Abstract

The measurement of the wavefront at the exit pupil of an optical system is a reliable method to investigate its imaging performance. It is sometimes necessary to compare the measurement obtained with a wavefront sensor to some known aberration function, which is for example given by a simulation or a gold standard measurement technique such as interferometry. For accurate measurement systems, the residual after direct subtraction of the two wavefronts is often partly due to a mismatch between coordinate systems. We present in this paper a method that uses the Zernike expansion of wavefronts to numerically cancel such small alignment errors. We use this algorithm to quantify the accuracy of a custom-built Shack-Hartmann wavefront sensor in the measurement of known aberration functions.

© 2010 Optical Society of America

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References

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  1. V. N. Mahajan, “Zernike Polynomial and Wavefront Fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 2007), pp. 498–546.
    [CrossRef]
  2. J. Pfund, N. Lindlein, and J. Schwider, “Misalignment effects of the Shack–Hartmann sensor,” Appl. Opt. 37, 22–27 (1998).
    [CrossRef]
  3. D. R. Neal, J. Copland, and D. Neal, “Shack–Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
    [CrossRef]
  4. P. Rodrguez, R. Navarro, J. Arines, and S. Bara, “A new calibration set of phase plates for ocular aberrometers,” J. Refract. Surg. 22, 275–284 (2006).
  5. K. M. Morzinski, K. B. W. Harpsoe, D. T. Gavel, and S. M. Ammons, “The open-loop control of MEMS: modeling and experimental results,” Proc. SPIE 6467, 64670G (2007).
    [CrossRef]
  6. L. Lundstrom, 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]
  7. S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
    [CrossRef]
  8. R. Navarro, E. Moreno-Barriuso, S. Bara, and T. Mancebo, “Phase plates for wave-aberration compensation in the human eye,” Opt. Lett. 25, 236–238 (2000).
    [CrossRef]
  9. H. Schreiber, and J. H. Bruning, “Phase Shifting Interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 2007), pp. 547–666.
    [CrossRef]

2007 (2)

K. M. Morzinski, K. B. W. Harpsoe, D. T. Gavel, and S. M. Ammons, “The open-loop control of MEMS: modeling and experimental results,” Proc. SPIE 6467, 64670G (2007).
[CrossRef]

L. Lundstrom, 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]

2006 (2)

S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
[CrossRef]

P. Rodrguez, R. Navarro, J. Arines, and S. Bara, “A new calibration set of phase plates for ocular aberrometers,” J. Refract. Surg. 22, 275–284 (2006).

2002 (1)

D. R. Neal, J. Copland, and D. Neal, “Shack–Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

2000 (1)

1998 (1)

Ammons, S. M.

K. M. Morzinski, K. B. W. Harpsoe, D. T. Gavel, and S. M. Ammons, “The open-loop control of MEMS: modeling and experimental results,” Proc. SPIE 6467, 64670G (2007).
[CrossRef]

Ares, J.

Arines, J.

P. Rodrguez, R. Navarro, J. Arines, and S. Bara, “A new calibration set of phase plates for ocular aberrometers,” J. Refract. Surg. 22, 275–284 (2006).

S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
[CrossRef]

Bara, S.

Copland, J.

D. R. Neal, J. Copland, and D. Neal, “Shack–Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

Gavel, D. T.

K. M. Morzinski, K. B. W. Harpsoe, D. T. Gavel, and S. M. Ammons, “The open-loop control of MEMS: modeling and experimental results,” Proc. SPIE 6467, 64670G (2007).
[CrossRef]

Harpsoe, K. B. W.

K. M. Morzinski, K. B. W. Harpsoe, D. T. Gavel, and S. M. Ammons, “The open-loop control of MEMS: modeling and experimental results,” Proc. SPIE 6467, 64670G (2007).
[CrossRef]

Lindlein, N.

Lundstrom, L.

Mancebo, T.

Moreno-Barriuso, E.

Morzinski, K. M.

K. M. Morzinski, K. B. W. Harpsoe, D. T. Gavel, and S. M. Ammons, “The open-loop control of MEMS: modeling and experimental results,” Proc. SPIE 6467, 64670G (2007).
[CrossRef]

Navarro, R.

P. Rodrguez, R. Navarro, J. Arines, and S. Bara, “A new calibration set of phase plates for ocular aberrometers,” J. Refract. Surg. 22, 275–284 (2006).

R. Navarro, E. Moreno-Barriuso, S. Bara, and T. Mancebo, “Phase plates for wave-aberration compensation in the human eye,” Opt. Lett. 25, 236–238 (2000).
[CrossRef]

Neal, D.

D. R. Neal, J. Copland, and D. Neal, “Shack–Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

Neal, D. R.

D. R. Neal, J. Copland, and D. Neal, “Shack–Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

Pfund, J.

Prado, P.

Rodrguez, P.

P. Rodrguez, R. Navarro, J. Arines, and S. Bara, “A new calibration set of phase plates for ocular aberrometers,” J. Refract. Surg. 22, 275–284 (2006).

Schwider, J.

Unsbo, P.

Appl. Opt. (1)

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

J. Refract. Surg. (1)

P. Rodrguez, R. Navarro, J. Arines, and S. Bara, “A new calibration set of phase plates for ocular aberrometers,” J. Refract. Surg. 22, 275–284 (2006).

Opt. Lett. (1)

Proc. SPIE (2)

K. M. Morzinski, K. B. W. Harpsoe, D. T. Gavel, and S. M. Ammons, “The open-loop control of MEMS: modeling and experimental results,” Proc. SPIE 6467, 64670G (2007).
[CrossRef]

D. R. Neal, J. Copland, and D. Neal, “Shack–Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

Other (2)

H. Schreiber, and J. H. Bruning, “Phase Shifting Interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 2007), pp. 547–666.
[CrossRef]

V. N. Mahajan, “Zernike Polynomial and Wavefront Fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 2007), pp. 498–546.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Optical layout of the measurements. Left: double-pass measurement with the commercial PSI. Right: single-pass measurement with the SHWFS.

Fig. 2
Fig. 2

Wavefront maps in microns, as measured by the PSI (left) and the SHWFS (middle). Right: Wavefront map as measured by the SHWFS, after correction of the pupil mismatch.

Fig. 3
Fig. 3

Comparison of the measured Zernike aberrations, for the four phase plates. The red bars show the SHWFS measurement after pupil correction, and are very comparable to the PSI measurements (blue bars).

Fig. 4
Fig. 4

PSI map of the first plate, measured over a 6.3 mm pupil diameter. To obtain the results of Figure 5, we have fitted the set of Zernike coefficients zref from this map, using a rotated and translated coordinate systems. The circle shows an example of modified pupil area over which we computed zref, for a shift [tx = 0.15 mm, ty = −0.15 mm] and a rotation θ = 2 degree. The pupil diameter that we use to compute zref is 5.25 mm, as in Table 1.

Fig. 5
Fig. 5

Estimated positions of the centre of the pupil (left) and the angle of rotation (right) obtained with the first plate, after modifying the coordinate system of zref as we illustrated in Figure 4.

Tables (1)

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Table 1 Experimental results

Equations (5)

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p ^ = argmin { | | z 1 z r e f | | }
z 1 = z 0 + M p × z 0
M p t x t x , 0 × M [ t x , 0 , 0 , 0 , 1 ] + t y t y , 0 × M [ 0 , t y , 0 , 0 , 1 ] + θ θ 0 × M [ 0 , 0 , θ 0 , 1 ] + γ 1 γ 0 1 × M [ 0 , 0 , 0 , γ 0 ]
A × p = z 0 z r e f
p i + 1 = p i + A + × ( z i z r e f ) , with : z i = z 0 + M p i × z 0

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