Abstract

We demonstrate a method with which to calibrate a Shack–Hartmann sensor for absolute wavefront measurement of collimated laser beams. Nearly perfect spherical wavefronts originating from a single-mode fiber were used as references. After the calibration, the uncertainty of the wavefront was less than λ/100 peak to valley across a diameter of 6 mm. For example, this method allowed us to balance aberrations and prepare collimated beams with wavefronts that are plane to λ/500 across 1 mm.

© 2005 Optical Society of America

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References

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  1. P. W. Milonni, “Resource letter AOA-1: adaptive optics for astronomy,” Am. J. Phys. 67, 476–485 (1999).
    [CrossRef]
  2. J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejer, R. L. Byer, D. Clubley, S. Yoshida, 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]
  3. T. Trebst, T. Binnewies, J. Helmcke, F. Riehle, “Suppression of spurious phase shifts in an optical frequency standard,” IEEE Trans. Instrum. Meas. 50, 535–538 (2001).
    [CrossRef]
  4. G. Wilpers, C. Degenhardt, T. Binnewies, A. Chernyshov, F. Riehle, J. Helmcke, U. Sterr, “Improvement of the fractional uncertainty of a neutral atom calcium optical frequency standard to 2 × 10−14,” Appl. Phys. B 76, 149–156 (2003).
    [CrossRef]
  5. J. Fils, F. Leduc, P. Bouyer, D. Holleville, N. Dimarcq, A. Clairon, A. Landragin, “Influence of optical aberrations in an atomic gyroscope,” Eur. J. Phys. D (to be published), and arXiv:physics/0507060.
  6. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1991), Chap. 14, pp. 501–598.
  7. M. V. R. K. Murty, “The use of a single plane-parallel plate as a lateral shearing interferometer with a visible gas laser source,” Appl. Opt. 4, 531–534 (1964).
    [CrossRef]
  8. O. Bryngdahl, “Applications of shearing interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1965), Vol. 4, pp. 37–83.
    [CrossRef]
  9. V. P. Linnik, “Simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. USSR 1, 208–210 (1933).
  10. R. N. Smartt, J. Strong, “Point-diffraction interferometer,” J. Opt. Soc. Am. 62, 737 (1972).
  11. R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, Suppl. 1–14, 351–357 (1975).
  12. J. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  13. V. Michau, G. Rousset, F. Mendez, B. Riou, “Hartmann–Shack wavefront sensor for laser diode testing,” in Optical Space Communication, G. Otrio, ed., Proc. SPIE1131, 160–167 (1989).
    [CrossRef]
  14. G. Artzner, “On the absolute calibration of Shack–Hartmann sensors and UV laboratory wavefront measurements,” Pure Appl. Opt. 3, 121–132 (1994).
    [CrossRef]
  15. G. Yoon, T. Jitsuno, M. Nakatsuka, S. Nakai, “Shack–Hartmann wave-front measurement with a large f-number plastic microlens array,” Appl. Opt. 35, 188–192 (1996).
    [CrossRef] [PubMed]
  16. 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. Whistler, S. E. Winters, B. W. Woods, “Experimental comparison of a Shack–Hartmann sensor and a phase-shifting interferometer for large-optics metrology applications,” Appl. Opt. 39, 4540–4544 (2000).
    [CrossRef]
  17. J. D. Mansell, E. K. Gustafson, “Focal plane position detection with a diffractive optic for Shack–Hartmann wavefront sensor fabrication,” Appl. Opt. 40, 1074–1079 (2001).
    [CrossRef]
  18. J. Pfund, N. Lindlein, J. Schwider, R. Burrow, T. Blümel, K.-E. Elsner, “Absolute sphericity measurement: a comparative study of the use of interferometry and a Shack–Hartmann sensor,” Opt. Lett. 23, 742–744 (1998).
    [CrossRef]
  19. J. Pfund, N. Lindlein, J. Schwider, “Misalignment effects of the Shack–Hartmann sensor,” Appl. Opt. 37, 22–27 (1998).
    [CrossRef]
  20. G. Cao, X. Yu, “Accuracy analysis of a Hartmann–Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994).
    [CrossRef]
  21. H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
    [CrossRef]
  22. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).
  23. J. Ares, T. Mancebo, S. Bará, “Position and displacement sensing with Shack–Hartmann wave-front sensors,” Appl. Opt. 39, 1511–1520 (2000).
    [CrossRef]

2003 (1)

G. Wilpers, C. Degenhardt, T. Binnewies, A. Chernyshov, F. Riehle, J. Helmcke, U. Sterr, “Improvement of the fractional uncertainty of a neutral atom calcium optical frequency standard to 2 × 10−14,” Appl. Phys. B 76, 149–156 (2003).
[CrossRef]

2001 (3)

2000 (2)

1999 (1)

P. W. Milonni, “Resource letter AOA-1: adaptive optics for astronomy,” Am. J. Phys. 67, 476–485 (1999).
[CrossRef]

1998 (2)

1996 (1)

1994 (3)

G. Artzner, “On the absolute calibration of Shack–Hartmann sensors and UV laboratory wavefront measurements,” Pure Appl. Opt. 3, 121–132 (1994).
[CrossRef]

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

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

1978 (1)

J. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1975 (1)

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, Suppl. 1–14, 351–357 (1975).

1972 (1)

R. N. Smartt, J. Strong, “Point-diffraction interferometer,” J. Opt. Soc. Am. 62, 737 (1972).

1964 (1)

1933 (1)

V. P. Linnik, “Simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. USSR 1, 208–210 (1933).

Ares, J.

Artzner, G.

G. Artzner, “On the absolute calibration of Shack–Hartmann sensors and UV laboratory wavefront measurements,” Pure Appl. Opt. 3, 121–132 (1994).
[CrossRef]

Bará, S.

Binnewies, T.

G. Wilpers, C. Degenhardt, T. Binnewies, A. Chernyshov, F. Riehle, J. Helmcke, U. Sterr, “Improvement of the fractional uncertainty of a neutral atom calcium optical frequency standard to 2 × 10−14,” Appl. Phys. B 76, 149–156 (2003).
[CrossRef]

T. Trebst, T. Binnewies, J. Helmcke, F. Riehle, “Suppression of spurious phase shifts in an optical frequency standard,” IEEE Trans. Instrum. Meas. 50, 535–538 (2001).
[CrossRef]

Bliss, E. S.

Blümel, T.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Bouyer, P.

J. Fils, F. Leduc, P. Bouyer, D. Holleville, N. Dimarcq, A. Clairon, A. Landragin, “Influence of optical aberrations in an atomic gyroscope,” Eur. J. Phys. D (to be published), and arXiv:physics/0507060.

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1991), Chap. 14, pp. 501–598.

Bryngdahl, O.

O. Bryngdahl, “Applications of shearing interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1965), Vol. 4, pp. 37–83.
[CrossRef]

Burrow, R.

Byer, R. L.

Cao, G.

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

Chernyshov, A.

G. Wilpers, C. Degenhardt, T. Binnewies, A. Chernyshov, F. Riehle, J. Helmcke, U. Sterr, “Improvement of the fractional uncertainty of a neutral atom calcium optical frequency standard to 2 × 10−14,” Appl. Phys. B 76, 149–156 (2003).
[CrossRef]

Clairon, A.

J. Fils, F. Leduc, P. Bouyer, D. Holleville, N. Dimarcq, A. Clairon, A. Landragin, “Influence of optical aberrations in an atomic gyroscope,” Eur. J. Phys. D (to be published), and arXiv:physics/0507060.

Clubley, D.

Dailey, M. J.

Degenhardt, C.

G. Wilpers, C. Degenhardt, T. Binnewies, A. Chernyshov, F. Riehle, J. Helmcke, U. Sterr, “Improvement of the fractional uncertainty of a neutral atom calcium optical frequency standard to 2 × 10−14,” Appl. Phys. B 76, 149–156 (2003).
[CrossRef]

Dimarcq, N.

J. Fils, F. Leduc, P. Bouyer, D. Holleville, N. Dimarcq, A. Clairon, A. Landragin, “Influence of optical aberrations in an atomic gyroscope,” Eur. J. Phys. D (to be published), and arXiv:physics/0507060.

Elsner, K.-E.

Falkenstörfer, O.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Fejer, M. M.

Feldman, M.

Fils, J.

J. Fils, F. Leduc, P. Bouyer, D. Holleville, N. Dimarcq, A. Clairon, A. Landragin, “Influence of optical aberrations in an atomic gyroscope,” Eur. J. Phys. D (to be published), and arXiv:physics/0507060.

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1991), Chap. 14, pp. 501–598.

Grey, A. A.

Gustafson, E. K.

Hardy, J.

J. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Helmcke, J.

G. Wilpers, C. Degenhardt, T. Binnewies, A. Chernyshov, F. Riehle, J. Helmcke, U. Sterr, “Improvement of the fractional uncertainty of a neutral atom calcium optical frequency standard to 2 × 10−14,” Appl. Phys. B 76, 149–156 (2003).
[CrossRef]

T. Trebst, T. Binnewies, J. Helmcke, F. Riehle, “Suppression of spurious phase shifts in an optical frequency standard,” IEEE Trans. Instrum. Meas. 50, 535–538 (2001).
[CrossRef]

Hennawi, J.

Holdener, F. R.

Holleville, D.

J. Fils, F. Leduc, P. Bouyer, D. Holleville, N. Dimarcq, A. Clairon, A. Landragin, “Influence of optical aberrations in an atomic gyroscope,” Eur. J. Phys. D (to be published), and arXiv:physics/0507060.

Jitsuno, T.

Koch, J. A.

Landragin, A.

J. Fils, F. Leduc, P. Bouyer, D. Holleville, N. Dimarcq, A. Clairon, A. Landragin, “Influence of optical aberrations in an atomic gyroscope,” Eur. J. Phys. D (to be published), and arXiv:physics/0507060.

Leduc, F.

J. Fils, F. Leduc, P. Bouyer, D. Holleville, N. Dimarcq, A. Clairon, A. Landragin, “Influence of optical aberrations in an atomic gyroscope,” Eur. J. Phys. D (to be published), and arXiv:physics/0507060.

Lindlein, N.

Linnik, V. P.

V. P. Linnik, “Simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. USSR 1, 208–210 (1933).

Mancebo, T.

Mansell, J. D.

Mendez, F.

V. Michau, G. Rousset, F. Mendez, B. Riou, “Hartmann–Shack wavefront sensor for laser diode testing,” in Optical Space Communication, G. Otrio, ed., Proc. SPIE1131, 160–167 (1989).
[CrossRef]

Michau, V.

V. Michau, G. Rousset, F. Mendez, B. Riou, “Hartmann–Shack wavefront sensor for laser diode testing,” in Optical Space Communication, G. Otrio, ed., Proc. SPIE1131, 160–167 (1989).
[CrossRef]

Milonni, P. W.

P. W. Milonni, “Resource letter AOA-1: adaptive optics for astronomy,” Am. J. Phys. 67, 476–485 (1999).
[CrossRef]

Murty, M. V. R. K.

Nakai, S.

Nakatsuka, M.

Pfund, J.

Presta, R. W.

Reitze, D. H.

Riehle, F.

G. Wilpers, C. Degenhardt, T. Binnewies, A. Chernyshov, F. Riehle, J. Helmcke, U. Sterr, “Improvement of the fractional uncertainty of a neutral atom calcium optical frequency standard to 2 × 10−14,” Appl. Phys. B 76, 149–156 (2003).
[CrossRef]

T. Trebst, T. Binnewies, J. Helmcke, F. Riehle, “Suppression of spurious phase shifts in an optical frequency standard,” IEEE Trans. Instrum. Meas. 50, 535–538 (2001).
[CrossRef]

Riou, B.

V. Michau, G. Rousset, F. Mendez, B. Riou, “Hartmann–Shack wavefront sensor for laser diode testing,” in Optical Space Communication, G. Otrio, ed., Proc. SPIE1131, 160–167 (1989).
[CrossRef]

Rousset, G.

V. Michau, G. Rousset, F. Mendez, B. Riou, “Hartmann–Shack wavefront sensor for laser diode testing,” in Optical Space Communication, G. Otrio, ed., Proc. SPIE1131, 160–167 (1989).
[CrossRef]

Sacks, R. A.

Salmon, J. T.

Schwider, J.

Seppala, L. G.

Sickinger, H.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Smartt, R. N.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, Suppl. 1–14, 351–357 (1975).

R. N. Smartt, J. Strong, “Point-diffraction interferometer,” J. Opt. Soc. Am. 62, 737 (1972).

Steel, W. H.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, Suppl. 1–14, 351–357 (1975).

Sterr, U.

G. Wilpers, C. Degenhardt, T. Binnewies, A. Chernyshov, F. Riehle, J. Helmcke, U. Sterr, “Improvement of the fractional uncertainty of a neutral atom calcium optical frequency standard to 2 × 10−14,” Appl. Phys. B 76, 149–156 (2003).
[CrossRef]

Strong, J.

R. N. Smartt, J. Strong, “Point-diffraction interferometer,” J. Opt. Soc. Am. 62, 737 (1972).

Toeppen, J. S.

Trebst, T.

T. Trebst, T. Binnewies, J. Helmcke, F. Riehle, “Suppression of spurious phase shifts in an optical frequency standard,” IEEE Trans. Instrum. Meas. 50, 535–538 (2001).
[CrossRef]

Van Atta, L.

Van Wonterghem, B. M.

Whistler, W. T.

Wilpers, G.

G. Wilpers, C. Degenhardt, T. Binnewies, A. Chernyshov, F. Riehle, J. Helmcke, U. Sterr, “Improvement of the fractional uncertainty of a neutral atom calcium optical frequency standard to 2 × 10−14,” Appl. Phys. B 76, 149–156 (2003).
[CrossRef]

Winters, S. E.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Woods, B. W.

Yoon, G.

Yoshida, S.

Yu, X.

G. Cao, 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.

Am. J. Phys. (1)

P. W. Milonni, “Resource letter AOA-1: adaptive optics for astronomy,” Am. J. Phys. 67, 476–485 (1999).
[CrossRef]

Appl. Opt. (7)

J. Ares, T. Mancebo, S. Bará, “Position and displacement sensing with Shack–Hartmann wave-front sensors,” Appl. Opt. 39, 1511–1520 (2000).
[CrossRef]

G. Yoon, T. Jitsuno, M. Nakatsuka, S. Nakai, “Shack–Hartmann wave-front measurement with a large f-number plastic microlens array,” Appl. Opt. 35, 188–192 (1996).
[CrossRef] [PubMed]

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. Whistler, S. E. Winters, B. W. Woods, “Experimental comparison of a Shack–Hartmann sensor and a phase-shifting interferometer for large-optics metrology applications,” Appl. Opt. 39, 4540–4544 (2000).
[CrossRef]

J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejer, R. L. Byer, D. Clubley, S. Yoshida, 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]

J. Pfund, N. Lindlein, J. Schwider, “Misalignment effects of the Shack–Hartmann sensor,” Appl. Opt. 37, 22–27 (1998).
[CrossRef]

J. D. Mansell, E. K. Gustafson, “Focal plane position detection with a diffractive optic for Shack–Hartmann wavefront sensor fabrication,” Appl. Opt. 40, 1074–1079 (2001).
[CrossRef]

M. V. R. K. Murty, “The use of a single plane-parallel plate as a lateral shearing interferometer with a visible gas laser source,” Appl. Opt. 4, 531–534 (1964).
[CrossRef]

Appl. Phys. B (1)

G. Wilpers, C. Degenhardt, T. Binnewies, A. Chernyshov, F. Riehle, J. Helmcke, U. Sterr, “Improvement of the fractional uncertainty of a neutral atom calcium optical frequency standard to 2 × 10−14,” Appl. Phys. B 76, 149–156 (2003).
[CrossRef]

C. R. Acad. Sci. USSR (1)

V. P. Linnik, “Simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. USSR 1, 208–210 (1933).

IEEE Trans. Instrum. Meas. (1)

T. Trebst, T. Binnewies, J. Helmcke, F. Riehle, “Suppression of spurious phase shifts in an optical frequency standard,” IEEE Trans. Instrum. Meas. 50, 535–538 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

R. N. Smartt, J. Strong, “Point-diffraction interferometer,” J. Opt. Soc. Am. 62, 737 (1972).

Jpn. J. Appl. Phys. (1)

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, Suppl. 1–14, 351–357 (1975).

Opt. Eng. (2)

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

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

J. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Pure Appl. Opt. (1)

G. Artzner, “On the absolute calibration of Shack–Hartmann sensors and UV laboratory wavefront measurements,” Pure Appl. Opt. 3, 121–132 (1994).
[CrossRef]

Other (5)

O. Bryngdahl, “Applications of shearing interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1965), Vol. 4, pp. 37–83.
[CrossRef]

J. Fils, F. Leduc, P. Bouyer, D. Holleville, N. Dimarcq, A. Clairon, A. Landragin, “Influence of optical aberrations in an atomic gyroscope,” Eur. J. Phys. D (to be published), and arXiv:physics/0507060.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1991), Chap. 14, pp. 501–598.

V. Michau, G. Rousset, F. Mendez, B. Riou, “Hartmann–Shack wavefront sensor for laser diode testing,” in Optical Space Communication, G. Otrio, ed., Proc. SPIE1131, 160–167 (1989).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

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Figures (9)

Fig. 1
Fig. 1

Schematic of a Shack–Hartmann wavefront sensor. Top, light-spot pattern in the plane of the CCD chip for a spherical wavefront. Bottom, light-spot position in a microlens subaperture for a locally tilted wavefront.

Fig. 2
Fig. 2

Setup for calibration: SMF, single-mode fiber; BS, beam splitter; M, mirror.

Fig. 3
Fig. 3

Initial differences δρ between the measured curvature and the curvature of the spherical reference for the nominal parameters of the sensor. Open triangles (filled squares) denote the x (y) direction.

Fig. 4
Fig. 4

Residual deviations between the reference curvature and the curvature measured by the SHS after calibration. Top and bottom, vertical and horizontal planes, respectively.

Fig. 5
Fig. 5

Left, residuals of a spherical reference wavefront (R0 = 1.54 m) measured with the SHS. Spacing between the lines is λ/500, and the aberrations are 0.04λ (PV) and 0.006λ (rms). The difference between consecutive measurements is shown at the right, with deviations of 0.003λ (PV) and 0.0004λ (rms).

Fig. 6
Fig. 6

Difference between two measurements with a lateral displacement of 4.5 cm of the spherical reference wavefront (R0 = 1.0 m; left) and with a displacement 14 cm and R0 = 6.0 m (right). The aberrations are 0.017λ (PV), 0.0025λ (rms) and 0.0035λ (PV), 0.0007λ (rms) for R0 = 1.0 m, R0 = 6.0 m, respectively. Spacing between the lines is λ/500.

Fig. 7
Fig. 7

Quality of the correction method for different radii. Top, measurement of a reference wave with R0 = 0.84 m: (a) uncorrected data with residual aberrations 0.04λ (PV) and 0.007λ (rms), (b) corrected wavefront with 0.01λ (PV) and 0.002λ (rms). Bottom, measurement of a reference wave with R0 = 6.0 m: (c) uncorrected data with 0.03λ (PV) and 0.006λ (rms), (d) corrected wavefront with 0.008λ (PV) and 0.001λ (rms). In both cases the same reference wavefront with Rref = 9.0 m was used and the spherical part was subtracted from the measurements. The spacing between the lines is λ/500.

Fig. 8
Fig. 8

Difference between two reference wavefronts that were written at an interval of 7 days with 0.007λ (PV) and 0.001λ (rms). Line spacing is λ/500.

Fig. 9
Fig. 9

Wavefront aberrations of a collimated beam: (a) with the smallest global curvature over the full measurement area of 6 mm diameter. The residual aberrations (mostly astigmatism) are 0.017λ (PV) and 0.0024λ (rms). (b) After balancing, the aberrations over the central part with a diameter of 1 mm can be reduced to λ/500. The separation between the lines corresponds to λ/500.

Tables (1)

Tables Icon

Table 1 Nominal Parameters of the Shack–Hartmann Sensor

Equations (15)

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σ ( r ) = f 0 Φ ( r ) ,
Φ ( r ) = r 2 2 R 0 + r 4 8 R 0 3 + .
σ ( r k ) = f 0 Φ r r k f 0 R 0 ,
Q = P 0 + σ 1 = P 0 ( 1 + f 0 R 0 ) .
ρ 0 = ( N S 0 P 0 - 1 ) 1 f 0 .
S = S 0 + δ S , P = P 0 + δ P , f = f 0 + δ f , δ S / S 1 , δ P / P 1 , δ f / f 1.
δ ρ = ρ meas - ρ 0 = - ρ 0 δ f f 0 - N S 0 P 0 f 0 ( δ P P 0 - δ S S 0 ) .
S 0 = P 0 + σ 1 N P 0 N .
δ ρ = - ρ 0 δ f f 0 - 1 f 0 ( δ P P 0 - δ S S 0 ) .
ρ 0 = 1 R ref - δ R ρ ref + ρ ref 2 δ R .
δ ρ = ρ ref 2 δ R - ρ ref δ f f 0 - 1 f 0 ( δ P P 0 - δ S S 0 ) .
f 0 R max σ min r det ,
δ ρ ( δ T ) = [ ρ 0 α ( Al ) + α ( FS ) f 0 - α ( Si ) f 0 ] δ T .
Φ cor = Φ shs - WF ref ( R ref ) + WF sph ( R ref ) .
Δ Φ < 2 π λ D d 0 8 R 0 Δ Φ 0

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