Abstract

Wave aberrations in the human eye are usually known with respect to the ideal spherical wavefront in the exit pupil. Using Kirchhoff’s diffraction theory, we have derived a diffraction integral to compute the optical field on the retina from the wave aberration data. We have proposed a numerical algorithm based on the Stamnes–Spjelkavik–Pedersen (SSP) method to solve that integral. We have shown which approximations are admissible to reduce the complexity of the diffraction integral. In addition, we have compared our results with those of the conventional procedure used to compute intensities on the retina. We have found significant differences between our results and the conventional ones.

© 2008 Optical Society of America

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References

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  1. B. Vohnsen, “Photoreceptor waveguides and effective retinal image quality,” J. Opt. Soc. Am. A 24, 597-607 (2007).
    [CrossRef]
  2. J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (IOP, 1986).
  3. J. Perez, D. Mas, C. Illueca, J. J. Miret, C. Vazquez, and C. Hernandez, “Complete algorithm for the calculation of light patterns inside the ocular media,” J. Opt. Soc. Am. A 52, 1161-1176 (2005).
  4. D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinemann, 2000).
  5. S. Chavez-Cerda, “Nonparaxial effects in visual imaging of the Emsley schematic eye,” presented at the Third European Meeting in Physiological Optics, City University, London, September 7-9, 2006.
  6. R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951-953 (1999).
    [CrossRef]
  7. E. Moreno-Barriuso and R. Navarro, “Laser ray tracing versus Hartmann-Shack sensor for measuring optical aberrations in the human eye,” J. Opt. Soc. Am. A 17, 974-985 (2000).
    [CrossRef]
  8. A. Castro, S. Barbero, and S. Marcos, “A reconstruction technique to estimate the gradient-index distribution of the crystalline lens using ray aberration data in vivo,” presented at the 2007 ARVO Annual Meeting, Fort Lauderdale, Florida, May 6-10, 2007.
  9. H. G. Kraus, “Huygens-Fresnel-Kirchhoff wavefront diffraction formulation: Spherical waves,” J. Opt. Soc. Am. A 6, 1196-1205 (1989).
    [CrossRef]
  10. P. Artal, “Calculations of two-dimensional foveal retinal images in real eyes,” J. Opt. Soc. Am. A 7, 1374-1380 (1990).
    [CrossRef] [PubMed]
  11. S. Marcos and S. A. Burns, “On the symmetry between eyes of wavefront aberration and cone directionality” Vision Res. 40, 2437-2447 (2000).
    [CrossRef] [PubMed]
  12. G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883).
    [CrossRef]
  13. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1980).
    [PubMed]
  14. E. Wolf and E. W. Marchand, “Comparison of Kirchhoff and the Rayleigh-Sommerfeld theories of diffraction at an aperture,” J. Opt. Soc. Am. 54, 587-594 (1964).
    [CrossRef]
  15. H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157-182 (1970).
    [CrossRef]
  16. J. J. Stamnes, B. Spjelkavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local and phase amplitude approximations,” Opt. Acta 30, 207-222 (1983).
    [CrossRef]
  17. S. Barbero and S. Marcos, “Analytical tools for customized design of monofocal intraocular lenses,” Opt. Express 15, 8576-8591 (2007).
    [CrossRef] [PubMed]
  18. X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” Aust. Math. Soc. Gaz. 4, 310-321 (2004).
  19. S. Marcos, S. Barbero, and I. Jimenez-Alfaro, “Optical quality and depth-of-field of eyes implanted with spherical and aspheric intraocular lenses,” J. Refract. Surg. 21, 223-235 (2005).
    [PubMed]

2007 (2)

2005 (2)

J. Perez, D. Mas, C. Illueca, J. J. Miret, C. Vazquez, and C. Hernandez, “Complete algorithm for the calculation of light patterns inside the ocular media,” J. Opt. Soc. Am. A 52, 1161-1176 (2005).

S. Marcos, S. Barbero, and I. Jimenez-Alfaro, “Optical quality and depth-of-field of eyes implanted with spherical and aspheric intraocular lenses,” J. Refract. Surg. 21, 223-235 (2005).
[PubMed]

2004 (1)

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” Aust. Math. Soc. Gaz. 4, 310-321 (2004).

2000 (2)

E. Moreno-Barriuso and R. Navarro, “Laser ray tracing versus Hartmann-Shack sensor for measuring optical aberrations in the human eye,” J. Opt. Soc. Am. A 17, 974-985 (2000).
[CrossRef]

S. Marcos and S. A. Burns, “On the symmetry between eyes of wavefront aberration and cone directionality” Vision Res. 40, 2437-2447 (2000).
[CrossRef] [PubMed]

1999 (1)

1990 (1)

1989 (1)

1983 (1)

J. J. Stamnes, B. Spjelkavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local and phase amplitude approximations,” Opt. Acta 30, 207-222 (1983).
[CrossRef]

1970 (1)

H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157-182 (1970).
[CrossRef]

1964 (1)

1883 (1)

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883).
[CrossRef]

Artal, P.

Atchison, D. A.

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

Barbero, S.

S. Barbero and S. Marcos, “Analytical tools for customized design of monofocal intraocular lenses,” Opt. Express 15, 8576-8591 (2007).
[CrossRef] [PubMed]

S. Marcos, S. Barbero, and I. Jimenez-Alfaro, “Optical quality and depth-of-field of eyes implanted with spherical and aspheric intraocular lenses,” J. Refract. Surg. 21, 223-235 (2005).
[PubMed]

A. Castro, S. Barbero, and S. Marcos, “A reconstruction technique to estimate the gradient-index distribution of the crystalline lens using ray aberration data in vivo,” presented at the 2007 ARVO Annual Meeting, Fort Lauderdale, Florida, May 6-10, 2007.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1980).
[PubMed]

Bradley, A.

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” Aust. Math. Soc. Gaz. 4, 310-321 (2004).

Burns, S. A.

S. Marcos and S. A. Burns, “On the symmetry between eyes of wavefront aberration and cone directionality” Vision Res. 40, 2437-2447 (2000).
[CrossRef] [PubMed]

Castro, A.

A. Castro, S. Barbero, and S. Marcos, “A reconstruction technique to estimate the gradient-index distribution of the crystalline lens using ray aberration data in vivo,” presented at the 2007 ARVO Annual Meeting, Fort Lauderdale, Florida, May 6-10, 2007.

Chavez-Cerda, S.

S. Chavez-Cerda, “Nonparaxial effects in visual imaging of the Emsley schematic eye,” presented at the Third European Meeting in Physiological Optics, City University, London, September 7-9, 2006.

Cheng, X.

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” Aust. Math. Soc. Gaz. 4, 310-321 (2004).

Hernandez, C.

J. Perez, D. Mas, C. Illueca, J. J. Miret, C. Vazquez, and C. Hernandez, “Complete algorithm for the calculation of light patterns inside the ocular media,” J. Opt. Soc. Am. A 52, 1161-1176 (2005).

Hopkins, H. H.

H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157-182 (1970).
[CrossRef]

Illueca, C.

J. Perez, D. Mas, C. Illueca, J. J. Miret, C. Vazquez, and C. Hernandez, “Complete algorithm for the calculation of light patterns inside the ocular media,” J. Opt. Soc. Am. A 52, 1161-1176 (2005).

Jimenez-Alfaro, I.

S. Marcos, S. Barbero, and I. Jimenez-Alfaro, “Optical quality and depth-of-field of eyes implanted with spherical and aspheric intraocular lenses,” J. Refract. Surg. 21, 223-235 (2005).
[PubMed]

Kirchhoff, G.

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883).
[CrossRef]

Kraus, H. G.

Marchand, E. W.

Marcos, S.

S. Barbero and S. Marcos, “Analytical tools for customized design of monofocal intraocular lenses,” Opt. Express 15, 8576-8591 (2007).
[CrossRef] [PubMed]

S. Marcos, S. Barbero, and I. Jimenez-Alfaro, “Optical quality and depth-of-field of eyes implanted with spherical and aspheric intraocular lenses,” J. Refract. Surg. 21, 223-235 (2005).
[PubMed]

S. Marcos and S. A. Burns, “On the symmetry between eyes of wavefront aberration and cone directionality” Vision Res. 40, 2437-2447 (2000).
[CrossRef] [PubMed]

A. Castro, S. Barbero, and S. Marcos, “A reconstruction technique to estimate the gradient-index distribution of the crystalline lens using ray aberration data in vivo,” presented at the 2007 ARVO Annual Meeting, Fort Lauderdale, Florida, May 6-10, 2007.

Mas, D.

J. Perez, D. Mas, C. Illueca, J. J. Miret, C. Vazquez, and C. Hernandez, “Complete algorithm for the calculation of light patterns inside the ocular media,” J. Opt. Soc. Am. A 52, 1161-1176 (2005).

Miret, J. J.

J. Perez, D. Mas, C. Illueca, J. J. Miret, C. Vazquez, and C. Hernandez, “Complete algorithm for the calculation of light patterns inside the ocular media,” J. Opt. Soc. Am. A 52, 1161-1176 (2005).

Moreno-Barriuso, E.

Navarro, R.

Pedersen, H. M.

J. J. Stamnes, B. Spjelkavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local and phase amplitude approximations,” Opt. Acta 30, 207-222 (1983).
[CrossRef]

Perez, J.

J. Perez, D. Mas, C. Illueca, J. J. Miret, C. Vazquez, and C. Hernandez, “Complete algorithm for the calculation of light patterns inside the ocular media,” J. Opt. Soc. Am. A 52, 1161-1176 (2005).

Smith, G.

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

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local and phase amplitude approximations,” Opt. Acta 30, 207-222 (1983).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, B. Spjelkavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local and phase amplitude approximations,” Opt. Acta 30, 207-222 (1983).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (IOP, 1986).

Thibos, L. N.

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” Aust. Math. Soc. Gaz. 4, 310-321 (2004).

Vazquez, C.

J. Perez, D. Mas, C. Illueca, J. J. Miret, C. Vazquez, and C. Hernandez, “Complete algorithm for the calculation of light patterns inside the ocular media,” J. Opt. Soc. Am. A 52, 1161-1176 (2005).

Vohnsen, B.

Wolf, E.

E. Wolf and E. W. Marchand, “Comparison of Kirchhoff and the Rayleigh-Sommerfeld theories of diffraction at an aperture,” J. Opt. Soc. Am. 54, 587-594 (1964).
[CrossRef]

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1980).
[PubMed]

Yzuel, M. J.

H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157-182 (1970).
[CrossRef]

Ann. Phys. (1)

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883).
[CrossRef]

Aust. Math. Soc. Gaz. (1)

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” Aust. Math. Soc. Gaz. 4, 310-321 (2004).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

J. Refract. Surg. (1)

S. Marcos, S. Barbero, and I. Jimenez-Alfaro, “Optical quality and depth-of-field of eyes implanted with spherical and aspheric intraocular lenses,” J. Refract. Surg. 21, 223-235 (2005).
[PubMed]

Opt. Acta (2)

H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157-182 (1970).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local and phase amplitude approximations,” Opt. Acta 30, 207-222 (1983).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Vision Res. (1)

S. Marcos and S. A. Burns, “On the symmetry between eyes of wavefront aberration and cone directionality” Vision Res. 40, 2437-2447 (2000).
[CrossRef] [PubMed]

Other (5)

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (IOP, 1986).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1980).
[PubMed]

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

S. Chavez-Cerda, “Nonparaxial effects in visual imaging of the Emsley schematic eye,” presented at the Third European Meeting in Physiological Optics, City University, London, September 7-9, 2006.

A. Castro, S. Barbero, and S. Marcos, “A reconstruction technique to estimate the gradient-index distribution of the crystalline lens using ray aberration data in vivo,” presented at the 2007 ARVO Annual Meeting, Fort Lauderdale, Florida, May 6-10, 2007.

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Figures (4)

Fig. 1
Fig. 1

Geometry showing the coordinates and distances used in the diffraction integral analysis. P, point at the ideal spherical wavefront ( Ω ) ; O, observation point; F, paraxial focal point

Fig. 2
Fig. 2

Normalized intensity distribution using a pseudoaphakic eye model [17]: (a) along the optical axis for different object vergences and (b) along the plane perpendicular to the optical axis at an image plane located 1 mm out of the paraxial focus. The intensities computed using the SSP algorithm and the Simpson method are represented by a solid curve and black circles, respectively. The pupil radius was set to 3.25 mm , the Zernike spherical aberration ( Z 40 ) was set to 1 μ m , and the wavelength was set to 0.555 μ m (photopic peak).

Fig. 3
Fig. 3

Normalized intensity distribution along the optical axis for different object vergences using a pseudoaphakic eye model [17] applying the constant amplitude approximation (solid curve), Fresnel approximation (circles), Fraunhofer approximation (triangles), and the conventional procedure (solid curve with circles) as explained in Section 6. The intensity was computed using the SSP algorithm. The pupil radius was set to 3.25 mm , the Zernike astigmatic aberration ( Z 22 ) was set to 1 μ m , and the wavelength was set to 0.555 μ m .

Fig. 4
Fig. 4

Intensity distribution at points located in a transverse plane in focus and 3 diopters (D) out of focus. The dimensions are a square of 0.1 mm . Shown are the constant amplitude approximation at focus (a) and out of focus (b); the Fresnel approximation at focus (c) and out of focus (d); and the. Fraunhofer approximation at focus (e) and out of focus (f). The intensity was computed using the two-dimensional SSP algorithm. The conventional procedure, using a FFT algorithm, was used to generate the intensity images at focus (g) and out of focus (h). The pupil radius was set to 3.25 mm , the Zernike astigmatic aberration ( Z 22 ) was set to 1 μ m , and the wavelength is set to 0.555 μ m .

Tables (1)

Tables Icon

Table 1 Percentage Errors and Computation Times (for One Observation Point) of the SSP Algorithm as a Function of the Number of Subdomains in the Radial ( N r ) and Angular Coordinates ( N θ )

Equations (17)

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u ( P ) = 1 4 π A { u n ( e i k s s ) ( e i k s s ) u n } d A ,
u = e i k ( W + n ) .
u ( P ) = 1 4 π A e i k ( W + n ) { e i k s s ( i k 1 s ) s n i k e i k s s ( 1 + W n ) } d A .
i k 1 s i k .
u ( P ) = i k 4 π A e i k ( W + n + s ) s { ( W n s n ) + 1 } d A .
u ( P ) = i k 4 π A e i k ( W + n + s ) s ( K + 1 ) d A ,
K = W n s n .
s = ρ o 2 + ρ 2 + ( z o z ) 2 2 ρ o ρ cos ( θ θ o ) ,
s ( ρ ) = ( ρ o ρ ) 2 + ( z o z ) 2 .
W ( ρ ) = W 40 ρ 4 .
K ( ρ , ρ o , z o ) = W ( ρ ) n s ( ρ ) n = ρ n ( W ( ρ ) ρ s ( ρ ) ρ ) .
z = z a z a 2 ρ 2 ,
ρ n = cos ( ρ , n ) = cos ( ρ , n ) = ρ ρ 2 + ( z a z ) 2 .
s ( ρ ) ρ = z a 2 ρ 2 ( ρ ρ o ) ρ ( z o z ) ( z a 2 ρ 2 ) ( ρ o ρ ) 2 + ( z a 2 ρ 2 ) ( z o z ) 2 .
u ( ρ o , z o ) = i k 4 π C e i k ( W ( ρ ) + s ( ρ ) ) s ( ρ ) ( K ( ρ , ρ o , z o ) + 1 ) d ρ ,
s ( z o z a ) ρ ρ o cos ( θ θ o ) ( z o z a ) + ρ 2 2 ( z o z a ) ( ρ ρ o cos ( θ θ o ) ) 2 2 ( z o z a ) 3 .
s ( z o z a ) ρ ρ o cos ( θ θ o ) ( z o z a ) .

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