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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 8 — Apr. 16, 2007
  • pp: 4435–4444
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Absolute three-dimensional coordinate measurement by the two-point diffraction interferometry

Hyug-Gyo Rhee, Jiyoung Chu, and Yun-Woo Lee  »View Author Affiliations


Optics Express, Vol. 15, Issue 8, pp. 4435-4444 (2007)
http://dx.doi.org/10.1364/OE.15.004435


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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

There have been many efforts towards absolute distance measurement, which determine the target position by reading interferometric phase values straightway with no need of continuous measurements between points. When interferometric techniques are applied to absolute distance measurements, one of the fundamental ways is to enlarge the equivalent wavelength by grazing incidence [1

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

] or multiple wavelengths synthesis [2

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

, 3

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]

]. Tunable sources by frequency modulation [4

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

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]

] are also available for this purpose. White light interferometry [6

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

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]

] can offer better resolutions below a single wavelength within the scanning range. The pulse fringe autocorrelation technique [8

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

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

] or the wavelength synthesis [10

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]

] of femtosecond pulse lasers are applicable to absolute distance measurements. However, it is hard to expand these approaches into absolute coordinate measurement. For absolute three-dimensional coordinates measuring, a technique so-called multilateration [11

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

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]

] was proposed. This method determines the absolute three-dimensional coordinates of the retro-reflector target by using a geometric model of the measured diagonal distances between the target and several fixed laser trackers. In this case the geometrical form error and material inhomogeneity of the target can increase the measuring uncertainty.

As part of the effort to overcome the problem, we already proposed the new concept of the two point-diffraction phase-measuring interferometer [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]

] and its side version based on the lateral shearing interferometry [16

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]

]. The two point-diffraction phase-measuring interferometer is intended to directly measure the xyz-coordinates of the target moving in three-dimensions. The interferometer consists of a fixed photo-detector array and a movable target. The target is made of two spherical wavefront sources, whose interference pattern is monitored by the photo-detector array. Phase shifting technique is applied to obtain the precise phase values. Then the measured phases are fitted to a geometric model so as to determine the xyz-coordinates of the target. The two point diffraction sources on the target are given the coordinates as (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

The interferometric intensity field from two spherical wavefronts, referred to as u1 and u2, 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]

]

I=u1+u22=Π+Γcos(Φ+Δϕ),
whereΠ=U12r12+U22r22,Γ=2U1U2r1r2,Φ=2πλ(r1r2),andΔϕ=(ϕ1ϕ2).
(1)

In Eq. (1), 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 u1 and u2, 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

Φxyz=2πλ[r1xyzr2xyz]
=2πλ[(x1x)2+(y1y)2+(z1z)2(x2x)2+(y2y)2+(z2z)2].
(2)

Eq. (2) indicates that the six unknowns (x1, y1, z1) and (x2, y2, z2) 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 Φk refers to the computed value of Φ at the location of (xk, yk, zk). All the measured values of Φk are processed to be unwrapped, starting from a particular reference principal phase value that is for convenience designated as Φ0 at location (x0, y0, z0). Then, we define a new geometric model Λk as

Λk=λ2π[(Φk+Δϕ)(Φ0+Δϕ)]=λ2π[ΦkΦ0].
(3)

Now, as the final step for absolute distance measurements, using the relationship of Eq. (3), the six unknowns (x1, y1, z1) and (x2, y2, z2) are determined so as to minimize the cost function that is defined as

E=k[λ2π(ΦkΦ0)Λ̑k]2,
(4)

Table 1. Performance comparison between the BFGS and the new algorithm. The stopping criterion was 0.006 μn2 and the average calculation times were estimated when the cost function was successfully converged.

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Table 2. Error budget of the two point-diffraction phase-measuring interferometer. X represents the wavelength of the laser source.

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In Table 2, we considered the worst cases that could possibly occur over the whole working volume. The expanded uncertainty [20

20. 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).

] was 418 nm for x-, 548 nm for y-, and 9.7 nm for z-directions when we use a 640 × 480 CCD camera (The coverage factor 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.

Fig. 1. Optical configuration of the two-point diffraction phase-measuring interferometer.

Figure 1 illustrates the optical configuration of the two-point diffraction phase-measuring interferometer. A beam of stabilized HeNe laser (wavelength: 632.8 nm) is fed into one fiber through an input coupler, which is then connected to a 2×2 coupler that divides the input beam by 50:50 into two output branches. Each branch fiber is wound around a tube type PZT that elongates the length of the fiber to induce phase shift. Only one of the two PZT extenders is operated for phase shifting, while the other is inserted as a dummy just for matching fiber length. The target is made of two single-mode optical fibers, whose exit ends are polished and aligned side by side with an inter-axis distance of nearly 125 μm. So the six unknowns (x1, y1, z1) and (x2, y2, z2) have the physical relationship such as

d2=(x1x2)2+(y1y2)2+(z1z2)2=constant,
(5)

where the constraint d is considered a fixed value, so that we compute the value once for all. By this method we solve only five unknowns instead of six, thereby accelerating its convergence. For the purpose of checking the phase shifting process, back-reflected intensity from the two fiber ends is monitored by a single photo-detector connected to an input end of the 2×2 coupler. An array of photo-detectors is constituted by a two-dimensional CCD with 640 × 480 size, whose spacing is 8.44 μm in the x direction and 9.78 μm in the y-direction in terms of mean values.

3. Performance test

3.1 One dimensional test

We set up the interferometer on a precision one-axis stage and repeatedly measured the target coordinates 35 times at various different positions within the working volume [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]

]. The repeatability in terms of ±2σ value was less than 0.05 μm for the x-coordinate, 0.1 μm for the y-coordinate and 0.01 μm for the z-coordinate. One-dimensional uncertainty was verified through a comparison test with a commercial heterodyne HeNe laser interferometer. The maximum deviation was measured 0.26 μm over a travel of 80 mm.

3.2 Two-dimensional test

For the two-dimensional performance test the interferometer has been set up on a two-dimensional precision stage as illustrated in Fig. 2.

Fig. 2. Photographic view of the two-dimensional performance test setup.

Fig. 3. Deviation map between the readings of the optical scale and the results of the proposed interferometer. The blue lines mean the intentionally exaggerated deviation.

Test results were obtained within 60mm × 60mm working area, in which the maximum deviations were measured 0.482 μm for x- and 1.308 μm for z-direction as shown in Fig. 3. However this deviation map does not mean the true error of the interferometer because it includes the error from misalignment between two metrology frames, the optical scale’s systematic error, the scale factor, the random measurement noise and so on.

3.3 Fourier self-calibration

Fig. 4. Designed artifact plate for self-calibration.

To extract the true error of the proposed interferometer, the employment of a standard specimen is considered. We used the artifact plate of 7 × 7 mark array as a standard specimen as illustrated in Fig. 4, which was made by an electron beam lithography system. Each mark consists of 8 lines. To obtain the center position (x- and z- coordinates) of the mark, we used a microscope as shown in Fig. 2 and a center of gravity algorithm [21

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]

]. However the quantified accuracy of the center position of each mark is not enough to verify the interferometer, because the uncertainty of the specimen is on the same level of the interferometer errors. Furthermore the center detection algorithm also brings some errors.

Fig. 5. Two-dimensional self-calibration procedure using an artifact.

Fourier 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]

] in precision metrology aims to extract/remove the systematic errors of the instrument without relying upon externally calibrated artifacts. The stage self-calibration is the procedure of calibrating the metrology frame of the stage by an artifact plate whose mark positions are not exactly known. By assuming rigidness of the plate, this method extracts the error map of the stage metrology frame from comparison of three different measurement views of the plate as illustrated in Fig. 5. We applied the Fourier self-calibration algorithm to our interferometer. When we obtained the coordinates of each mark of Fig. 4 for calibration, we used averaging technique to reduce the random noise effect. Figure 6(a) shows the systematic error of the two-dimensional optical scale, in which the absolute values of the maximum error were measured 1.472 μm for x- and 1.347 μm for z-direction. On the other hand, Fig. 6(b) shows the systematic error of the interferometer itself, in which the absolute values of the maximum error were measured 0.997 μm for x- and 0.999 μm for z-direction. Through a repeat test of self-calibration, we confirm that the reconstructed systematic error has reasonable repeatability.

Fig. 6. Extracted systematic error of (a) the conventional optical scale, and (b) the proposed interferometer. The measured y-coordinate is about −1.12 mm.

Moreover we expanded the test region to three-dimension. 10 mm thickness block gauges were installed to lift up and down the target along y-direction as shown in Fig. 7(a).

Fig. 7. Three-dimensional performance test scheme. (a) Target setup on block gauges, and (b) the actual test plane.
Fig. 8. Results of the Fourier self-calibration. (a) The error of the optical scale at y = −11.12, (b) the error of the proposed interferometer at y = −11.12, (c) the error of the optical scale at y = -21.12, and (d) the error of the proposed interferometer at y = −21.12.

Figure 7(b) shows three test planes; the y-coordinate of the target is about −21.12 mm, − 11.12 mm, and −1.12 mm. These coordinate values were obtained by our interferometer. To apply the self-calibration, we measured the three views at each plane. Thus the whole measured volume was 60 mm × 60 mm × 20 mm. Figure 8 shows the self-calibration results at the second and the third planes (The result of the first plane is already displayed in Fig. 6). Figure 6 and Fig. 8 demonstrate that the proposed interferometer is capable of measuring the xyz-coordinates with maximum 0.999 μm errors in terms of the absolute value, while the optical scale has maximum 2.005 μm errors.

4. Conclusion

Acknowledgments

The authors would like to thank professor Seung-Woo Kim (KAIST) for insightful discussions and suggestions concerning the direction of this research.

References and links

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]

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 40, 523–526 (1991). [CrossRef]

13.

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]

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]

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]

18.

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).

19.

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]

20.

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).

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]

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

  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. 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. Dändliker, "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]
  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 40, 523-526 (1991). [CrossRef]
  13. 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]
  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]
  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]
  18. 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).
  19. 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]
  20. 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).
  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]

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