Abstract

An ideal correcting method, such as a customized contact lens, laser refractive surgery, or adaptive optics, that corrects higher-order aberrations as well as defocus and astigmatism could improve vision. The benefit achieved with this ideal method will be limited by decentration. To estimate the significance of this potential limitation we studied the effect on image quality expected when an ideal correcting method translates or rotates with respect to the eye’s pupil. Actual wave aberrations were obtained from ten human eyes for a 7.3-mm pupil with a Shack–Hartmann sensor. We computed the residual aberrations that appear as a result of translation or rotation of an otherwise ideal correction. The model is valid for adaptive optics, contact lenses, and phase plates, but it constitutes only a first approximation to the laser refractive surgery case where tissue removal occurs. Calculations suggest that the typical decentrations will reduce only slightly the optical benefits expected from an ideal correcting method. For typical decentrations the ideal correcting method offers a benefit in modulation 2–4 times higher (1.5–2 times in white light) than with a standard correction of defocus and astigmatism. We obtained analytical expressions that show the impact of translation and rotation on individual Zernike terms. These calculations also reveal which aberrations are most beneficial to correct. We provided practical rules to implement a selective correction depending on the amount of decentration. An experimental study was performed with an aberrated artificial eye corrected with an adaptive optics system, validating the theoretical predictions. The results in a keratoconic subject, also corrected with adaptive optics, showed that important benefits are obtained despite decentrations in highly aberrated eyes.

© 2001 Optical Society of America

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  46. N. López-Gil, E. Villegas, A. Benito, E. Berrio, N. Chateau, P. Artal, “Are the aberrations introduced by soft contact lenses in the eye predictable?” Invest. Ophthalmol. Visual Sci. 41, S426 (2000).
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    [CrossRef]
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    [CrossRef]
  51. Liang et al. (see Ref. 25) found that aberrations beyond sixth order remained uncorrected after correction with adaptive optics and that the lower orders up to fourth order were significantly reduced. Although an explanation of this result is that the deformable mirror (37 actuators) could not correct those higher orders, a complementary reason may be that the effect of small decentration of observers makes ineffective a correction beyond the fourth-order.
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    [PubMed]
  54. N. Chateau, A. Blanchard, D. Baude, “Influence of myopia and aging on the optimal spherical aberration of soft contact lenses,” J. Opt. Soc. Am. A 15, 2589–2596 (1998).
    [CrossRef]
  55. P. Artal, A. Guirao, E. Villegas, C. González, N. Chateau, E. Berrio, “Image quality in eyes with spherical aberration induced by soft contact lenses,” Vision Science and Its Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 232–235.

2000 (9)

P. M. Prieto, F. Vargas-Martin, S. Goelz, P. Artal, “Analysis of the performance of the Shack–Hartmann sensor in the human eye,” J. Opt. Soc. Am. A 17, 1388–1398 (2000).
[CrossRef]

J. Porter, I. G. Cox, A. Guirao, R. J. Potvin, M. A. Lagana, D. R. Williams, “A compact description of the eye’s aberrations in a large population,” Invest. Ophthalmol. Visual Sci. 41, S428 (2000).

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

S. M. MacRae, J. Schwiegerling, R. Snyder, “Customized corneal ablation and super vision,” J. Refract. Surgery 16, S230–S235 (2000).

M. Mrochen, M. Kaemmerer, T. Seiler, “Wavefront-guided laser in situ keratomileusis: early results in three eyes,” J. Refract. Surgery 16, 116–121 (2000).

J. Schwiegerling, R. W. Snyder, “Eye movement during laser in situ keratomileusis,” J. Cataract Refractive Surgery 26, 345–351 (2000).
[CrossRef]

S. Bará, T. Mancebo, E. Moreno-Barriuso, “Positioning tolerances for phase plates compensating aberrations of the human eye,” Appl. Opt. 39, 3413–3420 (2000).
[CrossRef]

N. López-Gil, E. Villegas, A. Benito, E. Berrio, N. Chateau, P. Artal, “Are the aberrations introduced by soft contact lenses in the eye predictable?” Invest. Ophthalmol. Visual Sci. 41, S426 (2000).

H. Hofer, P. Artal, B. Singer, J. L. Aragon, D. R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A 18, 497–506 (2000).
[CrossRef]

1999 (3)

S. Marcos, S. A. Burns, E. Moreno-Barriuso, R. Navarro, “A new approach to the study of ocular chromatic aberrations,” Vision Res. 39, 4309–4323 (1999).
[CrossRef]

T. Oshika, S. D. Klyce, R. A. Applegate, H. C. Howland, M. A. El Danasoury, “Comparison of corneal wavefront aberrations after photorefractive keratectomy and laser in situ keratomileusis,” Am. J. Ophthalmol. 127, 1–7 (1999).
[CrossRef] [PubMed]

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

1998 (4)

1997 (5)

J. Schwiegerling, J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vision Sci. 74, 906–916 (1997).
[CrossRef]

J. Liang, D. R. Williams, D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
[CrossRef]

J. Liang, D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
[CrossRef]

R. Navarro, M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vision Sci. 74, 540–547 (1997).
[CrossRef]

P. Mierdel, H. E. Krinke, W. Wiegand, M. Kaemmerer, T. Seiler, “A measuring device for the assessment of monochromatic aberrations in human eyes,” Ophthalmologe 94, 441–445 (1997).
[CrossRef] [PubMed]

1996 (2)

A. J. Taylor, S. D. R. Wilson, “Centration mechanism of soft contact lenses,” Optom. Vision Sci. 73, 215–221 (1996).
[CrossRef]

N. Chateau, J. de Brabander, F. Bouchard, H. Molenaar, “Infrared pupillometry in presbyopes fitted with soft contact lenses,” Optom. Vision Sci. 73, 733–741 (1996).
[CrossRef]

1995 (2)

P. Artal, S. Marcos, R. Navarro, D. R. Williams, “Odd aberrations and double-pass measurements of retinal image quality,” J. Opt. Soc. Am. A 12, 195–201 (1995).
[CrossRef]

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, 36–42 (1995).

1994 (4)

P. Artal, R. Navarro, “Monochromatic modulation transfer function of the human eye for different pupil diameters: an analytical expression,” J. Opt. Soc. Am. A 11, 246–249 (1994).
[CrossRef]

A. Tomlinson, W. H. Ridder, R. Watanabe, “Blink-induced variations in visual performance with toric soft contact lenses,” Optom. Vision Sci. 71, 545–549 (1994).
[CrossRef]

J. Liang, B. Grimm, S. Goelz, J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Shack–Hartmann wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
[CrossRef]

S. A. Little, A. S. Bruce, “Hydrogel (Acuvue) lens movement is influenced by the postlens tear film,” Optom. Vision Sci. 71, 364–370 (1994).
[CrossRef]

1990 (1)

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

1987 (1)

1985 (3)

D. R. Williams, “Visibility of interference fringes near the resolution limit,” J. Opt. Soc. Am. A 2, 1087–1093 (1985).
[CrossRef] [PubMed]

P. Erickson, M. Robboy, “Performance characteristics of a hydrophilic concentric bifocal contact lens,” Am. J. Optom. Physiol. Opt. 62, 702–708 (1985).
[CrossRef] [PubMed]

D. R. Williams, “Aliasing in foveal human vision,” Vision Res. 25, 195–205 (1985).
[CrossRef]

1984 (1)

1983 (1)

A. Tomlinson, “Succeeding with toric soft lenses,” Rev. Optom. 120, 71–80 (1983).

1980 (1)

1978 (1)

L. Alvarez, “Development of variable-focus lenses and a new refractor,” J. Am. Optom. Assoc. 49, 24–29 (1978).
[PubMed]

1977 (1)

1976 (1)

1966 (1)

F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. 186, 558–578 (1966).
[PubMed]

1962 (1)

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 7, 766–795 (1962).

Alvarez, L.

L. Alvarez, “Development of variable-focus lenses and a new refractor,” J. Am. Optom. Assoc. 49, 24–29 (1978).
[PubMed]

Applegate, R.

Applegate, R. A.

T. Oshika, S. D. Klyce, R. A. Applegate, H. C. Howland, M. A. El Danasoury, “Comparison of corneal wavefront aberrations after photorefractive keratectomy and laser in situ keratomileusis,” Am. J. Ophthalmol. 127, 1–7 (1999).
[CrossRef] [PubMed]

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, 36–42 (1995).

Aragon, J. L.

Artal, P.

H. Hofer, P. Artal, B. Singer, J. L. Aragon, D. R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A 18, 497–506 (2000).
[CrossRef]

N. López-Gil, E. Villegas, A. Benito, E. Berrio, N. Chateau, P. Artal, “Are the aberrations introduced by soft contact lenses in the eye predictable?” Invest. Ophthalmol. Visual Sci. 41, S426 (2000).

P. M. Prieto, F. Vargas-Martin, S. Goelz, P. Artal, “Analysis of the performance of the Shack–Hartmann sensor in the human eye,” J. Opt. Soc. Am. A 17, 1388–1398 (2000).
[CrossRef]

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

I. Iglesias, M. E. Berrio, P. Artal, “Estimates of the ocular-wave aberration from pairs of double-pass retinal images,” J. Opt. Soc. Am. A 15, 2466–2476 (1998).
[CrossRef]

P. Artal, S. Marcos, R. Navarro, D. R. Williams, “Odd aberrations and double-pass measurements of retinal image quality,” J. Opt. Soc. Am. A 12, 195–201 (1995).
[CrossRef]

P. Artal, R. Navarro, “Monochromatic modulation transfer function of the human eye for different pupil diameters: an analytical expression,” J. Opt. Soc. Am. A 11, 246–249 (1994).
[CrossRef]

J. Santamarı́a, P. Artal, J. Bescós, “Determination of the point-spread function of human eyes using a hybrid optical–digital method,” J. Opt. Soc. Am. A 4, 1109–1114 (1987).
[CrossRef]

P. Artal, A. Guirao, E. Villegas, C. González, N. Chateau, E. Berrio, “Image quality in eyes with spherical aberration induced by soft contact lenses,” Vision Science and Its Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 232–235.

Bara, S.

Bará, S.

Baude, D.

Benito, A.

N. López-Gil, E. Villegas, A. Benito, E. Berrio, N. Chateau, P. Artal, “Are the aberrations introduced by soft contact lenses in the eye predictable?” Invest. Ophthalmol. Visual Sci. 41, S426 (2000).

Berrio, E.

N. López-Gil, E. Villegas, A. Benito, E. Berrio, N. Chateau, P. Artal, “Are the aberrations introduced by soft contact lenses in the eye predictable?” Invest. Ophthalmol. Visual Sci. 41, S426 (2000).

P. Artal, A. Guirao, E. Villegas, C. González, N. Chateau, E. Berrio, “Image quality in eyes with spherical aberration induced by soft contact lenses,” Vision Science and Its Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 232–235.

Berrio, M. E.

Bescós, J.

Bille, J. F.

Blanchard, A.

Bouchard, F.

N. Chateau, J. de Brabander, F. Bouchard, H. Molenaar, “Infrared pupillometry in presbyopes fitted with soft contact lenses,” Optom. Vision Sci. 73, 733–741 (1996).
[CrossRef]

Bradley, A.

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

Bruce, A. S.

S. A. Little, A. S. Bruce, “Hydrogel (Acuvue) lens movement is influenced by the postlens tear film,” Optom. Vision Sci. 71, 364–370 (1994).
[CrossRef]

Buettner, J.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, 36–42 (1995).

Burns, S. A.

S. Marcos, S. A. Burns, E. Moreno-Barriuso, R. Navarro, “A new approach to the study of ocular chromatic aberrations,” Vision Res. 39, 4309–4323 (1999).
[CrossRef]

J. C. He, S. Marcos, R. H. Webb, S. A. Burns, “Measurement of the wavefront aberration of the eye by a fast psychophysical procedure,” J. Opt. Soc. Am. A 15, 2449–2456 (1998).
[CrossRef]

Campbell, F. W.

F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. 186, 558–578 (1966).
[PubMed]

Charman, N.

Charman, W. N.

G. Walsh, W. N. Charman, H. C. Howland, “Objective technique for the determination of monochromatic aberra-tions of the human eye,” J. Opt. Soc. Am. A 1, 987–992 (1984).
[CrossRef] [PubMed]

W. N. Charman, “The retinal image in the human eye,” in Progress in Retinal Research, N. Osborne, G. Chader, eds. (Pergamon, Oxford, 1983), Vol. 2, Chap. 1.

Chateau, N.

N. López-Gil, E. Villegas, A. Benito, E. Berrio, N. Chateau, P. Artal, “Are the aberrations introduced by soft contact lenses in the eye predictable?” Invest. Ophthalmol. Visual Sci. 41, S426 (2000).

N. Chateau, A. Blanchard, D. Baude, “Influence of myopia and aging on the optimal spherical aberration of soft contact lenses,” J. Opt. Soc. Am. A 15, 2589–2596 (1998).
[CrossRef]

N. Chateau, J. de Brabander, F. Bouchard, H. Molenaar, “Infrared pupillometry in presbyopes fitted with soft contact lenses,” Optom. Vision Sci. 73, 733–741 (1996).
[CrossRef]

P. Artal, A. Guirao, E. Villegas, C. González, N. Chateau, E. Berrio, “Image quality in eyes with spherical aberration induced by soft contact lenses,” Vision Science and Its Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 232–235.

Cottingham, A. J.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, 36–42 (1995).

Cox, I. G.

J. Porter, I. G. Cox, A. Guirao, R. J. Potvin, M. A. Lagana, D. R. Williams, “A compact description of the eye’s aberrations in a large population,” Invest. Ophthalmol. Visual Sci. 41, S428 (2000).

de Brabander, J.

N. Chateau, J. de Brabander, F. Bouchard, H. Molenaar, “Infrared pupillometry in presbyopes fitted with soft contact lenses,” Optom. Vision Sci. 73, 733–741 (1996).
[CrossRef]

El Danasoury, M. A.

T. Oshika, S. D. Klyce, R. A. Applegate, H. C. Howland, M. A. El Danasoury, “Comparison of corneal wavefront aberrations after photorefractive keratectomy and laser in situ keratomileusis,” Am. J. Ophthalmol. 127, 1–7 (1999).
[CrossRef] [PubMed]

Erickson, P.

P. Erickson, M. Robboy, “Performance characteristics of a hydrophilic concentric bifocal contact lens,” Am. J. Optom. Physiol. Opt. 62, 702–708 (1985).
[CrossRef] [PubMed]

Geraghty, E.

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

Goelz, S.

Gonzalez, C.

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

González, C.

P. Artal, A. Guirao, E. Villegas, C. González, N. Chateau, E. Berrio, “Image quality in eyes with spherical aberration induced by soft contact lenses,” Vision Science and Its Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 232–235.

Goodman, J. W.

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

Greivenkamp, J. E.

J. Schwiegerling, J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vision Sci. 74, 906–916 (1997).
[CrossRef]

Grimm, B.

Gubisch, R. W.

F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. 186, 558–578 (1966).
[PubMed]

Guirao, A.

J. Porter, I. G. Cox, A. Guirao, R. J. Potvin, M. A. Lagana, D. R. Williams, “A compact description of the eye’s aberrations in a large population,” Invest. Ophthalmol. Visual Sci. 41, S428 (2000).

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

P. Artal, A. Guirao, E. Villegas, C. González, N. Chateau, E. Berrio, “Image quality in eyes with spherical aberration induced by soft contact lenses,” Vision Science and Its Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 232–235.

He, J. C.

Hofer, H.

Howarth, P. A.

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

Howland, B.

Howland, H. C.

T. Oshika, S. D. Klyce, R. A. Applegate, H. C. Howland, M. A. El Danasoury, “Comparison of corneal wavefront aberrations after photorefractive keratectomy and laser in situ keratomileusis,” Am. J. Ophthalmol. 127, 1–7 (1999).
[CrossRef] [PubMed]

N. López-Gil, H. C. Howland, B. Howland, N. Charman, R. Applegate, “Generation of third-order spherical and coma aberration using radially symmetric fourth-order lenses,” J. Opt. Soc. Am. A 15, 2563–2571 (1998).
[CrossRef]

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, 36–42 (1995).

G. Walsh, W. N. Charman, H. C. Howland, “Objective technique for the determination of monochromatic aberra-tions of the human eye,” J. Opt. Soc. Am. A 1, 987–992 (1984).
[CrossRef] [PubMed]

H. C. Howland, B. Howland, “A subjective method for the measurement of monochromatic aberrations of the eye,” J. Opt. Soc. Am. 67, 1508–1518 (1977).
[CrossRef] [PubMed]

Iglesias, I.

Kaemmerer, M.

M. Mrochen, M. Kaemmerer, T. Seiler, “Wavefront-guided laser in situ keratomileusis: early results in three eyes,” J. Refract. Surgery 16, 116–121 (2000).

P. Mierdel, H. E. Krinke, W. Wiegand, M. Kaemmerer, T. Seiler, “A measuring device for the assessment of monochromatic aberrations in human eyes,” Ophthalmologe 94, 441–445 (1997).
[CrossRef] [PubMed]

Kim, C-J.

C-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987), Vol. X, Chap. 4.

Klyce, S. D.

T. Oshika, S. D. Klyce, R. A. Applegate, H. C. Howland, M. A. El Danasoury, “Comparison of corneal wavefront aberrations after photorefractive keratectomy and laser in situ keratomileusis,” Am. J. Ophthalmol. 127, 1–7 (1999).
[CrossRef] [PubMed]

Krinke, H. E.

P. Mierdel, H. E. Krinke, W. Wiegand, M. Kaemmerer, T. Seiler, “A measuring device for the assessment of monochromatic aberrations in human eyes,” Ophthalmologe 94, 441–445 (1997).
[CrossRef] [PubMed]

Lagana, M. A.

J. Porter, I. G. Cox, A. Guirao, R. J. Potvin, M. A. Lagana, D. R. Williams, “A compact description of the eye’s aberrations in a large population,” Invest. Ophthalmol. Visual Sci. 41, S428 (2000).

Liang, J.

Little, S. A.

S. A. Little, A. S. Bruce, “Hydrogel (Acuvue) lens movement is influenced by the postlens tear film,” Optom. Vision Sci. 71, 364–370 (1994).
[CrossRef]

López-Gil, N.

N. López-Gil, E. Villegas, A. Benito, E. Berrio, N. Chateau, P. Artal, “Are the aberrations introduced by soft contact lenses in the eye predictable?” Invest. Ophthalmol. Visual Sci. 41, S426 (2000).

N. López-Gil, H. C. Howland, B. Howland, N. Charman, R. Applegate, “Generation of third-order spherical and coma aberration using radially symmetric fourth-order lenses,” J. Opt. Soc. Am. A 15, 2563–2571 (1998).
[CrossRef]

Losada, M. A.

R. Navarro, M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vision Sci. 74, 540–547 (1997).
[CrossRef]

MacRae, S. M.

S. M. MacRae, J. Schwiegerling, R. Snyder, “Customized corneal ablation and super vision,” J. Refract. Surgery 16, S230–S235 (2000).

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations (SPIE Press, Bellingham, Wash., 1998).

Mancebo, T.

Mandell, R. B.

R. B. Mandell, Contact Lens Practice, 4th ed. (Charles C. Thomas, Springfield, Ill., 1988).

Marcos, S.

Mierdel, P.

P. Mierdel, H. E. Krinke, W. Wiegand, M. Kaemmerer, T. Seiler, “A measuring device for the assessment of monochromatic aberrations in human eyes,” Ophthalmologe 94, 441–445 (1997).
[CrossRef] [PubMed]

Miller, D. T.

J. Liang, D. R. Williams, D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
[CrossRef]

D. R. Williams, J. Liang, D. T. Miller, A. Roorda, “Wavefront sensing and compensation for the human eye,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), Chap. 10.

Molenaar, H.

N. Chateau, J. de Brabander, F. Bouchard, H. Molenaar, “Infrared pupillometry in presbyopes fitted with soft contact lenses,” Optom. Vision Sci. 73, 733–741 (1996).
[CrossRef]

Moreno-Barriuso, E.

Mrochen, M.

M. Mrochen, M. Kaemmerer, T. Seiler, “Wavefront-guided laser in situ keratomileusis: early results in three eyes,” J. Refract. Surgery 16, 116–121 (2000).

Navarro, R.

Noll, R. J.

Norrby, S.

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

Oshika, T.

T. Oshika, S. D. Klyce, R. A. Applegate, H. C. Howland, M. A. El Danasoury, “Comparison of corneal wavefront aberrations after photorefractive keratectomy and laser in situ keratomileusis,” Am. J. Ophthalmol. 127, 1–7 (1999).
[CrossRef] [PubMed]

Porter, J.

J. Porter, I. G. Cox, A. Guirao, R. J. Potvin, M. A. Lagana, D. R. Williams, “A compact description of the eye’s aberrations in a large population,” Invest. Ophthalmol. Visual Sci. 41, S428 (2000).

Potvin, R. J.

J. Porter, I. G. Cox, A. Guirao, R. J. Potvin, M. A. Lagana, D. R. Williams, “A compact description of the eye’s aberrations in a large population,” Invest. Ophthalmol. Visual Sci. 41, S428 (2000).

Prieto, P. M.

Redondo, M.

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

Ridder, W. H.

A. Tomlinson, W. H. Ridder, R. Watanabe, “Blink-induced variations in visual performance with toric soft contact lenses,” Optom. Vision Sci. 71, 545–549 (1994).
[CrossRef]

Robboy, M.

P. Erickson, M. Robboy, “Performance characteristics of a hydrophilic concentric bifocal contact lens,” Am. J. Optom. Physiol. Opt. 62, 702–708 (1985).
[CrossRef] [PubMed]

Roorda, A.

D. R. Williams, J. Liang, D. T. Miller, A. Roorda, “Wavefront sensing and compensation for the human eye,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), Chap. 10.

Santamari´a, J.

Schwiegerling, J.

S. M. MacRae, J. Schwiegerling, R. Snyder, “Customized corneal ablation and super vision,” J. Refract. Surgery 16, S230–S235 (2000).

J. Schwiegerling, R. W. Snyder, “Eye movement during laser in situ keratomileusis,” J. Cataract Refractive Surgery 26, 345–351 (2000).
[CrossRef]

J. Schwiegerling, J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vision Sci. 74, 906–916 (1997).
[CrossRef]

Seiler, T.

M. Mrochen, M. Kaemmerer, T. Seiler, “Wavefront-guided laser in situ keratomileusis: early results in three eyes,” J. Refract. Surgery 16, 116–121 (2000).

P. Mierdel, H. E. Krinke, W. Wiegand, M. Kaemmerer, T. Seiler, “A measuring device for the assessment of monochromatic aberrations in human eyes,” Ophthalmologe 94, 441–445 (1997).
[CrossRef] [PubMed]

Shannon, R. R.

C-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987), Vol. X, Chap. 4.

Sharp, R. P.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, 36–42 (1995).

Silva, D. E.

Singer, B.

Smirnov, M. S.

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 7, 766–795 (1962).

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).

Snyder, R.

S. M. MacRae, J. Schwiegerling, R. Snyder, “Customized corneal ablation and super vision,” J. Refract. Surgery 16, S230–S235 (2000).

Snyder, R. W.

J. Schwiegerling, R. W. Snyder, “Eye movement during laser in situ keratomileusis,” J. Cataract Refractive Surgery 26, 345–351 (2000).
[CrossRef]

Still, D. L.

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

Taylor, A. J.

A. J. Taylor, S. D. R. Wilson, “Centration mechanism of soft contact lenses,” Optom. Vision Sci. 73, 215–221 (1996).
[CrossRef]

Thibos, L. N.

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

Tomlinson, A.

A. Tomlinson, W. H. Ridder, R. Watanabe, “Blink-induced variations in visual performance with toric soft contact lenses,” Optom. Vision Sci. 71, 545–549 (1994).
[CrossRef]

A. Tomlinson, “Succeeding with toric soft lenses,” Rev. Optom. 120, 71–80 (1983).

Vargas-Martin, F.

Villegas, E.

N. López-Gil, E. Villegas, A. Benito, E. Berrio, N. Chateau, P. Artal, “Are the aberrations introduced by soft contact lenses in the eye predictable?” Invest. Ophthalmol. Visual Sci. 41, S426 (2000).

P. Artal, A. Guirao, E. Villegas, C. González, N. Chateau, E. Berrio, “Image quality in eyes with spherical aberration induced by soft contact lenses,” Vision Science and Its Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 232–235.

von Helmholtz, H.

H. von Helmholtz, Physiological Optics, J. P. C. Southall, ed. (Dover, New York, 1896).

Walsh, G.

Wang, J. Y.

Watanabe, R.

A. Tomlinson, W. H. Ridder, R. Watanabe, “Blink-induced variations in visual performance with toric soft contact lenses,” Optom. Vision Sci. 71, 545–549 (1994).
[CrossRef]

Webb, R. H.

Wiegand, W.

P. Mierdel, H. E. Krinke, W. Wiegand, M. Kaemmerer, T. Seiler, “A measuring device for the assessment of monochromatic aberrations in human eyes,” Ophthalmologe 94, 441–445 (1997).
[CrossRef] [PubMed]

Williams, D. R.

J. Porter, I. G. Cox, A. Guirao, R. J. Potvin, M. A. Lagana, D. R. Williams, “A compact description of the eye’s aberrations in a large population,” Invest. Ophthalmol. Visual Sci. 41, S428 (2000).

H. Hofer, P. Artal, B. Singer, J. L. Aragon, D. R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A 18, 497–506 (2000).
[CrossRef]

J. Liang, D. R. Williams, D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
[CrossRef]

J. Liang, D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
[CrossRef]

P. Artal, S. Marcos, R. Navarro, D. R. Williams, “Odd aberrations and double-pass measurements of retinal image quality,” J. Opt. Soc. Am. A 12, 195–201 (1995).
[CrossRef]

D. R. Williams, “Aliasing in foveal human vision,” Vision Res. 25, 195–205 (1985).
[CrossRef]

D. R. Williams, “Visibility of interference fringes near the resolution limit,” J. Opt. Soc. Am. A 2, 1087–1093 (1985).
[CrossRef] [PubMed]

D. R. Williams, J. Liang, D. T. Miller, A. Roorda, “Wavefront sensing and compensation for the human eye,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), Chap. 10.

G. Y. Yoon, D. R. Williams, “Visual benefit of correcting the higher-order monochromatic aberrations and longitudinal chromatic aberration in the eye,” in Vision Science and Its Applications, 2000 OSA Technical Digest Series (Optical Society of America, Washington D.C., 2000) pp. PD5-1–PD5-4.

Wilson, S. D. R.

A. J. Taylor, S. D. R. Wilson, “Centration mechanism of soft contact lenses,” Optom. Vision Sci. 73, 215–221 (1996).
[CrossRef]

Yee, R. W.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, 36–42 (1995).

Yoon, G. Y.

G. Y. Yoon, D. R. Williams, “Visual benefit of correcting the higher-order monochromatic aberrations and longitudinal chromatic aberration in the eye,” in Vision Science and Its Applications, 2000 OSA Technical Digest Series (Optical Society of America, Washington D.C., 2000) pp. PD5-1–PD5-4.

Zhang, X.

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

Am. J. Ophthalmol. (1)

T. Oshika, S. D. Klyce, R. A. Applegate, H. C. Howland, M. A. El Danasoury, “Comparison of corneal wavefront aberrations after photorefractive keratectomy and laser in situ keratomileusis,” Am. J. Ophthalmol. 127, 1–7 (1999).
[CrossRef] [PubMed]

Am. J. Optom. Physiol. Opt. (1)

P. Erickson, M. Robboy, “Performance characteristics of a hydrophilic concentric bifocal contact lens,” Am. J. Optom. Physiol. Opt. 62, 702–708 (1985).
[CrossRef] [PubMed]

Appl. Opt. (2)

Biophysics (1)

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 7, 766–795 (1962).

Invest. Ophthalmol. Visual Sci. (4)

J. Porter, I. G. Cox, A. Guirao, R. J. Potvin, M. A. Lagana, D. R. Williams, “A compact description of the eye’s aberrations in a large population,” Invest. Ophthalmol. Visual Sci. 41, S428 (2000).

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottingham, R. P. Sharp, R. W. Yee, “Radial keratotomy (RK), corneal aberrations and visual performance,” Invest. Ophthalmol. Visual Sci. 36, 36–42 (1995).

N. López-Gil, E. Villegas, A. Benito, E. Berrio, N. Chateau, P. Artal, “Are the aberrations introduced by soft contact lenses in the eye predictable?” Invest. Ophthalmol. Visual Sci. 41, S426 (2000).

J. Am. Optom. Assoc. (1)

L. Alvarez, “Development of variable-focus lenses and a new refractor,” J. Am. Optom. Assoc. 49, 24–29 (1978).
[PubMed]

J. Cataract Refractive Surgery (1)

J. Schwiegerling, R. W. Snyder, “Eye movement during laser in situ keratomileusis,” J. Cataract Refractive Surgery 26, 345–351 (2000).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (14)

G. Walsh, W. N. Charman, H. C. Howland, “Objective technique for the determination of monochromatic aberra-tions of the human eye,” J. Opt. Soc. Am. A 1, 987–992 (1984).
[CrossRef] [PubMed]

J. Santamarı́a, P. Artal, J. Bescós, “Determination of the point-spread function of human eyes using a hybrid optical–digital method,” J. Opt. Soc. Am. A 4, 1109–1114 (1987).
[CrossRef]

J. Liang, B. Grimm, S. Goelz, J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Shack–Hartmann wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
[CrossRef]

P. Artal, S. Marcos, R. Navarro, D. R. Williams, “Odd aberrations and double-pass measurements of retinal image quality,” J. Opt. Soc. Am. A 12, 195–201 (1995).
[CrossRef]

J. Liang, D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
[CrossRef]

I. Iglesias, M. E. Berrio, P. Artal, “Estimates of the ocular-wave aberration from pairs of double-pass retinal images,” J. Opt. Soc. Am. A 15, 2466–2476 (1998).
[CrossRef]

J. C. He, S. Marcos, R. H. Webb, S. A. Burns, “Measurement of the wavefront aberration of the eye by a fast psychophysical procedure,” J. Opt. Soc. Am. A 15, 2449–2456 (1998).
[CrossRef]

P. Artal, R. Navarro, “Monochromatic modulation transfer function of the human eye for different pupil diameters: an analytical expression,” J. Opt. Soc. Am. A 11, 246–249 (1994).
[CrossRef]

P. M. Prieto, F. Vargas-Martin, S. Goelz, P. Artal, “Analysis of the performance of the Shack–Hartmann sensor in the human eye,” J. Opt. Soc. Am. A 17, 1388–1398 (2000).
[CrossRef]

J. Liang, D. R. Williams, D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
[CrossRef]

N. Chateau, A. Blanchard, D. Baude, “Influence of myopia and aging on the optimal spherical aberration of soft contact lenses,” J. Opt. Soc. Am. A 15, 2589–2596 (1998).
[CrossRef]

H. Hofer, P. Artal, B. Singer, J. L. Aragon, D. R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A 18, 497–506 (2000).
[CrossRef]

N. López-Gil, H. C. Howland, B. Howland, N. Charman, R. Applegate, “Generation of third-order spherical and coma aberration using radially symmetric fourth-order lenses,” J. Opt. Soc. Am. A 15, 2563–2571 (1998).
[CrossRef]

D. R. Williams, “Visibility of interference fringes near the resolution limit,” J. Opt. Soc. Am. A 2, 1087–1093 (1985).
[CrossRef] [PubMed]

J. Physiol. (1)

F. W. Campbell, R. W. Gubisch, “Optical quality of the human eye,” J. Physiol. 186, 558–578 (1966).
[PubMed]

J. Refract. Surgery (2)

S. M. MacRae, J. Schwiegerling, R. Snyder, “Customized corneal ablation and super vision,” J. Refract. Surgery 16, S230–S235 (2000).

M. Mrochen, M. Kaemmerer, T. Seiler, “Wavefront-guided laser in situ keratomileusis: early results in three eyes,” J. Refract. Surgery 16, 116–121 (2000).

Ophthalmologe (1)

P. Mierdel, H. E. Krinke, W. Wiegand, M. Kaemmerer, T. Seiler, “A measuring device for the assessment of monochromatic aberrations in human eyes,” Ophthalmologe 94, 441–445 (1997).
[CrossRef] [PubMed]

Opt. Lett. (1)

Optom. Vision Sci. (6)

J. Schwiegerling, J. E. Greivenkamp, “Using corneal height maps and polynomial decomposition to determine corneal aberrations,” Optom. Vision Sci. 74, 906–916 (1997).
[CrossRef]

R. Navarro, M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vision Sci. 74, 540–547 (1997).
[CrossRef]

A. Tomlinson, W. H. Ridder, R. Watanabe, “Blink-induced variations in visual performance with toric soft contact lenses,” Optom. Vision Sci. 71, 545–549 (1994).
[CrossRef]

N. Chateau, J. de Brabander, F. Bouchard, H. Molenaar, “Infrared pupillometry in presbyopes fitted with soft contact lenses,” Optom. Vision Sci. 73, 733–741 (1996).
[CrossRef]

A. J. Taylor, S. D. R. Wilson, “Centration mechanism of soft contact lenses,” Optom. Vision Sci. 73, 215–221 (1996).
[CrossRef]

S. A. Little, A. S. Bruce, “Hydrogel (Acuvue) lens movement is influenced by the postlens tear film,” Optom. Vision Sci. 71, 364–370 (1994).
[CrossRef]

Rev. Optom. (1)

A. Tomlinson, “Succeeding with toric soft lenses,” Rev. Optom. 120, 71–80 (1983).

Vision Res. (3)

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

S. Marcos, S. A. Burns, E. Moreno-Barriuso, R. Navarro, “A new approach to the study of ocular chromatic aberrations,” Vision Res. 39, 4309–4323 (1999).
[CrossRef]

D. R. Williams, “Aliasing in foveal human vision,” Vision Res. 25, 195–205 (1985).
[CrossRef]

Other (13)

Liang et al. (see Ref. 25) found that aberrations beyond sixth order remained uncorrected after correction with adaptive optics and that the lower orders up to fourth order were significantly reduced. Although an explanation of this result is that the deformable mirror (37 actuators) could not correct those higher orders, a complementary reason may be that the effect of small decentration of observers makes ineffective a correction beyond the fourth-order.

P. Artal, A. Guirao, E. Villegas, C. González, N. Chateau, E. Berrio, “Image quality in eyes with spherical aberration induced by soft contact lenses,” Vision Science and Its Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 232–235.

The exact factor depending on rotation in rule (R3) is 2[1-exp(m2σr2/2)], and approximately, 4 sin2(mσr/2), or m2σr2.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).

D. R. Williams, J. Liang, D. T. Miller, A. Roorda, “Wavefront sensing and compensation for the human eye,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), Chap. 10.

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

C-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987), Vol. X, Chap. 4.

This rms corresponds to a Strehl ratio lower than 0.05. This means that after a conventional correction of defocus and astigmatism, the optical quality is still poor in comparison with the diffraction limit.

V. N. Mahajan, Optical Imaging and Aberrations (SPIE Press, Bellingham, Wash., 1998).

R. B. Mandell, Contact Lens Practice, 4th ed. (Charles C. Thomas, Springfield, Ill., 1988).

G. Y. Yoon, D. R. Williams, “Visual benefit of correcting the higher-order monochromatic aberrations and longitudinal chromatic aberration in the eye,” in Vision Science and Its Applications, 2000 OSA Technical Digest Series (Optical Society of America, Washington D.C., 2000) pp. PD5-1–PD5-4.

W. N. Charman, “The retinal image in the human eye,” in Progress in Retinal Research, N. Osborne, G. Chader, eds. (Pergamon, Oxford, 1983), Vol. 2, Chap. 1.

H. von Helmholtz, Physiological Optics, J. P. C. Southall, ed. (Dover, New York, 1896).

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Figures (10)

Fig. 1
Fig. 1

Example of wave-aberration maps of the ideal correcting method, the eye, and the coupling. The correcting method is ideal in the sense that it corrects the monochromatic aberrations when it is centered. Decentration produces a mismatch.

Fig. 2
Fig. 2

Wave aberration of the correcting method (axis xy) decentered with respect to the eye (axis xy). The wave aberration added to that of the eye is the rotated (angle α) version and the translated (Δx, Δy) version of the wave aberration of the centered correcting method. The correcting method extends beyond the eye’s pupil.

Fig. 3
Fig. 3

Average MTF (monochromatic light) across the population of ten eyes after the correction of the higher-order aberrations of each eye with an ideal correcting method, as a function of (a) fixed rotations and (b) translations. Results are averaged across ±x and ±y axis for translation and ±α for rotation. The pupil is 6 mm in diameter. Also shown: the MTF of the average uncorrected eye, and the eye corrected for defocus and astigmatism (conventional correction). The conventional correction is considered perfectly centered.

Fig. 4
Fig. 4

Average MTF’s in the population for typical distributions of movement of the ideal correcting method in (a) monochromatic light and (b) white light. The MTF’s were averaged for each subject for a normally distributed translation and rotation with widths of 0.2 mm, 2°; 0.3 mm, 3°. Also shown: the average MTF for the uncorrected eye and the MTF for a centered conventional correction for a 6-mm pupil.

Fig. 5
Fig. 5

Rms (mean value in the ten-eye population) of the residual WA for a 6-mm pupil as a function of (a) fixed rotations and (b) translations, when the ideal correcting method corrects the higher-order aberrations up to second-, third-, fourth-, fifth- and sixth-order. Results for translation are averaged across ±x and ±y axis. Also shown in (a), the rms when only defocus and spherical aberration are corrected.

Fig. 6
Fig. 6

Rms (mean value in the 10-eye population) of the residual WA for a 6-mm pupil as a function of the width of the Gaussian distribution of movement, with correction of all six orders (thin curves), and optimum correction (thick curves) from the selection rule (R3). The selective correction has been computed for each interval of movement.

Fig. 7
Fig. 7

Theoretical values for the Zernike coefficients of the artificial eye versus experimental values measured for different rotations and translations after correction with adaptive optics.

Fig. 8
Fig. 8

MTF’s for the artificial eye after correcting with adaptive optics and applying different (a) rotations and (b) translations. Comparison is between experimental results (thick curves) and theoretical results (thin curves). Also shown: the MTF for a conventional centered correction.

Fig. 9
Fig. 9

MTF’s for the artificial eye for different rotations (thin curves) in comparison with the MTF’s obtained by correcting only astigmatism and coma (thick curves).

Fig. 10
Fig. 10

MTF’s for the keratoconic eye after the best correction with adaptive optics for a well-centered position (solid curve) and for a 0.3-mm horizontal translation (dashed curve). The MTF for the conventional correction is also shown for comparison.

Tables (4)

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Table 1 Zernike Polynomials up to Sixth Order

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Table 2 Conversion Matrix for Contact-Lens Aberration Coefficients after a Translation

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Table 3 Values of the Factor F for Fixed or Gaussian Translations

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Table 4 Maximum Decentrations to Have No Benefit (or a 50% Reduction) in the Contribution of the Terms Zn±m to the rms

Equations (16)

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WAeye(ρ, θ)=n,±man±mZn±m(ρ, θ),n=1, 2, , 6.
WAICM=-n,±man±mZn±m,
WAresidual=WAeye+WAICM(decentered)=n,±man±mZn±m-n,±mCn±mZn±m.
x=(x-Δx)cos α+(y-Δy)sin α,y=(y-Δy)cos α-(x-Δx)sin α,
Ci=j,kTijRjkak,
(1-TijRjk)ak.
WAresidual=noncorrectedaiZi+corrected(1-TijRjk)akZi.
cos mα-sin mαsin mαcos mα
1(2π)3/2σt2σr exp(-(Δx2+Δy2)/2σt2)exp(-α2/2σr2),
rms2=4n,m0[(an+m)2+(an-m)2]·sin2 mα/2.
mα<60°.
A2F<1,
F=(n+1)j(n-2j)Λj.
Λj=3ifm=11ifm=n-2j0ifm>n-2j2otherwise
m2σr2+σt2r02F<1,
Zn±m(x-Δx, y-Δy)=k=0 (-1)kk! Δx x+Δy yk[Zn±m(x, y)].

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