Both experiments and simulations have shown that for a sample containing fibrous scatterers along a particular direction, as we rotate the sample, therefore, change the orientation of cylindrical scatterers, some of the Mueller matrix elements display periodic intensity variations. In both the experiments and MC simulations, it has been concluded that as the direction of the fibrous structures varies from 0 to 360 deg, the $m22$, $m23$, $m32$, and $m33$ elements repeat four times, whereas the $m12$, $m13$, $m21$, and $m31$ repeat only twice. A closer examination of both the experimental and simulated results reveals that, for normal incidences, the $m12$, $m21$, $m13$, $m31$, $m22$, $m33$, $m23$, and $m32$ can be fitted to trigonometric forms as shown in Eq. (1).^{15} There are three independent parameters that can be extracted from Eq. (1): $t$ [$t1$ or $t2$ shown in Eq. (2)] is related to the amplitude of the trigonometric curves or the magnitude of anisotropy for the samples, $b$ [shown in Eq. (3)] is related to an offset that can be affected by both the structure and density of the scatterers, and $x$ represents the direction of the fibrous structure. There are two ways to obtain $x$ from the Mueller matrix using either Eqs. (4) or (5). However, $x$ extracted from Eqs. (4) or (5) has four or two periods in a $\pi $ range, and therefore allows us to determine the alignment angles in a $\pi /4$ range and $\pi /2$ range, respectively. To determine the orientation angle in a $\pi $ range, we can combine Eq. (5) with a positive or negative value of the $m13$ element (or other Mueller matrix elements) as a determination condition [shown in Eq. (6)]. Therefore, by measuring the $m12$ and $m13$ elements, we can calculate the orientation of the aligned fibrous microstructures Display Formula
${m22=t1\u2009cos\u20094x+bm33=\u2212t1\u2009cos\u20094x+bm23=m32=t1\u2009sin\u20094x{m12=m21=t2\u2009cos\u20092xm13=m31=t2\u2009sin\u20092x,$(1)
Display Formula${t1=(m22\u2212m33)2+(m23+m32)22t2=m122+m132,$(2)
Display Formula$b=m22+m332,$(3)
Display Formula$tan(4x)=m23+m32m22\u2212m33,$(4)
Display Formula$tan(2x)=m13m12,$(5)
Display Formula$x\u2208{[0,\pi 2]if\u2009\u2009(m13\u22650)(\pi 2,\pi )if\u2009\u2009(m13<0).$(6)