S16_MieScattering
Determining the Size of Micron-Sized Latex Spheres using Mie Scattering
Jacob Freyermuth and Vikram Nagarajan
Abstract
Light from a helium neon laser is scattered off of a suspension of micron-sized latex spheres and the scattered intensity is measured as a function of angle. The measured scattering intensity is fitted to a model using the theory of Mie scattering to determine the size of the spheres. The spheres are found to have a radius of 1.56 ± 0.03 μm, which varies from the manufacturer’s value of 1.50 μm by 2.0 standard deviations.
Introduction
Mie scattering refers to the phenomenon of light scattering off of spherical particles of approximately the same size as the wavelength of the incident light. This differs from Rayleigh scattering, which applies for particles whose size are much smaller than the wavelength of the incident light. For example, the air molecules in the Earth’s atmosphere are much smaller than the wavelength of incident light from the sun, so they exhibit Rayleigh scattering. The intensity of scattered light in the Rayleigh scattering regime exhibits a 1/λ4 dependence, so shorter wavelengths of incident light have a greater scattered intensity. Since blue light has a smaller wavelength than other visible light, we therefore see the sky as blue. On the other hand, the larger water droplets in clouds are about the same size as the wavelength of the incident light, so they exhibit Mie scattering. The intensity distribution for Mie scattering is approximately wavelength independent, so wavelengths of visible light are scattered an equal amount, giving clouds their white color. One application of Mie scattering theory is to accurately measure the size of very small particles, typically 50 microns or smaller. This is important for many applications where measuring the size of an object on the order of microns is required, including measuring the concentration of oil droplets in polluted water or studying the structure of certain microscopic pathogens [1,2]. In this experiment, Mie scattering is used to measure the radius of micron-sized dielectric spheres by using numerical techniques to fit the angular scattering distribution to the predictions of Mie theory.
Theory
When light scatters off of a particle whose size is on the same order as the wavelength of the incident light, the resulting pattern is governed by Mie scattering theory. To model a Mie Scattering distribution, Maxwell’s equations are solved for a plane wave scattering off of a dielectric sphere. The result is an infinite series of basis functions, akin to a Fourier series. The scattered electric field is [3]:
where M and N are vector spherical harmonics, and they encode the angular dependence of the scattering distribution. These vector spherical harmonics are the known basis functions which were mentioned before. The superscripts denote a specific radial dependence. The coefficients an and bn are [3]:
Parameters μ and p are material specific constants and functions j and h are known basis functions (primes denote derivatives). Parameter x is the size parameter, and is given as x = 2πNa/λ, where N, a, and λ are the refractive index of the medium, the radius of the sphere, and wavelength of the incident light, respectively. The shape and form of the theoretical scattering model depends solely on this size parameter. Several plots of scattering curves with different size parameters are shown in Figure 1 below.
Figure 1: Theoretical Mie scattering curves for size parameters of x = 5 (top left), x = 10 (top right), x = 20 (bottom left), and x = 25 (bottom right). Note the logarithmic scale of the intensity on the y-axis, which shows that the intensity drops fairly rapidly with increasing angle. All of these curves correspond to a wavelength of 633 nm, which means the sphere diameters are 0.76, 1.51, 3.03, and 3.79 μm, respectively. Original figures.
By fitting this model to an experimental scattering distribution, the correct size parameter can be determined, which allows the radius of the sphere to be ascertained.
Apparatus
The experimental setup is shown in Figure 2. A 3 mW 632.8 nm Helium-Neon laser was directed toward a rectangular cuvette containing a suspension of latex spheres in distilled water. The beam passed through a polarizer to control the polarization and an optical chopper spinning at some reference frequency, which was connected to a lock-in amplifier to reduce effects of noise. The cuvette was mounted over a stepper motor with a photodetector placed on the arm. This photodetector was connected to the lock-in amplifier, which measured any light signal that was chopped at the frequency of the chopper. The stepper motor stepped approximately from -45° to +45°, allowing the photodetector to measure the intensity of scattered light as a function of angle.
Figure 2: The setup of the experiment. Intensities will be recorded by the LIA as a function of , which is determined by the stepper motor. The two mirrors were used for ease of alignment. Original figure.
Data Collection
The raw data is show in Figure 3. Because our apparatus blocked some light on the right side of the pattern, the right side was discarded and only the left side was analyzed. The uncertainty for each point was found by taking several measurements at each angle and calculating the standard deviation. In order to reduce effects of background scattering by the water and the cuvette, only data points past 9° were used, due to the fact that a measurement of background light found the background to be significant for angles up to about 9°. This is apparent from the measurements of background, shown in Figure 4.
Figure 3: Intensity vs angle for one run of data. The very large peak in the middle is due to unscaterred light. The noisiness on the very right of the plot is from the scattering pattern being blocked by the mounting apparatus. Thus, only the left side was analyzed.
Figure 4: Background light intensity. Data run was taken with distilled water containing no spheres in the cuvette. Background was determined to be significant up to angles of about 10 degrees. Points before 10 degrees were thus excluded.
Numerical Methods and Analysis
To generate the theoretical scattering curves, the scattering coefficients were calculated, paired with the appropriate basis functions, and summed up to an appropriate number of terms (we found about 25 to be sufficient). These curves were generated using Mathematica. Due to refraction at the face of the cuvette, the theoretical curves also had to be corrected for Snell’s Law. Next, the appropriate size parameter was found by minimizing the reduced χ2 value of the theoretical curve with respect to the size parameter as well as a normalization coefficient, since the theory was not be properly normalized to experiment. Since the size parameter and normalization coefficient are correlated, the minimum reduced χ2 was found using a simplistic grid search, which was programmed for in Python. The size parameter which gave the smallest value for the χ2 was taken to be the correct experimental value.
Results and Conclusion
A plot of the reduced χ2 value vs the optimized size parameters (with respect to normalization) is shown in Figure 5. From this, it can be seen that a global minimum for the χ2 is found at x = 20.6 ± 0.4, which corresponds to a reduced χ2 value of about 48. This in turn corresponds to a sphere of radius a = 1.56 ± 0.03 μm, which is 2.0 standard deviations from the manufacturer’s value of 1.50 μm. A plot of the theoretical and experimental scattering curves are shown in Figure 6. The theoretical curve is the curve given by the optimized values of the normalization coefficient and the size parameter.
Figure 5: Reduced chi-squared vs the size parameter. A global minimum at x = 20.6 can be seen.
Figure 6: A comparison of the theoretical and experimental scattering distributions. While the theory seems to follow the general trend of the data, the fit is overall not extremely accurate.
It is readily apparent that our fit is not very accurate due to the reduced χ2 value of 48. However, it is worth noting that this value is on the order of values obtained in literature (which contained values of about 32) [3]. It is also worth noting that the fit does seem to follow the general trend of the data, and our experimental value for the sphere is only 4% away from the manufacturer's value. Several possible reasons for the large χ2 include: not taking polarization into account, broadening of peaks at the cuvette interface, and not perfectly aligning the cuvette with the pivot of the arm of the stepper motor. The last point is easily fixable, the first point may take some effort, but should be fixable, and the second point cannot be corrected for trivially based on the code that we wrote. Future projects may find it useful to delve deeper into these points to achieve more accurate results.
References
[1] H. Lindner, G. Fritz, and O. Glatter. "Measurements on Concentrated Oil in Water Emulsions Using Static Light Scattering". Journal of Colloid and Interface Science 242: 239 (2001).
[2] Y. M. Serebrennikova, J. Patel, and L. H. Garcia-Rubio. "Interpretation of the ultraviolet-visible spectra of malaria parasite Plasmodium falciparum". Applied Optics 49 (2): 180-8 (2010).
[3] I. Weiner, M. Rust and T. D. Donnelly. "Particle size determination: An undergraduate lab in Mie scattering". American Journal of Physics 69, 129 (2001).