3.1.1.

Anisotropic orientation of nonbirefringent cylindrical samples and pure birefringent samples

Figure 5 shows the anisotropic orientation of pure anisotropic scattering and pure birefringence by measuring glass fibers and air-dried tendons, respectively. In Fig. 5(a), all three parameters can track the orientation of the fibers, indicating that nonbirefringent cylindrical scatters produces both diattenuation and form birefringence. ${\alpha }_P$ is perpendicular to the fiber long axis, meaning that the electric field vector component with the vibration direction perpendicular to the glass fiber axis has a higher transmittance. $\theta$ is consistent with the glass fiber axis, which means that the fast axis corresponding to the form birefringence induced by anisotropic scattering is parallel to the fiber long axis. ${\alpha }_q$ is parallel to the fiber long axis, also indicating that the scattering by cylinders causes an LR. In Fig. 5(b), the tendon becomes nearly transparent after air-drying, and its anisotropy is dominated by the intrinsic birefringence of the collagen fibers, while the anisotropic scattering effect is reduced. Therefore, ${\alpha }_P$ cannot track the orientation of collagen fibers. The fast-axis orientation by $\theta$ is perpendicular to the collagen fibers’ long axis, which also confirms that collagen fibers have a positive birefringence effect and is in agreement with the conclusions of previous researchers.^18,19 ${\alpha }_q$ is perpendicular to the fiber long axis, meaning that the intrinsic birefringence causes an LR. For details of quantitative values of diattenuation and LR, refer to Tables S1 and S2 in the Supplementary Material.

Fig. 5

Mueller matrix images and anisotropic orientation parameters of samples at different spatial orientations: (a) well-aligned glass fibers, (b) air-dried tendon (thickness: $300\mu \mathrm{m}$). Each subimage is divided into $4\times 4$ areas by an orange grid, and the top to bottom rows correspond to the four spatial orientations of the sample at 0 deg, 45 deg, 90 deg, and 135 deg; the left to right columns correspond to the Mueller matrix images of the measured sample and the FDH of the three parameters $\theta$, ${\alpha }_P$, and ${\alpha }_q$ in polar coordinates. All Mueller matrix images share the same color bar.

3.1.2.

Anisotropic orientation of birefringent cylindrical samples

The phantom is prepared by neatly winding two layers of silk along the same orientation on the alloy scaffold. We measured their Mueller matrix images and calculated the FDH of three anisotropy parameters $\theta$, ${\alpha }_P$, and ${\alpha }_q$, as shown in Fig. 6. First, the FDH of $\theta$ shows a bimodal distribution with two peak angular orientations perpendicular to each other, which confirms that silk is a fiber with intrinsic birefringence. The fiber axial peak is due to cylindrical scattering, and the peak perpendicular to the fiber axis is due to intrinsic birefringence of the silk. By comparing the probability densities of the two peaks, we can see that the intrinsic birefringence of this two-layer silk sample contributes more to the anisotropy than the cylindrical scattering. Second, the FDH of ${\alpha }_P$ shows a unimodal distribution, implying that this parameter is only related to the diattenuation caused by cylindrical scattering. The peak angle of ${\alpha }_P$ is perpendicular to the long axis of silk. Compared with the ${\alpha }_P$ of glass fiber in Fig. 5(a), the ${\alpha }_P$ of both glass fiber and silk is perpendicular to the fiber axis. So ${\alpha }_P$ is not affected by the intrinsic birefringence and robust in characterizing fiber orientation. Third, compared with Fig. 5(a), the ${\alpha }_q$ of glass fiber and silk is not consistent for the same fiber orientation, implying that ${\alpha }_q$ is not robust in characterizing fiber orientation. For fibers without intrinsic birefringence, ${\alpha }_q$ is only related to the form birefringence caused by cylindrical scattering, and the peak angle of ${\alpha }_q$ is parallel to the fiber axis. For fibers with intrinsic birefringence, ${\alpha }_q$ will be affected by both form and intrinsic birefringence. When multiple anisotropic effects are superimposed in the sample, the parameter ${\alpha }_q$ becomes unstable and cannot effectively characterize the anisotropic orientation. Therefore, we preferentially use $\theta$ and ${\alpha }_P$ when discriminating anisotropy of biological tissues. The fast axis corresponding to form birefringence is parallel to the fiber axis, which can be determined by the orientation perpendicular to ${\alpha }_P$. Then we can distinguish the peak due to form birefringence in the bimodal FDH of $\theta$, and the other peak corresponds to intrinsic birefringence. In addition, the fluctuation range of peak angle of FDH for $\theta$ and ${\alpha }_P$ is given in Table 1. For details of quantitative values of diattenuation and LR, refer to Table S3 in the Supplementary Material.

Fig. 6

Mueller matrix images and anisotropic orientation parameters of silk at different spatial orientations. This figure is divided into $4\times 4$ areas by an orange grid, and the top to bottom rows correspond to the four spatial orientations of the sample at 0 deg, 45 deg, 90 deg, and 135 deg; the left to right columns correspond to the Mueller matrix images of the measured silk and the FDH of the three parameters $\theta$, ${\alpha }_P$, and ${\alpha }_q$ in polar coordinates. All Mueller matrix images share the same color bar.

Table 1

Fluctuation range of peak angle of FDH for anisotropic orientation parameters of silk.

3.2.1.

Evaluating tissue mechanical properties of skeletal muscle

Skeletal muscle tissue is widely distributed in mammals, and its anatomical structure is shown in Fig. 7(a). The basic unit of skeletal muscle is the myofibrils, and the fast axis corresponding to intrinsic birefringence is oriented perpendicular to its long axis. There are periodic sarcomeres in myofibrils,^15,16 which are composed of actin and myosin. Myosin forms the thick myofilaments with birefringence, actin forms the thin myofilaments without birefringence, and the two types of myofilaments form isotropic and anisotropic bands in sarcomere. When the muscle contracts, the sarcomere becomes shorter, and when the muscle is relaxed, the sarcomere becomes longer. Here we use transverse and longitudinal stress stretching to achieve sarcomere length variation.¹²

Fig. 7

Experimental results of chicken breast under different stretching states: (a) schematic diagram of skeletal muscle anatomy. (b) Schematic diagram of muscle stretching state, NS corresponding to the initial state without stretching, CS1 and CS2 corresponding to transverse stretching to 3.5 and 4 cm, LS1 and LS2 corresponding to longitudinal stretching to 3.5 and 4 cm, respectively. (c)–(g) FDH of $\theta$ corresponding to different muscle stretching states. (h)–(l) FDH of ${\alpha }_P$ corresponding to different muscle stretching states. (m) The $\text{peak}({\theta }_b)$, $\text{peak}({\theta }_c)$ (corresponds to the right ordinate), and $\gamma$ value (corresponds to the left ordinate) corresponding to different muscle stretching states.

A chicken breast slice (size: $3\mathrm{cm}\times 3\mathrm{cm}$, thickness: $800\mu \mathrm{m}$) was used for the experiment considering its lower fat content and less fascia between the muscle fibers. As shown in Fig. 7(b), the actual orientation of muscle fibers is in the vertical direction. Figures 7(h)–7(l) show the FDHs of ${\alpha }_P$ corresponding to different stretching states. The peak center angle is always 0 deg, perpendicular to the actual orientation of muscle fibers, indicating that external stretching does not affect the accuracy of ${\alpha }_P$ in fiber orientation tracking. Figures 7(c)–7(g) show the bimodal distribution characteristic in the FDHs of $\theta$ corresponding to different stretching states. Here 90 deg is the fast-axis angle corresponding to form birefringence, and 0 deg is the fast-axis angle corresponding to intrinsic birefringence. Figure 7(e) corresponds to the initial state (NS) without stretching. In the first case, from NS to CS2, the transverse stretch is strengthened, with the peak at 90 deg increasing and the peak at 0 deg decreasing. In the second case, the longitudinal stretch is strengthened, with the peak of 90 deg decreasing and the peak of 0 deg increasing. This is also confirmed by the trend of $\text{peak}({\theta }_b)$, $\text{peak}({\theta }_c)$, and $\gamma$ in Fig. 7(m). These results are consistent with the myofilament sliding theory.¹⁵ The birefringence of muscle tissue varies with sarcomere length because some of the contractile proteins change their molecular conformation or orientation. When skeletal muscle is stretched laterally, the sarcomere becomes shorter, more actin filaments are mixed into myosin filaments, the number of overlapping fibers per unit length increases, and the form birefringence caused by anisotropic scattering is enhanced. When skeletal muscle is stretched longitudinally, actin filaments are separated from myosin and the number of overlapping fibers per unit length will decrease as the length of the sarcomere increases, reducing the anisotropy caused by cylindrical scattering. Due to sarcomere stretching, the lattice spacing of myosin molecules becomes smaller, which can increase the anisotropy caused by molecular birefringence. Obviously, this experiment also confirms the potential of the orientation analysis of polarization parameters applied to evaluate stress state of skeletal muscles. For details of quantitative values of diattenuation and LR, refer to Table S5 in the Supplementary Material.

3.2.2.

Evaluating anisotropy differences from different tissue types

Tendons are the connective tissue that links skeletal muscle to bone. Tendon is composed of collagen fibers, different from myofibrils in molecular structure. Collagen fibers have uniformly distributed positive birefringence effects, while myofibrils contain sarcomeres that periodically appear as “A band” and “I band.” Using the anisotropic orientation parameters to distinguish these two kinds of tissues can provide a valuable reference to further understand the optical properties of the microstructure of complex biological tissues.

Comparison between bovine skeletal muscle and bovine tendon with the same thickness

Figures 8(a) and 8(b) show Mueller matrices of bovine skeletal muscle and bovine tendon with both $600\mu \mathrm{m}$ thickness. The $3\times 3$ elements in the lower right corner of the matrix have obvious differences. The absolute values of M22, M33, and M44 of the bovine tendon are closer to 0, due to the stronger depolarization by the anisotropic scattering of the bovine tendon. In Figs. 8(i) and 8(j), both the center angle of the ${\alpha }_p$ FDH of bovine skeletal muscle and bovine tendon is 90 deg, perpendicular to the actual long axis of the fiber. The ${\alpha }_p$ FDH of skeletal muscle is wide and has a lower peak value around 0.08, whereas the ${\alpha }_p$ FDH of tendon is narrow and has a higher peak value around 0.16. These phenomena suggest that the alignment of fibers in tendons is better than in skeletal muscle. Figures 8(e) and 8(f) show the $\theta$ FDH of bovine skeletal muscle and bovine tendon, whose distribution characteristics are obviously different. Skeletal muscle’s bimodal FDH confirms that it contains both form birefringence and intrinsic birefringence. In contrast, the tendon only shows a single peak at 0 deg, corresponding to the form birefringence caused by anisotropic scattering, while the intrinsic birefringence is obliterated by multiple scattering.

Fig. 8

Experimental results of bovine skeletal muscle and bovine tendon. In each measurement, the actual long axis of the sample was 0 deg. The measured samples were bovine skeletal muscle with a thickness of $600\mu \mathrm{m}$, and bovine tendon with a thickness of 600, 300, and $200\mu \mathrm{m}$, respectively. (a)–(d) The corresponding Mueller matrix; (e)–(h) FDH of the corresponding $\theta$; and (i)–(l) FDH of the corresponding ${\alpha }_P$. For details of quantitative values of diattenuation and LR, refer to Table S6 in the Supplementary Material.

4.2.1.

Difference between $\theta$ and ${\alpha }_q$ in detecting fast axis

$\theta$ and ${\alpha }_q$ are both parameters to extract the fast-axis information of biological fibers. For pure cylindrical samples [e.g., glass fiber in Fig. 5(a)] or pure birefringent samples [e.g., air-dried tendon in Fig. 5(b)], the fast-axis tracking ability of both parameters is the same. However, for a sample composed of birefringent fibrous scatterers (e.g., silk fiber in Fig. 6), the fast-axis tracking ability of these two parameters is not the same. The bimodal distribution of $\theta$ can more effectively reflect the variation of anisotropy within the sample (two perpendicular peaks correspond to the intrinsic birefringence and form birefringence), while the FDH of ${\alpha }_q$ is unstable in tracking the fast axis. The reason for this difference can be found in the mathematical calculation theory of the two parameters.

For the calculation of $\theta$, the measured Mueller matrix $M$ was decomposed into the product of scattering depolarization matrix $M_{\mathrm{\Delta }}$, diattenuation matrix $M_D$, and phase retardance matrix $M_R$ according to MMPD, as shown in Eq. (7). This separates the scattering and diattenuation information in the original Muller matrix $M$, and the birefringence information is contained in $M_R$. Then the parameter $\theta$ is calculated from $M_R$ according to Eq. (1). For a birefringent cylindrical sample, the vertical bimodal distribution in $\theta$ FDH can be seen

For the calculation of ${\alpha }_q$, Eq. (3) directly calculate ${\alpha }_q$ using M42 and M43 elements in the measured Muller matrix $M$ according to MMT, without considering the influence of scattering and diattenuation on these two matrix elements. For a pure birefringent sample (without scattering depolarization and diattenuation), such as QWP, the theoretical Muller matrix of QWP is

In Eq. (8), $m_{42}=\sin (2\theta )$, $m_{43}=-\cos (2\theta )$. And through Eq. (3), we can clearly see that $\theta$ and ${\alpha }_q$ are equivalent. However, when there is multiple polarization information in the sample, $m_{42}$ and $m_{43}$ are the coupling of multiple effects (e.g., scattering and birefringence), and the physical meaning of the two elements is no longer as clear as in Eq. (8). ${\alpha }_q$ will have errors when Mueller matrix contains strong anisotropic linear depolarization,²⁹ and it is not reasonable to calculate ${\alpha }_q$ directly through Eq. (3). Therefore, for birefringent cylindrical samples (as shown in Fig. 9), the parameter $\theta$ derived from MMPD should be used preferentially to evaluate the fast-axis orientation. Meanwhile, when the Mueller matrix contains weak depolarization and weak diattenuation, ${\alpha }_q$ still can track the fast-axis orientation (as shown in Fig. 5).²⁹

4.2.2.

Analysis of measurement error of ${\alpha }_P$ and ${\alpha }_q$

Based on the experimental results in Sec. 4.1, we analyzed the error sources of ${\alpha }_P$ and ${\alpha }_q$ from the perspective of numerical calculation. In Eqs. (2) and (3), $m_{21}$ and $m_{43}$ are used as denominators respectively, and the closer their absolute values are to zero, the more likely they are to cause large errors. Figure 10 shows the statistical boxplot graph for $m_{21}$ and $m_{43}$ in the Muller matrix corresponding to Figs. 9(a)–9(e). It is obvious that the value of $m_{43}$ is closer to 0, which indicates that ${\alpha }_q$ has a large error. In contrast, the value of $m_{21}$ is far from zero, so ${\alpha }_P$ has relatively high reliability. In biological fibers, ${\alpha }_P$ is mainly associated with cylindrical scattering. As shown in Figs. 5(a) and 6, no matter the cylinder with or without birefringence is, there will be diattenuation caused by cylinder scattering, and ${\alpha }_P$ correspondingly presents a unimodal distribution with its peak angle perpendicular to the actual long axis of the fiber. In Figs. 7(h)–7(l) and Figs. 8(i)–8(l), ${\alpha }_P$ of muscle fibers and tendons also show a robust effect.

Fig. 10

Boxplot of values of $m_{21}$ and $m_{43}$ of neatly arranged silk. The thicknesses of the silk are 2, 4, 6, 8, and 10 layers, respectively.

4.3.1.

Fast-axis orientation of PCM

As to PCM, the light scattering and transmission process in the silk sample occur simultaneously, and the FDH of $\theta$ shows a vertical bimodal distribution, as shown in Fig. 6. However, previous work tended to average all data of $\theta$ to obtain the overall fast-axis orientation of the sample, ignoring the bimodal characteristics of the statistical distribution. If the FDH of $\theta$ is a unimodal distribution, the global mean can represent the principal orientation. However, if the bimodal distribution also takes the global average, it may lead to errors in orientation evaluation. Therefore, for birefringent cylinders, it is more accurate to extract information from bimodal statistical distribution characteristics. The experimental results in Secs. 3.2 and 4.1 show that the changes in the two peaks in the bimodal distribution can be used to track the changes in sample microstructure.

4.3.2.

Fast-axis orientation of SCM

As to SCM, the incident light passes through the glass fiber layers with different orientations in sequence, as shown in Fig. 11(a). In Fig. 11(b), two layers of glass fibers aligned at 60 deg are stacked. The solid red line is the overlapping area of the two layers of glass fibers, and the dotted green line is the overall fiber area, including the overlapping area and two single-layer areas. In Fig. 11(c), the $\theta$ FDH in the green dashed line corresponds to the overall fiber region surrounded by the green dashed line in Fig. 11(b), which is reflected as three peaks near 0 deg, 30 deg, and 60 deg. The $\theta$ FDH in the solid red line corresponds to the stacked fiber region within the solid red line in Fig. 11(b), which is reflected as a unimodal distribution with a central angle $\sim 30\mathrm{deg}$. These results show the fast-axis orientation differences of SCM and PCM based on polarization analysis. For a layered stacking of nonbirefringent fibers, such as glass fibers, fast axis is suitable for characterizing the general orientation after the integration of multiorientation anisotropy. When the scattering intensity of two layers of fibers is comparable, the final equivalent fast axis is the angular bisector of the fast-axis orientation of the two layers.

Fig. 11

(a) Schematic diagram of light passing through two layers of nonbirefringent cylinders with different orientations. (b) M11 image with two layers of glass fibers stacked at an angle of 60 deg. (c) FDH of $\theta$, where FDH of the green dashed line corresponds to the area surrounded by the green dashed line in (b). FDH of the red solid line corresponds to the area surrounded by the red solid line in (b).

4.3.3.

Fast-axis orientation of SPCM

Few nonbirefringent fibers in biological tissue are as perfect as glass fibers. Given that most fibers are birefringent cylinders (such as muscle and tendon), SPCM is more common in biological tissues. As shown in Fig. 12(a), for birefringent fibrous scatterers like the silk sample, the simultaneous existence of two anisotropic sources in the same layer, mixed with stacked fiber layers with different orientations, can make it difficult to distinguish the specific fast-axis orientation. As shown in Fig. 12(b), we measured two layers of silk stacked at 45 deg. The $\theta$ FDH of the sample is shown in Fig. 12(c), and the stacked silk generates an intricate multipetal pattern. This multipetal distribution cannot track the actual fiber orientations, but it can be used as an indicator to assess whether the order of the fiber is broken.

Fig. 12

(a) Schematic diagram of light passing through two layers of birefringent cylinders with different orientations; (b) M11 image with two layers of silk fibers stacked at an angle of 45 deg; and (c) FDH of $\theta$, corresponds to the area surrounded by the red solid line in (b).