Abstract

We propose a complete methodology to develop custom monofocal Intraocular Lens (IOL) designs and evaluate their performance on-axis based on an analytical formulation. The analytical formulation was based on Gaussian and primary aberration theory applied to custom (individual biometric data) and realistic (multilayer cornea and thick IOL) pseudoaphakic eye models. Gradient-based optimization algorithms were performed to search for optimal designs. Using two parameters, the best design was obtained by directly minimizing the wavefront variance. We showed, in a case example, that custom designs achieved better final performance than generic IOL designs. Tolerances analysis allowed an evaluation of the implications of the manufacturing errors of the different parameters.

© 2007 Optical Society of America

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  1. J. C. Erie, M. H. Bandhauer, and J. W. McLaren, “Analysis of postoperative glare and intraocular lens design,” J. Cataract. Refract. Surg. 27, 614–621 (2001).
    [Crossref] [PubMed]
  2. C. W. Lu and G. Smith, “The Aspherizing Of Intraocular Lenses,” Ophthalmic. Physiol. Opt. 10, 54–66 (1990).
    [Crossref] [PubMed]
  3. G. E. MacKenzie, “Compensation of aniseikonia in astigmatic pseudophakic eyes,” Ophthalmic Physiol. Opt. 25, 576–581 (2005).
    [Crossref] [PubMed]
  4. A. Guirao, J. Tejedor, and P. Artal, “Corneal aberrations before and after small-incision cataract surgery,” Invest. Ophthalmol. Visual Sci. 45, 4312–4319 (2004).
    [Crossref]
  5. S. Marcos, P. Rosales, L. Llorente, and I. Jimenez-Alfaro, “Change in corneal aberrations after cataract surgery with 2 types of aspherical intraocular lenses,” J. Cataract. Refract. Surg. 33, 217–226 (2007).
    [Crossref] [PubMed]
  6. M. J. Simpson, “Optical-Quality Of Intraocular Lenses,” J. Cataract. Refract. Surg. 18, 86–94 (1992).
    [PubMed]
  7. S. Norrby, P. Artal, P. A. Piers, and M. Van der Mooren, “Methods of obtaining ophthalmic lenses providing the eye with reduced aberrations,” N. Pharmacia Groningen BV (2000).
  8. H. H. J. Shammas, Intraocular lens power calculations (Slack Incorporated2004).
  9. P. R. Preussner, J. Wahl, H. Lahdo, B. Dick, and O. Findl, “Ray tracing for intraocular lens calculation,” J. Cataract. Refract. Surg. 28, 1412–1419 (2002).
    [Crossref] [PubMed]
  10. P. R. Preussner and J. Wahl, “Consistent numerical calculation of the optics of the pseudophakie eye,” Ophthal-mologe 97, 126–141 (2000).
    [Crossref]
  11. G. Smith and C. W. Lu, “The Spherical-Aberration Of Intra-Ocular Lenses,” Ophthalmic Physiol. Opt. 8, 287–294 (1988).
    [Crossref] [PubMed]
  12. D. A. Atchison, “3rd-Order Aberrations Of Pseudophakic Eyes,” Ophthalmic Physiol. Opt. 9, 205–211 (1989).
    [Crossref] [PubMed]
  13. D. A. Atchison, “Design Of Aspheric Intraocular Lenses,” Ophthalmic Physiol. Opt. 11, 137–146 (1991).
    [Crossref] [PubMed]
  14. E. R. Villegas, L. Carretero, and A. Fimia, “Optimum bending factor of intraocular lenses in pseudophakic eyes with high myopia,” J. Mod. Opt. 44, 941–952 (1997).
    [Crossref]
  15. C. Gonzalez, I. Pascual, A. Bacete, and A. Fimia, “Elimination and minimization of the spherical aberration of intraocular lenses in high myopia,” Ophthalmic. Physiol. Opt. 16, 19–30 (1996).
    [Crossref] [PubMed]
  16. D. A. Atchison, “Optical Design Of Intraocular Lenses .2. Off-Axis Performance,” Optom. Vision Sci. 66, 579–590 (1989).
    [Crossref]
  17. G. Smith and C. W. Lu, “Peripheral Power Errors And Astigmatism Of Eyes Corrected With Intraocular Lenses,” Optom. Vision Sci. 68, 12–21 (1991).
    [Crossref]
  18. S. Barbero, “Refractive power of a multilayer rotationally symmetric model of the human cornea and tear film,” J. Opt. Soc. Am. A 23, 1578–1585 (2006).
    [Crossref]
  19. J. Tabernero, P. Piers, and P. Artal, “Intraocular lens to correct corneal coma,” Opt. Lett. 32, 406–408 (2007).
    [Crossref] [PubMed]
  20. D. A. Atchison, “Optical Design Of Intraocular Lenses .1. On-Axis Performance,” Optom. Vision Sci. 66, 492–506 (1989).
    [Crossref]
  21. G. M. Dai, “Optical surface optimization for the correction of presbyopia,” Appl. Opt. 45, 4184–4195 (2006).
    [Crossref] [PubMed]
  22. D. A. Atchison, “Optical Design Of Intraocular Lenses .3. On-Axis Performance In The Presence Of Lens Displacement,” Optom. Vision Sci. 66, 671–681 (1989).
    [Crossref]
  23. S. Barbero, S. Marcos, and I. Jimenez-Alfaro, “Optical aberrations of intraocular lenses measured in vivo and in vitro,” J. Opt. Soc. Am. A 20, 1841–1851 (2003).
    [Crossref]
  24. P. Rosales and S. Marcos, “Customized computer models of eyes with intraocular lenses,” Opt. Express ,,  15, 2204–2218 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-5-2204.
    [Crossref] [PubMed]
  25. A. de Castro, P. Rosales, and S. Marcos, “Tilt and decentration of intraocular lenses in vivo from Purkinje and Scheimpflug imaging,” J. Cataract. Refract. Surg. 33, 418–429 (2007).
    [Crossref] [PubMed]
  26. L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
    [Crossref]
  27. D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. 22, 29–37 (2005).
    [Crossref]
  28. B. Winn, D. Whitaker, D. B. Elliott, and N. J. Phillips, “Factors affecting ligtht-adapted pupil size in normal human subjects,” Invest. Ophthalmol. Visual Sci. 35, 1132–1137 (1994).
  29. G. Smith, D. A. Atchison, and S. Barbero, “Effect of defocus on on-axis wave aberration of a centered optical system,” J. Opt. Soc. Am. A 23, 2686–2689 (2006).
    [Crossref]
  30. M. Born, E. Wolf, and A. Joint, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, (New York, Oxford, 1980).
    [PubMed]
  31. G. O. Smith and D. A. Atchison, The eye and visual optical instruments (Cambridge University Press, 1997).
    [Crossref]
  32. V. V. N. Mahajan, Optical imaging and aberrations (SPIE Optical Engineering Press, 1998).
  33. A. Stockman and L. T. Sharpe, “The spectral sensitivities of the middle- and long-wavelength-sensitive cones derived from measurements in observers of known genotype,” Vision Res. 40, 1711 (2000).
    [Crossref] [PubMed]
  34. R. R. Fletcher, Practical methods of optimization (John Wiley & Sons Ltd, 1987).
  35. K. S. Kamal, “Intraocular lens manufacturing process,” B. L. Incorporated, ed. (USA, 2002).
  36. S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the estimation of corneal aberrations from videokeratography in keratoconus,” J. Refractive Surg. 18, 263–270 (2002).
  37. S. Kasthurirangan and A. Glasser, “Age related changes in accommodative dynamics in humans,” Vision Res. 46, 1507–1519 (2006).
    [Crossref]
  38. T. Olsen, “Sources Of Error In Intraocular-Lens Power Calculation,” J. Cataract. Refract. Surg. 18, 125–129 (1992).
    [PubMed]
  39. T. Olsen, “Prediction of the effective postoperative (intraocular lens) anterior chamber depth,” J. Cataract. Refract. Surg. 32, 419–424 (2006).
    [Crossref] [PubMed]
  40. W. M. Rosenblum and D. L. Shealy, “Caustic Analysis Of Interocular Lens Implants In Humans,” J. Opt. Soc. Am. 67, 1427–1427 (1977).
  41. D. L. Shealy and W. M. Rosenblum, “Caustic And Analytical Illuminance Calculations For A Model Of Human Eye,” Opt. Eng. 14, 237–240 (1975).
  42. S. Marcos and S. A. Burns, “Cone spacing and waveguide properties from cone directionality measurements,” J. Opt. Soc. Am. 16, 995–1004 (1999).
    [Crossref]
  43. A. Guirao and D. R. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vision Sci. 80, 36–42 (2003).
    [Crossref]
  44. X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310–321 (2004).
    [Crossref]
  45. A. Franchini, “Compromise between spherical and chromatic aberration and depth of focus in aspheric intraocular lenses,” J. Cataract. Refract. Surg. 33, 497–509 (2007).
    [Crossref] [PubMed]
  46. J. S. McLellan, S. Marcos, P. M. Prieto, and S. A. Burns, “Imperfect optics may be the eye’s defence against chromatic blur,” Nature 417, 174–176 (2002).
    [Crossref] [PubMed]

2007 (5)

S. Marcos, P. Rosales, L. Llorente, and I. Jimenez-Alfaro, “Change in corneal aberrations after cataract surgery with 2 types of aspherical intraocular lenses,” J. Cataract. Refract. Surg. 33, 217–226 (2007).
[Crossref] [PubMed]

A. de Castro, P. Rosales, and S. Marcos, “Tilt and decentration of intraocular lenses in vivo from Purkinje and Scheimpflug imaging,” J. Cataract. Refract. Surg. 33, 418–429 (2007).
[Crossref] [PubMed]

A. Franchini, “Compromise between spherical and chromatic aberration and depth of focus in aspheric intraocular lenses,” J. Cataract. Refract. Surg. 33, 497–509 (2007).
[Crossref] [PubMed]

J. Tabernero, P. Piers, and P. Artal, “Intraocular lens to correct corneal coma,” Opt. Lett. 32, 406–408 (2007).
[Crossref] [PubMed]

P. Rosales and S. Marcos, “Customized computer models of eyes with intraocular lenses,” Opt. Express ,,  15, 2204–2218 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-5-2204.
[Crossref] [PubMed]

2006 (5)

2005 (2)

D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. 22, 29–37 (2005).
[Crossref]

G. E. MacKenzie, “Compensation of aniseikonia in astigmatic pseudophakic eyes,” Ophthalmic Physiol. Opt. 25, 576–581 (2005).
[Crossref] [PubMed]

2004 (2)

A. Guirao, J. Tejedor, and P. Artal, “Corneal aberrations before and after small-incision cataract surgery,” Invest. Ophthalmol. Visual Sci. 45, 4312–4319 (2004).
[Crossref]

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310–321 (2004).
[Crossref]

2003 (2)

S. Barbero, S. Marcos, and I. Jimenez-Alfaro, “Optical aberrations of intraocular lenses measured in vivo and in vitro,” J. Opt. Soc. Am. A 20, 1841–1851 (2003).
[Crossref]

A. Guirao and D. R. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vision Sci. 80, 36–42 (2003).
[Crossref]

2002 (4)

S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the estimation of corneal aberrations from videokeratography in keratoconus,” J. Refractive Surg. 18, 263–270 (2002).

J. S. McLellan, S. Marcos, P. M. Prieto, and S. A. Burns, “Imperfect optics may be the eye’s defence against chromatic blur,” Nature 417, 174–176 (2002).
[Crossref] [PubMed]

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[Crossref]

P. R. Preussner, J. Wahl, H. Lahdo, B. Dick, and O. Findl, “Ray tracing for intraocular lens calculation,” J. Cataract. Refract. Surg. 28, 1412–1419 (2002).
[Crossref] [PubMed]

2001 (1)

J. C. Erie, M. H. Bandhauer, and J. W. McLaren, “Analysis of postoperative glare and intraocular lens design,” J. Cataract. Refract. Surg. 27, 614–621 (2001).
[Crossref] [PubMed]

2000 (2)

P. R. Preussner and J. Wahl, “Consistent numerical calculation of the optics of the pseudophakie eye,” Ophthal-mologe 97, 126–141 (2000).
[Crossref]

A. Stockman and L. T. Sharpe, “The spectral sensitivities of the middle- and long-wavelength-sensitive cones derived from measurements in observers of known genotype,” Vision Res. 40, 1711 (2000).
[Crossref] [PubMed]

1999 (1)

S. Marcos and S. A. Burns, “Cone spacing and waveguide properties from cone directionality measurements,” J. Opt. Soc. Am. 16, 995–1004 (1999).
[Crossref]

1997 (1)

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

1996 (1)

C. Gonzalez, I. Pascual, A. Bacete, and A. Fimia, “Elimination and minimization of the spherical aberration of intraocular lenses in high myopia,” Ophthalmic. Physiol. Opt. 16, 19–30 (1996).
[Crossref] [PubMed]

1994 (1)

B. Winn, D. Whitaker, D. B. Elliott, and N. J. Phillips, “Factors affecting ligtht-adapted pupil size in normal human subjects,” Invest. Ophthalmol. Visual Sci. 35, 1132–1137 (1994).

1992 (2)

M. J. Simpson, “Optical-Quality Of Intraocular Lenses,” J. Cataract. Refract. Surg. 18, 86–94 (1992).
[PubMed]

T. Olsen, “Sources Of Error In Intraocular-Lens Power Calculation,” J. Cataract. Refract. Surg. 18, 125–129 (1992).
[PubMed]

1991 (2)

G. Smith and C. W. Lu, “Peripheral Power Errors And Astigmatism Of Eyes Corrected With Intraocular Lenses,” Optom. Vision Sci. 68, 12–21 (1991).
[Crossref]

D. A. Atchison, “Design Of Aspheric Intraocular Lenses,” Ophthalmic Physiol. Opt. 11, 137–146 (1991).
[Crossref] [PubMed]

1990 (1)

C. W. Lu and G. Smith, “The Aspherizing Of Intraocular Lenses,” Ophthalmic. Physiol. Opt. 10, 54–66 (1990).
[Crossref] [PubMed]

1989 (4)

D. A. Atchison, “3rd-Order Aberrations Of Pseudophakic Eyes,” Ophthalmic Physiol. Opt. 9, 205–211 (1989).
[Crossref] [PubMed]

D. A. Atchison, “Optical Design Of Intraocular Lenses .2. Off-Axis Performance,” Optom. Vision Sci. 66, 579–590 (1989).
[Crossref]

D. A. Atchison, “Optical Design Of Intraocular Lenses .1. On-Axis Performance,” Optom. Vision Sci. 66, 492–506 (1989).
[Crossref]

D. A. Atchison, “Optical Design Of Intraocular Lenses .3. On-Axis Performance In The Presence Of Lens Displacement,” Optom. Vision Sci. 66, 671–681 (1989).
[Crossref]

1988 (1)

G. Smith and C. W. Lu, “The Spherical-Aberration Of Intra-Ocular Lenses,” Ophthalmic Physiol. Opt. 8, 287–294 (1988).
[Crossref] [PubMed]

1977 (1)

W. M. Rosenblum and D. L. Shealy, “Caustic Analysis Of Interocular Lens Implants In Humans,” J. Opt. Soc. Am. 67, 1427–1427 (1977).

1975 (1)

D. L. Shealy and W. M. Rosenblum, “Caustic And Analytical Illuminance Calculations For A Model Of Human Eye,” Opt. Eng. 14, 237–240 (1975).

Artal, P.

J. Tabernero, P. Piers, and P. Artal, “Intraocular lens to correct corneal coma,” Opt. Lett. 32, 406–408 (2007).
[Crossref] [PubMed]

A. Guirao, J. Tejedor, and P. Artal, “Corneal aberrations before and after small-incision cataract surgery,” Invest. Ophthalmol. Visual Sci. 45, 4312–4319 (2004).
[Crossref]

S. Norrby, P. Artal, P. A. Piers, and M. Van der Mooren, “Methods of obtaining ophthalmic lenses providing the eye with reduced aberrations,” N. Pharmacia Groningen BV (2000).

Atchison, D. A.

G. Smith, D. A. Atchison, and S. Barbero, “Effect of defocus on on-axis wave aberration of a centered optical system,” J. Opt. Soc. Am. A 23, 2686–2689 (2006).
[Crossref]

D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. 22, 29–37 (2005).
[Crossref]

D. A. Atchison, “Design Of Aspheric Intraocular Lenses,” Ophthalmic Physiol. Opt. 11, 137–146 (1991).
[Crossref] [PubMed]

D. A. Atchison, “Optical Design Of Intraocular Lenses .2. Off-Axis Performance,” Optom. Vision Sci. 66, 579–590 (1989).
[Crossref]

D. A. Atchison, “3rd-Order Aberrations Of Pseudophakic Eyes,” Ophthalmic Physiol. Opt. 9, 205–211 (1989).
[Crossref] [PubMed]

D. A. Atchison, “Optical Design Of Intraocular Lenses .1. On-Axis Performance,” Optom. Vision Sci. 66, 492–506 (1989).
[Crossref]

D. A. Atchison, “Optical Design Of Intraocular Lenses .3. On-Axis Performance In The Presence Of Lens Displacement,” Optom. Vision Sci. 66, 671–681 (1989).
[Crossref]

G. O. Smith and D. A. Atchison, The eye and visual optical instruments (Cambridge University Press, 1997).
[Crossref]

Bacete, A.

C. Gonzalez, I. Pascual, A. Bacete, and A. Fimia, “Elimination and minimization of the spherical aberration of intraocular lenses in high myopia,” Ophthalmic. Physiol. Opt. 16, 19–30 (1996).
[Crossref] [PubMed]

Bandhauer, M. H.

J. C. Erie, M. H. Bandhauer, and J. W. McLaren, “Analysis of postoperative glare and intraocular lens design,” J. Cataract. Refract. Surg. 27, 614–621 (2001).
[Crossref] [PubMed]

Barbero, S.

Born, M.

M. Born, E. Wolf, and A. Joint, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, (New York, Oxford, 1980).
[PubMed]

Bradley, A.

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310–321 (2004).
[Crossref]

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[Crossref]

Burns, S. A.

J. S. McLellan, S. Marcos, P. M. Prieto, and S. A. Burns, “Imperfect optics may be the eye’s defence against chromatic blur,” Nature 417, 174–176 (2002).
[Crossref] [PubMed]

S. Marcos and S. A. Burns, “Cone spacing and waveguide properties from cone directionality measurements,” J. Opt. Soc. Am. 16, 995–1004 (1999).
[Crossref]

Carretero, L.

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

Cheng, X.

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310–321 (2004).
[Crossref]

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[Crossref]

Dai, G. M.

de Castro, A.

A. de Castro, P. Rosales, and S. Marcos, “Tilt and decentration of intraocular lenses in vivo from Purkinje and Scheimpflug imaging,” J. Cataract. Refract. Surg. 33, 418–429 (2007).
[Crossref] [PubMed]

Dick, B.

P. R. Preussner, J. Wahl, H. Lahdo, B. Dick, and O. Findl, “Ray tracing for intraocular lens calculation,” J. Cataract. Refract. Surg. 28, 1412–1419 (2002).
[Crossref] [PubMed]

Elliott, D. B.

B. Winn, D. Whitaker, D. B. Elliott, and N. J. Phillips, “Factors affecting ligtht-adapted pupil size in normal human subjects,” Invest. Ophthalmol. Visual Sci. 35, 1132–1137 (1994).

Erie, J. C.

J. C. Erie, M. H. Bandhauer, and J. W. McLaren, “Analysis of postoperative glare and intraocular lens design,” J. Cataract. Refract. Surg. 27, 614–621 (2001).
[Crossref] [PubMed]

Fimia, A.

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

C. Gonzalez, I. Pascual, A. Bacete, and A. Fimia, “Elimination and minimization of the spherical aberration of intraocular lenses in high myopia,” Ophthalmic. Physiol. Opt. 16, 19–30 (1996).
[Crossref] [PubMed]

Findl, O.

P. R. Preussner, J. Wahl, H. Lahdo, B. Dick, and O. Findl, “Ray tracing for intraocular lens calculation,” J. Cataract. Refract. Surg. 28, 1412–1419 (2002).
[Crossref] [PubMed]

Fletcher, R. R.

R. R. Fletcher, Practical methods of optimization (John Wiley & Sons Ltd, 1987).

Franchini, A.

A. Franchini, “Compromise between spherical and chromatic aberration and depth of focus in aspheric intraocular lenses,” J. Cataract. Refract. Surg. 33, 497–509 (2007).
[Crossref] [PubMed]

Glasser, A.

S. Kasthurirangan and A. Glasser, “Age related changes in accommodative dynamics in humans,” Vision Res. 46, 1507–1519 (2006).
[Crossref]

Gonzalez, C.

C. Gonzalez, I. Pascual, A. Bacete, and A. Fimia, “Elimination and minimization of the spherical aberration of intraocular lenses in high myopia,” Ophthalmic. Physiol. Opt. 16, 19–30 (1996).
[Crossref] [PubMed]

Guirao, A.

A. Guirao, J. Tejedor, and P. Artal, “Corneal aberrations before and after small-incision cataract surgery,” Invest. Ophthalmol. Visual Sci. 45, 4312–4319 (2004).
[Crossref]

A. Guirao and D. R. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vision Sci. 80, 36–42 (2003).
[Crossref]

Hong, X.

Jimenez-Alfaro, I.

S. Marcos, P. Rosales, L. Llorente, and I. Jimenez-Alfaro, “Change in corneal aberrations after cataract surgery with 2 types of aspherical intraocular lenses,” J. Cataract. Refract. Surg. 33, 217–226 (2007).
[Crossref] [PubMed]

S. Barbero, S. Marcos, and I. Jimenez-Alfaro, “Optical aberrations of intraocular lenses measured in vivo and in vitro,” J. Opt. Soc. Am. A 20, 1841–1851 (2003).
[Crossref]

Joint, A.

M. Born, E. Wolf, and A. Joint, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, (New York, Oxford, 1980).
[PubMed]

Kamal, K. S.

K. S. Kamal, “Intraocular lens manufacturing process,” B. L. Incorporated, ed. (USA, 2002).

Kasthurirangan, S.

S. Kasthurirangan and A. Glasser, “Age related changes in accommodative dynamics in humans,” Vision Res. 46, 1507–1519 (2006).
[Crossref]

Lahdo, H.

P. R. Preussner, J. Wahl, H. Lahdo, B. Dick, and O. Findl, “Ray tracing for intraocular lens calculation,” J. Cataract. Refract. Surg. 28, 1412–1419 (2002).
[Crossref] [PubMed]

Llorente, L.

S. Marcos, P. Rosales, L. Llorente, and I. Jimenez-Alfaro, “Change in corneal aberrations after cataract surgery with 2 types of aspherical intraocular lenses,” J. Cataract. Refract. Surg. 33, 217–226 (2007).
[Crossref] [PubMed]

Lu, C. W.

G. Smith and C. W. Lu, “Peripheral Power Errors And Astigmatism Of Eyes Corrected With Intraocular Lenses,” Optom. Vision Sci. 68, 12–21 (1991).
[Crossref]

C. W. Lu and G. Smith, “The Aspherizing Of Intraocular Lenses,” Ophthalmic. Physiol. Opt. 10, 54–66 (1990).
[Crossref] [PubMed]

G. Smith and C. W. Lu, “The Spherical-Aberration Of Intra-Ocular Lenses,” Ophthalmic Physiol. Opt. 8, 287–294 (1988).
[Crossref] [PubMed]

MacKenzie, G. E.

G. E. MacKenzie, “Compensation of aniseikonia in astigmatic pseudophakic eyes,” Ophthalmic Physiol. Opt. 25, 576–581 (2005).
[Crossref] [PubMed]

Mahajan, V. V. N.

V. V. N. Mahajan, Optical imaging and aberrations (SPIE Optical Engineering Press, 1998).

Marcos, S.

S. Marcos, P. Rosales, L. Llorente, and I. Jimenez-Alfaro, “Change in corneal aberrations after cataract surgery with 2 types of aspherical intraocular lenses,” J. Cataract. Refract. Surg. 33, 217–226 (2007).
[Crossref] [PubMed]

P. Rosales and S. Marcos, “Customized computer models of eyes with intraocular lenses,” Opt. Express ,,  15, 2204–2218 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-5-2204.
[Crossref] [PubMed]

A. de Castro, P. Rosales, and S. Marcos, “Tilt and decentration of intraocular lenses in vivo from Purkinje and Scheimpflug imaging,” J. Cataract. Refract. Surg. 33, 418–429 (2007).
[Crossref] [PubMed]

S. Barbero, S. Marcos, and I. Jimenez-Alfaro, “Optical aberrations of intraocular lenses measured in vivo and in vitro,” J. Opt. Soc. Am. A 20, 1841–1851 (2003).
[Crossref]

J. S. McLellan, S. Marcos, P. M. Prieto, and S. A. Burns, “Imperfect optics may be the eye’s defence against chromatic blur,” Nature 417, 174–176 (2002).
[Crossref] [PubMed]

S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the estimation of corneal aberrations from videokeratography in keratoconus,” J. Refractive Surg. 18, 263–270 (2002).

S. Marcos and S. A. Burns, “Cone spacing and waveguide properties from cone directionality measurements,” J. Opt. Soc. Am. 16, 995–1004 (1999).
[Crossref]

McLaren, J. W.

J. C. Erie, M. H. Bandhauer, and J. W. McLaren, “Analysis of postoperative glare and intraocular lens design,” J. Cataract. Refract. Surg. 27, 614–621 (2001).
[Crossref] [PubMed]

McLellan, J. S.

J. S. McLellan, S. Marcos, P. M. Prieto, and S. A. Burns, “Imperfect optics may be the eye’s defence against chromatic blur,” Nature 417, 174–176 (2002).
[Crossref] [PubMed]

Merayo-Lloves, J.

S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the estimation of corneal aberrations from videokeratography in keratoconus,” J. Refractive Surg. 18, 263–270 (2002).

Mooren, M. Van der

S. Norrby, P. Artal, P. A. Piers, and M. Van der Mooren, “Methods of obtaining ophthalmic lenses providing the eye with reduced aberrations,” N. Pharmacia Groningen BV (2000).

Moreno-Barriuso, E.

S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the estimation of corneal aberrations from videokeratography in keratoconus,” J. Refractive Surg. 18, 263–270 (2002).

Norrby, S.

S. Norrby, P. Artal, P. A. Piers, and M. Van der Mooren, “Methods of obtaining ophthalmic lenses providing the eye with reduced aberrations,” N. Pharmacia Groningen BV (2000).

Olsen, T.

T. Olsen, “Prediction of the effective postoperative (intraocular lens) anterior chamber depth,” J. Cataract. Refract. Surg. 32, 419–424 (2006).
[Crossref] [PubMed]

T. Olsen, “Sources Of Error In Intraocular-Lens Power Calculation,” J. Cataract. Refract. Surg. 18, 125–129 (1992).
[PubMed]

Pascual, I.

C. Gonzalez, I. Pascual, A. Bacete, and A. Fimia, “Elimination and minimization of the spherical aberration of intraocular lenses in high myopia,” Ophthalmic. Physiol. Opt. 16, 19–30 (1996).
[Crossref] [PubMed]

Phillips, N. J.

B. Winn, D. Whitaker, D. B. Elliott, and N. J. Phillips, “Factors affecting ligtht-adapted pupil size in normal human subjects,” Invest. Ophthalmol. Visual Sci. 35, 1132–1137 (1994).

Piers, P.

Piers, P. A.

S. Norrby, P. Artal, P. A. Piers, and M. Van der Mooren, “Methods of obtaining ophthalmic lenses providing the eye with reduced aberrations,” N. Pharmacia Groningen BV (2000).

Preussner, P. R.

P. R. Preussner, J. Wahl, H. Lahdo, B. Dick, and O. Findl, “Ray tracing for intraocular lens calculation,” J. Cataract. Refract. Surg. 28, 1412–1419 (2002).
[Crossref] [PubMed]

P. R. Preussner and J. Wahl, “Consistent numerical calculation of the optics of the pseudophakie eye,” Ophthal-mologe 97, 126–141 (2000).
[Crossref]

Prieto, P. M.

J. S. McLellan, S. Marcos, P. M. Prieto, and S. A. Burns, “Imperfect optics may be the eye’s defence against chromatic blur,” Nature 417, 174–176 (2002).
[Crossref] [PubMed]

Rosales, P.

A. de Castro, P. Rosales, and S. Marcos, “Tilt and decentration of intraocular lenses in vivo from Purkinje and Scheimpflug imaging,” J. Cataract. Refract. Surg. 33, 418–429 (2007).
[Crossref] [PubMed]

P. Rosales and S. Marcos, “Customized computer models of eyes with intraocular lenses,” Opt. Express ,,  15, 2204–2218 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-5-2204.
[Crossref] [PubMed]

S. Marcos, P. Rosales, L. Llorente, and I. Jimenez-Alfaro, “Change in corneal aberrations after cataract surgery with 2 types of aspherical intraocular lenses,” J. Cataract. Refract. Surg. 33, 217–226 (2007).
[Crossref] [PubMed]

Rosenblum, W. M.

W. M. Rosenblum and D. L. Shealy, “Caustic Analysis Of Interocular Lens Implants In Humans,” J. Opt. Soc. Am. 67, 1427–1427 (1977).

D. L. Shealy and W. M. Rosenblum, “Caustic And Analytical Illuminance Calculations For A Model Of Human Eye,” Opt. Eng. 14, 237–240 (1975).

Shammas, H. H. J.

H. H. J. Shammas, Intraocular lens power calculations (Slack Incorporated2004).

Sharpe, L. T.

A. Stockman and L. T. Sharpe, “The spectral sensitivities of the middle- and long-wavelength-sensitive cones derived from measurements in observers of known genotype,” Vision Res. 40, 1711 (2000).
[Crossref] [PubMed]

Shealy, D. L.

W. M. Rosenblum and D. L. Shealy, “Caustic Analysis Of Interocular Lens Implants In Humans,” J. Opt. Soc. Am. 67, 1427–1427 (1977).

D. L. Shealy and W. M. Rosenblum, “Caustic And Analytical Illuminance Calculations For A Model Of Human Eye,” Opt. Eng. 14, 237–240 (1975).

Simpson, M. J.

M. J. Simpson, “Optical-Quality Of Intraocular Lenses,” J. Cataract. Refract. Surg. 18, 86–94 (1992).
[PubMed]

Smith, G.

G. Smith, D. A. Atchison, and S. Barbero, “Effect of defocus on on-axis wave aberration of a centered optical system,” J. Opt. Soc. Am. A 23, 2686–2689 (2006).
[Crossref]

D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. 22, 29–37 (2005).
[Crossref]

G. Smith and C. W. Lu, “Peripheral Power Errors And Astigmatism Of Eyes Corrected With Intraocular Lenses,” Optom. Vision Sci. 68, 12–21 (1991).
[Crossref]

C. W. Lu and G. Smith, “The Aspherizing Of Intraocular Lenses,” Ophthalmic. Physiol. Opt. 10, 54–66 (1990).
[Crossref] [PubMed]

G. Smith and C. W. Lu, “The Spherical-Aberration Of Intra-Ocular Lenses,” Ophthalmic Physiol. Opt. 8, 287–294 (1988).
[Crossref] [PubMed]

Smith, G. O.

G. O. Smith and D. A. Atchison, The eye and visual optical instruments (Cambridge University Press, 1997).
[Crossref]

Stockman, A.

A. Stockman and L. T. Sharpe, “The spectral sensitivities of the middle- and long-wavelength-sensitive cones derived from measurements in observers of known genotype,” Vision Res. 40, 1711 (2000).
[Crossref] [PubMed]

Tabernero, J.

Tejedor, J.

A. Guirao, J. Tejedor, and P. Artal, “Corneal aberrations before and after small-incision cataract surgery,” Invest. Ophthalmol. Visual Sci. 45, 4312–4319 (2004).
[Crossref]

Thibos, L. N.

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310–321 (2004).
[Crossref]

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[Crossref]

Villegas, E. R.

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

Wahl, J.

P. R. Preussner, J. Wahl, H. Lahdo, B. Dick, and O. Findl, “Ray tracing for intraocular lens calculation,” J. Cataract. Refract. Surg. 28, 1412–1419 (2002).
[Crossref] [PubMed]

P. R. Preussner and J. Wahl, “Consistent numerical calculation of the optics of the pseudophakie eye,” Ophthal-mologe 97, 126–141 (2000).
[Crossref]

Whitaker, D.

B. Winn, D. Whitaker, D. B. Elliott, and N. J. Phillips, “Factors affecting ligtht-adapted pupil size in normal human subjects,” Invest. Ophthalmol. Visual Sci. 35, 1132–1137 (1994).

Williams, D. R.

A. Guirao and D. R. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vision Sci. 80, 36–42 (2003).
[Crossref]

Winn, B.

B. Winn, D. Whitaker, D. B. Elliott, and N. J. Phillips, “Factors affecting ligtht-adapted pupil size in normal human subjects,” Invest. Ophthalmol. Visual Sci. 35, 1132–1137 (1994).

Wolf, E.

M. Born, E. Wolf, and A. Joint, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, (New York, Oxford, 1980).
[PubMed]

Appl. Opt. (1)

Invest. Ophthalmol. Visual Sci. (2)

B. Winn, D. Whitaker, D. B. Elliott, and N. J. Phillips, “Factors affecting ligtht-adapted pupil size in normal human subjects,” Invest. Ophthalmol. Visual Sci. 35, 1132–1137 (1994).

A. Guirao, J. Tejedor, and P. Artal, “Corneal aberrations before and after small-incision cataract surgery,” Invest. Ophthalmol. Visual Sci. 45, 4312–4319 (2004).
[Crossref]

J. Cataract. Refract. Surg. (8)

S. Marcos, P. Rosales, L. Llorente, and I. Jimenez-Alfaro, “Change in corneal aberrations after cataract surgery with 2 types of aspherical intraocular lenses,” J. Cataract. Refract. Surg. 33, 217–226 (2007).
[Crossref] [PubMed]

M. J. Simpson, “Optical-Quality Of Intraocular Lenses,” J. Cataract. Refract. Surg. 18, 86–94 (1992).
[PubMed]

P. R. Preussner, J. Wahl, H. Lahdo, B. Dick, and O. Findl, “Ray tracing for intraocular lens calculation,” J. Cataract. Refract. Surg. 28, 1412–1419 (2002).
[Crossref] [PubMed]

J. C. Erie, M. H. Bandhauer, and J. W. McLaren, “Analysis of postoperative glare and intraocular lens design,” J. Cataract. Refract. Surg. 27, 614–621 (2001).
[Crossref] [PubMed]

A. de Castro, P. Rosales, and S. Marcos, “Tilt and decentration of intraocular lenses in vivo from Purkinje and Scheimpflug imaging,” J. Cataract. Refract. Surg. 33, 418–429 (2007).
[Crossref] [PubMed]

T. Olsen, “Sources Of Error In Intraocular-Lens Power Calculation,” J. Cataract. Refract. Surg. 18, 125–129 (1992).
[PubMed]

T. Olsen, “Prediction of the effective postoperative (intraocular lens) anterior chamber depth,” J. Cataract. Refract. Surg. 32, 419–424 (2006).
[Crossref] [PubMed]

A. Franchini, “Compromise between spherical and chromatic aberration and depth of focus in aspheric intraocular lenses,” J. Cataract. Refract. Surg. 33, 497–509 (2007).
[Crossref] [PubMed]

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. (3)

D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. 22, 29–37 (2005).
[Crossref]

S. Marcos and S. A. Burns, “Cone spacing and waveguide properties from cone directionality measurements,” J. Opt. Soc. Am. 16, 995–1004 (1999).
[Crossref]

W. M. Rosenblum and D. L. Shealy, “Caustic Analysis Of Interocular Lens Implants In Humans,” J. Opt. Soc. Am. 67, 1427–1427 (1977).

J. Opt. Soc. Am. A (4)

J. Refractive Surg. (1)

S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the estimation of corneal aberrations from videokeratography in keratoconus,” J. Refractive Surg. 18, 263–270 (2002).

J. Vision (1)

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310–321 (2004).
[Crossref]

Nature (1)

J. S. McLellan, S. Marcos, P. M. Prieto, and S. A. Burns, “Imperfect optics may be the eye’s defence against chromatic blur,” Nature 417, 174–176 (2002).
[Crossref] [PubMed]

Ophthal-mologe (1)

P. R. Preussner and J. Wahl, “Consistent numerical calculation of the optics of the pseudophakie eye,” Ophthal-mologe 97, 126–141 (2000).
[Crossref]

Ophthalmic Physiol. Opt. (4)

G. Smith and C. W. Lu, “The Spherical-Aberration Of Intra-Ocular Lenses,” Ophthalmic Physiol. Opt. 8, 287–294 (1988).
[Crossref] [PubMed]

D. A. Atchison, “3rd-Order Aberrations Of Pseudophakic Eyes,” Ophthalmic Physiol. Opt. 9, 205–211 (1989).
[Crossref] [PubMed]

D. A. Atchison, “Design Of Aspheric Intraocular Lenses,” Ophthalmic Physiol. Opt. 11, 137–146 (1991).
[Crossref] [PubMed]

G. E. MacKenzie, “Compensation of aniseikonia in astigmatic pseudophakic eyes,” Ophthalmic Physiol. Opt. 25, 576–581 (2005).
[Crossref] [PubMed]

Ophthalmic. Physiol. Opt. (2)

C. W. Lu and G. Smith, “The Aspherizing Of Intraocular Lenses,” Ophthalmic. Physiol. Opt. 10, 54–66 (1990).
[Crossref] [PubMed]

C. Gonzalez, I. Pascual, A. Bacete, and A. Fimia, “Elimination and minimization of the spherical aberration of intraocular lenses in high myopia,” Ophthalmic. Physiol. Opt. 16, 19–30 (1996).
[Crossref] [PubMed]

Opt. Eng. (1)

D. L. Shealy and W. M. Rosenblum, “Caustic And Analytical Illuminance Calculations For A Model Of Human Eye,” Opt. Eng. 14, 237–240 (1975).

Opt. Express (1)

Opt. Lett. (1)

Optom. Vision Sci. (5)

D. A. Atchison, “Optical Design Of Intraocular Lenses .1. On-Axis Performance,” Optom. Vision Sci. 66, 492–506 (1989).
[Crossref]

D. A. Atchison, “Optical Design Of Intraocular Lenses .2. Off-Axis Performance,” Optom. Vision Sci. 66, 579–590 (1989).
[Crossref]

G. Smith and C. W. Lu, “Peripheral Power Errors And Astigmatism Of Eyes Corrected With Intraocular Lenses,” Optom. Vision Sci. 68, 12–21 (1991).
[Crossref]

D. A. Atchison, “Optical Design Of Intraocular Lenses .3. On-Axis Performance In The Presence Of Lens Displacement,” Optom. Vision Sci. 66, 671–681 (1989).
[Crossref]

A. Guirao and D. R. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vision Sci. 80, 36–42 (2003).
[Crossref]

Vision Res. (2)

S. Kasthurirangan and A. Glasser, “Age related changes in accommodative dynamics in humans,” Vision Res. 46, 1507–1519 (2006).
[Crossref]

A. Stockman and L. T. Sharpe, “The spectral sensitivities of the middle- and long-wavelength-sensitive cones derived from measurements in observers of known genotype,” Vision Res. 40, 1711 (2000).
[Crossref] [PubMed]

Other (7)

R. R. Fletcher, Practical methods of optimization (John Wiley & Sons Ltd, 1987).

K. S. Kamal, “Intraocular lens manufacturing process,” B. L. Incorporated, ed. (USA, 2002).

M. Born, E. Wolf, and A. Joint, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, (New York, Oxford, 1980).
[PubMed]

G. O. Smith and D. A. Atchison, The eye and visual optical instruments (Cambridge University Press, 1997).
[Crossref]

V. V. N. Mahajan, Optical imaging and aberrations (SPIE Optical Engineering Press, 1998).

S. Norrby, P. Artal, P. A. Piers, and M. Van der Mooren, “Methods of obtaining ophthalmic lenses providing the eye with reduced aberrations,” N. Pharmacia Groningen BV (2000).

H. H. J. Shammas, Intraocular lens power calculations (Slack Incorporated2004).

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Figures (5)

Fig. 1.
Fig. 1.

Contour plots of the equivalent defocus (Me) as function of the two radii of curvature of: (a) Biconvex IOL design (b) Meniscus IOL design. The colorbar represents Me in Diopters. Contour lines are plotted every 0.25 D. The scale of the x-axis and y-axis are different to allow better visualization. Results are for the custom pseudoaphakic eye model of Table 2. Pupil radius is set to 2 mm.

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Fig. 2.
Fig. 2.

Contour plots of the equivalent defocus (Me) of a biconvex IOL design as function of: (a) The anterior radius of curvature and asphericity (b) The posterior radius of curvature and asphericity. The colorbar represents Me in Diopters. Contour lines are plotted every 0.25 D. The scale of the x-axis and y-axis are different to allow better visualization. Results are for the custom pseudoaphakic eye model of Table 2. Pupil radius is set to 2 mm.

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Fig. 3.
Fig. 3.

Scheme of cascade optimization steps, using different targets and different design parameters in each step. Radii of curvatures (Ra and Rp) and asphericities (Qa and Qp) were used as design parameters. IOL thickness was set as fixed parameter (custom value of Table 2). The merit function, with the target and design parameters used, is shown in the circular shaped rectangular boxes, whereas the designs obtained in each step appear in rectangular boxes and are labelled with IOL followed by a number code. See text for details.

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Fig. 4.
Fig. 4.

Defocus term (W20), spherical aberration (W40) and the equivalent defocus in Diopters units as function of pupil radius (mm) using the pseudoaphakic eye model of Table 2.

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Fig. 5.
Fig. 5.

Defocus term (W20), spherical aberration (W40) and the equivalent defocus in Diopters as function of the anterior chamber depth (mm) using the pseudoaphakic eye model of Table 2.

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Tables (6)

Tables Icon

Table 1. Parameters of a generic pseudoaphakic eye model. C denotes a custom parameter. R denotes the apical radius. Q denotes the asphericity (or deformation factor)defined by the conic explicit formula: Y 2 = 2Rz-(1+Q)z 2. n(λ) a , n(λ) v and n(λ) c denote the refractive index dispersion formulae for the aqueous, vitreous and corneal media derived by Atchison et al[26]. I1: Air-Tear. I2: Tear-Epithelium. I3: Epithelium-Stroma. I4: Stroma-Aqueous. I5: Aqueous-IOL. I6: IOL-Vitreous

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Table 2. Parameters of a case example customized pseudoaphakic eye model (patient AA with an 22 D Tecnis Z9000 IOL). R denotes the apical radius. Q denotes the asphericity (or deformation factor)defined by the conic explicit formula: Y 2 = 2Rz-(1+Q)z 2. n(λ) a , n(λ) v , n(λ) c and n(λ) s denote the refractive index dispersion formulae for the aqueous, vitreous, corneal and silicon media. I1: Air-Tear. I2: Tear-Epithelium. I3: Epithelium-Stroma. I4: Stroma-Aqueous. I5: Aqueous-IOL. I6: IOL-Vitreous

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Table 3. Design specifications and efficiency of the optimization algorithm. Ra and Rp denote the anterior and posterior radii of curvature of the IOL. Qa and Qp denote the anterior and posterior asphericities of the IOL. Me denotes the equivalent defocus for the IOL design in the pseudoaphakic for the eye model of Table 2. N iterations denotes for the number of iterations needed by the optimization searching algorithm and N functions the number of functions evaluated

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Table 4. Design specifications and optical performance of the the IOL designs obtained by different procedures. Ra and Rp denote the anterior and posterior radii of curvature of the IOL. Qa and Qp denote the anterior and posterior asphericities of the IOL. W20, W40 and Me denote the equivalent defocus (D) due to the defocus term, spherical aberration and the combination of both for several IOL designs.

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Table 5. Tolerance limits of the pseudoaphakic eye model of Table 2 for the design parameters: anterior and posterior radii and asphericities, thickness and refractive index.

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Table 6. Comparison of optical performance of the custom design and the Tecnis IOL design in the more realistic pseudoaphakic eye model of Table 2. The different metrics were computed through a ray tracing analysis using Zemax (Zemax, Optima Research, 2006 Tucson). Mes: Me contribution of symmetric terms. Mea: Me contribution of asymmetric terms

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Equations (4)

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

M t = R 1 * T 1 * R 2 * T 2 * R 3 * T 3 * R 4 * T 4 * R 5 * T 5 * R 6 * T 6 ,
W 20 = f ( R i , ti , λ ) ,
W 40 = g ( P , R i , Q i , ti , λ ) ,
Mew = Me ( 420.7 ) + 0.5 * ( Me ( 530.3 ) + Me ( 558.9 ) )

Metrics