Abstract

In this paper we demonstrate how Bessel beam interferometry can be used to characterize the curvature of a reflecting surface. The approach is based on the fact that the intensity distribution produced by the coherent superposition of Bessel beams is a sensitive function of the relative phases between the constituting beams. We show how this phase sensitivity can translate into accurate measurements of the curvature of a wavefront. Experimental tests were made with a liquid mirror. We have also used Bessel beams to measure the precession angle of the liquid mirror. Our results show that Bessel beam interferometry is a very accurate tool for the optical testing of non-stationary surfaces and that it could be used as a general method of real-time, non-contact sensing. Bessel beam interferometry has the advantage of not requiring any reference arm that needs to be stabilized.

© 2004 Optical Society of America

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References

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  1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1986).
    [Crossref]
  2. J. Durnin, J.J. Miceli, and J.H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  3. S. Fujiwara, “Optical properties of conic surfaces. I. Reflecting cone,” J. Opt. Soc. 52, 287–292 (1962).
    [Crossref]
  4. A. Vasara, J. Turunen, and A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [Crossref] [PubMed]
  5. L. Niggel, T. Lanz, and M. Maier, “Properties of Bessel beams generated by periodic grating of circular symmetry,” J. Opt. Soc. Am. A 14, 27–33 (1997).
    [Crossref]
  6. P.L. Overfelt and C.S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel-Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–745 (1991).
    [Crossref]
  7. R.M. Herman and T.A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991).
    [Crossref]
  8. V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. 184, 245–255 (2000).
    [Crossref]
  9. G. Gadonas, V. Jarutis, R. Paskauskas, V. Smilgevicius, A. Stabinis, and V. Vaicaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun. 196, 309–316 (2001).
    [Crossref]
  10. C. Conti and S. Trillo, “X waves generated at the second harmonic,” Opt. Lett. 28, 1251–1253 (2003).
    [Crossref] [PubMed]
  11. M. Florjanczyk and R. Tremblay, “Guiding of atoms in a travelling-wave laser traps formed by the axicon,” Opt. Commun. 73, 448–451 (1989).
    [Crossref]
  12. V. Garces-Chavez, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
    [Crossref] [PubMed]
  13. D. McGloin, V. Garces-Chavez, D, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28, 657–659 (2003).
    [Crossref] [PubMed]
  14. Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. 27, 243–245 (2002).
    [Crossref]
  15. J. Amako, D. Sawaki, and E. Fujii, “Microstructuring transparent materials by use of nondiffracting ultrashort pulse beams generated by diffractive optics,” J. Opt. Soc. Am. B 20, 2562–2568 (2003).
    [Crossref]
  16. A. Marcinkevicius, S. Juodkazis, V. Mizeikis, S. Matsuo, and H. Misawa, “Application of femtosecond Bessel-Gauss beam in microstructuring of transparent materials,” Proc. SPIE 4271, 150–157 (2001).
    [Crossref]
  17. B. Bélanger, X. Zhu, and et M. Piché, “Auto-imagerie à l’aide de faisceaux non-diffractants,” Phys. Can. 52, 160–161 (1996).
  18. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [Crossref]
  19. S. Chavez-Cerda, M.A. Meneses-Nava, and J. M. Hickmann, “Interference of traveling nondiffracting beams,” Opt. Lett. 23, 1871–1873 (1998).
    [Crossref]
  20. A. E. Siegman, Lasers, chapters 15 and 20. (University Science Books, Mill Valley, 1986).
  21. E. F. Borra, “The liquid-mirror telescope as a viable astronomical tool,” Astron. Soc. Can. 76, 245–256 (1982).
  22. E. F. Borra, “Liquid mirrors,” Can. J. Phys. 73, 109–125 (1995).
    [Crossref]
  23. E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay, and E. Boily, “Liquid mirrors: optical shop tests and contributions to the technology,” Astrophys. J. 393, 829–847 (1992)
    [Crossref]
  24. L. Girard and E. F. Borra, “Optical tests of a 2.5-m-diameter liquid mirror: behavior under external perturbations and scattered light measurements,” Appl. Opt. 36, 25, 6278–6288 (1997).
    [Crossref]
  25. G. Tremblay and E. F. Borra, “Optical Tests of a 3.7-m-diameter liquid mirror: behavior under external perturbations,” Appl. Opt. 39, 5651–5662 (2000).
    [Crossref]
  26. R. J. Sica, S. Sargoytchev, E. F. Borra, L. Girard, S. Argall, C.T. Sarrow, and S. Flatt, “Lidar measurements taken with a large aperture liquid mirror:1.The Rayleigh-scatter system,” Appl. Opt.,  34, 6925–6936 (1995).
    [Crossref] [PubMed]
  27. P. Hickson and M. Mulrooney, University of British Columbia-NASA Multi-Narrowband Survey. I. Description and photometric properties of the survey,” Astrophys. J. Suppl. 115, 35–42 (1998).
    [Crossref]
  28. R. Cabanac, E. F. Borra, and M. Beauchemin, “A search for peculiar objects with the NASA orbital debris observatory 3-m liquid mirror Telescope,” Astrophys. J. 509, 309–323, (1998).
    [Crossref]

2003 (3)

2002 (2)

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. 27, 243–245 (2002).
[Crossref]

V. Garces-Chavez, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

2001 (2)

A. Marcinkevicius, S. Juodkazis, V. Mizeikis, S. Matsuo, and H. Misawa, “Application of femtosecond Bessel-Gauss beam in microstructuring of transparent materials,” Proc. SPIE 4271, 150–157 (2001).
[Crossref]

G. Gadonas, V. Jarutis, R. Paskauskas, V. Smilgevicius, A. Stabinis, and V. Vaicaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun. 196, 309–316 (2001).
[Crossref]

2000 (2)

V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. 184, 245–255 (2000).
[Crossref]

G. Tremblay and E. F. Borra, “Optical Tests of a 3.7-m-diameter liquid mirror: behavior under external perturbations,” Appl. Opt. 39, 5651–5662 (2000).
[Crossref]

1998 (3)

S. Chavez-Cerda, M.A. Meneses-Nava, and J. M. Hickmann, “Interference of traveling nondiffracting beams,” Opt. Lett. 23, 1871–1873 (1998).
[Crossref]

P. Hickson and M. Mulrooney, University of British Columbia-NASA Multi-Narrowband Survey. I. Description and photometric properties of the survey,” Astrophys. J. Suppl. 115, 35–42 (1998).
[Crossref]

R. Cabanac, E. F. Borra, and M. Beauchemin, “A search for peculiar objects with the NASA orbital debris observatory 3-m liquid mirror Telescope,” Astrophys. J. 509, 309–323, (1998).
[Crossref]

1997 (2)

1996 (1)

B. Bélanger, X. Zhu, and et M. Piché, “Auto-imagerie à l’aide de faisceaux non-diffractants,” Phys. Can. 52, 160–161 (1996).

1995 (2)

1992 (1)

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay, and E. Boily, “Liquid mirrors: optical shop tests and contributions to the technology,” Astrophys. J. 393, 829–847 (1992)
[Crossref]

1991 (2)

1989 (1)

M. Florjanczyk and R. Tremblay, “Guiding of atoms in a travelling-wave laser traps formed by the axicon,” Opt. Commun. 73, 448–451 (1989).
[Crossref]

1988 (1)

1987 (1)

J. Durnin, J.J. Miceli, and J.H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

1986 (1)

1982 (1)

E. F. Borra, “The liquid-mirror telescope as a viable astronomical tool,” Astron. Soc. Can. 76, 245–256 (1982).

1967 (1)

1962 (1)

S. Fujiwara, “Optical properties of conic surfaces. I. Reflecting cone,” J. Opt. Soc. 52, 287–292 (1962).
[Crossref]

Amako, J.

Argall, S.

Beauchemin, M.

R. Cabanac, E. F. Borra, and M. Beauchemin, “A search for peculiar objects with the NASA orbital debris observatory 3-m liquid mirror Telescope,” Astrophys. J. 509, 309–323, (1998).
[Crossref]

Bélanger, B.

B. Bélanger, X. Zhu, and et M. Piché, “Auto-imagerie à l’aide de faisceaux non-diffractants,” Phys. Can. 52, 160–161 (1996).

Boily, E.

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay, and E. Boily, “Liquid mirrors: optical shop tests and contributions to the technology,” Astrophys. J. 393, 829–847 (1992)
[Crossref]

Borra, E. F.

G. Tremblay and E. F. Borra, “Optical Tests of a 3.7-m-diameter liquid mirror: behavior under external perturbations,” Appl. Opt. 39, 5651–5662 (2000).
[Crossref]

R. Cabanac, E. F. Borra, and M. Beauchemin, “A search for peculiar objects with the NASA orbital debris observatory 3-m liquid mirror Telescope,” Astrophys. J. 509, 309–323, (1998).
[Crossref]

L. Girard and E. F. Borra, “Optical tests of a 2.5-m-diameter liquid mirror: behavior under external perturbations and scattered light measurements,” Appl. Opt. 36, 25, 6278–6288 (1997).
[Crossref]

R. J. Sica, S. Sargoytchev, E. F. Borra, L. Girard, S. Argall, C.T. Sarrow, and S. Flatt, “Lidar measurements taken with a large aperture liquid mirror:1.The Rayleigh-scatter system,” Appl. Opt.,  34, 6925–6936 (1995).
[Crossref] [PubMed]

E. F. Borra, “Liquid mirrors,” Can. J. Phys. 73, 109–125 (1995).
[Crossref]

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay, and E. Boily, “Liquid mirrors: optical shop tests and contributions to the technology,” Astrophys. J. 393, 829–847 (1992)
[Crossref]

E. F. Borra, “The liquid-mirror telescope as a viable astronomical tool,” Astron. Soc. Can. 76, 245–256 (1982).

Cabanac, R.

R. Cabanac, E. F. Borra, and M. Beauchemin, “A search for peculiar objects with the NASA orbital debris observatory 3-m liquid mirror Telescope,” Astrophys. J. 509, 309–323, (1998).
[Crossref]

Chavez-Cerda, S.

Chen, Z.

Content, R.

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay, and E. Boily, “Liquid mirrors: optical shop tests and contributions to the technology,” Astrophys. J. 393, 829–847 (1992)
[Crossref]

Conti, C.

D,

Dholakia, K.

D. McGloin, V. Garces-Chavez, D, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28, 657–659 (2003).
[Crossref] [PubMed]

V. Garces-Chavez, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Ding, Z.

Durnin, J.

J. Durnin, J.J. Miceli, and J.H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1986).
[Crossref]

Eberly, J.H.

J. Durnin, J.J. Miceli, and J.H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Flatt, S.

Florjanczyk, M.

M. Florjanczyk and R. Tremblay, “Guiding of atoms in a travelling-wave laser traps formed by the axicon,” Opt. Commun. 73, 448–451 (1989).
[Crossref]

Friberg, A. T.

Fujii, E.

Fujiwara, S.

S. Fujiwara, “Optical properties of conic surfaces. I. Reflecting cone,” J. Opt. Soc. 52, 287–292 (1962).
[Crossref]

Gadonas, G.

G. Gadonas, V. Jarutis, R. Paskauskas, V. Smilgevicius, A. Stabinis, and V. Vaicaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun. 196, 309–316 (2001).
[Crossref]

Garces-Chavez, V.

D. McGloin, V. Garces-Chavez, D, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28, 657–659 (2003).
[Crossref] [PubMed]

V. Garces-Chavez, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Girard, L.

Herman, R.M.

Hickmann, J. M.

Hickson, P.

P. Hickson and M. Mulrooney, University of British Columbia-NASA Multi-Narrowband Survey. I. Description and photometric properties of the survey,” Astrophys. J. Suppl. 115, 35–42 (1998).
[Crossref]

Jarutis, V.

G. Gadonas, V. Jarutis, R. Paskauskas, V. Smilgevicius, A. Stabinis, and V. Vaicaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun. 196, 309–316 (2001).
[Crossref]

Juodkazis, S.

A. Marcinkevicius, S. Juodkazis, V. Mizeikis, S. Matsuo, and H. Misawa, “Application of femtosecond Bessel-Gauss beam in microstructuring of transparent materials,” Proc. SPIE 4271, 150–157 (2001).
[Crossref]

Kenney, C.S.

Lanz, T.

Magni, V.

V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. 184, 245–255 (2000).
[Crossref]

Maier, M.

Marcinkevicius, A.

A. Marcinkevicius, S. Juodkazis, V. Mizeikis, S. Matsuo, and H. Misawa, “Application of femtosecond Bessel-Gauss beam in microstructuring of transparent materials,” Proc. SPIE 4271, 150–157 (2001).
[Crossref]

Matsuo, S.

A. Marcinkevicius, S. Juodkazis, V. Mizeikis, S. Matsuo, and H. Misawa, “Application of femtosecond Bessel-Gauss beam in microstructuring of transparent materials,” Proc. SPIE 4271, 150–157 (2001).
[Crossref]

McGloin, D.

Melville, H.

V. Garces-Chavez, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Meneses-Nava, M.A.

Miceli, J.J.

J. Durnin, J.J. Miceli, and J.H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Misawa, H.

A. Marcinkevicius, S. Juodkazis, V. Mizeikis, S. Matsuo, and H. Misawa, “Application of femtosecond Bessel-Gauss beam in microstructuring of transparent materials,” Proc. SPIE 4271, 150–157 (2001).
[Crossref]

Mizeikis, V.

A. Marcinkevicius, S. Juodkazis, V. Mizeikis, S. Matsuo, and H. Misawa, “Application of femtosecond Bessel-Gauss beam in microstructuring of transparent materials,” Proc. SPIE 4271, 150–157 (2001).
[Crossref]

Montgomery, W. D.

Mulrooney, M.

P. Hickson and M. Mulrooney, University of British Columbia-NASA Multi-Narrowband Survey. I. Description and photometric properties of the survey,” Astrophys. J. Suppl. 115, 35–42 (1998).
[Crossref]

Nelson, J. S.

Niggel, L.

Overfelt, P.L.

Paskauskas, R.

G. Gadonas, V. Jarutis, R. Paskauskas, V. Smilgevicius, A. Stabinis, and V. Vaicaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun. 196, 309–316 (2001).
[Crossref]

Piché, et M.

B. Bélanger, X. Zhu, and et M. Piché, “Auto-imagerie à l’aide de faisceaux non-diffractants,” Phys. Can. 52, 160–161 (1996).

Ren, H.

Sargoytchev, S.

Sarrow, C.T.

Sawaki, D.

Sibbett, W.

V. Garces-Chavez, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Sica, R. J.

Siegman, A. E.

A. E. Siegman, Lasers, chapters 15 and 20. (University Science Books, Mill Valley, 1986).

Smilgevicius, V.

G. Gadonas, V. Jarutis, R. Paskauskas, V. Smilgevicius, A. Stabinis, and V. Vaicaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun. 196, 309–316 (2001).
[Crossref]

Stabinis, A.

G. Gadonas, V. Jarutis, R. Paskauskas, V. Smilgevicius, A. Stabinis, and V. Vaicaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun. 196, 309–316 (2001).
[Crossref]

Szapiel, S.

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay, and E. Boily, “Liquid mirrors: optical shop tests and contributions to the technology,” Astrophys. J. 393, 829–847 (1992)
[Crossref]

Tremblay, G.

Tremblay, L. M.

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay, and E. Boily, “Liquid mirrors: optical shop tests and contributions to the technology,” Astrophys. J. 393, 829–847 (1992)
[Crossref]

Tremblay, R.

M. Florjanczyk and R. Tremblay, “Guiding of atoms in a travelling-wave laser traps formed by the axicon,” Opt. Commun. 73, 448–451 (1989).
[Crossref]

Trillo, S.

Turunen, J.

Vaicaitis, V.

G. Gadonas, V. Jarutis, R. Paskauskas, V. Smilgevicius, A. Stabinis, and V. Vaicaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun. 196, 309–316 (2001).
[Crossref]

Vasara, A.

Wiggins, T.A.

Zhao, Y.

Zhu, X.

B. Bélanger, X. Zhu, and et M. Piché, “Auto-imagerie à l’aide de faisceaux non-diffractants,” Phys. Can. 52, 160–161 (1996).

Appl. Opt. (4)

Astron. Soc. Can. (1)

E. F. Borra, “The liquid-mirror telescope as a viable astronomical tool,” Astron. Soc. Can. 76, 245–256 (1982).

Astrophys. J. (2)

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay, and E. Boily, “Liquid mirrors: optical shop tests and contributions to the technology,” Astrophys. J. 393, 829–847 (1992)
[Crossref]

R. Cabanac, E. F. Borra, and M. Beauchemin, “A search for peculiar objects with the NASA orbital debris observatory 3-m liquid mirror Telescope,” Astrophys. J. 509, 309–323, (1998).
[Crossref]

Astrophys. J. Suppl. (1)

P. Hickson and M. Mulrooney, University of British Columbia-NASA Multi-Narrowband Survey. I. Description and photometric properties of the survey,” Astrophys. J. Suppl. 115, 35–42 (1998).
[Crossref]

Can. J. Phys. (1)

E. F. Borra, “Liquid mirrors,” Can. J. Phys. 73, 109–125 (1995).
[Crossref]

J. Opt. Soc. (1)

S. Fujiwara, “Optical properties of conic surfaces. I. Reflecting cone,” J. Opt. Soc. 52, 287–292 (1962).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

Nature (1)

V. Garces-Chavez, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Opt. Commun. (3)

V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. 184, 245–255 (2000).
[Crossref]

G. Gadonas, V. Jarutis, R. Paskauskas, V. Smilgevicius, A. Stabinis, and V. Vaicaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun. 196, 309–316 (2001).
[Crossref]

M. Florjanczyk and R. Tremblay, “Guiding of atoms in a travelling-wave laser traps formed by the axicon,” Opt. Commun. 73, 448–451 (1989).
[Crossref]

Opt. Lett. (4)

Phys. Can. (1)

B. Bélanger, X. Zhu, and et M. Piché, “Auto-imagerie à l’aide de faisceaux non-diffractants,” Phys. Can. 52, 160–161 (1996).

Phys. Rev. Lett. (1)

J. Durnin, J.J. Miceli, and J.H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Proc. SPIE (1)

A. Marcinkevicius, S. Juodkazis, V. Mizeikis, S. Matsuo, and H. Misawa, “Application of femtosecond Bessel-Gauss beam in microstructuring of transparent materials,” Proc. SPIE 4271, 150–157 (2001).
[Crossref]

Other (1)

A. E. Siegman, Lasers, chapters 15 and 20. (University Science Books, Mill Valley, 1986).

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Figures (7)

Fig 1.
Fig 1.

(a) Single Bessel beam generated by using a mask with one thin annular slit. (b) Constructive interference of two Bessel beams produced by using a mask with two concentric annular slits. (c) Destructive interference of two Bessel beams produced by using a mask with two concentric annular slits. (d) Off-axis Bessel beam generated by an incident beam with a tilted wavefront.

Fig. 2.
Fig. 2.

Coherent superposition of two Bessel beams. Upper left: intensity at the center of the interference pattern produced with a mask having two annular slits, as a function of the curvature of the beam incident on the mask. Lower left: interference pattern for the three values of curvature indicated above by points i, ii and iii. Upper right: intensity at the center of the interference pattern produced with a mask having one annular slit and one hole at center. Lower right: interference pattern for the three values of curvature indicated above by points iv, v and vi.

Fig. 3.
Fig. 3.

Experimental setup used for the characterization of the curvature of a liquid mirror based on Bessel beam interferometry.

Fig. 4.
Fig. 4.

Variation of the intensity at the center of the Bessel beam interference pattern as a function of the radius of curvature of the liquid mirror. Experimental data points (crosses) were obtained with the mask having two annular slits. The continuous curve is the theoretical prediction. Measurements were made in varying the value of Rm by steps of 200 µm.

Fig. 5.
Fig. 5.

Interference patterns observed with a two-slit mask, for values of the radius of curvature of the liquid mirror Rm between 1.9996 m and 2.0167 m.

Fig. 6.
Fig. 6.

Cardioid motion of the center of curvature of the liquid mirror during one revolution. Eight measurements were made, by angular steps of 45 degrees.

Fig. 7.
Fig. 7.

Interference patterns produced by the superposition of a Bessel beam and an Airy pattern, with R1=15 cm, for different phases during a revolution of the liquid mirror.

Equations (2)

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E tot ( r , z = 2 f ) n = 1 N A n exp ( i k d n 2 4 R 1 ) J 0 ( α n r )
I tot ( r = 0 , z = 2 f ) B 1 2 + B 2 2 + 2 B 1 B 2 cos ( k ( d 2 2 d 1 2 ) 8 R 1 )

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