Abstract

Form-biattenuance (Δχ) in biological tissue arises from anisotropic light scattering by regularly oriented cylindrical fibers and results in a differential attenuation (diattenuation) of light amplitudes polarized parallel and perpendicular to the fiber axis (eigenpolarizations). Form-biattenuance is complimentary to form-birefringence (Δn) which results in a differential delay (phase retardation) between eigenpolarizations. We justify the terminology and motivate the theoretical basis for form-biattenuance in depth-resolved polarimetry. A technique to noninvasively and accurately quantify form-biattenuance which employs a polarization-sensitive optical coherence tomography (PS-OCT) instrument in combination with an enhanced sensitivity algorithm is demonstrated on ex vivo rat tail tendon (mean Δχ=5.3·10-4, N=111), rat Achilles tendon (Δχ=1.3·10-4, N=45), chicken drumstick tendon (Δχ=2.1·10-4, N=57), and in vivo primate retinal nerve fiber layer (Δχ=0.18·10-4, N=6). A physical model is formulated to calculate the contributions of Δχ and Δn to polarimetric transformations in anisotropic media.

© 2005 Optical Society of America

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References

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Abh. Math.-Phys. Klasse Koniglich Sachsi (1)

O. Wiener, "Die Theorie des Mischkorpers fur das Feld der stationaren Stromung," Abh. Math.-Phys. Klasse Koniglich Sachsischen Des. Wiss. 32, 509-604 (1912).

Acta. Crystallogr. (1)

W. L. Bragg and A. B. Pippard, "The form birefringence of macromolecules," Acta. Crystallogr. 6, 865-867 (1953).
[CrossRef]

Appl. Opt. (1)

Biophys. J. (1)

R. Oldenbourg and T. Ruiz, "Birefringence of macromolecules: Wiener's theory revisited, with applications to DNA and tobacco mosaic virus," Biophys. J. 56, 195-205 (1989).
[CrossRef] [PubMed]

Connect. Tissue Res. (2)

R. W. D. Rowe, "The structure of rat tail tendon," Connect. Tissue Res. 14, 9-20 (1985).
[CrossRef] [PubMed]

J. Kastelic, A. Galeski and E. Baer, "The multicomposite structure of tendon," Connect. Tissue Res. 6, 11-23 (1978).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

B. Huttner, C. Geiser and N. Gisin, "Polarization-Induced Distortions in Optical Fiber Networks with Polarization-Mode Dispersion and Polarization-Dependent Losses," IEEE J. Sel. Top. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

Int. J. Biol. Macromol. (1)

S. P. Nicholls, L. J. Gathercole, A. Keller and J. S. Shah, "Crimping in rat tail tendon collagen: morphology and transverse mechanical anisotropy," Int. J. Biol. Macromol. 5, 283-88 (1983).
[CrossRef]

J. Biomed. Opt. (2)

B. H. Park, C. Saxer, T. Chen, S. M. Srinivas, J. S. Nelson and J. F. de Boer, "In vivo burn depth determination by high-speed fiber-based polarization sensitive optical coherence tomography," J. Biomed. Opt. 6, 474-479 (2001).
[CrossRef] [PubMed]

S. Jiao and L. V. Wang, "Jones-matrix imaging of biological tissues with quadruple-channel optical coherence tomography," J. Biomed. Opt. 7, 350-358 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

Lasers in Surgery and Medicine (1)

G. Vargas, E. K. Chan, J. K. Barton, H. G. Rylander and A. J. Welch, "Use of an agent to reduce scattering in skin," Lasers in Surgery and Medicine 24, 133-141 (1999).
[CrossRef] [PubMed]

Opt. Eng. (1)

R. A. Chipman, "Polarization analysis of optical systems," Opt. Eng. 28, 90-99 (1989).

Opt. Express (2)

Opt. Lett. (3)

Pure App. Chem. (1)

J. W. Verhoeven, "Glossary of terms used in photochemistry," Pure App. Chem. 68, 2228 (1996).

Other (5)

R. M. Craig, S. L. Gilbert and P. D. Hale, "Accurate Polarization Dependent Loss Measurement and Calibration Standard Development," Symposium on Optical Fiber Measurements NIST Special Publication 930, 5-8 (1998).

J. Park, N. J. Kemp, H. N. Zaatari, H. G. Rylander III, and T. E. Milner, Dept. of Biomedical Engineering, University of Texas, 1 University Station, #C0800, Austin, TX 78712 are preparing a manuscript to be called �??Differential geometry of the normalized Stokes vector trajectories in anisotropic media.�??

H. G. Rylander, N. J. Kemp, J. Park, H. N. Zaatari and T. E. Milner, "Birefringence of the primate retinal nerve fiber layer," Exp. Eye Res. In Press, (2005).

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd Ed. (Cambridge University Press, United Kingdom, 1992).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1959).

Supplementary Material (4)

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» Media 2: MOV (1144 KB)     
» Media 3: MOV (1184 KB)     
» Media 4: MOV (1272 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

Noise-free model polarization arc [P(z), black] and eigen-axis ( β ^ , green) on the Poincaré sphere (left) and corresponding normalized Stokes parameters [Q(z), U(z), V(z)] vs. depth (right). Polarizations at the front [P(0)] and rear [Pz)] specimen surfaces are represented by red and blue dots respectively. (a) Pure form-birefringence causes rotation of P(z) around β ^ in plane Π1 which is normal to β ^ . (b) Pure form-biattenuance causes translation of P(z) toward β ^ in plane Π2. (c) Combined birefringence and biattenuance cause P(z) to spiral toward β ^ and orthogonal planes Π1 and Π2 are therefore functions of depth [Π1(z) and Π2(z)]. Movies showing 3D nature of Poincaré sphere: a (1.21 MB); b (1.17 MB); c (1.21 MB).

Fig. 2.
Fig. 2.

Depth-resolved polarization data [S 1(z), orange] and associated noise-free model polarization arc [P 1(z), black] and eigen-axis ( β ^ , green) determined by the multistate nonlinear algorithm in rat tail tendon are shown on the Poincaré sphere (left). Corresponding normalized Stokes parameters [Q(z), U(z), V(z)] and associated nonlinear fits (black) are shown on the right. A single incident polarization state (m=1) is shown for simplicity. (a) S m (z) for tendon with relatively high form-biattenuance (Δχ=8.0·10-4) collapses toward β ^ faster than that for (b) tendon with relatively low form-biattenuance (Δχ=3.0·10-4).

Fig. 3.
Fig. 3.

S1(z) (orange) and associated P 1(z) (black) and β ^ (green) determined by the multistate nonlinear algorithm in rat Achilles tendon are shown on the Poincaré sphere (left). A single incident polarization state (m=1) is shown for simplicity. Form-biattenuance in this specimen (Δχ=3.2 °/100µm) is lower than for specimens shown in Figures 2(a) and 2(b) and spiral collapse toward β ^ is correspondingly slower.

Fig. 4.
Fig. 4.

S m (z) (colored) and associated P m (z) (black) and β ^ (green) for in vivo primate RNFL shown on the Poincaré sphere for M=6 (left). Corresponding normalized Stokes parameters [Q(z), U(z), V(z)] and associated nonlinear fits (black) are shown for a single incident polarization state (m=1, right). Notice the RNFL exhibits only a fraction of a wave of phase retardation compared to multiple waves exhibited by tendon specimens in Figures 2(a), 2(b), and 6. Movie showing 3D nature of Poincaré sphere (1.30 MB).

Fig. 5.
Fig. 5.

A model for form-biattenuance consisting of alternating anisotropic and isotropic layers.

Equations (28)

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J = [ exp ( ( Δ χ + i Δ n ) π Δ z λ 0 ) 0 0 exp ( ( Δ χ i Δ n ) π Δ z λ 0 ) ]
= [ ξ 1 exp ( i arg ( ξ 1 ) ) 0 0 ξ 2 exp ( i arg ( ξ 2 ) ) ] ,
J = [ ξ 1 exp ( i δ 2 ) 0 0 ξ 2 exp ( i δ 2 ) ] .
D = T 1 T 2 T 1 + T 2 = ξ 1 2 ξ 2 2 ξ 1 2 + ξ 2 2 0 D 1 ,
Δ n = λ 0 2 π δ Δ z = n s n f ,
Δ χ = χ s χ f ,
ε = 2 π λ 0 Δ z Δ χ ,
J = [ exp ( ε + i δ 2 ) 0 0 exp ( ε i δ 2 ) ] ,
D = e ε e ε e ε + e ε = tanh ( ε ) .
d P ( z ) dz + ( P ( z ) × β re ) + P ( z ) × ( P ( z ) × β im ) = 0 ,
β = β re + i β im = ( β re + i β im ) β ̂ .
β = β re + i β im = 2 π λ 0 ( Δ n + i Δ χ ) ,
2 δ = 2 β re Δ z ,
γ ( z ) = 2 tan 1 [ tan ( γ ( 0 ) 2 ) exp ( 2 β im z ) ] 0 γ < π ,
2 ε = 2 β im Δ z .
PSNR = l arc σ speckle ,
d l arc = 2 ( β re 2 + β im 2 ) 1 2 sin [ γ ( z ) ] dz ,
l arc = [ 1 + ( δ ε ) 2 ] 1 2 [ γ ( 0 ) γ ( Δ z ) ] .
l arc 2 ( δ 2 + ε 2 ) 1 2 sin [ γ ( 0 ) ] .
S m ( z ) = ( Q ( z ) U ( z ) V ( z ) ) = ( E h , m ( z ) 2 E v , m ( z ) 2 N A 2 E h , m ( z ) E v , m ( z ) cos [ Δ ϕ c , m ( z ) ] N A 2 E h , m ( z ) E v , m ( z ) sin [ Δ ϕ c , m ( z ) ] N A ) E h , m ( z ) 2 + E v , m ( z ) 2 N A .
W m ( z ) = ( E h , m ( z ) 2 E v , m ( z ) 2 2 E h , m ( z ) E v , m ( z ) cos [ Δ ϕ c , m ( z ) ] 2 E h , m ( z ) E v , m ( z ) sin [ Δ ϕ c , m ( z ) ] ) E h , m ( z ) 2 + E v , m ( z ) 2 N A .
R M = m = 1 M R o [ S m ( z j ) , W m ( z j ) ; 2 ε , 2 δ , β ̂ , P m ( 0 ) ] ,
R o = j = 1 J { W ( z j ) [ S ( z j ) P [ z j ; 2 ε , 2 δ , β ̂ , P ( 0 ) ] } 2 ,
n p 2 = ( h 1 a ) n f 2 + ( 1 h 1 a ) n w 2 and
n s 2 = n w 2 + ( h 1 a ) ( n f 2 n w 2 ) 1 + 1 2 ( 1 h 1 a ) ( n f 2 n w 2 n w 2 ) .
Δ n = h 1 h ( n p n s ) .
t p t s ( z ) = [ n p ( n s + n w ) 2 n s ( n p + n w ) 2 ] z h ,
Δ χ = λ 0 2 π h ln ( n p ( n s + n w ) 2 n s ( n p + n w ) 2 ) .

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