Abstract

The propagation of light through a slowly twisting anisotropic medium is described by a coupled-mode theory; expressions are derived for the electric field for the case of a birefringent dichroic medium with a constant rate of twist. The method provides a simple and intuitive means for determining the effect of twisting on the linear birefringence and dichroic absorption of the medium, particularly when the light is initially linearly polarized. The theory is well suited to the analysis of light absorption in twisting insect photoreceptors, such as found in bees and ants. We provide full expressions and useful approximations for the polarization sensitivity and the initial direction of polarization to give maximum absorption for several types of photoreceptors.

© 1978 Optical Society of America

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  1. A. R. Stokes, The Theory of the Optical Properties of Inhomogenous Materials (Spon Press, London, 1963).
  2. H. L. deVries, "Rotatory power and other optical properties of certain liquid crystals, " Acta Crystallogr. 4, 219 (1951).
  3. A. S. Marathay, "Matrix-operator description of the propagation of polarized light through cholesteric liquid cyrstals, " J. Opt. Soc. Am. 61, 1363 (1971).
  4. D. W. Berreman, "Optics in stratified and anisotropic media: 4×4 matrix formulation, " J. Opt. Soc. Am. 62, 502 (1972); D. W. Berreman, "Optics in smoothly varying planar structures: Applications to liquid-crystal twist cells, " J. Opt. Soc. Am. 63, 1374 (1973).
  5. R. M. A. Azzam and N. M. Bashara, "Simplified approach to the propagation of polarized light in anisotropic media—application to liquid crystals, " J. Opt. Soc. Am. 62, 1252 (1972).
  6. P. D. McIntyre, "Crosstalk between optical waveguides with applications to visual photoreceptors, " Ph. D. thesis (Australian National University, Canberra, Australia, 1976).
  7. A. W. Snyder and P. D. McIntyre, "Polarization sensitivity of twisted fused rhabdoms, " in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).
  8. R. Wehner, G. D. Bernard, and E. Geiger, "Twisted and non-twisted rhabdoms and their significance for polarization detection in the bee, " J. Comp. Physiol. 104, 225 (1975).
  9. C. Mauguin, Bull. Soc. Fr. Mineral. 34, 71 (1911), quoted in Ref. 2.
  10. C. Hu and J. R. Whinery, "Losses of a nematic liquid-crystal optical waveguide, " J. Opt. Soc. Am. 64, 1424 (1974).
  11. W. H. Louisell, Coupled Modes and Parametric Electronics (Wiley, New York, 1960).
  12. A. W. Snyder, "Coupling of modes on a tapered dielectric cylinder, " IEEE Trans. Microwave Theory Tech. MTT-18, 383 (1970).
  13. A. Reisinger, "Characteristics of optical guided modes in lossy waveguides, " Appl. Opt. 12, 1015 (1973).
  14. A. W. Snyder, "Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide," IEEE Trans. Microwave Theory Tech. MTT-17, 1130 (1969).
  15. p. D. McIntyre and A. W. Snyder, "Power transfer between optical fibers, " J. Opt. Soc. Am. 63, 1518 (1973).
  16. A. W. Snyder, "Power loss on optical fibers, " Proc. IEEE 60, 757 (1972).
  17. A. W. Snyder, "Leaky-ray theory of optical waveguides of circular cross-section, " Appl. Phys. 4, 273 (1974).
  18. The ellipticity is defined as the ratio of the lengths of the major and minor axes of the polarization ellipse. The usual symbol is η but here we use σ to avoid confusion with the waveguide parameter η.
  19. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Perga mon, New York, 1970).
  20. D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, 1971).
  21. O. J. Grundler, "Elektronenmikroskopische Untersuchungen am Auge der Honigbiene (Apis mellifera), " Cytobiol. 9, 203 (1974).
  22. R. Menzel, "Polarisation Sensitivity in Insect Eyes with Fused Rhabdoms," in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).
  23. R. Menzel and M. Blakers, "Functional organisation of an insect ommatidium with fused rhabdom, " Cytobiol. 11, 279 (1975).
  24. J. N. Israelachvili, R. A. Sammut, and A. W. Snyder, "Birefringence and dichroism of photoreceptors, " Vis. Res. 16, 47 (1976).
  25. A. W. Snyder and S. B. Laughlin, "Dichroism and absorption by photoreceptors, " J. Comp. Physiol. 100, 101 (1975).
  26. K. Kirschfeld and A. W. Snyder, "Waveguide Mode Effects, Birefringence and Dichroism in Fly Photoreceptors, " in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).
  27. A. W. Snyder, "Polarization sensitivity of individual retinula cells, " J. Comp. Physiol. 83, 331 (1973).
  28. K. Kirschfeld, "The Visual System of Musca: Studies on Optics, Structure and Function, " in Information Processing in the Visual System of Arthropods, edited by R. Wehner (Springer-Verlag, Berlin, 1972).
  29. F. G. Gribakin, "Cellular basis of colour vision in the honey bee, " Nature 223, 639 (1969).
  30. A. W. Snyder and R. A. Sammut, "Direction of E for maximum response of a retinula cell," J. Comp. Physiol. 85, 37 (1973).
  31. R. Menzel, "The Fine Structure of the Compound Eye of Formica polyctena-Functional Morphology of a Hymenopteran Eye, " in Information Processing in the Visual System of Arthropods, edited by R. Wehner (Springer-Verlag, Berlin, 1972).
  32. G. A. Horridge, "The Retina of the Locust, " in The Functional Organization of the Compound Eye, edited by C. G. Bernhard (Pergamon, Oxford, 1966).
  33. A. W. Snyder, "Light absorption in visual photoreceptors," J. Opt. Soc. Am. 64, 216 (1974).
  34. R. Menzel and A. W. Snyder, "Polarized light detection in the bee, Apis mellifera, " J. Comp. Physiol. 88, 247 (1974).
  35. R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics (Addison-Wesley, Reading, Mass., 1963), Vol. 1, p. 33. S. G. Lipson and H. Lipson, Optical Physics (Cambridge U. P., Cambridge, England, 1962). L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1963).

1976

P. D. McIntyre, "Crosstalk between optical waveguides with applications to visual photoreceptors, " Ph. D. thesis (Australian National University, Canberra, Australia, 1976).

J. N. Israelachvili, R. A. Sammut, and A. W. Snyder, "Birefringence and dichroism of photoreceptors, " Vis. Res. 16, 47 (1976).

1975

A. W. Snyder and S. B. Laughlin, "Dichroism and absorption by photoreceptors, " J. Comp. Physiol. 100, 101 (1975).

K. Kirschfeld and A. W. Snyder, "Waveguide Mode Effects, Birefringence and Dichroism in Fly Photoreceptors, " in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).

R. Menzel, "Polarisation Sensitivity in Insect Eyes with Fused Rhabdoms," in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).

R. Menzel and M. Blakers, "Functional organisation of an insect ommatidium with fused rhabdom, " Cytobiol. 11, 279 (1975).

A. W. Snyder and P. D. McIntyre, "Polarization sensitivity of twisted fused rhabdoms, " in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).

R. Wehner, G. D. Bernard, and E. Geiger, "Twisted and non-twisted rhabdoms and their significance for polarization detection in the bee, " J. Comp. Physiol. 104, 225 (1975).

1974

C. Hu and J. R. Whinery, "Losses of a nematic liquid-crystal optical waveguide, " J. Opt. Soc. Am. 64, 1424 (1974).

A. W. Snyder, "Leaky-ray theory of optical waveguides of circular cross-section, " Appl. Phys. 4, 273 (1974).

O. J. Grundler, "Elektronenmikroskopische Untersuchungen am Auge der Honigbiene (Apis mellifera), " Cytobiol. 9, 203 (1974).

A. W. Snyder, "Light absorption in visual photoreceptors," J. Opt. Soc. Am. 64, 216 (1974).

R. Menzel and A. W. Snyder, "Polarized light detection in the bee, Apis mellifera, " J. Comp. Physiol. 88, 247 (1974).

1973

A. W. Snyder and R. A. Sammut, "Direction of E for maximum response of a retinula cell," J. Comp. Physiol. 85, 37 (1973).

A. W. Snyder, "Polarization sensitivity of individual retinula cells, " J. Comp. Physiol. 83, 331 (1973).

p. D. McIntyre and A. W. Snyder, "Power transfer between optical fibers, " J. Opt. Soc. Am. 63, 1518 (1973).

A. Reisinger, "Characteristics of optical guided modes in lossy waveguides, " Appl. Opt. 12, 1015 (1973).

1972

A. W. Snyder, "Power loss on optical fibers, " Proc. IEEE 60, 757 (1972).

D. W. Berreman, "Optics in stratified and anisotropic media: 4×4 matrix formulation, " J. Opt. Soc. Am. 62, 502 (1972); D. W. Berreman, "Optics in smoothly varying planar structures: Applications to liquid-crystal twist cells, " J. Opt. Soc. Am. 63, 1374 (1973).

R. M. A. Azzam and N. M. Bashara, "Simplified approach to the propagation of polarized light in anisotropic media—application to liquid crystals, " J. Opt. Soc. Am. 62, 1252 (1972).

K. Kirschfeld, "The Visual System of Musca: Studies on Optics, Structure and Function, " in Information Processing in the Visual System of Arthropods, edited by R. Wehner (Springer-Verlag, Berlin, 1972).

R. Menzel, "The Fine Structure of the Compound Eye of Formica polyctena-Functional Morphology of a Hymenopteran Eye, " in Information Processing in the Visual System of Arthropods, edited by R. Wehner (Springer-Verlag, Berlin, 1972).

1971

1970

A. W. Snyder, "Coupling of modes on a tapered dielectric cylinder, " IEEE Trans. Microwave Theory Tech. MTT-18, 383 (1970).

1969

A. W. Snyder, "Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide," IEEE Trans. Microwave Theory Tech. MTT-17, 1130 (1969).

F. G. Gribakin, "Cellular basis of colour vision in the honey bee, " Nature 223, 639 (1969).

1966

G. A. Horridge, "The Retina of the Locust, " in The Functional Organization of the Compound Eye, edited by C. G. Bernhard (Pergamon, Oxford, 1966).

1951

H. L. deVries, "Rotatory power and other optical properties of certain liquid crystals, " Acta Crystallogr. 4, 219 (1951).

1911

C. Mauguin, Bull. Soc. Fr. Mineral. 34, 71 (1911), quoted in Ref. 2.

Azzam, R. M. A.

Bashara, N. M.

Bernard, G. D.

R. Wehner, G. D. Bernard, and E. Geiger, "Twisted and non-twisted rhabdoms and their significance for polarization detection in the bee, " J. Comp. Physiol. 104, 225 (1975).

Berreman, D. W.

Blakers, M.

R. Menzel and M. Blakers, "Functional organisation of an insect ommatidium with fused rhabdom, " Cytobiol. 11, 279 (1975).

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Perga mon, New York, 1970).

Clarke, D.

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, 1971).

deVries, H. L.

H. L. deVries, "Rotatory power and other optical properties of certain liquid crystals, " Acta Crystallogr. 4, 219 (1951).

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics (Addison-Wesley, Reading, Mass., 1963), Vol. 1, p. 33. S. G. Lipson and H. Lipson, Optical Physics (Cambridge U. P., Cambridge, England, 1962). L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1963).

Geiger, E.

R. Wehner, G. D. Bernard, and E. Geiger, "Twisted and non-twisted rhabdoms and their significance for polarization detection in the bee, " J. Comp. Physiol. 104, 225 (1975).

Grainger, J. F.

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, 1971).

Gribakin, F. G.

F. G. Gribakin, "Cellular basis of colour vision in the honey bee, " Nature 223, 639 (1969).

Grundler, O. J.

O. J. Grundler, "Elektronenmikroskopische Untersuchungen am Auge der Honigbiene (Apis mellifera), " Cytobiol. 9, 203 (1974).

Horridge, G. A.

G. A. Horridge, "The Retina of the Locust, " in The Functional Organization of the Compound Eye, edited by C. G. Bernhard (Pergamon, Oxford, 1966).

Hu, C.

Israelachvili, J. N.

J. N. Israelachvili, R. A. Sammut, and A. W. Snyder, "Birefringence and dichroism of photoreceptors, " Vis. Res. 16, 47 (1976).

Kirschfeld, K.

K. Kirschfeld and A. W. Snyder, "Waveguide Mode Effects, Birefringence and Dichroism in Fly Photoreceptors, " in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).

K. Kirschfeld, "The Visual System of Musca: Studies on Optics, Structure and Function, " in Information Processing in the Visual System of Arthropods, edited by R. Wehner (Springer-Verlag, Berlin, 1972).

Laughlin, S. B.

A. W. Snyder and S. B. Laughlin, "Dichroism and absorption by photoreceptors, " J. Comp. Physiol. 100, 101 (1975).

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics (Addison-Wesley, Reading, Mass., 1963), Vol. 1, p. 33. S. G. Lipson and H. Lipson, Optical Physics (Cambridge U. P., Cambridge, England, 1962). L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1963).

Louisell, W. H.

W. H. Louisell, Coupled Modes and Parametric Electronics (Wiley, New York, 1960).

Marathay, A. S.

Mauguin, C.

C. Mauguin, Bull. Soc. Fr. Mineral. 34, 71 (1911), quoted in Ref. 2.

McIntyre, P. D.

P. D. McIntyre, "Crosstalk between optical waveguides with applications to visual photoreceptors, " Ph. D. thesis (Australian National University, Canberra, Australia, 1976).

A. W. Snyder and P. D. McIntyre, "Polarization sensitivity of twisted fused rhabdoms, " in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).

p. D. McIntyre and A. W. Snyder, "Power transfer between optical fibers, " J. Opt. Soc. Am. 63, 1518 (1973).

Menzel, R.

R. Menzel, "Polarisation Sensitivity in Insect Eyes with Fused Rhabdoms," in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).

R. Menzel and M. Blakers, "Functional organisation of an insect ommatidium with fused rhabdom, " Cytobiol. 11, 279 (1975).

R. Menzel and A. W. Snyder, "Polarized light detection in the bee, Apis mellifera, " J. Comp. Physiol. 88, 247 (1974).

R. Menzel, "The Fine Structure of the Compound Eye of Formica polyctena-Functional Morphology of a Hymenopteran Eye, " in Information Processing in the Visual System of Arthropods, edited by R. Wehner (Springer-Verlag, Berlin, 1972).

Reisinger, A.

Sammut, R. A.

J. N. Israelachvili, R. A. Sammut, and A. W. Snyder, "Birefringence and dichroism of photoreceptors, " Vis. Res. 16, 47 (1976).

A. W. Snyder and R. A. Sammut, "Direction of E for maximum response of a retinula cell," J. Comp. Physiol. 85, 37 (1973).

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics (Addison-Wesley, Reading, Mass., 1963), Vol. 1, p. 33. S. G. Lipson and H. Lipson, Optical Physics (Cambridge U. P., Cambridge, England, 1962). L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1963).

Snyder, A. W.

J. N. Israelachvili, R. A. Sammut, and A. W. Snyder, "Birefringence and dichroism of photoreceptors, " Vis. Res. 16, 47 (1976).

A. W. Snyder and S. B. Laughlin, "Dichroism and absorption by photoreceptors, " J. Comp. Physiol. 100, 101 (1975).

A. W. Snyder and P. D. McIntyre, "Polarization sensitivity of twisted fused rhabdoms, " in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).

K. Kirschfeld and A. W. Snyder, "Waveguide Mode Effects, Birefringence and Dichroism in Fly Photoreceptors, " in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).

A. W. Snyder, "Light absorption in visual photoreceptors," J. Opt. Soc. Am. 64, 216 (1974).

R. Menzel and A. W. Snyder, "Polarized light detection in the bee, Apis mellifera, " J. Comp. Physiol. 88, 247 (1974).

A. W. Snyder, "Leaky-ray theory of optical waveguides of circular cross-section, " Appl. Phys. 4, 273 (1974).

p. D. McIntyre and A. W. Snyder, "Power transfer between optical fibers, " J. Opt. Soc. Am. 63, 1518 (1973).

A. W. Snyder, "Polarization sensitivity of individual retinula cells, " J. Comp. Physiol. 83, 331 (1973).

A. W. Snyder and R. A. Sammut, "Direction of E for maximum response of a retinula cell," J. Comp. Physiol. 85, 37 (1973).

A. W. Snyder, "Power loss on optical fibers, " Proc. IEEE 60, 757 (1972).

A. W. Snyder, "Coupling of modes on a tapered dielectric cylinder, " IEEE Trans. Microwave Theory Tech. MTT-18, 383 (1970).

A. W. Snyder, "Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide," IEEE Trans. Microwave Theory Tech. MTT-17, 1130 (1969).

Stokes, A. R.

A. R. Stokes, The Theory of the Optical Properties of Inhomogenous Materials (Spon Press, London, 1963).

Wehner, R.

R. Wehner, G. D. Bernard, and E. Geiger, "Twisted and non-twisted rhabdoms and their significance for polarization detection in the bee, " J. Comp. Physiol. 104, 225 (1975).

Whinery, J. R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Perga mon, New York, 1970).

Acta Crystallogr.

H. L. deVries, "Rotatory power and other optical properties of certain liquid crystals, " Acta Crystallogr. 4, 219 (1951).

Appl. Opt.

Appl. Phys.

A. W. Snyder, "Leaky-ray theory of optical waveguides of circular cross-section, " Appl. Phys. 4, 273 (1974).

Cytobiol.

O. J. Grundler, "Elektronenmikroskopische Untersuchungen am Auge der Honigbiene (Apis mellifera), " Cytobiol. 9, 203 (1974).

R. Menzel and M. Blakers, "Functional organisation of an insect ommatidium with fused rhabdom, " Cytobiol. 11, 279 (1975).

IEEE Trans. Microwave Theory Tech.

A. W. Snyder, "Coupling of modes on a tapered dielectric cylinder, " IEEE Trans. Microwave Theory Tech. MTT-18, 383 (1970).

A. W. Snyder, "Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide," IEEE Trans. Microwave Theory Tech. MTT-17, 1130 (1969).

J. Comp. Physiol.

R. Wehner, G. D. Bernard, and E. Geiger, "Twisted and non-twisted rhabdoms and their significance for polarization detection in the bee, " J. Comp. Physiol. 104, 225 (1975).

A. W. Snyder and S. B. Laughlin, "Dichroism and absorption by photoreceptors, " J. Comp. Physiol. 100, 101 (1975).

A. W. Snyder, "Polarization sensitivity of individual retinula cells, " J. Comp. Physiol. 83, 331 (1973).

A. W. Snyder and R. A. Sammut, "Direction of E for maximum response of a retinula cell," J. Comp. Physiol. 85, 37 (1973).

R. Menzel and A. W. Snyder, "Polarized light detection in the bee, Apis mellifera, " J. Comp. Physiol. 88, 247 (1974).

J. Opt. Soc. Am.

Nature

F. G. Gribakin, "Cellular basis of colour vision in the honey bee, " Nature 223, 639 (1969).

Vis. Res.

J. N. Israelachvili, R. A. Sammut, and A. W. Snyder, "Birefringence and dichroism of photoreceptors, " Vis. Res. 16, 47 (1976).

Other

K. Kirschfeld and A. W. Snyder, "Waveguide Mode Effects, Birefringence and Dichroism in Fly Photoreceptors, " in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).

A. R. Stokes, The Theory of the Optical Properties of Inhomogenous Materials (Spon Press, London, 1963).

R. Menzel, "Polarisation Sensitivity in Insect Eyes with Fused Rhabdoms," in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).

R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics (Addison-Wesley, Reading, Mass., 1963), Vol. 1, p. 33. S. G. Lipson and H. Lipson, Optical Physics (Cambridge U. P., Cambridge, England, 1962). L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1963).

R. Menzel, "The Fine Structure of the Compound Eye of Formica polyctena-Functional Morphology of a Hymenopteran Eye, " in Information Processing in the Visual System of Arthropods, edited by R. Wehner (Springer-Verlag, Berlin, 1972).

G. A. Horridge, "The Retina of the Locust, " in The Functional Organization of the Compound Eye, edited by C. G. Bernhard (Pergamon, Oxford, 1966).

K. Kirschfeld, "The Visual System of Musca: Studies on Optics, Structure and Function, " in Information Processing in the Visual System of Arthropods, edited by R. Wehner (Springer-Verlag, Berlin, 1972).

A. W. Snyder, "Power loss on optical fibers, " Proc. IEEE 60, 757 (1972).

The ellipticity is defined as the ratio of the lengths of the major and minor axes of the polarization ellipse. The usual symbol is η but here we use σ to avoid confusion with the waveguide parameter η.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Perga mon, New York, 1970).

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, 1971).

P. D. McIntyre, "Crosstalk between optical waveguides with applications to visual photoreceptors, " Ph. D. thesis (Australian National University, Canberra, Australia, 1976).

A. W. Snyder and P. D. McIntyre, "Polarization sensitivity of twisted fused rhabdoms, " in Photoreceptor Optics, edited by A. W. Snyder and R. Menzel (Springer-Verlag, Heidelberg, 1975).

W. H. Louisell, Coupled Modes and Parametric Electronics (Wiley, New York, 1960).

C. Mauguin, Bull. Soc. Fr. Mineral. 34, 71 (1911), quoted in Ref. 2.

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