Abstract

To model the light-guiding properties of a hexagonal array of dielectric cylinders, we have numerically solved Maxwell’s equations with the finite-difference time-domain technique. The sizes and refractive indices of the cylinders are representative of those of the outer segments of the cone photoreceptors in the human central retina. In the array, light propagates predominantly as a “slow” mode, with a noticeable contribution of a “fast” mode, with the optical field localized in the intra- and inter-cylinder spaces, respectively. Interference between these modes leads to substantial (up to approximately 60%) axial oscillations in optical power within the cylinders. Our numerical model offered approximate dependence of the optical intensity distribution within the cylinders on their radii and separations.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. M. Enoch, “Waveguide modes: are they present, and what is their possible role in the visual mechanism?” J. Opt. Soc. Am. 50, 1025-1026 (1960).
    [CrossRef] [PubMed]
  2. A.W.Snyder and R.Menzel, eds., Photoreceptor Optics (Springer-Verlag, 1975).
    [CrossRef]
  3. J.M.Enoch and F.L.Tobey, Jr., eds., Vertebrate Photoreceptor Optics (Springer-Verlag, 1981).
  4. V. Lakshminarayanan and M. L. Calvo, “Initial field and energy flux in absorbing optical waveguides. II. Implications,” J. Opt. Soc. Am. A 4, 2133-2140 (1987).
    [CrossRef] [PubMed]
  5. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  6. B. Vohnsen, “Photoreceptor waveguides and effective retinal image quality,” J. Opt. Soc. Am. A 24, 597-607 (2007).
    [CrossRef]
  7. L. Eyges and P. Wintersteiner, “Modes of an array of dielectric waveguides,” J. Opt. Soc. Am. 71, 1351-1360 (1981).
  8. W. Wijngaard, “Some normal modes of an infinite hexagonal array of identical circular dielectric rods,” J. Opt. Soc. Am. 64, 1136-1144 (1974).
    [CrossRef] [PubMed]
  9. A. W. Snyder and R. L. Kyhl, “Surface mode propagation along an array of dielectric rods with all elements excited identically,” IEEE Trans. Antennas Propag. AP14, 510-511 (1966).
    [CrossRef]
  10. R. Vanclooster and P. Phariseau, “Light propagation in fiber bundles,” Physica 49, 493-501 (1970).
    [CrossRef]
  11. R. Vanclooster and P. Phariseau, “Diffraction of an electromagnetic wave by a fiber bundle,” Physica 50, 308-316 (1970).
    [CrossRef]
  12. W. Wijngaard, “Guided normal modes of two parallel circular dielectric rods,” J. Opt. Soc. Am. 63, 944-950 (1973).
    [CrossRef]
  13. J. Limeres, M. L. Calvo, J. M. Enoch, and V. Lakshminarayanan, “Light scattering by an array of birefringent optical waveguides: theoretical foundations,” J. Opt. Soc. Am. B 20, 1542-1549 (2003).
    [CrossRef]
  14. M. L. Calvo and V. Lakshminarayanan, “Birefringent optical waveguides,” in Optical Waveguides: From Theory to Applied Technologies, M.L.Calvo and R.F.Alvarez-Estrada, eds. (CRC, 2007), pp. 38-96.
  15. R. Sammut, C. Pask, and A. W. Snyder, “Excitation and power of unbound modes within a circular dielectric waveguide,” Proc. Inst. Electr. Eng. 122, 25-33 (1975).
    [CrossRef]
  16. R. Sammut, “Effect of leaky modes on response of short optical fibers,” J. Opt. Soc. Am. 67, 1284-1285 (1977).
    [CrossRef]
  17. B. Vohnsen, I. Iglesias, and P. Artal, “Guided light and diffraction model of human-eye photoreceptors,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 2318-2328 (2005).
    [CrossRef] [PubMed]
  18. E. Snitzer, “Cylindrical dielectric waveguide modes,” J. Opt. Soc. Am. 51, 491-498 (1961).
    [CrossRef]
  19. J. A. Besley and J. D. Love, “Supermode analysis of fibre transmission,” IEE Proc.: Optoelectron. 144, 411-419 (1997).
    [CrossRef]
  20. A. W. Snyder and M. Hamer, “Light-capture area of a photoreceptor,” Vision Res. 12, 1749-1753 (1972).
    [CrossRef] [PubMed]
  21. A. W. Snyder and C. Pask, “The Stiles-Crawford effect: explanation and consequences,” Vision Res. 13, 1115-1137 (1973).
    [CrossRef] [PubMed]
  22. W. Fischer and R. Röhler, “Absorption of light in an idealized photoreceptor on basis of waveguide theory. I. Infinite dielectric cylinder,” Vision Res. 14, 1013-1019 (1974).
    [CrossRef] [PubMed]
  23. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  24. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
    [CrossRef]
  25. M. J. Piketmay, A. Taflove, and J. B. Troy, “Electrodynamics of visible light interactions with the vertebrate retinal rod,” Opt. Lett. 18, 568-570 (1993).
    [CrossRef]
  26. A. M. Pozo, F. Perez-Ocon, and J. R. Jimenez, “FDTD analysis of the light propagation in the cones of the human retina: an approach to the Stiles-Crawford effect of the first kind,” J. Opt. A, Pure Appl. Opt. 7, 357-363 (2005).
    [CrossRef]
  27. J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363-379 (1996).
    [CrossRef]
  28. M. Vorobyev, L. Fischer, A. Zvyagin, and T. Plakhotnik, “Polarization sensitivity in vertebrates: a computational approach,” in Polarization Conference, Heron Island (2008), p. 33.
  29. A. Hendrickson and D. Drucker, “The development of parafoveal and mid-peripheral human retina,” Behav. Brain Res. 49, 21-31 (1992).
    [CrossRef] [PubMed]
  30. R. Barer, “Refractometry and interferometry of living cells,” J. Opt. Soc. Am. 47, 545-556 (1957).
    [CrossRef] [PubMed]
  31. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).
  32. C. A. Curcio, K. R. Sloan, R. E. Kalina, and A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497-523 (1990).
    [CrossRef] [PubMed]
  33. W. Wijngaard, “Mode interference patterns in retinal receptor outer segments,” Vision Res. 14, 889-893 (1974).
    [CrossRef] [PubMed]
  34. M. L. Calvo and V. Lakshminarayanan, “Initial field and energy flux in absorbing optical waveguides. I. Theoretical formalism,” J. Opt. Soc. Am. A 4, 1037-1042 (1987).
    [CrossRef] [PubMed]
  35. A. Ghatak and K. Thyagarajan, An Introduction to Fiber Optics (Cambridge Univ. Press, 1998).

2007

2005

A. M. Pozo, F. Perez-Ocon, and J. R. Jimenez, “FDTD analysis of the light propagation in the cones of the human retina: an approach to the Stiles-Crawford effect of the first kind,” J. Opt. A, Pure Appl. Opt. 7, 357-363 (2005).
[CrossRef]

B. Vohnsen, I. Iglesias, and P. Artal, “Guided light and diffraction model of human-eye photoreceptors,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 2318-2328 (2005).
[CrossRef] [PubMed]

2003

1997

J. A. Besley and J. D. Love, “Supermode analysis of fibre transmission,” IEE Proc.: Optoelectron. 144, 411-419 (1997).
[CrossRef]

1996

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363-379 (1996).
[CrossRef]

1993

1992

A. Hendrickson and D. Drucker, “The development of parafoveal and mid-peripheral human retina,” Behav. Brain Res. 49, 21-31 (1992).
[CrossRef] [PubMed]

1990

C. A. Curcio, K. R. Sloan, R. E. Kalina, and A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497-523 (1990).
[CrossRef] [PubMed]

1987

1981

1977

1975

R. Sammut, C. Pask, and A. W. Snyder, “Excitation and power of unbound modes within a circular dielectric waveguide,” Proc. Inst. Electr. Eng. 122, 25-33 (1975).
[CrossRef]

1974

W. Fischer and R. Röhler, “Absorption of light in an idealized photoreceptor on basis of waveguide theory. I. Infinite dielectric cylinder,” Vision Res. 14, 1013-1019 (1974).
[CrossRef] [PubMed]

W. Wijngaard, “Mode interference patterns in retinal receptor outer segments,” Vision Res. 14, 889-893 (1974).
[CrossRef] [PubMed]

W. Wijngaard, “Some normal modes of an infinite hexagonal array of identical circular dielectric rods,” J. Opt. Soc. Am. 64, 1136-1144 (1974).
[CrossRef] [PubMed]

1973

A. W. Snyder and C. Pask, “The Stiles-Crawford effect: explanation and consequences,” Vision Res. 13, 1115-1137 (1973).
[CrossRef] [PubMed]

W. Wijngaard, “Guided normal modes of two parallel circular dielectric rods,” J. Opt. Soc. Am. 63, 944-950 (1973).
[CrossRef]

1972

A. W. Snyder and M. Hamer, “Light-capture area of a photoreceptor,” Vision Res. 12, 1749-1753 (1972).
[CrossRef] [PubMed]

1970

R. Vanclooster and P. Phariseau, “Light propagation in fiber bundles,” Physica 49, 493-501 (1970).
[CrossRef]

R. Vanclooster and P. Phariseau, “Diffraction of an electromagnetic wave by a fiber bundle,” Physica 50, 308-316 (1970).
[CrossRef]

1966

A. W. Snyder and R. L. Kyhl, “Surface mode propagation along an array of dielectric rods with all elements excited identically,” IEEE Trans. Antennas Propag. AP14, 510-511 (1966).
[CrossRef]

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

1961

1960

1957

Artal, P.

B. Vohnsen, I. Iglesias, and P. Artal, “Guided light and diffraction model of human-eye photoreceptors,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 2318-2328 (2005).
[CrossRef] [PubMed]

Barer, R.

Berenger, J. P.

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363-379 (1996).
[CrossRef]

Besley, J. A.

J. A. Besley and J. D. Love, “Supermode analysis of fibre transmission,” IEE Proc.: Optoelectron. 144, 411-419 (1997).
[CrossRef]

Calvo, M. L.

Curcio, C. A.

C. A. Curcio, K. R. Sloan, R. E. Kalina, and A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497-523 (1990).
[CrossRef] [PubMed]

Drucker, D.

A. Hendrickson and D. Drucker, “The development of parafoveal and mid-peripheral human retina,” Behav. Brain Res. 49, 21-31 (1992).
[CrossRef] [PubMed]

Enoch, J. M.

Eyges, L.

Fischer, L.

M. Vorobyev, L. Fischer, A. Zvyagin, and T. Plakhotnik, “Polarization sensitivity in vertebrates: a computational approach,” in Polarization Conference, Heron Island (2008), p. 33.

Fischer, W.

W. Fischer and R. Röhler, “Absorption of light in an idealized photoreceptor on basis of waveguide theory. I. Infinite dielectric cylinder,” Vision Res. 14, 1013-1019 (1974).
[CrossRef] [PubMed]

Ghatak, A.

A. Ghatak and K. Thyagarajan, An Introduction to Fiber Optics (Cambridge Univ. Press, 1998).

Hamer, M.

A. W. Snyder and M. Hamer, “Light-capture area of a photoreceptor,” Vision Res. 12, 1749-1753 (1972).
[CrossRef] [PubMed]

Hendrickson, A.

A. Hendrickson and D. Drucker, “The development of parafoveal and mid-peripheral human retina,” Behav. Brain Res. 49, 21-31 (1992).
[CrossRef] [PubMed]

Hendrickson, A. E.

C. A. Curcio, K. R. Sloan, R. E. Kalina, and A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497-523 (1990).
[CrossRef] [PubMed]

Iglesias, I.

B. Vohnsen, I. Iglesias, and P. Artal, “Guided light and diffraction model of human-eye photoreceptors,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 2318-2328 (2005).
[CrossRef] [PubMed]

Jimenez, J. R.

A. M. Pozo, F. Perez-Ocon, and J. R. Jimenez, “FDTD analysis of the light propagation in the cones of the human retina: an approach to the Stiles-Crawford effect of the first kind,” J. Opt. A, Pure Appl. Opt. 7, 357-363 (2005).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Kalina, R. E.

C. A. Curcio, K. R. Sloan, R. E. Kalina, and A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497-523 (1990).
[CrossRef] [PubMed]

Kyhl, R. L.

A. W. Snyder and R. L. Kyhl, “Surface mode propagation along an array of dielectric rods with all elements excited identically,” IEEE Trans. Antennas Propag. AP14, 510-511 (1966).
[CrossRef]

Lakshminarayanan, V.

Limeres, J.

Love, J. D.

J. A. Besley and J. D. Love, “Supermode analysis of fibre transmission,” IEE Proc.: Optoelectron. 144, 411-419 (1997).
[CrossRef]

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Pask, C.

R. Sammut, C. Pask, and A. W. Snyder, “Excitation and power of unbound modes within a circular dielectric waveguide,” Proc. Inst. Electr. Eng. 122, 25-33 (1975).
[CrossRef]

A. W. Snyder and C. Pask, “The Stiles-Crawford effect: explanation and consequences,” Vision Res. 13, 1115-1137 (1973).
[CrossRef] [PubMed]

Perez-Ocon, F.

A. M. Pozo, F. Perez-Ocon, and J. R. Jimenez, “FDTD analysis of the light propagation in the cones of the human retina: an approach to the Stiles-Crawford effect of the first kind,” J. Opt. A, Pure Appl. Opt. 7, 357-363 (2005).
[CrossRef]

Phariseau, P.

R. Vanclooster and P. Phariseau, “Diffraction of an electromagnetic wave by a fiber bundle,” Physica 50, 308-316 (1970).
[CrossRef]

R. Vanclooster and P. Phariseau, “Light propagation in fiber bundles,” Physica 49, 493-501 (1970).
[CrossRef]

Piketmay, M. J.

Plakhotnik, T.

M. Vorobyev, L. Fischer, A. Zvyagin, and T. Plakhotnik, “Polarization sensitivity in vertebrates: a computational approach,” in Polarization Conference, Heron Island (2008), p. 33.

Pozo, A. M.

A. M. Pozo, F. Perez-Ocon, and J. R. Jimenez, “FDTD analysis of the light propagation in the cones of the human retina: an approach to the Stiles-Crawford effect of the first kind,” J. Opt. A, Pure Appl. Opt. 7, 357-363 (2005).
[CrossRef]

Röhler, R.

W. Fischer and R. Röhler, “Absorption of light in an idealized photoreceptor on basis of waveguide theory. I. Infinite dielectric cylinder,” Vision Res. 14, 1013-1019 (1974).
[CrossRef] [PubMed]

Sammut, R.

R. Sammut, “Effect of leaky modes on response of short optical fibers,” J. Opt. Soc. Am. 67, 1284-1285 (1977).
[CrossRef]

R. Sammut, C. Pask, and A. W. Snyder, “Excitation and power of unbound modes within a circular dielectric waveguide,” Proc. Inst. Electr. Eng. 122, 25-33 (1975).
[CrossRef]

Sloan, K. R.

C. A. Curcio, K. R. Sloan, R. E. Kalina, and A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497-523 (1990).
[CrossRef] [PubMed]

Snitzer, E.

Snyder, A. W.

R. Sammut, C. Pask, and A. W. Snyder, “Excitation and power of unbound modes within a circular dielectric waveguide,” Proc. Inst. Electr. Eng. 122, 25-33 (1975).
[CrossRef]

A. W. Snyder and C. Pask, “The Stiles-Crawford effect: explanation and consequences,” Vision Res. 13, 1115-1137 (1973).
[CrossRef] [PubMed]

A. W. Snyder and M. Hamer, “Light-capture area of a photoreceptor,” Vision Res. 12, 1749-1753 (1972).
[CrossRef] [PubMed]

A. W. Snyder and R. L. Kyhl, “Surface mode propagation along an array of dielectric rods with all elements excited identically,” IEEE Trans. Antennas Propag. AP14, 510-511 (1966).
[CrossRef]

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Taflove, A.

Thyagarajan, K.

A. Ghatak and K. Thyagarajan, An Introduction to Fiber Optics (Cambridge Univ. Press, 1998).

Troy, J. B.

Vanclooster, R.

R. Vanclooster and P. Phariseau, “Diffraction of an electromagnetic wave by a fiber bundle,” Physica 50, 308-316 (1970).
[CrossRef]

R. Vanclooster and P. Phariseau, “Light propagation in fiber bundles,” Physica 49, 493-501 (1970).
[CrossRef]

Vohnsen, B.

B. Vohnsen, “Photoreceptor waveguides and effective retinal image quality,” J. Opt. Soc. Am. A 24, 597-607 (2007).
[CrossRef]

B. Vohnsen, I. Iglesias, and P. Artal, “Guided light and diffraction model of human-eye photoreceptors,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 2318-2328 (2005).
[CrossRef] [PubMed]

Vorobyev, M.

M. Vorobyev, L. Fischer, A. Zvyagin, and T. Plakhotnik, “Polarization sensitivity in vertebrates: a computational approach,” in Polarization Conference, Heron Island (2008), p. 33.

Wijngaard, W.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Wintersteiner, P.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Zvyagin, A.

M. Vorobyev, L. Fischer, A. Zvyagin, and T. Plakhotnik, “Polarization sensitivity in vertebrates: a computational approach,” in Polarization Conference, Heron Island (2008), p. 33.

Behav. Brain Res.

A. Hendrickson and D. Drucker, “The development of parafoveal and mid-peripheral human retina,” Behav. Brain Res. 49, 21-31 (1992).
[CrossRef] [PubMed]

IEE Proc.: Optoelectron.

J. A. Besley and J. D. Love, “Supermode analysis of fibre transmission,” IEE Proc.: Optoelectron. 144, 411-419 (1997).
[CrossRef]

IEEE Trans. Antennas Propag.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

A. W. Snyder and R. L. Kyhl, “Surface mode propagation along an array of dielectric rods with all elements excited identically,” IEEE Trans. Antennas Propag. AP14, 510-511 (1966).
[CrossRef]

J. Comp. Neurol.

C. A. Curcio, K. R. Sloan, R. E. Kalina, and A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497-523 (1990).
[CrossRef] [PubMed]

J. Comput. Phys.

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363-379 (1996).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

A. M. Pozo, F. Perez-Ocon, and J. R. Jimenez, “FDTD analysis of the light propagation in the cones of the human retina: an approach to the Stiles-Crawford effect of the first kind,” J. Opt. A, Pure Appl. Opt. 7, 357-363 (2005).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A Opt. Image Sci. Vis

B. Vohnsen, I. Iglesias, and P. Artal, “Guided light and diffraction model of human-eye photoreceptors,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 2318-2328 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B

Opt. Lett.

Physica

R. Vanclooster and P. Phariseau, “Light propagation in fiber bundles,” Physica 49, 493-501 (1970).
[CrossRef]

R. Vanclooster and P. Phariseau, “Diffraction of an electromagnetic wave by a fiber bundle,” Physica 50, 308-316 (1970).
[CrossRef]

Proc. Inst. Electr. Eng.

R. Sammut, C. Pask, and A. W. Snyder, “Excitation and power of unbound modes within a circular dielectric waveguide,” Proc. Inst. Electr. Eng. 122, 25-33 (1975).
[CrossRef]

Vision Res.

A. W. Snyder and M. Hamer, “Light-capture area of a photoreceptor,” Vision Res. 12, 1749-1753 (1972).
[CrossRef] [PubMed]

A. W. Snyder and C. Pask, “The Stiles-Crawford effect: explanation and consequences,” Vision Res. 13, 1115-1137 (1973).
[CrossRef] [PubMed]

W. Fischer and R. Röhler, “Absorption of light in an idealized photoreceptor on basis of waveguide theory. I. Infinite dielectric cylinder,” Vision Res. 14, 1013-1019 (1974).
[CrossRef] [PubMed]

W. Wijngaard, “Mode interference patterns in retinal receptor outer segments,” Vision Res. 14, 889-893 (1974).
[CrossRef] [PubMed]

Other

A. Ghatak and K. Thyagarajan, An Introduction to Fiber Optics (Cambridge Univ. Press, 1998).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

M. Vorobyev, L. Fischer, A. Zvyagin, and T. Plakhotnik, “Polarization sensitivity in vertebrates: a computational approach,” in Polarization Conference, Heron Island (2008), p. 33.

A.W.Snyder and R.Menzel, eds., Photoreceptor Optics (Springer-Verlag, 1975).
[CrossRef]

J.M.Enoch and F.L.Tobey, Jr., eds., Vertebrate Photoreceptor Optics (Springer-Verlag, 1981).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

M. L. Calvo and V. Lakshminarayanan, “Birefringent optical waveguides,” in Optical Waveguides: From Theory to Applied Technologies, M.L.Calvo and R.F.Alvarez-Estrada, eds. (CRC, 2007), pp. 38-96.

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 (8)

Fig. 1
Fig. 1

Schematic layout of the infinite hexagonal array of dielectric cylinders. Parameter values correspond to the human central retina: λ, free-space wavelength of the excitation source; L, cylinder length; a, cylinder radius; d, intra-cylinder spacing. n 1 and n 2 are cylinder and surrounding medium refractive indices, respectively. The coordinate system is shown with the positive z-axis directing into the page, along the cylinder axis.

Fig. 2
Fig. 2

Time-averaged electric field intensity obtained from the hexagonal array for a unitary amplitude excitation source. A longitudinal cross-section through the cylinder center in the array is shown in the top panel. The average intensities over the cylinder cross-section are shown in the four bottom panels for various spacing designated with d / λ , a / λ = 1 . Note that the spatial period of the oscillations along the z-direction increases versus d / λ .

Fig. 3
Fig. 3

Electric field amplitude profile of the two dominant modes, as excited by an x-polarized plane wave (Cartesian components are shown). The left hand side demonstrates the dominant fast mode, while the right hand side demonstrates the dominant slow mode. Panels have different scales. Note the symmetry with respect to the x - z plane of the mode components, which is determined by the symmetry of the excitation source. For example, the y component of the electric field is equal to zero along the y = 0 line, and accordingly the y component of electric field changes its direction, when the x-axis is crossed.

Fig. 4
Fig. 4

Co-polarized component of the electric field for the fundamental bound mode alongside two examples of the numerically obtained fundamental slow mode. The analytical expression is shown as compared to the numerical results for the sparse array (dots) as well as the hexagonal array with a cylinder spacing of d / λ = 30 (crosses). The dashed line indicates the cylinder boundary at x = 0.5 μ m . Numerically computed field profiles were obtained via mode deconvolution and thus do not represent the actual field distribution throughout the simulation region. Note that the analytical solution accurately describes the slow mode profile in the sparse array.

Fig. 5
Fig. 5

Comparison of numerical and approximate analytical results (see Appendix A) for the axial beat wavenumber Δ k , occurring as a result of interaction of the two dominant modes within a single cylinder in the sparse array. Note that the period of oscillations is related to wavenumber as 2 π / Δ k . Examples are also shown for the hexagonal array, where the beat frequency as a function of a / λ approaches the single cylinder regime for the large d / λ . For the sparse array, ξ remains close to the wavenumber in the surrounding medium. β varies versus a / λ similarly to that of the single cylinder.

Fig. 6
Fig. 6

Propagation constant as a function of the cylinder spacing for the hexagonal array. Markers indicate the numerical result obtained directly from fast Fourier transform, while the corresponding thick solid lines indicate the approximate result attained from applying Eqs. (2, 3, 4a, 4b) to the extracted modes. A good fit is obtained for the majority of the data set, with the approximate result for the fast mode being omitted at the small d / λ due to increased numerical error. The dashed line indicates the value of the wavenumber in the surrounding medium. An additional thin solid line shown for the slow mode demonstrates the approximate result obtained by the use of the analytical expression [31] for the fundamental bound mode of the single cylinder to represent ψ ( x , y ) in Eq. (4).

Fig. 7
Fig. 7

Average intensity over the total cross-section. Markers indicate average intensity for the slow (dots) and fast (crosses) modes. The corresponding dashed lines were obtained from the estimation of the mode intensities based on the overlap integral [[5], p. 425]. The energy distribution predicted by Eqs. (5, 6) is shown as solid lines. Note the good agreement with numerical results achieved at larger d / λ , while this approximation is not valid at small d / λ .

Fig. 8
Fig. 8

(a) The fractional contribution of the fast mode to the total energy propagating within a cylinder. Crosses indicate the direct numerical result, while a solid line demarcates the estimate obtained via Eq. (7). (b) The modulation depth of the axial oscillations as compared to the average intensity within the cylinder. A solid-line-marked plot is obtained by considering the intensity of each of the modes as obtained by mode deconvolution (see Section 4). A dot-marked plot shows the modulation depth determined from the calculation of the overlap modal integral [Eq. (8)], while a dot-marked plot shows the result obtained directly from comparing peaks and troughs in the optical intensity.

Equations (20)

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

Δ k = β ξ .
β 2 = ( k 1 2 k 2 2 ) b s + k 2 2 ,
ξ 2 = ( k 1 2 k 2 2 ) b f + k 2 2 ,
b s = A in ψ s ( x , y ) d A A in ψ s ( x , y ) d A + A out ψ s ( x , y ) d A ,
b f = A in ψ f ( x , y ) d A A in ψ f ( x , y ) d A + A out ψ f ( x , y ) d A ,
e s = 2 E s A 3 d 2 ,
e f = 1 2 E s A 3 d 2 .
e f in E f in ( E f in + E s in ) = e f f g s A .
D = 2 A in ψ F ψ S d A A in ψ F 2 d A + A in ψ S 2 d A ,
D 2 e f in ( 1 e f in ) .
Δ k = β k 2 ,
κ J ν + 1 ( κ a ) J ν ( κ a ) = i γ H ν + 1 ( 1 ) ( i γ a ) H ν ( 1 ) ( i γ a ) ,
Δ t ψ I = ( β 2 k 1 2 ) ψ I ,
Δ t ψ I I = ( β 2 k 2 2 ) ψ I I ,
A in Δ t ψ I d A = c ψ I d l = ( β 2 k 1 2 ) A in ψ I d A ,
A out Δ t ψ I I d A = c ψ I I d l = ( β 2 k 2 2 ) A out ψ I I d A ,
c ψ I d l = c ψ I I d l .
( β 2 k 1 2 ) A in ψ I d A = ( β 2 k 2 2 ) A out ψ I I d A .
β 2 = ( k 1 2 k 2 2 ) b + k 2 2 ,
b = A in ψ ( x , y ) d A A in ψ ( x , y ) d A + A out ψ ( x , y ) d A .

Metrics