Abstract

Retinal birefringence scanning (RBS) is a new technique that is used to detect the fixation of the eye remotely and noninvasively. The method is based on analysis of polarization changes induced by the retina. In this study, the principles of RBS were mathematically modeled to facilitate a better understanding of the origins of the signals obtained. Stokes vector analysis and Mueller matrix multiplication were augmented with Poincaré sphere representation. The cornea was modeled as a linear retarder. The foveal area was modeled as a radially symmetric birefringent medium. The model accurately predicted the frequency and phase of RBS signals obtained during central and paracentral fixation. The signal that indicates central fixation during RBS likely results from a combination of the radial birefringence of the Henle fibers and the overlying corneal birefringence.

© 1999 Optical Society of America

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References

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  1. D. G. Hunter, S. N. Patel, D. L. Guyton, “Automated detection of foveal fixation by use of retinal birefringence scanning,” Appl. Opt. 38, 1273–1277 (1999).
    [CrossRef]
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    [CrossRef]
  17. R. N. Weinreb, S. Shakiba, L. Zangwill, “Scanning laser polarimetry to measure the nerve fiber layer of normal and glaucomatous eyes,” Am. J. Ophthalmol. 119, 626–36 (1995).
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1999 (1)

1998 (1)

1997 (1)

T. J. T. P. van den Berg, H. Spekreijse, “Near infrared light absorption in the human eye media,” Vision Res. 37, 249–253 (1997).
[CrossRef] [PubMed]

1995 (2)

D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “Numerical modeling of the cornea’s lamellar structure and birefringence properties,” J. Opt. Soc. Am. A 12, 1425–1438 (1995).
[CrossRef]

R. N. Weinreb, S. Shakiba, L. Zangwill, “Scanning laser polarimetry to measure the nerve fiber layer of normal and glaucomatous eyes,” Am. J. Ophthalmol. 119, 626–36 (1995).

1991 (1)

1990 (1)

R. N. Weinreb, A. W. Dreher, A. Coleman, H. Quigley, B. Shaw, K. Reiter, “Histopathologic validation of Fourier-ellipsometry measurements of retinal nerve fiber layer thickness,” Arch. Ophthalmol. (Chicago) 108, 557–560 (1990).
[CrossRef]

1989 (1)

1988 (1)

1987 (2)

1986 (1)

D. van Norren, L. F. Tiemeijer, “Spectral reflectance of the human eye,” Vision Res. 26, 313–320 (1986).
[CrossRef] [PubMed]

1985 (1)

1979 (1)

R. A. Weale, “Sex, age and the birefringence of the human crystalline lens,” Exp. Eye Res. 29, 449–461 (1979).
[CrossRef] [PubMed]

1966 (1)

F. Vrabec, “The temporal raphe of the human retina,” Am. J. Ophthalmol. 62, 926–938 (1966).
[PubMed]

1954 (1)

Allen, J.

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, New York, 1980), pp. 554–555.

Coleman, A.

R. N. Weinreb, A. W. Dreher, A. Coleman, H. Quigley, B. Shaw, K. Reiter, “Histopathologic validation of Fourier-ellipsometry measurements of retinal nerve fiber layer thickness,” Arch. Ophthalmol. (Chicago) 108, 557–560 (1990).
[CrossRef]

Delori, F. C.

Donohue, D. J.

Dreher, A. W.

R. N. Weinreb, A. W. Dreher, A. Coleman, H. Quigley, B. Shaw, K. Reiter, “Histopathologic validation of Fourier-ellipsometry measurements of retinal nerve fiber layer thickness,” Arch. Ophthalmol. (Chicago) 108, 557–560 (1990).
[CrossRef]

Farrell, R. A.

Fry, R. L.

D. L. Guyton, D. G. Hunter, J. C. Sandruck, S. N. Patel, R. L. Fry, “Eye fixation monitor and tracker,” U.S. patent application (October21, 1997).

Guyton, D. L.

Hunter, D. G.

D. G. Hunter, S. N. Patel, D. L. Guyton, “Automated detection of foveal fixation by use of retinal birefringence scanning,” Appl. Opt. 38, 1273–1277 (1999).
[CrossRef]

D. L. Guyton, D. G. Hunter, J. C. Sandruck, S. N. Patel, R. L. Fry, “Eye fixation monitor and tracker,” U.S. patent application (October21, 1997).

Jerrard, H. G.

klein Brink, H. B.

McCally, R. L.

Patel, S. N.

D. G. Hunter, S. N. Patel, D. L. Guyton, “Automated detection of foveal fixation by use of retinal birefringence scanning,” Appl. Opt. 38, 1273–1277 (1999).
[CrossRef]

D. L. Guyton, D. G. Hunter, J. C. Sandruck, S. N. Patel, R. L. Fry, “Eye fixation monitor and tracker,” U.S. patent application (October21, 1997).

Pflibsen, K. P.

Pierscionek, B. K.

Quigley, H.

R. N. Weinreb, A. W. Dreher, A. Coleman, H. Quigley, B. Shaw, K. Reiter, “Histopathologic validation of Fourier-ellipsometry measurements of retinal nerve fiber layer thickness,” Arch. Ophthalmol. (Chicago) 108, 557–560 (1990).
[CrossRef]

Reiter, K.

R. N. Weinreb, A. W. Dreher, A. Coleman, H. Quigley, B. Shaw, K. Reiter, “Histopathologic validation of Fourier-ellipsometry measurements of retinal nerve fiber layer thickness,” Arch. Ophthalmol. (Chicago) 108, 557–560 (1990).
[CrossRef]

Sandruck, J. C.

D. L. Guyton, D. G. Hunter, J. C. Sandruck, S. N. Patel, R. L. Fry, “Eye fixation monitor and tracker,” U.S. patent application (October21, 1997).

Scattergood, K. D.

Shakiba, S.

R. N. Weinreb, S. Shakiba, L. Zangwill, “Scanning laser polarimetry to measure the nerve fiber layer of normal and glaucomatous eyes,” Am. J. Ophthalmol. 119, 626–36 (1995).

Shaw, B.

R. N. Weinreb, A. W. Dreher, A. Coleman, H. Quigley, B. Shaw, K. Reiter, “Histopathologic validation of Fourier-ellipsometry measurements of retinal nerve fiber layer thickness,” Arch. Ophthalmol. (Chicago) 108, 557–560 (1990).
[CrossRef]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light: Production and Use (Harvard U. Press, Cambridge, Mass., 1962), p. 169.

Simons, K.

Sliney, D.

D. Sliney, M. Wolbarsht, Safety with Lasers and Other Optical Sources (Plenum, New York, 1980), pp. 261–283.

Spekreijse, H.

T. J. T. P. van den Berg, H. Spekreijse, “Near infrared light absorption in the human eye media,” Vision Res. 37, 249–253 (1997).
[CrossRef] [PubMed]

Stoyanov, B. J.

Tiemeijer, L. F.

D. van Norren, L. F. Tiemeijer, “Spectral reflectance of the human eye,” Vision Res. 26, 313–320 (1986).
[CrossRef] [PubMed]

van Blokland, G. J.

van den Berg, T. J. T. P.

T. J. T. P. van den Berg, H. Spekreijse, “Near infrared light absorption in the human eye media,” Vision Res. 37, 249–253 (1997).
[CrossRef] [PubMed]

van Norren, D.

D. van Norren, L. F. Tiemeijer, “Spectral reflectance of the human eye,” Vision Res. 26, 313–320 (1986).
[CrossRef] [PubMed]

Verhelst, S. C.

Vrabec, F.

F. Vrabec, “The temporal raphe of the human retina,” Am. J. Ophthalmol. 62, 926–938 (1966).
[PubMed]

Weale, R. A.

Weinreb, R. N.

R. N. Weinreb, S. Shakiba, L. Zangwill, “Scanning laser polarimetry to measure the nerve fiber layer of normal and glaucomatous eyes,” Am. J. Ophthalmol. 119, 626–36 (1995).

R. N. Weinreb, A. W. Dreher, A. Coleman, H. Quigley, B. Shaw, K. Reiter, “Histopathologic validation of Fourier-ellipsometry measurements of retinal nerve fiber layer thickness,” Arch. Ophthalmol. (Chicago) 108, 557–560 (1990).
[CrossRef]

Wolbarsht, M.

D. Sliney, M. Wolbarsht, Safety with Lasers and Other Optical Sources (Plenum, New York, 1980), pp. 261–283.

Wolf, E.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, New York, 1980), pp. 554–555.

Zangwill, L.

R. N. Weinreb, S. Shakiba, L. Zangwill, “Scanning laser polarimetry to measure the nerve fiber layer of normal and glaucomatous eyes,” Am. J. Ophthalmol. 119, 626–36 (1995).

Am. J. Ophthalmol. (2)

R. N. Weinreb, S. Shakiba, L. Zangwill, “Scanning laser polarimetry to measure the nerve fiber layer of normal and glaucomatous eyes,” Am. J. Ophthalmol. 119, 626–36 (1995).

F. Vrabec, “The temporal raphe of the human retina,” Am. J. Ophthalmol. 62, 926–938 (1966).
[PubMed]

Appl. Opt. (4)

Arch. Ophthalmol. (Chicago) (1)

R. N. Weinreb, A. W. Dreher, A. Coleman, H. Quigley, B. Shaw, K. Reiter, “Histopathologic validation of Fourier-ellipsometry measurements of retinal nerve fiber layer thickness,” Arch. Ophthalmol. (Chicago) 108, 557–560 (1990).
[CrossRef]

Exp. Eye Res. (1)

R. A. Weale, “Sex, age and the birefringence of the human crystalline lens,” Exp. Eye Res. 29, 449–461 (1979).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Vision Res. (2)

T. J. T. P. van den Berg, H. Spekreijse, “Near infrared light absorption in the human eye media,” Vision Res. 37, 249–253 (1997).
[CrossRef] [PubMed]

D. van Norren, L. F. Tiemeijer, “Spectral reflectance of the human eye,” Vision Res. 26, 313–320 (1986).
[CrossRef] [PubMed]

Other (4)

D. Sliney, M. Wolbarsht, Safety with Lasers and Other Optical Sources (Plenum, New York, 1980), pp. 261–283.

W. A. Shurcliff, Polarized Light: Production and Use (Harvard U. Press, Cambridge, Mass., 1962), p. 169.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, New York, 1980), pp. 554–555.

D. L. Guyton, D. G. Hunter, J. C. Sandruck, S. N. Patel, R. L. Fry, “Eye fixation monitor and tracker,” U.S. patent application (October21, 1997).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of RBS. When a subject fixates on the target (T), the target is imaged onto the fovea of the eye (*). The scanned laser diode source (SLD) is imaged onto the radial arrangement of nerve fibers (NF) surrounding fovea as an annulus of polarized light. Returning light is imaged by means of a nonpolarizing plate beam splitter (BS1) onto a differential polarization analyzer (DPA) to detect changes in the polarization state of returning light as the retina is scanned. The DPA consists of a polarizing beam splitter cube (BS2) and silicon photodetectors X and Y. The differential polarization signal XY is the output of the RBS device.

Fig. 2
Fig. 2

Polarization ellipse: a, minor semiaxis; b, major semiaxis; α, polarization azimuth. The diagram can be considered a one-cycle, time-exposure snapshot of the polarization of the light seen by an observer positioned in the path of the beam looking toward the source. In the most general case (shown), the pattern of polarization is an ellipse. The polarization azimuth of the ellipse (α) is the angle between the major semiaxis and the x axis. Ellipticity (ε) is a function of the relative lengths of the major and minor semiaxes.

Fig. 3
Fig. 3

The Poincaré sphere. S, input Stokes vector located by plotting normalized Stokes parameters s1, s2, and s3 or by placing the vector at longitude 2α and latitude 2ε; M, Muller matrix of the retarder; S*, output Stokes vector. S* can be determined by rotating S about vector R [located at twice the linear retarder azimuth (2θM)] through an angle equal to the retardance (δM). The curved, dotted arrow shows the direction and amount of rotation. The normalized incident irradiance s0 is unrelated to the state of polarization but can be considered equal to the unit radius of the sphere.

Fig. 4
Fig. 4

Retinal retardation azimuth calculation during scan of radially symmetric retinal fibers (paracentral fixation). F, fovea. In both diagrams, the large circle is the annulus of the scanned light, the asterisk is the center of the scan annulus, and the small solid circle is the momentary location of the laser spot during the scan. In the detailed view on the right, θr is the momentary orientation of the retinal retardation azimuth during the scan, ϕ is the angular displacement of the laser from the scan origin, and the curved arrows represent the direction of the scan. [R sin(ϕ),R cos(ϕ)] is the momentary displacement of the scanning laser from the center of the annulus, and (xret, yret) is the distance of the center of the annulus from the fovea. The laser spot advances through 360 deg during one retinal scan.  

Fig. 5
Fig. 5

Changes in polarization during double pass of light through the eye. A, First pass through the cornea (step 1). P0, circularly polarized input. The curved, dotted arrow represents rotation of the polarization vector about the axis at 2θc through angle δc. P1, resulting state of polarization. B, First pass through the retina (step 2). P1, input polarization state (output of step 1). Inset, change in retinal retardance azimuth as polarized light scans the circular path about the fovea. Note the different orientations of retinal fibers at points a–d. Corresponding points inscribe an ellipse when they are plotted on the Poincaré sphere. C, Reflection by the fundus (step 3), represented on the sphere as reflection across planes s2=0 and s3=0. Note that the reflected ellipse is inscribed on the rear surface of the Poincaré sphere. D, Second passes through the retina (step 4) and the cornea (step 5). The ellipse is larger after the second pass through retinal retardance. The second pass through the cornea causes the entire ellipse to rotate about axis -2θc through angle δc. All three ellipses are located on the rear surface of the Poincaré sphere.

Fig. 6
Fig. 6

Output of the retinal birefringence scanner. Left side of each panel, voltage output in the time domain; right side of each panel, power output in the frequency domain. A, B, Predicted output: left, elliptical inscription on the surface of the Poincaré sphere as projected onto the s1 axis and plotted versus time; right, Fourier power spectrum of the time-domain signal (relative power units). C, D, Output measured from a representative normal subject: left, signal voltage versus time; right, power (in V2 RMS×10-6) obtained from discrete Fourier power spectrum analysis of time signal versus frequency.

Fig. 7
Fig. 7

Changes in retinal birefringence scan output as a function of horizontal fixation position (relative power units). The measured output in B was obtained from a normal volunteer (power in V2 RMS×10-6).

Fig. 8
Fig. 8

Phase shift of the differential polarization signal during paracentral fixation. A, predicted output. Each labeled asterisk indicates a predicted phase shift for the gaze direction indicated. Each unlabeled point asterisk indicates an intermediate gaze direction. B, Measured output obtained from a normal volunteer. Labeled clusters indicate five phase-shift measurements at the indicated gaze directions. Unlabeled clusters indicate intermediate gaze directions. Tick marks represent fixation targets 1.5 deg away from the target.

Fig. 9
Fig. 9

Proposed origin of the 88-Hz signal in the nerve fiber layer between the optic nerve head and the fovea. F, fovea; ONH, optic nerve head. Circle, path of the retinal birefringence scan during paracentral fixation, crossing nerve fiber orientations that have a slight periodic variation (a–b–a–b) at twice the scanning frequency.

Tables (1)

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Table 1 Summary of Key Parameters Used in Modeling Polarization Changes a

Equations (16)

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s0=incidentirradiance
s1=s0 cos(2ε)cos(2α),
s2=s0 cos(2ε)sin(2α),
s3=s0 sin(2ε).
s0=I(0°, 0)+I(90°, 0),
s1=I(0°, 0)-I(90°, 0),
s2=I(45°, 0)-I(135°, 0),
s3=I(45°, π/2)-I(135°, π/2).
S0*S1*S2*S3*=10000d2-e2+g22de-2eg02de-d2+e2+g22dg02eg-2dg2g2-1S0S1S2S3.
d=sin(δ/2)cos(2 θ),
e=sin(δ/2)sin(2 θ),
g=cos(δ/2).
θr=tan-1R sin(ϕ)+yretR cos(ϕ)+xret.
xret(mm)=-0.454.2xfix[cm],
yret(mm)=0.454.2yfix[cm].
θr=tan-1R cos(ϕ)R sin(ϕ)=ϕ.

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