An apparatus for birefringence and extinction angle distributions measurements in cone and plate geometry by polarization imaging method

Abstract

A new apparatus for flow visualization based on Polarization Imaging Method (PIM) is developed. In this method, the polarization imaging camera, which is composed of a photonic crystal and CCD image sensor, is used to measure anisotropy of refractive index tensor. The photonic crystal placed in the front of CCD image sensor works as an assembly of analyzer plates for each of photo detectors of the image sensor, which provides four polarization components of image. By analysis of the polarization imaging data, the spatial distribution of birefringence and extinction angle can be obtained simultaneously. Since birefringence and extinction angle have strong relationship with stress and orientation angle due to Stress-Optical Rule (SOR), their distribution is appropriate to understand non-uniform flow. The advantage of the apparatus was examined by imaging flow of wormlike micellar solution in uniform and non-uniform shear flows. In the uniform shear rate geometry of cone and plate, we successfully obtained spatial distribution of birefringence and extinction angle of flow-induced structure of threadlike micelles.

Appendix

Correction method for blur of polarizer array

In order to correct blur of polarizer array, we considered expansion of each transmitted light at the front of each detector as shown in Fig. 2. For such a case, the measured intensity of I(i,j) at position (i,j) could be smeared with the leaked light from the adjacent polarizing plates such as (i ± 1, j ± 1). Let us suppose that the true light intensity, I'(i,j), representing the intensity of light just after passing through the polarizer. We further assume that this light is uniformly expanded from 1 to 1 + 2a in ratio in both x and y directions in the front of each detector. The smeared intensity, I', can be related to the true intensities, I', as follows

$$ \begin{array}{l}I\left(i,j\right)=\left[{a}^2I^{\prime}\left(i\hbox{--} 1,j\hbox{--} 1\right) + aI^{\prime}\left(i\hbox{--} 1,j\right)+{a}^2I^{\prime}\left(i\hbox{--} 1,j+1\right)\right.\hfill \\ {}+\kern-1.90em aI^{\prime}\kern-1.50em \left(i,j\hbox{--} 1\right)\kern-1.90em +\kern-1.90em I^{\prime}\kern-1.80em \left(i,j\right)\kern-2.15em +\kern-1.80em aI^{\prime}\kern-1.80em \left(i,j+1\right)\hfill \\ {} + {a}^2I^{\prime}\left(i+1,j\hbox{--} 1\right)+ aI^{\prime}\left(i+1,j\right)+{a}^2I^{\prime}\left(i+1,j+1\right)\Big]/\ {\left(1+2a\right)}^2\hfill \end{array} $$

(A-1)

By using Eq. (16), we obtain

$$ \begin{array}{l}J\left(n,m,1\right) = \Big[{a}^2J^{\prime}\left(n\hbox{--} 1,m\hbox{--} 1,3\right)+aJ^{\prime}\left(n\hbox{--} 1,m,2\right)+{a}^2J^{\prime}\left(n\hbox{--} 1,m,3\right)\hfill \\ {}+\kern-1.20em a\kern-0.95em J^{\prime}\kern-0.95em \left(n,m\hbox{--} 1,4\right)\kern-0.95em +\kern-0.95em J^{\prime}\kern-0.75em \left(n,m,1\right)\kern-0.75em +\kern-0.75em a\kern-0.75em J^{\prime}\kern-0.75em \left(n,m,4\right)\hfill \\ {}\left. + {a}^2J^{\prime}\left(n,m\hbox{--} 1,3\right)+aJ^{\prime}\left(n,m,2\right)+{a}^2J^{\prime}\left(n,m,3\right)\right]/{\left(1+2a\right)}^2\hfill \end{array} $$

(A-2)

If the pixel size is enough small to the overall image, then the gradient of light could be negligibly small. For such a case, we can assume

$$ J^{\prime}\left(n\hbox{--} 1,m\hbox{--} 1,k\right) = J^{\prime}\left(n,m,k\right) $$

(A-3)

Thus, we obtain

$$ J\left(n,m,1\right) = {\left(1+2a\right)}^{\hbox{--} 2}\left[J^{\prime}\left(n,m,1\right)+2aJ^{\prime}\left(n,m,2\right) + 2aJ^{\prime}\left(n,m,4\right)+4{a}^2J^{\prime}\left(n,m,3\right)\right] $$

(A-4)

This indicates that J(n,m,1) can be related to Jʹ(n,m,k). Similar equations can be derived for other k = 2, 3, and 4. Thus, we obtain

$$ \mathrm{J}\left(n,m\right)=\left(\begin{array}{c}\hfill J\left(n,m,1\right)\hfill \\ {}\hfill J\left(n,m,2\right)\hfill \\ {}\hfill J\left(n,m,3\right)\hfill \\ {}\hfill J\left(n,m,4\right)\hfill \end{array}\right)=\frac{1}{{\left(1+2a\right)}^2}\left(\begin{array}{llll}1\hfill & 2a\hfill & 4{a}^2\hfill & 2a\hfill \\ {}2a\hfill & 1\hfill & 2a\hfill & 4{a}^2\hfill \\ {}4{a}^2\hfill & 2a\hfill & 1\hfill & 2a\hfill \\ {}2a\hfill & 4{a}^2\hfill & 2a\hfill & 1\hfill \end{array}\right)\left(\begin{array}{l}J^{\prime}\left(n,m,1\right)\hfill \\ {}J^{\prime}\left(n,m,2\right)\hfill \\ {}J^{\prime}\left(n,m,3\right)\hfill \\ {}J^{\prime}\left(n,m,4\right)\hfill \end{array}\right)=\mathbf{A}(a){\mathbf{J}}^{\mathbf{\prime}}\left(n,m\right) $$

(A-5)

The desmeared intensity, J′(n,m), can be evaluated from J(n,m) by solving the above simultaneous equations, J′(n,m) = A ^–1 J(n,m). For our PIC, a was found to be 0.0692.