Birefringence offers an intrinsic contrast mechanism related to the microstructure and arrangement of fibrillary tissue components. Here we present a reconstruction strategy to recover not only the scalar amount of birefringence, but also its optic axis orientation as a function of depth in tissue from measurements with catheter-based polarization-sensitive optical coherence tomography. A polarization symmetry constraint, intrinsic to imaging in the backscatter direction, facilitates the required compensation for wavelength-dependent transmission through the system elements, the rotating catheter, and overlying tissue layers. Applied to the intravascular imaging of coronary atherosclerosis in human patients, the optic axis affords refined interpretation of plaque architecture.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Fibrous tissues are prevalent in nature and their directional fiber organization is critical to their physiological function. Many such tissues exhibit birefringence and cause a differential delay, or retardation, between light polarized parallel and orthogonal to the fibrillar tissue components. Assessing the birefringence and its optic axis orientation can furnish crucial insight into the make-up and physical orientations of these tissue elements. A prominent method to assess tissue polarization effects is polarization-sensitive optical coherence tomography (PS-OCT), which measures the polarization state of light backscattered in a tissue sample as a function of its path length traveled into the sample [1–3]. Because OCT measures the round-trip of light from the tissue surface to a given depth and back, the measured polarization states correspond to the cumulative effect of the transmission through all these tissue layers. The scalar amount of birefringence relates to the variation of the polarization states along depth and can be computed independently of the overlying tissue layers [4,5]. In contrast, the reconstruction of the optic axis orientation requires careful compensation for all overlying tissue layers in order to obtain intelligible optic axis measurements, as demonstrated using free-space PS-OCT instruments by Todorovic et al. , and more recently by Fan and Yao .
In fiber-based systems, and even more so in catheter-based OCT systems, transmission through the fiber and system components alters the polarization states, hindering the reconstruction of depth-resolved optic axis maps. Moreover, many fiber-based components exhibit wavelength-dependent retardance, resulting in polarization mode dispersion (PMD), which limits the ability to obtain meaningful birefringence measurements  and can lead to birefringence artifacts . We have previously developed spectral binning as a reconstruction strategy to mitigate these artifacts and obtain improved maps of scalar tissue birefringence . This has enabled improved assessment of coronary atherosclerosis in human patients with intravascular OCT [11,12].
Here, we demonstrate depth-resolved optic axis imaging through a rotating intravascular catheter, compensating for the transmission through all system elements, including correction for PMD. The reconstruction builds on the intrinsic symmetry present in polarization measurements performed with identical illumination and detection paths. It also leverages the linear birefringence induced by total internal reflection at the distal tip of the fiber probe, and uses the weak birefringence in the transparent sheath encapsulating the probe as a reference to render the relative optic axis measurements absolute. We present optic axis imaging in the coronary arteries of human patients and validate the method with phantom experiments.
OCT measures the round-trip or double-pass transmission through tissue. In general, it is not possible to estimate the effect on the polarization states due to the forward transmission through an optical element based on the measurements of the double-pass transmission. However, in the case of tissue, the linear birefringence induced by the fibrillar geometry of many tissue building blocks dominates and exceeds the optical activity that may typically be present in biological samples. From the double-pass transmission of a homogeneous linearly birefringent (and diattenuating) element, the single-pass transmission can be obtained by taking its matrix root. The optic axis of a linear retarder lies in the QU plane of the Poincaré sphere, and the transmission in the reverse direction through the same polarization element is identical to the forward transmission, resulting in a specific matrix symmetry. However, a stack of differently oriented linear retarders corresponds to a general retarder without any symmetry. Yet, by virtue of the imaging geometry, its double-pass transmission always appears as a linear element. Reconstructing the depth-resolved optic axis orientation from PS-OCT measurements relies on the assumption that the axial resolution is sufficient to resolve homogenously retarding tissue layers and estimate their forward transmission, one layer at a time [6,7].
Figure 1 outlines the necessary processing steps to retrieve the depth-resolved optic axis orientation from the measurements with our intravascular PS-OCT system and explains the catheter geometry. The goal is to reconstruct the optic axis orientation within the plane orthogonal to the imaging beam at each angular position of the fiber probe. Whereas the measurements are performed in a static laboratory frame, the optic axis orientation is expressed within the local coordinate system of the rotating probe beam, corresponding to the longitudinal and circumferential directions of the vessel lumen. We demonstrate the rich optic axis orientation contrast by processing the in vivo measurements of coronary arteries previously acquired during a clinical pilot trial of intravascular polarimetry .
A. Experimental Methods
All imaging was performed with a custom-built polarization-sensitive optical frequency-domain imaging console, as described previously . In short, the system operated at a center wavelength of 1300 nm with a repetition rate of 103.6 kHz and with a wavelength scanning range of 110 nm, corresponding to a depth resolution of 9.4 μm in tissue, assuming a refractive index of 1.34. The system employed a polarization modulation scheme, whereby the source polarization state was alternated for subsequent A-lines between two states located orthogonal to each other on the Poincaré sphere. The polarization-diverse receiver measured both the horizontal () and vertical () polarization components of the sample light for polarization-sensitive detection. The imaging console interfaced with commercial intravascular catheters (FastView, Terumo), rotating at 100 revolutions per second, acquiring 1024 depth scans in each frame, and featuring a lateral resolution of . Clinical imaging used a pullback speed of 20 mm/s, during the injection of a non-ionic contrast solution at a rate of 1–3 ml/s. From the 36 coronary arteries imaged in 30 patients, we identified 344 independent cross sections of varying lesion architecture that we used to evaluate the robustness of the individual processing steps.
B. Reconstruction of Optic Axis Orientation
All processing steps were implemented in Matlab 2017b (Mathworks Inc., Natick, Massachusetts) and processed with a 2.9 GHz Intel Core i7 processor and 16 GB of RAM.
1. Signal Formalism
The two input polarization states, , probed the Jones matrix , describing the sequential transmission through the system components, the sample, and the detection optics. Measured at each wavenumber and at each angular orientation of the illumination beam, the complex-valued interference signal can be written as10] to reconstruct the Jones vectors as functions of depth, , and spectral bin, :
The average of the degrees of polarization (DOPs) across all spectral bins and input polarization states served as a measure of depolarization  to judge the reliability of the measured polarization states:
The matrix , belonging to the special orthogonal group, or the rotation group, SO(3), that best describes the measured transmission through the combined system and sample components, was then constructed by orthogonalizing the normalized Stokes vectors:
The recovered transmission matrix can be described by7), as indicated in Fig. 1(b). and are assumed to be independent of the wavenumber within each bin, but to vary between spectral bins due to PMD. describes the transmission through the symmetric part of the system that the light traverses on both the illumination and detection paths, including the rotating catheter, and is assumed to be free of PMD. is the single-pass transmission from the sample surface to depth , where the sample is modeled as consisting of individual layers , with locally constant polarization properties, and a thickness corresponding to one axial depth pixel . is the reverse transmission from depth to the sample surface, where ), with the identity matrix . This corresponds to the transpose in Jones formalism and follows the convention to use a left-handed coordinate system on the reverse path . Each is presumed to act as a linear retarder, which makes them -transpose symmetric , and the goal is to characterize each layer .
2. Recovering Symmetry
The presence of distinct illumination and detection components and , respectively, breaks the symmetry intrinsic to measurements along identical illumination and detection paths [15,16]. The symmetry can be recovered by left-multiplying with a correction matrix , as demonstrated in Fig. 2. Estimating can be cast as a quadratic problem that minimizes the non-symmetric signal components. Reverting to SU(2) and writing instead of , and for , the correction matrix minimizes the following Frobenius norm:8) can be rewritten as 17,18], removing any ambiguity related to the global phase of the Jones matrices. Due to the intrinsic symmetry imposed by measuring in the backscatter direction, the -matrix should, theoretically, have , and the solution to the minimization would be obtained by aligning the complex conjugate of with its nullspace. In practice, the presence of noise results in finite eigenvalues, and aligns with the smallest eigenvalue of . corresponds to a general retarder and Fig. 2(b) shows the , , and components of its retardation vector.
A perfectly -transpose symmetric SO(3) matrix exhibits only linear retardation, and its eigenvector lies in the QU plane. After applying to , we enforced strict symmetry by setting the residual components of its retardation vectors to zero (see Supplement 1 for detailed derivations), to obtain
3. Compensating for Wavelength Dependence of System Elements
The difference between the spectral bins of is attributed to . In order to align all bins to the central bin , we first matched the cumulative retardation by minimizing3.
To compute the Euclidean mean across the spectral bins, the arithmetic mean of the SO(3) matrices was decomposed into , the average retarding element, and , a symmetric, positive-semidefinite matrix corresponding to a depolarizer [19,20]:21], and used the Caley–Hamilton theorem, to directly compute the eigenvalues and eigenvectors of , and to retrieve :
4. System and Catheter Transmissions
We modeled the system and catheter transmissions and compared them with the measured double-pass transmission, obtained by isolating the signal reflected at the ball lens or the inner surface of the catheter sheath where corresponds to the identity matrix:Supplement 1) to obtain , , and . To ensure that the correction matrices cancel all retardation at the sheath interface, rather than using the fitted , we used 4.
5. Reconstructing Depth-Resolved Optic Axis Orientation
To obtain the depth-resolved optic axis orientation, the effect of each tissue layer on the subsequent layers needs to be considered. Due to the -transpose symmetry of the linearly birefringent tissue layers, the isolated sample signal for each depth is described by7]:
A. Measurements are Intrinsically Symmetric
In the case of fiber- and catheter-based PS-OCT systems, the presence of fiber-based elements exclusively in the illumination or detection path breaks the symmetry intrinsic to double-pass measurements. Comparing the off-diagonal elements of the Jones matrices reconstructed from intravascular PS-OCT measurements in the coronary artery of a human patient indeed reveals the presence of substantial amounts of energy in the asymmetric components, although they vary strongly with catheter rotation [Fig. 2(a)]. Because the symmetry-breaking elements are static, they can be estimated efficiently and accurately. Correcting for their effect dramatically reduced the energy in the asymmetric elements [Fig. 2(c)]. To accommodate for the presence of wavelength dependence, or PMD, in the system components, the correction was performed independently for various spectral bins [Fig. 2(b)]. To validate the robustness of recovering the intrinsic symmetry, we processed 344 independent coronary artery cross sections, imaged in a total of 30 patients. Whereas the original amount of energy in the asymmetric components varied greatly across this set of cross sections, it was reduced after correction to an average value of 1.1%, integrating over all points within the cross section that feature sufficient DOP, i.e., [Fig. 2(d)]. This convincingly demonstrates the intrinsic symmetry present in the measured data and the ability to recover them reliably. This symmetry is an important reduction in dimensionality, and is essential for further estimation of system elements and for the reconstruction of the depth-resolved birefringence properties.
B. Correction for Wavelength Dependence
Although the symmetrization removed part of the wavelength dependence of the system components, the symmetric measurements remained impacted by PMD. Comparing the cumulative retardance of the first, middle, and last spectral bins in Fig. 3(a) reveals a noticeable discrepancy, introduced by the spectral dependence of the system components. The angle between the (linear) cumulative optic axes of the first and the central bin further highlights an angular offset within the QU plane [Fig. 3(b)]. Owing to the established symmetry of the measurements, estimating the difference in the system components from the central bin becomes a tractable optimization problem. As presented in Figs. 3(c) and 3(d), the signal of the various spectral bins could be accurately aligned. The correction terms exhibit close-to-linear variation across the spectrum [Fig. 3(e)]. The correction for the wavelength dependence of system components worked reliably on all evaluated cross sections [Fig. 3(f)] and resulted in an average discrepancy of only 0.277, evaluated as the squared Frobenius norm of the difference between the rotation matrices of the various spectral bins and the central bin, and averaged across each of the 344 cross sections in areas with . This corresponds to an average retardation of 0.37 rad.
Once the spectral bins were aligned to depict identical sample information, they were averaged, which benefited the accuracy of the polarization measurements. The data then appeared to originate from measurements with a perfectly symmetric system, devoid of any wavelength-dependent polarization effects in the system components.
C. System and Catheter Compensation
The remaining system transmission contained both static elements and the spinning catheter. The signal from the ball lens or the interface at the inside of the catheter sheath, which provide prominent reflections at the tip of the catheter, reveals the double-pass transmission through these elements and describes a periodic evolution in the QU plane, if properly unwrapped (Fig. 4). Because the angle in the QU plane is twice the physical angle between the linear polarization states, a single rotation of the catheter traces out a double circle in the QU plane.
At the probe tip, total internal reflection directs the probing beam through the ball lens interface and the transparent sheath into the tissue, where we used the longitudinal and azimuthal directions as the local reference frame [Fig. 1(e)]. The catheter transmission corresponds to the coordinate transformation from the static laboratory frame to the rotating reference frame of the proximal end of the catheter, equivalent to a circular retarder with linearly increasing retardance. Due to the Fresnel coefficients, the total internal reflection acts as a linear retarder. However, static stress in the fiber probe may contribute additional retardance. Owing to the combined forward and backward transmissions, the overall appearance is that of a linear retarder, rotating around an apparent center in the QU plane defined by the remaining static system components. It was thus possible to independently estimate the transmission through the static elements and the rotating catheter and correct for their effect on the tissue signal. Even so, there remained a common global angular offset in the absolute orientation of the recovered cumulative tissue retardation vectors within the QU plane.
To also estimate this offset angle and obtain absolute optic axis orientations, we used the birefringence of the catheter sheath, similarly to Adams et al. . Manufactured by extrusion, the polyethylene sheaths exhibit weak birefringence, presumably due to a combination of flow-induced polymer-chain alignment and cooling-induced stress birefringence. The optic axis induced by both of these effects is aligned longitudinally or circumferentially to the catheter, corresponding to a straight line in the Poincaré sphere, and defines a convenient reference, as illustrated in Fig. 4(d).
Although the previous correction steps worked robustly and reliably on all evaluated cross sections, the estimation of the catheter transmission depends on a catheter evolution that is free of non-uniform rotational distortions, which can dynamically induce additional shear stress. Two-thirds of the cross sections resulted in a ball-fitting error equivalent to a retardation error smaller than 0.4 rad, which is the residual error obtained during PMD compensation, and generated credible optic axis maps. However, larger distortions can result in impaired optic axis maps, and would require improved strategies for estimating the catheter transmission. Conveniently, the fitting error of the catheter-transmission signal identifies questionable optic axis maps.
D. Optic Axis Imaging in Coronary Arteries
After the sequential corrections of the previous processing steps, we recovered the retardation matrices equivalent to a measurement performed in free space in the local sample coordinates. This enabled retrieval of the local, depth-resolved birefringence, , and optic axis orientation, , following the recursive procedure of Fan and Yao . Advancing layer by layer, the linear retardation vector of each layer was recovered, and its orientation within the QU plane was corrected with the cumulative effect of all overlying layers. Only layers with were considered in computing the cumulative correction signal, in order to avoid the detrimental impact of random retardation in signal-void or depolarized regions. Figure 5 shows typical optic axis orientations in the coronary artery cross sections of four patients. The optic axis is mapped using a periodic color map  to indicate the physical direction of the optic axis orientation. There is a -ambiguity, since polarization has no directionality. The brightness of the color is adjusted by the amount of birefringence to mask areas of low birefringence with ill-defined optic axis orientation. Regions with are grayed out.
Figure 5(a) shows a cross section of the right coronary artery in a 69-year-old male patient who presented with unstable angina. The conventional backscatter signal reveals a fibrotic lesion with a large calcification. The orientation map reveals distinct tissue layers within the homogeneously scattering fibrous tissue areas. With the aid of the optic axis orientation, it is possible to identify the media layer in the short segment from 10 to 11 o’clock, which exhibits a circumferential orientation, consistent with the orientation of the densely aligned smooth muscle in this layer . The bulk of the fibrous tissue appears to be aligned longitudinally. Interestingly, the most intimal tissue layer again features circumferential orientation, strongly contrasting it from the subsequent layer. In comparison, the entire intimal region results in a uniform backscatter signal.
The second example [Fig. 5(b)], from the right coronary artery of a 73-year-old male patient with stable angina, depicts very similar features. The tunica media is clearly visible in the entire cross section, except behind the calcification from 4 to 8 o’clock. The innermost layer from 3 to 6 o’clock exhibits a very distinct circumferential orientation. The atherosclerotic plaque in Fig. 5(c), imaged in the right coronary artery of a 69-year-old male patient with stable angina, is less stenotic but features distinct neovascularization. The most intimal layer at 9 o’clock and from 11 to 2 o’clock shows distinct circumferential orientation, contrasting with the longitudinal orientation of the majority of the intimal area. The vessel in Fig. 5(d) has minimal features of disease, but instead of the typical adipose-rich peri-adventitial tissue, it is embedded in a homogenously scattering tissue with high birefringence and a circumferential optic axis orientation, similar to that of the tunica media.
E. Validation with a Birefringence Phantom
To validate the reconstruction of depth-resolved optic axis orientation and compare the computed orientation maps with a known orientation, we assembled a two-layer birefringence phantom, consisting of stretched ribbons cut from a scattering polycarbonate film . Held together at different relative angles ranging from to 90˚ in steps of 22.5˚, both layers were rotated together in steps of 22.5˚ over a total angle of 180˚, yielding a total of 81 distinct combinations of axis orientations. The reconstructed optic axis maps appear very homogeneous, and quantitative evaluation of the axis orientations in a central region of interest demonstrates accurate recovery of the orientations for both layers, independent of each other, as demonstrated in Fig. 6.
Supplementary Fig. S1 presents the axis maps for all angles of orientation, whereas Fig. S2 shows the maps that result when the last processing step that corrects for overlying tissue layers is ignored, instead evaluating the apparent optic axis orientation by computing . Using the square root of the cumulative matrix of the previous layers on each side results in a linear retarder, albeit with an imprecise orientation. Figure 6(d) shows that the orientation of the first layer remains mostly unaltered in this case, whereas the optic axis orientation of the second layer is substantially distorted, and results in orientations far from the true orientation angle. This clearly illustrates the necessity for accurate depth correction in order to properly interpret the optic axis orientation in a sample consisting of multiple layers.
Tissues with a fibrillar architecture exhibit birefringence through a combination of birefringence on the molecular level and form birefringence induced by the regular arrangement of individual fibrils on a sub-resolution scale. While the scalar amount of birefringence simply indicates their presence, the optic axis orientation reveals the physical orientation of the fibrillar tissue elements. Our processing strategy enables the recovery of the optic axis orientation at each depth in the tissue from measurements with catheter-based OCT for the first time to the best of our knowledge. Depth-resolved optic axis mapping has previously been performed with PS-OCT, but only using bench-top systems with illumination and collection paths that preserve the polarization states of the illumination and scattered light.
The optic axis imaging of atherosclerotic plaques in the coronary arteries of human patients clearly revealed distinct tissue layers that appear continuous in conventional tomograms and that also feature remarkably uniform scalar birefringence. Collagen is the primary source of birefringence in the intimal layer of fibrous plaques [11,25], and plays a crucial role in the pathophysiology of atherosclerosis. We found that areas of minimal disease feature a longitudinal optic axis orientation, consistent with previous reports using polarized light microscopy . In contrast, the smooth muscle cells in the tunica media align circumferentially.
The distinct intimal layers with circumferential orientation may correspond to areas of healed plaque rupture or organized thrombus, where the wound healing response leads to a distinct orientation of the deposited collagen. This signature could help improve our mechanistic understanding of plaque progression, which is known to be associated with silent rupture and subsequent healing [27,28].
Although birefringence offers meaningful intrinsic contrast, it has to be interpreted in the context of the specific tissue being imaged. The observed birefringence is the combined effect of the various tissue elements and depends on their quantity, physical arrangement, and intrinsic polarization properties. Owing to the fibrillar nature of the constituent elements, tissue acts as a linear retarder locally and defines a slow and a fast eigen-polarization state, parallel and orthogonal to the fiber direction, respectively, in the case of positively birefringent collagen and muscle . Yet, only the component of the optic axis orthogonal to the probing beam can be measured. The birefringence of the fibrils that are oriented outside this plane will be underestimated, and retrieving the full three-dimensional optic axis orientation would require additional measurements at different probing beam angles [30,31].
The specific implementation of PS-OCT used in this work assumes absence of diattenuation in the system and throughout the sample. Because two input polarization states are used consecutively, limited phase stability and the continued rotation of the catheter prevent coherent interpretation of the measurements that would be required to obtain the full Jones matrix, including retardation and diattenuation. Instead, the measurements are directly converted to Stokes space, spatially averaged to improve their accuracy and to remove speckle noise, and then used to retrieve the retardation matrix that best explains these measurements in the least-square sense . With measurements of the full Jones matrix, it would also become possible to recover the loss of axial resolution due to spectral binning .
Without specific efforts to speed up the processing, reconstructing a cross-sectional image consisting of 1024 depth scans took 8.4 s with a 2.9 GHz CPU. Except for the catheter transmission, all correction terms are computed for an entire cross section, which makes the underlying optimization problem to be largely overdetermined. Accordingly, for the compensation for wavelength dependence, computation can be alleviated by considering only a subset of points, without noticeably impacting the accuracy of the result. The correction terms for the symmetrization and wavelength dependence do not change within a pullback and could be recycled between cross sections to accelerate the processing. Implementation on a GPU would offer additional reduction of the reconstruction time to approach real-time processing.
The key to reconstructing the depth-resolved optic axis orientation from catheter-based PS-OCT measurements is to recover the intrinsic symmetry expected in measurements performed in the backscattering direction. Although this symmetry has been assumed implicitly in previous free-space PS-OCT work [6,7], this report is the first to investigate it in detail . The symmetry enables correction for system PMD and estimation of the system component transmission without assumptions on the sample birefringence. PMD manifested only in the static fiber components in the illumination and detection arms, but was absent from the catheter and sample signals. This allowed averaging the cumulative retardation matrices among the spectral bins before recovery of the catheter transmission. The symmetrization and PMD compensation can be applied blindly on any data set, as long as the system transmission features some amount of retardation. For bench-top imaging, where the sample fiber remains static, these first processing steps suffice to retrieve the depth-resolved relative optic axis orientation, as the remaining system transmission can be estimated from the surface reflection signal, up to an unknown circular retarder. However, to recover the transmission through the rotating catheter, we rely on its retardation. The intravascular catheters employed in this work use total internal reflection at the probe tip, which conveniently generates a suitable amount of retardance. We further used the birefringence properties of the sheath to render the optic axis orientation absolute.
Adams et al. previously used sheath birefringence and the reconstruction of the apparent (i.e., uncorrected) optic axis orientation for imaging of airway in animals and humans . Assessing the alignment of the apparent optic axis of the airway wall with the optic axis of the sheath enabled accurate quantification of airway smooth muscle in vivo and opened the unique opportunity to research its role in diseases of the airway. The orthogonal alignment of the smooth muscle (circumferential) and cartilage (longitudinal), which corresponds to collinear optic axes with opposite directions, likely limited the generally detrimental impact of the overlying tissue layers on the apparent optic axis in that work. The rigorous reconstruction of the depth-resolved optic axis presented in the current paper no longer requires assumptions on the sample structure and offers a general framework for optic axis imaging.
Propagation of light through a single-mode fiber, fiber-based components, and spinning catheters impacts the transmitted polarization states. Our experimental results demonstrate that it is possible to accurately estimate this transmission and correct for its impact on tissue measurements. Illumination and detection through identical optical paths imposes a strict symmetry on the measurements, which can be readily recovered, and reduces the dimensionality of further correction steps. This makes depth-resolved optic axis imaging through a catheter possible. Optic axis orientation in coronary atherosclerotic lesions in human patients revealed an elaborate variation in their collagen organization, imperceptible to conventional OCT or scalar birefringence imaging. Disease often corresponds to a disruption of the normal tissue organization, which makes this intrinsic contrast mechanism potentially relevant in many clinical and preclinical applications of fiber- and catheter-based OCT.
National Institutes of Health (NIH) (P41EB-015903, R01HL-119065); Terumo Corporation.
See Supplement 1 for supporting content.
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