Figure 5.7: Glare points of reflexion (right) and first order refraction ( n = 1,333 and dp = 400 µm). Pulsed illumination with tp = 200 fs.
phase relationship between the contributing wavelengths in the according spectral interval. Hence this study offers the experimental and the numerical foundation for future applications of the feedback-source in Rainbow refractometry.
Figure 5.8: Smoothing of the glare point intensity ratio by ultrashort laser pulse illumination. Pulsed illumination leads to the addition of intensities of various scattering orders while cw illumination results in the Mie sum.
the particle is obviously also a function of droplet size. The refractive index of the particle also influences this parameters. Glare points of higher scattering orders have far lower intensities and are therefore difficult to observe. According to Schaller (2000), the ratio of glare point intensities can be used to estimate particle size and refractive index. The ratio of the reflective and first order refraction glare point is depicted in fig. 5.8. For CW illumination the ratio varies strongly for different particle diameters. This is due to interference between scattering orders.
The diagram also highlights the striking difference when pulsed illumination is used. Pulsed illumination yields a linear relationship between the ratio of the two glare point intensities and particle diameter. For femtosecond pulse illumination the different scattering orders arrive at the camera lens at different times and therefore can not interfere with one another. This lack of interference results in a smooth linear function.
To experimentally verify this prediction a spray nozzle was chosen, resulting in a polydispersed distribution of particle sizes. The main obstacles encountered while imaging a spray was to identify and assign proper intensity values to sets of glare point pairs. Due to the large number of illuminated droplets per picture, it was necessary to find a method of identifying multiple glare point pairs per single picture to calculate the ratio of the detected intensities. The highly convenient experimental setup used for data collection is depicted in fig. 5.9. An example of a picture that was used for analysis is shown in fig. 5.10. Note the relative
Figure 5.9: Experimental Setup for glare point detection.
weakness of the imaged points, this is due to the high shutter speed required to avoid a blurry picture of the spray.
The Ti:Sa laser provided either CW or tp = 200 fs laser pulse illumination of the droplets leaving the atomizer nozzle at a central wavelength of λ = 780 nm. The Sensicam QE cooled digital 12bit CCD-detector was equipped with a microscope.
The experiment was performed using an ultrasonic atomizer US2 water nozzle.
The polarization of the light was parallel to the scattering plane because then the intensities of reflection and first-order refraction are of comparable magnitude. The obtained images were stored in bitmap format. An application specific software uses a function named "subdivider", which scans images in discrete 24*24 pixel squares. These squares are each saved in a 2D array and passed to a second function, which scans the square for both the pixel with the highest grayscale value (most white), which corresponds to the glare point of first order refraction, and the pixel with the second highest grayscale value, for the reflection glare point. The function then returns the positions of both the highest and second highest peaks in intensity as well as their position in the image. The function also checks to see if the pair of points are, in fact, associated glare point pairs. It does this first by taking the mean value of the 24*24 pixel square and checking if both intensity peaks found are above this. It then checks wether the highest peak is at least 20 arbitrary grayscale points above the mean. If the intensity peaks match these criteria and are far enough in x-separation and close enough in y-separation to actually constitute an associated pair of glare points, the function returns that the two peaks are indeed glare points.
actual droplet size
Figure 5.10: Detail of an applied image of glare points in the illuminated spray.
Since pixel data is discrete, the determined peaks had to be refined by fitting the area around the peak to a Gaussian. The fitting of two 1D Gaussians curves, horizontally and vertically, along the intensity peak was sufficient to accurately estimate the correct value of the peak. In order to fit the 1D Gaussians, the value at the peak was used as well as the two surrounding points (either horizontally or vertically). Accordingly, the natural logarithm of the values was calculated.
Therefore only a quadratic fitting function was necessary. Gaussian curves are generally expressed as:
Aexp[−a(x−b)2] (5.1)
Hence we can form a fit on the data by treating it as a linear combination of fitting functions:
f(t) =
m
X
k=1
akfk(t) (5.2)
Figure 5.11: Histograms of glare point ratios for (left) pulsed illumination and (right) CW illumination.
fk(t) are the individual fitting functions (1, x, x2) with the coefficients ak. This setup can be treated as a linear system defined by:
F~a = ~b (5.3)
In order to solve it, a pivotless Gauss-Jordan elimination was implemented. The Gaussian amplitude, and therefore the intensity of the glarepoint was retrieved by reverting a quadratic to a Gaussian function:
exp[−ax2 +bx+c] = exp[−a(x+k1)2+k2] (5.4)
Therefore the intensity of the glare point is given by exp(c+b2/4a), and the choice of whether to use the horizontal or vertical fit comes from whichever gives us the greatest yield. In other words, both the vertical and horizontal fits were performed and the highest estimated peak was selected.
The plots in fig. 5.11 have been determined from multiple stacks of data. Nu-merous pictures of the illuminated spray have been analysed with the described
software (fig 5.10). The identified glare point ratios are depicted by absolute value (x-axis) and quantity, for pulsed and CW illumination. An overestimation of ab-solute values for CW illumination was expected (fig 5.8). Many more glare point pairs are detected in the pulsed illumination case. This is because the CW-ratio tends to drop down to zero for numerous particle diameters (fig 5.8). For the CW case many more pairs with high absolute value are detected. The plot for the pulsed case resembles the expected poisson shape of the nozzle particle size distri-bution. In contrast the plot for the CW case is not useful in determining the size distribution. The absolute value of the peak position in the pulsed case deviates slightly from the expected mean particle size ofdp = 30µm. This deviation is most likely due to a high signal-to-noise ratio and the neglected influence of overlaps and layering in the spray, which is another source of error.