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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 21 — Oct. 21, 2002
  • pp: 1179–1189
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Analytical methods for scanning laser polarimetry

Robert W. Knighton and Xiang-Run Huang  »View Author Affiliations


Optics Express, Vol. 10, Issue 21, pp. 1179-1189 (2002)
http://dx.doi.org/10.1364/OE.10.001179


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Abstract

Scanning laser polarimetry (SLP), a technology for glaucoma diagnosis, uses imaging polarimetry to detect the birefringence of the retinal nerve fiber layer. A simple model of SLP suggests an algorithm for calculating birefringence that, unlike previous methods, uses all of the data available in the images to achieve better signal-to-noise ratio and lower sensitivity to depolarization. The uncertainty of the calculated retardance is estimated and an appropriate averaging strategy to reduce uncertainty is demonstrated. Averaging over a large area of the macula of the eye is used in a new method for determining anterior segment birefringence.

© 2002 Optical Society of America

1. Introduction

Scanning laser polarimetry (SLP), a technology for glaucoma diagnosis, incorporates polarimetry into a scanning laser ophthalmoscope in order to detect the birefringence of the retinal nerve fiber layer (RNFL) [1

1. A. W. Dreher and K. Reiter, “Scanning laser polarimetry of the retinal nerve fiber layer,” Proc. of SPIE 1746, 34–41 (1992). [CrossRef]

,2

2. A. W. Dreher and K. Reiter, “Retinal laser ellipsometry - a new method for measuring the retinal nerve-fiber layer thickness distribution,” Clin. Vision Sci. 7, 481–488 (1992).

]. SLP is attractive because it provides rapid assessment, a visual image of the RNFL, and the potential for reproducible, objective measures of RNFL loss. A confounding variable, birefringence of the anterior segment of the eye (chiefly corneal birefringence) [3

3. D. S. Greenfield, R. W. Knighton, and X. R. Huang, “Effect of corneal polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,” Am. J. Ophthalmol. 129, 715–722 (2000). [CrossRef] [PubMed]

,4

4. R. W. Knighton, X. R. Huang, and D. S. Greenfield, “Analytical model of scanning laser polarimetry for retinal nerve fiber layer assessment,” Invest. Ophthalmol. Visual Sci. 43, 383–392 (2002).

], is being addressed by newer designs for individualized anterior segment compensation that use the radial birefringence of Henle’s fiber layer in the macula as an “intraocular polarimeter” [5

5. Q. Zhou and R. N. Weinreb, “Individualized compensation of anterior segment birefringence during scanning laser polarimetry,” Invest. Ophthalmol. Visual Sci. 43, 2221–2228 (2002).

]. Now that the possibility exists to portray accurately the RNFL birefringence, methods that improve this accuracy can enhance the diagnostic capabilities of SLP.

This paper presents algorithms for the analysis of SLP signals that can reduce variability and susceptibility to noise, increase measurement sensitivity, increase the speed of data collection, and decrease the sensitivity to depolarization in the optical pathway. In addition, a method is presented for determining anterior segment birefringence even when macular pathology disrupts Henle’s fiber layer.

Fig. 1. Schematic diagram of scanning laser polarimetry (SLP).
  • P: Linear polarizer with azimuth θ.
  • A: Linear analyzers perpendicular and parallel to P.
  • D: Partial depolarizer.
  • R: Retarder to be measured.
Azimuth θ varies from 0° to 90°. The output of D is analyzed as two components, one completely polarized with intensity K (red line) and the other completely depolarized with intensity 2d (blue line), that are analyzed separately and then added to produce the output intensities I × and I p.

2. SLP measurement model

The diagram in Fig. 1 shows the elements of SLP. The specimen to be measured (e.g., the eye) was modeled as a partial depolarizer (D) followed by a retarder (R) with slow axis θ R and retardance δ. In the case of the eye, the corneal and retinal birefringence are in series and are measured in reflection, with the light beam often assumed to make a double pass through both structures [6

6. A. W. Dreher and K. Reiter, inventors, Laser Diagnostic Technologies, Inc., assignee. “Retinal eye disease diagnostic system,” United States Patent No. 5,303,709, April 19, 1994.

,7

7. K. Reiter and A. W. Dreher, inventors, Laser Diagnostic Technologies, Inc., assignee. “Eye examination apparatus employing polarized light probe,” United States Patent No. 5,787,890, August 4, 1998.

]. Depolarization can occur at reflection and by scatter as the beam passes through ocular tissue. Depolarization may be low in normal eyes, but could be problematic in various ocular pathologies. Although depolarization within the eye can occur anywhere relative to the birefringent structures, any system of polarization components can be replaced by a diattenuator, a retarder, and a depolarizer in series [8

8. S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996). [CrossRef]

]. Because the eye at long wavelengths exhibits low diattenuation [9

9. R. A. Bone, “The role of macular pigment in the detection of polarized light,” Vision Res. 20, 213–220 (1980). [CrossRef] [PubMed]

,10

10. A. W. Dreher, K. Reiter, and R. N. Weinreb, “Spatially resolved birefringence of the retinal nerve-fiber layer assessed with a retinal laser ellipsometer,” Appl. Opt. 31, 3730–3735 (1992). [CrossRef] [PubMed]

], only a depolarizer and a retarder were used in this model. The goal of an SLP measurement is to determine δ. In the case of an eye with a well-compensated anterior segment, δ is attributed to the RNFL and is considered to be proportional to its thickness [5

5. Q. Zhou and R. N. Weinreb, “Individualized compensation of anterior segment birefringence during scanning laser polarimetry,” Invest. Ophthalmol. Visual Sci. 43, 2221–2228 (2002).

,11

11. R. N. Weinreb, A. W. Dreher, A. Coleman, H. Quigley, B. Shaw, and K. Reiter, “Histopathologic validation of Fourier-ellipsometry measurements of retinal nerve fiber layer thickness,” Arch. Ophthalmol. 108, 557–60 (1990). [CrossRef] [PubMed]

].

An SLP measurement (Fig. 1) uses linearly polarized light from polarizer P(θ). Light returning from the eye is separated into two channels by linear analyzers A(θ) polarized parallel and A(θ+90°) polarized perpendicular to the incident beam. The plane of polarization and the two analyzers rotate through 90° to produce a series of images with different input polarization states. (For simplicity the scanning system that produces an image is not shown, but its location is indicated in Fig. 1 by vertical dashed lines. The analysis applies to each image pixel separately and, as will be presented in a later section, also to averages across pixels.)

Fig. 2. Theoretical outputs I × and I P vary sinusoidally with θ.
  • F 0: Mean of I ×
  • P ave: Mean of I p
  • F 1: Amplitude of the sinusoidal output component common to both channels

Mueller calculus [12

12. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, Inc., New York, 1990).

] expresses the system in Fig. 1 as S out = M A M R M D M P S in, where S in and S out are the input and output Stokes vectors, respectively, and M X is the Mueller matrix of optical element X. The Stokes vector after the depolarizer (M D M P S in) can be split into two components, one completely polarized with intensity K and one completely depolarized with intensity 2d [12

12. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, Inc., New York, 1990).

]. Solving each channel and recognizing that intensity is given by the first element of a Stokes vector yields

I×=d+F1[1cos4(θRθ)]
(1)

and

IP=d+KF1[1cos4(θRθ)]
(2)

where

F1=K4(1cosδ).
(3)

The terms in Eqs. (1) and (2) may be understood as follows. The depolarized component passes the retarder unchanged to produce constant intensity d at the output of each analyzer. The polarized component passes through the retarder and becomes elliptically polarized, still with intensity K, but with azimuth and ellipticity that vary with θ, θ R, and δ. The components of the elliptically polarized beam are sampled by the analyzers to produce sinusoidal intensity variations that go through one period for 90° of rotation (Fig. 2) [1

1. A. W. Dreher and K. Reiter, “Scanning laser polarimetry of the retinal nerve fiber layer,” Proc. of SPIE 1746, 34–41 (1992). [CrossRef]

]. The minimum of I × and the maximum of I p occur when the polarization is aligned with either the slow or fast axis of R (θ = θ R,θ R±90°). The completely polarized component contributes zero to the minimum of I × and K to the maximum of I p.

Fig. 3. (785 KB) Movie of an image series of the optic disc and surrounding RNFL from a normal right eye. The movie runs at 40% of actual speed. Small shifts were due to residual image misalignment. The bright artifacts near the center were due to internal reflections in the instrument. The contrast of the crossed channel was enhanced to better reveal the interaction of the linearly polarized incident beam with the RNFL birefringence. This interaction forms a four-armed cross that spins as the plane of polarization rotates. The dark arms of the cross correspond to the minima of I ×. The circle encloses the pixel from which the data in Fig. 4 were extracted.

3. Methods

SLP images were obtained with a GDxTM Nerve Fiber Analyzer (Laser Diagnostic Technologies, Inc., San Diego, CA) modified to incorporate a variable corneal compensator [5

5. Q. Zhou and R. N. Weinreb, “Individualized compensation of anterior segment birefringence during scanning laser polarimetry,” Invest. Ophthalmol. Visual Sci. 43, 2221–2228 (2002).

]. The instrument uses a 785 nm laser diode, fixed polarizers, and beam splitters to implement the system in Fig. 1. A rotating half-wave plate, driven by a stepper motor and placed in a portion of the optical path common to both the incident and reflected beams, serves as a polarization rotator [6

6. A. W. Dreher and K. Reiter, inventors, Laser Diagnostic Technologies, Inc., assignee. “Retinal eye disease diagnostic system,” United States Patent No. 5,303,709, April 19, 1994.

,7

7. K. Reiter and A. W. Dreher, inventors, Laser Diagnostic Technologies, Inc., assignee. “Eye examination apparatus employing polarized light probe,” United States Patent No. 5,787,890, August 4, 1998.

]. A scanning system synchronized with the polarization steps produces one image frame per step. Analog-to-digital (A/D) conversion of the outputs of the two channels yields two digital images per frame. Twenty steps are used to cover the range from 0°-90°, resulting in a series of 40 images, 20 from the parallel channel and 20 from the crossed channel. Software in the instrument brings the images into register by using the prominent blood vessel pattern in the parallel images and then writes the series to a computer file. Subsequent analyses were performed in MATLAB (The MathWorks, Natick, MA).

4. SLP measurement data

Fig. 4. Data from the single pixel at the center of the red circle in Fig. 3 (dots) and the smooth curves used to characterize them (lines). I × (upper graph) was approximated with the first and second terms of its Fourier series (mean and fundamental); I p was described by its mean (P ave). (Note: Because I ×(θ) is theoretically the same at 0° and 90°, the average of images number 1 and 20 was used for the value at 0° when calculating the Fourier series.)

5. Analysis of SLP data

5.1 Previous analysis algorithms

The SLP data in Fig. 4 reveal a number of ways in which previously published analysis algorithms are not optimum. First, these algorithms neglect depolarization, i.e., to determine δ they implicitly assume that the maximum of I × equals 2F 1 (d = 0) [1

1. A. W. Dreher and K. Reiter, “Scanning laser polarimetry of the retinal nerve fiber layer,” Proc. of SPIE 1746, 34–41 (1992). [CrossRef]

]. Second, several algorithms choose the maximum for each pixel as the highest value reached by I × [1

1. A. W. Dreher and K. Reiter, “Scanning laser polarimetry of the retinal nerve fiber layer,” Proc. of SPIE 1746, 34–41 (1992). [CrossRef]

,6

6. A. W. Dreher and K. Reiter, inventors, Laser Diagnostic Technologies, Inc., assignee. “Retinal eye disease diagnostic system,” United States Patent No. 5,303,709, April 19, 1994.

,7

7. K. Reiter and A. W. Dreher, inventors, Laser Diagnostic Technologies, Inc., assignee. “Eye examination apparatus employing polarized light probe,” United States Patent No. 5,787,890, August 4, 1998.

], a measure that, to avoid missing the maximum, requires the acquisition of images at many polarization angles and that, because it is derived from a single image, is subject to the full standard deviation of the instrument noise in the crossed channel, including the quantization error due to A/D conversion. Third, some algorithms individually normalize each point in I × by the sum I × + I p, a quantity that ideally remains constant as P rotates (see Fig. 2) and that, in principle, can be used to compensate for image-to-image intensity variations that might result from accommodation or eye movements [13

13. Q. Zhou, inventor, Laser Diagnostic Technologies, Inc., assignee. “System and method for determining birefringence of anterior segment of a patient’s eye,” United States Patent No. 6,356,036, March 12, 2002.

]. Unfortunately, although the expected sinusoidal variation of I × is clearly seen, the complementary sinusoid in I p is obscured by random variation that is not shared with I ×. This noise probably represents small changes in the optical path that occur with eye movements. Any normalization that uses individual values of I p causes this additional noise to enter directly into the calculation of δ. Finally, algorithms that depend on single measurements provide no way to estimate the uncertainty in the derived value.

5.2 Improved analysis algorithm

A comparison of the theoretical output of the SLP measurement model (Fig. 2) and the actual SLP data (Fig. 4) suggested an improved algorithm for measuring R, as follows. The parameters F 0 and F 1 were obtained by Fourier analysis of I ×, with F 0 the first term of the Fourier series (mean of I ×) and F 1 the amplitude of the fundamental. The minimum of I × (and thus the fast or slow axis of R) was determined from the phase of the fundamental. The contribution from depolarization was assumed, from Fig. 2, to be d = F 0-F 1. The average of I p (i.e., P ave) provided a measure of incident intensity that smoothed the variation in the parallel channel (Fig. 4). The polarized amplitude K was found by assuming that I × and I P behaved as shown in Fig. 2, i.e., K = P ave + F 1-d. Rearranging Eq. (3) yielded

δ=cos1(14F1Pave+2F1F0)
(4)

as the final expression for retardance.

Fig. 5. Retardance image derived from the image series in Fig. 3 by applying Eq. (4). Calculated retardance (nm) is shown for five positions (crosses) around the optic disc. The crosses were aligned with the calculated axis of retardance. In a typical clinical application this quantitative image would serve as input to more extensive analyses of RNFL integrity.

Figure 5 shows the retardance image derived from the image series of Fig. 3 by applying Eq. (4) separately to each pixel. Retardance values are given for five arbitrary pixels, marked with crosses. The arms of each cross show the calculated axes of the birefringence, including the 90° ambiguity inherent in using linearly polarized light to measure birefringence. At each pixel one arm of the cross corresponded closely with the known anatomic orientation of axons in the RNFL [14

14. S. L. Polyak, The Retina (The University of Chicago Press, Chicago, 1941).

,15

15. S. C. Pollock and N. R. Miller, “The retinal nerve fiber layer,” Int. Ophthalmol. Clin. 26, 201–221 (1986). [CrossRef] [PubMed]

], as expected if the slow axis of birefringence lies along the axon direction [10

10. A. W. Dreher, K. Reiter, and R. N. Weinreb, “Spatially resolved birefringence of the retinal nerve-fiber layer assessed with a retinal laser ellipsometer,” Appl. Opt. 31, 3730–3735 (1992). [CrossRef] [PubMed]

].

The algorithm that leads to Eq. (4) has several advantages over existing methods for measuring the properties of R. First, because they are derived from data obtained over the entire set of images and thus represent the action of R on the entire series of input polarizations, the three parameters F 0, F 1 and P ave are less susceptible to noise in I × and I p than parameters based on any single value. Second, the algorithm explicitly recognizes depolarization of the measuring beam and attempts to exclude it from the retardance calculation. This may improve measurement accuracy in eyes with various light-scattering pathologies. Third, the Fourier fit to I × can determine the parameters F 0 and F 1 with many fewer data points (i.e., polarization steps) than have been used previously. This could translate into a faster measurement that is less affected by eye movements and blinking.

5.3 Estimating uncertainty

Deriving birefringence parameters from the entire data set rather than from single points also provided a means to estimate the error in the result. Although the uncertainty in δ can be used to determine the probability that δ is different from zero, a simpler approach was to neglect the error in P ave and examine the significance of the Fourier fit to I ×. For each pixel the fraction of variance explained by the Fourier fit (r 2) was tested for significance with the F-statistic

F2,n3=r2/2(1r2)/(n3),
(5)

where n was the total number of data points [16

16. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (WCB/McGraw-Hill, Boston, 1992).

]. The degrees of freedom reflect that the fit involved three parameters and used two of them, the amplitude (F 1) and phase of the Fourier fundamental, to explain the variance around the third, the mean F 0. An example of significance testing is shown in Fig. 6A, where non-significant pixels (p > 0.01; r 2 < 0.42 for n = 20 ) are colored red. In analyses aimed at detecting RNFL loss such pixels can be treated separately (e.g., ignored or set to zero) from pixels that clearly contain retardance information.

Fig. 6. Significance testing and image averaging. (A) Retardance image of Fig. 5 with the nonsignificant pixels (p > 0.01) colored red. (B) Retardance image formed by first smoothing each image in the image series of Fig. 3 with a 5×5 averaging window before applying Eq. (4).

5.4 Averaging to reduce noise

Given the underlying orderly anatomy of the RNFL [14

14. S. L. Polyak, The Retina (The University of Chicago Press, Chicago, 1941).

,15

15. S. C. Pollock and N. R. Miller, “The retinal nerve fiber layer,” Int. Ophthalmol. Clin. 26, 201–221 (1986). [CrossRef] [PubMed]

], it seemed likely that the birefringence of adjacent areas was similar. This suggested that spatial averaging could improve the signal-to-noise ratio of the retardance calculation and increase the sensitivity to weak birefringence (at the expense of spatial resolution). Averaging pixels in the calculated retardance image is not appropriate because it ignores axis variations between pixels; therefore, each image in a series was smoothed separately before applying Eq. (4). Figure 6B shows the result of applying a 5×5 uniform square averaging filter to the image series of Fig. 3, then calculating retardance as in Fig. 5, followed by significance testing as in Fig. 6A. Note the decreased number of non-significant pixels between Figs. 6A and 6B.

5.5 Extension of algorithm to continuously rotating polarization components

The considerations above can be extended to other methods of imaging ellipsometry that use rotating components. For example, if imaging ellipsometry by the rotating compensator method [17

17. P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978). [CrossRef]

,18

18. R. M. A. Azzam, inventor, The Board of Regents of the University of Nebraska, assignee. “Polarimeter,” United States Patent No. 4,306,809, Dec. 22, 1981.

] were implemented with continuous rotation, the higher Fourier terms used in the analysis (up to the 12th term) would be especially susceptible to the jitter introduced by image registration. These terms can be extracted from the long, non-evenly spaced data series by the Lomb-Scargle algorithm [19

19. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannerly, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1992).

], which is essentially a least squares procedure, or by interpolation onto an evenly spaced series.

6. Determination of anterior segment birefringence

6.1 Published “bow tie” method

An essential feature of the quantitative RNFL retardance image in Fig. 5 is that anterior segment birefringence was substantially neutralized before the image series was acquired. Neutralization was achieved by means of a variable retarder in the optical path of the polarimeter with its retardance set equal to the corneal retardance and its fast axis oriented parallel to the slow axis of the cornea [5

5. Q. Zhou and R. N. Weinreb, “Individualized compensation of anterior segment birefringence during scanning laser polarimetry,” Invest. Ophthalmol. Visual Sci. 43, 2221–2228 (2002).

]. In order to set this “corneal compensator”, one must first measure the axis and retardance of the anterior segment. Zhou and Weinreb present a method that uses the radial birefringence of Henle’s fiber layer in the macula [5

5. Q. Zhou and R. N. Weinreb, “Individualized compensation of anterior segment birefringence during scanning laser polarimetry,” Invest. Ophthalmol. Visual Sci. 43, 2221–2228 (2002).

,13

13. Q. Zhou, inventor, Laser Diagnostic Technologies, Inc., assignee. “System and method for determining birefringence of anterior segment of a patient’s eye,” United States Patent No. 6,356,036, March 12, 2002.

].

Figure 7 illustrates the basis for their method; it shows SLP images of a normal macula obtained without corneal compensation (i.e., with the corneal compensator set to zero retardance). The P ave image had a normal pattern of blood vessels surrounding the avascular fovea. The retardance image, however, had a “bow tie” pattern centered on the fovea, even though normal macular anatomy is approximately symmetric around the fovea. The bright arms of the bow tie occurred where the slow axis of Henle’s fiber layer and the slow axis of the cornea were parallel (retardances added). Similarly, the dark arms occurred where the slow axes were perpendicular (retardances cancel). Assuming that Henle’s fiber layer has circular symmetry permits direct readout of the corneal axis. Analysis of the retardance profile on a circle around the fovea provides a measure of corneal retardance [5

5. Q. Zhou and R. N. Weinreb, “Individualized compensation of anterior segment birefringence during scanning laser polarimetry,” Invest. Ophthalmol. Visual Sci. 43, 2221–2228 (2002).

].

6.2 New “screen” method

Although robust in normal eyes, the use of macular birefringence as a means to measure anterior segment birefringence cannot work in all clinical situations. Many patients with glaucoma or suspected glaucoma (the group for which RNFL assessment is intended) also have macular disease (maculopathy) that can disrupt or destroy Henle’s fiber layer and distort or eliminate macular birefringence. Another method for measuring anterior segment birefringence would be desirable for these patients. Such a method was suggested by the patient illustrated in Fig. 8.

Fig. 7. SLP images of a normal macula (right eye) obtained without corneal compensation. The schematic diagram depicts a theoretical specimen comprising a fixed axis linear retarder representing the cornea (C) and a radial axis linear retarder representing Henle’s fiber layer (R), with reflection from a polarization preserving reflector (PPR) at the back of the eye. The arrows show the slow axis of R. The scanning beam is shown as a single dotted line but is understood to produce an image of R+PPR. In the retardance image, a bow tie pattern centered at the fovea arises from the interaction of the corneal and macular birefringence.
Fig. 8. SLP images of a diseased macula (right eye) obtained without corneal compensation. The diagram postulates the absence of the radial retarder, causing PPR to function as a screen onto which the corneal properties were projected. The crosses in the retardance image show the calculated retardance and axis at six locations on the fundus.

Figure 8 shows uncompensated SLP images of the macular area of an eye with cystoid macular edema, an epiretinal membrane, and poor ability to fixate on a target. No macular bow tie pattern was apparent in the retardance image. The six marked pixels show that retardance was high and similarly oriented over the entire image. Furthermore, the axes of the retardance were everywhere similar to the corneal axis measured independently by a method based on Purkinje images [3

3. D. S. Greenfield, R. W. Knighton, and X. R. Huang, “Effect of corneal polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,” Am. J. Ophthalmol. 129, 715–722 (2000). [CrossRef] [PubMed]

]. This and similar images led to the working hypothesis that, in the absence of macular birefringence, the fundus of the eye acts as a polarization-preserving screen that displays the anterior segment birefringence.

In order to extract birefringence information from the irregularities of the screen itself, the averaging procedure described in the preceding section was applied to a large portion of each image in the image series to produce the results shown in Fig. 9. The highly smoothed outputs I × and I p (Fig. 9) were then used in Eq. (4) to compute retardance, which was ascribed entirely to a double pass through the anterior segment. The corneal compensator, therefore, would be set to one-half this value. The choice of slow axis was based on the knowledge that the slow axis of most corneas is oriented nasally downward [3

3. D. S. Greenfield, R. W. Knighton, and X. R. Huang, “Effect of corneal polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,” Am. J. Ophthalmol. 129, 715–722 (2000). [CrossRef] [PubMed]

,20

20. R. W. Knighton and X. R. Huang, “Linear birefringence of the central human cornea,” Invest. Ophthalmol. Visual Sci. 43, 82–86 (2002).

]. The success of the screen method in normal subjects and patients with maculopathy is the subject of another paper (H. Bagga, D.S. Greenfield, and R.W. Knighton, submitted).

Fig. 9. Averaging over a large area (43,847 pixels; shaded gray) of the macula in Fig. 8 produced a single value for the projected birefringence. A mask excluded from the average the central area of instrumental artifact and also pixels in the parallel channel that were saturated. The cross, placed arbitrarily, applies to the entire shaded area. The graphs, in the same format as Fig. 4, show the very smooth averaged outputs of the crossed and parallel channels.

7. Conclusion

A simple model of SLP suggested an improved formula for calculating the retardance of the RNFL. The three parameters in the formula, F 0, F 1, and P ave, were derived from the entire set of data in an image series, which reduced the variability of the result. This improvement can be translated into increased measurement sensitivity and/or decreased time for data collection. The formula also explicitly accounted for depolarization in the optical path. The uncertainty of the calculated retardance was estimated and an appropriate averaging strategy for reducing this uncertainty was demonstrated. Finally, averaging over a large area of macular images provides a new method for determining anterior segment birefringence in eyes with macular pathology.

Acknowledgements

Laser Diagnostic Technologies, Inc., generously provided a modified GDxTM Nerve Fiber Analyzer for this work. William J. Feuer and Qienyuan Zhou provided useful discussions. The work was supported by a grant from the National Eye Institute, Bethesda, MD.

References and links

1.

A. W. Dreher and K. Reiter, “Scanning laser polarimetry of the retinal nerve fiber layer,” Proc. of SPIE 1746, 34–41 (1992). [CrossRef]

2.

A. W. Dreher and K. Reiter, “Retinal laser ellipsometry - a new method for measuring the retinal nerve-fiber layer thickness distribution,” Clin. Vision Sci. 7, 481–488 (1992).

3.

D. S. Greenfield, R. W. Knighton, and X. R. Huang, “Effect of corneal polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,” Am. J. Ophthalmol. 129, 715–722 (2000). [CrossRef] [PubMed]

4.

R. W. Knighton, X. R. Huang, and D. S. Greenfield, “Analytical model of scanning laser polarimetry for retinal nerve fiber layer assessment,” Invest. Ophthalmol. Visual Sci. 43, 383–392 (2002).

5.

Q. Zhou and R. N. Weinreb, “Individualized compensation of anterior segment birefringence during scanning laser polarimetry,” Invest. Ophthalmol. Visual Sci. 43, 2221–2228 (2002).

6.

A. W. Dreher and K. Reiter, inventors, Laser Diagnostic Technologies, Inc., assignee. “Retinal eye disease diagnostic system,” United States Patent No. 5,303,709, April 19, 1994.

7.

K. Reiter and A. W. Dreher, inventors, Laser Diagnostic Technologies, Inc., assignee. “Eye examination apparatus employing polarized light probe,” United States Patent No. 5,787,890, August 4, 1998.

8.

S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996). [CrossRef]

9.

R. A. Bone, “The role of macular pigment in the detection of polarized light,” Vision Res. 20, 213–220 (1980). [CrossRef] [PubMed]

10.

A. W. Dreher, K. Reiter, and R. N. Weinreb, “Spatially resolved birefringence of the retinal nerve-fiber layer assessed with a retinal laser ellipsometer,” Appl. Opt. 31, 3730–3735 (1992). [CrossRef] [PubMed]

11.

R. N. Weinreb, A. W. Dreher, A. Coleman, H. Quigley, B. Shaw, and K. Reiter, “Histopathologic validation of Fourier-ellipsometry measurements of retinal nerve fiber layer thickness,” Arch. Ophthalmol. 108, 557–60 (1990). [CrossRef] [PubMed]

12.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, Inc., New York, 1990).

13.

Q. Zhou, inventor, Laser Diagnostic Technologies, Inc., assignee. “System and method for determining birefringence of anterior segment of a patient’s eye,” United States Patent No. 6,356,036, March 12, 2002.

14.

S. L. Polyak, The Retina (The University of Chicago Press, Chicago, 1941).

15.

S. C. Pollock and N. R. Miller, “The retinal nerve fiber layer,” Int. Ophthalmol. Clin. 26, 201–221 (1986). [CrossRef] [PubMed]

16.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (WCB/McGraw-Hill, Boston, 1992).

17.

P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978). [CrossRef]

18.

R. M. A. Azzam, inventor, The Board of Regents of the University of Nebraska, assignee. “Polarimeter,” United States Patent No. 4,306,809, Dec. 22, 1981.

19.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannerly, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1992).

20.

R. W. Knighton and X. R. Huang, “Linear birefringence of the central human cornea,” Invest. Ophthalmol. Visual Sci. 43, 82–86 (2002).

OCIS Codes
(170.3880) Medical optics and biotechnology : Medical and biological imaging
(170.4460) Medical optics and biotechnology : Ophthalmic optics and devices
(260.1440) Physical optics : Birefringence

ToC Category:
Research Papers

History
Original Manuscript: August 15, 2002
Revised Manuscript: October 11, 2002
Published: October 21, 2002

Citation
Robert Knighton and Xiang-Run Huang, "Analytical methods for scanning laser polarimetry," Opt. Express 10, 1179-1189 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-21-1179


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References

  1. A. W. Dreher and K. Reiter, "Scanning laser polarimetry of the retinal nerve fiber layer," Proc. of SPIE 1746, 34-41 (1992). [CrossRef]
  2. A. W. Dreher and K. Reiter, "Retinal laser ellipsometry - a new method for measuring the retinal nervefiber layer thickness distribution," Clin. Vision Sci. 7, 481-488 (1992).
  3. D. S. Greenfield, R. W. Knighton and X. R. Huang, "Effect of corneal polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry," Am. J. Ophthalmol. 129, 715-722 (2000). [CrossRef] [PubMed]
  4. R. W. Knighton, X. R. Huang and D. S. Greenfield, "Analytical model of scanning laser polarimetry for retinal nerve fiber layer assessment," Invest. Ophthalmol. Visual Sci. 43, 383-392 (2002).
  5. Q. Zhou and R. N. Weinreb, "Individualized compensation of anterior segment birefringence during scanning laser polarimetry," Invest. Ophthalmol. Visual Sci. 43, 2221-2228 (2002).
  6. A. W. Dreher and K. Reiter, inventors, Laser Diagnostic Technologies, Inc., assignee. "Retinal eye disease diagnostic system," United States Patent No. 5,303,709, April 19, 1994.
  7. K. Reiter and A. W. Dreher, inventors, Laser Diagnostic Technologies, Inc., assignee. "Eye examination apparatus employing polarized light probe," United States Patent No. 5,787,890, August 4, 1998.
  8. S.-Y. Lu and R. A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106-1113 (1996). [CrossRef]
  9. R. A. Bone, "The role of macular pigment in the detection of polarized light," Vision Res. 20, 213-220 (1980). [CrossRef] [PubMed]
  10. A. W. Dreher, K. Reiter and R. N. Weinreb, "Spatially resolved birefringence of the retinal nerve-fiber layer assessed with a retinal laser ellipsometer," Appl. Opt. 31, 3730-3735 (1992). [CrossRef] [PubMed]
  11. R. N. Weinreb, A. W. Dreher, A. Coleman, H. Quigley, B. Shaw and K. Reiter, "Histopathologic validation of Fourier-ellipsometry measurements of retinal nerve fiber layer thickness," Arch. Ophthalmol. 108, 557-60 (1990). [CrossRef] [PubMed]
  12. D. S. Kliger, J. W. Lewis and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, Inc., New York, 1990).
  13. Q. Zhou, inventor, Laser Diagnostic Technologies, Inc., assignee. "System and method for determining birefringence of anterior segment of a patient's eye," United States Patent No. 6,356,036, March 12, 2002.
  14. S. L. Polyak, The Retina (The University of Chicago Press, Chicago, 1941).
  15. S. C. Pollock and N. R. Miller, "The retinal nerve fiber layer," Int. Ophthalmol. Clin. 26, 201-221 (1986). [CrossRef] [PubMed]
  16. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (WCB/McGraw-Hill, Boston, 1992).
  17. P. S. Hauge, "Mueller matrix ellipsometry with imperfect compensators," J. Opt. Soc. Am. 68, 1519-1528 (1978). [CrossRef]
  18. R. M. A. Azzam, inventor, The Board of Regents of the University of Nebraska, assignee. "Polarimeter," United States Patent No. 4,306,809, Dec. 22, 1981.
  19. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannerly, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1992).
  20. R. W. Knighton and X. R. Huang, "Linear birefringence of the central human cornea," Invest. Ophthalmol. Visual Sci. 43, 82-86 (2002).

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