMed]

## 2.2. Absolute distance determination

_{c},y

_{c},z

_{c}), the reconstructed master wavefront W(x,y) has the Euclidian geometrical relation of

_{0}λ/2π where φ

_{0}is the initial phase formed at the exit of the point-diffraction source and λ is the wavelength of the source light. The absolute distance of the transmitter to be measured from the receiver is represented as

_{c}is not much larger than x

_{c}and y

_{c}. Eq. (4) is then rearranged into the form of W=

_{i}U

_{i}, in which U

_{i}are the Zernike polynomials and Ai are their corresponding coefficients that are derived in terms of x

_{c}, y

_{c}, z

_{c}, and R as listed in Table 1. This mathematical manipulation implies that the unknown R can be obtained deterministically once the coefficients A

_{i}are computed by transforming the measured W(x,y) into the Zernike form. Specifically, the distance R is obtained using A

_{4}, A

_{5}, and A

_{6}as

## 2.3. Measurement range

_{c}<z

_{c}tanθ and y

_{c}<z

_{c}tanθ. Secondly, the longitudinal range regarding z

_{c}is associated with the fringe sampling capability of the receiver. The average spacing of the shearing fringes observed in ΔW

_{x}and ΔW

_{y}becomes dense with the relation of λz

_{c}/S as z

_{c}decreases. Therefore, the lower bound of z

_{c}is imposed by the Nyquist sampling limit such as z

_{c}>2dS/λ, in which d is the spatial resolution of the photodetector array comprising the receiver. Similarly, the upper bound of z

_{c}is given as z

_{c}<2DS/λ in which D denotes the overall size of the photodetector array so that the minimum measurable fringe spacing is limited to 2D. Another factor to be considered for the upper bound of z

_{c}is the total power of the source, of which irradiance reduces, in proportion to 1/R

^{2}, below the level of electric noise when z

_{c}reaches a certain threshold. The overall performance is expected to improve gradually if the size of the photodetector array in the receiver is further increased or the shearing offset between the fibers in the transmitter is made larger.

## 3. Experiments and discussions

### 3.1. Wavefront reconstruction using Fourier-transform

_{x}and ΔW

_{y}are observed simultaneously being overlapped. Figure 3(b) depicts the Fourier-transformed frequency spectrum, in which the two peaks indicated as A and B correspond to ΔW

_{y}and ΔW

_{x}, respectively. Each peak is isolated by a low-pass filter of finite width and then inverse Fourier-transformed with subsequent phase determination of ΔW

_{x}and ΔW

_{y}as illustrated in Fig. 3(c). Finally, W(x,y) is reconstructed following the computational procedure of Eq. (1) and Eq. (2), of which result is drawn in Fig. 3(d).

## 3.2. Experiment results

^{5}without compensation of temperature, pressure, and humidity. The main causes of the measurement errors are considered the temperature variation in air, electrical noise encountered in interferograms sampling, and external vibration. These sources of errors tend to be more significant as the distance increases.

## 4. Conclusion

^{-5}when measuring a distance up to 1200 mm, being disturbed by the presence of temperature fluctuation, electrical noise, and vibration which deteriorate the temporal and spatial stability of the spherical wavefronts emitted from the transmitter fibers.

## References and links

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

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

3. | F. Bien, M. Camac, H. J. Caulfield, and S. Ezekiel, “Absolute distance measurements by variable wavelength interferometry,” Appl. Opt. |

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

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

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

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

8. | 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. |

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

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

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

12. | M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a Lateral Shear Interferometer having variable shear,” Appl. Opt. |

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3940) Instrumentation, measurement, and metrology : Metrology

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 12, 2006

Revised Manuscript: June 10, 2006

Manuscript Accepted: June 14, 2006

Published: June 26, 2006

**Citation**

Jiyoung Chu and Seung-Woo Kim, "Absolute distance measurement by lateral shearing interferometry of point-diffracted spherical waves," Opt. Express **14**, 5961-5967 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-5961

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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]
- F. Bien, M. Camac, H. J. Caulfield, and S. Ezekiel, "Absolute distance measurements by variable wavelength interferometry," Appl. Opt. 20, 400-403 (1981). [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]
- U. Schnell, R. Dändliker, and S. Gray, "Dispersive white-light interferometry for absolute distance measurement with dielectric multilayer systems on the target," Opt. Lett. 21, 528-530 (1996). [CrossRef] [PubMed]
- H.-G. Rhee and S.-W. Kim, "Absolute distance measurement by two-point-diffraction interferometry," Appl. Opt. 41, 5921-5928 (2002). [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]
- 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]
- J. Ye, "Absolute measurement of a long, arbitrary distance to less than an optical fringe," Opt. Lett. 29, 1153-1155 (2004). [CrossRef] [PubMed]
- 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]
- 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]
- M. P. Rimmer and J. C. Wyant, "Evaluation of large aberrations using a Lateral Shear Interferometer having variable shear," Appl. Opt. 14, 142-150 (1975). [PubMed]

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