## 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

Polarization-sensitive optical coherence tomography (PS-OCT) incorporates polarimetric sensitivity with the ranging capability of optical coherence tomography (OCT) to characterize the depth-resolved polarization state of light backscattered from turbid media such as biological tissue. Using requisite processing methods, diagnostically-relevant polarization properties such as tissue form-birefringence and optic axis orientation can be decoded from the detected transformation in light polarization state versus propagation depth [1–4]. Accurate quantification of tissue polarization properties in thin specimens exhibiting weak form-birefringence necessitates an approach which minimizes the effects of speckle noise and exploits the known behavior of polarization transformations in a noise-free model [5].

Form-birefringence (Δ*n*) in tissue arises from anisotropic light scattering by ordered submicroscopic cylindrical structures (e.g., microtubules or collagen fibrils) whose diameter is smaller than the wavelength of incident light but larger than the dimension of molecules [6–8]. Inasmuch as form-birefringence describes the effect of differential phase velocities between light polarized parallel- and perpendicular-to the fiber axis (eigenpolarizations), we introduce the term *form-biattenuance* (Δ*χ*) to describe the related effect of differential attenuation on eigenpolarization amplitudes.

In the eigenpolarization coordinate frame, the polarization-transforming properties of a non-depolarizing, homogeneous optical medium such as anisotropic fibrous tissue are described by the Jones matrix

$$=\left[\begin{array}{cc}\mid {\xi}_{1}\mid \mathrm{exp}\left(i\mathrm{arg}\left({\xi}_{1}\right)\right)& 0\\ 0& \mid {\xi}_{2}\mid \mathrm{exp}\left(i\mathrm{arg}\left({\xi}_{2}\right)\right)\end{array}\right],$$

where *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.

The phase retardation (*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

Chipman [9,10] introduced the polarimetric parameter *diattenuation* (*D*) given quantitatively by

where *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

where *n _{s}* and

*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*).

To the authors’ knowledge, the field of polarimetry lacks a related term for the analogous phenomenon responsible for diattenuation in an anisotropically scattering element. Although *dichroism* describes the phenomenon of diattenuation in an anisotropically absorbing element (such as that exhibited by a sheet polarizer), the term is also used to describe differential transmission or reflection between spectral components (such as that exhibited by a dichroic beam splitter), leading to confusion if taken in the incorrect context. More importantly for work reported here, neither dichroism nor diattenuation nor polarization dependent loss (PDL) [11,12] can be expressed on a per-unit-depth basis and are thus unsuitable quantities for depth-resolved polarimetry in scattering media. Inasmuch as the term *attenuance* has come to describe the loss of transmittance by either absorption or scattering [13], we propose a new term, *biattenuance*, to describe differential loss of transmittance between two eigenpolarization states by either absorption (dichroism) or scattering. Importantly, we show that form-biattenuance is an experimentally (Sec. 4) and theoretically (Sec. 5.2) relevant term that can be expressed on a per-unit-depth basis. Numerically, biattenuance (Δ*χ*) is given by

where *χs* and *χf* are attenuation coefficients of the slow and fast eigenpolarizations. For absorbing (dichroic) media, *χ _{s}* and

*χf*are simply imaginary-valued refractive indices.

As noted previously, the phase retardation (*δ*) and thickness (Δ*z*) of an element are linearly related by its birefringence (Δ*n*). However, the relationship between an element’s diattenuation [*D*, Eq. (3)] and thickness (Δ*z*) is nonlinear. This nonlinear relationship complicates expression of an element’s form-biattenuance: one *cannot* generally and without approximation refer to a diattenuation-per-unit-depth as one can refer to form-birefringence as a phase retardation-per-unit-depth. For example, if optical element *A* has thickness Δ*z _{A}*=1 mm, diattenuation

*D*=0.4, and phase retardation

_{A}*δ*=

_{A}*π*/4 radians and element

*B*is made of the same material but has twice the thickness Δ

*z*=2 mm, element

_{B}*B*will have twice the phase retardation

*δ*=2

_{B}*δ*=

_{A}*π*/2 radians but will not have twice the diattenuation

*D*=0.69≠2

_{B}*D*.

_{A}For depth-resolved polarimetry in scattering media (i.e. PS-OCT) [14], expression of an element’s form-biattenuance on a per-unit-depth basis is desirable both theoretically and experimentally. We define the *relative-attenuation* (*ε*) experienced by light propagating to a depth Δ*z* in an anisotropic element as

and form-biattenuance (Δ*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*).

The Jones matrix of the anisotropic medium from Eqs. (1) and (2) becomes

and the “anisotropic damping” effect of the relative-attenuation (*ε*) becomes apparent. Diattenuation (*D*) is related to relative-attenuation (*ε*) by

We note that for small relative-attenuation, a small-angle approximation is valid and *D*≈*ε*.

The problem with the nonlinear diattenuation-thickness relationship in depth-resolved polarimetry was recognized by Todorovic et al [15]. They defined dual attenuation coefficients (*µ _{ax}* and

*µ*) to represent Beer’s law attenuation for each eigenpolarization and related them to diattenuation (

_{ay}*D*) using Eq. (8) where relative-attenuation (

*ε*) is related to dual attenuation coefficients by

*ε*=|

*µ*-

_{ax}*µ*|Δ

_{ay}*z*/2.

Phase retardation (*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.

Several investigators have reported PS-OCT measurements of diattenuation. Park et al [16] recently reported an approximate single-pass “diattenuation-per-unit-depth” in chicken tendon of *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] reported single-pass *D*/Δ*z*=0.13/mm for a sample with thickness Δ*z*=0.3 mm and Todorovic et al [15] reported single-pass *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.

We present a novel technique for measuring form-biattenuance (Δ*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

The continuous transformation of perfectly polarized light propagating through a scattering medium with Jones matrix given in Eq. (7) is geometrically represented by normalized Stokes parameters vs. depth [*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],

where the complex differential wavevector *$\stackrel{\rightharpoonup}{\beta}$* is a property of the medium and is defined as

*β _{re}* and

*β*are real and imaginary parts of the complex differential wavenumber

_{im}*β*,

and eigen-axis *$\widehat{\beta}$* is a unit-vector on the Poincaré sphere representing the fast eigenpolarization state which propagates through the medium without transformation.

The vector differential equation [Eq. (9)] indicates that differential movement of the **P**(*z*) arc is orthogonal to P(*z*) for all *z*. In fibrous media, form-birefringence (Δ*n*) and form-biattenuance (Δ*χ*) arise from the same anisotropic scattering structures so that $\stackrel{\rightharpoonup}{\beta}$
* _{re}* and $\stackrel{\rightharpoonup}{\beta}$

*share a common eigen-axis (*

_{im}*$\widehat{\beta}$*) and differential movement of

**P**(

*z*) on the Poincaré sphere due to form-birefringence

**P**×$\stackrel{\rightharpoonup}{\beta}$

*) is orthogonal to movement due to form-biattenuance [*

_{re}dz**P**×(

**P**×$\stackrel{\rightharpoonup}{\beta}$

*)*

_{im}*dz*]. At each depth

*z*in the medium, the instantaneous trajectory of

**P**(

*z*) is resolved into two orthogonal planes: 1) a plane [Π

_{1}(

*z*)] with normal

*$\widehat{\beta}$*which encompasses differential movement of

**P**(

*z*) due to Δ

*n*, and 2) a plane [Π

_{2}(

*z*)] containing both

**P**(

*z*) and

*$\widehat{\beta}$*which encompasses differential movement of

**P**(

*z*) due to Δ

*χ*.

Cumulative movement of **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

and therefore 2*δ* is the total double-pass phase retardation of the specimen. Separation-angle [γ(*z*)] is defined as the angle between **P**(*z*) and *$\widehat{\beta}$* and moves within Π_{2}(*z*) according to

where *γ*(0)=cos^{-1}[**β**̂·**P**(0)] is the initial separation-angle between *$\widehat{\beta}$* and the incident polarization state [**P**(0)]. At *γ*(Δ*z*), the total double-pass relative-attenuation (2*ε*) is given by

Form-birefringence (Δ*n*) gives a circular rotation [Fig. 1(a)] of **P**(*z*) about *$\widehat{\beta}$* with rotation angle equal to the double-pass phase retardation (2*δ*) while form-biattenuance (Δ*χ*) produces a collapse [Fig. 1(b)] of **P**(*z*) toward *$\widehat{\beta}$* which is related to the double-pass relative attenuation (2*ε*) by Eqs. (13) and (14). For media exhibiting both Δ*n* and Δ*χ*, the trajectory is a spiral converging toward *$\widehat{\beta}$* [Fig. 1(c)].

Polarimetric signal-to-noise ratio (PSNR) characterizes the ability of PS-OCT to estimate *δ* and *ε* in the presence of polarimetric speckle noise with standard deviation *σ*
_{speckle} (Eq. 7 of Ref [5]),

where *l _{arc}* is the arc length of the noise-free model polarization arc [

**P**(

*z*)]. Differential arc length (

*dl*) is given by

_{arc}which is integrated to provide arc length of **P**(*z*),

When relative-attenuation is small (*ε*<0.1), *l*
_{arc} can be approximated to the first order,

## 3. Methods

#### 3.1 Instrumentation

Our PS-OCT instrumentation has been previously reported in detail [5] so we provide only a brief summary here. The source is a mode-locked Ti:Al_{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

Detected photocurrents representing Γ* _{h}*(

*z*) and Γ

*(*

_{v}*z*) are preamplified, bandpass filtered, and digitized. Coherent demodulation of Γ

*(*

_{h}*z*) and Γ

*(*

_{v}*z*) yields signals proportional to the horizontal and vertical electric field amplitudes [

*E*(

_{h}*z*) and

*E*(

_{v}*z*)] and relative phase [Δ

*ϕ*(

*z*)] of light backscattered from the specimen at each depth

*z*within the A-scan. An ensemble (

*N*) of A-scans representing uncorrelated speckle fields are acquired on a grid within a small square region (50 µm×50 µm) at each location of interest on the specimen. Acquisition of an ensemble of

_{A}*N*A-scans at each location is repeated for

_{A}*M*incident polarization states distributed uniformly in a great circle on the Poincaré sphere by

*M*phase shifts (

*δ*

_{LCVR,m}) of the LCVR. For each

*M*, the calibrated LCVR phase shift (

*δ*

_{LCVR,m}) is subtracted from the demodulated relative phase [Δ

*ϕ*(

_{m}*z*)] to compensate for the light’s return propagation through the LCVR. This yields

*M*sets of horizontal and vertical electric field amplitudes [

*E*(

_{h,m}*z*) and

*E*(

_{v,m}*z*)] and compensated relative phase [Δ

*ϕ*(

_{c,m}*z*)].

Non-normalized Stokes vectors are calculated from *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,

When Stokes vectors are first normalized and then ensemble-averaged over *NA*, the resulting Stokes vectors [**W**
* _{m}*(

*z*)] have magnitude [0 ≤ |

**W**

*(*

_{m}*z*)|=

*W*(

_{m}*z*)≤1] which is directly related to the extent of the distribution of pre-averaged normalized Stokes vectors on the Poincaré sphere within the ensemble at each depth

*z*,

*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

High sensitivity quantification of form-biattenuance (Δ*χ*) is accomplished using a nonlinear fitting algorithm based on the approach introduced by Kemp et al [5] for determining form-birefringence (Δ*n*) with PS-OCT. A modified multistate residual function (*R _{M}*) has been implemented which gives the composite squared deviation between

*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*),

where *R _{o}* is the weighted single-state residual function,

and the subscript “*j*” is used to denote the discrete nature of sampled data versus depth (*z*). Model parameters [2*ε*, 2*δ*, *$\widehat{\beta}$*, and **P**
* _{m}*(0)] are estimated by minimizing

*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).

Uncertainty in estimates of any single **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 (

*$\widehat{\beta}$*) at the

*rate (2*

**same***ε*) and

*must*rotate around

*$\widehat{\beta}$*by the

*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

Four mature, freshly-euthanized Sprague-Dawley rats were obtained from an unrelated study. To collect tail tendon specimens, each tail was cut from the body and a longitudinal incision the length of the tail was made in the skin on the dorsal side. Skin was peeled back and tertiary fascicle groups were extracted with tweezers and placed in phosphate buffered saline solution to prevent dehydration before imaging. Anatomical terminology used in this paper is consistent with that given by Rowe [20]. Tertiary fascicle groups were teased apart into individual fascicles with tweezers and placed in a modified cuvette in the sample path of the PS-OCT instrument. The cuvette maintained saline solution around the fascicle, prevented mechanical deformation in the radial direction, and allowed 20 g weights to be attached at each end of the fascicle. Weights provided minimal longitudinal loading in order to flatten the collagen fibril crimp structure present in rat tail tendon [21,22]. A total of 111 different fascicle locations were imaged from the four rats, each at the location of maximum diameter across its transverse cross-section (as determined by an OCT B-scan image). Achilles tendon specimens from the same rats were harvested in a straightforward manner and imaged in the modified cuvette with the same loading conditions. Four different Achilles tendons were imaged in 45 different randomly chosen locations. S* _{m}*(

*z*) was recorded (

*N*=64,

_{A}*M*=3) at all locations and the multistate nonlinear algorithm estimated

*ε*and

*δ*for each location.

To investigate the effect of the relative depth of light focus within the tissue on the estimated values for *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).

To investigate the effect of different initial separation-angles [*γ _{m}*(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}*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

We imaged 57 randomly chosen locations in tendons extracted from the proximal end of chicken thighs obtained at a local grocery store. Temperature variations (freezing/thawing or refrigeration) and postmortem time prior to characterization by PS-OCT were unknown. Extracted tendon specimens were kept hydrated in the modified cuvette and imaged without mechanical loading. S* _{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

Details of the animal protocol for PS-OCT characterization of the in vivo primate retinal nerve fiber layer (RNFL) were given in Ref [18]. S* _{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

For 111 locations in rat tail tendon, mean ± standard deviation and [range] in rat tail tendon form-biattenuance were Δ*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

For 10 different displacements between the rear principal plane of the *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

For 12 orientations of the 1/6-wave retarder axis, mean±standard deviation in relative-attenuation were *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 _{m}*(0)=0.81 rad up to

*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

For 45 locations in rat Achilles tendon, mean ± standard deviation and [range] in rat Achilles tendon form-biattenuance were Δ*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

For 57 locations in chicken drumstick tendon, mean ± standard deviation and [range] in chicken drumstick tendon form-biattenuance were Δ*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

We report mean ± standard deviation and [range] in form-biattenuance for the six locations within a 100 µm region around a point 1 mm inferior to the ONH center: Δ*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

Uncertainty in phase retardation (*u _{δ}*) was analyzed previously [5] and is predominantly due to polarimetric speckle noise (

*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.

The range of systematic variation in measurements of *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 _{m}*(0)] was higher than variation in

*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

Since the pioneering work of O. Wiener [7], many investigators have examined the origin of form-birefringence in anisotropic media. Notably, Bragg and Pippard [23] derived an expression for the form-birefringence of a suspension of aligned ellipsoidal particles and compared their result to that derived by Wiener. More recently, Oldenbourg and Ruiz [8] reexamined Wiener’s original theory and applied their model to compute form-birefringence of macromolecules including DNA and the tobacco mosaic virus. None of these investigations address the phenomenon of form-biattenuance nor introduce a model predicting the relative contribution of Δ*n* and Δ*χ* to transformations in polarization state of light propagating in anisotropic media.

In the following, we introduce a rudimentary model for form-biattenuance (Δ*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 _{f}*) imbedded in water (

*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}The second layer has thickness *h* - *h*
_{1} and contains the same isotropic fluid (water, *n _{w}*) found between fibers in the adjacent anisotropic layers.

When *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*),

Relative amplitude (*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,

where multiple reflections between layers are neglected. The form-biattenuance (Δ*χ*) is computed directly from Eqs. (6) and (26) by computing the logarithm,

A notable difference between the functional dependence of Δ*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 _{f}*=1.51) in water (

*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.

Recently Louis-Dorr et al [24] reported corneal form-biattenuance measurements. That a transparent fibrous biological material such as cornea exhibits form-biattenuance is consistent with the model presented here. Although this model does not include effects such as differential polarimetric scattering and evanescent field propagation which may also lead to form-biattenuance, the computed value (Δ*χ*=1.2·10^{-4}) for 0.12 µm collagen fibers (*n _{f}*=1.51) in water (

*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

The form-biattenuance values we measured in tendon are significantly higher than those approximated from *D*/Δ*z* values reported by Park et al [16] in chicken tendon (Δ*χ*=0.8·10^{-4}) or by Jiao and Wang [4] in porcine tendon (Δ*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.

Our motivation for loading tendon specimens with 20 g weights was two-fold. First, extracted tendon exhibits a well-known crimp structure [21,22] in which the constituent collagen fibers are not regularly aligned. Applying a small load on the tendon effectively flattens the crimp, providing a reproducible specimen which can be modeled using the Jones matrix in Eq. (7) for a homogeneous linear retarder/diattenuator and producing results that can be objectively compared. Second, the in vivo state of tendon is arguably more similar to our slightly loaded state than to a completely relaxed or non-loaded state, especially considering that even minimal muscle tone would cause slight tension in connected tendons. Consistent with these motivations, variance in measurements of rat tail tendon biattenuance and birefringence was substantially reduced by loading, but mean Δ*χ* and Δ*n* was not noticeably affected. For the purpose of comparison with previous results on chicken tendon [16], we did not mechanically load our chicken tendon specimens. Our measured values of form-biattenuance of chicken tendon (Δ*χ*≥1.4·10^{-4}) are a factor of two higher than previously reported (Δ*χ*=0.8·10^{-4}) [16].

Because the small-angle approximation introduces only minimal error in previously reported values of *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*.

The validity of using diattenuation-per-unit-depth (*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.

Previous discussion of diattenuation by PS-OCT investigators has been primarily concerned with its effect on estimates of phase retardation or form-birefringence. Park et al [16] concluded that reasonable estimates of phase retardation can be made in tendon and muscle even if diattenuation is physically present but is ignored in the model. Indeed, one can discern from the “damped” sinusoidal nature of the normalized Stokes parameters vs. depth [Figs. 1(c), 2, and 6] that the frequency of sinusoidal oscillation (proportional to form-birefringence) can be estimated without considering the “damped” amplitude variation (due to form-biattenuance) when multiple periods of oscillation (multiple waves of phase retardation) are present. Our results show that relative contribution to polarimetric transformations from Δ*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.

In Sec. 4.3 we noted that polarimetric speckle noise (*σ*
_{speckle}) depends on the initial separation-angle [*γ _{m}*(0)]. Based on this observation, we conclude that the number of incident polarization states (

*M*) and the selection of those states [S

*(0)] relative to*

_{m}*$\widehat{\beta}$*employed by a particular PS-OCT approach will affect one’s ability to accurately distinguish between Δ

*and Δ*

_{n}*χ*. Previously reported approaches [2,4] use

*M*=2 incident polarization states which are positioned orthogonally to each other in their representation on the Poincaré sphere and are best suited for detecting

*δ*and Δ

*n*. Alternatively,

*M*=2 polarization states which are oriented parallel and perpendicular to the optic axis in physical space (and opposite to each other in their representation on the Poincaré sphere) may provide the best estimates of

*ε*and Δ

*χ*. These considerations suggest that future studies should select at least

*M*=3 incident polarization states for optimal determination of both form-birefringence and form-biattenuance, regardless of the particular PS-OCT approach used. Our multistate nonlinear algorithm discriminates between Δ

*n*and Δ

*χ*by restricting contributions to movement of S

*(*

_{m}*z*) on the Poincaré sphere from each phenomenon into two orthogonal planes (Sec. 2) and seamlessly incorporates

*M*≥3 incident polarization states while avoiding issues related to overdetermined Jones matrices.

#### 5.4 Relevance and motivation for form-biattenuance

The ability of PS-OCT and the multistate nonlinear algorithm to noninvasively measure form-biattenuance is demonstrated by in vivo imaging of the primate RNFL. We acknowledge that the value we report for form-biattenuance (Δ*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.

Because form-birefringence and form-biattenuance arise from light scattering by nanometer-sized anisotropic structures, development of sophisticated models relating Δ*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.

Although the small-angle approximation introduces minimal error for previously reported values of diattenuation-per-unit-depth (*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], future studies using PS-OCT may probe significantly deeper into tissue specimens than 1–2 mm, likely making the small-angle approximation invalid even in tissues with low biattenuance. Fifth, PS-OCT may be applied to characterize non-biological samples [26] which may have high *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

A summary of the terms and symbols used in this paper is appropriate here. Biattenuance (Δ*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).

Novel work reported in this paper includes: 1) theoretical and experimental motivation for a new term in optical polarimetry, *biattenuance* (Δ*χ*), which describes the phenomenon of anisotropic or polarization-dependent attenuation of light amplitudes due to absorption (dichroism) or scattering; 2) detailed mathematical formulation of Δ*χ* and relative-attenuation (*ε*) in a manner consistent with established polarimetry (i.e., birefringence and phase retardation), and mathematical relationships to related polarimetric terms diattenuation [9] and dual attenuation coefficients [15]; 3) analytic expression for trajectory of normalized Stokes vectors on the Poincaré sphere in the presence of both birefringence and biattenuance; 4) expression for arc length (*l*
_{arc}) and PSNR of normalized Stokes vector arcs on the Poincaré sphere in the presence of both birefringence and biattenuance; 5) modification of a multistate nonlinear algorithm [5] to provide sensitive and accurate estimates of *ε* and Δ*χ* in addition to *δ* and Δ*n*; 6) incorporation of a scalar weighting factor [*W _{m}(z)*] into the multistate nonlinear algorithm [5]; 7) substantial ex vivo and in vivo experimental data in several species and two tissue types to demonstrate large variation in Δ

*χ*, as well as interpretation of this data in the context of previously reported values; 8) description of the expected uncertainty in our measurements of

*ε*and Δ

*χ*; and 9) introduction of a physical model for birefringent and biattenuating optical media.

The experimental results and physical model introduced here demonstrate that form-biattenuance and form-birefringence are closely related *but physically independent* phenomena which convey different information about tissue microstructure. Although the diagnostic relevance of form-biattenuance remains unknown, this work motivates further investigation into how high sensitivity determination of form-biattenuance and form-birefringence might be concurrently used in biomedical research or clinical diagnostics. Further studies are also required to quantify the effect of tendon crimp on Δ*n* and Δ*χ* and to establish the acceptable uncertainty in biattenuance for diagnosis of various pathological tissue states. Additional experimentation in collagenous, neural, and other fibrous tissues at multiple imaging wavelengths is necessary to refine the physical model and establish a comprehensive anatomical range for biattenuance.

## Acknowledgments

The authors gratefully acknowledge the Optics Express editor and reviewers for helpful suggestions to improve this paper, and the Texas Higher Education Coordinating Board Advanced Technology Program (003658-0359-1999), the National Eye Institute at NIH (R24EY12877-02), and the NSF-IGERT program for financial support of this research. Thanks also to Jennifer Cassaday for procurement of rat specimens.

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