Abstract

We offer an analysis that shows that the approximations made for the ablation depth during practical refractive surgery, in which the square-root terms are replaced by the first two terms of the series expansion, can limit the visual function of the observer by reducing the modulation transfer function (MTF). To simulate the refractive-surgical operation, we considered two groups of myopic patients with different ametropia who were emmetropized with different ablation profiles. We made the MTF calculations by taking the spherical aberration into account. In addition, a fuller analysis showed that these approximations limit the possibility of considering surfaces that are aspherical for reshaping the anterior cornea to optimize the observer’s visual function.

© 2001 Optical Society of America

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References

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  1. J. T. Lin, “Critical review on refractive surgical lasers,” Opt. Eng. 34, 668–675 (1995).
    [CrossRef]
  2. J.-H. Shen, P. Soderberg, T. Matsui, F. Manns, J.-M. Parel, “Development of an algorithm for corneal reshaping with a scanning laser beam,” Appl. Opt. 34, 4600–4608 (1995).
    [CrossRef] [PubMed]
  3. J. Schwiegerling, R. W. Snyder, “Custom photorefractive keratectomy ablations for the correction of spherical and cylindrical refractive error and higher-order aberration,” J. Opt. Soc. Am. A 15, 2572–2579 (1998).
    [CrossRef]
  4. J. F. Fleming, “Corneal asphericity and visual function after radial keratotomy,” Cornea 12, 233–240 (1993).
    [CrossRef] [PubMed]
  5. L. G. Carney, J. C. Mainstone, B. A. Henderson, “Corneal topography and myopia,” Invest. Ophthalmol. Visual Sci. 38, 311–318 (1997).
  6. J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, L. Carretero, “Retinal image quality in myopic subjects after refractive surgery,” J. Mod. Opt. 47, 1587–1598 (2000).
  7. D. A. Atchison, G. Smith, Optics of the Human Eye (Butterworth-Heinemann, Oxford, 2000), pp. 11 and 167.
    [CrossRef]
  8. J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, “Influence of aspheric of the anterior corneal surface in visual performance,” Optik (Stuttgart) 111, 429–434 (2000).
  9. R. Navarro, J. Santamaría, J. Bescós, “Accommodation-dependent model of the human eye with aspherics,” J. Opt. Soc. Am. A 2, 1273–1281 (1985).
    [CrossRef] [PubMed]
  10. E. R. Villegas, L. Carretero, A. Fimia, “Image quality in pseudophakic eyes with two different types of intraocular lenses ranging in the degree of high myopia,” J. Biomed. Opt. 2, 1–7 (1997).
  11. E. R. Villegas, L. Carretero, A. Fimia, “Optimum bending factor of intraocular lenses in pseudophakic eyes with high miopia,” J. Mod. Opt. 44, 941–952 (1997).
    [CrossRef]
  12. I. Escudero-Sanz, R. Navarro, “Off-axis aberrations of a wide-angle schematic eye model,” J. Opt. Soc. Am. A 16, 1881–1891 (1999).
    [CrossRef]
  13. H. L. Liou, N. A. Brennan, “Anatomically accurate, finite model eye for optical modelling,” J. Opt. Soc. Am. A 14, 1684–1695 (1997).
    [CrossRef]
  14. R. Barakat, “Computation of the transfer function of an optical system from the design data for rotationally symmetric aberrations. I. Theory,” J. Opt. Soc. Am. 52, 985–991 (1962).
    [CrossRef]
  15. R. Barakat, M. V. Morello, “Computation of the transfer function of an optical system from the design data for rotationally symmetric aberrations. II. Programming and numerical results,” J. Opt. Soc. Am 52, 992–997 (1962).
    [CrossRef]
  16. D. Malacara, Z. Malacara, Handbook of Lens Design (Marcel Dekker, New York, 1994).
  17. F. Eghbali, K. K. Yeung, R. K. Maloney, “Topographic determination of corneal asphericity and its lack of effect on the refractive outcome of radial keratotomy,” Am. J. Ophthalmol. 119, 275–280 (1995).
    [PubMed]
  18. S. Patel, J. Marshall, F. W. Fitzke, “Model for predicting the optical performance of the eye in refractive surgery,” Refract. Corneal Surg. 9, 366–375 (1993).
    [PubMed]

2000

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, L. Carretero, “Retinal image quality in myopic subjects after refractive surgery,” J. Mod. Opt. 47, 1587–1598 (2000).

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, “Influence of aspheric of the anterior corneal surface in visual performance,” Optik (Stuttgart) 111, 429–434 (2000).

1999

1998

1997

E. R. Villegas, L. Carretero, A. Fimia, “Image quality in pseudophakic eyes with two different types of intraocular lenses ranging in the degree of high myopia,” J. Biomed. Opt. 2, 1–7 (1997).

E. R. Villegas, L. Carretero, A. Fimia, “Optimum bending factor of intraocular lenses in pseudophakic eyes with high miopia,” J. Mod. Opt. 44, 941–952 (1997).
[CrossRef]

H. L. Liou, N. A. Brennan, “Anatomically accurate, finite model eye for optical modelling,” J. Opt. Soc. Am. A 14, 1684–1695 (1997).
[CrossRef]

L. G. Carney, J. C. Mainstone, B. A. Henderson, “Corneal topography and myopia,” Invest. Ophthalmol. Visual Sci. 38, 311–318 (1997).

1995

F. Eghbali, K. K. Yeung, R. K. Maloney, “Topographic determination of corneal asphericity and its lack of effect on the refractive outcome of radial keratotomy,” Am. J. Ophthalmol. 119, 275–280 (1995).
[PubMed]

J. T. Lin, “Critical review on refractive surgical lasers,” Opt. Eng. 34, 668–675 (1995).
[CrossRef]

J.-H. Shen, P. Soderberg, T. Matsui, F. Manns, J.-M. Parel, “Development of an algorithm for corneal reshaping with a scanning laser beam,” Appl. Opt. 34, 4600–4608 (1995).
[CrossRef] [PubMed]

1993

J. F. Fleming, “Corneal asphericity and visual function after radial keratotomy,” Cornea 12, 233–240 (1993).
[CrossRef] [PubMed]

S. Patel, J. Marshall, F. W. Fitzke, “Model for predicting the optical performance of the eye in refractive surgery,” Refract. Corneal Surg. 9, 366–375 (1993).
[PubMed]

1985

1962

R. Barakat, “Computation of the transfer function of an optical system from the design data for rotationally symmetric aberrations. I. Theory,” J. Opt. Soc. Am. 52, 985–991 (1962).
[CrossRef]

R. Barakat, M. V. Morello, “Computation of the transfer function of an optical system from the design data for rotationally symmetric aberrations. II. Programming and numerical results,” J. Opt. Soc. Am 52, 992–997 (1962).
[CrossRef]

Anera, R. G.

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, L. Carretero, “Retinal image quality in myopic subjects after refractive surgery,” J. Mod. Opt. 47, 1587–1598 (2000).

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, “Influence of aspheric of the anterior corneal surface in visual performance,” Optik (Stuttgart) 111, 429–434 (2000).

Atchison, D. A.

D. A. Atchison, G. Smith, Optics of the Human Eye (Butterworth-Heinemann, Oxford, 2000), pp. 11 and 167.
[CrossRef]

Barakat, R.

R. Barakat, “Computation of the transfer function of an optical system from the design data for rotationally symmetric aberrations. I. Theory,” J. Opt. Soc. Am. 52, 985–991 (1962).
[CrossRef]

R. Barakat, M. V. Morello, “Computation of the transfer function of an optical system from the design data for rotationally symmetric aberrations. II. Programming and numerical results,” J. Opt. Soc. Am 52, 992–997 (1962).
[CrossRef]

Bescós, J.

Brennan, N. A.

Carney, L. G.

L. G. Carney, J. C. Mainstone, B. A. Henderson, “Corneal topography and myopia,” Invest. Ophthalmol. Visual Sci. 38, 311–318 (1997).

Carretero, L.

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, L. Carretero, “Retinal image quality in myopic subjects after refractive surgery,” J. Mod. Opt. 47, 1587–1598 (2000).

E. R. Villegas, L. Carretero, A. Fimia, “Image quality in pseudophakic eyes with two different types of intraocular lenses ranging in the degree of high myopia,” J. Biomed. Opt. 2, 1–7 (1997).

E. R. Villegas, L. Carretero, A. Fimia, “Optimum bending factor of intraocular lenses in pseudophakic eyes with high miopia,” J. Mod. Opt. 44, 941–952 (1997).
[CrossRef]

Eghbali, F.

F. Eghbali, K. K. Yeung, R. K. Maloney, “Topographic determination of corneal asphericity and its lack of effect on the refractive outcome of radial keratotomy,” Am. J. Ophthalmol. 119, 275–280 (1995).
[PubMed]

Escudero-Sanz, I.

Fimia, A.

E. R. Villegas, L. Carretero, A. Fimia, “Optimum bending factor of intraocular lenses in pseudophakic eyes with high miopia,” J. Mod. Opt. 44, 941–952 (1997).
[CrossRef]

E. R. Villegas, L. Carretero, A. Fimia, “Image quality in pseudophakic eyes with two different types of intraocular lenses ranging in the degree of high myopia,” J. Biomed. Opt. 2, 1–7 (1997).

Fitzke, F. W.

S. Patel, J. Marshall, F. W. Fitzke, “Model for predicting the optical performance of the eye in refractive surgery,” Refract. Corneal Surg. 9, 366–375 (1993).
[PubMed]

Fleming, J. F.

J. F. Fleming, “Corneal asphericity and visual function after radial keratotomy,” Cornea 12, 233–240 (1993).
[CrossRef] [PubMed]

Henderson, B. A.

L. G. Carney, J. C. Mainstone, B. A. Henderson, “Corneal topography and myopia,” Invest. Ophthalmol. Visual Sci. 38, 311–318 (1997).

Jiménez, J. R.

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, L. Carretero, “Retinal image quality in myopic subjects after refractive surgery,” J. Mod. Opt. 47, 1587–1598 (2000).

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, “Influence of aspheric of the anterior corneal surface in visual performance,” Optik (Stuttgart) 111, 429–434 (2000).

Jiménez del Barco, L.

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, “Influence of aspheric of the anterior corneal surface in visual performance,” Optik (Stuttgart) 111, 429–434 (2000).

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, L. Carretero, “Retinal image quality in myopic subjects after refractive surgery,” J. Mod. Opt. 47, 1587–1598 (2000).

Lin, J. T.

J. T. Lin, “Critical review on refractive surgical lasers,” Opt. Eng. 34, 668–675 (1995).
[CrossRef]

Liou, H. L.

Mainstone, J. C.

L. G. Carney, J. C. Mainstone, B. A. Henderson, “Corneal topography and myopia,” Invest. Ophthalmol. Visual Sci. 38, 311–318 (1997).

Malacara, D.

D. Malacara, Z. Malacara, Handbook of Lens Design (Marcel Dekker, New York, 1994).

Malacara, Z.

D. Malacara, Z. Malacara, Handbook of Lens Design (Marcel Dekker, New York, 1994).

Maloney, R. K.

F. Eghbali, K. K. Yeung, R. K. Maloney, “Topographic determination of corneal asphericity and its lack of effect on the refractive outcome of radial keratotomy,” Am. J. Ophthalmol. 119, 275–280 (1995).
[PubMed]

Manns, F.

Marshall, J.

S. Patel, J. Marshall, F. W. Fitzke, “Model for predicting the optical performance of the eye in refractive surgery,” Refract. Corneal Surg. 9, 366–375 (1993).
[PubMed]

Matsui, T.

Morello, M. V.

R. Barakat, M. V. Morello, “Computation of the transfer function of an optical system from the design data for rotationally symmetric aberrations. II. Programming and numerical results,” J. Opt. Soc. Am 52, 992–997 (1962).
[CrossRef]

Navarro, R.

Parel, J.-M.

Patel, S.

S. Patel, J. Marshall, F. W. Fitzke, “Model for predicting the optical performance of the eye in refractive surgery,” Refract. Corneal Surg. 9, 366–375 (1993).
[PubMed]

Santamaría, J.

Schwiegerling, J.

Shen, J.-H.

Smith, G.

D. A. Atchison, G. Smith, Optics of the Human Eye (Butterworth-Heinemann, Oxford, 2000), pp. 11 and 167.
[CrossRef]

Snyder, R. W.

Soderberg, P.

Villegas, E. R.

E. R. Villegas, L. Carretero, A. Fimia, “Image quality in pseudophakic eyes with two different types of intraocular lenses ranging in the degree of high myopia,” J. Biomed. Opt. 2, 1–7 (1997).

E. R. Villegas, L. Carretero, A. Fimia, “Optimum bending factor of intraocular lenses in pseudophakic eyes with high miopia,” J. Mod. Opt. 44, 941–952 (1997).
[CrossRef]

Yeung, K. K.

F. Eghbali, K. K. Yeung, R. K. Maloney, “Topographic determination of corneal asphericity and its lack of effect on the refractive outcome of radial keratotomy,” Am. J. Ophthalmol. 119, 275–280 (1995).
[PubMed]

Am. J. Ophthalmol.

F. Eghbali, K. K. Yeung, R. K. Maloney, “Topographic determination of corneal asphericity and its lack of effect on the refractive outcome of radial keratotomy,” Am. J. Ophthalmol. 119, 275–280 (1995).
[PubMed]

Appl. Opt.

Cornea

J. F. Fleming, “Corneal asphericity and visual function after radial keratotomy,” Cornea 12, 233–240 (1993).
[CrossRef] [PubMed]

Invest. Ophthalmol. Visual Sci.

L. G. Carney, J. C. Mainstone, B. A. Henderson, “Corneal topography and myopia,” Invest. Ophthalmol. Visual Sci. 38, 311–318 (1997).

J. Biomed. Opt.

E. R. Villegas, L. Carretero, A. Fimia, “Image quality in pseudophakic eyes with two different types of intraocular lenses ranging in the degree of high myopia,” J. Biomed. Opt. 2, 1–7 (1997).

J. Mod. Opt.

E. R. Villegas, L. Carretero, A. Fimia, “Optimum bending factor of intraocular lenses in pseudophakic eyes with high miopia,” J. Mod. Opt. 44, 941–952 (1997).
[CrossRef]

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, L. Carretero, “Retinal image quality in myopic subjects after refractive surgery,” J. Mod. Opt. 47, 1587–1598 (2000).

J. Opt. Soc. Am

R. Barakat, M. V. Morello, “Computation of the transfer function of an optical system from the design data for rotationally symmetric aberrations. II. Programming and numerical results,” J. Opt. Soc. Am 52, 992–997 (1962).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

J. T. Lin, “Critical review on refractive surgical lasers,” Opt. Eng. 34, 668–675 (1995).
[CrossRef]

Optik (Stuttgart)

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, “Influence of aspheric of the anterior corneal surface in visual performance,” Optik (Stuttgart) 111, 429–434 (2000).

Refract. Corneal Surg.

S. Patel, J. Marshall, F. W. Fitzke, “Model for predicting the optical performance of the eye in refractive surgery,” Refract. Corneal Surg. 9, 366–375 (1993).
[PubMed]

Other

D. A. Atchison, G. Smith, Optics of the Human Eye (Butterworth-Heinemann, Oxford, 2000), pp. 11 and 167.
[CrossRef]

D. Malacara, Z. Malacara, Handbook of Lens Design (Marcel Dekker, New York, 1994).

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Figures (4)

Fig. 1
Fig. 1

MTF corresponding to the group of patients with -2D myopia when the surgery is performed with the theoretical ablation depth (solid curve) and when it is performed with the approximation used in practical surgery (dotted curve). Wave-aberration coefficients are included.

Fig. 2
Fig. 2

MTF corresponding to the group of patients with -2D myopia when the surgery is performed with the theoretical ablation depth (solid curve) and when it is performed with an approximation that affects the corneal profile (dotted curve). Wave-aberration coefficients are included.

Fig. 3
Fig. 3

MTF corresponding to the group of patients with -4D myopia when the surgery is performed with the theoretical ablation depth (solid curve) and when it is performed with an approximation that affects the corneal profile (dotted curve). Wave-aberration coefficients are included.

Fig. 4
Fig. 4

Schematic of the corneal profile before and after refractive surgery, assuming corneal asphericity. (R 1, Q 1) and (R 2, Q 2) represent radius and corneal asphericity before and after surgery, respectively. Ablation-zone diameter, d; ablation depth, s.

Tables (1)

Tables Icon

Table 1 Parameters of Anterior Surface of the Cornea Calculated As an Average of Data from Myopic Subjects Provided by a Topographic Keratometera

Equations (17)

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

D=ΔnR2-ΔnR1,
sy=R12-y21/2-R22-y21/2+R22-d241/2-R12-d241/2.
1-a1/21-a/2,  a  1.
sy4Dy23-Dd23.
x2+y2+1+Qz2-2zR=0,
sy=R1-R2-Dy22Δn+R22-d241/2-R12-d241/2.
sy=11+Q1R12-y21+Q11/2-11+Q2×R22-y21+Q21/2+11+Q2R22-d24×1+Q21/2-11+Q1R12-d241+Q11/2.
11+QiRi2-1+Qix21/2Ri1+Qi1-1+Qix22Ri2=Ri1+Qi-x22Ri.
sy=4Dy23-d21+Q18R1-d21+Q28R2.
sy=4Dy23-d28R1-d28R2=4Dy23-Dd28Δn1=4Dy23-Dd23,
sy=4Dy23-d21+Q18R1-d21+Q18R2=4Dy23-Dd231+Q1.
OTFfx, fy=- Pξ+λdifx2, η+λdify2Pξ-λdifx2, η-λdify2dξdη- Pξ, ηdξdη,
OTFw=MRp2w/2Rp0Rp1-ξ2 PRpξ, Rpη×P*Rpξ-w, Rpηdξdη,
Pξ, η=exp-0.1151Rp2ξ2+η21/2 expikW20×ξ2+η2+W40ξ2+η22+W60ξ2+η23.
δr=-RwnWξ,  δs=-RwnWη,
W20λ=1/2Rp2C20λ,  W40λ=1/4Rp4C40λ, W60λ=1/4Rp6C60λ,
C20λη+C40λη3+C60λη5=-nδs/Rw.

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