## Deviations of Lambert-Beer’s law affect corneal refractive parameters after refractive surgery

Optics Express, Vol. 14, Issue 12, pp. 5411-5417 (2006)

http://dx.doi.org/10.1364/OE.14.005411

Acrobat PDF (140 KB)

### 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

## 1. Introduction

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]

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]

13. A. Vogel and V. Venugopalan, “Mechanisms of pulsed laser ablation of biological tissues,” Chem. Rev. **103**, 577–644 (2003). [CrossRef] [PubMed]

13. A. Vogel and V. Venugopalan, “Mechanisms of pulsed laser ablation of biological tissues,” Chem. Rev. **103**, 577–644 (2003). [CrossRef] [PubMed]

*d*is the ablation depth per pulse,

_{p}*m*is the slope efficiency of the ablation,

*F*is the incident exposure of the laser (energy per illuminated area) pulse and

_{inc}*F*is the threshold exposure for the ablation. The quantification of this law would not be very important if the incident exposure at the cornea did not vary during ablation. Reflection losses and nonnormal incidence on the cornea are two factors [5

_{th}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]

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]

*F*, varies during the ablation.

_{inc}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]

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]

## 2. Method

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

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]

*y*=

*mx*, with

*y*being the ablation depth per pulse and

*x*=ln(

*F*/

_{inc}*F*). As an example, Fig. 1 shows experimental data from Krueger et al. [11

_{th}11. R. R. Krueger and S. L. Trokel, “Quantization of corneal ablation by ultraviolet laser light,” Arch. Ophthalmol. **103**, 1741–1742 (1985). [CrossRef] [PubMed]

*y=mx*and a non-linear fit by including a quadratic term that quantifies the linear deviation-that is,

*y*=

*ax*+

*bx*

^{2}. As can be seen in Fig. 1, the correlation is higher when including a non-linear term. This tendency is similar with other experimental data [10

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

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]

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]

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]

*ρ*in Eq. (3) provides the deviation in the ablation when assuming Lambert-Beer’s law while the real ablation depth more closely approaches a non-linear quadratic fit. The parameters

*a, b*and

*m*are determined from the fits of the particular experimental data of the corneal ablation for each laser system. To deduce a general expression that can be evaluated in practice, we need [3

**29**, 962–965 (2004). [CrossRef] [PubMed]

**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]

*p*=1+

*Q*, with

*Q*being the corneal asphericity). To determine

*ρ*, we will introduce into the term

*F*of the numerator the variations being due to reflection losses and non-normal incidence on the cornea [5

_{inc}**81**, 1521–1523 (2002). [CrossRef]

*F*

_{0}is a constant that indicates the maximum exposure, the factor (1-

*R*̃) provides the information concerning the reflection losses [5

**81**, 1521–1523 (2002). [CrossRef]

*cosα*concerns non-normal incidence on the cornea [5

**81**, 1521–1523 (2002). [CrossRef]

**29**, 962–965 (2004). [CrossRef] [PubMed]

**81**, 1521–1523 (2002). [CrossRef]

*R*̃) and

*cosα*depend on the incidence height of the laser from the optical axis [3

**29**, 962–965 (2004). [CrossRef] [PubMed]

*ρ*in Eq. (3), depending on the variables that the ablation algorithm takes into account. Lambert-Beer’s law is assumed for ablation depth but we do not know whether corrections for reflection losses and non-normal incidence are also applied in ablation algorithms (they are proprietary). Therefore, if in the denominator of Eq. (3) we take

*F*as given by Eq. (4), the factor

_{inc}*ρ*would compute the deviation with an algorithm that takes into account reflection losses and non-normal incidence but not any deviation of Lambert-Beer’s law. If in the denominator of

*ρ*we assume

*F*=

_{inc}*F*, the parameter

_{0}*ρ*will enable us to compare the deviation with algorithms that assume the Lambert-Beer’s law and the incidence exposure on the cornea to be constant for a laser device. In the mathematical procedure described in this paper, we will consider this second option,

*F*=

_{inc}*F*, since, as we will discuss below, the first possibility implies the same procedure but mathematically simpler.

_{0}*ρ*, we will apply a numerical procedure (described extensively in different papers [3

**29**, 962–965 (2004). [CrossRef] [PubMed]

**81**, 1521–1523 (2002). [CrossRef]

**13**, 336–342 (2005). [CrossRef] [PubMed]

**29**, 962–965 (2004). [CrossRef] [PubMed]

**81**, 1521–1523 (2002). [CrossRef]

**13**, 336–342 (2005). [CrossRef] [PubMed]

*ρ*as a function of laser parameters and the distance from the optical axis [3

**29**, 962–965 (2004). [CrossRef] [PubMed]

**81**, 1521–1523 (2002). [CrossRef]

*ρ*can be expressed analytically as a series expansion in the variable

*y/R*up to order 4 [3

**29**, 962–965 (2004). [CrossRef] [PubMed]

*y*indicates the distance from the optical axis and

*R*is the corneal radius. Thus, it is necessary to compute the coefficients of the following factor:

**29**, 962–965 (2004). [CrossRef] [PubMed]

*R*and

*p*being the pre-surgery radius and p-factor, respectively, and

*t*=ln(

*F*

_{0}/

*F*). Although this factor allows us to evaluate a point-to-point correction, we will evaluate its effect on refractive corneal parameters. We will calculate the post-surgical radius,

_{th}*R*’ and p-factor,

*p*’, by applying the standard paraxial Munnerlyn formula for ablation depth corrected (multiplied) by the factor

*ρ*, given by Eq. (3) [5

**81**, 1521–1523 (2002). [CrossRef]

**13**, 336–342 (2005). [CrossRef] [PubMed]

*ρ*. The paraxial Munnerlyn formula,

*c(y)*, used in non-customized refractive surgery [1

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]

**81**, 1521–1523 (2002). [CrossRef]

*s*is the ablation diameter and

*D*the number of diopters to correct. Other equations could be tested but algorithms are proprietary and cannot be explicity known. We used the paraxial formula, as it is usual in different works, given that most non-customized algorithms are based on the paraxial formula. We also computed the procedure shown here with the non-paraxial Munnerlyn formula [1

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]

**21**, 98–103 (2004). [CrossRef]

**29**, 962–965 (2004). [CrossRef] [PubMed]

**81**, 1521–1523 (2002). [CrossRef]

**13**, 336–342 (2005). [CrossRef] [PubMed]

## 3. Results and discussion

*R’*and

_{Munn}*p’*are given by [5

_{Munn}**81**, 1521–1523 (2002). [CrossRef]

*a, b*and

*m*from Krueger et al. [11

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

**29**, 962–965 (2004). [CrossRef] [PubMed]

*d*=7 mm,

*R*=7.7 mm,

*p*=0.74,

*t*=0.69. We calculated the corneal-power difference Δφ

*=φ’-φ*obtained from Eqs. (8) and (10). Corneal power is calculated as

_{Munn}*φ=Δn/R*, with

*R*being the corneal radius and Δ

*n*=0.375 the refraction-index difference between the air and cornea. After computations, we get Δ

*φ*=0.14

*D*. The results show that the differences are significant for visual performance. For example, from

*D*=-2 (diopters) of initial ametropia, the difference is greater than 0.28D (diopters), a value that clearly reduces the effective visual acuity [6

**13**, 336–342 (2005). [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]

**13**, 336–342 (2005). [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]

*t*=0.38 (120mJ/cm

^{2}) to

*t*=0.90 (400mJ/cm

^{2}), the difference in corneal power ranged from Δ

*φ*=0.14

*D*to Δ

*φ*=0.064

*D*, respectively, being significant for visual performance for a wide range of laser fluences.

*c(y)*, the ablation profile corrected by the factor ρ is given by ρ

*·c(y)*, and therefore the ablation profile difference is Δ

*c*(y)=(ρ-1)·

*c*(y). Δ

*c*(y), as a function of

*t*and

*D*is :

*p*’=

*p*’-

*p’*, calculated from Eqs. (9) and (10) is higher than Δ

_{Munn}*p*’=0.01 from 1D (diopters) of initial myopia. Values of Δ

*p*’ higher than 0.01 diminish significantly contrast-sensitivity. Figure 2 shows the corneal primary spherical-aberration difference as a function of the initial degree of myopia computed from the post-surgery p-factor differences. Primary spherical-aberration,

*S*, is given by [14

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

*S*=((

*p*-1)y

^{4}Δn)/

*R*

^{3}. Spherical-aberration differences exceed the quarter-wavelength criterion [8

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]

*d*, is given by

_{p}*d*=a(

_{p}*x*-1)/

*x*, with

*x*=

*F*and

_{inc}/F_{th}*a*constant, and a model that unifies the blow-off and the steady-state model for which [13

**103**, 577–644 (2003). [CrossRef] [PubMed]

*d*=

_{p}*a*+

*b*ln((

*F*)+c) with

_{inc}/F_{th}*a*,

*b*and

*c*as constants. We computed fits from experimental data [10

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

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]

**29**, 962–965 (2004). [CrossRef] [PubMed]

**81**, 1521–1523 (2002). [CrossRef]

**13**, 336–342 (2005). [CrossRef] [PubMed]

*ρ*’, is given by a linear equation type

*pe ρ’=cte*(cosα·(1-

_{1}+cte_{2}*R*̃)) with

*cte*and

_{1}*cte*being constants and depending on laser parameters. For obtaining

_{2}*ρ*’, using the same procedure shown in methods, we get:

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

## Acknowledgments

## References and links

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 |

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. |

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. |

4. | C. Roberts, “Biomechanical customization,” J. Cataract Refract. Surg. |

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. |

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 |

7. | M. Mrochen and T. Seiler, “Influence of corneal curvature on calculation of ablation patterns used in photorefractive laser surgery,” J. Refract. Surg. |

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. |

9. | D. Huang and M. Arif, “Spot size and quality of scanning laser correction of higher-order wavefront aberrations,” J. Cataract Refract. Surg. |

10. | G. H. Pettit, M. Ediger, and R. P. Weiblinger, “Excimer laser ablation of the cornea,” Opt. Eng. |

11. | R. R. Krueger and S. L. Trokel, “Quantization of corneal ablation by ultraviolet laser light,” Arch. Ophthalmol. |

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. |

13. | A. Vogel and V. Venugopalan, “Mechanisms of pulsed laser ablation of biological tissues,” Chem. Rev. |

14. | D. A. Atchison and G. Smith, |

**OCIS Codes**

(170.1020) Medical optics and biotechnology : Ablation of tissue

(170.4470) Medical optics and biotechnology : Ophthalmology

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: March 2, 2006

Revised Manuscript: May 22, 2006

Manuscript Accepted: May 23, 2006

Published: June 12, 2006

**Virtual Issues**

Vol. 1, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

José R. Jiménez, Francisco Rodríguez-Marín, Rosario G. Anera, and Luis Jiménez del Barco, "Deviations of Lambert-Beer’s law affect corneal refractive parameters after refractive surgery," Opt. Express **14**, 5411-5417 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-12-5411

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### References

- 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]
- 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]
- 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]
- C. Roberts, "Biomechanical customization," J. Cataract Refract. Surg. 31, 2-5 (2005). [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]
- 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]
- 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]
- 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]
- 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]
- G. H. Pettit, M. Ediger and R. P. Weiblinger, "Excimer laser ablation of the cornea," Opt. Eng. 34, 661-667 (1995). [CrossRef]
- R. R. Krueger and S. L. Trokel, "Quantization of corneal ablation by ultraviolet laser light," Arch. Ophthalmol. 103, 1741-1742 (1985). [CrossRef] [PubMed]
- 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]
- A. Vogel and V. Venugopalan, "Mechanisms of pulsed laser ablation of biological tissues," Chem. Rev. 103, 577-644 (2003). [CrossRef] [PubMed]
- 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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