Accuracy optimization of combined multiparameter measuring systems with application to polarized light microscopy

We study the accuracy of combined multiparameter measuring systems (CMPMSs) that determine several unknown quantities from measurements of a single variable at different preprogrammed conditions determined by control parameters. To reduce inaccuracies of determined quantities, we propose a mathematical method for selection of control parameters that are optimal for all possible values of determined quantities. Using the submultiplicativity of the spectral and Frobenius matrix norms, we construct the upper bound of the error function and determine the set of control parameters by minimizing this bound. To demonstrate the capability of our method for CMPMSs, we apply it to the polarized light microscopy technique called LC-PolScope, which is used for determining inhomogeneous two-dimensional fields of optical retardation and orientation of the slow optic axis in thin organic and inorganic samples. We compare the computed set of control parameters with other sets, including the one used in the PolScope, and demonstrate that our choice of control parameters works very well even though it does not take into account any specific features of the PolScope. We expect that our method will be successful in various CMPMSs, as it is applicable to any error distribution of the control parameters and measured values.

Figure 1

Scheme of the PolScope microscope. Liquid crystal plates LCA and LCB have variable values of retardances, ${\alpha }_1$ and ${\alpha }_2$; $\lambda /4$ is a quarter-wave plate with phase shift between polarization components equal to $\pi /2$; $P$ and $A$ are the linear polarizer and analyzer which transmit only the horizontal component of the beam.

Figure 2

Representation of the control parameters of measurement ${\mathit{A}}_i=[{\alpha }_{i1},{\alpha }_{i2}]$ through the Jones vector (36) on the Poincaré sphere in the space of the Stokes parameters, $S_1,S_2,S_3$. $\chi$ is the ellipticity angle and $\psi$ defines the orientation of the major axis of the polarization ellipse.

Figure 3

Sets of control parameters ${\mathit{A}}^{\mathit{B}}$, ${\mathit{A}}^{\mathbf{90}}$, ${\mathit{A}}^{\mathbf{11}}$, ${\mathit{A}}^{\mathit{T}}$ shown on the Poincaré sphere.

Figure 4

Normalized standard deviations $\sigma \left(\delta {\mathrm{\Delta }}_M\right)/{\sigma }_{\alpha }$ and $\sigma \left(\delta {\varphi }_M\right)/{\sigma }_{\alpha }$ vs $\mathrm{\Delta }$ for different values of $\varphi$ and the set $\mathit{A}={\mathit{A}}^{\mathit{B}}$.

Figure 5

Normalized standard deviations $\sigma \left(\delta {\mathrm{\Delta }}_B\right)/{\sigma }_{\mathrm{\Xi }}$ and $\sigma \left(\delta {\varphi }_B\right)/{\sigma }_{\mathrm{\Xi }}$ vs $\mathrm{\Delta }$ for different values of $\varphi$, set $\mathit{A}={\mathit{A}}^{\mathit{B}}$.

Figure 6

$\varphi$-averaged standard deviations $\overline{\delta {\mathrm{\Delta }}_M}$, $\overline{\delta {\varphi }_M}$, $\overline{\delta {\mathrm{\Delta }}_B}$, $\overline{\delta {\varphi }_B}$ vs $\mathrm{\Delta }$ for sets ${\mathit{A}}^{\mathit{B}}$, ${\mathit{A}}^{\mathbf{90}}$, ${\mathit{A}}^{\mathbf{11}}$, ${\mathit{A}}^{\mathit{T}}$. The inset shows the same plots for small values of $\mathrm{\Delta }\in [0,\pi /180]$.

Figure 7

Effect of noise contamination on retardation: (a) original (noise-free) image ${\mathrm{\Delta }}^{*}(x,y)$; (b) image with noise in control parameters ${\mathrm{\Delta }}_M(x,y)$ when $\mathit{A}={\mathit{A}}^{\mathit{B}}$; (c) ${\mathrm{\Delta }}_M(x,y)$ when $\mathit{A}={\mathit{A}}^{\mathbf{90}}$; (d) image with relative noise in measured values ${\mathrm{\Delta }}_B(x,y)$ when $\mathit{A}={\mathit{A}}^{\mathit{B}}$; (e) ${\mathrm{\Delta }}_B(x,y)$ when $\mathit{A}={\mathit{A}}^{\mathbf{90}}$. The gray-scale bar represents values of the optical retardation.

Figure 8

Effect of noise contamination on the slow axis orientation: (a) original (noise-free) image ${\varphi }^{*}(x,y)$; (b) image with noise in control parameters ${\varphi }_M(x,y)$ when $\mathit{A}={\mathit{A}}^{\mathit{B}}$; (c) ${\varphi }_M(x,y)$ when $\mathit{A}={\mathit{A}}^{\mathbf{90}}$; (d) image with relative noise in measured values ${\varphi }_B(x,y)$ when $\mathit{A}={\mathit{A}}^{\mathit{B}}$; (e) ${\varphi }_B(x,y)$ when $\mathit{A}={\mathit{A}}^{\mathbf{90}}$. The gray-scale bar represents values of the slow axis azimuthal angle.