Abstract

We calculate whether deviations of Lambert-Beer’s law, which regulates depth ablation during corneal ablation, significantly influence corneal refractive parameters after refractive surgery and whether they influence visual performance. For this, we compute a point-to-point correction on the cornea while assuming a non-linear (including a quadratic term) fit for depth ablation. Post-surgical equations for refractive parameters using a non-linear fit show significant differences with respect to parameters obtained from a linear fit (Lambert-Beer’s law). Differences were also significant for corneal aberrations. These results show that corneal-ablation algorithms should include analytical information on deviations from Lambert-Beer’s law for achieving an accurate eye correction

© 2006 Optical Society of America

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  1. J. R. Jiménez, R. G. Anera, J. A. Díaz and F. Pérez-Ocón, "Corneal asphericity after refractive surgery when the Munnerlyn formula is applied," J. Opt. Soc. Am. A 21, 98-103 (2004).
    [CrossRef]
  2. R. G. Anera, C. Villa, J. R. Jiménez, R. Gutiérrez and L. Jiménez del Barco, "Differences between real and predicted corneal shapes after aspherical corneal ablation," Appl. Opt. 44, 4528-4532 (2005).
    [CrossRef] [PubMed]
  3. J. R. Jiménez, R. G. Anera, L. Jiménez del Barco and E. Hita, "Influence of laser polarization on ocular refractive parameters after refractive surgery," Opt. Lett. 29, 962-965 (2004).
    [CrossRef] [PubMed]
  4. C. Roberts, "Biomechanical customization," J. Cataract Refract. Surg. 31, 2-5 (2005).
    [CrossRef] [PubMed]
  5. J. R. Jiménez, R. G. Anera, L. Jiménez del Barco and E. Hita, "Effect on laser-ablation algorithms of reflection losses and nonnormal incidence on the anterior cornea," Appl. Phys. Lett. 81, 1521-1523 (2002).
    [CrossRef]
  6. J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, E. Hita and F. Pérez-Ocón, "Correction factor for ablation algorithms used in corneal refractive surgery with gaussian-profile beams," Opt. Express 13, 336-342 (2005).
    [CrossRef] [PubMed]
  7. M. Mrochen and T. Seiler, "Influence of corneal curvature on calculation of ablation patterns used in photorefractive laser surgery," J. Refract. Surg. 17, S584-S587 (2001).
    [PubMed]
  8. F. Manns, J. Shen, P. Soderberg, T. Matsui and J. Parel, "Development of an algorithm for corneal reshaping with a scanning laser beam," Appl. Opt. 21, 4600-4608 (1995).
    [CrossRef]
  9. D. Huang and M. Arif, "Spot size and quality of scanning laser correction of higher-order wavefront aberrations," J. Cataract Refract. Surg. 28, 407-416 (2002).
    [CrossRef] [PubMed]
  10. G. H. Pettit, M. Ediger and R. P. Weiblinger, "Excimer laser ablation of the cornea," Opt. Eng. 34, 661-667 (1995).
    [CrossRef]
  11. R. R. Krueger and S. L. Trokel, "Quantization of corneal ablation by ultraviolet laser light," Arch. Ophthalmol. 103, 1741-1742 (1985).
    [CrossRef] [PubMed]
  12. Z. Bor, B. Hopp, B. Racz, G. Szabo, Z. Marton, I. Ratkay, J. Mohay, I. Suveges and A. Fust, "Physical problems of excimer laser cornea ablation," Opt. Eng. 32, 2481-2486 (1993).
    [CrossRef]
  13. A. Vogel and V. Venugopalan, "Mechanisms of pulsed laser ablation of biological tissues," Chem. Rev. 103, 577-644 (2003).
    [CrossRef] [PubMed]
  14. D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth Heinemann, Oxford, 2000) p. 11-20, 167, 195-210.
    [CrossRef]

2005 (3)

2004 (2)

2003 (1)

A. Vogel and V. Venugopalan, "Mechanisms of pulsed laser ablation of biological tissues," Chem. Rev. 103, 577-644 (2003).
[CrossRef] [PubMed]

2002 (2)

D. Huang and M. Arif, "Spot size and quality of scanning laser correction of higher-order wavefront aberrations," J. Cataract Refract. Surg. 28, 407-416 (2002).
[CrossRef] [PubMed]

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco and E. Hita, "Effect on laser-ablation algorithms of reflection losses and nonnormal incidence on the anterior cornea," Appl. Phys. Lett. 81, 1521-1523 (2002).
[CrossRef]

2001 (1)

M. Mrochen and T. Seiler, "Influence of corneal curvature on calculation of ablation patterns used in photorefractive laser surgery," J. Refract. Surg. 17, S584-S587 (2001).
[PubMed]

1995 (2)

F. Manns, J. Shen, P. Soderberg, T. Matsui and J. Parel, "Development of an algorithm for corneal reshaping with a scanning laser beam," Appl. Opt. 21, 4600-4608 (1995).
[CrossRef]

G. H. Pettit, M. Ediger and R. P. Weiblinger, "Excimer laser ablation of the cornea," Opt. Eng. 34, 661-667 (1995).
[CrossRef]

1993 (1)

Z. Bor, B. Hopp, B. Racz, G. Szabo, Z. Marton, I. Ratkay, J. Mohay, I. Suveges and A. Fust, "Physical problems of excimer laser cornea ablation," Opt. Eng. 32, 2481-2486 (1993).
[CrossRef]

1985 (1)

R. R. Krueger and S. L. Trokel, "Quantization of corneal ablation by ultraviolet laser light," Arch. Ophthalmol. 103, 1741-1742 (1985).
[CrossRef] [PubMed]

Anera, R. G.

Arif, M.

D. Huang and M. Arif, "Spot size and quality of scanning laser correction of higher-order wavefront aberrations," J. Cataract Refract. Surg. 28, 407-416 (2002).
[CrossRef] [PubMed]

Bor, Z.

Z. Bor, B. Hopp, B. Racz, G. Szabo, Z. Marton, I. Ratkay, J. Mohay, I. Suveges and A. Fust, "Physical problems of excimer laser cornea ablation," Opt. Eng. 32, 2481-2486 (1993).
[CrossRef]

Díaz, J. A.

Ediger, M.

G. H. Pettit, M. Ediger and R. P. Weiblinger, "Excimer laser ablation of the cornea," Opt. Eng. 34, 661-667 (1995).
[CrossRef]

Fust, A.

Z. Bor, B. Hopp, B. Racz, G. Szabo, Z. Marton, I. Ratkay, J. Mohay, I. Suveges and A. Fust, "Physical problems of excimer laser cornea ablation," Opt. Eng. 32, 2481-2486 (1993).
[CrossRef]

Gutiérrez, R.

Hita, E.

Hopp, B.

Z. Bor, B. Hopp, B. Racz, G. Szabo, Z. Marton, I. Ratkay, J. Mohay, I. Suveges and A. Fust, "Physical problems of excimer laser cornea ablation," Opt. Eng. 32, 2481-2486 (1993).
[CrossRef]

Huang, D.

D. Huang and M. Arif, "Spot size and quality of scanning laser correction of higher-order wavefront aberrations," J. Cataract Refract. Surg. 28, 407-416 (2002).
[CrossRef] [PubMed]

Jiménez, J. R.

Jiménez del Barco, L.

Krueger, R. R.

R. R. Krueger and S. L. Trokel, "Quantization of corneal ablation by ultraviolet laser light," Arch. Ophthalmol. 103, 1741-1742 (1985).
[CrossRef] [PubMed]

Manns, F.

F. Manns, J. Shen, P. Soderberg, T. Matsui and J. Parel, "Development of an algorithm for corneal reshaping with a scanning laser beam," Appl. Opt. 21, 4600-4608 (1995).
[CrossRef]

Marton, Z.

Z. Bor, B. Hopp, B. Racz, G. Szabo, Z. Marton, I. Ratkay, J. Mohay, I. Suveges and A. Fust, "Physical problems of excimer laser cornea ablation," Opt. Eng. 32, 2481-2486 (1993).
[CrossRef]

Matsui, T.

F. Manns, J. Shen, P. Soderberg, T. Matsui and J. Parel, "Development of an algorithm for corneal reshaping with a scanning laser beam," Appl. Opt. 21, 4600-4608 (1995).
[CrossRef]

Mohay, J.

Z. Bor, B. Hopp, B. Racz, G. Szabo, Z. Marton, I. Ratkay, J. Mohay, I. Suveges and A. Fust, "Physical problems of excimer laser cornea ablation," Opt. Eng. 32, 2481-2486 (1993).
[CrossRef]

Mrochen, M.

M. Mrochen and T. Seiler, "Influence of corneal curvature on calculation of ablation patterns used in photorefractive laser surgery," J. Refract. Surg. 17, S584-S587 (2001).
[PubMed]

Parel, J.

F. Manns, J. Shen, P. Soderberg, T. Matsui and J. Parel, "Development of an algorithm for corneal reshaping with a scanning laser beam," Appl. Opt. 21, 4600-4608 (1995).
[CrossRef]

Pérez-Ocón, F.

Pettit, G. H.

G. H. Pettit, M. Ediger and R. P. Weiblinger, "Excimer laser ablation of the cornea," Opt. Eng. 34, 661-667 (1995).
[CrossRef]

Racz, B.

Z. Bor, B. Hopp, B. Racz, G. Szabo, Z. Marton, I. Ratkay, J. Mohay, I. Suveges and A. Fust, "Physical problems of excimer laser cornea ablation," Opt. Eng. 32, 2481-2486 (1993).
[CrossRef]

Ratkay, I.

Z. Bor, B. Hopp, B. Racz, G. Szabo, Z. Marton, I. Ratkay, J. Mohay, I. Suveges and A. Fust, "Physical problems of excimer laser cornea ablation," Opt. Eng. 32, 2481-2486 (1993).
[CrossRef]

Roberts, C.

C. Roberts, "Biomechanical customization," J. Cataract Refract. Surg. 31, 2-5 (2005).
[CrossRef] [PubMed]

Seiler, T.

M. Mrochen and T. Seiler, "Influence of corneal curvature on calculation of ablation patterns used in photorefractive laser surgery," J. Refract. Surg. 17, S584-S587 (2001).
[PubMed]

Shen, J.

F. Manns, J. Shen, P. Soderberg, T. Matsui and J. Parel, "Development of an algorithm for corneal reshaping with a scanning laser beam," Appl. Opt. 21, 4600-4608 (1995).
[CrossRef]

Soderberg, P.

F. Manns, J. Shen, P. Soderberg, T. Matsui and J. Parel, "Development of an algorithm for corneal reshaping with a scanning laser beam," Appl. Opt. 21, 4600-4608 (1995).
[CrossRef]

Suveges, I.

Z. Bor, B. Hopp, B. Racz, G. Szabo, Z. Marton, I. Ratkay, J. Mohay, I. Suveges and A. Fust, "Physical problems of excimer laser cornea ablation," Opt. Eng. 32, 2481-2486 (1993).
[CrossRef]

Szabo, G.

Z. Bor, B. Hopp, B. Racz, G. Szabo, Z. Marton, I. Ratkay, J. Mohay, I. Suveges and A. Fust, "Physical problems of excimer laser cornea ablation," Opt. Eng. 32, 2481-2486 (1993).
[CrossRef]

Trokel, S. L.

R. R. Krueger and S. L. Trokel, "Quantization of corneal ablation by ultraviolet laser light," Arch. Ophthalmol. 103, 1741-1742 (1985).
[CrossRef] [PubMed]

Venugopalan, V.

A. Vogel and V. Venugopalan, "Mechanisms of pulsed laser ablation of biological tissues," Chem. Rev. 103, 577-644 (2003).
[CrossRef] [PubMed]

Villa, C.

Vogel, A.

A. Vogel and V. Venugopalan, "Mechanisms of pulsed laser ablation of biological tissues," Chem. Rev. 103, 577-644 (2003).
[CrossRef] [PubMed]

Weiblinger, R. P.

G. H. Pettit, M. Ediger and R. P. Weiblinger, "Excimer laser ablation of the cornea," Opt. Eng. 34, 661-667 (1995).
[CrossRef]

Appl. Opt. (2)

F. Manns, J. Shen, P. Soderberg, T. Matsui and J. Parel, "Development of an algorithm for corneal reshaping with a scanning laser beam," Appl. Opt. 21, 4600-4608 (1995).
[CrossRef]

R. G. Anera, C. Villa, J. R. Jiménez, R. Gutiérrez and L. Jiménez del Barco, "Differences between real and predicted corneal shapes after aspherical corneal ablation," Appl. Opt. 44, 4528-4532 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

J. R. Jiménez, R. G. Anera, L. Jiménez del Barco and E. Hita, "Effect on laser-ablation algorithms of reflection losses and nonnormal incidence on the anterior cornea," Appl. Phys. Lett. 81, 1521-1523 (2002).
[CrossRef]

Arch. Ophthalmol. (1)

R. R. Krueger and S. L. Trokel, "Quantization of corneal ablation by ultraviolet laser light," Arch. Ophthalmol. 103, 1741-1742 (1985).
[CrossRef] [PubMed]

Chem. Rev. (1)

A. Vogel and V. Venugopalan, "Mechanisms of pulsed laser ablation of biological tissues," Chem. Rev. 103, 577-644 (2003).
[CrossRef] [PubMed]

J. Cataract Refract. Surg. (2)

C. Roberts, "Biomechanical customization," J. Cataract Refract. Surg. 31, 2-5 (2005).
[CrossRef] [PubMed]

D. Huang and M. Arif, "Spot size and quality of scanning laser correction of higher-order wavefront aberrations," J. Cataract Refract. Surg. 28, 407-416 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

J. Refract. Surg. (1)

M. Mrochen and T. Seiler, "Influence of corneal curvature on calculation of ablation patterns used in photorefractive laser surgery," J. Refract. Surg. 17, S584-S587 (2001).
[PubMed]

Opt. Eng. (2)

G. H. Pettit, M. Ediger and R. P. Weiblinger, "Excimer laser ablation of the cornea," Opt. Eng. 34, 661-667 (1995).
[CrossRef]

Z. Bor, B. Hopp, B. Racz, G. Szabo, Z. Marton, I. Ratkay, J. Mohay, I. Suveges and A. Fust, "Physical problems of excimer laser cornea ablation," Opt. Eng. 32, 2481-2486 (1993).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (1)

D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth Heinemann, Oxford, 2000) p. 11-20, 167, 195-210.
[CrossRef]

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Figures (2)

Fig. 1.
Fig. 1.

Linear and quadratic fit of experimental data on ablation depth corresponding to Krueger et al. [11]. F and Fth indicate the intensity exposure on the cornea and the threshold exposure, respectively.

Fig. 2.
Fig. 2.

Post-surgical corneal spherical-aberration difference (ΔS) between computations when considering or not a quadratic deviation from Lambert-Beer’s law as a function of the initial degree of myopia.

Equations (15)

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d p = m · ln ( F i n c F t h )
d p N L = a · ln ( F i n c F t h ) + b · ( ln ( F i n c F t h ) ) 2
ρ = d p N L d p = a · ln ( F i n c F t h ) + b · ( ln ( F i n c F t h ) ) 2 m · ln ( F i n c F t h )
F i n c = F 0 ( cos α · ( 1 R ˜ ) )
ρ = ρ 0 + ρ 1 ( y R ) 2 + ρ 2 ( y R ) 4
ρ 0 = a ( 0.0435 + t ) + b ( 0.0018 0.087 t + t 2 ) m t ρ 1 = 0.5 a + b ( 0.0435 t ) m t
ρ 2 = a ( 0.23424 0.5 p ) + b ( 0.223 + p ( 0.0435 t ) + 0.4648 t ) m t
c ( y ) = 4 D y 2 3 D s 2 3
1 R = ( 8 b D 3 ) · ( 0.053 + t ) · ( + 0.034 + t ) + 129.87 m t + a D · ( 0.116 + 8 t 3 ) m t
p = a m 2 t 2 D R 3 ( 144 + s 2 ( 16.7328 + 36 p ) ) + 27 p m 3 t 3 { 3 m t + D R [ a ( s 2 0.348 + 8 t ) + b ( 0.144 0.87 s 2 + ( 2 s 2 0.696 ) t + 8 t 2 ) ] } 3 +
+ b s m 2 t 2 R 3 ( 12.528 288 t + s 2 ( 16.538 33.4656 t + p ( 72 t 3.132 ) ) ) { 3 m t + D R [ a ( s 2 0.348 + 8 t ) + b ( 0.144 0.87 s 2 + ( 2 s 2 0.696 ) t + 8 t 2 ) ] } 3
8 D 3 = 1 R M u n n 1 R p M u n n = R ' M u n n 3 R 3 p
Δ c ( y ) = 0.476 D + 3.06 D y 2 t + ( 5.61 D 22.384 D y 2 ) + ( 4.76 D + 23.06 D y 2 ) t
ρ = ( α 0.043 β ) 0.5 β y 2 + ( 0.232 0.5 p ) β y 4
with α = 1 m [ a + b t ] and β = b m

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