Junzhong Liang, Bernhard Grimm, Stefan Goelz, and Josef F. Bille, "Objective measurement of wave aberrations of the human eye with the use of a Hartmann–Shack wave-front sensor," J. Opt. Soc. Am. A 11, 1949-1957 (1994)
A Hartmann–Shack wave-front sensor is used to measure the wave aberrations of the human eye by sensing the wave front emerging from the eye produced by the retinal reflection of a focused light spot on the fovea. Since the test involves the measurements of the local slopes of the wave front, the actual wave front is reconstructed by the use of wave-front estimation with Zernike polynomials. From the estimated Zernike coefficients of the tested wave front the aberrations of the eye are evaluated. It is shown that with this method, using a Hartmann–Shack wave-front sensor, one can obtain a fast, precise, and objective measurement of the aberrations of the eye.
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Ref. 20.
Here we adopt the meanings usually used in optical metrology for the individual terms of the Zernike polynomials, although they might be different from those of primary aberrations.
Table 2
Estimated Zernike Coefficients for the Tested Eyesa
L1
L2
B1
B2
Lens
Meaning of the Coefficients
C1
−2.52
−3.91
−1.84
−2.48
−7.239 ± 0.014
Tilt in x direction
C2
17.32
18.94
−5.15
−5.31
0.721 ± 0.004
Tilt in y direction
C3
0.89
1.02
0.60
0.87
−0.072 ± 0.009
Astigmatism with axis at ±45°
C4
2.78
2.75
1.15
1.17
0.841 ± 0.002
Focus shift
C5
0.43
0.33
−0.15
−0.52
0.032 ± 0.003
Astigmatism with axis at 0° or 90°
C6
0.20
0.21
−0.17
−0.20
−0.017 ± 0.001
C7
−0.32
−0.26
−0.41
−0.39
−0.010 ± 0.002
Third-order coma along x axis
C8
0.23
0.03
0.52
0.47
0
Third-order coma along y axis
C9
−0.29
−0.34
0.40
0.35
0.015 ± 0.002
C10
−0.09
−0.12
0.06
0.06
0.004 ± 0.003
C11
0.10
0.05
−0.04
0.03
−0.002 ± 0.001
C12
0.15
0.19
−0.04
0
−0.003 ± 0.001
Third-order spherical aberration
C13
−0.17
−0.19
−0.15
−0.09
−0.011 ± 0.002
C14
−0.03
0.15
−0.06
−0.10
−0.001 ± 0.002
C1–C14 are the estimated Zernike coefficients of the tested eyes in micrometers. L1 and L2 apply to tests of subject L, and B1 and B2 apply to tests of subject B. As a comparison, the sixth column provides the test of a spherical wave produced by a lens (f = 2000) focusing a plane wave. No standard deviation of the Zernike coefficients of the tested eyes is given, since the ocular optics is a dynamic system with accommodation and microfluctuation of accommodation.20
Table 3
Measured Defocus and Astigmatism of the Tested Eyesa
L1
L2
B1
B2
Lens
ΦD (diopters)
−1.25
−1.21
−0.46
−0.36
−0.44
ΦA (diopters)
−0.54
−0.59
−0.34
−0.56
−0.04
α
58°
54°
38°
30°
33°
ΦA is the measured astigmatism of the tested eyes that is derived from the Zernike coefficients C3 and C5 by ΦA = ±16 (C32 + C52)1/2/D2. D is the diameter of the tested pupil size and in all our cases was 5.4 mm. We chose negative values for ΦA. The axis of astigmatism is given by α = 90° + 0.5 tan−1(−C3/C5) for −C3/C5 smaller than zero and α = 0.5 tan−1 (−C3/C5) for C3/C5 less than or equal to zero. ΦD is the measured defocus of the tested eyes that is derived from the Zernike coefficients by ΦD = −16C4/D2 − 0.54ΦA. L1 and L2 apply to tests of subject L’s eye, and B1 and B2 apply to tests of subject B’s eye. The last column provides the measured values of defocus and astigmatism when a plane wave is deformed by a 2000-mm focal-length lens.
Ref. 20.
Here we adopt the meanings usually used in optical metrology for the individual terms of the Zernike polynomials, although they might be different from those of primary aberrations.
Table 2
Estimated Zernike Coefficients for the Tested Eyesa
L1
L2
B1
B2
Lens
Meaning of the Coefficients
C1
−2.52
−3.91
−1.84
−2.48
−7.239 ± 0.014
Tilt in x direction
C2
17.32
18.94
−5.15
−5.31
0.721 ± 0.004
Tilt in y direction
C3
0.89
1.02
0.60
0.87
−0.072 ± 0.009
Astigmatism with axis at ±45°
C4
2.78
2.75
1.15
1.17
0.841 ± 0.002
Focus shift
C5
0.43
0.33
−0.15
−0.52
0.032 ± 0.003
Astigmatism with axis at 0° or 90°
C6
0.20
0.21
−0.17
−0.20
−0.017 ± 0.001
C7
−0.32
−0.26
−0.41
−0.39
−0.010 ± 0.002
Third-order coma along x axis
C8
0.23
0.03
0.52
0.47
0
Third-order coma along y axis
C9
−0.29
−0.34
0.40
0.35
0.015 ± 0.002
C10
−0.09
−0.12
0.06
0.06
0.004 ± 0.003
C11
0.10
0.05
−0.04
0.03
−0.002 ± 0.001
C12
0.15
0.19
−0.04
0
−0.003 ± 0.001
Third-order spherical aberration
C13
−0.17
−0.19
−0.15
−0.09
−0.011 ± 0.002
C14
−0.03
0.15
−0.06
−0.10
−0.001 ± 0.002
C1–C14 are the estimated Zernike coefficients of the tested eyes in micrometers. L1 and L2 apply to tests of subject L, and B1 and B2 apply to tests of subject B. As a comparison, the sixth column provides the test of a spherical wave produced by a lens (f = 2000) focusing a plane wave. No standard deviation of the Zernike coefficients of the tested eyes is given, since the ocular optics is a dynamic system with accommodation and microfluctuation of accommodation.20
Table 3
Measured Defocus and Astigmatism of the Tested Eyesa
L1
L2
B1
B2
Lens
ΦD (diopters)
−1.25
−1.21
−0.46
−0.36
−0.44
ΦA (diopters)
−0.54
−0.59
−0.34
−0.56
−0.04
α
58°
54°
38°
30°
33°
ΦA is the measured astigmatism of the tested eyes that is derived from the Zernike coefficients C3 and C5 by ΦA = ±16 (C32 + C52)1/2/D2. D is the diameter of the tested pupil size and in all our cases was 5.4 mm. We chose negative values for ΦA. The axis of astigmatism is given by α = 90° + 0.5 tan−1(−C3/C5) for −C3/C5 smaller than zero and α = 0.5 tan−1 (−C3/C5) for C3/C5 less than or equal to zero. ΦD is the measured defocus of the tested eyes that is derived from the Zernike coefficients by ΦD = −16C4/D2 − 0.54ΦA. L1 and L2 apply to tests of subject L’s eye, and B1 and B2 apply to tests of subject B’s eye. The last column provides the measured values of defocus and astigmatism when a plane wave is deformed by a 2000-mm focal-length lens.