Abstract

Physical constraints and peculiarities of the wavefront sensing technique, based on the Talbot effect, are discussed. The limitation on the curvature of the measurable wavefront is derived. The requirements to the Fourier spectrum of the periodic mask are formulated. Two kinds of masks are studied for their performance in the wavefront sensor. It is shown that the boundary part of the mask aperture does not contribute to the initial data for wavefront restoration. It is verified by experiment and computer simulation that the performance of the Talbot sensor, which meets established conditions, is similar to that of the Shack–Hartmann sensor.

© 2014 Optical Society of America

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References

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  2. M. Rocktäschel and H. J. Tiziani, “Limitations of the Shack-Hartmann sensor for testing optical aspherics,” Opt. Laser Technol. 34, 631–637 (2002).
    [CrossRef]
  3. A. A. Goloborodko, V. I. Grygoruk, M. M. Kotov, V. N. Kurashov, D. V. Podanchuk, and N. S. Sutyagina, “Determination of local surface defects using a Shack-Hartmann wavefront sensor,” Ukr. J. Phys. 53, 946–951 (2008).
  4. O. Azucena, J. Crest, S. Kotadia, W. Sullivan, X. Tao, M. Reinig, D. Gavel, S. Olivier, and J. Kubby, “Adaptive optics wide-field microscopy using direct wavefront sensing,” Opt. Lett. 36, 825–827 (2011).
    [CrossRef]
  5. V. V. Molebny, V. N. Kurashov, D. V. Podanchuk, A. V. Kovalenko, I. G. Pallikaris, and L. P. Naoumidis, “Aberration mapping for sight correction,” Proc. SPIE 3246, 238–248 (1998).
    [CrossRef]
  6. S. Manzanera, C. Canovas, P. M. Prieto, and P. Artal, “A wavelength tunable wavefront sensor for the human eye,” Opt. Express 16, 7748–7755 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. C. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
    [CrossRef]
  12. R. Sekine, T. Shibuya, K. Ukai, S. Komatsu, M. Hattori, T. Mihashi, N. Nakazawa, and Y. Hirihara, “Measurement of wavefront aberration of human eye using Talbot image of two-dimensional grating,” Opt. Rev. 13, 207–211 (2006).
    [CrossRef]
  13. D. Podanchuk, V. Kurashov, A. Goloborodko, V. Dan’ko, M. Kotov, and N. Goloborodko, “Wavefront sensor based on the Talbot effect with the precorrected holographic grating,” Appl. Opt. 51, C125–C132 (2012).
    [CrossRef]
  14. D. V. Podanchuk, A. A. Goloborodko, and M. M. Kotov, “Features of the wavefront sensor based on the Talbot effect,” in Proceedings of the International Conference on Advanced Optoelectronics & Lasers (CAOL), O. V. Shulika and I. A. Sukhoivanov, eds. (IEEE, 2013), pp. 337–339.
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    [CrossRef]
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  19. S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
    [CrossRef]
  20. L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24, 569–577 (2007).
    [CrossRef]
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    [CrossRef]

2013 (1)

2012 (1)

2011 (1)

2010 (1)

2008 (2)

A. A. Goloborodko, V. I. Grygoruk, M. M. Kotov, V. N. Kurashov, D. V. Podanchuk, and N. S. Sutyagina, “Determination of local surface defects using a Shack-Hartmann wavefront sensor,” Ukr. J. Phys. 53, 946–951 (2008).

S. Manzanera, C. Canovas, P. M. Prieto, and P. Artal, “A wavelength tunable wavefront sensor for the human eye,” Opt. Express 16, 7748–7755 (2008).
[CrossRef]

2007 (1)

2006 (3)

R. Sekine, T. Shibuya, K. Ukai, S. Komatsu, M. Hattori, T. Mihashi, N. Nakazawa, and Y. Hirihara, “Measurement of wavefront aberration of human eye using Talbot image of two-dimensional grating,” Opt. Rev. 13, 207–211 (2006).
[CrossRef]

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

D. V. Podanchuk, V. P. Danko, M. M. Kotov, J.-Y. Son, and Y.-J. Choi, “Extended-range Shack-Hartmann wavefront sensor with nonlinear holographic lenslet array,” Opt. Eng. 45, 053605 (2006).
[CrossRef]

2005 (1)

J. F. Barrera, R. Henao, Z. Jaroszewicz, and A. Kolodziejczyk, “Talbot effect for periodical objects limited by finite apertures: a new interpretation,” Optik 116, 144–148 (2005).
[CrossRef]

2003 (1)

2002 (1)

M. Rocktäschel and H. J. Tiziani, “Limitations of the Shack-Hartmann sensor for testing optical aspherics,” Opt. Laser Technol. 34, 631–637 (2002).
[CrossRef]

2001 (1)

C. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

1999 (1)

N. Salama, D. Patrignani, L. De Pasquale, and E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
[CrossRef]

1998 (1)

V. V. Molebny, V. N. Kurashov, D. V. Podanchuk, A. V. Kovalenko, I. G. Pallikaris, and L. P. Naoumidis, “Aberration mapping for sight correction,” Proc. SPIE 3246, 238–248 (1998).
[CrossRef]

1992 (1)

Artal, P.

Azucena, O.

Balmer, J. E.

C. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

Barrera, J. F.

J. F. Barrera, R. Henao, Z. Jaroszewicz, and A. Kolodziejczyk, “Talbot effect for periodical objects limited by finite apertures: a new interpretation,” Optik 116, 144–148 (2005).
[CrossRef]

Burge, J. H.

Campbell, C. E.

Canovas, C.

Choi, Y.-J.

D. V. Podanchuk, V. P. Danko, M. M. Kotov, J.-Y. Son, and Y.-J. Choi, “Extended-range Shack-Hartmann wavefront sensor with nonlinear holographic lenslet array,” Opt. Eng. 45, 053605 (2006).
[CrossRef]

Crest, J.

Crouse, R. F.

Dan’ko, V.

Danko, V. P.

D. V. Podanchuk, V. P. Danko, M. M. Kotov, J.-Y. Son, and Y.-J. Choi, “Extended-range Shack-Hartmann wavefront sensor with nonlinear holographic lenslet array,” Opt. Eng. 45, 053605 (2006).
[CrossRef]

De Pasquale, L.

N. Salama, D. Patrignani, L. De Pasquale, and E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
[CrossRef]

Fusco, T.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Gavel, D.

Goloborodko, A.

Goloborodko, A. A.

A. A. Goloborodko, V. I. Grygoruk, M. M. Kotov, V. N. Kurashov, D. V. Podanchuk, and N. S. Sutyagina, “Determination of local surface defects using a Shack-Hartmann wavefront sensor,” Ukr. J. Phys. 53, 946–951 (2008).

D. V. Podanchuk, A. A. Goloborodko, and M. M. Kotov, “Features of the wavefront sensor based on the Talbot effect,” in Proceedings of the International Conference on Advanced Optoelectronics & Lasers (CAOL), O. V. Shulika and I. A. Sukhoivanov, eds. (IEEE, 2013), pp. 337–339.

Goloborodko, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Grygoruk, V. I.

A. A. Goloborodko, V. I. Grygoruk, M. M. Kotov, V. N. Kurashov, D. V. Podanchuk, and N. S. Sutyagina, “Determination of local surface defects using a Shack-Hartmann wavefront sensor,” Ukr. J. Phys. 53, 946–951 (2008).

Hattori, M.

R. Sekine, T. Shibuya, K. Ukai, S. Komatsu, M. Hattori, T. Mihashi, N. Nakazawa, and Y. Hirihara, “Measurement of wavefront aberration of human eye using Talbot image of two-dimensional grating,” Opt. Rev. 13, 207–211 (2006).
[CrossRef]

Henao, R.

J. F. Barrera, R. Henao, Z. Jaroszewicz, and A. Kolodziejczyk, “Talbot effect for periodical objects limited by finite apertures: a new interpretation,” Optik 116, 144–148 (2005).
[CrossRef]

Hirihara, Y.

R. Sekine, T. Shibuya, K. Ukai, S. Komatsu, M. Hattori, T. Mihashi, N. Nakazawa, and Y. Hirihara, “Measurement of wavefront aberration of human eye using Talbot image of two-dimensional grating,” Opt. Rev. 13, 207–211 (2006).
[CrossRef]

Jaroszewicz, Z.

J. F. Barrera, R. Henao, Z. Jaroszewicz, and A. Kolodziejczyk, “Talbot effect for periodical objects limited by finite apertures: a new interpretation,” Optik 116, 144–148 (2005).
[CrossRef]

Kolodziejczyk, A.

J. F. Barrera, R. Henao, Z. Jaroszewicz, and A. Kolodziejczyk, “Talbot effect for periodical objects limited by finite apertures: a new interpretation,” Optik 116, 144–148 (2005).
[CrossRef]

Komatsu, S.

R. Sekine, T. Shibuya, K. Ukai, S. Komatsu, M. Hattori, T. Mihashi, N. Nakazawa, and Y. Hirihara, “Measurement of wavefront aberration of human eye using Talbot image of two-dimensional grating,” Opt. Rev. 13, 207–211 (2006).
[CrossRef]

Kotadia, S.

Kotov, M.

Kotov, M. M.

A. A. Goloborodko, V. I. Grygoruk, M. M. Kotov, V. N. Kurashov, D. V. Podanchuk, and N. S. Sutyagina, “Determination of local surface defects using a Shack-Hartmann wavefront sensor,” Ukr. J. Phys. 53, 946–951 (2008).

D. V. Podanchuk, V. P. Danko, M. M. Kotov, J.-Y. Son, and Y.-J. Choi, “Extended-range Shack-Hartmann wavefront sensor with nonlinear holographic lenslet array,” Opt. Eng. 45, 053605 (2006).
[CrossRef]

D. V. Podanchuk, A. A. Goloborodko, and M. M. Kotov, “Features of the wavefront sensor based on the Talbot effect,” in Proceedings of the International Conference on Advanced Optoelectronics & Lasers (CAOL), O. V. Shulika and I. A. Sukhoivanov, eds. (IEEE, 2013), pp. 337–339.

Kovalenko, A. V.

V. V. Molebny, V. N. Kurashov, D. V. Podanchuk, A. V. Kovalenko, I. G. Pallikaris, and L. P. Naoumidis, “Aberration mapping for sight correction,” Proc. SPIE 3246, 238–248 (1998).
[CrossRef]

Kubby, J.

Kurashov, V.

Kurashov, V. N.

A. A. Goloborodko, V. I. Grygoruk, M. M. Kotov, V. N. Kurashov, D. V. Podanchuk, and N. S. Sutyagina, “Determination of local surface defects using a Shack-Hartmann wavefront sensor,” Ukr. J. Phys. 53, 946–951 (2008).

V. V. Molebny, V. N. Kurashov, D. V. Podanchuk, A. V. Kovalenko, I. G. Pallikaris, and L. P. Naoumidis, “Aberration mapping for sight correction,” Proc. SPIE 3246, 238–248 (1998).
[CrossRef]

Latimer, P.

Li, F.

Loewenthal, F.

C. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

Lundström, L.

Manzanera, S.

Michau, V.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Mihashi, T.

R. Sekine, T. Shibuya, K. Ukai, S. Komatsu, M. Hattori, T. Mihashi, N. Nakazawa, and Y. Hirihara, “Measurement of wavefront aberration of human eye using Talbot image of two-dimensional grating,” Opt. Rev. 13, 207–211 (2006).
[CrossRef]

Molebny, V. V.

V. V. Molebny, V. N. Kurashov, D. V. Podanchuk, A. V. Kovalenko, I. G. Pallikaris, and L. P. Naoumidis, “Aberration mapping for sight correction,” Proc. SPIE 3246, 238–248 (1998).
[CrossRef]

Nakazawa, N.

R. Sekine, T. Shibuya, K. Ukai, S. Komatsu, M. Hattori, T. Mihashi, N. Nakazawa, and Y. Hirihara, “Measurement of wavefront aberration of human eye using Talbot image of two-dimensional grating,” Opt. Rev. 13, 207–211 (2006).
[CrossRef]

Naoumidis, L. P.

V. V. Molebny, V. N. Kurashov, D. V. Podanchuk, A. V. Kovalenko, I. G. Pallikaris, and L. P. Naoumidis, “Aberration mapping for sight correction,” Proc. SPIE 3246, 238–248 (1998).
[CrossRef]

Nicolle, M.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Olivier, S.

Pallikaris, I. G.

V. V. Molebny, V. N. Kurashov, D. V. Podanchuk, A. V. Kovalenko, I. G. Pallikaris, and L. P. Naoumidis, “Aberration mapping for sight correction,” Proc. SPIE 3246, 238–248 (1998).
[CrossRef]

Patrignani, D.

N. Salama, D. Patrignani, L. De Pasquale, and E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
[CrossRef]

Podanchuk, D.

Podanchuk, D. V.

A. A. Goloborodko, V. I. Grygoruk, M. M. Kotov, V. N. Kurashov, D. V. Podanchuk, and N. S. Sutyagina, “Determination of local surface defects using a Shack-Hartmann wavefront sensor,” Ukr. J. Phys. 53, 946–951 (2008).

D. V. Podanchuk, V. P. Danko, M. M. Kotov, J.-Y. Son, and Y.-J. Choi, “Extended-range Shack-Hartmann wavefront sensor with nonlinear holographic lenslet array,” Opt. Eng. 45, 053605 (2006).
[CrossRef]

V. V. Molebny, V. N. Kurashov, D. V. Podanchuk, A. V. Kovalenko, I. G. Pallikaris, and L. P. Naoumidis, “Aberration mapping for sight correction,” Proc. SPIE 3246, 238–248 (1998).
[CrossRef]

D. V. Podanchuk, A. A. Goloborodko, and M. M. Kotov, “Features of the wavefront sensor based on the Talbot effect,” in Proceedings of the International Conference on Advanced Optoelectronics & Lasers (CAOL), O. V. Shulika and I. A. Sukhoivanov, eds. (IEEE, 2013), pp. 337–339.

Prieto, P. M.

Reinig, M.

Rocktäschel, M.

M. Rocktäschel and H. J. Tiziani, “Limitations of the Shack-Hartmann sensor for testing optical aspherics,” Opt. Laser Technol. 34, 631–637 (2002).
[CrossRef]

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

Rousset, G.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Salama, N.

N. Salama, D. Patrignani, L. De Pasquale, and E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
[CrossRef]

Sekine, R.

R. Sekine, T. Shibuya, K. Ukai, S. Komatsu, M. Hattori, T. Mihashi, N. Nakazawa, and Y. Hirihara, “Measurement of wavefront aberration of human eye using Talbot image of two-dimensional grating,” Opt. Rev. 13, 207–211 (2006).
[CrossRef]

Shibuya, T.

R. Sekine, T. Shibuya, K. Ukai, S. Komatsu, M. Hattori, T. Mihashi, N. Nakazawa, and Y. Hirihara, “Measurement of wavefront aberration of human eye using Talbot image of two-dimensional grating,” Opt. Rev. 13, 207–211 (2006).
[CrossRef]

Sicre, E.

N. Salama, D. Patrignani, L. De Pasquale, and E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
[CrossRef]

Siegel, C.

C. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

Son, J.-Y.

D. V. Podanchuk, V. P. Danko, M. M. Kotov, J.-Y. Son, and Y.-J. Choi, “Extended-range Shack-Hartmann wavefront sensor with nonlinear holographic lenslet array,” Opt. Eng. 45, 053605 (2006).
[CrossRef]

Sullivan, W.

Sutyagina, N. S.

A. A. Goloborodko, V. I. Grygoruk, M. M. Kotov, V. N. Kurashov, D. V. Podanchuk, and N. S. Sutyagina, “Determination of local surface defects using a Shack-Hartmann wavefront sensor,” Ukr. J. Phys. 53, 946–951 (2008).

Tao, X.

Teng, S.

Thomas, S.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Tiziani, H. J.

M. Rocktäschel and H. J. Tiziani, “Limitations of the Shack-Hartmann sensor for testing optical aspherics,” Opt. Laser Technol. 34, 631–637 (2002).
[CrossRef]

Tokovinin, A.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Tyson, R.

R. Tyson, Principles of Adaptive Optics, 3rd ed. (CRC Press, 2010).

Ukai, K.

R. Sekine, T. Shibuya, K. Ukai, S. Komatsu, M. Hattori, T. Mihashi, N. Nakazawa, and Y. Hirihara, “Measurement of wavefront aberration of human eye using Talbot image of two-dimensional grating,” Opt. Rev. 13, 207–211 (2006).
[CrossRef]

Unsbo, P.

Wang, J.

Zhang, C.

Zhang, W.

Zhou, P.

Appl. Opt. (4)

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

Mon. Not. R. Astron. Soc. (1)

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Opt. Commun. (1)

C. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

Opt. Eng. (1)

D. V. Podanchuk, V. P. Danko, M. M. Kotov, J.-Y. Son, and Y.-J. Choi, “Extended-range Shack-Hartmann wavefront sensor with nonlinear holographic lenslet array,” Opt. Eng. 45, 053605 (2006).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (2)

M. Rocktäschel and H. J. Tiziani, “Limitations of the Shack-Hartmann sensor for testing optical aspherics,” Opt. Laser Technol. 34, 631–637 (2002).
[CrossRef]

N. Salama, D. Patrignani, L. De Pasquale, and E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
[CrossRef]

Opt. Lett. (1)

Opt. Rev. (1)

R. Sekine, T. Shibuya, K. Ukai, S. Komatsu, M. Hattori, T. Mihashi, N. Nakazawa, and Y. Hirihara, “Measurement of wavefront aberration of human eye using Talbot image of two-dimensional grating,” Opt. Rev. 13, 207–211 (2006).
[CrossRef]

Optik (1)

J. F. Barrera, R. Henao, Z. Jaroszewicz, and A. Kolodziejczyk, “Talbot effect for periodical objects limited by finite apertures: a new interpretation,” Optik 116, 144–148 (2005).
[CrossRef]

Proc. SPIE (1)

V. V. Molebny, V. N. Kurashov, D. V. Podanchuk, A. V. Kovalenko, I. G. Pallikaris, and L. P. Naoumidis, “Aberration mapping for sight correction,” Proc. SPIE 3246, 238–248 (1998).
[CrossRef]

Ukr. J. Phys. (1)

A. A. Goloborodko, V. I. Grygoruk, M. M. Kotov, V. N. Kurashov, D. V. Podanchuk, and N. S. Sutyagina, “Determination of local surface defects using a Shack-Hartmann wavefront sensor,” Ukr. J. Phys. 53, 946–951 (2008).

Other (4)

R. Tyson, Principles of Adaptive Optics, 3rd ed. (CRC Press, 2010).

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

D. V. Podanchuk, A. A. Goloborodko, and M. M. Kotov, “Features of the wavefront sensor based on the Talbot effect,” in Proceedings of the International Conference on Advanced Optoelectronics & Lasers (CAOL), O. V. Shulika and I. A. Sukhoivanov, eds. (IEEE, 2013), pp. 337–339.

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Figures (9)

Fig. 1.
Fig. 1.

Fragments of the diffraction image in the Talbot plane for gratings with different profile shape: (a) binary and (b) cosine. Incident wavefronts are plane, spherical (C2;0=4; a=3.5mm), or astigmatic (C2;2=4; a=3.5mm).

Fig. 2.
Fig. 2.

Dependence of the wavefront reconstruction error on the normalized curvature of the incident wavefront with defocus for cosine (1) and binary (2) gratings. The intensity threshold is chosen equal to 100 for the cosine grating and 30 for the binary one.

Fig. 3.
Fig. 3.

Dependence of the error of the wavefront reconstruction on the normalized curvature of the incident astigmatic wavefront for cosine (1,2) and binary (3) gratings. The threshold equals 100 for curve (2) and 30 for curves (1,3).

Fig. 4.
Fig. 4.

Deviation of the estimated defocus coefficient from the initial value (3.24 rad) for the gratings with different aperture.

Fig. 5.
Fig. 5.

Experimental setup for testing the sensor based on the Talbot effect.

Fig. 6.
Fig. 6.

Dependence of the defocus coefficient (C2;0) (normalized to the 1 mm radius circle) of the spherical wavefront on the power (D) of the test lens. Data were received by the Shack–Hartmann sensor and the Talbot sensor with the cosine grating (○) and the grating with square holes (×).

Fig. 7.
Fig. 7.

Dependence of the wavefront reconstruction error on a size of the grating aperture (in number of periods N) for (a) cosine grating and (b) grating with square holes. The normalized wavefront curvature zT/R is 0.16.

Fig. 8.
Fig. 8.

Fragments of the grating image in the Talbot plane: the grating has (a) 25×25 or (b) 9×9 square holes. Dotted frames show the subarea of wavefront reconstruction.

Fig. 9.
Fig. 9.

Zernike coefficients obtained by the Talbot sensor with different size of the grating for two tested astigmatic wavefronts. The lines show the reference values obtained by the Shack–Hartmann sensor.

Equations (28)

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u(x,y,0)=uin(x,y,0)τ(x,y)=n,mqn,mej2πd(nx+my)j2πd(nx+my)usph(x,y,0),
usph(x,y,z)=14π(z+R)ej2πλ(z+R)ejπ(x2+y2)λ(z+R).
PSF(x,y;z)=1jλzej2πzλejπ(x2+y2)λz,
u(x,y,z)=usph(x,y,z)n,mqn,m×ej2πRd(z+R)(nx+my)ejπλzRd2(z+R)(n2+m2).
Zl=zTl(1lzTR)1,l=1,2,,
u(x,y,Zl)=usph(x,y,Zl)τ(RxZl+R,RyZl+R).
I(x,y,Zl)=|u(x,y,Zl)|22=|usph|22|τ(xs,ys)|2.
|ZlzT|=|zT(1zTR)1zT|zT2|R|zT,
u(x,y,zT)=usph(x,y,zT)n,mqn,m×ej2πRd(zT+R)(nx+my)ej2πzTR(n2+m2).
uin(x,y,0)=ejφ(x,y).
φ(x,y)=φ0+2πfx,0x+2πfy,0y+ψ(x,y),
uin,ψ(x,y,0)=ejψ(x,y).
uψ(x,y,zT)=τ(ξ,η)uin,ψ(ξ,η,0)×PSF(xξ,yη;zT)dξdη.
uψ(x,y,zT)=ej2πzTλjλzTτ(ξ,η)ejΦ(ξ,η;x,y)dξdη,
Φ(ξ,η;x,y)=ψ(ξ,η)+π(xξ)2+(yη)22d2.
ψ(ξ,η)ψ(x,y)+ψξ(x,y)(ξx)+ψη(x,y)(ηy)+12(ψξξ(x,y)(ξx)2+ψηη(x,y)(ηy)2).
|R1(x,y)|,|R2(x,y)|zT,
1R1(x,y)=λ2πψξξ(x,y),1R2(x,y)=λ2πψηη(x,y).
uψ(x,y,zT)τ(ξ,η)ej(ξψξ(x,y)+ηψη(x,y))×PSF(xξ,yη;zT)dξdη,
Iψ(x,y,zT)|τ(xψx(x,y)λzT2π,yψy(x,y)λzT2π)|2.
I(x,y,zT)=Const|τ(xφx(x,y)λzT2π,yφy(x,y)λzT2π)|2.
τ(x,y)=m,n=(N1)/2(N1)/2t(xnd,ymd),
t(1)(x,y)={1;|x|,|y|b20;otherwise,t(2)(x,y)={cos2(πxd)cos2(πyd);|x|,|y|d20;otherwise.
φ(x,y)=C2;mZ2;m(2xa,2ya).
Z2;0=X2+Y2,Z2;2=X2Y2.
R1=4λπa2C2;m.
σΔφ2=|ΔφΔϕ|2,
Δφ(x,y)=φ(x,y)φ(x,y),Δϕ(x,y)=ϕ(x,y)ϕ(x,y).

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