3.1.

Birefringence Axis Perpendicular to the Plane Surface

Multiple scattering of polarized light was simulated for an anisotropic infinite plane medium with birefringence axis oriented vertical to the plane surface. The reduced effective scattering matrix was hardly dependent on the azimuth around the illumination point but only dependent on the radial distance. Hence, the matrix elements were azimuthally averaged at varying radial distances. The averaged matrix elements are shown as a function of the radial distance in Fig. 1. Small but considerable values are found for $m_{02}$, $m_{12}$, $m_{13}$, $m_{20}$, $m_{21}$, and $m_{31}$. These elements exactly satisfy reciprocity; $m_{20}=-m_{02}$; $m_{21}=-m_{12}$; $m_{31}=m_{13}$. Some elements nearly satisfy reciprocity; $m_{10}\approx m_{01}$; $m_{32}\approx -m_{23}$. The simulation was also conducted for the acceptance angle of 10 deg for exiting photons and the total photon number of ${10}^9$ with the other conditions being kept the same. It was found that the reduced matrix exactly fulfills reciprocity in this case, namely $m_{10}=m_{01}$ and $m_{32}=-m_{23}$ as well as the other nondiagonal elements. The exact reciprocity may require that the direction of beam exit from the infinite plane medium is approximately normal to the plane surface.

Fig. 1

Elements of the reduced effective scattering Mueller matrix as a function of the radial distance for anisotropic media with birefringence axis perpendicular to the plane surface. The matrix elements as the ordinates are normalized by $m_{00}$ and dimensionless.

The reduced matrix averaged azimuthally was decomposed to provide the polarization parameters as a function of the radial distance, as shown in Fig. 2. Note that the $|\delta |$ value becomes approximately zero for distances in the 0.5 to 2.2-mm range. Meanwhile, ${\alpha }_2$ and ${\alpha }_3$ values exhibit jumps at distances of 0.5 and 2.2 mm. These facts indicate that the decomposition fails in the 0.5 to 2.2-mm range, whereas it yields physically reasonable matrix factors for distances longer than 2.2 mm, similar to the case of isotropic infinite plane media.²¹ Such a trend was also seen for the anisotropic media with the birefringence axis parallel or inclined to the plane surface, which are described in the succeeding sections.

Fig. 2

Polarization parameters as a function of the radial distance for anisotropic media with birefringence axis perpendicular to the plane surface. The ordinates are dimensionless except $|\delta |$ whose unit is radian.

The ${\alpha }_3$ value is reduced by 0.3 at maximum in the distance range longer than 10 mm relative to the case for the isotropic media.²¹ The ${\alpha }_1$ value is also reduced by 0.1 at maximum, whereas the ${\alpha }_2$ value barely changes in the same distance range, which results in ${\alpha }_1$ being smaller than the ${\alpha }_2$ value. The two coefficients are nearly the same in the long distance range due to randomization of linear polarizations for isotropic media. It is likely that the greater depolarization for the horizontally and circularly polarized light comes from selective depolarization due to the birefringence perpendicular to the plane surface.

The $D_{\mathrm{H}}$ value displays negative value that decreases with decreasing the distance, whereas $D_{45}$ and $D_{\mathrm{C}}$ have negligible values over the whole distance. The $D_{\mathrm{H}}$ value is $-0.26$ at the distance of 2.2 mm, which is close to the value for an isotropic medium.²¹ Probably, the horizontal linear diattenuation arising from the nature of the single scattering increases with the distance approaching the illumination point.

A double-scattering model has been proposed to rationalize the simulation results for a birefringent infinite plane medium.²² Briefly, although the photon propagates along many possible curved trajectories, photon trajectories may be approximated by three straight lines from the entrance $O$ to the exit $P$ via the intermediate points $A$ and $B$, as seen in Fig. 3(a). It is likely that the direction of the electric vector of the photon wave as well as the photon’s local reference frame is fairly preserved after repeated scatterings close to the forward direction, as seen in Fig. 3(b).

Fig. 3

Double scattering model. (a) Approximation of the photon trajectory by three lines. (b) Configuration of the local reference frame during the photon propagation.

According to the double-scattering model, the vertically propagating photon does not produce the retardation because the propagation direction is parallel to the birefringence axis. On the other hand, linearly polarized photons propagate laterally in the birefringent medium as shown in Fig. 4. Apparently, the $\delta$ value is positive during the lateral propagation of the photon because the horizontal axis of the local reference frame of the photon ${\mathbf{e}}_{\parallel }$ is parallel to the birefringence axis. In the mean time, a left-handed circular polarization of a photon is converted to the left-handed elliptical polarization with the major axis orientation $\mathbf{E}$ inclined at 45 deg to the birefringence axis, as seen in Fig. 4. Evidently, the positive value for $\delta$ is also generated. Then let us examine the variation of $|\delta |$ with the distance in Fig. 2. The $|\delta |$ value decreases to zero and increases up to $\sim 2.1$ when increasing the distance. This variation indicates that the $\delta$ value increases from a negative value to a positive value. The negative value at distances shorter than 4.7 mm is of the linear retardance, which may reflect from the nature of the single scattering, and decreases with increasing the distance, as observed for an isotropic medium.²¹ The positive value at distances longer than 4.7 mm is of the retardance, which comes from the medium birefringence, and almost linearly increases with the distance. The slope of the linear variation of $|\delta |$ is $1.35{\mathrm{cm}}^{-1}$, which is lower than the value, $1.64{\mathrm{cm}}^{-1}$, as estimated in Sec. 3.2, for the anisotropic medium with the birefringence axis parallel to the plane surface and the same birefringence amplitude. This is probably because the vertical propagation does not contribute to the retardation for the case in which the birefringence axis is perpendicular to the plane surface.

Fig. 4

Local reference frame and the polarization orientation during the lateral propagation of a polarized photon.

3.2.

Birefringence Axis Parallel to the Plane Surface

Multiple scattering of polarized light has been described in detail for an anisotropic infinite plane medium with birefringence axis oriented parallel to the plane surface.²² In this study, simulations were performed for anisotropic media that have the same birefringence orientation but have different birefringence amplitudes. Because the reduced effective scattering matrices were dependent on the azimuth and the radial distance around the illumination point, the matrices were factorized in two dimensions to obtain the polarization parameters. The retardations for these media are shown, as a function of the azimuth at the radial distance of 5 mm, in Fig. 5. The $|\delta |$ value shows periodic changes with the azimuth; it displays two larger maxima at 0 deg and 180 deg and two smaller maxima at 90 deg and 270 deg for all media. The retardations are also shown along the radial distance at the azimuth of 0 deg in Fig. 6. For the birefringence of $0.68\times {10}^{-5}$, the $|\delta |$ value continuously increases almost linearly up to $\sim 2.0$ with distance except for the range smaller than 2.4 mm, where the polarization parameters are not realistic, as noted in Sec. 3.1. In contrast, after the $|\delta |$ value increases to $\pi$, it turns into a decrease for the two cases with larger values of the birefringence. The retardance $\delta$ is negative and increases in its absolute value with the distance at the azimuth of 0 deg, as described previously.²² Hence, the $\delta$ value decreases almost linearly in the distance range longer than $\sim 3\mathrm{mm}$ for the two larger values of birefringence. The corresponding slopes of the linear variation are $-0.641$, $-1.64$, and $-3.23{\mathrm{cm}}^{-1}$ in the distance range of 10 to 20 mm for the birefringence values of $0.68\times {10}^{-5}$, $1.36\times {10}^{-5}$, and $2.72\times {10}^{-5}$, respectively. Thus, the slope for the $\delta$ variation becomes greater nearly in proportion to the birefringence value at the azimuth of 0 deg.

Fig. 5

Absolute values of the retardance $|\delta |$ as a function of the azimuth for anisotropic media with birefringence of $0.68\times {10}^{-5}$ (solid), $1.36\times {10}^{-5}$ (gray), and $2.72\times {10}^{-5}$ (dotted).

Fig. 6

Absolute values of the retardance $|\delta |$ along the radial distance. See the caption to Fig. 5.

When the birefringence slow axis is parallel to the plane surface, the retardation $\delta$ exhibits characteristic variations with the azimuth ($\phi$) depending on whether the photon propagates laterally or vertically, as shown in Fig. 7.²² When the slow axis is along the $y$ axis, the $\delta$ value becomes the largest negative along the $x$ axis ($\phi =0\mathrm{deg}$) and increases up to 0 with $\phi$ for the laterally propagating photon. Meanwhile, it is also the largest negative along the $x$ axis, increases with increasing $\phi$, and becomes the largest positive along the $y$ axis ($\phi =90\mathrm{deg}$) for the vertically propagating photon. This may be true for linearly and circularly polarized photons. It is also assumed that photons propagating laterally and vertically independently contribute to the $\delta$ value. It is likely that photons with the horizontal–vertical polarization generate no retardation and that the $|\delta |$ value becomes larger, as the orientation of polarization approaches $\pm 45\mathrm{deg}$ from 0 deg or 90 deg. On the other hand, the circularly polarized photons may be transformed by the birefringence to the elliptical polarized photons with the orientation of the major axis at $\pm 45\mathrm{deg}$ during both lateral and vertical propagations. It is suggested that the two main dependences due to the lateral and vertical propagations of photons are dominant and that the actual variations of $\delta$ are made from mixing of the two dependences. These assumptions are required to generate the characteristic azimuthal variations of the $|\delta |$ value, as shown in Fig. 5; the $|\delta |$ value takes larger maxima at 0 deg and 180 deg (largest negative $\delta$) and takes smaller maxima at 90 deg and 270 deg (largest positive $\delta$).

Fig. 7

Azimuthal dependence of the relative retardation produced during the lateral and vertical propagations of light. The ordinate unit is arbitrary. The two main contributions are mixed with varying their ratio, which produces intermediate dependences.

3.3.

Birefringence Axis Inclined to the Plane Surface

Multiple scattering of polarized light is simulated for anisotropic media with the birefringence axis that is inclined at 45 deg to the plane surface along the $y$ axis. The polarization parameters are shown as two-dimensional maps in Fig. 8 (the matrix elements were not averaged). Each image shows a circular 2 cm radius area centered at the point of beam incidence. The parameters are symmetric or antisymmetric with respect to the $y\text{-}z$ plane but are not either with respect to the $x\text{-}z$ plane. In contrast, the parameters show second- or fourth-order rotational symmetry when the birefringence axis is parallel to the plane surface.²² This symmetric feature is remarkable for the ${\alpha }_2$ value, as seen in the two-dimensional maps of Fig. 9, for media with the birefringence axis inclined at varying angles to the plane surface. The nonsymmetry with respect to the $x\text{-}z$ plane is obvious even for the inclination angle of only 9.9 deg. Moreover, ${\alpha }_2$ is larger along the $x$ axis than along the $y$ axis for the birefringence parallel to the plane surface, whereas the ${\alpha }_2$ variation becomes more prominent between the positive and the negative $y$ directions as the inclination angle increases.

Fig. 8

Two-dimensional maps of the polarization characteristics. The scale bars are dimensionless except $|\delta |$, $|\psi |$, and $\theta$ whose unit is radian.

Fig. 9

Depolarization coefficient for the 45 deg linearly polarized light ${\alpha }_2$ for media with the birefringence axis inclined at the designated angles. The scale bar is dimensionless.

The parameters are also shown as a function of the azimuth around the illumination point at the radial distance of 15 mm in Fig. 10. Evidently, $D_{\mathrm{H}}$, $|\delta |$, and $\theta$ are nonsymmetric between the two ranges of 0 deg to 180 deg and 180 deg to 360 deg. Then, let us pay attention to the dependence of $|\delta |$. There are two large maxima at the azimuths of 0 deg and 180 deg and two small minima at the azimuths of 90 deg and 270 deg. The $|\delta |$ value shows shallow valleys between these maxima, of which the bottom has offset values of $\sim 1.2$. This is also in contrast to the case in which the birefringence axis is parallel to the plane surface.

Fig. 10

Polarization characteristics along the azimuthal direction at 15-mm distance. The ordinates are dimensionless except $|\delta |$, $|\psi |$, and $\theta$ whose unit is radian.

Let us examine using the double-scattering model how the birefringence affects the retardance according to the propagation direction of photons. When a photon propagates vertically to the surface plane, it propagates forward at the origin into the page and, after it travels laterally along the azimuth $\phi$, it propagates backward out of the page, as seen in Fig. 11(a). The local reference frame ${\mathbf{e}}_{\parallel }$ and ${\mathbf{e}}_{\perp }$ is aligned along the azimuth for the forward and backward propagations; the horizontal axis ${\mathbf{e}}_{\parallel }$ becomes reversed in direction between both propagations. Because the birefringence slow axis for the photon ${\mathbf{b}}^{\prime }$ is along the $y$ axis, the photon may behave in the same manner as the case in which the slow axis itself is along the $y$ axis. Next, examine a photon propagating laterally along the surface plane. Assume that the birefringence slow axis $\mathbf{b}$ is inclined at 45 deg to the surface plane. When the horizontal axis of the laboratory reference frame ${\mathbf{e}}_{\parallel }^0$ is aligned along the $x$ axis ($\phi =0\mathrm{deg}$), as seen in Fig. 11(b), the birefringence slow axis $\mathbf{b}$ itself is parallel to the local reference frame ${\mathbf{e}}_{\perp }$ and ${\mathbf{e}}_{\parallel }$ and is inclined at an orientation $\theta$ of 45 deg to the horizontal axis ${\mathbf{e}}_{\parallel }$. As the axis ${\mathbf{e}}_{\parallel }^0$ makes an azimuth $\phi$ to the $x$ axis, the slow axis for photon ${\mathbf{b}}^{\prime }$ is oriented at an angle smaller than 45 deg to the axis ${\mathbf{e}}_{\parallel }$, as seen in Fig. 11(c). Because the orientation of the slow axis for the photon changes with the azimuth, the linearly polarized photon behaves differently depending on its orientation of polarization. In addition, the circularly polarized photon is converted to the elliptically polarized photon with different orientations of the major axis depending on the azimuth.

Fig. 11

Configuration of the local reference frame when a photon propagates (a) vertically, and laterally along (b) the x axis and (c) an azimuth φ.

The retardation $\delta$ is shown as a function of the azimuth for the birefringence whose axis is inclined at varying angles to the plane surface in Fig. 12. As the inclination angle becomes larger, the amplitudes of the maxima decrease, the valleys between the maxima are broadened, and the offset values become larger. This clearly reveals that, the more inclined the birefringence axis, photons propagating laterally produce a more complex mixture of the scalar and vector retardances, which leads to less characteristic dependence of the $\delta$ value on the azimuth. Although the vertical propagation still keeps its characteristic feature in the $\delta$ value, its contribution is more reduced. Thus, the total contribution from the lateral and vertical propagations to the $\delta$ value shows a broader variation with a larger offset.

Fig. 12

Absolute values of the retardance $|\delta |$ along the azimuthal direction at the 10-mm distance. The birefringence axis is inclined at the designated angles.

Next, let us inspect the azimuthal dependence of the orientation of the fast axis $\theta$ as a linear retarder of the birefringent medium, as seen in Fig. 13. For the birefringence with its axis parallel to the plane surface, as the azimuth increases from 0 deg to 90 deg, $\theta$ decreases from 0 to $-\pi /4$, turns its sign at an intermediate azimuth, and decreases to 0. This azimuthal dependence corresponds to the orientation of the fast axis relative to the horizontal axis of the laboratory reference frame ${\mathbf{e}}_{\parallel }^0$; the $\theta$ change corresponds to the orientation change from 0 deg to $-45\mathrm{deg}$ below the intermediate azimuth and from $-45\mathrm{deg}$ to $-90\mathrm{deg}$ above the intermediate azimuth. Because the orientation of the fast axis is constant at 0 deg during the lateral propagation of photons, it is mainly determined by the vertical propagation. However, when the birefringence is inclined at 9.9 deg, $\theta$ has a value $\sim 0.1$ at the azimuth of 0 deg. As the azimuth increases to 90 deg, the orientation $\theta$ increases to $\pi /4$ until an intermediate azimuth, turns its sign, and then increases to 0. This azimuthal dependence is partly influenced by the lateral propagation; the $\theta$ change corresponds to the orientation change of the fast axis from $-9.9\mathrm{deg}$ to $-45\mathrm{deg}$ below the intermediate azimuth and from $-45\mathrm{deg}$ to $-90\mathrm{deg}$ above the intermediate azimuth. Note that the sign of $\theta$ in the range of 0 deg–180 deg is reversed between the cases in which the birefringence axis is parallel and inclined to the plane surface. This may originate from the different orientation changes of the fast axis during the lateral and vertical propagations of photons. The orientation turns its sign around the azimuths of 0 deg and 180 deg during the vertical propagation, whereas it does not during the lateral propagation. Even though the inclination of the birefringence increases to 45 deg, the fundamental variation of $\theta$ is the same; the value at the azimuth 0 deg increases and the smallest intermediate azimuth shifts to a smaller azimuth.

Fig. 13

Fast-axis orientation as a linear retarder along the azimuthal direction at the 10-mm distance. The birefringence axis is inclined at the designated angles.

Finally, let us examine why the polarization parameters are symmetric or antisymmetric with respect to the $y\text{-}z$ plane, but why they are not either with respect to the $x\text{-}z$ plane for anisotropic media with birefringence directions that are inclined to the surface plane. The distributions of the polarization parameters are different due to the nonsymmetry below and above the $x$ axis for only a 9.9 deg inclination of the birefringence direction, as noted above. This suggests that the nonsymmetry does not come from the birefringence itself but comes from the nature of the multiple scattering in infinite plane media. A possible explanation is as follows. The photon that is injected normal to the plane surface vertically travels into the medium up to a fairly deep depth due to the first vertical propagation and the following near-forward scattering events. Then the photon migrates away from the central area to the radial distance, as shown in Fig. 14(a). Such a photon trajectory may be well approximated by a double-scattering model in which the lateral propagation after the first scattering is oblique upward at a small angle $\chi$ with respect to the plane surface, as is represented by a line AB in Fig. 14(b). This obliqueness is originated from the nature of the multiple scattering but is not influenced by the direction and amplitude of the birefringence. The effective birefringence for the propagating photon becomes larger in the positive $y$ direction than in the negative $y$ direction due to the oblique lateral propagation because the angle between the birefringence and the propagation direction of the photon is closer to 90 deg along the positive $y$ direction, as seen in Fig. 14(c). Apparently, the effective birefringence is identical between the positive and negative directions along the $x$ axis in spite of the oblique propagation. Thus, the difference in the effective birefringence causes the nonsymmetry of the polarization parameters with respect to the $x\text{-}z$ plane in anisotropic infinite plane media with the birefringence direction that is inclined to the plane surface. It is likely that the two-dimensional distribution of the polarization parameters is not affected very much for media whose birefringence axis is parallel to the plane surface, when the lateral propagation is oblique at angles smaller than $\sim 10\mathrm{deg}$. This is supported by the fact that the $|\delta |$ variation for the case in which the birefringence axis is inclined at 9.9 deg to the plane surface is not very different from the case of zero inclination of the birefringence axis, as seen in Fig. 12.

Fig. 14

Modification of the double-scattering model in which the lateral propagation direction is oblique at a small angle $\chi$ with respect to the plane surface. (a) A photon trajectory, (b) its approximation by three lines, and (c) the configuration of the local reference frame during the photon propagation.