Physically Realizable Space for the Purity-Depolarization Plane for Polarized Light Scattering Media

We propose a physically realizable space for the polarized light scattering measurement using the Stokes-Mueller formalism by a purity-index–depolarization-index ($\mathrm{PI}-P_{\mathrm{\Delta }}$) plane. The parameter PI is defined from indices of polarimetric purity (IPP), which exhibits the overall magnitude of the polarimetric randomness of a medium, while the depolarization index ($P_{\mathrm{\Delta }}$) delineates a proper global degree of polarimetric purity and may also refer to the average measure of depolarization power of the scattering medium. Subregions and curves connecting the edge points in the plane are obtained by imposing certain constraints on the IPP; consequently any point on the subregion indicates the information related to a decomposition of the Mueller matrix into its components as a convex sum. From the same set of constraints, complete information about the depolarization index versus the entropy [$S(\mathbf{M})-P_{\mathrm{\Delta }}$] diagram is recovered. This work provides a simple geometric representation and a deeper perceptivity of the light scattering media comprising depolarization.

Figure 1

Feasible spaces with curves for (a) the $\mathrm{PI}-P_{\mathrm{\Delta }}$ plane with the edge points $A$ (0, 0), $B$ ($1/3$, $1/\surd 3$), $C$ ($1/\surd 3$,$\surd (2/3)$), and $D$ (1,1). (b) The $S(\mathbf{M})-P_{\mathrm{\Delta }}$ plane with the cusp points $A$ (0, 1), $B$ [$1/3$,${\log }_4(3)$], $C$ ($1/\surd 3$,$1/2$), and $D$ (1, 0). The regions I, II, III, and IV are specified by the areas $BCD$, $ABC$, $ABD$, and $ACD$, respectively. ${\mathbf{M}}_a$ is the experimentally measured matrix, Ref. [27], whereas ${\mathbf{M}}_b$ and ${\mathbf{M}}_c$ are generated from the formulations given in Refs. [6, 27, 28].

Figure 2

MC simulations for spherical scatterers in forward- (red points) and backscattering (blue points) in the Mie regime with diameters increased from 1.02 to $1.46\mu \mathrm{m}$ in steps of 0.03. The scattering coefficients are 5 (square), 10 (triangle), and 15 (asterisk) ${\mathrm{cm}}^{-1}$. The increase in the size of the scatterer is shown by increasing the size of the points.