Abstract

Some species of fish are able to discriminate, in addition to intensity and wavelength (color), the direction of polarization of visible light. Optical experiments on axially oriented retinal cones from trout and sunfish with use of two types of polarization microscope indicate anisotropic light transmission through paired cones. The measured linear birefringence of paired cone ellipsoids is consistent with the presence of membranous partitions. It is proposed that the partition between the two members of a paired cone, which often appears extensive and flat, functions as a dielectric mirror and that polarization-dependent reflection and refraction at this partition constitutes the underlying mechanism in the transduction of polarization into intensity variation at the photoreceptor’s outer segments. We support this hypothesis with linear birefringence and linear dichroism measurements, histological evidence, large-scale optical model measurements, and theoretical calculations based on Fresnel’s formulas.

© 1998 Optical Society of America

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1997 (1)

I. Novales Flamarique and C. W. Hawryshyn, “No evidence of polarization sensitivity in freshwater sunfish from multi-unit optic nerve recordings,” Vision Res. 37, 967–973 (1997).
[CrossRef] [PubMed]

1996 (1)

I. Novales Flamarique and C. W. Hawryshyn, “Retinal development and visual sensitivity of young Pacific sockeye salmon (Oncorhynchus nerka),” J. Exp. Biol. 199, 869–882 (1996).

1995 (3)

D. J. Coughlin and C. W. Hawryshyn, “A cellular basis for polarized-light vision in rainbow trout,” J. Comp. Physiol. A 176, 261–272 (1995).
[CrossRef]

J. J. Vos Hzn, M. A. J. M. Coemans, and J. F. W. Nuboer, “No evidence for polarization sensitivity in the pigeon electroretinogram,” J. Exp. Biol. 198, 325–335 (1995).

R. Oldenbourg and G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

1994 (2)

C. W. Hawryshyn and F. I. Hárosi, “Spectral characteristics of visual pigments in rainbow trout (Oncorhynchus mykiss),” Vision Res. 34, 1385–1392 (1994).
[CrossRef] [PubMed]

M. P. Rowe, N. Engheta, S. S. Easter, Jr., and E. N. Pugh, Jr., “Graded-index model of a fish double cone exhibits differential polarization sensitivity,” J. Opt. Soc. Am. A 11, 55–70 (1994).
[CrossRef]

1993 (1)

D. C. Parkyn and C. W. Hawryshyn, “Polarized light sensitivity in rainbow trout (Oncorhynchus mykiss): characterization from multiunit ganglion cell responses in the optic nerve fibres,” J. Comp. Physiol. A 172, 493–500 (1993).
[CrossRef]

1991 (2)

S. M. Goddard and R. B. Forward, Jr., “The role of the underwater polarized light pattern in sun compass navigation of the grass shrimp Palaemonetes vulgaris,” J. Comp. Physiol. A 169, 479–491 (1991).
[CrossRef]

D. A. Cameron and E. N. Pugh, Jr., “Double cones as a basis for a new type of polarization vision in vertebrates,” Nature 353, 161–164 (1991).
[CrossRef] [PubMed]

1990 (1)

C. W. Hawryshyn, M. G. Arnold, E. Bowering, and R. L. Cole, “Spatial orientation of rainbow trout to plane-polarized light: the ontogeny of E-vector discrimination and spectral sensitivity characteristics,” J. Comp. Physiol. A 166, 565–574 (1990).
[CrossRef]

1988 (1)

T. Labhart, “Polarization interneurons in the insect visual system,” Nature 331, 435–437 (1988).
[CrossRef]

1987 (4)

C. W. Hawryshyn and W. N. McFarland, “Cone photoreceptor mechanisms and the detection of polarized light in fish,” J. Comp. Physiol. A 160, 459–465 (1987).
[CrossRef]

D.-E. Nilsson, T. Labhart, and E. Meyer, “Photoreceptor design and optical properties affecting polarization sensitivity in ants and crickets,” J. Comp. Physiol. A 161, 645–658 (1987).
[CrossRef]

J. K. Bowmaker and Y. W. Kunz, “Ultraviolet receptors, tetrachromatic colour vision, and retinal mosaics in brown trout (Salmo trutta),” Vision Res. 27, 2101–2108 (1987).
[CrossRef]

A. Dearry and R. B. Barlow, “Circadian rhythms in the green sunfish retina,” J. Gen. Physiol. 89, 745–770 (1987).
[CrossRef] [PubMed]

1985 (1)

K. Adler and J. B. Phillips, “Orientation in a desert lizard (Uma notata): Time-compensated compass movement and polarotaxis,” J. Comp. Physiol. A 156, 547–552 (1985).
[CrossRef]

1984 (1)

S. R. Young and G. R. Martin, “Optics of retinal oil droplets: a model of light detection and polarization detection in the avian retina,” Vision Res. 24, 129–137 (1984).
[CrossRef]

1982 (1)

K. P. Able, “Skylight polarization patterns at dusk influence migratory orientations in birds,” Nature 299, 550–551 (1982).
[CrossRef]

1978 (1)

B. A. Fineran and J. J. A. C. Nicol, “Studies on the photoreceptors of Anchoa mitchilli and A. hepsetus (Engraulidae) with particular references to the cones,” Philos. Trans. R. Soc. London, Ser. B 283, 25–60 (1978).
[CrossRef]

1977 (1)

T. H. Goldsmith and R. Wehner, “Restrictions on rotational and translational diffusion of pigment in the membranes of a rhabdomeric photoreceptor,” J. Gen. Physiol. 70, 453–490 (1977).
[CrossRef] [PubMed]

1976 (1)

W. K. Stell and F. I. Hárosi, “Cone structure and visual pigment content in the retina of goldfish,” Vision Res. 16, 647–657 (1976).
[CrossRef]

1974 (2)

P. A. Liebman and G. Entine, “Lateral diffusion of visual pigment in photoreceptor disk membranes,” Science 185, 457–459 (1974).
[CrossRef] [PubMed]

P. A. Liebman, W. S. Jagger, M. W. Kaplan, and F. G. Bargoot, “Membrane structure changes in rod outer segments associated with rhodopsin bleaching,” Nature 251, 31–36 (1974).
[CrossRef] [PubMed]

1973 (2)

W. N. Charman and J. Tucker, “The optical system of the goldfish eye,” Vision Res. 13, 1–8 (1973).
[CrossRef] [PubMed]

D. H. Taylor and K. Adler, “Spatial orientation by salamanders using plane polarized light,” Science 181, 285–287 (1973).
[CrossRef] [PubMed]

1972 (2)

R. A. Cone, “Rotational diffusion of rhodopsin in the visual receptor membrane,” Nature 236, 39–43 (1972).

M. F. Land, “The physics and biology of animal reflectors,” Prog. Biophys. Mol. Biol. 24, 75–106 (1972).
[CrossRef] [PubMed]

1970 (1)

T. H. Waterman and R. B. Forward, Jr., “Field evidence for polarized light sensitivity in the fish Zenarchopterus,” Nature 288, 85–87 (1970).
[CrossRef]

1969 (2)

S. R. Shaw, “Sense-cell structure and interspecies comparison of polarized-light absorption in arthropod compound eyes,” Vision Res. 9, 1031–1040 (1969).
[CrossRef] [PubMed]

R. J. Cherry and D. Chapman, “Optical properties of black lecithin films,” J. Mol. Biol. 40, 19–32 (1969).
[CrossRef] [PubMed]

1959 (1)

F. S. Sjöstrand, “Fine structure of cytoplasm: the organization of membraneous layers,” Rev. Mod. Phys. 31, 301–318 (1959).
[CrossRef]

1957 (1)

R. L. Sidman, “The structure and concentration of solids in photoreceptor cells studied by refractometry and interference microscopy,” J. Biophys. Biochem. Cytol. 3, 15–33 (1957).
[CrossRef] [PubMed]

Able, K. P.

K. P. Able, “Skylight polarization patterns at dusk influence migratory orientations in birds,” Nature 299, 550–551 (1982).
[CrossRef]

Adler, K.

K. Adler and J. B. Phillips, “Orientation in a desert lizard (Uma notata): Time-compensated compass movement and polarotaxis,” J. Comp. Physiol. A 156, 547–552 (1985).
[CrossRef]

D. H. Taylor and K. Adler, “Spatial orientation by salamanders using plane polarized light,” Science 181, 285–287 (1973).
[CrossRef] [PubMed]

Arnold, M. G.

C. W. Hawryshyn, M. G. Arnold, E. Bowering, and R. L. Cole, “Spatial orientation of rainbow trout to plane-polarized light: the ontogeny of E-vector discrimination and spectral sensitivity characteristics,” J. Comp. Physiol. A 166, 565–574 (1990).
[CrossRef]

Bargoot, F. G.

P. A. Liebman, W. S. Jagger, M. W. Kaplan, and F. G. Bargoot, “Membrane structure changes in rod outer segments associated with rhodopsin bleaching,” Nature 251, 31–36 (1974).
[CrossRef] [PubMed]

Barlow, R. B.

A. Dearry and R. B. Barlow, “Circadian rhythms in the green sunfish retina,” J. Gen. Physiol. 89, 745–770 (1987).
[CrossRef] [PubMed]

Bowering, E.

C. W. Hawryshyn, M. G. Arnold, E. Bowering, and R. L. Cole, “Spatial orientation of rainbow trout to plane-polarized light: the ontogeny of E-vector discrimination and spectral sensitivity characteristics,” J. Comp. Physiol. A 166, 565–574 (1990).
[CrossRef]

Bowmaker, J. K.

J. K. Bowmaker and Y. W. Kunz, “Ultraviolet receptors, tetrachromatic colour vision, and retinal mosaics in brown trout (Salmo trutta),” Vision Res. 27, 2101–2108 (1987).
[CrossRef]

Cameron, D. A.

D. A. Cameron and E. N. Pugh, Jr., “Double cones as a basis for a new type of polarization vision in vertebrates,” Nature 353, 161–164 (1991).
[CrossRef] [PubMed]

Chapman, D.

R. J. Cherry and D. Chapman, “Optical properties of black lecithin films,” J. Mol. Biol. 40, 19–32 (1969).
[CrossRef] [PubMed]

Charman, W. N.

W. N. Charman and J. Tucker, “The optical system of the goldfish eye,” Vision Res. 13, 1–8 (1973).
[CrossRef] [PubMed]

Cherry, R. J.

R. J. Cherry and D. Chapman, “Optical properties of black lecithin films,” J. Mol. Biol. 40, 19–32 (1969).
[CrossRef] [PubMed]

Coemans, M. A. J. M.

J. J. Vos Hzn, M. A. J. M. Coemans, and J. F. W. Nuboer, “No evidence for polarization sensitivity in the pigeon electroretinogram,” J. Exp. Biol. 198, 325–335 (1995).

Cole, R. L.

C. W. Hawryshyn, M. G. Arnold, E. Bowering, and R. L. Cole, “Spatial orientation of rainbow trout to plane-polarized light: the ontogeny of E-vector discrimination and spectral sensitivity characteristics,” J. Comp. Physiol. A 166, 565–574 (1990).
[CrossRef]

Cone, R. A.

R. A. Cone, “Rotational diffusion of rhodopsin in the visual receptor membrane,” Nature 236, 39–43 (1972).

Coughlin, D. J.

D. J. Coughlin and C. W. Hawryshyn, “A cellular basis for polarized-light vision in rainbow trout,” J. Comp. Physiol. A 176, 261–272 (1995).
[CrossRef]

Dearry, A.

A. Dearry and R. B. Barlow, “Circadian rhythms in the green sunfish retina,” J. Gen. Physiol. 89, 745–770 (1987).
[CrossRef] [PubMed]

Easter, Jr., S. S.

Engheta, N.

Entine, G.

P. A. Liebman and G. Entine, “Lateral diffusion of visual pigment in photoreceptor disk membranes,” Science 185, 457–459 (1974).
[CrossRef] [PubMed]

Fineran, B. A.

B. A. Fineran and J. J. A. C. Nicol, “Studies on the photoreceptors of Anchoa mitchilli and A. hepsetus (Engraulidae) with particular references to the cones,” Philos. Trans. R. Soc. London, Ser. B 283, 25–60 (1978).
[CrossRef]

Forward, Jr., R. B.

S. M. Goddard and R. B. Forward, Jr., “The role of the underwater polarized light pattern in sun compass navigation of the grass shrimp Palaemonetes vulgaris,” J. Comp. Physiol. A 169, 479–491 (1991).
[CrossRef]

T. H. Waterman and R. B. Forward, Jr., “Field evidence for polarized light sensitivity in the fish Zenarchopterus,” Nature 288, 85–87 (1970).
[CrossRef]

Goddard, S. M.

S. M. Goddard and R. B. Forward, Jr., “The role of the underwater polarized light pattern in sun compass navigation of the grass shrimp Palaemonetes vulgaris,” J. Comp. Physiol. A 169, 479–491 (1991).
[CrossRef]

Goldsmith, T. H.

T. H. Goldsmith and R. Wehner, “Restrictions on rotational and translational diffusion of pigment in the membranes of a rhabdomeric photoreceptor,” J. Gen. Physiol. 70, 453–490 (1977).
[CrossRef] [PubMed]

Hárosi, F. I.

C. W. Hawryshyn and F. I. Hárosi, “Spectral characteristics of visual pigments in rainbow trout (Oncorhynchus mykiss),” Vision Res. 34, 1385–1392 (1994).
[CrossRef] [PubMed]

W. K. Stell and F. I. Hárosi, “Cone structure and visual pigment content in the retina of goldfish,” Vision Res. 16, 647–657 (1976).
[CrossRef]

Hawryshyn, C. W.

I. Novales Flamarique and C. W. Hawryshyn, “No evidence of polarization sensitivity in freshwater sunfish from multi-unit optic nerve recordings,” Vision Res. 37, 967–973 (1997).
[CrossRef] [PubMed]

I. Novales Flamarique and C. W. Hawryshyn, “Retinal development and visual sensitivity of young Pacific sockeye salmon (Oncorhynchus nerka),” J. Exp. Biol. 199, 869–882 (1996).

D. J. Coughlin and C. W. Hawryshyn, “A cellular basis for polarized-light vision in rainbow trout,” J. Comp. Physiol. A 176, 261–272 (1995).
[CrossRef]

C. W. Hawryshyn and F. I. Hárosi, “Spectral characteristics of visual pigments in rainbow trout (Oncorhynchus mykiss),” Vision Res. 34, 1385–1392 (1994).
[CrossRef] [PubMed]

D. C. Parkyn and C. W. Hawryshyn, “Polarized light sensitivity in rainbow trout (Oncorhynchus mykiss): characterization from multiunit ganglion cell responses in the optic nerve fibres,” J. Comp. Physiol. A 172, 493–500 (1993).
[CrossRef]

C. W. Hawryshyn, M. G. Arnold, E. Bowering, and R. L. Cole, “Spatial orientation of rainbow trout to plane-polarized light: the ontogeny of E-vector discrimination and spectral sensitivity characteristics,” J. Comp. Physiol. A 166, 565–574 (1990).
[CrossRef]

C. W. Hawryshyn and W. N. McFarland, “Cone photoreceptor mechanisms and the detection of polarized light in fish,” J. Comp. Physiol. A 160, 459–465 (1987).
[CrossRef]

Jagger, W. S.

P. A. Liebman, W. S. Jagger, M. W. Kaplan, and F. G. Bargoot, “Membrane structure changes in rod outer segments associated with rhodopsin bleaching,” Nature 251, 31–36 (1974).
[CrossRef] [PubMed]

Kaplan, M. W.

P. A. Liebman, W. S. Jagger, M. W. Kaplan, and F. G. Bargoot, “Membrane structure changes in rod outer segments associated with rhodopsin bleaching,” Nature 251, 31–36 (1974).
[CrossRef] [PubMed]

Kunz, Y. W.

J. K. Bowmaker and Y. W. Kunz, “Ultraviolet receptors, tetrachromatic colour vision, and retinal mosaics in brown trout (Salmo trutta),” Vision Res. 27, 2101–2108 (1987).
[CrossRef]

Labhart, T.

T. Labhart, “Polarization interneurons in the insect visual system,” Nature 331, 435–437 (1988).
[CrossRef]

D.-E. Nilsson, T. Labhart, and E. Meyer, “Photoreceptor design and optical properties affecting polarization sensitivity in ants and crickets,” J. Comp. Physiol. A 161, 645–658 (1987).
[CrossRef]

Land, M. F.

M. F. Land, “The physics and biology of animal reflectors,” Prog. Biophys. Mol. Biol. 24, 75–106 (1972).
[CrossRef] [PubMed]

Liebman, P. A.

P. A. Liebman, W. S. Jagger, M. W. Kaplan, and F. G. Bargoot, “Membrane structure changes in rod outer segments associated with rhodopsin bleaching,” Nature 251, 31–36 (1974).
[CrossRef] [PubMed]

P. A. Liebman and G. Entine, “Lateral diffusion of visual pigment in photoreceptor disk membranes,” Science 185, 457–459 (1974).
[CrossRef] [PubMed]

Martin, G. R.

S. R. Young and G. R. Martin, “Optics of retinal oil droplets: a model of light detection and polarization detection in the avian retina,” Vision Res. 24, 129–137 (1984).
[CrossRef]

McFarland, W. N.

C. W. Hawryshyn and W. N. McFarland, “Cone photoreceptor mechanisms and the detection of polarized light in fish,” J. Comp. Physiol. A 160, 459–465 (1987).
[CrossRef]

Mei, G.

R. Oldenbourg and G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

Meyer, E.

D.-E. Nilsson, T. Labhart, and E. Meyer, “Photoreceptor design and optical properties affecting polarization sensitivity in ants and crickets,” J. Comp. Physiol. A 161, 645–658 (1987).
[CrossRef]

Nicol, J. J. A. C.

B. A. Fineran and J. J. A. C. Nicol, “Studies on the photoreceptors of Anchoa mitchilli and A. hepsetus (Engraulidae) with particular references to the cones,” Philos. Trans. R. Soc. London, Ser. B 283, 25–60 (1978).
[CrossRef]

Nilsson, D.-E.

D.-E. Nilsson, T. Labhart, and E. Meyer, “Photoreceptor design and optical properties affecting polarization sensitivity in ants and crickets,” J. Comp. Physiol. A 161, 645–658 (1987).
[CrossRef]

Novales Flamarique, I.

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I. Novales Flamarique and C. W. Hawryshyn, “Retinal development and visual sensitivity of young Pacific sockeye salmon (Oncorhynchus nerka),” J. Exp. Biol. 199, 869–882 (1996).

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[CrossRef]

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[CrossRef]

Science (2)

P. A. Liebman and G. Entine, “Lateral diffusion of visual pigment in photoreceptor disk membranes,” Science 185, 457–459 (1974).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Vision Res. (7)

S. R. Young and G. R. Martin, “Optics of retinal oil droplets: a model of light detection and polarization detection in the avian retina,” Vision Res. 24, 129–137 (1984).
[CrossRef]

I. Novales Flamarique and C. W. Hawryshyn, “No evidence of polarization sensitivity in freshwater sunfish from multi-unit optic nerve recordings,” Vision Res. 37, 967–973 (1997).
[CrossRef] [PubMed]

S. R. Shaw, “Sense-cell structure and interspecies comparison of polarized-light absorption in arthropod compound eyes,” Vision Res. 9, 1031–1040 (1969).
[CrossRef] [PubMed]

W. N. Charman and J. Tucker, “The optical system of the goldfish eye,” Vision Res. 13, 1–8 (1973).
[CrossRef] [PubMed]

W. K. Stell and F. I. Hárosi, “Cone structure and visual pigment content in the retina of goldfish,” Vision Res. 16, 647–657 (1976).
[CrossRef]

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The principal indices for a positive uniaxial crystal, such as quartz, would be larger for ne (slow axis) and smaller for no (fast axis) and would yield a positive retardance.

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M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40. Equation (21) describes the two orthogonally polarized components of the reflected waves. We eliminated in these equations the angle of refraction with the aid of Snell’s law and then expressed them as reflection coefficients, as indicated in the text. Note that the parallel (∥) and perpendicular (⊥) polarization components are defined with respect to the plane of incidence. In the large-scale model, these correspond to the horizontal and vertical directions, respectively.

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We consider the median angle to be the inclination of the median ray in a homogeneous cone of rays. For the case of numerical aperture 0.45 and medium refractive index of 1.334, the median angle is 14° and that of the extreme ray is ~20°.

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Figures (8)

Fig. 1
Fig. 1

A, Microspectrophotometrically recorded signal traces for a pumpkinseed sunfish twin cone at 0° and 90° orientations (parallel and perpendicular polarizations with respect to the partition); the top trace was obtained with a quartz standard at 2-nm retardance. Average values of retardance varied between 0.76 and 0.85 for orientations from 0°–180°, but the differences between orientations were not statistically significant. B, New Pol-Scope birefringence image of pumpkinseed sunfish twin-cone mosaic. The arrow indicates the slow axis of birefringence at the twin-cone partition; retardance scale goes from 0 nm (black) to 2 nm (white). Adapted with permission from the Biological Bulletin.29 Similar results were also obtained with green sunfish and rainbow trout.

Fig. 2
Fig. 2

Photomicrographs of photoreceptors from the centro-temporal retina (near the optic nerve) of light-adapted parr rainbow trout (A, B) (weight, 12.2 g; total length, 11.1 cm) and young pumpkinseed (C, D) (weight, 9.8 g; total length, 8.5 cm). A, Transverse section of retinal mosaic in which double-cone (DC) inner segments and their partitions (arrowhead) were cut perpendicular to their long dimensions. Center cones (CC’s) are located at partition intersections and define the centers of mosaic units. Accessory corner cones (AC’s) form the corners of the mosaic and face the DC partitions. B, Radial section showing partition tilt along the length of DC inner segments. Note the mitochondria that are larger and stain darker in one member (see also A) than the smaller mitochondria to be found in the other member. Unequal staining of DC outer segments (*) indicate further differences between the two members.32-34 The AC outer segments (α) are located at the level of neighboring DC inner segments. C, Transverse section showing twin cones (TC’s) surrounding center cones (CC’s) in the sunfish mosaic. D, Radial section through the photoreceptor layer showing a nearly straight partition separating TC members; asterisks label the similar outer segments. Calibration bars correspond to (A) 7.2 µm, (B) 5.6 µm, (C) 6.43 µm, and (D) 3.75 µm.

Fig. 3
Fig. 3

Schematic drawing of the large-scale optical model used in this study to mimic anisotropic transmission of polarized light in paired-cone inner segments. The device permits photocurrent measurements of D1 and D2 as functions of partition tilt and polarization direction of the rotatable linear polarizer. Parallel (horizontal) and perpendicular (vertical) polarizations are defined with respect to the partition’s plane of incidence.

Fig. 4
Fig. 4

Reflectivity coefficient variations with angle of incidence at an air–glass interface as described by Fresnel’s formulas. Note the large difference (dashed–dotted curve) between the perpendicular and the parallel components at large angles of incidence.

Fig. 5
Fig. 5

Experimental and theoretical contrast variations with partition tilt angle +τ (clockwise rotation) for the large-scale model depicted in Fig. 3. The glass partition was treated as a thick slab with air–glass–air interfaces. Reflectivity and transmissivity coefficients were summed at discrete one-deg increments within the collection angles of the diodes.

Fig. 6
Fig. 6

Theoretical polarization contrast variations (A) between two members of an individual double cone and (B) between members of two adjacent double cones with orthogonally oriented partitions in the trout retinal cone mosaic. Assumed refractive indices were 1.475 (partition) and 1.365 (cytoplasm). The contrast traces for C23, not shown, are the negative equivalents of C14.

Fig. 7
Fig. 7

Schematic representation of anisotropic transfer of linearly polarized light, assuming that the partitions act as dielectric mirrors. A, double- and B, twin-cone mosaic units. Letter designations indicate long (L), medium (M), short (S), and ultraviolet (UV) wavelength sensitivity regions for the sensory pigments residing in the cone outer segments. Double arrows indicate the predominant polarization direction after reflection from the partition. The TC dimensions indicated in 7 B are approximately valid for the corresponding DC ones in 7 A.

Fig. 8
Fig. 8

Calculated fractions of light loss from the right member of a double cone (toward which the partition tilts) as a function of tilt angle. Fractions are normalized for all light received (reflected+transmitted) by the same outer segment at τ=0 (straight partition). Assumed refractive indices were 1.475 (partition) and 1.365 (cytoplasm).

Tables (1)

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Table 1 Percent Contrast (±SD), Calculated From Transmitted Polarized Light Fluxes Averaged Over Various Spectral Intervalsa

Equations (6)

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

ρ(ϕ)=[n22 cos ϕ-n1(n22-n12 sin2 ϕ)1/2]/[n22 cos ϕ+n1(n22-n12 sin2 ϕ)1/2],
ρ(ϕ)=[n1 cos ϕ-(n22-n12 sin2 ϕ)1/2]/[n1 cos ϕ+(n22-n12 sin2 ϕ)1/2],
R(ϕ)=2|ρ(ϕ)|2/(1+|ρ(ϕ)|2),
R(ϕ)=2|ρ(ϕ)|2/(1+|ρ(ϕ)|2),
T(ϕ)=1-R(ϕ),
T(ϕ)=1-R(ϕ).

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