Abstract

Ray tracing, a computational method for tracing the trajectories of rays of light through matter, is often used to characterize mechanical or biological visual systems with aberrations that are larger than the effect of diffraction inherent in the system. For example, ray tracing may be used to calculate geometric point spread functions (PSFs), which describe the image of a point source after it passes through an optical system. Calculating a geometric PSF is useful because it gives an estimate of the detail and quality of the image formed by a given optical system. However, when using ray tracing to calculate a PSF, the accuracy of the estimated PSF directly depends on the number of discrete rays used in the calculation; higher accuracies may require more computational power. Furthermore, adding optical components to a modeled system will increase its complexity and require critical modifications so that the model will describe the system correctly, sometimes necessitating a completely new model. Here, we address these challenges by developing a method that represents rays of light as a continuous function that depends on the light’s initial direction. By utilizing Chebyshev approximations (via the chebfun toolbox in MATLAB) for the implementation of this method, we greatly simplified the calculations for the location and direction of the rays. This method provides high precision and fast calculation speeds that allow the characterization of any symmetrical optical system (with a centered point source) in an analytical-like manner. Next, we demonstrate our methods by showing how they can easily calculate PSFs for complicated optical systems that contain multiple refractive and/or reflective interfaces.

© 2014 Optical Society of America

PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. L. Gagnon, R. H. H. Kröger, and B. Söderberg, “Adjusting a light dispersion model to fit measurements from vertebrate ocular media as well as ray-tracing in fish lenses,” Vis. Res. 50, 850–853 (2010).
    [CrossRef]
  2. L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vis. Res. 41, 973–979 (2001).
    [CrossRef]
  3. A. Sharma, D. V. Kumar, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. 21, 984–987 (1982).
    [CrossRef]
  4. W. S. Jagger, “The optics of the spherical fish lens,” Vis. Res. 32, 1271–1284 (1992).
    [CrossRef]
  5. Y. L. Gagnon, T. T. Sutton, and S. Johnsen, “Visual acuity in pelagic fishes and mollusks,” Vis. Res. 92, 1–9 (2013).
    [CrossRef]
  6. D. Y. C. Chan, “Determination and modeling of the 3-D gradient refractive-indexes in crystalline lenses,” Appl. Opt. 27, 926–931 (1988).
    [CrossRef]
  7. P. L. Chu, “Nondestructive measurement of index profile of an optical-fibre preform,” Electron. Lett. 13, 736–738 (1977).
    [CrossRef]
  8. Y. L. Gagnon and R. H. H. Kröger, “Gradient index models of monofocal and multifocal spherical fish lenses,” Investig. Ophthalmol. Vis. Sci. 47, 1211 (2006).
  9. Y. L. Gagnon, B. Söderberg, and R. H. H. Kröger, “Optical advantages and function of multifocal spherical fish lenses,” J. Opt. Soc. Am. A 29, 1786–1793 (2012).
    [CrossRef]
  10. Y. L. Gagnon, B. Söderberg, and R. H. H. Kröger, “Effects of the peripheral layers on the optical properties of spherical fish lenses,” J. Opt. Soc. Am. A 25, 2468–2475 (2008).
    [CrossRef]
  11. R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, and H. J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
    [CrossRef]
  12. R. H. H. Kröger, M. C. W. Campbell, R. Munger, and R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vis. Res. 34, 1815–1822 (1994).
    [CrossRef]
  13. O. E. Lind, A. Kelber, and R. H. H. Kröger, “Multifocal optical systems and pupil dynamics in birds,” J. Exp. Biol. 211, 2752–2758 (2008).
    [CrossRef]
  14. L. Matthiessen, “Ueber die beziehungen, welche zwischen dem brechungsindex des kerncentrums der krystalllinse und den dimensionen des auges bestehen,” Pflüger’s Archiv 27, 510–523 (1882).
  15. J. M. Schartau, B. Sjögreen, Y. L. Gagnon, and R. H. H. Kröger, “Optical plasticity in the crystalline lenses of the cichlid fish Aequidens pulcher,” Curr. Biol. 19, 122–126 (2009).
    [CrossRef]
  16. J. G. Sivak and R. O. Kreuzer, “Spherical aberration of the crystalline lens,” Vis. Res. 23, 59–70 (1983).
    [CrossRef]
  17. P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford University, 1997).
  18. E. Hecht, Optics (Addison-Wesley, 2002).
  19. J. Arasa and J. Alda, Real Ray Tracing (Marcel Dekker, 2004).
  20. J. Portilla and S. Barbero, “Accuracy of geometric point spread function estimation using the ray-counting method,” Proc. SPIE 8550, 855003 (2012).
    [CrossRef]
  21. O. N. Stavroudis and D. P. Feder, “Automatic computation of spot diagrams,” J. Opt. Soc. Am. 44, 163–164 (1954).
    [CrossRef]
  22. C.-S. Liu and P. D. Lin, “Computational method for deriving the geometric point spread function of an optical system,” Appl. Opt. 49, 126–136 (2010).
    [CrossRef]
  23. L. N. Trefethen, “Chebfun Version 4.2,” The Chebfun Development Team (2011), http://www.chebfun.org/ .
  24. Y. L. Gagnon, “chebRay,” (2014), https://github.com/yakir12/chebRay .
  25. M. F. Land, “Activity in the optic nerve of Pecten maximus in response to changes in light intensity, and to pattern and movement in the optical environment,” J. Exp. Biol. 45, 83–99 (1966).

2013 (1)

Y. L. Gagnon, T. T. Sutton, and S. Johnsen, “Visual acuity in pelagic fishes and mollusks,” Vis. Res. 92, 1–9 (2013).
[CrossRef]

2012 (2)

J. Portilla and S. Barbero, “Accuracy of geometric point spread function estimation using the ray-counting method,” Proc. SPIE 8550, 855003 (2012).
[CrossRef]

Y. L. Gagnon, B. Söderberg, and R. H. H. Kröger, “Optical advantages and function of multifocal spherical fish lenses,” J. Opt. Soc. Am. A 29, 1786–1793 (2012).
[CrossRef]

2010 (2)

C.-S. Liu and P. D. Lin, “Computational method for deriving the geometric point spread function of an optical system,” Appl. Opt. 49, 126–136 (2010).
[CrossRef]

Y. L. Gagnon, R. H. H. Kröger, and B. Söderberg, “Adjusting a light dispersion model to fit measurements from vertebrate ocular media as well as ray-tracing in fish lenses,” Vis. Res. 50, 850–853 (2010).
[CrossRef]

2009 (1)

J. M. Schartau, B. Sjögreen, Y. L. Gagnon, and R. H. H. Kröger, “Optical plasticity in the crystalline lenses of the cichlid fish Aequidens pulcher,” Curr. Biol. 19, 122–126 (2009).
[CrossRef]

2008 (2)

O. E. Lind, A. Kelber, and R. H. H. Kröger, “Multifocal optical systems and pupil dynamics in birds,” J. Exp. Biol. 211, 2752–2758 (2008).
[CrossRef]

Y. L. Gagnon, B. Söderberg, and R. H. H. Kröger, “Effects of the peripheral layers on the optical properties of spherical fish lenses,” J. Opt. Soc. Am. A 25, 2468–2475 (2008).
[CrossRef]

2006 (1)

Y. L. Gagnon and R. H. H. Kröger, “Gradient index models of monofocal and multifocal spherical fish lenses,” Investig. Ophthalmol. Vis. Sci. 47, 1211 (2006).

2001 (1)

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vis. Res. 41, 973–979 (2001).
[CrossRef]

1999 (1)

R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, and H. J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
[CrossRef]

1994 (1)

R. H. H. Kröger, M. C. W. Campbell, R. Munger, and R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vis. Res. 34, 1815–1822 (1994).
[CrossRef]

1992 (1)

W. S. Jagger, “The optics of the spherical fish lens,” Vis. Res. 32, 1271–1284 (1992).
[CrossRef]

1988 (1)

1983 (1)

J. G. Sivak and R. O. Kreuzer, “Spherical aberration of the crystalline lens,” Vis. Res. 23, 59–70 (1983).
[CrossRef]

1982 (1)

1977 (1)

P. L. Chu, “Nondestructive measurement of index profile of an optical-fibre preform,” Electron. Lett. 13, 736–738 (1977).
[CrossRef]

1966 (1)

M. F. Land, “Activity in the optic nerve of Pecten maximus in response to changes in light intensity, and to pattern and movement in the optical environment,” J. Exp. Biol. 45, 83–99 (1966).

1954 (1)

1882 (1)

L. Matthiessen, “Ueber die beziehungen, welche zwischen dem brechungsindex des kerncentrums der krystalllinse und den dimensionen des auges bestehen,” Pflüger’s Archiv 27, 510–523 (1882).

Alda, J.

J. Arasa and J. Alda, Real Ray Tracing (Marcel Dekker, 2004).

Arasa, J.

J. Arasa and J. Alda, Real Ray Tracing (Marcel Dekker, 2004).

Augusteyn, R. C.

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vis. Res. 41, 973–979 (2001).
[CrossRef]

Barbero, S.

J. Portilla and S. Barbero, “Accuracy of geometric point spread function estimation using the ray-counting method,” Proc. SPIE 8550, 855003 (2012).
[CrossRef]

Campbell, M. C. W.

R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, and H. J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
[CrossRef]

R. H. H. Kröger, M. C. W. Campbell, R. Munger, and R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vis. Res. 34, 1815–1822 (1994).
[CrossRef]

Chan, D. Y. C.

Chu, P. L.

P. L. Chu, “Nondestructive measurement of index profile of an optical-fibre preform,” Electron. Lett. 13, 736–738 (1977).
[CrossRef]

Feder, D. P.

Fernald, R. D.

R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, and H. J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
[CrossRef]

R. H. H. Kröger, M. C. W. Campbell, R. Munger, and R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vis. Res. 34, 1815–1822 (1994).
[CrossRef]

Gagnon, Y. L.

Y. L. Gagnon, T. T. Sutton, and S. Johnsen, “Visual acuity in pelagic fishes and mollusks,” Vis. Res. 92, 1–9 (2013).
[CrossRef]

Y. L. Gagnon, B. Söderberg, and R. H. H. Kröger, “Optical advantages and function of multifocal spherical fish lenses,” J. Opt. Soc. Am. A 29, 1786–1793 (2012).
[CrossRef]

Y. L. Gagnon, R. H. H. Kröger, and B. Söderberg, “Adjusting a light dispersion model to fit measurements from vertebrate ocular media as well as ray-tracing in fish lenses,” Vis. Res. 50, 850–853 (2010).
[CrossRef]

J. M. Schartau, B. Sjögreen, Y. L. Gagnon, and R. H. H. Kröger, “Optical plasticity in the crystalline lenses of the cichlid fish Aequidens pulcher,” Curr. Biol. 19, 122–126 (2009).
[CrossRef]

Y. L. Gagnon, B. Söderberg, and R. H. H. Kröger, “Effects of the peripheral layers on the optical properties of spherical fish lenses,” J. Opt. Soc. Am. A 25, 2468–2475 (2008).
[CrossRef]

Y. L. Gagnon and R. H. H. Kröger, “Gradient index models of monofocal and multifocal spherical fish lenses,” Investig. Ophthalmol. Vis. Sci. 47, 1211 (2006).

Garner, L. F.

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vis. Res. 41, 973–979 (2001).
[CrossRef]

Ghatak, A. K.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002).

Jagger, W. S.

W. S. Jagger, “The optics of the spherical fish lens,” Vis. Res. 32, 1271–1284 (1992).
[CrossRef]

Johnsen, S.

Y. L. Gagnon, T. T. Sutton, and S. Johnsen, “Visual acuity in pelagic fishes and mollusks,” Vis. Res. 92, 1–9 (2013).
[CrossRef]

Kelber, A.

O. E. Lind, A. Kelber, and R. H. H. Kröger, “Multifocal optical systems and pupil dynamics in birds,” J. Exp. Biol. 211, 2752–2758 (2008).
[CrossRef]

Kreuzer, R. O.

J. G. Sivak and R. O. Kreuzer, “Spherical aberration of the crystalline lens,” Vis. Res. 23, 59–70 (1983).
[CrossRef]

Kröger, R. H. H.

Y. L. Gagnon, B. Söderberg, and R. H. H. Kröger, “Optical advantages and function of multifocal spherical fish lenses,” J. Opt. Soc. Am. A 29, 1786–1793 (2012).
[CrossRef]

Y. L. Gagnon, R. H. H. Kröger, and B. Söderberg, “Adjusting a light dispersion model to fit measurements from vertebrate ocular media as well as ray-tracing in fish lenses,” Vis. Res. 50, 850–853 (2010).
[CrossRef]

J. M. Schartau, B. Sjögreen, Y. L. Gagnon, and R. H. H. Kröger, “Optical plasticity in the crystalline lenses of the cichlid fish Aequidens pulcher,” Curr. Biol. 19, 122–126 (2009).
[CrossRef]

O. E. Lind, A. Kelber, and R. H. H. Kröger, “Multifocal optical systems and pupil dynamics in birds,” J. Exp. Biol. 211, 2752–2758 (2008).
[CrossRef]

Y. L. Gagnon, B. Söderberg, and R. H. H. Kröger, “Effects of the peripheral layers on the optical properties of spherical fish lenses,” J. Opt. Soc. Am. A 25, 2468–2475 (2008).
[CrossRef]

Y. L. Gagnon and R. H. H. Kröger, “Gradient index models of monofocal and multifocal spherical fish lenses,” Investig. Ophthalmol. Vis. Sci. 47, 1211 (2006).

R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, and H. J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
[CrossRef]

R. H. H. Kröger, M. C. W. Campbell, R. Munger, and R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vis. Res. 34, 1815–1822 (1994).
[CrossRef]

Kumar, D. V.

Land, M. F.

M. F. Land, “Activity in the optic nerve of Pecten maximus in response to changes in light intensity, and to pattern and movement in the optical environment,” J. Exp. Biol. 45, 83–99 (1966).

Lin, P. D.

Lind, O. E.

O. E. Lind, A. Kelber, and R. H. H. Kröger, “Multifocal optical systems and pupil dynamics in birds,” J. Exp. Biol. 211, 2752–2758 (2008).
[CrossRef]

Liu, C.-S.

Macdonald, J.

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford University, 1997).

Matthiessen, L.

L. Matthiessen, “Ueber die beziehungen, welche zwischen dem brechungsindex des kerncentrums der krystalllinse und den dimensionen des auges bestehen,” Pflüger’s Archiv 27, 510–523 (1882).

Mouroulis, P.

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford University, 1997).

Munger, R.

R. H. H. Kröger, M. C. W. Campbell, R. Munger, and R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vis. Res. 34, 1815–1822 (1994).
[CrossRef]

Portilla, J.

J. Portilla and S. Barbero, “Accuracy of geometric point spread function estimation using the ray-counting method,” Proc. SPIE 8550, 855003 (2012).
[CrossRef]

Schartau, J. M.

J. M. Schartau, B. Sjögreen, Y. L. Gagnon, and R. H. H. Kröger, “Optical plasticity in the crystalline lenses of the cichlid fish Aequidens pulcher,” Curr. Biol. 19, 122–126 (2009).
[CrossRef]

Sharma, A.

Sivak, J. G.

J. G. Sivak and R. O. Kreuzer, “Spherical aberration of the crystalline lens,” Vis. Res. 23, 59–70 (1983).
[CrossRef]

Sjögreen, B.

J. M. Schartau, B. Sjögreen, Y. L. Gagnon, and R. H. H. Kröger, “Optical plasticity in the crystalline lenses of the cichlid fish Aequidens pulcher,” Curr. Biol. 19, 122–126 (2009).
[CrossRef]

Smith, G.

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vis. Res. 41, 973–979 (2001).
[CrossRef]

Söderberg, B.

Stavroudis, O. N.

Sutton, T. T.

Y. L. Gagnon, T. T. Sutton, and S. Johnsen, “Visual acuity in pelagic fishes and mollusks,” Vis. Res. 92, 1–9 (2013).
[CrossRef]

Wagner, H. J.

R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, and H. J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
[CrossRef]

Yao, S.

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vis. Res. 41, 973–979 (2001).
[CrossRef]

Appl. Opt. (3)

Curr. Biol. (1)

J. M. Schartau, B. Sjögreen, Y. L. Gagnon, and R. H. H. Kröger, “Optical plasticity in the crystalline lenses of the cichlid fish Aequidens pulcher,” Curr. Biol. 19, 122–126 (2009).
[CrossRef]

Electron. Lett. (1)

P. L. Chu, “Nondestructive measurement of index profile of an optical-fibre preform,” Electron. Lett. 13, 736–738 (1977).
[CrossRef]

Investig. Ophthalmol. Vis. Sci. (1)

Y. L. Gagnon and R. H. H. Kröger, “Gradient index models of monofocal and multifocal spherical fish lenses,” Investig. Ophthalmol. Vis. Sci. 47, 1211 (2006).

J. Comp. Physiol. A (1)

R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, and H. J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
[CrossRef]

J. Exp. Biol. (2)

O. E. Lind, A. Kelber, and R. H. H. Kröger, “Multifocal optical systems and pupil dynamics in birds,” J. Exp. Biol. 211, 2752–2758 (2008).
[CrossRef]

M. F. Land, “Activity in the optic nerve of Pecten maximus in response to changes in light intensity, and to pattern and movement in the optical environment,” J. Exp. Biol. 45, 83–99 (1966).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Pflüger’s Archiv (1)

L. Matthiessen, “Ueber die beziehungen, welche zwischen dem brechungsindex des kerncentrums der krystalllinse und den dimensionen des auges bestehen,” Pflüger’s Archiv 27, 510–523 (1882).

Proc. SPIE (1)

J. Portilla and S. Barbero, “Accuracy of geometric point spread function estimation using the ray-counting method,” Proc. SPIE 8550, 855003 (2012).
[CrossRef]

Vis. Res. (6)

R. H. H. Kröger, M. C. W. Campbell, R. Munger, and R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vis. Res. 34, 1815–1822 (1994).
[CrossRef]

J. G. Sivak and R. O. Kreuzer, “Spherical aberration of the crystalline lens,” Vis. Res. 23, 59–70 (1983).
[CrossRef]

W. S. Jagger, “The optics of the spherical fish lens,” Vis. Res. 32, 1271–1284 (1992).
[CrossRef]

Y. L. Gagnon, T. T. Sutton, and S. Johnsen, “Visual acuity in pelagic fishes and mollusks,” Vis. Res. 92, 1–9 (2013).
[CrossRef]

Y. L. Gagnon, R. H. H. Kröger, and B. Söderberg, “Adjusting a light dispersion model to fit measurements from vertebrate ocular media as well as ray-tracing in fish lenses,” Vis. Res. 50, 850–853 (2010).
[CrossRef]

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vis. Res. 41, 973–979 (2001).
[CrossRef]

Other (5)

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford University, 1997).

E. Hecht, Optics (Addison-Wesley, 2002).

J. Arasa and J. Alda, Real Ray Tracing (Marcel Dekker, 2004).

L. N. Trefethen, “Chebfun Version 4.2,” The Chebfun Development Team (2011), http://www.chebfun.org/ .

Y. L. Gagnon, “chebRay,” (2014), https://github.com/yakir12/chebRay .

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.


Metrics