Abstract

In displacement measurements by two-beam interferometers, the wavefront curvature of a laser beam causes a systematic increase of the fringe period. This increase depends on beam collimation: It is null for a plane wave and proportional to the squared divergence of the beam. With interfering beams not perfectly recombined, an additional fringe-period error is caused, with the effect of counteracting and also of compensating for and prevailing over the usual error. We describe this hitherto unsuspected effect and give a correction equation.

© 2006 Optical Society of America

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References

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  1. E. R. Wlliams, R. L. Steiner, D. B. Newell, and P. T. Olsen, "Accurate measurement of the Planck constant," Phys. Rev. Lett. 81, 2404-2408 (1998).
    [CrossRef]
  2. G. W. Small, B. W. Ricketts, P. C. Coogan, B. J. Pritchard, and M. M. R. Sovierzoski, "A new determination of the quantized Hall resistance in terms of the NML calculable cross capacitor," Metrologia 34, 241-243 (1997).
    [CrossRef]
  3. E. Krüger, W. Nistler, and W. Weirauch, "Determination of the fine-structure constant by a precise measurement of h/mn," Metrologia 32, 117-128 (1995).
    [CrossRef]
  4. G. Mana and G. Zosi, "The Avogadro constant," Riv. Nuovo Cimento 18, 1-23 (1995).
  5. D. van Westrum and T. M. Niebauer, "The diffraction correction for absolute gravimeters," Metrologia 40, 258-263 (2003).
  6. G. Basile, A. Bergamin, G. Cavagnero, G. Mana, E. Vittone, and G. Zosi, "Measurement of the silicon (220) lattice spacing," Phys. Rev. Lett. 72, 3133-3137 (1994).
    [CrossRef]
  7. K. Nakayama and H. Fujimoto, "Progress in the measurement of lattice spacing d(220) of silicon," IEEE Trans. Instrum. Meas. 46, 580-583 (1997).
    [CrossRef]
  8. G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 56-64 (2004).
    [CrossRef]
  9. G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Erratum: measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 445-446 (2004).
    [CrossRef]
  10. P. Giacomo, "Length measurement standards" in Proceedings of the International School "E. Fermi", Course LXVIII, Metrology and Fundamental Constants, A.Ferro Milone and P.Giacomo, eds. (North Holland, 1980).
    [PubMed]
  11. W. J. Tango and R. Q. Twiss, "Diffraction effects in long path interferometers," Appl. Opt. 13, 1814-1819 (1974).
    [CrossRef] [PubMed]
  12. I. P. Monchalin, M. J. Kelly, I. E. Thomas, N. A. Kurnit, A. Szoke, 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]
  13. K. Dorenwendt and G. Bonsch, "Ber den Einfluss der Beugung auf die interferentielle Langenmessung," Metrologia 12, 57-60 (1976).
    [CrossRef]
  14. 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]
  15. G. Mana, "Diffraction effects in optical interferometers illuminated by laser sources," Metrologia 26, 87-93 (1989).
    [CrossRef]
  16. 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]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  18. S. Wolfram, The Mathematica Book (Wolfram Media/Cambridge U. Press, 1999).
  19. H. Kogelnik and T. Li, "Laser beams and resonators," Appl. Opt. , 5, 1550-1566 (1966).
    [CrossRef]
  20. A. Bergamin, G. Cavagnero. G. Mana, and G. Zosi, "Scanning x-ray interferometry and the silicon lattice parameter: towards 10-9 relative uncertainty?" Eur. Phys. J. B 9, 225-232 (1999).
    [CrossRef]

2004 (2)

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 56-64 (2004).
[CrossRef]

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Erratum: measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 445-446 (2004).
[CrossRef]

2003 (1)

D. van Westrum and T. M. Niebauer, "The diffraction correction for absolute gravimeters," Metrologia 40, 258-263 (2003).

1999 (2)

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. G. Mana, and G. Zosi, "Scanning x-ray interferometry and the silicon lattice parameter: towards 10-9 relative uncertainty?" Eur. Phys. J. B 9, 225-232 (1999).
[CrossRef]

1998 (1)

E. R. Wlliams, R. L. Steiner, D. B. Newell, and P. T. Olsen, "Accurate measurement of the Planck constant," Phys. Rev. Lett. 81, 2404-2408 (1998).
[CrossRef]

1997 (2)

G. W. Small, B. W. Ricketts, P. C. Coogan, B. J. Pritchard, and M. M. R. Sovierzoski, "A new determination of the quantized Hall resistance in terms of the NML calculable cross capacitor," Metrologia 34, 241-243 (1997).
[CrossRef]

K. Nakayama and H. Fujimoto, "Progress in the measurement of lattice spacing d(220) of silicon," IEEE Trans. Instrum. Meas. 46, 580-583 (1997).
[CrossRef]

1995 (2)

E. Krüger, W. Nistler, and W. Weirauch, "Determination of the fine-structure constant by a precise measurement of h/mn," Metrologia 32, 117-128 (1995).
[CrossRef]

G. Mana and G. Zosi, "The Avogadro constant," Riv. Nuovo Cimento 18, 1-23 (1995).

1994 (2)

G. Basile, A. Bergamin, G. Cavagnero, G. Mana, E. Vittone, and G. Zosi, "Measurement of the silicon (220) lattice spacing," Phys. Rev. Lett. 72, 3133-3137 (1994).
[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]

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. Bonsch, "Ber den Einfluss der Beugung auf die interferentielle Langenmessung," Metrologia 12, 57-60 (1976).
[CrossRef]

1974 (1)

1966 (1)

Basile, G.

G. Basile, A. Bergamin, G. Cavagnero, G. Mana, E. Vittone, and G. Zosi, "Measurement of the silicon (220) lattice spacing," Phys. Rev. Lett. 72, 3133-3137 (1994).
[CrossRef]

Bergamin, A.

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. G. Mana, and G. Zosi, "Scanning x-ray interferometry and the silicon lattice parameter: towards 10-9 relative uncertainty?" Eur. Phys. J. B 9, 225-232 (1999).
[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]

G. Basile, A. Bergamin, G. Cavagnero, G. Mana, E. Vittone, and G. Zosi, "Measurement of the silicon (220) lattice spacing," Phys. Rev. Lett. 72, 3133-3137 (1994).
[CrossRef]

Bonsch, G.

K. Dorenwendt and G. Bonsch, "Ber den Einfluss der Beugung auf die interferentielle Langenmessung," Metrologia 12, 57-60 (1976).
[CrossRef]

Cavagnero, G.

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 56-64 (2004).
[CrossRef]

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Erratum: measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 445-446 (2004).
[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, and G. Mana, "Observation of Fresnel diffraction in a two-beam laser interferometer," Phys. Rev. A 49, 2167-2173 (1994).
[CrossRef] [PubMed]

G. Basile, A. Bergamin, G. Cavagnero, G. Mana, E. Vittone, and G. Zosi, "Measurement of the silicon (220) lattice spacing," Phys. Rev. Lett. 72, 3133-3137 (1994).
[CrossRef]

Coogan, P. C.

G. W. Small, B. W. Ricketts, P. C. Coogan, B. J. Pritchard, and M. M. R. Sovierzoski, "A new determination of the quantized Hall resistance in terms of the NML calculable cross capacitor," Metrologia 34, 241-243 (1997).
[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]

Dorenwendt, K.

K. Dorenwendt and G. Bonsch, "Ber den Einfluss der Beugung auf die interferentielle Langenmessung," Metrologia 12, 57-60 (1976).
[CrossRef]

Fujimoto, H.

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Erratum: measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 445-446 (2004).
[CrossRef]

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 56-64 (2004).
[CrossRef]

K. Nakayama and H. Fujimoto, "Progress in the measurement of lattice spacing d(220) of silicon," IEEE Trans. Instrum. Meas. 46, 580-583 (1997).
[CrossRef]

Giacomo, P.

P. Giacomo, "Length measurement standards" in Proceedings of the International School "E. Fermi", Course LXVIII, Metrology and Fundamental Constants, A.Ferro Milone and P.Giacomo, eds. (North Holland, 1980).
[PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Javan, A.

Kelly, M. J.

Kogelnik, H.

Krüger, E.

E. Krüger, W. Nistler, and W. Weirauch, "Determination of the fine-structure constant by a precise measurement of h/mn," Metrologia 32, 117-128 (1995).
[CrossRef]

Kurnit, N. A.

Lee, P. H.

Li, T.

Mana, G.

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 56-64 (2004).
[CrossRef]

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Erratum: measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 445-446 (2004).
[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]

G. Mana and G. Zosi, "The Avogadro constant," Riv. Nuovo Cimento 18, 1-23 (1995).

G. Basile, A. Bergamin, G. Cavagnero, G. Mana, E. Vittone, and G. Zosi, "Measurement of the silicon (220) lattice spacing," Phys. Rev. Lett. 72, 3133-3137 (1994).
[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]

G. Mana, "Diffraction effects in optical interferometers illuminated by laser sources," Metrologia 26, 87-93 (1989).
[CrossRef]

Mana, G. Cavagnero. G.

A. Bergamin, G. Cavagnero. G. Mana, and G. Zosi, "Scanning x-ray interferometry and the silicon lattice parameter: towards 10-9 relative uncertainty?" Eur. Phys. J. B 9, 225-232 (1999).
[CrossRef]

Massa, E.

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Erratum: measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 445-446 (2004).
[CrossRef]

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 56-64 (2004).
[CrossRef]

Monchalin, I. P.

Nakayama, K.

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 56-64 (2004).
[CrossRef]

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Erratum: measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 445-446 (2004).
[CrossRef]

K. Nakayama and H. Fujimoto, "Progress in the measurement of lattice spacing d(220) of silicon," IEEE Trans. Instrum. Meas. 46, 580-583 (1997).
[CrossRef]

Newell, D. B.

E. R. Wlliams, R. L. Steiner, D. B. Newell, and P. T. Olsen, "Accurate measurement of the Planck constant," Phys. Rev. Lett. 81, 2404-2408 (1998).
[CrossRef]

Niebauer, T. M.

D. van Westrum and T. M. Niebauer, "The diffraction correction for absolute gravimeters," Metrologia 40, 258-263 (2003).

Nistler, W.

E. Krüger, W. Nistler, and W. Weirauch, "Determination of the fine-structure constant by a precise measurement of h/mn," Metrologia 32, 117-128 (1995).
[CrossRef]

Olsen, P. T.

E. R. Wlliams, R. L. Steiner, D. B. Newell, and P. T. Olsen, "Accurate measurement of the Planck constant," Phys. Rev. Lett. 81, 2404-2408 (1998).
[CrossRef]

Pritchard, B. J.

G. W. Small, B. W. Ricketts, P. C. Coogan, B. J. Pritchard, and M. M. R. Sovierzoski, "A new determination of the quantized Hall resistance in terms of the NML calculable cross capacitor," Metrologia 34, 241-243 (1997).
[CrossRef]

Ricketts, B. W.

G. W. Small, B. W. Ricketts, P. C. Coogan, B. J. Pritchard, and M. M. R. Sovierzoski, "A new determination of the quantized Hall resistance in terms of the NML calculable cross capacitor," Metrologia 34, 241-243 (1997).
[CrossRef]

Small, G. W.

G. W. Small, B. W. Ricketts, P. C. Coogan, B. J. Pritchard, and M. M. R. Sovierzoski, "A new determination of the quantized Hall resistance in terms of the NML calculable cross capacitor," Metrologia 34, 241-243 (1997).
[CrossRef]

Sovierzoski, M. M. R.

G. W. Small, B. W. Ricketts, P. C. Coogan, B. J. Pritchard, and M. M. R. Sovierzoski, "A new determination of the quantized Hall resistance in terms of the NML calculable cross capacitor," Metrologia 34, 241-243 (1997).
[CrossRef]

Steiner, R. L.

E. R. Wlliams, R. L. Steiner, D. B. Newell, and P. T. Olsen, "Accurate measurement of the Planck constant," Phys. Rev. Lett. 81, 2404-2408 (1998).
[CrossRef]

Szoke, A.

Tango, W. J.

Thomas, I. E.

Twiss, R. Q.

van Westrum, D.

D. van Westrum and T. M. Niebauer, "The diffraction correction for absolute gravimeters," Metrologia 40, 258-263 (2003).

Vittone, E.

G. Basile, A. Bergamin, G. Cavagnero, G. Mana, E. Vittone, and G. Zosi, "Measurement of the silicon (220) lattice spacing," Phys. Rev. Lett. 72, 3133-3137 (1994).
[CrossRef]

Weirauch, W.

E. Krüger, W. Nistler, and W. Weirauch, "Determination of the fine-structure constant by a precise measurement of h/mn," Metrologia 32, 117-128 (1995).
[CrossRef]

Wlliams, E. R.

E. R. Wlliams, R. L. Steiner, D. B. Newell, and P. T. Olsen, "Accurate measurement of the Planck constant," Phys. Rev. Lett. 81, 2404-2408 (1998).
[CrossRef]

Wolfram, S.

S. Wolfram, The Mathematica Book (Wolfram Media/Cambridge U. Press, 1999).

Zernike, F.

Zosi, G.

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Erratum: measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 445-446 (2004).
[CrossRef]

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 56-64 (2004).
[CrossRef]

A. Bergamin, G. Cavagnero. G. Mana, and G. Zosi, "Scanning x-ray interferometry and the silicon lattice parameter: towards 10-9 relative uncertainty?" Eur. Phys. J. B 9, 225-232 (1999).
[CrossRef]

G. Mana and G. Zosi, "The Avogadro constant," Riv. Nuovo Cimento 18, 1-23 (1995).

G. Basile, A. Bergamin, G. Cavagnero, G. Mana, E. Vittone, and G. Zosi, "Measurement of the silicon (220) lattice spacing," Phys. Rev. Lett. 72, 3133-3137 (1994).
[CrossRef]

Appl. Opt. (3)

Eur. Phys. J. B (1)

A. Bergamin, G. Cavagnero. G. Mana, and G. Zosi, "Scanning x-ray interferometry and the silicon lattice parameter: towards 10-9 relative uncertainty?" Eur. Phys. J. B 9, 225-232 (1999).
[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. (1)

K. Nakayama and H. Fujimoto, "Progress in the measurement of lattice spacing d(220) of silicon," IEEE Trans. Instrum. Meas. 46, 580-583 (1997).
[CrossRef]

Metrologia (7)

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 56-64 (2004).
[CrossRef]

G. Cavagnero, H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, "Erratum: measurement repetitions of the Si(220) lattice spacing," Metrologia 41, 445-446 (2004).
[CrossRef]

G. W. Small, B. W. Ricketts, P. C. Coogan, B. J. Pritchard, and M. M. R. Sovierzoski, "A new determination of the quantized Hall resistance in terms of the NML calculable cross capacitor," Metrologia 34, 241-243 (1997).
[CrossRef]

E. Krüger, W. Nistler, and W. Weirauch, "Determination of the fine-structure constant by a precise measurement of h/mn," Metrologia 32, 117-128 (1995).
[CrossRef]

D. van Westrum and T. M. Niebauer, "The diffraction correction for absolute gravimeters," Metrologia 40, 258-263 (2003).

K. Dorenwendt and G. Bonsch, "Ber den Einfluss der Beugung auf die interferentielle Langenmessung," 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]

Phys. Rev. Lett. (2)

E. R. Wlliams, R. L. Steiner, D. B. Newell, and P. T. Olsen, "Accurate measurement of the Planck constant," Phys. Rev. Lett. 81, 2404-2408 (1998).
[CrossRef]

G. Basile, A. Bergamin, G. Cavagnero, G. Mana, E. Vittone, and G. Zosi, "Measurement of the silicon (220) lattice spacing," Phys. Rev. Lett. 72, 3133-3137 (1994).
[CrossRef]

Riv. Nuovo Cimento (1)

G. Mana and G. Zosi, "The Avogadro constant," Riv. Nuovo Cimento 18, 1-23 (1995).

Other (3)

P. Giacomo, "Length measurement standards" in Proceedings of the International School "E. Fermi", Course LXVIII, Metrology and Fundamental Constants, A.Ferro Milone and P.Giacomo, eds. (North Holland, 1980).
[PubMed]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

S. Wolfram, The Mathematica Book (Wolfram Media/Cambridge U. Press, 1999).

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Figures (6)

Fig. 1
Fig. 1

Schematic of the interferometer. The detector perceives two phase-locked beams coming from different directions. Beam waists are indicated by bullets. Axis z is orthogonal to the detection plane; β 1 γ is the beam intersection angle, γ is detector inclination, x 0 is beam offset, and s is the interval between the beam waists.

Fig. 2
Fig. 2

Geometry of interferometer operation. When the test beam slides on its axis by s o , the interval between beam waists changes by s = s o cos γ and the beam offset by s 0 tan γ .

Fig. 3
Fig. 3

Electric fields Re ( E 1 ) and Re ( E 2 ) on the detector plane with beams parallel but not coaxial. Solid and dashed curves correspond to reference E 1 and test E 2 beams, respectively; the shaded plot shows the interference pattern. Simulation parameters are λ = 633 nm , θ = 1 mrad , x 0 = 0.3 mm , 2 α = γ = 0 mrad , z = 2 m , and s 0 = 5 mm .

Fig. 4
Fig. 4

Electric fields Re ( E 1 ) and Re ( E 2 ) on the detector plane with beams intersecting at an angle 2 α = 0.5 mrad . Solid and dashed curves correspond to reference E 1 and test E 2 beams, respectively; the shaded plot shows the interference pattern. Simulation parameters are λ = 633 nm , θ = 1 mrad , x 0 = 0.3 mm , 2 α = 0.5 mrad , γ = 0 mrad , z = 2 m , and s 0 = 5 mm .

Fig. 5
Fig. 5

Extra phase of the interference fringes. All graphs have been shifted to display a zero phase when ξ = 0 . The solid curve corresponds to ideal geometry ( x 0 w 0 = 0 and 2 α = 0 rad ), the dashed–dotted curve to noncoaxial beams ( x 0 w 0 = 1 , 2 α = 0 rad ), and the dashed curve to misaligned beams ( x 0 w 0 = 1 and 2 α = 0.5 mrad ).

Fig. 6
Fig. 6

Diffraction correction (parts per billion) for the effective wavelength of the interference fringes. The solid curve corresponds to ideal geometry ( x 0 w 0 = 0 and 2 α = 0 rad ), the dashed–dotted curve to noncoaxial beams ( x 0 w 0 = 1 and 2 α = 0 rad ), and the dashed curve to misaligned beams ( x 0 w 0 = 1 and 2 α = 0.5 rad ).

Tables (1)

Tables Icon

Table 1 Analytical and Numerical Calculations of the Correction for Diffraction

Equations (75)

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

E 1 ( x ; 0 ) = A 1 ( x ) = exp ( x 2 + i k β 1 x ) ,
E 2 ( x ; s ) = A 2 ( x ) = exp [ ( x x 0 ) 2 + i k γ x ] ,
E ̃ ( p ; z ) = + E ( x ; z ) exp ( i p x ) d x ,
E ̃ 1 ( p ; z ) = U ( p ; z ) A ̃ 1 ( p ) ,
E ̃ 2 ( p ; z ) = U ( p ; z + s ) A ̃ 2 ( p ) ,
U ( p ; z ) exp [ i k ( 1 p 2 2 k 2 ) z ] .
E ̃ 2 ( p ; z ) = exp [ i k ( 1 p 2 2 k 2 ) s ] U ( p ; z ) A ̃ 2 ( p ) .
S ( x ) = E 1 ( x ; z ) + E 2 ( x ; z ) 2
I = + S ( x ) d x = + S ( p ) d p ,
S ( p ) = E ̃ 1 ( p ; z ) + E ̃ 2 ( p ; z ) 2 .
I = 2 G [ 1 + Γ cos ( k s Φ ) ] ,
2 G = + E ̃ 1 ( p ; z ) 2 d p + + E ̃ 2 ( p ; z ) 2 d p
Ξ = + A ̃ 1 ( p ) A ̃ 2 * ( p ) exp ( i s p 2 2 k ) d p
λ e = λ ( 1 γ 2 2 + 1 k d Φ d s o ) .
d Φ d s o = s Φ + s x 0 0 Φ + s α α Φ ,
λ e = λ [ 1 γ 2 2 + ( s γ 0 ) Φ k ] .
λ e λ λ = Δ λ λ = ( s γ 0 ) Φ k γ 2 2 .
A ̃ 1 ( p ) = A ̃ ( p ) ,
A ̃ 2 ( p ) = A ̃ ( p ) e i p x 0 ,
exp ( i s p 2 2 k ) = 1 + i k ψ 2 s 2 ,
Ξ = + ( 1 + i k ψ 2 s 2 ) A ̃ ( ψ ) 2 e i k ψ x 0 d ψ ,
λ e = λ ( 1 + ψ 2 2 ) ,
ψ 2 = 2 s arg ( Ξ ) k = + ψ 2 A ̃ ( ψ ) 2 cos ( i k ψ x 0 ) d ψ + A ̃ ( ψ ) 2 cos ( i k ψ x 0 ) d ψ .
A 1 ( x ) e i k γ x A 1 ( x ) ,
A 2 ( x ) e i k γ x A 2 ( x + γ s ) ,
A ̃ 1 ( p ) A ̃ 1 ( p k γ )
A ̃ 2 ( p ) A ̃ 2 ( p k γ ) e i ( p k γ ) s .
Ξ + A ̃ 1 ( p ) A ̃ 2 * ( p ) e i s ( p + k γ ) 2 2 k e i p γ s d p = e i k γ 2 s 2 + A ̃ ( p 2 α k ) A ̃ * ( p ) exp [ i ( s p 2 2 k p x 0 ) ] d p , = e i k γ 2 s 2 Ξ 0 ,
lim s ± Ξ 0 e ± i π 4 A ̃ 1 ( 2 α k ) A ̃ 2 * ( 0 ) .
E ( x ; 0 ) = exp ( x 2 )
E ̃ ( p , 0 ) = π exp ( p 2 4 )
E ( x ; z ) = π 2 π + exp [ i k ( 1 p 2 2 k 2 ) z ] exp ( p 2 4 ) exp ( i p x ) d p = exp [ i k z x 2 ( 1 + 2 i z b ) ] 1 + 2 i z b ,
E ( x ; z ) = w 0 w exp ( x 2 w 2 ) exp [ i ( k z + 2 z x 2 b w 2 φ 2 ) ] ,
E ( x ; z ) = w 0 w exp [ ( x 2 α z ) 2 w 2 ] exp { i [ k z + 2 k α ( x α z ) u 2 + 2 z x 2 b w 2 φ 2 ] } ,
E ( x ; z ) exp { i [ k ( 1 2 α 2 ) z + 2 k α x ] } ,
E ( x ; z ) exp [ i ( k z + 2 k α x ̱ + k x ̱ 2 2 z ) ] ,
S ( x ; z ) = E 1 ( x ; z ) + E 2 ( x ; z ) 2 exp [ 2 ( x 2 a ) x w 2 ] cos ( k s ξ u 2 + σ x ρ ξ x 2 ) ,
a = ( x 0 + 2 α z ) 2 = ( x 0 + 2 α u 2 1 ) 2
σ = 2 ( u 2 1 x 0 2 α ) u 2
ρ ξ 2 = ( u 2 2 ) ξ u 2
S ( x ; z ) exp [ 2 ( x x 0 ) x w 2 ] cos ( k s s b 2 k α x + 2 s x 2 b ) .
S ( x ; z ) cos ( k s 2 s x 2 b ) .
A 1 ( x ) = A 2 ( x ) = exp ( x 2 ) ,
A ̃ 1 ( p ) = A ̃ 2 ( p ) = π exp ( p 2 4 ) ,
Ξ = 2 π 3 1 i ξ ,
lim ξ 0 Δ λ λ = Δ λ λ 0 = θ 2 8 ,
+ ( θ x w ) 2 exp ( 2 x 2 w 2 ) d x 2 + exp ( 2 x 2 w 2 ) d x = θ 2 8 .
S ( x ; z ) exp ( 2 x 2 w 2 ) cos [ k s ξ u 2 2 ( u 2 2 ) x 2 ξ u 2 w 2 ] ,
ζ ( x ; z ) = λ [ 1 + 1 k b u 2 + 2 ( u 2 2 ) x 2 k b u 2 w 2 ] ,
λ e = + ζ ( x ; z ) exp ( 2 x 2 w 2 ) d x + exp ( 2 x 2 w 2 ) d x = λ ( 1 + θ 2 8 ) .
A 1 ( x ) = exp ( x 2 ) ,
A 2 ( x ) = exp [ ( x x 0 ) 2 ] ,
A ̃ 1 ( p ) = π exp ( p 2 4 ) ,
A ̃ 2 ( p ) = π exp [ ( p 2 4 + i x 0 p ) ] ,
Ξ = 2 π 3 1 i ξ exp x 0 2 2 ( 1 i ξ ) .
Φ = 1 2 arctan ξ x 0 2 ξ 2 ( 1 + ξ 2 ) ,
Δ λ λ 0 = ( 1 x 0 2 w 0 2 ) θ 2 8 .
A 1 ( x ) = exp ( x 2 + 2 i k α x ) ,
A 2 ( x ) = exp ( x 2 ) ,
A ̃ 1 ( p ) = π exp [ ( p 2 k α ) 2 4 ] ,
A ̃ 2 ( p ) = π exp [ p 2 4 ] ,
Ξ = 2 π 3 1 i ξ exp 2 ( 2 i ξ 1 ) α 2 ( 1 i ξ ) θ 2 .
Φ = 1 2 arctan ξ + 2 α 2 ξ ( 1 + ξ 2 ) θ 2
Δ λ λ 0 = θ 2 8 + α 2 2 .
A 1 ( x ) = exp ( x 2 + i k β 1 x ) ,
A 2 ( x ) = exp [ ( x x 0 ) 2 + i k γ x ] ,
A ̃ 1 ( p ) = π exp [ ( p β 1 k ) 2 4 ] ,
A ̃ 2 ( p ) = π exp [ ( p γ k ) 2 4 i x 0 ( p γ k ) ] ,
Ξ = 2 π 3 1 i ξ exp 4 i ( α 2 + β 2 ) ξ + 4 i α x 0 x 0 2 + 4 ( α + β ) x 0 ξ 4 α 2 2 ( 1 i ξ ) ,
Φ = 1 2 arctan ξ + 2 α x 0 + ( 4 β 2 + 4 β x 0 ξ x 0 2 ) ξ 2 ( 1 + ξ 2 ) ,
( s γ 0 ) Φ s = 0 = 1 + 4 β 2 x 0 2 2 b 2 γ α .
Δ λ λ 0 = ( 1 x 0 2 w 0 2 ) θ 2 8 + α 2 2 ,
Δ λ λ 0 = [ 1 x 0 2 ( 2 w 0 2 ) ] θ 2 4 + α 2 2 .
Ξ = e i k s o + E 1 ( x ; z ) E 2 * ( x ; z ) d x ,
λ e = λ ( 1 + 1 k d Φ d s o ) .

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