## Form-biattenuance in fibrous tissues measured with polarization-sensitive optical coherence tomography (PS-OCT)

Optics Express, Vol. 13, Issue 12, pp. 4611-4628 (2005)

http://dx.doi.org/10.1364/OPEX.13.004611

Acrobat PDF (722 KB)

### Abstract

Form-biattenuance (Δ*χ*) 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 (Δ*n*) 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 Δ*χ*=5.3·10^{-4}, *N*=111), rat Achilles tendon (Δ*χ*=1.3·10^{-4}, *N*=45), chicken drumstick tendon (Δ*χ*=2.1·10^{-4}, *N*=57), and in vivo primate retinal nerve fiber layer (Δ*χ*=0.18·10^{-4}, *N*=6). A physical model is formulated to calculate the contributions of Δ*χ* and Δ*n* to polarimetric transformations in anisotropic media.

© 2005 Optical Society of America

## 1. Introduction

1. C. K. Hitzenberger, E. Gotzinger, M. Sticker, M. Pircher, and A. F. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express **9**, 780–790 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-780. [CrossRef] [PubMed]

4. 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]

5. N. J. Kemp, J. Park, H. N. Zaatari, H. G. Rylander, and T. E. Milner, “High sensitivity determination of birefringence in turbid media using enhanced polarization-sensitive OCT,” J. Opt. Soc. Am. A **22**, 552–560 (2005). [CrossRef]

*ξ*

_{1}and

*ξ*

_{2}are the complex eigenvalues representing changes in amplitude and phase for orthogonal eigenpolarization states with free-space wavelength λ

_{0}propagating a distance Δ

*z*through the medium. Attenuation common to both eigenpolarizations does not affect the light polarization state and is neglected here.

*δ*, expressed in radians) between eigenpolarization states after propagation through the medium is the difference between the arguments of the eigenvalues,

*δ*=arg(

*ξ*

_{1})-arg(

*ξ*

_{2}), which allows simplification of the Jones matrix to

10. 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]

*diattenuation*(

*D*) given quantitatively by

*T*

_{1}and

*T*

_{2}are the intensity transmittances for the two orthogonal eigenpolarizations and the attenuation can be a consequence of

*either*anisotropic absorption

*or*anisotropic scattering of light out of the detected field.

*Birefringence*(Δ

*n*) is the phenomenon responsible for phase retardation (

*δ*) of light propagating a distance Δ

*z*in an anisotropic element and is given by

*n*and

_{s}*n*are the real-valued refractive indices experienced by the slow and fast eigenpolarizations, respectively. Note form-birefringence (Δ

_{f}*n*) is proportional to and given experimentally by the phase retardation-per-unit-depth (

*δ*/Δ

*z*).

*χs*and

*χf*are attenuation coefficients of the slow and fast eigenpolarizations. For absorbing (dichroic) media,

*χ*and

_{s}*χf*are simply imaginary-valued refractive indices.

14. J. F. de Boer, T. E. Milner, and J. S. Nelson, “Determination of the depth-resolved Stokes parameters of light backscattered from turbid media using Polarization Sensitive Optical Coherence Tomography,” Opt. Lett. **24**, 300–302 (1999). [CrossRef]

*relative-attenuation*(

*ε*) experienced by light propagating to a depth Δ

*z*in an anisotropic element as

*χ*) can now be meaningfully expressed on a relative-attenuation-per-unit- depth basis (

*ε*/Δ

*z*). Relative-attenuation (

*ε*) is the complimentary term to phase retardation [

*δ*, Eq. (4)], just as biattenuance (Δ

*χ*) is the complementary term to birefringence (Δ

*n*).

*D*≈

*ε*.

*δ*) is in the argument of an exponential [Eq. (7)] and therefore has units of radians, but is also commonly expressed in units of degrees (180·

*δ*/

*π*), fractions of waves (

*δ/2π*), or length (λ

_{0}

*·δ/2π*). Similarly, relative-attenuation (

*ε*) is in the argument of an exponential [Eq. (7)] and has units of radians. Regrettably, expression of relative-attenuation in units of radians (or degrees, fractions of waves, or length) is less intuitive than for phase retardation.

16. B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, “Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components,” Opt. Lett. **29**, 2512–2514 (2004). [CrossRef] [PubMed]

*D*/Δ

*z*=0.39/mm using a PS-OCT instrument with 1310 nm light. For their sample thickness (Δ

*z*=0.4 mm), the small-angle approximation is valid and

*ε*/Δ

*z*=0.39 rad/mm (Δ

*χ*=0.82·10

^{-4}). Unfortunately, reporting

*D*/Δ

*z*=0.39/mm mistakenly implies that a 3-mm-thick specimen of the same tendon would have diattenuation

*D*=1.17 [0≤

*D*≤1, Eq. (3)]. Jiao and Wang [4

4. 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]

*D*/Δ

*z*=0.13/mm for a sample with thickness Δ

*z*=0.3 mm and Todorovic et al [15

15. M. Todorovic, S. Jiao, L. V. Wang, and G. Stoica, “Determination of local polarization properties of biological samples in the presence of diattenuation by use of Mueller optical coherence tomography,” Opt. Lett. **29**, 2402–2404 (2004). [CrossRef] [PubMed]

*D*=0.1 for a sample with thickness Δ

*z*=1 mm. Again, the small-angle approximation is valid for these cases (

*ε*/Δ

*z*=0.13 rad/mm or Δ

*χ*=0.17·10

^{-4};

*ε*/Δ

*z*=0.1 rad/mm or Δ

*χ*=0.13·10

^{-4}), but for specimens with higher diattenuation,

*D*/Δ

*z*≠

*ε*/Δ

*z*[see Eq. (8)]. Expression of a specimen’s polarization-dependent attenuation in terms of form-biattenuance (Δ

*χ*) or corresponding relative-attenuation-per-unit-depth (

*ε*/Δ

*z*) overcomes all ambiguity associated with the nonlinear diattenuation-thickness relationship and no approximations are necessary in depth-resolved polarimetry.

*χ*) in highly scattering biological media. In Section 2, we give the trajectory of noise-free model depth-resolved normalized Stokes vector arcs on the Poincaré sphere for light propagation through media exhibiting both form-birefringence and form-biattenuance and give an expression for the polarimetric signal-to-noise ratio (PSNR). Section 3 describes a PS-OCT instrument and multistate nonlinear fitting algorithm used to estimate values of form-biattenuance (Δ

*χ*) given in Section 4 for a variety of specimens. In Sections 5 and 6, we discuss the expected uncertainty in measurements of Δ

*χ*, present a rudimentary physical model of form-biattenuance based on structural properties, compare measured values given here with previously reported results, and conclude with a discussion of this work’s relevance in biomedical diagnostics.

## 2. Theory

*Q*(

*z*),

*U*(

*z*),

*V*(

*z*)] which trace a noise-free model polarization arc [P(

*z*)] on the Poincaré sphere surface (Fig. 1). Trajectory of the P(

*z*) arc is governed by a vector differential equation [17

17. 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.”

*is a property of the medium and is defined as*β ⇀

*β*and

_{re}*β*are real and imaginary parts of the complex differential wavenumber

_{im}*β*,

*is a unit-vector on the Poincaré sphere representing the fast eigenpolarization state which propagates through the medium without transformation.*β ^

**P**(

*z*) on the Poincaré sphere represents the entire transformation experienced by polarized light propagating into and back from the medium (double-pass) and can be found by direct integration of Eq. (9) in the mutually orthogonal planes Π

_{1}(

*z*) and Π

_{2}(

*z*). The cumulative angle between Π

_{2}(0) and Π

_{2}(Δ

*z*) is given by

*δ*is the total double-pass phase retardation of the specimen. Separation-angle [γ(

*z*)] is defined as the angle between

**P**(

*z*) and

*and moves within Π*β ^

_{2}(

*z*) according to

*γ*(0)=cos

^{-1}[

**β**̂·

**P**(0)] is the initial separation-angle between

*and the incident polarization state [*β ^

**P**(0)]. At

*γ*(Δ

*z*), the total double-pass relative-attenuation (2

*ε*) is given by

*δ*and

*ε*in the presence of polarimetric speckle noise with standard deviation

*σ*

_{speckle}(Eq. 7 of Ref [5

5. N. J. Kemp, J. Park, H. N. Zaatari, H. G. Rylander, and T. E. Milner, “High sensitivity determination of birefringence in turbid media using enhanced polarization-sensitive OCT,” J. Opt. Soc. Am. A **22**, 552–560 (2005). [CrossRef]

*l*is the arc length of the noise-free model polarization arc [

_{arc}**P**(

*z*)]. Differential arc length (

*dl*) is given by

_{arc}**P**(

*z*),

*ε*<0.1),

*l*

_{arc}can be approximated to the first order,

## 3. Methods

### 3.1 Instrumentation

5. N. J. Kemp, J. Park, H. N. Zaatari, H. G. Rylander, and T. E. Milner, “High sensitivity determination of birefringence in turbid media using enhanced polarization-sensitive OCT,” J. Opt. Soc. Am. A **22**, 552–560 (2005). [CrossRef]

_{2}O

_{3}laser (λ

_{0}=830 nm and Δλ

_{FWHM}=55 nm). Source light is linearly polarized at 45° to insure that incident horizontal and vertical electric field amplitudes are equal. A 50/50 non-polarizing beam splitter divides the source beam into reference and sample paths. Longitudinal scanning (1 mm deep in air, 30 Hz) is provided by a corner-cube retroreflector mounted to a loudspeaker diaphragm in the reference path. Sample path optics include a liquid-crystal variable retarder (LCVR) oriented with its fast axis horizontally (0°),

*x*- and

*y*- lateral scanning galvanometers, and scanning optics. For tendon specimens, an achromatic lens (

*f*=25 mm) focuses light onto the specimen (power=3 mW). For retinal specimens, the intact cornea focuses light onto the retinal nerve fiber layer (RNFL) (power=1.8 mW) [18]. A polarizing beam splitter divides horizontal and vertical polarization components returning from sample and reference paths. Dual silicon photoreceivers measure horizontal and vertical interference fringe intensities versus depth [Γ

*(*

_{h}*z*) and Γ

*(*

_{v}*z*)].

### 3.2 Signal conditioning

*E*(

_{h,m}*z*),

*E*(

_{v,m}*z*), and Δ

*ϕ*(

_{c,m}*z*) for each of

*N*A-scans in the ensemble and for each

_{A}*M*. Ensemble-averaging over

*N*at each depth

_{A}*z*(denoted by 〈 〉

*N*) reduces

_{A}*σ*

_{speckle}by a factor of approximately

*N*

^{1/2}

*A*and then normalization yields

*M*sets of depth-resolved polarization data [

**S**

*(*

_{m}*z*)] for each location,

*W*(

_{m}*z*) is used as a scalar weighting factor in the multistate nonlinear algorithm to estimate phase retardation (

*δ*) and relative-attenuation (

*ε*).

### 3.3 Multistate nonlinear algorithm to determine form-biattenuance

*χ*) is accomplished using a nonlinear fitting algorithm based on the approach introduced by Kemp et al [5

**22**, 552–560 (2005). [CrossRef]

*n*) with PS-OCT. A modified multistate residual function (

*R*) has been implemented which gives the composite squared deviation between

_{M}*M*sets of depth-resolved polarization data [S

*(*

_{m}*z*)] and corresponding

*M*noise-free model polarization arcs [

**P**

*(*

_{m}*z*)] weighted by

*W*(

_{m}*z*),

*R*is the weighted single-state residual function,

_{o}*j*” is used to denote the discrete nature of sampled data versus depth (

*z*). Model parameters [2

*ε*, 2

*δ*,

*, and*β ^

**P**

*(0)] are estimated by minimizing*

_{m}*R*using a Levenberg-Marquardt algorithm [19] and represent the best estimate of

_{M}**P**

*(*

_{m}*z*). At increased penetration depths (lower electrical signal-to-noise ratio) or large initial separation-angles [

*γ*(0), as discussed in Section 5.1],

_{m}*W*(

_{m}*z*) decreases and S

*(*

_{m}*z*) are given less weight. Δ

*χ*and Δ

*n*are calculated using Eqs. (6) and (4) from estimates of

*ε*and

*δ*provided by the multistate nonlinear algorithm. Δ

*z*is measured by subtracting the front and rear specimen boundaries in the OCT intensity image and dividing by the bulk refractive index (

*n*

_{tendon}=1.4 and

*n*

_{RNFL}=1.38).

**P**

*(*

_{m}*z*) arc is offset through constraints placed upon the other

*M*- 1 arcs by the modified residual function [Eq. (21)]. All

*M*noise-free model polarization arcs [

**P**

*(*

_{m}*z*)]

*must*collapse toward the

*same*eigen-axis (

*) at the*β ^

*rate (2*

**same***ε*) and

*must*rotate around

*by the*β ^

*angle (2*

**same***δ*) regardless of the incident polarization state. Discrimination between arc movements on the Poincaré sphere due to either Δ

*n*or Δ

*χ*is accomplished by restricting contributions from each into orthogonal planes (Sec. 2).

### 3.4 Ex vivo rat tendon measurements

*δ*and

*ε*, we imaged the same location on a single rat tail tendon fascicle for a range of axial displacements between the rear principal plane of the

*f*=25 mm focusing lens and the fascicle surface. We recorded S

*(*

_{m}*z*) (

*N*=64,

_{A}*M*=3) for ten 50-µm-steps from 0 (focused at surface) to 450 µm (focused deep within fascicle).

*γ*(0)] on the PS-OCT-estimated values for δ and ε, we placed a 1/6-wave retarder in the sample path between the LCVR and scanning optics and recorded S

_{m}*(*

_{m}*z*) (

*N*=64,

_{A}*M*=5) from the same location on a single rat tail tendon fascicle for 12 uniformly spaced orientations (between 0° and 165°) of the 1/6-wave retarder axis.

### 3.5 Ex vivo chicken tendon measurements

*(*

_{m}*z*) was recorded (

*N*=64,

_{A}*M*=3) at all locations and the multistate nonlinear algorithm was used to estimate

*ε*and

*δ*for each location.

### 3.6 In vivo primate retinal nerve fiber layer measurements

*(*

_{m}*z*) was recorded (

*N*=36,

_{A}*M*=6) on two different days for six locations distributed in a 100 µm region around a point 1 mm inferior to the optic nerve head (ONH) center and six locations distributed in a 100 µm region around a point 1 mm nasal to the ONH center.

## 4. Results

### 4.1 Rat tail tendon

*χ*=5.3·10

^{-4}± 1.3·10

^{-4}[3.0·10

^{-4}, 8.0·10-4] and in form-birefringence were Δ

*n*=51.7·10

^{-4}± 2.6·10

^{-4}[46.8·10

^{-4}, 56.3·10

^{-4}]. Figures 2(a) and 2(b) show S

*(*

_{m}*z*) and

**P**

*(*

_{m}*z*) plotted on the Poincaré sphere for two different rat tail tendon fascicles with the largest (Δ

*χ*=8.0·10

^{-4}) and smallest (Δ

*χ*=3.0·10

^{-4}) form-biattenuances detected. Form-birefringence for the two fascicles shown in Figures 2(a) and 2(b) were Δ

*n*=47.4·10

^{-4}and Δ

*n*=55.2·10

^{-4}respectively. Polarimetric signal-to-noise ratio (PSNR) ranged from 51 to 155 and standard deviation of polarimetric speckle noise (

*σ*

_{speckle}) was approximately 0.22 rad for the 111 rat tail tendon locations measured.

### 4.2 Variation in form-biattenuance versus relative focal depth

*f*=25 mm focusing lens and the fascicle surface,

*σ*

_{speckle}≈0.22 rad and mean±standard deviation in relative-attenuation were

*ε*=1.42±0.022 rad and in phase retardation were

*δ*=11.5±0.034 rad. Thickness of the tendon specimen was Δ

*z*=360 µm.

### 4.3 Variation in form-biattenuance versus 1/6-wave retarder axis orientation

*ε*=1.54±0.096 rad and in phase retardation were

*δ*=13.1±0.046 rad.

*σ*

_{speckle}increased exponentially from 0.066 rad for a small initial separation-angle of

*γ*(0)=0.81 rad up to

_{m}*σ*

_{speckle}=0.69 rad for a large

*γ*(0)=3.0 rad. Thickness of the tendon specimen was Δ

_{m}*z*=383 µm.

### 4.4 Rat Achilles tendon

*χ*=1.3·10

^{-4}±0.53·10

^{-4}[0.74·10

^{-4}, 3.2·10

^{-4}] and in form-birefringence were Δ

*n*=46.9·10

^{-4}± 5.9·10

^{-4}[32.9·10

^{-4}, 56.3·10

^{-4}]. Figure 3 shows S

*(*

_{m}*z*) and

**P**

*(*

_{m}*z*) plotted on the Poincaré sphere for the location in which the form-biattenuance was the lowest of all tendon specimens studied (Δ

*χ*=0.74·10

^{-4}). PSNR ranged from 76 to 175 and

*σ*

_{speckle}≈0.20 rad for the 45 rat Achilles tendon locations measured.

### 4.5 Chicken drumstick tendon

*χ*=2.1·10

^{-4}±0.3·10

^{-4}[1.4·10

^{-4}, 3.1·10

^{-4}] and in form-birefringence were Δ

*n*=44.4·10

^{-4}±1.9·10

^{-4}[38.4·10

^{-4}, 48.4·10

^{-4}]. PSNR ranged from 42 to 96 and

*σ*

_{speckle}≈0.28 rad for the 57 chicken drumstick tendon locations measured.

### 4.6 In vivo primate RNFL

*χ*=0.18·10

^{-4}±0.09·10

^{-4}[0.07·10

^{-4}, 0.33·10

^{-4}] on day 1 and Δ

*χ*=0.18·10

^{-4}± 0.13·10

^{-4}[0.06·10

^{-4}, 0.42·10

^{-4}] on day 2. Average RNFL thickness in this region was 166 µm and average relative-attenuation (

*ε*) was 0.023 radians. Figure 4 shows typical S

*(*

_{m}*z*) and

**P**

*(*

_{m}*z*) plotted on the Poincaré sphere for the region 1 mm inferior to the ONH center in the primate RNFL. PSNR ranged from 3 to 16 and

*σ*

_{speckle}≈ 0.06 rad for the six inferior locations measured. In the region 1 mm nasal to the ONH center, RNFL thickness averaged 50 µm and PSNR was too low for reliable estimates of Δ

*χ*in the nasal region of the primate RNFL.

## 5. Discussion

### 5.1 Variation in measurements of form-biattenuance

*u*) was analyzed previously [5

_{δ}**22**, 552–560 (2005). [CrossRef]

*σ*

_{speckle}) which lingers after ensemble-averaging. Arc length (

*l*

_{arc}) has approximately the same functional dependence on

*δ*and

*ε*[Eqs. (17) and (18)]; therefore we expect uncertainty in relative-attenuation (

*u*) to be similar to

_{ε}*u*for a given

_{δ}*σ*

_{speckle}, though additional experiments in a controlled model are necessary to verify the relationship between

*u*,

_{ε}*u*, and

_{δ}*σ*

_{speckle}. Uncertainties in form-birefringence (

*u*) or form-biattenuance (

_{Δn}*uΔ*) are dependent on

_{χ}*u*or

_{δ}*u*as well as the specimen thickness (

_{ε}*Δ*), which complicates comparison of

_{z}*uΔ*or

_{n}*uΔ*between specimens or between other variations of PS-OCT. For the rat and chicken tendon specimens we studied (

_{χ}*N*=64),

_{A}*σ*

_{speckle}ranged from 0.20 to 0.28 rad, giving uncertainties (

*u*and

_{δ}*u*) due to polarimetric speckle noise no higher than ± 0.07 rad. Corresponding uncertainty in form-biattenuance for a Δ

_{ε}*z*=160-µm-thick specimen is

*uΔ*≈±0.57·10

_{χ}^{-4}. In primate RNFL (

*N*=36),

_{A}*σ*

_{speckle}≈0.06 rad corresponds to

*u*≈ ± 0.015 rad or

_{ε}*uΔ*≈±0.12·10

_{χ}^{-4}for an RNFL thickness of Δ

*z*=166 µm.

*δ*and

*ε*due to placement of the beam focus was negligible (Sec. 4.2). Variation in measured

*ε*(6.2%) due to different initial separation-angles [

*γ*(0)] was higher than variation in

_{m}*δ*(0.35%, Sec. 4.3). Interestingly,

*σ*

_{speckle}has a roughly exponential dependence on

*γ*(0). Large initial separation angles [

_{m}*γ*(0)≈

_{m}*π*] correspond to incident polarization states [S

*(0)] near the preferentially attenuated eigenpolarization; therefore, we expect these S*

_{m}*(0) to have lower detected intensity and relatively higher noise variation on the Poincaré sphere than S*

_{m}*(0) with lower*

_{m}*γ*(0). Additional experiments are needed to characterize completely the dependence of

_{m}*σ*

_{speckle}on

*γ*(0). Because

_{m}*W*(

_{m}*z*) [Eq. (20)] decreases with increasing

*σ*

_{speckle}, states with large

*γ*(0) are weighted less by our multistate nonlinear algorithm when estimating

_{m}*δ*and

*ε*. Inspection of the right sides of Figs. 2(a), 2(b), 3, and 4 reveals that

*σ*

_{speckle}does not increase significantly versus depth (

*z*) for the limited tissue thicknesses studied. Therefore, we do not expect that reduced collection of light backscattered from deeper in the tissue significantly affects our estimates of

*ε*for the range of depths probed.

### 5.2 Model for form-biattenuance and form-birefringence

23. W. L. Bragg and A. B. Pippard, “The form birefringence of macromolecules,” Acta. Crystallogr. **6**, 865–867 (1953). [CrossRef]

8. 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]

*n*and Δ

*χ*to transformations in polarization state of light propagating in anisotropic media.

*χ*). Consider light normally incident on a fibrous material that consists of alternating anisotropic and isotropic layers (Fig. 5). The anisotropic layer with thickness

*h*

_{1}is composed of cylindrical fibers (

*n*) imbedded in water (

_{f}*n*) with center-to-center spacing (

_{w}*a*) and with diameters (

*h*

_{1}) much less than the wavelength of incident light (

*h*

_{1}≤

*a*≪

*λ*

_{0}).

*Effective*refractive indices parallel (

*n*) and perpendicular (

_{p}*n*) to the fibers are [8

_{s}8. 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]

*h*-

*h*

_{1}and contains the same isotropic fluid (water,

*n*) found between fibers in the adjacent anisotropic layers.

_{w}*n*-

_{p,s}*n*≈0, the Fresnel relations can be used to find expressions for the form-birefringence (Δ

_{w}*n*) and form-biattenuance (Δ

*χ*). The form-birefringence (Δ

*n*) is that of the anisotropic layer reduced by the layer fill factor (

*h*

_{1}/

*h*),

*t*/

_{p}*t*) between

_{s}*p*and

*s*components of normally incident light transmitted to depth

*z*is reduced by transmission through

*z/h*layer-pairs,

*χ*) is computed directly from Eqs. (6) and (26) by computing the logarithm,

*n*and Δ

*χ*in this model is apparent in Eqs. (25) and (27). Whereas form-biattenuance (Δ

*χ*) depends directly on a wavelength-relative structural dimension (

*h*/λ

_{0}), the form of Δ

*n*is wavelength independent (ignoring dispersion). Consider for example

*h*

_{1}=0.12 µm collagen fibers (

*n*=1.51) in water (

_{f}*n*=1.33) with fill factors

_{w}*h*

_{1}/

*a*=0.8 and

*h*

_{1}/

*h*=0.8, Eqs. (25) and (27) give Δ

*n*=31·10

^{-4}and Δ

*χ*=1.2·10

^{-4}. Increasing the collagen fiber diameter to

*h*

_{1}=0.18 µm while preserving the fill-factors and material properties gives an identical form-birefringence (Δ

*n*=31·10

^{-4}) while reducing the form-biattenuance (Δ

*χ*=0.81·10

^{-4}) by one-third. Interestingly, in this model the ratio of form-biattenuance to form-birefringence (Δ

*χ*/Δ

*n*) is dependent on the structural dimensions of the fibers comprising the material.

24. V. Louis-Dorr, K. Naoun, P. Alle, A. Benoit, and A. Raspiller, “Linear dichroism of the cornea,” Appl. Opt. **43**, 1515–1521 (2004). [CrossRef] [PubMed]

*χ*=1.2·10

^{-4}) for 0.12 µm collagen fibers (

*n*=1.51) in water (

_{f}*n*=1.33) is comparable to that measured experimentally in rat Achilles tendon (Δ

_{w}*χ*=1.3·10

^{-4}). Determining the contribution of differential polarimetric scattering and evanescent field propagation and other candidate mechanisms to the form-biattenuance will require further studies in a model system that allows independent variation of these mechanisms.

### 5.3 Comparison with previously reported values

*D*/Δ

*z*values reported by Park et al [16

16. B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, “Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components,” Opt. Lett. **29**, 2512–2514 (2004). [CrossRef] [PubMed]

*χ*=0.8·10

^{-4}) or by Jiao and Wang [4

4. 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]

*χ*=0.17·10

^{-4}). The range of Δ

*χ*values we measured in a substantial number of specimens of rat tail tendon (3.0·10

^{-4}to 8.0·10

^{-4}for

*N*=111), rat Achilles tendon (0.74·10

^{-4}to 3.2·10

^{-4}for

*N*=45), and chicken drumstick tendon (1.4·10

^{-4}to 3.1·10

^{-4}for

*N*=57) demonstrate that a sizable inter-species and intra-species variation is present in tendon form-biattenuance.

21. J. Kastelic, A. Galeski, and E. Baer, “The multicomposite structure of tendon,” Connect. Tissue Res. **6**, 11–23 (1978). [CrossRef] [PubMed]

22. 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]

*χ*and Δ

*n*was not noticeably affected. For the purpose of comparison with previous results on chicken tendon [16

16. B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, “Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components,” Opt. Lett. **29**, 2512–2514 (2004). [CrossRef] [PubMed]

*χ*≥1.4·10

^{-4}) are a factor of two higher than previously reported (Δ

*χ*=0.8·10

^{-4}) [16

**29**, 2512–2514 (2004). [CrossRef] [PubMed]

*D*/Δ

*z*, discrepancy with Δ

*χ*values reported here is not due to conversion from diattenuation (

*D*) to relative-attenuation (

*ε*). Difference in values may be due to wide inherent anatomical variation in form-biattenuance, nonstandard tissue extraction and preparation, or large uncertainty in the methodologies. Details such as tissue freshness, anatomical origin of the harvested specimens, and detailed description of the expected uncertainty are not available in previous reports; therefore direct comparison with results reported here is not possible. Additionally, crimp structure present in non-loaded tendon specimens could cause spatial variations in collagen fiber orientation over the sample beam diameter, resulting in poor agreement with a homogeneous linear retarder/diattenuator model [Eq. (7)] and artifacts in measurements of Δ

*χ*.

*D*/Δ

*z*) as an approximation for

*ε*/Δ

*z*(or Δ

*χ*) is dependent on the acceptable uncertainty for a particular application. For example, using the rat tail tendon results presented in Sec. 4.3 (

*ε*=1.54±0.096 rad), we calculate the percentage error as 6.2% (due to 1/6-wave retarder orientation). Using Eq. (8), the corresponding diattenuation is

*D*=tanh(1.54)=0.91. Because

*ε*increases linearly with depth, we can say this tendon (Δ

*z*=383 µm) has relative-attenuation-per-unit-depth of

*ε*/Δ

*z*=1.54 rad/383 µm=0.004 rad/µm or form-biattenuance Δ

*χ*=5.3·10

^{-4}. Expressing this as diattenuation-per-unit-depth

*D*/Δ

*z*=0.91/383 µm=0.0024/µm results in an error of 40%, which is much higher than the next largest error source (6.2%) and is unacceptable for many applications. Importantly, additional reduction in

*σ*

_{speckle}will allow more sensitive determination of

*ε*. For arbitrarily large PSNR, the small-angle approximation is invalid for any specimen.

**29**, 2512–2514 (2004). [CrossRef] [PubMed]

*n*and Δ

*χ*varies largely. In rat tail tendon, we measured Δ

*χ*/Δ

*n*as high as 0.17 and in Achilles tendon as low as 0.017. In instances where either 1) Δ

*χ*/Δ

*n*is high, 2) multiple periods of oscillation are not present, or 3) PSNR is low, accuracy in estimates of Δ

*n*(Δ

*χ*) will be reduced if form-biattenuance (form-birefringence) is ignored.

### 5.4 Relevance and motivation for form-biattenuance

*χ*=0.18·10

^{-4}) in the region 1 mm inferior to the optic nerve head center is marginally higher than our sensitivity (0.12·10

^{-4}); however, we can conclude that the reported value represents an upper limit on RNFL form-biattenuance in this region.

*n*and Δ

*χ*to underlying microstructure will allow use of PS-OCT for noninvasively quantifying fibrous constituents (e.g., neurotubules in the RNFL or collagen fibers in tendon) which are smaller than the resolution limit of light microscopy. Although the term

*form*-birefringence is used throughout this paper, PS-OCT is not directly capable of discriminating between form and intrinsic effects; therefore a portion of the reported tendon birefringence may be due to intrinsic birefringence on the molecular scale. Because we expect biattenuance in tendon or RNFL to arise from interactions on the nanometer scale,

*form-biattenuance*and

*biattenuance*are used interchangeably.

*D*/Δ

*z*≈

*ε*/Δ

*z*) observed in thin tissue specimens (Δ

*z*< 1 mm), we motivate and justify the introduction of a new term, biattenuance (Δ

*χ*). First, substantial measurements on tissues studied here have a diattenuation (

*D*) that is outside the range of the small-angle approximation and cannot meaningfully be reported on a diattenuation-per-unit-depth (

*D*/Δ

*z*) basis. Second, biattenuance (Δ

*χ*) requires no approximation and is analogous and complementary to a well-understood term, birefringence (Δ

*n*). Use of the term biattenuance overcomes the need to clumsily specify when a diattenuation-per-unit-depth approximation is valid or not. Third, consistency in definitions between birefringence (Δ

*n*) and biattenuance (Δ

*χ*) or between phase retardation (

*δ*) and relative-attenuation (

*ε*) allow a meaningful and intuitive comparison of the relative values (i.e. Δ

*χ*/Δ

*n*,

*ε/δ*) of amplitude and phase anisotropy in any optical medium or specimen. Fourth, availability of narrow line-width swept-source lasers may allow construction of Fourier-domain PS-OCT instruments having scan depths far longer than current PS-OCT instruments. By using these sources and hyperosmotic agents to reduce scattering in tissue [25

25. 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]

26. K. Wiesauer, M. Pircher, E. Goetzinger, S. Bauer, R. Engelke, G. Ahrens, G. Grutzner, C. K. Hitzenberger, and D. Stifter, “En-face scanning optical coherence tomography with ultra-high resolution for material investigation,” Opt. Express **13**, 1015–1024 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-1015. [CrossRef] [PubMed]

*D*and not satisfy the small-angle approximation. Sixth, biattenuance may be useful to investigators employing other polarimetric optical characterization techniques which can detect anisotropically

*scattered*light and for which dichroism is therefore inappropriate. Finally, although the term “depth-resolved” is frequently used in OCT literature in the context of either “measured in the depth dimension” or “local variation in a parameter versus depth [e.g., Δ

*χ*(

*z*)]”, motivation for biattenuance is independent of the particular interpretation. The first interpretation is assumed for work reported here, but the multistate nonlinear algorithm can be extended in a straightforward manner to provide local variation in biattenuance versus depth [Δ

*χ*(

*z*)].

## 6. Conclusion

*χ*) is an intrinsic physical property responsible for polarization-dependent amplitude attenuation, just as birefringence (Δ

*n*) is the physical property responsible for polarization-dependent phase delay. Diattenuation (

*D*) gives the quantity of accumulated anisotropic attenuation over a given depth (Δ

*z*) by a given optical element. The nonlinear dependence of diattenuation on depth motivated our introduction of relative-attenuation (

*ε*), which depends linearly on depth, maintains parallelism and consistency with phase retardation (

*δ*) in Eq. (7), and is a natural parameter in depth-resolved polarimetry such as PS-OCT. The mathematical relationships between these parameters were given in Eqs. (4), (6), and (8).

## Acknowledgments

## References and links

1. | C. K. Hitzenberger, E. Gotzinger, M. Sticker, M. Pircher, and A. F. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express |

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. |

3. | M. G. Ducros, J. D. Marsack, H. G. Rylander, S. L. Thomsen, and T. E. Milner, “Primate retinal imaging with polarization-sensitive optical coherence tomography,” J. Opt. Soc. Am. A |

4. | S. Jiao and L. V. Wang, “Jones-matrix imaging of biological tissues with quadruple-channel optical coherence tomography,” J. Biomed. Opt. |

5. | N. J. Kemp, J. Park, H. N. Zaatari, H. G. Rylander, and T. E. Milner, “High sensitivity determination of birefringence in turbid media using enhanced polarization-sensitive OCT,” J. Opt. Soc. Am. A |

6. | M. Born and E. Wolf, |

7. | O. Wiener, “Die Theorie des Mischkorpers fur das Feld der stationaren Stromung,” Abh. Math.-Phys. Klasse Koniglich Sachsischen Des. Wiss. |

8. | R. Oldenbourg and T. Ruiz, “Birefringence of macromolecules: Wiener’s theory revisited, with applications to DNA and tobacco mosaic virus,” Biophys. J. |

9. | R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. |

10. | S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A |

11. | R. M. Craig, S. L. Gilbert, and P. D. Hale, “Accurate Polarization Dependent Loss Measurement and Calibration Standard Development,” Symposium on Optical Fiber Measurements |

12. | 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. |

13. | J. W. Verhoeven, “Glossary of terms used in photochemistry,” Pure App. Chem. |

14. | J. F. de Boer, T. E. Milner, and J. S. Nelson, “Determination of the depth-resolved Stokes parameters of light backscattered from turbid media using Polarization Sensitive Optical Coherence Tomography,” Opt. Lett. |

15. | M. Todorovic, S. Jiao, L. V. Wang, and G. Stoica, “Determination of local polarization properties of biological samples in the presence of diattenuation by use of Mueller optical coherence tomography,” Opt. Lett. |

16. | B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, “Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components,” Opt. Lett. |

17. | 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.” |

18. | 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. |

19. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

20. | R. W. D. Rowe, “The structure of rat tail tendon,” Connect. Tissue Res. |

21. | J. Kastelic, A. Galeski, and E. Baer, “The multicomposite structure of tendon,” Connect. Tissue Res. |

22. | 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. |

23. | W. L. Bragg and A. B. Pippard, “The form birefringence of macromolecules,” Acta. Crystallogr. |

24. | V. Louis-Dorr, K. Naoun, P. Alle, A. Benoit, and A. Raspiller, “Linear dichroism of the cornea,” Appl. Opt. |

25. | 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 |

26. | K. Wiesauer, M. Pircher, E. Goetzinger, S. Bauer, R. Engelke, G. Ahrens, G. Grutzner, C. K. Hitzenberger, and D. Stifter, “En-face scanning optical coherence tomography with ultra-high resolution for material investigation,” Opt. Express |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(170.4500) Medical optics and biotechnology : Optical coherence tomography

(260.1440) Physical optics : Birefringence

(260.5430) Physical optics : Polarization

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 22, 2005

Revised Manuscript: April 22, 2005

Published: June 13, 2005

**Citation**

Nate Kemp, Haitham Zaatari, Jesung Park, H. Grady Rylander III, and Thomas Milner, "Form-biattenuance in fibrous tissues measured with polarization-sensitive optical coherence tomography (PS-OCT)," Opt. Express **13**, 4611-4628 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-12-4611

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### References

- C. K. Hitzenberger, E. Gotzinger, M. Sticker, M. Pircher and A. F. Fercher, "Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography," Opt. Express 9, 780-790 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-780.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-780.</a> [CrossRef] [PubMed]
- 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]
- M. G. Ducros, J. D. Marsack, H. G. Rylander, S. L. Thomsen and T. E. Milner, "Primate retinal imaging with polarization-sensitive optical coherence tomography," J. Opt. Soc. Am. A 18, 2945-2956 (2001). [CrossRef]
- 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]
- N. J. Kemp, J. Park, H. N. Zaatari, H. G. Rylander and T. E. Milner, "High sensitivity determination of birefringence in turbid media using enhanced polarization-sensitive OCT," J. Opt. Soc. Am. A 22, 552-560 (2005). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1959).
- O. Wiener, "Die Theorie des Mischkorpers fur das Feld der stationaren Stromung," Abh. Math.-Phys. Klasse Koniglich Sachsischen Des. Wiss. 32, 509-604 (1912).
- 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]
- R. A. Chipman, "Polarization analysis of optical systems," Opt. Eng. 28, 90-99 (1989).
- 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]
- 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).
- 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]
- J. W. Verhoeven, "Glossary of terms used in photochemistry," Pure App. Chem. 68, 2228 (1996).
- J. F. de Boer, T. E. Milner and J. S. Nelson, "Determination of the depth-resolved Stokes parameters of light backscattered from turbid media using Polarization Sensitive Optical Coherence Tomography," Opt. Lett. 24, 300-302 (1999). [CrossRef]
- M. Todorovic, S. Jiao, L. V. Wang and G. Stoica, "Determination of local polarization properties of biological samples in the presence of diattenuation by use of Mueller optical coherence tomography," Opt. Lett. 29, 2402-2404 (2004). [CrossRef] [PubMed]
- B. H. Park, M. C. Pierce, B. Cense and J. F. de Boer, "Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components," Opt. Lett. 29, 2512-2514 (2004). [CrossRef] [PubMed]
- 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).
- 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]
- 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]
- W. L. Bragg and A. B. Pippard, "The form birefringence of macromolecules," Acta. Crystallogr. 6, 865-867 (1953). [CrossRef]
- V. Louis-Dorr, K. Naoun, P. Alle, A. Benoit and A. Raspiller, "Linear dichroism of the cornea," Appl. Opt. 43, 1515-1521 (2004). [CrossRef] [PubMed]
- 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]
- K. Wiesauer, M. Pircher, E. Goetzinger, S. Bauer, R. Engelke, G. Ahrens, G. Grutzner, C. K. Hitzenberger and D. Stifter, "En-face scanning optical coherence tomography with ultra-high resolution for material investigation," Opt. Express 13, 1015-1024 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-1015.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-1015.</a> [CrossRef] [PubMed]

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