Abstract

This paper proposes a modified Shack–Hartmann wavefront sensor for phase retrieval. The sensor is revamped by placing a detector at a defocused plane before the focal plane of the lenslet array of the Shack–Hartmann sensor. The algorithm for phase retrieval is an optimization with initial Zernike coefficients calculated by the conventional phase reconstruction of the Shack–Hartmann sensor. Numerical simulations show that the proposed sensor permits sensitive, accurate phase retrieval. Furthermore, experiments tested the feasibility of phase retrieval using the proposed sensor. The surface irregularity for a flat mirror was measured by the proposed method and a Veeco interferometer, respectively. The irregularity for the mirror measured by the proposed method is in very good agreement with that measured using the Veeco interferometer.

© 2014 Optical Society of America

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  1. B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).
  2. J. Gonglewski, D. Voelz, J. Fender, D. Dayton, B. Spielbusch, and R. Pierson, “First astronomical application of postdetection turbulence compensation: images of α Aurigae, ν Ursae Majoris, and α Geminorum using self-referenced speckle holography,” Appl. Opt. 29, 4527–4529 (1990).
    [CrossRef]
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    [CrossRef]
  4. J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejer, R. L. Byer, D. Clubley, S. Yoshida, and D. H. Reitze, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack–Hartmann wave-front sensor,” Appl. Opt. 40, 366–374 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. J. Lee, R. V. Shack, and M. R. Descour, “Sorting method to extend the dynamic range of the Shack–Hartmann wavefront sensor,” Appl. Opt. 44, 4838–4845 (2005).
    [CrossRef]
  8. G. Yoon, S. Pantanelli, and L. J. Nagy, “Large-dynamic-range Shack–Hartmann wavefront sensor for highly aberrated eyes,” J. Biomed. Opt. 11, 030502 (2006).
    [CrossRef]
  9. 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]
  10. J. Ares and J. Arines, “Influence of thresholding on centroid statistics: full analytical description,” Appl. Opt. 43, 5796–5805 (2004).
    [CrossRef]
  11. R. Irwan and R. G. Lane, “Analysis of optimal centroid estimation applied to Shack–Hartmann sensing,” Appl. Opt. 38, 6737–6743 (1999).
    [CrossRef]
  12. G. Cao and X. Yu, “Accuracy analysis of a Hartmann–Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994).
    [CrossRef]
  13. S. Ríos and D. López, “Modified Shack–Hartmann wavefront sensor using an array of superresolution pupil filters,” Opt. Express 17, 9669–9679 (2009).
    [CrossRef]
  14. D. López and S. Ríos, “Interferometric Shack–Hartmann wavefront sensor with an array of four-hole apertures,” Appl. Opt. 49, 2334–2338 (2010).
    [CrossRef]
  15. A. Polo, V. Kutchoukov, F. Bociort, S. Pereira, and H. Urbach, “Determination of wavefront structure for a Hartmann wavefront sensor using a phase-retrieval method,” Opt. Express 20, 7822–7832 (2012).
    [CrossRef]
  16. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
    [CrossRef]
  17. B. H. Dean and C. W. Bowers, “Diversity selection for phase-diverse phase retrieval,” J. Opt. Soc. Am. A 20, 1490–1504 (2003).
    [CrossRef]
  18. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).
  19. M. Vorontsov and V. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. 15, 2745–2758 (1998).
    [CrossRef]
  20. M. Vorontsov, G. Carhart, and J. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997).
    [CrossRef]
  21. G. Cauwenberghs, “A fast stochastic error-descent algorithm for supervised learning and optimization,” in Advances in Neural Information Processing Systems, S. J. Hanson, J. D. Cowan, and C. L. Giles, eds., Vol. 5 (Morgan Kaufman, 1993), pp. 244–251.

2012 (1)

2010 (1)

2009 (1)

2006 (2)

G. Yoon, S. Pantanelli, and L. J. Nagy, “Large-dynamic-range Shack–Hartmann wavefront sensor for highly aberrated eyes,” J. Biomed. Opt. 11, 030502 (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]

2005 (1)

2004 (1)

2003 (1)

2001 (2)

2000 (2)

1999 (1)

1998 (2)

G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

M. Vorontsov and V. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. 15, 2745–2758 (1998).
[CrossRef]

1997 (1)

1994 (1)

G. Cao and X. Yu, “Accuracy analysis of a Hartmann–Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994).
[CrossRef]

1990 (1)

1982 (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[CrossRef]

Ares, J.

Arines, J.

Bliss, E. S.

Bociort, F.

Bowers, C. W.

Byer, R. L.

Cao, G.

G. Cao and X. Yu, “Accuracy analysis of a Hartmann–Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994).
[CrossRef]

Carhart, G.

Cauwenberghs, G.

G. Cauwenberghs, “A fast stochastic error-descent algorithm for supervised learning and optimization,” in Advances in Neural Information Processing Systems, S. J. Hanson, J. D. Cowan, and C. L. Giles, eds., Vol. 5 (Morgan Kaufman, 1993), pp. 244–251.

Chanan, G.

Clubley, D.

Dailey, M. J.

Dayton, D.

Dean, B. H.

Dekens, F.

Descour, M. R.

Fejer, M. M.

Feldman, M.

Fender, J.

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]

Gonglewski, J.

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Grey, A. A.

Gustafson, E. K.

Hennawi, J.

Holdener, F. R.

Irwan, R.

Kirkman, D.

Koch, J. A.

Kutchoukov, V.

Lane, R. G.

Lee, J.

López, D.

Mansell, J. D.

Mast, T.

Michaels, 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]

Moreno-Barriuso, E.

Nagy, L. J.

G. Yoon, S. Pantanelli, and L. J. Nagy, “Large-dynamic-range Shack–Hartmann wavefront sensor for highly aberrated eyes,” J. Biomed. Opt. 11, 030502 (2006).
[CrossRef]

Navarro, R.

Nelson, J.

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]

Pantanelli, S.

G. Yoon, S. Pantanelli, and L. J. Nagy, “Large-dynamic-range Shack–Hartmann wavefront sensor for highly aberrated eyes,” J. Biomed. Opt. 11, 030502 (2006).
[CrossRef]

Pereira, S.

Pierson, R.

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).

Polo, A.

Presta, R. W.

Reitze, D. H.

Ricklin, J.

Ríos, S.

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]

Sacks, R. A.

Salmon, J. T.

Seppala, L. G.

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).

Shack, R. V.

Sivokon, V.

M. Vorontsov and V. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. 15, 2745–2758 (1998).
[CrossRef]

Spielbusch, B.

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]

Toeppen, J. S.

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]

Troy, M.

Urbach, H.

Van Atta, L.

Van Wonterghem, B. M.

Voelz, D.

Vorontsov, M.

M. Vorontsov and V. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. 15, 2745–2758 (1998).
[CrossRef]

M. Vorontsov, G. Carhart, and J. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997).
[CrossRef]

Wayne, W. T.

Winters, S. E.

Woods, B. W.

Yoon, G.

G. Yoon, S. Pantanelli, and L. J. Nagy, “Large-dynamic-range Shack–Hartmann wavefront sensor for highly aberrated eyes,” J. Biomed. Opt. 11, 030502 (2006).
[CrossRef]

Yoshida, S.

Yu, X.

G. Cao and X. Yu, “Accuracy analysis of a Hartmann–Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994).
[CrossRef]

Zacharias, R. A.

Appl. Opt. (8)

J. Gonglewski, D. Voelz, J. Fender, D. Dayton, B. Spielbusch, and R. Pierson, “First astronomical application of postdetection turbulence compensation: images of α Aurigae, ν Ursae Majoris, and α Geminorum using self-referenced speckle holography,” Appl. Opt. 29, 4527–4529 (1990).
[CrossRef]

R. Irwan and R. G. Lane, “Analysis of optimal centroid estimation applied to Shack–Hartmann sensing,” Appl. Opt. 38, 6737–6743 (1999).
[CrossRef]

J. A. Koch, R. W. Presta, R. A. Sacks, R. A. Zacharias, E. S. Bliss, M. J. Dailey, M. Feldman, A. A. Grey, F. R. Holdener, J. T. Salmon, L. G. Seppala, J. S. Toeppen, L. Van Atta, B. M. Van Wonterghem, W. T. Wayne, S. E. Winters, and B. W. Woods, “Experimental comparison of a Shack–Hartmann sensor and a phase-shifting interferometer for large-optics metrology applications,” Appl. Opt. 39, 4540–4546 (2000).
[CrossRef]

J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejer, R. L. Byer, D. Clubley, S. Yoshida, and D. H. Reitze, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack–Hartmann wave-front sensor,” Appl. Opt. 40, 366–374 (2001).
[CrossRef]

G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

J. Ares and J. Arines, “Influence of thresholding on centroid statistics: full analytical description,” Appl. Opt. 43, 5796–5805 (2004).
[CrossRef]

J. Lee, R. V. Shack, and M. R. Descour, “Sorting method to extend the dynamic range of the Shack–Hartmann wavefront sensor,” Appl. Opt. 44, 4838–4845 (2005).
[CrossRef]

D. López and S. Ríos, “Interferometric Shack–Hartmann wavefront sensor with an array of four-hole apertures,” Appl. Opt. 49, 2334–2338 (2010).
[CrossRef]

J. Biomed. Opt. (1)

G. Yoon, S. Pantanelli, and L. J. Nagy, “Large-dynamic-range Shack–Hartmann wavefront sensor for highly aberrated eyes,” J. Biomed. Opt. 11, 030502 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

M. Vorontsov and V. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. 15, 2745–2758 (1998).
[CrossRef]

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

J. Refractive Surg. (1)

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).

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. Eng. (2)

G. Cao and X. Yu, “Accuracy analysis of a Hartmann–Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994).
[CrossRef]

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Other (2)

G. Cauwenberghs, “A fast stochastic error-descent algorithm for supervised learning and optimization,” in Advances in Neural Information Processing Systems, S. J. Hanson, J. D. Cowan, and C. L. Giles, eds., Vol. 5 (Morgan Kaufman, 1993), pp. 244–251.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

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Figures (11)

Fig. 1.
Fig. 1.

Optical layout of the MSH sensor; L is the depth of defocus.

Fig. 2.
Fig. 2.

(a) Wavefront used to test the accuracy of phase retrieval; (b) the residual wavefront error by phase retrieval using the MSH sensor with subaperture number 8 × 8 , scales in waves.

Fig. 3.
Fig. 3.

RMS (a) and PV (b) value of the residual wavefront errors by phase retrieval using the MSH and SH sensors with lenslet array having different subaperture number.

Fig. 4.
Fig. 4.

Wavefront composed of Zernike polynomials from term 35 to 55, scale in waves.

Fig. 5.
Fig. 5.

Residual wavefront errors by phase retrieval using the MSH (a); and SH (b) sensor with subaperture number 8 × 8 , scales in waves.

Fig. 6.
Fig. 6.

Comparison of Zernike coefficients retrieved using both the MSH (green) and the SH (red) sensors with those of the input wavefront (blue).

Fig. 7.
Fig. 7.

Layout of optical system for phase retrieval using the MSH sensor, L 1 L 4 , lenses, CBS, cubic beam splitter, TM, mirror for testing, MSH, and the MSH sensor.

Fig. 8.
Fig. 8.

Wavefront retrieved by phase retrieval using the MSH sensor, scale in waves.

Fig. 9.
Fig. 9.

Wavefront measured by the Veeco commercial interferometer.

Fig. 10.
Fig. 10.

Evolution curves of cost functions with the calculated initial values (the red curve) and with initial values set to 0 (the black curve).

Fig. 11.
Fig. 11.

Zernike coefficients of the calculated initial values (red square) and the retrieved (blackdot).

Equations (7)

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

W defocus = L 8 ( f # ) 2 ,
E in ( x , y ) = A exp [ i ϕ ( x , y ) ] ,
ϕ ( x , y ) = j = 1 J a j Z j ( x , y ) ,
E in ( x , y ) = E in ( x , y ) t ( x , y ) ,
t ( x , y ) = p ( x , y ) exp [ i φ ( x , y ) ] ,
E out ( x , y ) = F 1 { F ( E in ( x , y ) ) exp [ i 2 π λ ( f L ) 1 ( λ f x ) 2 ( λ f y ) 2 ] } ,
CF = i , j | I ( x i , y j ) | E out ( x i , y j ) | 2 | 2 ,

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