Wave One Gold

Histograms of the images corresponding to the three different techniques (bright field –BF, phase contrast – PC, and dark field - DF) are plotted in Fig. 2.2.3.5 with no filter inserted. The horizontal axis represents the gray levels between 0 and 255 and the vertical axis represents the percentage of the total number of pixels in the images with a given gray level. The original image size was 240 x 240 pixels but because the moving average filter distorts the borders of the final image it was cropped again to eliminate the distortions so that the final size of the image was 222 x 222 pixels.

Figure 2.2.3.5. Histogram of gray level intensities without any filter for the incident light. Points are BF (filled square), PC (red circle), and DF (green triangles).The horizontal axis is the gray level with values between 0 and 255 and the vertical axis represents the percentage of the total number of pixels with a given gray level.

The widths of the distributions for the case of “nothing” (Fig. 2.2.3.5) were σi,BF = 10.23 ± 0.03

(BF), σi,PC = 10.57 ± 0.04 (PC), and σi,DF = 18.3 ± 0.2 (DF) and the corresponding statistical tests

for the Gaussian fit were:χBF = 1.35 10-7, RBF2 = 0.998; χPC = 2.23 10-7, RPC2 = 0.998, and χDF =

9.36 10-7, RDF2 = 0.982. According to the histograms in Fig. 2.2.3.5, there is a slight difference

between the BF and PC, in the sense that the contrast increases in the latter. Both BF and PC histograms follow a Gaussian distribution, which is consistent with the interference mechanism for the formation of the fluctuating image. The small shift in intensity between BF and PC may be caused by absorption in the mica wave plate that is slightly opaque, the wave plate wedges and the mounting pieces for the wave-plate. Overall, there is a clear increase in the contrast of the PC image as can be seen in the Fig. 2.2.3.2. The DF histogram is shifted toward the lower

0 0.025 0.050

100 200

gray level intensity

Number of pix

histograms. This difference lies in the fact that the transmitted light is blocked for the DF case. The direct visualization of the DF images suggests that an extinction process from scattering must contribute to the formation of the images in this case.

For the case of red filter treatment (Fig. 2.2.3.3), the shift between the two histograms corresponding to BF and PC in the original image Fig. 2.2.3.2 disappears and the two histograms overlap (Fig. 2.2.3.6). At the same time, the amplitude and the contrast of the DF histogram increases and approaches the BF and PC images contrast. The histograms corresponding to the BF and PC maintain the same Gaussian shape while the DF histogram has a shape closer to an exponential distribution.

Figure 2.2.3.6. Histograms

corresponding to images recorded after the treatment of the incident light with a red

filter. The BF and PC histogram

have the same contrast and overlap (black squares and red circles), while the DF (green triangles) histogram has a different shape than in the normal incident light.

The widths of the distributions for the case of red filter (Fig. 2.2.3.6) were σi,BF = 11.75 ± 0.05

(BF), σi,PC = 11.75 ± 0.05 (PC), and σi,DF = 15.9 ± 0.3 (DF) and the corresponding statistical tests

for the Gaussian fit were:χBF = 1.98 10-7, RBF2 = 0.998; χPC = 1.98 10-7, RPC2 = 0.998, and χDF =

3.06 10-6, RDF2 = 0.947. According to the histograms in Fig. 2.2.3.6, the widths of the

distributions have a slight increase when the red filter is applied.

A third set of experiments was performed by using a circular polarizer placed in front of the incident light instead of either nothing (Fig. 2.2.3.2) or a red filter (Fig. 2.2.3.3). The widths of the distributions for the case of circular polarizer filter (Fig. 2.2.3.7) were σi,BF = 10.88 ± 0.04

(BF), σi,PC = 10.96 ± 0.03 (PC), and σi,DF = 13.9 ± 0.4 (DF) and the corresponding statistical tests

for the Gaussian fit were:χBF = 1.61 10-7, RBF2 = 0.998; χPC = 9.70 10-8, RPC2 = 0.999, and χDF =

0.025 0.050

Number of pix

els (%)

0 _{100 200}

4.68 10-6, RDF2 = 0.906. The fluctuation images obtained after using the moving average filter are

shown in Fig. 2.2.3.4. The contrast in these images changed in comparison with the experiment with the red filter. The corresponding histograms for the circular polarized treatment of the incident light are shown in Fig. 2.2.3.7.

Figure 2.2.3.7. Histograms

corresponding to images recorded after the treatment of the incident

light with a circular polarizer. The

histograms of images from Fig. 2.2.3.4 show a Gaussian distribution for BF (filled squares) and PC (red circles). The DF histogram (green triangles) has a slightly different distribution.

A side by side comparison between the widths of the Gaussian distribution using the three different methods for BF, PC and DF is presented in Fig. 2.2.3.8. These plots show the variance of the histograms (vertical axis) versus the corresponding image index (horizontal axis) for all images we analyzed. The widths were found by fitting the histograms of the images to Gaussians. 0 10 20 1000 2000 Image index V a

riance of the histog

ra m A 0 1000 2000 Image index B 0 1000 2000 Image index C 3000

Figure 2.2.3.8. The variances of the histograms depend on the special filter used. For the

BF (A), PC (B), and DF (C) using the three different methods to treat the incident light: “nothing”-black squares, the red filter-red circles and circular polarizer-green triangles. The variance of the histograms varies significantly.

0 0.025 0.050

100 200

gray level intensity

Number of pix

For BF (Fig. 2.2.3.8.A) in the case of the red filter, the average width of the histograms over the

total number of available images was σred = 11.86 ± 0.01 gray levels, which is a bit larger but

with a narrower range compared to the circular polarized case with σcpol = 11.72 ± 0.02. The

widths of the histograms for the BF without any filter are smaller, σnothing = 10.71 ± 0.01 gray

levels compared to BF with the red and circular polarizer filters. The widths of the histograms

for the PC (Fig 2.2.3.8.B) circular polarized is narrower σcpol = 11.476 ± 0.007 gray levels,

compared to the PC with the red filter, σred = = 11.63 ± 0.01 gray levels. The widths of the

histograms have even smaller values for the PC with nothing, σnothing = 11.19 ± 0.001 gray levels.

The distinctive feature of the widths of the histograms for the DF is the large dispersion (Fig. 2.2.3.8.C). In the case of the DF circular polarized experiment, the widths have the smallest range σcpol = 10.94 ± 0.3 gray levels, compared to the DF with nothing, σnothing = 15.70 ± 0.04