Abstract

A general expression for the form birefringence of a medium of parallel cylinders of arbitrary sizes and separations has been developed. It is necessary to assume only that the refractive index fluctuations through the medium are sufficiently small. This expression has been evaluated for a model of retinal nerve fiber layers. The birefringence of these layers has considerable potential as a diagnostic tool since it is expected to be sensitive to structural changes associated with retinal pathologies. Although accurate measurements of this birefringence have not yet been carried out they appear to be feasible or may soon be so.

© 1989 Optical Society of America

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References

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  1. B. F. Hochheimer, H. A. Kues, “Retinal Polarization Effects,” Appl. Opt. 21, 3811–3818 (1982).
    [Crossref] [PubMed]
  2. A. Sommer, H. A. Kues, S. A. D'Anna, S. Arkell, A. Robin, H. A. Quigley, “Cross-polarization Photography of the Nerve Fiber Layer,” Arch. Ophthalmol. 102, 864–869 (1984).
    [Crossref] [PubMed]
  3. A. Plesch, V. Klingbeil, J. Bille, “Digital Laser Scanning Fundus Camera,” Appl. Opt. 26, 1480–1486 (1987).
    [Crossref] [PubMed]
  4. A. Dreher, K. Reiter, J. Bille, “Assessment of Nerve Fiber Layer Thickness with the LTS Laser Tomographic Scanner,” Invest. Ophthalmol. Vis. Sci. (ARVO Suppl.) 29, 355 (1988).
  5. H. B. klein Brink, G. J. van Blokland, “Birefringence of the Human Foveal Area Assessed in vivo with Mueller-Matrix Ellipsometry,” J. Opt. Soc. Am. A 5, 49–57 (1988).
    [Crossref]
  6. M. J. Hogan, J. A. Alvarado, J. E. Weddell, Histology of the Human Eye (Saunders, Philadelphia, PA, 1971).
  7. O. Wiener, “Die Theorie des MischKorpers fur das Feld der Stationaren Stromung,” abh, Math. Phys. Kl. Konigl. Sachs. Ges. Wiss. 32, 507–604 (1912)[as cited in J. M. Enoch, F. L. Tobey, Eds., Vertebrate Photoreceptor Optics (Springer-Verlag, New York, 1981), p. 393.]
  8. C. Acquista, “Light Scattering by Tenuous Particles: a Generalization of the Rayleigh-Gans-Rocard Approach,” Appl. Opt. 15, 2932–2936 (1976).
    [Crossref] [PubMed]
  9. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 32–34.
  10. P. Debye, A. M. Bueche, “Scattering by an Inhomogeneous Solid,” J. Appl. Phys. 20, 518–525 (1949).
    [Crossref]

1988 (2)

A. Dreher, K. Reiter, J. Bille, “Assessment of Nerve Fiber Layer Thickness with the LTS Laser Tomographic Scanner,” Invest. Ophthalmol. Vis. Sci. (ARVO Suppl.) 29, 355 (1988).

H. B. klein Brink, G. J. van Blokland, “Birefringence of the Human Foveal Area Assessed in vivo with Mueller-Matrix Ellipsometry,” J. Opt. Soc. Am. A 5, 49–57 (1988).
[Crossref]

1987 (1)

1984 (1)

A. Sommer, H. A. Kues, S. A. D'Anna, S. Arkell, A. Robin, H. A. Quigley, “Cross-polarization Photography of the Nerve Fiber Layer,” Arch. Ophthalmol. 102, 864–869 (1984).
[Crossref] [PubMed]

1982 (1)

1976 (1)

1949 (1)

P. Debye, A. M. Bueche, “Scattering by an Inhomogeneous Solid,” J. Appl. Phys. 20, 518–525 (1949).
[Crossref]

1912 (1)

O. Wiener, “Die Theorie des MischKorpers fur das Feld der Stationaren Stromung,” abh, Math. Phys. Kl. Konigl. Sachs. Ges. Wiss. 32, 507–604 (1912)[as cited in J. M. Enoch, F. L. Tobey, Eds., Vertebrate Photoreceptor Optics (Springer-Verlag, New York, 1981), p. 393.]

Acquista, C.

Alvarado, J. A.

M. J. Hogan, J. A. Alvarado, J. E. Weddell, Histology of the Human Eye (Saunders, Philadelphia, PA, 1971).

Arkell, S.

A. Sommer, H. A. Kues, S. A. D'Anna, S. Arkell, A. Robin, H. A. Quigley, “Cross-polarization Photography of the Nerve Fiber Layer,” Arch. Ophthalmol. 102, 864–869 (1984).
[Crossref] [PubMed]

Bille, J.

A. Dreher, K. Reiter, J. Bille, “Assessment of Nerve Fiber Layer Thickness with the LTS Laser Tomographic Scanner,” Invest. Ophthalmol. Vis. Sci. (ARVO Suppl.) 29, 355 (1988).

A. Plesch, V. Klingbeil, J. Bille, “Digital Laser Scanning Fundus Camera,” Appl. Opt. 26, 1480–1486 (1987).
[Crossref] [PubMed]

Bueche, A. M.

P. Debye, A. M. Bueche, “Scattering by an Inhomogeneous Solid,” J. Appl. Phys. 20, 518–525 (1949).
[Crossref]

D'Anna, S. A.

A. Sommer, H. A. Kues, S. A. D'Anna, S. Arkell, A. Robin, H. A. Quigley, “Cross-polarization Photography of the Nerve Fiber Layer,” Arch. Ophthalmol. 102, 864–869 (1984).
[Crossref] [PubMed]

Debye, P.

P. Debye, A. M. Bueche, “Scattering by an Inhomogeneous Solid,” J. Appl. Phys. 20, 518–525 (1949).
[Crossref]

Dreher, A.

A. Dreher, K. Reiter, J. Bille, “Assessment of Nerve Fiber Layer Thickness with the LTS Laser Tomographic Scanner,” Invest. Ophthalmol. Vis. Sci. (ARVO Suppl.) 29, 355 (1988).

Hochheimer, B. F.

Hogan, M. J.

M. J. Hogan, J. A. Alvarado, J. E. Weddell, Histology of the Human Eye (Saunders, Philadelphia, PA, 1971).

klein Brink, H. B.

Klingbeil, V.

Kues, H. A.

A. Sommer, H. A. Kues, S. A. D'Anna, S. Arkell, A. Robin, H. A. Quigley, “Cross-polarization Photography of the Nerve Fiber Layer,” Arch. Ophthalmol. 102, 864–869 (1984).
[Crossref] [PubMed]

B. F. Hochheimer, H. A. Kues, “Retinal Polarization Effects,” Appl. Opt. 21, 3811–3818 (1982).
[Crossref] [PubMed]

Plesch, A.

Quigley, H. A.

A. Sommer, H. A. Kues, S. A. D'Anna, S. Arkell, A. Robin, H. A. Quigley, “Cross-polarization Photography of the Nerve Fiber Layer,” Arch. Ophthalmol. 102, 864–869 (1984).
[Crossref] [PubMed]

Reiter, K.

A. Dreher, K. Reiter, J. Bille, “Assessment of Nerve Fiber Layer Thickness with the LTS Laser Tomographic Scanner,” Invest. Ophthalmol. Vis. Sci. (ARVO Suppl.) 29, 355 (1988).

Robin, A.

A. Sommer, H. A. Kues, S. A. D'Anna, S. Arkell, A. Robin, H. A. Quigley, “Cross-polarization Photography of the Nerve Fiber Layer,” Arch. Ophthalmol. 102, 864–869 (1984).
[Crossref] [PubMed]

Sommer, A.

A. Sommer, H. A. Kues, S. A. D'Anna, S. Arkell, A. Robin, H. A. Quigley, “Cross-polarization Photography of the Nerve Fiber Layer,” Arch. Ophthalmol. 102, 864–869 (1984).
[Crossref] [PubMed]

van Blokland, G. J.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 32–34.

Weddell, J. E.

M. J. Hogan, J. A. Alvarado, J. E. Weddell, Histology of the Human Eye (Saunders, Philadelphia, PA, 1971).

Wiener, O.

O. Wiener, “Die Theorie des MischKorpers fur das Feld der Stationaren Stromung,” abh, Math. Phys. Kl. Konigl. Sachs. Ges. Wiss. 32, 507–604 (1912)[as cited in J. M. Enoch, F. L. Tobey, Eds., Vertebrate Photoreceptor Optics (Springer-Verlag, New York, 1981), p. 393.]

abh, Math. Phys. Kl. Konigl. Sachs. Ges. Wiss. (1)

O. Wiener, “Die Theorie des MischKorpers fur das Feld der Stationaren Stromung,” abh, Math. Phys. Kl. Konigl. Sachs. Ges. Wiss. 32, 507–604 (1912)[as cited in J. M. Enoch, F. L. Tobey, Eds., Vertebrate Photoreceptor Optics (Springer-Verlag, New York, 1981), p. 393.]

Appl. Opt. (3)

Arch. Ophthalmol. (1)

A. Sommer, H. A. Kues, S. A. D'Anna, S. Arkell, A. Robin, H. A. Quigley, “Cross-polarization Photography of the Nerve Fiber Layer,” Arch. Ophthalmol. 102, 864–869 (1984).
[Crossref] [PubMed]

Invest. Ophthalmol. Vis. Sci. (1)

A. Dreher, K. Reiter, J. Bille, “Assessment of Nerve Fiber Layer Thickness with the LTS Laser Tomographic Scanner,” Invest. Ophthalmol. Vis. Sci. (ARVO Suppl.) 29, 355 (1988).

J. Appl. Phys. (1)

P. Debye, A. M. Bueche, “Scattering by an Inhomogeneous Solid,” J. Appl. Phys. 20, 518–525 (1949).
[Crossref]

J. Opt. Soc. Am. A (1)

Other (2)

M. J. Hogan, J. A. Alvarado, J. E. Weddell, Histology of the Human Eye (Saunders, Philadelphia, PA, 1971).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 32–34.

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Figures (5)

Fig. 1
Fig. 1

Coordinate system used in calculations. The incident light is propagated in the direction of the positive z-axis. The y-axis is parallel to the fiber axes and to the optic axis of the uniaxial birefringent medium.

Fig. 2
Fig. 2

Schematic drawing which illustrates the rationale for assigning a refractive index with real and imaginary parts to a scattering but non-dissipative medium. The real part of the index accounts for the phase change of the forward beam while the imaginary part accounts for light lost from the forward beam due to scatter.

Fig. 3
Fig. 3

Drawing of the cross section of a simple model of the retinal nerve fiber layer in which the nerve fibers are assumed uniform with a refractive index different from that of the spaces between fibers. The largest fibers have diameters of 3 μm. This model is used to estimate the form and extent of the correlation function (see text).

Fig. 4
Fig. 4

Overlap of the spaces between fibers (see Fig. 3) as a function of lateral displacement determined by overlaying two transparencies of Fig. 3 (open circles). By normalizing the results so that the maximum overlap is unity and the minimum is zero the results correspond to a correlation function. The solid curve is a plot of the exponential, exp(−r/a), where r is the displacement in micrometers and the constant, a = 0.18 μm.

Fig. 5
Fig. 5

Wavelength dependent coefficient, C, in Eq. (12) as a function of the correlation length, a, divided by the wavelength inside the birefringent medium, λin = λ/navg.

Equations (15)

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n e 2 n o 2 = f 1 f 2 ( n 1 2 n 2 2 ) 2 / [ ( 1 + f 1 ) n 2 2 + f 2 n 1 2 ] .
E s ( 2 ) ( z o ) = k 2 exp ( i k z o ) / ( 2 π 2 z o ) × d 3 q u ( q + k z ) u ( q k z ̂ ) / ( q 2 k 2 ) × [ ( 2 k 2 / 3 + q 2 / 3 ) E i ( q E i ) q r ] .
n avg = n i f i
k = 2 π ( n avg ) / λ
u ( p ) = α ( r ) exp ( i p r ) d 3 r ,
α i = 3 ( n i 2 / n avg 2 1 ) / [ 4 π ( n i 2 / n avg 2 + 2 ) ] .
α i η i / ( 2 π )
u ( p ) u ( p ) = ( 2 π ) 2 d 3 r d 3 r η ( r ) η ( r ) exp [ i p ( r r ) ] V η 2 ¯ ( 2 π ) 2 d 3 r γ ( r ) exp ( i p r ) ,
γ ( r ) = ( V η 2 ¯ ) 1 d 3 r η ( r + r ) η ( r ) , η 2 ¯ = V 1 d 3 r η 2 ( r ) ,
d y exp [ i ( q y y ) ] = 2 π δ ( q y )
( n e n o ) / n avg = η 2 ¯ / ( 2 π 2 ) d 2 r γ ( r ) d 2 q × exp [ i ( q + k z ̂ ) r ] q x 2 / ( q 2 k 2 ) ,
( n e n o ) = ( n e n o ) W π 1 rdr γ ( r ) × d 2 q J o ( r | q + k z ̂ | ) q x 2 / ( q 2 k 2 ) ,
( n e n o ) W = n avg η 2 ¯ = ( 2 n avg ) 1 i j ( n i n j ) 2 f i f j .
( n e n o ) = ( n e n o ) W C ( k a )
C ( k a ) = ( π k a ) 1 0 2 π d ψ sin 2 ψ 0 d q q 3 ( q 2 1 ) 1 × [ ( k 2 a 2 ) 1 + 1 + q 2 + 2 q cos ψ ] 3 / 2

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