Abstract

We developed a method to convert aberrometry data obtained in one wavelength to the corresponding data in another wavelength using an eye model. A single map of aberrometry data is used to construct a free-form one-surface eye model. A general algorithm for the surface construction is described and implemented for real aberrometry data. Our method can handle varying conjugate distances of the measurement plane of the aberrometer and can also manage the chief ray prism that may be present. The algorithm is validated with the aid of an artificial plastic eye. The wavefronts in different wavelengths are compared through the Zernike analysis not only for lower-order aberrations, but also for higher-order aberrations. The results show that the changes of the Zernike aberration coefficients due to wavelengths are non-uniform. The defocus term has the highest effect from wavelength changes, which is consistent with the previous literature. Our method is compared with two approximate semi-analytical algorithms. The wavelength adjustments from a multi-surface eye model are contrasted with our method. We prove analytically that the conventional method of wavelength adjustment is based on paraxial analysis. In addition, we provide a method of finding the chief ray using back-projection in some cases and discuss different meanings of prism.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinnemann, 2000).
  2. R. A. Applegate, L. N. Thibos, M. D. Twa, and E. J. Sarver, “Importance of fixation, pupil center, and reference axis in ocular wavefront sensing, videokeratography, and retinal image quality,” J. Cataract Refractive Surg. 35, 139–152 (2009).
    [CrossRef]
  3. ANSI, “American National Standards for Ophthalmics-Methods for reporting optical aberrations of eyes,” American National Standards Institute ANSI Z80.28 (2004).
  4. L. Lundstrom, P. Unsbo, and J. Gustafsson, “Off-axis wave front measurements for optical correction in eccentric viewing,” J. Biomed. Opt. 10, 034002 (2005).
    [CrossRef] [PubMed]
  5. J. Nam and J. Rubinstein, “Weighted Zernike expansion with applications to the optical aberration of the human eye,” J. Opt. Soc. Am. A 22, 1709–1716 (2005).
    [CrossRef]
  6. J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
    [CrossRef]
  7. V. I. Arnold, Ordinary Differential Equations, 3rd ed. (Springer-Verlag, 1992).
  8. M. Rynders, B. Lidkea, W. Chisholm, and L. N. Thibos, “Statistical distribution of foveal transverse chromatic aberration, pupil centration, and angle psi in a population of young adult eyes,” J. Opt. Soc. Am. A 12, 2348–2357 (1995).
    [CrossRef]
  9. L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, and P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
    [CrossRef] [PubMed]
  10. X. Cheng, N. L. Himebaugh, P. S. Kollbaum, L. N. Thibos, and A. Bradley, “Validation of a clinical Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 587–595 (2003).
    [CrossRef]
  11. L. N. Thibos and A. Bradley, “Modeling the refractive and neurosensor systems of the eye,” in Visual Instrumentation, P.Moroulis, ed. (McGraw Hill, 1999).
  12. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
    [PubMed]
  13. J. B. Keller and H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Am. Optom. Assoc. 40, 48–52 (1950).
    [CrossRef]
  14. J. A. Kneisly, “Local curvature of wavefront in an optical system,” J. Am. Optom. Assoc. 54, 229–235 (1964).
    [CrossRef]
  15. J. Rubinstein and G. Wolansky, “Eikonal functions: Old and new,” in Applied Mathematics Celebration, D.Givoli, ed. (Kluwer, 2004).
  16. T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, “Measurement of refractive errors in young myopes using the COAS Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 6–14 (2003).
    [CrossRef]
  17. J. Copeland and D. Neal, “Combined objective and subjective optometer,” presented at the 7th International Congress on Wavefront Sensing and Optimized Refractive Correction, January 26–29, 2006. http://voi.opt.uh.edu/VOI/WavefrontCongress/2006/index.html.
  18. L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, “Aberrations of the human eye in visible and near infrared illumination,” Optom. Vision Sci. 80, 26–35 (2003).
    [CrossRef]
  19. S. Ravikumar, L. N. Thibos, and A. Bradley, “Calculation of retinal image quality for polychromatic light,” J. Opt. Soc. Am. A 25, 2395–2407 (2008).
    [CrossRef]
  20. R. Navarro, J. Santamaria, and J. Bescos, “Accommodation-dependent model of the human eye with aspherics,” J. Opt. Soc. Am. A 2, 1273–1281 (1985).
    [CrossRef] [PubMed]
  21. L. N. Thibos, M. Ye, X. Zhang, and A. Bradley, “The chromatic eye: a new reduced-eye model of ocular chromatic aberration in humans,” Appl. Opt. 31, 3594–3600 (1992).
    [CrossRef] [PubMed]

2009

R. A. Applegate, L. N. Thibos, M. D. Twa, and E. J. Sarver, “Importance of fixation, pupil center, and reference axis in ocular wavefront sensing, videokeratography, and retinal image quality,” J. Cataract Refractive Surg. 35, 139–152 (2009).
[CrossRef]

2008

2005

J. Nam and J. Rubinstein, “Weighted Zernike expansion with applications to the optical aberration of the human eye,” J. Opt. Soc. Am. A 22, 1709–1716 (2005).
[CrossRef]

L. Lundstrom, P. Unsbo, and J. Gustafsson, “Off-axis wave front measurements for optical correction in eccentric viewing,” J. Biomed. Opt. 10, 034002 (2005).
[CrossRef] [PubMed]

2004

ANSI, “American National Standards for Ophthalmics-Methods for reporting optical aberrations of eyes,” American National Standards Institute ANSI Z80.28 (2004).

J. Rubinstein and G. Wolansky, “Eikonal functions: Old and new,” in Applied Mathematics Celebration, D.Givoli, ed. (Kluwer, 2004).

2003

T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, “Measurement of refractive errors in young myopes using the COAS Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 6–14 (2003).
[CrossRef]

L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, “Aberrations of the human eye in visible and near infrared illumination,” Optom. Vision Sci. 80, 26–35 (2003).
[CrossRef]

X. Cheng, N. L. Himebaugh, P. S. Kollbaum, L. N. Thibos, and A. Bradley, “Validation of a clinical Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 587–595 (2003).
[CrossRef]

2002

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
[PubMed]

2001

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
[CrossRef]

2000

D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinnemann, 2000).

1999

L. N. Thibos and A. Bradley, “Modeling the refractive and neurosensor systems of the eye,” in Visual Instrumentation, P.Moroulis, ed. (McGraw Hill, 1999).

1995

1992

1990

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

1985

1964

J. A. Kneisly, “Local curvature of wavefront in an optical system,” J. Am. Optom. Assoc. 54, 229–235 (1964).
[CrossRef]

1950

J. B. Keller and H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Am. Optom. Assoc. 40, 48–52 (1950).
[CrossRef]

Applegate, R. A.

R. A. Applegate, L. N. Thibos, M. D. Twa, and E. J. Sarver, “Importance of fixation, pupil center, and reference axis in ocular wavefront sensing, videokeratography, and retinal image quality,” J. Cataract Refractive Surg. 35, 139–152 (2009).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
[PubMed]

Arnold, V. I.

V. I. Arnold, Ordinary Differential Equations, 3rd ed. (Springer-Verlag, 1992).

Atchison, D. A.

D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinnemann, 2000).

Bescos, J.

Bradley, A.

S. Ravikumar, L. N. Thibos, and A. Bradley, “Calculation of retinal image quality for polychromatic light,” J. Opt. Soc. Am. A 25, 2395–2407 (2008).
[CrossRef]

X. Cheng, N. L. Himebaugh, P. S. Kollbaum, L. N. Thibos, and A. Bradley, “Validation of a clinical Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 587–595 (2003).
[CrossRef]

L. N. Thibos and A. Bradley, “Modeling the refractive and neurosensor systems of the eye,” in Visual Instrumentation, P.Moroulis, ed. (McGraw Hill, 1999).

L. N. Thibos, M. Ye, X. Zhang, and A. Bradley, “The chromatic eye: a new reduced-eye model of ocular chromatic aberration in humans,” Appl. Opt. 31, 3594–3600 (1992).
[CrossRef] [PubMed]

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

Cheng, X.

X. Cheng, N. L. Himebaugh, P. S. Kollbaum, L. N. Thibos, and A. Bradley, “Validation of a clinical Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 587–595 (2003).
[CrossRef]

Chisholm, W.

Copeland, J.

J. Copeland and D. Neal, “Combined objective and subjective optometer,” presented at the 7th International Congress on Wavefront Sensing and Optimized Refractive Correction, January 26–29, 2006. http://voi.opt.uh.edu/VOI/WavefrontCongress/2006/index.html.

Diaz-Santana, L.

L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, “Aberrations of the human eye in visible and near infrared illumination,” Optom. Vision Sci. 80, 26–35 (2003).
[CrossRef]

Gasser, W.

T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, “Measurement of refractive errors in young myopes using the COAS Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 6–14 (2003).
[CrossRef]

Gustafsson, J.

L. Lundstrom, P. Unsbo, and J. Gustafsson, “Off-axis wave front measurements for optical correction in eccentric viewing,” J. Biomed. Opt. 10, 034002 (2005).
[CrossRef] [PubMed]

Himebaugh, N. L.

X. Cheng, N. L. Himebaugh, P. S. Kollbaum, L. N. Thibos, and A. Bradley, “Validation of a clinical Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 587–595 (2003).
[CrossRef]

Howarth, P. A.

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

Keller, H. B.

J. B. Keller and H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Am. Optom. Assoc. 40, 48–52 (1950).
[CrossRef]

Keller, J. B.

J. B. Keller and H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Am. Optom. Assoc. 40, 48–52 (1950).
[CrossRef]

Kenmore, T.

T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, “Measurement of refractive errors in young myopes using the COAS Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 6–14 (2003).
[CrossRef]

Kneisly, J. A.

J. A. Kneisly, “Local curvature of wavefront in an optical system,” J. Am. Optom. Assoc. 54, 229–235 (1964).
[CrossRef]

Kollbaum, P. S.

X. Cheng, N. L. Himebaugh, P. S. Kollbaum, L. N. Thibos, and A. Bradley, “Validation of a clinical Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 587–595 (2003).
[CrossRef]

Lara-Saucedo, D.

L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, “Aberrations of the human eye in visible and near infrared illumination,” Optom. Vision Sci. 80, 26–35 (2003).
[CrossRef]

Lidkea, B.

Llorente, L.

L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, “Aberrations of the human eye in visible and near infrared illumination,” Optom. Vision Sci. 80, 26–35 (2003).
[CrossRef]

Lundstrom, L.

L. Lundstrom, P. Unsbo, and J. Gustafsson, “Off-axis wave front measurements for optical correction in eccentric viewing,” J. Biomed. Opt. 10, 034002 (2005).
[CrossRef] [PubMed]

Marcos, S.

L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, “Aberrations of the human eye in visible and near infrared illumination,” Optom. Vision Sci. 80, 26–35 (2003).
[CrossRef]

Nam, J.

Navarro, R.

Neal, D.

J. Copeland and D. Neal, “Combined objective and subjective optometer,” presented at the 7th International Congress on Wavefront Sensing and Optimized Refractive Correction, January 26–29, 2006. http://voi.opt.uh.edu/VOI/WavefrontCongress/2006/index.html.

Ravikumar, S.

Rubinstein, J.

J. Nam and J. Rubinstein, “Weighted Zernike expansion with applications to the optical aberration of the human eye,” J. Opt. Soc. Am. A 22, 1709–1716 (2005).
[CrossRef]

J. Rubinstein and G. Wolansky, “Eikonal functions: Old and new,” in Applied Mathematics Celebration, D.Givoli, ed. (Kluwer, 2004).

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
[CrossRef]

Rynders, M.

Salmon, T. O.

T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, “Measurement of refractive errors in young myopes using the COAS Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 6–14 (2003).
[CrossRef]

Santamaria, J.

Sarver, E. J.

R. A. Applegate, L. N. Thibos, M. D. Twa, and E. J. Sarver, “Importance of fixation, pupil center, and reference axis in ocular wavefront sensing, videokeratography, and retinal image quality,” J. Cataract Refractive Surg. 35, 139–152 (2009).
[CrossRef]

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
[PubMed]

Smith, G.

D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinnemann, 2000).

Still, D. L.

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

Thibos, L. N.

R. A. Applegate, L. N. Thibos, M. D. Twa, and E. J. Sarver, “Importance of fixation, pupil center, and reference axis in ocular wavefront sensing, videokeratography, and retinal image quality,” J. Cataract Refractive Surg. 35, 139–152 (2009).
[CrossRef]

S. Ravikumar, L. N. Thibos, and A. Bradley, “Calculation of retinal image quality for polychromatic light,” J. Opt. Soc. Am. A 25, 2395–2407 (2008).
[CrossRef]

X. Cheng, N. L. Himebaugh, P. S. Kollbaum, L. N. Thibos, and A. Bradley, “Validation of a clinical Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 587–595 (2003).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
[PubMed]

L. N. Thibos and A. Bradley, “Modeling the refractive and neurosensor systems of the eye,” in Visual Instrumentation, P.Moroulis, ed. (McGraw Hill, 1999).

M. Rynders, B. Lidkea, W. Chisholm, and L. N. Thibos, “Statistical distribution of foveal transverse chromatic aberration, pupil centration, and angle psi in a population of young adult eyes,” J. Opt. Soc. Am. A 12, 2348–2357 (1995).
[CrossRef]

L. N. Thibos, M. Ye, X. Zhang, and A. Bradley, “The chromatic eye: a new reduced-eye model of ocular chromatic aberration in humans,” Appl. Opt. 31, 3594–3600 (1992).
[CrossRef] [PubMed]

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

Twa, M. D.

R. A. Applegate, L. N. Thibos, M. D. Twa, and E. J. Sarver, “Importance of fixation, pupil center, and reference axis in ocular wavefront sensing, videokeratography, and retinal image quality,” J. Cataract Refractive Surg. 35, 139–152 (2009).
[CrossRef]

Unsbo, P.

L. Lundstrom, P. Unsbo, and J. Gustafsson, “Off-axis wave front measurements for optical correction in eccentric viewing,” J. Biomed. Opt. 10, 034002 (2005).
[CrossRef] [PubMed]

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
[PubMed]

West, R. W.

T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, “Measurement of refractive errors in young myopes using the COAS Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 6–14 (2003).
[CrossRef]

Wolansky, G.

J. Rubinstein and G. Wolansky, “Eikonal functions: Old and new,” in Applied Mathematics Celebration, D.Givoli, ed. (Kluwer, 2004).

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
[CrossRef]

Ye, M.

Zhang, X.

L. N. Thibos, M. Ye, X. Zhang, and A. Bradley, “The chromatic eye: a new reduced-eye model of ocular chromatic aberration in humans,” Appl. Opt. 31, 3594–3600 (1992).
[CrossRef] [PubMed]

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

Appl. Opt.

J. Am. Optom. Assoc.

J. B. Keller and H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Am. Optom. Assoc. 40, 48–52 (1950).
[CrossRef]

J. A. Kneisly, “Local curvature of wavefront in an optical system,” J. Am. Optom. Assoc. 54, 229–235 (1964).
[CrossRef]

J. Biomed. Opt.

L. Lundstrom, P. Unsbo, and J. Gustafsson, “Off-axis wave front measurements for optical correction in eccentric viewing,” J. Biomed. Opt. 10, 034002 (2005).
[CrossRef] [PubMed]

J. Cataract Refractive Surg.

R. A. Applegate, L. N. Thibos, M. D. Twa, and E. J. Sarver, “Importance of fixation, pupil center, and reference axis in ocular wavefront sensing, videokeratography, and retinal image quality,” J. Cataract Refractive Surg. 35, 139–152 (2009).
[CrossRef]

J. Opt. Soc. Am. A

J. Refract. Surg.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
[PubMed]

Opt. Rev.

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8, 281–283 (2001).
[CrossRef]

Optom. Vision Sci.

X. Cheng, N. L. Himebaugh, P. S. Kollbaum, L. N. Thibos, and A. Bradley, “Validation of a clinical Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 587–595 (2003).
[CrossRef]

T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, “Measurement of refractive errors in young myopes using the COAS Shack–Hartmann aberrometer,” Optom. Vision Sci. 80, 6–14 (2003).
[CrossRef]

L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, “Aberrations of the human eye in visible and near infrared illumination,” Optom. Vision Sci. 80, 26–35 (2003).
[CrossRef]

Vision Res.

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

Other

L. N. Thibos and A. Bradley, “Modeling the refractive and neurosensor systems of the eye,” in Visual Instrumentation, P.Moroulis, ed. (McGraw Hill, 1999).

J. Copeland and D. Neal, “Combined objective and subjective optometer,” presented at the 7th International Congress on Wavefront Sensing and Optimized Refractive Correction, January 26–29, 2006. http://voi.opt.uh.edu/VOI/WavefrontCongress/2006/index.html.

J. Rubinstein and G. Wolansky, “Eikonal functions: Old and new,” in Applied Mathematics Celebration, D.Givoli, ed. (Kluwer, 2004).

D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinnemann, 2000).

ANSI, “American National Standards for Ophthalmics-Methods for reporting optical aberrations of eyes,” American National Standards Institute ANSI Z80.28 (2004).

V. I. Arnold, Ordinary Differential Equations, 3rd ed. (Springer-Verlag, 1992).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Schematic diagram for a general aberrometry setting. (a) Scenario 1: the probe beam enters the pupil at the center following the path of the chief ray. (b) Scenario 2: the probe beam enters the pupil at some eccentric location but intersects the retina at the same location as in (a). In both scenarios the solid line is the path of the probe beam and the dashed line is the path of the reflected chief ray. The reflected wavefront and chief ray are identical in both scenarios. The chief ray is parallel to the probe beam in (a) but not in (b). If the measurement axis of the aberrometer coincides with the probe beam, then the measured tilt of the chief ray will be different in the two scenarios. The gray area indicates the beam profile.

Fig. 2
Fig. 2

Comparison of the chief ray prism, the Zernike prism, and the prismatic displacement of the beam profile. (a) Wavefront of Seidel coma, z = ρ 3 cos θ in the pupil plane. (b) Expected wavefront slopes for z = ρ 3 cos θ in the pupil plane. (c) Zernike coefficients before ( t = 0 mm ) and after propagation ( t = 500 mm ) . (d) Wavefront slopes for z = ρ 3 cos θ after propagating 50 cm . The arithmetic average of ( x , y ) coordinates of the light patch is denoted by the + sign. The third definition of the prism can be quantified by comparing the centers of the beam profiles before and after the propagation, i.e., the origin in (b) and the location with the + sign in (d). The Zernike prism for the wavefront after propagating 50 cm was calculated over the region enclosed by the circle of radius 3 mm centered at (0,0).

Fig. 3
Fig. 3

Models depending on the location of the measurement plane and the location of the probe beam relative to the chief ray. The surfaces shown are refracting surfaces. No aperture is shown in the diagram. The aberrometer is designed such that the plane is perpendicular to the measurement axis (the solid line with the right arrow entering the eye model). (a) Scenario 1 in the pupil plane. (b) Scenario 2 in the pupil plane. (c) Scenario 1 out of the pupil plane. (d) Scenario 2 out of the pupil plane. In all cases the origin of the model surface coincides with the center of the eye’s entrance pupil. In (a) and (b) the measurement plane contains the center of the eye’s entrance pupil.

Fig. 4
Fig. 4

Schematic diagram of the eye model construction. Σ 0 is the refracted wavefront, Σ 1 is the model cornea, z = z h is the observation plane. The right figure is the enlarged version of a part in the left figure to help understand Eq. (4).

Fig. 5
Fig. 5

Plot of the vector fields s—the projection of the refracted rays on the observation plane, which is placed at 3.8 mm from the pupil plane. The pupil radius is 3.28 mm . Note that s does not vanish at the center of the measurement plane.

Fig. 6
Fig. 6

Vector fields on a basic cell. (a) Degree: 1, (b) Degree: 0.

Fig. 7
Fig. 7

Schematic diagram for transverse chromatic aberrations (TCA) for two wavelengths based on the chief ray prism. If the measurement axis coincides with the chief ray in λ 1 , then the chief ray prism in λ 2 defines TCA in terms of chief rays. If the measurement axis does not align with either of the chief rays, then the difference between the chief ray prisms in λ 1 and λ 2 measures TCA. An alternative definition of TCA could be formulated as the difference in the propagation directions for the best fitting plane waves for the two wavelengths. The text refers to this alternative definition as Zernike TCA.

Fig. 8
Fig. 8

Example of a wavefront and model cornea for an individual eye. The measurement was made at λ = 850 nm . (a) A wavefront Σ 0 ( 850 nm ) . (b) Contour plot of Σ 0 ( 850 nm ) . (c) A model cornea Σ 1 . (d) Contour plot of Σ 1 .

Fig. 9
Fig. 9

Validation of the algorithm with a schematic eye (a PMMA eye). (a) Schematic drawing of the plastic eye. (b) Error of z with the ray height r is shown in micrometers. z = 0 is the apex of the surface, which corresponds to r = 0 . The error is measured as the difference between the exact surface Σ 1 (or Σ p ) and an estimate for it. The errors are always within a few micrometers.

Fig. 10
Fig. 10

Wavelength adjustments for four real human eyes using the OPD method. The comparison criterion is the ratio C n m ( 550 ) C n m ( 850 ) 1 . Pupil radii are (a) 2.764 mm , (b) 2.810 mm , (c) 2.672 mm , (d) 2.940 mm . Positive values represent the (percentage) increase of C n m ( 550 ) relative to C n m ( 850 ) . Negative values represent the (percentage) decrease of C n m ( 550 ) relative to C n m ( 850 ) . The changes of the aberration coefficients from the OPD method are not monotonic. In contrast, our analysis shows that the paraxial method finds a single scaling factor 1.02 for wavelength adjustments as shown in Eq. (12). Since we are comparing the adjustments relative to 1 (no change at all), we denote the paraxial method at the level 0.02 in each of the subfigures below. The horizontal axis shows the Zernike coefficients in the single indexing order [12].

Fig. 11
Fig. 11

Schematic drawing for the CO method.

Fig. 12
Fig. 12

Comparison of the global eye models with the OPD method and the CO method. (a) C 2 0 ( 850 ) = 5 μ m , (b) C 4 0 ( 850 ) = 0.2 μ m . For (c) and (d), C 1 1 ( 850 ) = 0.283 μ m and C 3 1 ( 850 ) = 0.1 μ m are initially given to construct the models. (c) shows C 1 1 ( λ ) . (d) shows C 3 1 ( λ ) . The pupil radius is 3 mm . Although we test three conditions, since the last condition involves two aberration coefficients, the results are shown in four panels.

Fig. 13
Fig. 13

Longitudinal chromatic aberration (LCA) of the eye models from the OPD method and the CO method. The pupil radius is 3 mm . The wavefront aberrations specified in the box are provided initially to the models at 850 nm . Both the OPD method and the CO method generate the changes Δ C 2 0 ( λ ) = C 2 0 ( λ ) C 2 0 ( 850 ) of the Zernike defocus in wavelength, which we convert to LCA by multiplying by the factor 4 3 3 2 . For each case, LCAs of the eye models from the two methods are indistinguishable and appear as the same curves. When aberrations are zero in 850 nm , LCAs from either of the models are the same as LCAs from the two cases C 4 0 ( 850 ) = 0.2 μ m or C 1 1 ( 850 ) = 0.283 μ m and C 3 1 ( 850 ) = 0.1 μ m .

Fig. 14
Fig. 14

Comparison of the 4-surface eye model with the reduced eye model by the OPD method and the CO method. We use the aberrations computed with the Navarro’s 4-surface eye model and construct the reduced eye models. We compare the lower and higher aberrations in new wavelength λ between the three methods. (a) C 2 0 ( λ ) , (b) C 4 0 ( λ ) . The pupil radius is 3 mm .

Equations (33)

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

W p ( x , y ) = C 1 1 Z 1 1 ( x , y ) = C 1 1 2 x R .
C 1 1 Z 1 1 x = 2 C 1 1 R = 2 × 0.33 μ m 3 mm = 0.22 × 10 3 rad .
F = ( F x , F y ) = ( s 1 , s 2 ) .
2 F x y = 2 F y x s 1 y = s 2 x .
d o = n 0 | f 0 O | + n 1 | O f 1 | .
n 0 | P 0 P 1 | + n 1 | P 1 f 1 | = d 0 .
a = n 1 2 n 0 2 ,
b = n 1 2 ( q , N 0 ) + n 0 d 0 ,
c = n 1 2 | q | 2 d 0 2 .
n 0 | P 0 P 1 | = n 0 [ | Q P 1 | + F ( Q ) ] .
( 1.0503 , 0.2626 ) , ( 1.0503 , 0.1313 ) ,
( 1.1815 , 0.2626 ) , ( 1.1815 , 0.1313 ) .
P ¯ = ( 1.1406 , 0.1806 , 3.7994 ) .
n ( λ ) = a + b λ c ,
x 2 + y 2 = 2 r z p z 2 ,
κ r i = α κ s + ( 1 α ) κ c i , i = 1 , 2 ,
κ c i = 1 1 n 1 κ r i n 1 1 n 1 κ s , i = 1 , 2 .
z c = 1 2 ( κ c 1 x 2 + κ c 2 y 2 ) .
κ r 1 = 0.00105874 , κ r 2 = 0.00106543 .
κ c 1 ( QRF ) = 0.17196979 , κ c 2 ( QRF ) = 0.17199021 .
κ c 1 = 0.17194404 , κ c 2 = 0.17197230 .
κ ¯ r 1 ( QRF ) = 0.00191615 , κ ¯ r 2 ( QRF ) = 0.00192298 .
κ ¯ r 1 = 0.00192548 , κ ¯ r 2 = 0.00193234 .
κ ¯ r i = n ¯ κ ¯ s + ( 1 n ¯ ) κ ¯ c i , i = 1 , 2 .
κ ¯ r i = ( n ¯ n 1 n ) κ s + ( 1 n ¯ 1 n ) κ r i , i = 1 , 2 .
D = 1 2 ( κ r 1 + κ r 2 ) , A = κ 1 r κ r 2 .
A ¯ = Γ A , D ¯ = Γ D + ( n ¯ n 1 n ) | O f 1 | 1 , Γ = 1 n ¯ 1 n .
C ¯ 2 0 = Γ C 2 0 + 10 3 R 2 4 3 ( n ¯ n 1 n ) | O f 1 | 1 ,
C ¯ 2 2 = Γ C 2 2 , C ¯ 2 2 = Γ C 2 2 .
| f 0 O | + n 1 | O f 1 | = | g 2 g 1 | + n 1 | g 1 f 1 | .
| f 0 O | + n 1 | O f 1 | = | g 3 g 5 | + n 1 | g 5 f 1 | .
n 1 cos θ | g 5 g 1 | = F + | g 5 g 1 | ,
L model ( x , y ) = L Cartesian ( x , y ) + F ( x , y ) ( 1 n 1 cos θ ) .

Metrics