Abstract

High-accuracy dimensional measurements by laser interferometers require corrections because of diffraction, which makes the effective fringe-period different from the wavelength of a plane (or spherical) wave λ0. By using a combined X-ray and optical interferometer as a tool to investigate diffraction across a laser beam, we observed wavelength variations as large as 10−8λ0. We show that they originate from the wavefront evolution under paraxial propagation in the presence of wavefront- and intensity-profile perturbations.

© 2016 Optical Society of America

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References

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  1. H. Siegert and P. Becker, Precision Measurement and Fundamental Constants IISpec. Publ. 617B. N. Taylor and W. D. Phillips, eds. (Natl. Bur. Stand. USA, 1984), p. 625.
  2. A. Bergamin, G. Cavagnero, and G. Mana, “Observation of Fresnel diffraction in a two-beam laser interferometer,” Phys. Rev. A 49, 2167–2173 (1994).
    [Crossref] [PubMed]
  3. A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instrum. Meas. 46, 196–200 (1997).
    [Crossref]
  4. E. Massa, C. P. Sasso, G. Mana, and C. Palmisano, “A more accurate measurement of the 28Si lattice parameter,” J. Phys. Chem. Ref. Data 44, 031208 (2015).
    [Crossref]
  5. D. van Westrum and T. M. Niebauer, “The diffraction correction for absolute gravimeters,” Metrologia 40, 258–263 (2003).
    [Crossref]
  6. L. Robertsson, “On the diffraction correction in absolute gravimetry,” Metrologia 44, 35–39 (2007).
    [Crossref]
  7. G. D’Agostino and L. Robertsson, “Relative beam misalignment errors in high accuracy displacement interferometers: calculation and detection,” Appl. Phys. B 103, 357–361 (2011).
    [Crossref]
  8. R. A. Nicolaus and G. Boensch, “Aperture correction for a sphere interferometer,” Metrologia 46, 668–673 (2009).
    [Crossref]
  9. G. Bartl, H. Bettin, M. Krystek, T. Mai, A. Nicolaus, and A. Peter, “Volume determination of the Avogadro spheres of highly enriched 28Si with a spherical Fizeau interferometer,” Metrologia 48, S96–S103 (2011).
    [Crossref]
  10. N. Kuramoto, K. Fujii, and K. Yamazawa, “Volume measurements of 28Si spheres using an interferometer with a flat etalon to determine the Avogadro constant,” Metrologia 48, S83–S95 (2011).
    [Crossref]
  11. B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
    [Crossref]
  12. B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
    [Crossref]
  13. B. Andreas, G. Mana, and C. Palmisano, “Vectorial ray-based diffraction integral,” J. Opt. Soc. Am. A 32, 1403–1424 (2015).
    [Crossref]
  14. J.-P. Monchalin, M. J. Kelly, J. E. Thomas, N. A. Kurnit, A. Szöke, F. Zernike, P. H. Lee, and A. Javan, “Accurate laser wavelength measurement with a precision two-beam scanning Michelson interferometer,” Appl. Opt. 20, 736–757 (1981).
    [Crossref] [PubMed]
  15. K. Dorenwendt and G. Bönsch, “Über den Einfluß der Beugung auf die interferentielle Längenmessung,” Metrologia 12, 57–60 (1976).
    [Crossref]
  16. G. Mana, “Diffraction effects in optical interferometers illuminated by laser sources,” Metrologia 26, 87–93 (1989).
    [Crossref]
  17. A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
    [Crossref]
  18. G. Cavagnero, G. Mana, and E. Massa, “Aberration effects in two-beam laser interferometers,” J. Opt. Soc. Am. A 23, 1951–1959 (2006).
    [Crossref]
  19. A. Bergamin, G. Cavagnero, and G. Mana, “A displacement and angle interferometer with subatomic resolution,” Rev. Sci. Instrum. 64, 3076–3081 (1993).
    [Crossref]
  20. A. Bergamin, G. Cavagnero, G. Durando, G. Mana, and E. Massa, “A two-axis tip-tilt platform for x-ray interferometry,” Meas. Sci. Technol. 14, 717–723 (2003).
    [Crossref]
  21. H. Fujimoto, G. Mana, and K. Nakayama, “A possible solution for the discrepancy between the INRIM and NMIJ values of the Si lattice-parameter,” IEEE Trans. Instrum. Meas. 56, 351–355 (2007).
    [Crossref]
  22. A. Balsamo, G. Cavagnero, G. Mana, and E. Massa, “Retrieval of the phase profile of digitized interferograms,” J. Opt. A 5, 418–424 (2003).
    [Crossref]

2015 (2)

E. Massa, C. P. Sasso, G. Mana, and C. Palmisano, “A more accurate measurement of the 28Si lattice parameter,” J. Phys. Chem. Ref. Data 44, 031208 (2015).
[Crossref]

B. Andreas, G. Mana, and C. Palmisano, “Vectorial ray-based diffraction integral,” J. Opt. Soc. Am. A 32, 1403–1424 (2015).
[Crossref]

2012 (1)

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[Crossref]

2011 (4)

G. D’Agostino and L. Robertsson, “Relative beam misalignment errors in high accuracy displacement interferometers: calculation and detection,” Appl. Phys. B 103, 357–361 (2011).
[Crossref]

G. Bartl, H. Bettin, M. Krystek, T. Mai, A. Nicolaus, and A. Peter, “Volume determination of the Avogadro spheres of highly enriched 28Si with a spherical Fizeau interferometer,” Metrologia 48, S96–S103 (2011).
[Crossref]

N. Kuramoto, K. Fujii, and K. Yamazawa, “Volume measurements of 28Si spheres using an interferometer with a flat etalon to determine the Avogadro constant,” Metrologia 48, S83–S95 (2011).
[Crossref]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

2009 (1)

R. A. Nicolaus and G. Boensch, “Aperture correction for a sphere interferometer,” Metrologia 46, 668–673 (2009).
[Crossref]

2007 (2)

L. Robertsson, “On the diffraction correction in absolute gravimetry,” Metrologia 44, 35–39 (2007).
[Crossref]

H. Fujimoto, G. Mana, and K. Nakayama, “A possible solution for the discrepancy between the INRIM and NMIJ values of the Si lattice-parameter,” IEEE Trans. Instrum. Meas. 56, 351–355 (2007).
[Crossref]

2006 (1)

2003 (3)

D. van Westrum and T. M. Niebauer, “The diffraction correction for absolute gravimeters,” Metrologia 40, 258–263 (2003).
[Crossref]

A. Balsamo, G. Cavagnero, G. Mana, and E. Massa, “Retrieval of the phase profile of digitized interferograms,” J. Opt. A 5, 418–424 (2003).
[Crossref]

A. Bergamin, G. Cavagnero, G. Durando, G. Mana, and E. Massa, “A two-axis tip-tilt platform for x-ray interferometry,” Meas. Sci. Technol. 14, 717–723 (2003).
[Crossref]

1999 (1)

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
[Crossref]

1997 (1)

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instrum. Meas. 46, 196–200 (1997).
[Crossref]

1994 (1)

A. Bergamin, G. Cavagnero, and G. Mana, “Observation of Fresnel diffraction in a two-beam laser interferometer,” Phys. Rev. A 49, 2167–2173 (1994).
[Crossref] [PubMed]

1993 (1)

A. Bergamin, G. Cavagnero, and G. Mana, “A displacement and angle interferometer with subatomic resolution,” Rev. Sci. Instrum. 64, 3076–3081 (1993).
[Crossref]

1989 (1)

G. Mana, “Diffraction effects in optical interferometers illuminated by laser sources,” Metrologia 26, 87–93 (1989).
[Crossref]

1981 (1)

1976 (1)

K. Dorenwendt and G. Bönsch, “Über den Einfluß der Beugung auf die interferentielle Längenmessung,” Metrologia 12, 57–60 (1976).
[Crossref]

Andreas, B.

B. Andreas, G. Mana, and C. Palmisano, “Vectorial ray-based diffraction integral,” J. Opt. Soc. Am. A 32, 1403–1424 (2015).
[Crossref]

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[Crossref]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

Balsamo, A.

A. Balsamo, G. Cavagnero, G. Mana, and E. Massa, “Retrieval of the phase profile of digitized interferograms,” J. Opt. A 5, 418–424 (2003).
[Crossref]

Bartl, G.

G. Bartl, H. Bettin, M. Krystek, T. Mai, A. Nicolaus, and A. Peter, “Volume determination of the Avogadro spheres of highly enriched 28Si with a spherical Fizeau interferometer,” Metrologia 48, S96–S103 (2011).
[Crossref]

Becker, P.

H. Siegert and P. Becker, Precision Measurement and Fundamental Constants IISpec. Publ. 617B. N. Taylor and W. D. Phillips, eds. (Natl. Bur. Stand. USA, 1984), p. 625.

Bergamin, A.

A. Bergamin, G. Cavagnero, G. Durando, G. Mana, and E. Massa, “A two-axis tip-tilt platform for x-ray interferometry,” Meas. Sci. Technol. 14, 717–723 (2003).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instrum. Meas. 46, 196–200 (1997).
[Crossref]

A. Bergamin, G. Cavagnero, and G. Mana, “Observation of Fresnel diffraction in a two-beam laser interferometer,” Phys. Rev. A 49, 2167–2173 (1994).
[Crossref] [PubMed]

A. Bergamin, G. Cavagnero, and G. Mana, “A displacement and angle interferometer with subatomic resolution,” Rev. Sci. Instrum. 64, 3076–3081 (1993).
[Crossref]

Bettin, H.

G. Bartl, H. Bettin, M. Krystek, T. Mai, A. Nicolaus, and A. Peter, “Volume determination of the Avogadro spheres of highly enriched 28Si with a spherical Fizeau interferometer,” Metrologia 48, S96–S103 (2011).
[Crossref]

Boensch, G.

R. A. Nicolaus and G. Boensch, “Aperture correction for a sphere interferometer,” Metrologia 46, 668–673 (2009).
[Crossref]

Bönsch, G.

K. Dorenwendt and G. Bönsch, “Über den Einfluß der Beugung auf die interferentielle Längenmessung,” Metrologia 12, 57–60 (1976).
[Crossref]

Cavagnero, G.

G. Cavagnero, G. Mana, and E. Massa, “Aberration effects in two-beam laser interferometers,” J. Opt. Soc. Am. A 23, 1951–1959 (2006).
[Crossref]

A. Bergamin, G. Cavagnero, G. Durando, G. Mana, and E. Massa, “A two-axis tip-tilt platform for x-ray interferometry,” Meas. Sci. Technol. 14, 717–723 (2003).
[Crossref]

A. Balsamo, G. Cavagnero, G. Mana, and E. Massa, “Retrieval of the phase profile of digitized interferograms,” J. Opt. A 5, 418–424 (2003).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instrum. Meas. 46, 196–200 (1997).
[Crossref]

A. Bergamin, G. Cavagnero, and G. Mana, “Observation of Fresnel diffraction in a two-beam laser interferometer,” Phys. Rev. A 49, 2167–2173 (1994).
[Crossref] [PubMed]

A. Bergamin, G. Cavagnero, and G. Mana, “A displacement and angle interferometer with subatomic resolution,” Rev. Sci. Instrum. 64, 3076–3081 (1993).
[Crossref]

Cordiali, L.

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instrum. Meas. 46, 196–200 (1997).
[Crossref]

D’Agostino, G.

G. D’Agostino and L. Robertsson, “Relative beam misalignment errors in high accuracy displacement interferometers: calculation and detection,” Appl. Phys. B 103, 357–361 (2011).
[Crossref]

Dorenwendt, K.

K. Dorenwendt and G. Bönsch, “Über den Einfluß der Beugung auf die interferentielle Längenmessung,” Metrologia 12, 57–60 (1976).
[Crossref]

Durando, G.

A. Bergamin, G. Cavagnero, G. Durando, G. Mana, and E. Massa, “A two-axis tip-tilt platform for x-ray interferometry,” Meas. Sci. Technol. 14, 717–723 (2003).
[Crossref]

Ferroglio, L.

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

Fujii, K.

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[Crossref]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

N. Kuramoto, K. Fujii, and K. Yamazawa, “Volume measurements of 28Si spheres using an interferometer with a flat etalon to determine the Avogadro constant,” Metrologia 48, S83–S95 (2011).
[Crossref]

Fujimoto, H.

H. Fujimoto, G. Mana, and K. Nakayama, “A possible solution for the discrepancy between the INRIM and NMIJ values of the Si lattice-parameter,” IEEE Trans. Instrum. Meas. 56, 351–355 (2007).
[Crossref]

Javan, A.

Kelly, M. J.

Krystek, M.

G. Bartl, H. Bettin, M. Krystek, T. Mai, A. Nicolaus, and A. Peter, “Volume determination of the Avogadro spheres of highly enriched 28Si with a spherical Fizeau interferometer,” Metrologia 48, S96–S103 (2011).
[Crossref]

Kuramoto, N.

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[Crossref]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

N. Kuramoto, K. Fujii, and K. Yamazawa, “Volume measurements of 28Si spheres using an interferometer with a flat etalon to determine the Avogadro constant,” Metrologia 48, S83–S95 (2011).
[Crossref]

Kurnit, N. A.

Lee, P. H.

Mai, T.

G. Bartl, H. Bettin, M. Krystek, T. Mai, A. Nicolaus, and A. Peter, “Volume determination of the Avogadro spheres of highly enriched 28Si with a spherical Fizeau interferometer,” Metrologia 48, S96–S103 (2011).
[Crossref]

Mana, G.

E. Massa, C. P. Sasso, G. Mana, and C. Palmisano, “A more accurate measurement of the 28Si lattice parameter,” J. Phys. Chem. Ref. Data 44, 031208 (2015).
[Crossref]

B. Andreas, G. Mana, and C. Palmisano, “Vectorial ray-based diffraction integral,” J. Opt. Soc. Am. A 32, 1403–1424 (2015).
[Crossref]

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[Crossref]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

H. Fujimoto, G. Mana, and K. Nakayama, “A possible solution for the discrepancy between the INRIM and NMIJ values of the Si lattice-parameter,” IEEE Trans. Instrum. Meas. 56, 351–355 (2007).
[Crossref]

G. Cavagnero, G. Mana, and E. Massa, “Aberration effects in two-beam laser interferometers,” J. Opt. Soc. Am. A 23, 1951–1959 (2006).
[Crossref]

A. Bergamin, G. Cavagnero, G. Durando, G. Mana, and E. Massa, “A two-axis tip-tilt platform for x-ray interferometry,” Meas. Sci. Technol. 14, 717–723 (2003).
[Crossref]

A. Balsamo, G. Cavagnero, G. Mana, and E. Massa, “Retrieval of the phase profile of digitized interferograms,” J. Opt. A 5, 418–424 (2003).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
[Crossref]

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instrum. Meas. 46, 196–200 (1997).
[Crossref]

A. Bergamin, G. Cavagnero, and G. Mana, “Observation of Fresnel diffraction in a two-beam laser interferometer,” Phys. Rev. A 49, 2167–2173 (1994).
[Crossref] [PubMed]

A. Bergamin, G. Cavagnero, and G. Mana, “A displacement and angle interferometer with subatomic resolution,” Rev. Sci. Instrum. 64, 3076–3081 (1993).
[Crossref]

G. Mana, “Diffraction effects in optical interferometers illuminated by laser sources,” Metrologia 26, 87–93 (1989).
[Crossref]

Massa, E.

E. Massa, C. P. Sasso, G. Mana, and C. Palmisano, “A more accurate measurement of the 28Si lattice parameter,” J. Phys. Chem. Ref. Data 44, 031208 (2015).
[Crossref]

G. Cavagnero, G. Mana, and E. Massa, “Aberration effects in two-beam laser interferometers,” J. Opt. Soc. Am. A 23, 1951–1959 (2006).
[Crossref]

A. Balsamo, G. Cavagnero, G. Mana, and E. Massa, “Retrieval of the phase profile of digitized interferograms,” J. Opt. A 5, 418–424 (2003).
[Crossref]

A. Bergamin, G. Cavagnero, G. Durando, G. Mana, and E. Massa, “A two-axis tip-tilt platform for x-ray interferometry,” Meas. Sci. Technol. 14, 717–723 (2003).
[Crossref]

Monchalin, J.-P.

Nakayama, K.

H. Fujimoto, G. Mana, and K. Nakayama, “A possible solution for the discrepancy between the INRIM and NMIJ values of the Si lattice-parameter,” IEEE Trans. Instrum. Meas. 56, 351–355 (2007).
[Crossref]

Nicolaus, A.

G. Bartl, H. Bettin, M. Krystek, T. Mai, A. Nicolaus, and A. Peter, “Volume determination of the Avogadro spheres of highly enriched 28Si with a spherical Fizeau interferometer,” Metrologia 48, S96–S103 (2011).
[Crossref]

Nicolaus, R. A.

R. A. Nicolaus and G. Boensch, “Aperture correction for a sphere interferometer,” Metrologia 46, 668–673 (2009).
[Crossref]

Niebauer, T. M.

D. van Westrum and T. M. Niebauer, “The diffraction correction for absolute gravimeters,” Metrologia 40, 258–263 (2003).
[Crossref]

Palmisano, C.

E. Massa, C. P. Sasso, G. Mana, and C. Palmisano, “A more accurate measurement of the 28Si lattice parameter,” J. Phys. Chem. Ref. Data 44, 031208 (2015).
[Crossref]

B. Andreas, G. Mana, and C. Palmisano, “Vectorial ray-based diffraction integral,” J. Opt. Soc. Am. A 32, 1403–1424 (2015).
[Crossref]

Peter, A.

G. Bartl, H. Bettin, M. Krystek, T. Mai, A. Nicolaus, and A. Peter, “Volume determination of the Avogadro spheres of highly enriched 28Si with a spherical Fizeau interferometer,” Metrologia 48, S96–S103 (2011).
[Crossref]

Robertsson, L.

G. D’Agostino and L. Robertsson, “Relative beam misalignment errors in high accuracy displacement interferometers: calculation and detection,” Appl. Phys. B 103, 357–361 (2011).
[Crossref]

L. Robertsson, “On the diffraction correction in absolute gravimetry,” Metrologia 44, 35–39 (2007).
[Crossref]

Sasso, C. P.

E. Massa, C. P. Sasso, G. Mana, and C. Palmisano, “A more accurate measurement of the 28Si lattice parameter,” J. Phys. Chem. Ref. Data 44, 031208 (2015).
[Crossref]

Siegert, H.

H. Siegert and P. Becker, Precision Measurement and Fundamental Constants IISpec. Publ. 617B. N. Taylor and W. D. Phillips, eds. (Natl. Bur. Stand. USA, 1984), p. 625.

Szöke, A.

Thomas, J. E.

van Westrum, D.

D. van Westrum and T. M. Niebauer, “The diffraction correction for absolute gravimeters,” Metrologia 40, 258–263 (2003).
[Crossref]

Yamazawa, K.

N. Kuramoto, K. Fujii, and K. Yamazawa, “Volume measurements of 28Si spheres using an interferometer with a flat etalon to determine the Avogadro constant,” Metrologia 48, S83–S95 (2011).
[Crossref]

Zernike, F.

Appl. Opt. (1)

Appl. Phys. B (1)

G. D’Agostino and L. Robertsson, “Relative beam misalignment errors in high accuracy displacement interferometers: calculation and detection,” Appl. Phys. B 103, 357–361 (2011).
[Crossref]

Eur. Phys. J. D (1)

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “A Fourier optics model of two-beam scanning laser interferometers,” Eur. Phys. J. D 5, 433–440 (1999).
[Crossref]

IEEE Trans. Instrum. Meas. (2)

A. Bergamin, G. Cavagnero, L. Cordiali, and G. Mana, “Beam-astigmatism in laser interferometry,” IEEE Trans. Instrum. Meas. 46, 196–200 (1997).
[Crossref]

H. Fujimoto, G. Mana, and K. Nakayama, “A possible solution for the discrepancy between the INRIM and NMIJ values of the Si lattice-parameter,” IEEE Trans. Instrum. Meas. 56, 351–355 (2007).
[Crossref]

J. Opt. A (1)

A. Balsamo, G. Cavagnero, G. Mana, and E. Massa, “Retrieval of the phase profile of digitized interferograms,” J. Opt. A 5, 418–424 (2003).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Phys. Chem. Ref. Data (1)

E. Massa, C. P. Sasso, G. Mana, and C. Palmisano, “A more accurate measurement of the 28Si lattice parameter,” J. Phys. Chem. Ref. Data 44, 031208 (2015).
[Crossref]

Meas. Sci. Technol. (1)

A. Bergamin, G. Cavagnero, G. Durando, G. Mana, and E. Massa, “A two-axis tip-tilt platform for x-ray interferometry,” Meas. Sci. Technol. 14, 717–723 (2003).
[Crossref]

Metrologia (9)

D. van Westrum and T. M. Niebauer, “The diffraction correction for absolute gravimeters,” Metrologia 40, 258–263 (2003).
[Crossref]

L. Robertsson, “On the diffraction correction in absolute gravimetry,” Metrologia 44, 35–39 (2007).
[Crossref]

R. A. Nicolaus and G. Boensch, “Aperture correction for a sphere interferometer,” Metrologia 46, 668–673 (2009).
[Crossref]

G. Bartl, H. Bettin, M. Krystek, T. Mai, A. Nicolaus, and A. Peter, “Volume determination of the Avogadro spheres of highly enriched 28Si with a spherical Fizeau interferometer,” Metrologia 48, S96–S103 (2011).
[Crossref]

N. Kuramoto, K. Fujii, and K. Yamazawa, “Volume measurements of 28Si spheres using an interferometer with a flat etalon to determine the Avogadro constant,” Metrologia 48, S83–S95 (2011).
[Crossref]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, “Phase corrections in the optical interferometer for Si sphere volume measurements at NMIJ,” Metrologia 48, S104–S111 (2011).
[Crossref]

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[Crossref]

K. Dorenwendt and G. Bönsch, “Über den Einfluß der Beugung auf die interferentielle Längenmessung,” Metrologia 12, 57–60 (1976).
[Crossref]

G. Mana, “Diffraction effects in optical interferometers illuminated by laser sources,” Metrologia 26, 87–93 (1989).
[Crossref]

Phys. Rev. A (1)

A. Bergamin, G. Cavagnero, and G. Mana, “Observation of Fresnel diffraction in a two-beam laser interferometer,” Phys. Rev. A 49, 2167–2173 (1994).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

A. Bergamin, G. Cavagnero, and G. Mana, “A displacement and angle interferometer with subatomic resolution,” Rev. Sci. Instrum. 64, 3076–3081 (1993).
[Crossref]

Other (1)

H. Siegert and P. Becker, Precision Measurement and Fundamental Constants IISpec. Publ. 617B. N. Taylor and W. D. Phillips, eds. (Natl. Bur. Stand. USA, 1984), p. 625.

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Figures (6)

Fig. 1
Fig. 1 Schematics of the combined X-ray and optical interferometer. The analyzer displacement is simultaneously measured in terms of X-ray and optical fringes; the quadrant detector monitors the alignment of the interfering wavefronts.
Fig. 2
Fig. 2 Curvature of the wavelength profile across a Gaussian beam.
Fig. 3
Fig. 3 Wavelength surveys across the beams whose parameters are given in Table 1. Left: wavelength profiles. Right: residuals after the best-fit parabolas were removed. In 2b, for the sake of clearness, residuals are shown upsidedown. The (5 × 5) pixel2 images were scaled down to take the 3× magnification into account. The first line shows the scatter of 50 subsequent profiles spaced by about 2 mm. The beam parameters are given in Table 1.
Fig. 4
Fig. 4 Comparison of the observed and simulated phase noises. Left: observed phase-noise across the interfering beam # 2 in Table 1. Right: simulated phase noise. The color maps range from −λ0/30 to +λ0/30, the standard deviation is λ0/200, the correlation length is 0.3 mm.
Fig. 5
Fig. 5 Comparison of the observed and simulated intensity profiles. Left: residuals of the best Gaussian fit of the intensity profile of beam # 2 in Table 1. Right: simulated residuals. The color maps range from −10% (blue) to 10% (red) of the maximum intensity.
Fig. 6
Fig. 6 Simulation results. Residuals after the best-fit parabolas were removed from the wavelength-profiles across the top-hat beam shown in Fig. 5 (left, upsidedown) and a Gaussian beam where the noise shown in Fig. 4 was added to its phase (right).

Tables (1)

Tables Icon

Table 1 Beam parameters and curvature of the best-fit approximations of the wavelength profiles in Fig. 3. F is the focal length of the fiber collimator, wD is the beam radius at the detection plane, zD is the detector distance from the beam waist, κ is the observed curvature.

Equations (9)

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φ ( x , y ; z ) = k 0 ( x 2 + y 2 ) 2 R ( z ) ,
Δ λ λ 0 = z φ k 0 = ( z 2 z R 2 ) ( x 2 + y 2 ) 2 ( z 2 + z R 2 ) 2 ,
E ˜ ( p , q ; s ) = U ( p , q ; s ) u ˜ ( p , q ; 0 ) e i k 0 s ,
u ˜ ( p , q ; 0 ) = 1 2 π + u ( x , y ; 0 ) e i ( p x + q y ) d x d y .
U ( p , q ; s ) = exp [ i ( p 2 + q 2 ) s 2 k 0 ]
u ( x , y ; s ) = 1 2 π + u ˜ ( p , q ; s ) e i ( p x + q y ) d p d q ,
Δ ϕ ( x , y ) = k 0 s + arg [ u ( x , y ; s ) ] arg [ u ( x , y ; 0 ) ] .
λ ( x , y ) = λ 0 { 1 arg [ u ( x , y ; s ) ] arg [ u ( x , y ; 0 ) ] k 0 s } ,
u ( x , y ; 0 ) = [ 1 + α ( x , y ) ] exp [ r 2 a / w D 2 + i k 0 r 2 / ( 2 R D ) + i ϕ ( x , y ) ] ,

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