Abstract

Wavelength-scanning interferometry permits the simultaneous measurement of variations in surface shape and optical thickness of a nearly parallel plate. Interference signals from both surfaces of the test plate can be separated in frequency space; however, these frequencies are shifted from the expected frequency by the refractive-index dispersion of the test plate and any nonlinearity that is due to wavelength scanning. Conventional Fourier analysis is sensitive to this detuning of the signal frequency and suffers from multiple-beam interference noise. We propose new wavelength-scanning algorithms that permit a large tolerance for dispersion of the test plate and nonlinearity of scanning. Two 19-sample algorithms that suppress multiple-interference noise up to the second order of the reflectance of the test plate are presented. Experimental results show that the variation in surface shape and optical thickness of a glass parallel plate of 250-mm diameter was measured with a resolution of 1–2 nm rms.

© 2004 Optical Society of America

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References

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  1. M. V. Mantravadi, “Testing nearly parallel plates,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 22.
  2. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  3. K. Freischlad, “Large flat panel profiler,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers and Flat Panel Displays, J. C. Stover, ed. Proc. SPIE2862, 163–171 (1996).
    [CrossRef]
  4. P. de Groot, “Grating interferometer for metrology of transparent flats,” in Optical Fabrication and Testing, Vol. 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 28–30.
  5. P. G. Dewa, A. W. Kulawiec, “Grazing incidence interferometry for measuring transparent plane-parallel plates,” U.S. patent5,923,425 (13July1999).
  6. Y. Ishii, J. Chen, K. Murata, “Digital phase-measuring interferometry with a tunable laser diode,” Opt. Lett. 12, 233–235 (1987).
    [CrossRef] [PubMed]
  7. K. Okada, H. Sakuta, T. Ose, J. Tsujiuchi, “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable-source phase shifting interferometry,” Appl. Opt. 29, 3280–3285 (1990).
    [CrossRef] [PubMed]
  8. A. Sekine, I. Minegishi, H. Koizumi, “Axial eye-length measurement by wavelength-shift interferometry,” J. Opt. Soc. Am. A 10, 1651–1655 (1993).
    [CrossRef] [PubMed]
  9. M. Takeda, H. Yamamoto, “Fourier-transform speckle profilometry: three dimensional shape measurements of diffuse objects with large height steps and/or spatially isolated surfaces,” Appl. Opt. 33, 7829–7837 (1994).
    [CrossRef] [PubMed]
  10. H. J. Tiziani, B. Franze, P. Haible, “Wavelength shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
    [CrossRef]
  11. F. Lexer, C. K. Hitzenberger, A. F. Fercher, M. Kulhavy, “Wavelength-tuning interferometry of intraocular distances,” Appl. Opt. 36, 6548–6552 (1997).
    [CrossRef]
  12. H. Hiratsuka, E. Kido, T. Yoshimura, “Simultaneous measurements of three-dimensional reflectivity distributions in scattering media based on optical frequency-domain reflectometry,” Opt. Lett. 23, 1420–1422 (1998).
    [CrossRef]
  13. P. J. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000).
    [CrossRef]
  14. L. L. Deck, “Multiple surface phase shifting interferometry,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 424–431 (2001).
    [CrossRef]
  15. K. Hibino, T. Takatsuji, “Suppression of multiple-beam interference noise in testing an optical parallel plate by wavelength-scanning interferometry,” Opt. Rev. 9, 60–65 (2002).
    [CrossRef]
  16. A. Yamamoto, I. Yamaguchi, “Profilometry of sloped plane surfaces by wavelength scanning interferometry,” Opt. Rev. 9, 112–121 (2002).
    [CrossRef]
  17. P. Hariharan, “Phase-stepping interferometry with laser diodes: effect of changes in laser power with output wavelength,” Appl. Opt. 28, 27–29 (1989).
    [CrossRef] [PubMed]
  18. Y. Ishii, R. Onodera, “Phase-extraction algorithm in laser-diode phase-shifting interferometry,” Opt. Lett. 20, 1883–1885 (1995).
    [CrossRef] [PubMed]
  19. P. Sandoz, T. Gharbi, G. Tribillon, “Phase-shifting methods for interferometers using laser-diode frequency-modulation,” Opt. Commun. 132, 227–231 (1996).
    [CrossRef]
  20. G. Coppola, P. Ferraro, M. Iodice, S. D. Nicola, “Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear, wavelength-scanning interferometer,” Appl. Opt. 42, 3882–3887 (2003).
    [CrossRef] [PubMed]
  21. K. Hibino, B. F. Oreb, P. S. Fairman, “Wavelength-scanning interferometry of a transparent parallel plate with refractive index dispersion,” Appl. Opt. 42, 3888–3895 (2003).
    [CrossRef] [PubMed]
  22. D. W. Phillion, “General method for generating phase-shifting interferometry algorithms,” Appl. Opt. 36, 8098–8115 (1997).
    [CrossRef]
  23. K. Freischald, C. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  24. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  25. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [CrossRef]
  26. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for spatially nonuniform phase-shifts,” J. Opt. Soc. Am. A 14, 918–930 (1997).
    [CrossRef]
  27. K. Hibino, “Susceptibility of error-compensating phase-shifting algorithms to random noise,” Appl. Opt. 36, 2084–2093 (1997).
    [CrossRef] [PubMed]
  28. P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
    [CrossRef]
  29. B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, P. S. Fairman, “Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. 39, 5161–5171 (2000).
    [CrossRef]

2003 (2)

2002 (2)

K. Hibino, T. Takatsuji, “Suppression of multiple-beam interference noise in testing an optical parallel plate by wavelength-scanning interferometry,” Opt. Rev. 9, 60–65 (2002).
[CrossRef]

A. Yamamoto, I. Yamaguchi, “Profilometry of sloped plane surfaces by wavelength scanning interferometry,” Opt. Rev. 9, 112–121 (2002).
[CrossRef]

2000 (2)

1999 (1)

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

1998 (1)

1997 (5)

1996 (1)

P. Sandoz, T. Gharbi, G. Tribillon, “Phase-shifting methods for interferometers using laser-diode frequency-modulation,” Opt. Commun. 132, 227–231 (1996).
[CrossRef]

1995 (2)

1994 (1)

1993 (1)

1992 (1)

1990 (2)

1989 (1)

1987 (1)

1983 (1)

Burow, R.

Chen, J.

Coppola, G.

de Groot, P. J.

Deck, L. L.

L. L. Deck, “Multiple surface phase shifting interferometry,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 424–431 (2001).
[CrossRef]

Dewa, P. G.

P. G. Dewa, A. W. Kulawiec, “Grazing incidence interferometry for measuring transparent plane-parallel plates,” U.S. patent5,923,425 (13July1999).

Elssner, K. E.

Fairman, P. S.

Farrant, D. I.

Fercher, A. F.

Ferraro, P.

Forbes, G.

Franze, B.

H. J. Tiziani, B. Franze, P. Haible, “Wavelength shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

Freischald, K.

Freischlad, K.

K. Freischlad, “Large flat panel profiler,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers and Flat Panel Displays, J. C. Stover, ed. Proc. SPIE2862, 163–171 (1996).
[CrossRef]

Freund, C. H.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Gharbi, T.

P. Sandoz, T. Gharbi, G. Tribillon, “Phase-shifting methods for interferometers using laser-diode frequency-modulation,” Opt. Commun. 132, 227–231 (1996).
[CrossRef]

Gilliand, Y.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Groot, P. de

P. de Groot, “Grating interferometer for metrology of transparent flats,” in Optical Fabrication and Testing, Vol. 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 28–30.

Grzanna, J.

Haible, P.

H. J. Tiziani, B. Franze, P. Haible, “Wavelength shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

Hariharan, P.

Hibino, K.

Hiratsuka, H.

Hitzenberger, C. K.

Iodice, M.

Ishii, Y.

Kido, E.

Koizumi, H.

Koliopoulos, C.

Kulawiec, A. W.

P. G. Dewa, A. W. Kulawiec, “Grazing incidence interferometry for measuring transparent plane-parallel plates,” U.S. patent5,923,425 (13July1999).

Kulhavy, M.

Larkin, K. G.

Leistner, A. J.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Lexer, F.

Mantravadi, M. V.

M. V. Mantravadi, “Testing nearly parallel plates,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 22.

Merkel, K.

Minegishi, I.

Murata, K.

Nicola, S. D.

Okada, K.

Onodera, R.

Oreb, B. F.

Ose, T.

Phillion, D. W.

Sakuta, H.

Sandoz, P.

P. Sandoz, T. Gharbi, G. Tribillon, “Phase-shifting methods for interferometers using laser-diode frequency-modulation,” Opt. Commun. 132, 227–231 (1996).
[CrossRef]

Schwider, J.

Seckold, J. A.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Sekine, A.

Spolaczyk, R.

Takatsuji, T.

K. Hibino, T. Takatsuji, “Suppression of multiple-beam interference noise in testing an optical parallel plate by wavelength-scanning interferometry,” Opt. Rev. 9, 60–65 (2002).
[CrossRef]

Takeda, M.

Tiziani, H. J.

H. J. Tiziani, B. Franze, P. Haible, “Wavelength shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

Tribillon, G.

P. Sandoz, T. Gharbi, G. Tribillon, “Phase-shifting methods for interferometers using laser-diode frequency-modulation,” Opt. Commun. 132, 227–231 (1996).
[CrossRef]

Tsujiuchi, J.

Walsh, C. J.

B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, P. S. Fairman, “Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. 39, 5161–5171 (2000).
[CrossRef]

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Ward, B. K.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Yamaguchi, I.

A. Yamamoto, I. Yamaguchi, “Profilometry of sloped plane surfaces by wavelength scanning interferometry,” Opt. Rev. 9, 112–121 (2002).
[CrossRef]

Yamamoto, A.

A. Yamamoto, I. Yamaguchi, “Profilometry of sloped plane surfaces by wavelength scanning interferometry,” Opt. Rev. 9, 112–121 (2002).
[CrossRef]

Yamamoto, H.

Yoshimura, T.

Appl. Opt. (11)

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

K. Okada, H. Sakuta, T. Ose, J. Tsujiuchi, “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable-source phase shifting interferometry,” Appl. Opt. 29, 3280–3285 (1990).
[CrossRef] [PubMed]

M. Takeda, H. Yamamoto, “Fourier-transform speckle profilometry: three dimensional shape measurements of diffuse objects with large height steps and/or spatially isolated surfaces,” Appl. Opt. 33, 7829–7837 (1994).
[CrossRef] [PubMed]

F. Lexer, C. K. Hitzenberger, A. F. Fercher, M. Kulhavy, “Wavelength-tuning interferometry of intraocular distances,” Appl. Opt. 36, 6548–6552 (1997).
[CrossRef]

P. J. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000).
[CrossRef]

G. Coppola, P. Ferraro, M. Iodice, S. D. Nicola, “Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear, wavelength-scanning interferometer,” Appl. Opt. 42, 3882–3887 (2003).
[CrossRef] [PubMed]

K. Hibino, B. F. Oreb, P. S. Fairman, “Wavelength-scanning interferometry of a transparent parallel plate with refractive index dispersion,” Appl. Opt. 42, 3888–3895 (2003).
[CrossRef] [PubMed]

D. W. Phillion, “General method for generating phase-shifting interferometry algorithms,” Appl. Opt. 36, 8098–8115 (1997).
[CrossRef]

P. Hariharan, “Phase-stepping interferometry with laser diodes: effect of changes in laser power with output wavelength,” Appl. Opt. 28, 27–29 (1989).
[CrossRef] [PubMed]

K. Hibino, “Susceptibility of error-compensating phase-shifting algorithms to random noise,” Appl. Opt. 36, 2084–2093 (1997).
[CrossRef] [PubMed]

B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, P. S. Fairman, “Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. 39, 5161–5171 (2000).
[CrossRef]

J. Mod. Opt. (1)

H. J. Tiziani, B. Franze, P. Haible, “Wavelength shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

P. Sandoz, T. Gharbi, G. Tribillon, “Phase-shifting methods for interferometers using laser-diode frequency-modulation,” Opt. Commun. 132, 227–231 (1996).
[CrossRef]

Opt. Eng. (1)

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371–1380 (1999).
[CrossRef]

Opt. Lett. (3)

Opt. Rev. (2)

K. Hibino, T. Takatsuji, “Suppression of multiple-beam interference noise in testing an optical parallel plate by wavelength-scanning interferometry,” Opt. Rev. 9, 60–65 (2002).
[CrossRef]

A. Yamamoto, I. Yamaguchi, “Profilometry of sloped plane surfaces by wavelength scanning interferometry,” Opt. Rev. 9, 112–121 (2002).
[CrossRef]

Other (5)

L. L. Deck, “Multiple surface phase shifting interferometry,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 424–431 (2001).
[CrossRef]

M. V. Mantravadi, “Testing nearly parallel plates,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 22.

K. Freischlad, “Large flat panel profiler,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers and Flat Panel Displays, J. C. Stover, ed. Proc. SPIE2862, 163–171 (1996).
[CrossRef]

P. de Groot, “Grating interferometer for metrology of transparent flats,” in Optical Fabrication and Testing, Vol. 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 28–30.

P. G. Dewa, A. W. Kulawiec, “Grazing incidence interferometry for measuring transparent plane-parallel plates,” U.S. patent5,923,425 (13July1999).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the optical setup for testing a nearly parallel plate in a wavelength-scanning Fizeau interferometer. MOs, microscope objectives.

Fig. 2
Fig. 2

Sampling functions of the 19-sample algorithm for phase detection of the fundamental frequency.

Fig. 3
Fig. 3

Sampling functions of (a) the new 19-sample algorithm, (b) a 19-sample Fourier algorithm, and (c) a 128-sample Fourier algorithm for third-harmonic detection.

Fig. 4
Fig. 4

Peak-to-valley (PV) phase errors for measurements of surface and optical thickness of a glass plate caused by multiple-beam interference noise as a function of geometrical parameter g: OPD, optical path difference.

Fig. 5
Fig. 5

Interference fringes observed for a nearly parallel BK7 plate.

Fig. 6
Fig. 6

Surface shape of the BK7 plate as measured by the 30° 19-sample algorithm.

Fig. 7
Fig. 7

Variation in optical thickness of the BK7 glass plate measured by the 90° 19-sample algorithm.

Tables (1)

Tables Icon

Table 1 Frequencies of Interference Signals in the Wavelength-Scanning Fizeau Interferometer

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

D=pL+qnT,
φ=2πDλ+δp,q,
v=-12πdφdt,=pL+qn0Tλ02dλdt+qTλ0-dndλ0dλdt,
cg=cn01+λ0n0dndλ0,
v1=2Lλ02dλdt.
vk=v1p+qg1-λ0n0dndλ0,
g=n0T/L.
Ix, y, t=s0+k=19 sk cosφkx, y+vkt,
v1=2Lλ02dλdt1+ε1+ε2v1tπ+,
φx, y=arctanr=119 brIx, y, trr=119 arIx, y, tr,
F1v=r=119 br exp-ivtr,
F2v=r=119 ar exp-ivtr.
φ=arctan- F1vJv¯dv- F2vJv¯dv=arctan×s1 sin φ1+k=0,2,39 F1vkJvk¯-Jvks1 cos φ1+k=0,2,39 F2vkJvk¯+Jvk,
F1v1+iF2v1=0.
F1kv1=F2kv1=0, k=0, 2, 3,, 10.
ddvF1vvk=ddvF2vvk=0, vk=3v1, 4v1,
|F1,2v|R2,  3v1<v<5v1.
ddvF1v+iF2vv=v1=0,
ar=1483-1, 1-23, -5, -43, -7, -1-23, 1, 43, 12, 83, 12, 43, 1, -1-23, -7, -43, -5, 1-23, -1,br=14833-1, 1, 1, -2, 1-43,-11, -173, -12, -43, 0, 43, 12, 1+73, 11, -1+43, 2, -1, -1, -3+1.
F13v1+iF23v1=0,
F1kv1=F2kv1=0, k=0, 1, 2, 4,, 8.
F1-v=-F1v,
F2-v=F2v,
F1,2v+12v1=F1,2v.
ddvF1vvk=ddvF2vvk=0,  vk=v1, 4v1, 5v1.
ddvF1v+iF2vv=3v1=0,
ar=1361, 1, -2, -3, 2, 5, -1, -6, 0, 6, 0,-6, -1, 5, 2, -3, -2, 1, 1,br=1360, 2, 2, -2, -4, 2, 6, 0, -6, 0, 6, 0,-6, -2, 4, 2, -2, -2, 0.
ar=2mcos6πmr-m+12,br=2msin6πmr-m+12, k=1, 2,, m,
Δφ1/4π=ε28π2k=1,2,3 kiF12kv1-F22kv1×cosφ1+φk+k=1,2,3 kiF12kv1+F22×kv1cosφ1-φk+oε12+oε2Rε20.018 cos 2φ1-0.008 cosφ1+φ2-0.005 cosφ1-φ2-0.006 cosφ1+φ3+0.005 cosφ1-φ3,
Δφ2/4π=ε28π2k=1,2,3 kiF12kv1-F22kv1×cosφ2+φk+k=1,2,3 kiF12kv1+F22×kv1cosφ2-φk+oε12+oε2Rε20.01 cos 2φ2-0.038 cosφ1-φ2-0.013 cosφ1+φ2+0.12 cosφ2-φ3-0.009 cosφ2+φ3.
S=12r=119ar2+br21/2,

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