Introduction

Mie theory describes the scattering of light incident upon particles with a diameter similar in magnitude to the wavelength. Sunlight is scattered in this way by the water particles that make up clouds. Since the intensity distribution of Mie scattering is independent of wavelength, clouds appear to be white. Mie theory can be used to determine the size of particles with diameters on the order of microns via analysis of the intensity and angles at which all wavelengths of light diffract¹. Practical applications of particle size determination via Mie scattering include structure analysis in biological tissue and cell research ² , observation of cosmic dust³, and investigation of meteorological phenomena⁴. We measured the Mie scattering of light by spherical latex particles with diameters of 3 μm suspended in mixtures of glycerol and distilled water. This was achieved by directing a 16 mW He-Ne laser onto a fixed glass cuvette containing the particles of interest in the suspension. A lock-in amplifier connected to a chopper regulated the frequency of the signal which a photodiode observed. Through analysis of the diffraction patterns using a Mie theory algorithm published by Matzler,⁵ the size of the particles were measured. Further investigation explored the impact of altering the viscosity of the solution containing the particles. Altering viscosity changed the index of refraction, which the scattering distribution is dependent on.

Theory

Mie theory predicts the angular scattering distribution of light incident upon a dielectric sphere with a diameter on the order of microns. A full mathematical derivation for Mie scattering in spherical coordinates can be found in Weiner, Rust, and Donnelly. Using a Mie theory calculator that applies the preceding theory, a preliminary plot was created.

Scattering patterns for different size parameters. The intensities are in arbitrary units, and all three curves were normalized.⁵

It is clear from Figure 2 that different size parameters result in different scattering patterns and for larger size parameters, more minima and maxima intensities occur. Experimental data can therefore be fit to theory easily by comparison to different size parameters. Changes in viscosity of the solution containing the spheres result in different indexes of refractions. A more viscous solution will theoretically have a larger index of refraction and thus a larger size parameter. Expanding on the work of Weiner, Rust, and Donnelly1 this experiment investigated Mie scattering resulting from spheres suspended in solutions containing several different percentages by volume of glycerol and distilled water.

Experimental Setup

A 1-mW polarized He-Ne laser with a wavelength of 632.8 nm served as the light source in this experiment. The beam was initially sent through a polarizer and then incident onto a mirror which could easily be adjusted to control the outgoing laser’s path. The scattering distribution is dependent on polarization of the light source, and so the polarizer fixed the laser at a 45 degree polarization. The laser was then sent through a chopper to regulate the frequency at which a photodiode detected the intensity of the laser. A lock-in amplifier was connected to the photodiode to measure the intensity only for the frequency set by the chopper. The chopper and lock-in, therefore, increased the signal to noise ratio of the data and better-mitigated background noise. After passing through the chopper, the beam was directed off another mirror and sent through a glass cuvette containing the solution of spheres. The photodiode was fastened to a metal arm on top of a stepper motor and panned along a circular arc to ensure the scattered light was always perpendicular to its face. The lock-in amplifier and stepper motor were interfaced with a computer where a LabView program controlled the stepper motor, regulated the sensitivity and timing constant of the lock-in amplifier and also plotted the scattering intensity with respect to angle. The photodiode was allowed to span a range of angles of about 180 degrees behind the cuvette. The experimental apparatus can be seen in the figure below.

Experimental set up. A 16 mW He-Ne laser was sent through a polarizer and chopper before passing through a cuvette containing the scattering spheres and detected by the panning photodiode. Both the chopper and photodiode were connected to a lock-in amplifier.

The cuvette was fixed over the stepper motor’s pivot point and had flat, square cross sections so the incoming laser would normal to the surface and not refract due to the change of mediums. The scattered light did however refract when exiting the cuvette, and so Snell’s law was considered in data analysis. The cuvette contained filtered water and spheres made out of latex with diameters of 3μm. The water was filtered so scattering would result solely from the suspended spheres. The solution had a concentration of 1.8 x 10-4 spheres per cubic micrometer and the concentration was assumed to be low enough to approximate the collective scattering as the combination of many single scattering events. Additional trials for 25 and 50 percent glycerol solutions were also taken along with pure water. All other components of the experimental set up, including sphere size and concentration of spheres in the medium remained unchanged.

Results

Future Considerations

The glass cuvettes we used initially had a layer of spheres coated onto the inside surfaces and could not be removed no matter how many isopropyl alcohol baths and scrubbings were done. This resulted in background intensity data sets producing scattering patterns, and so new glass cuvettes needed to be purchased. To avoid this difficulty the cuvettes should be cleaned and soaked in isopropyl alcohol after taking data.

More thought should be given to correcting with Snell’s law than what was done in this experiment. Wiener, Rust, and Donnelly set forth a numerical method for correcting for refraction but it was considered underdetermined in this experiment and unsolvable. Drake and Gordon also briefly mention a correction for scattering intensities due to Snell’s law, but this correction was not pursued and it was not determined how significant it was. Thought should also be given to the percentage of light which gets transmitted through the glass. It was assumed all light was transmitted through the cuvette, but this in reality is not true. The Fresnel equations for reflection and transmission could be used to make further corrections to the scattering angles and intensities.

The effect of concentration was not considered in this experiment. Some theoretical models, such as that produced by Prahl specifically incorporate concentration, but the model we used by Matzler did not explicitly require it.

The polarization of the laser was also given little thought in this experiment. The diffraction pattern is dependent upon the polarization of the light and it was not considered how reflection off the mirrors altered this polarization.

The models proposed by Prahl and Matzler also differed and so it is important to determine which one, if either of them, properly quantify Mie theory.

References

I. Weiner, M. Rust, and T. D. Donnelly, Particle Size Determination: An undergraduate lab in Mie scattering. American Journal of Physics. 69, February 2001.

K.J. Chalut, M.G. Giacomelli, and A. Wax, Application of Mie theory to assess structure of spheroidal scattering in backscattering geometries. Journal of the Optical Society of America A 25, 1866 (2008).

J. Escobar-Cerezo, C. Palmer, O. Munoz, F. Moreno, A. Penttila, and K. Muinonen, Scattering Properties of Large Irregular Cosmic Dust Particles at Visible Wavelengths. The Astrophysical Journal. 74, 1 (2017).

Gula Mehreen, Muneera Tariq, Kambezidisb Harry. Models for Obtaining Solar Radiation from Other Meteorological Data. Solar Energy. 64, 99 (2017)

Matzler, Christian. "MATLAB Functions for Mie Scattering and Absorption." Research Report. 2002.

Zdunkowski Wilford, Trautmann Thomas, and Bott Andreas. Radiation in the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press, 2007.

Prahl, Scott. Mie Scattering Calculator. Oregon Medical Laser Center 2007. Web.

"Refractive Index of Glycerine-Water Solutions at 20 C (69 F)." 29 July 2016. Rochester Institute of Technology Multidisciplinary Engineering. Web. 4 April 2018.