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

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References

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  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)

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]

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

2010 (2)

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]

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]

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 .

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Figures (5)

Fig. 1.
Fig. 1.

Optical setup of the simple case study. The y axis is the system’s optical axis and the aperture is located on the x axis. The grid lines represent 0.5 unit increments. The system is defined across the domain of x, x:X=[1,1], and includes z1, z2, and zf with refractive indices of 1, 1.45, and 1.3, respectively. The point source is located at p=0+4i, and an example ray (green line) with exit angle β (red highlighted angular area) has direction I. The domain of β can vary within the blue highlighted angular area (i.e., β:B=[0,12.7°]). The example ray intersects the first interface, z1, at a point that is dz1 from the optical axis. After following an unrealistic hypothetical trajectory (chosen only for explanatory reasons), the ray intersects the image plane at “direct” distance dzf and arc length L from the optical axis. See further specifications in the code at [24].

Fig. 2.
Fig. 2.

Deviation of rays from the optical axis in the simple case study. (a) Deviations of rays from the optical axis as a function of β (the initial angle between the optical axis and an exiting ray of light) in 2D (black) and 3D (red). The x axis is β across its domain. The y axis is the arc length (along the image plane) between the optical axis and the point where each ray intersected with the image plane [calculated as L in Eq. (17)]. (b) PSFs of the optical system in 2D (black) and 3D (red). The x axis is the deviation sampled from (a). The (log-scale) y axis describes the corresponding probability each deviation has in the system.

Fig. 3.
Fig. 3.

Traced rays through the scallop eye. The y axis is the system’s optical axis and the aperture is located on the x axis. The grid lines represent 100 μm increments. The system includes a cornea, lens, distal retina (in blue), proximal retina (in red), and mirror. The refractive indices of the surrounding medium, cornea, and lens are 1.34, 1.37, and 1.52, respectively, while 1.36, 1.38, and 1.36 are the refractive indices outside the distal retina, proximal retina, and the mirror, respectively. The point source is located 500 μm from the aperture (see more size specifications in [24]).

Fig. 4.
Fig. 4.

Deviation of rays from the optical axis for a scallop’s eye. (a) Deviations of rays from the optical axis as a function of β in the distal retina (blue) and the proximal retina (red) for both incoming (solid) and exiting (dashed) light. The x axis is β across its domain. The y axis is the arc length (along the image plane) between each ray–plane intersection point and the optical axis [calculated as L in Eq. (17)]. (b) PSFs of the scallop eye at the distal retina (blue) and proximal retina (red). The x axis is the deviation sampled from (a). The (log-scale) y axis describes the corresponding probability each deviation has in the system.

Fig. 5.
Fig. 5.

Comparisons of the algorithm from this study to other methods. (a) Comparison of a spot diagram calculated by ZEMAX (blue cross) and our algorithm (red dot). (b) Comparison of a PSF calculated by Liu and Lin [22] (blue line) and our algorithm (red-dashed line).

Equations (18)

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

β:B=[0,arctan(r/L)],
p(β)=0+Li,
I(β)=exp(i(βπ/2)).
z=[z1z2zf],
z:X=[2r,2r],
z=zz1(r).
a=I(Iβ1)/R(Iβ1),
b=I(pβ1)aR(pβ1),
f=ax+b,
zif=0.
N=arctan(dzdx)+π2,
α1=N(R(p))arctanI(I)/R(I),
α2=arcsin(n1sin(α1)/n2),
I=exp(i(N+α2π)).
l(x)=X|d(x+iz3)dx|dx,
l(x)=l(x)l(0).
L=|lR(p)|.
L2=LdF.

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