# Light Scattering

### Introduction

The diffusion of light, also known as scattering, is a phenomenon which consists in the re-emission in many directions (usually non-random) of a beam of light that strikes a set of dispersed particles of variable size present in a system that can be in a solid, liquid or gaseous state.
Depending on the size of the dispersed particles and depending on the wavelength of the incident radiation, the angular distribution of the diffused light is different; the intensity of the diffused light is therefore dependent on the diffusion angle.
There are different ways of light scattering, but the most important are the Rayleigh scattering and the Mie scattering.

The image below shows the angular distribution of the scattered light for the different types of scattering.

#### Rayleigh Scattering

Rayleigh Scattering is the elastic scattering of a light wave caused by particles which have small dimensions compared to the wavelength of the wave itself, which occurs when light passes through a turbid medium, especially gases and liquids but also solids with impurities or inclusions. Since the diffusion is elastic, the scattered radiation has the same frequency (and wavelength) as that incident. Diffuse radiation is also called Rayleigh radiation.

The equations that describe the diffusion are very complex and, especially when this phenomenon is repeated many times, impossible to solve exactly in the general case. A widely used approximate solution is the so-called Rayleigh solution: in the case in which the particles responsible for the diffusion have dimensions much smaller than the wavelength of the incident light, the light dispersion is isotropic and the diffusion coefficient is given by the formula:

where n is the number of diffusion centers present, d their diameter, m their refractive index and the wavelength of the incident light. We highlight the strong dependence of the diffusion from the inverse of the fourth power of the wavelength, which explains why the radiation with small wavelengths are more effectively diffused.

#### Mie Scattering

Mie scattering, also known as Lorenz-Mie scattering, is a complete and mathematically rigorous solution to the problem of scattering of an electromagnetic wave on a sphere or cylinder. The theory that describes this type of scattering is named after the German physicist Gustav Mie who published the complete solution in 1908.
Mie scattering is valid for scattering centers of all sizes and, in the limit in which these are much smaller than the incident wavelength, the Rayleigh Scattering (which is valid only for point-like scatterer) is obtained, Mie scattering applies therefore in the cases in which the dimensions of the scattering centers are greater than the wavelength of the incident radiation. For this reason, Mie scattering is applied both in the optical study of colloids and in meteorology; in fact the drops of water that make up the clouds often have larger (or even much larger) dimensions of the wavelength of visible light.

### Experimental Setup

The availability in the PhysicsOpenLab laboratory of instrumentation for the detection and measurement of light intensity led us to carry out scattering measurement tests, in order to experimentally verify the different types of scattering : Rayleigh and Mie.
We relied on the equipment already described in the post : PSoC based Photometer.
The sensor is a Hamamatsu S1337-66BR photodiode. This photodiode is characterized by a sensitive area of 5.8×5.8 mm, it has a spectral range from 340nm to 1100nm with a peak sensitivity of 960nm, it is therefore suitable for use in the detection of low intensity light emissions from UV to near IR .
The images below show the photodiode mounted in a support, with space for a filter (for example a 633 nm interference filter to select only the radiation emitted by a He-Ne laser).

A thin slit (∼ 1mm) is placed in front of the sensitive area of ​​the photodiode in order to increase the angular resolution of the sensor. The sensor is positioned on a rotating plate, driven by a servo : the rotation of the servo allows the sensor to perform a scan around the point where the light diffusion originates. The movement range is from a minimum of 30° to a maximum of 140°, where 90° corresponds to the position of the sensor in axis with the light beam.
The image below shows the sensor on the rotating plate, the servo, the polarizing filter and the He-Ne laser.

The image below shows a sample subjected to measurement : the cuvette is placed along the direction of the beam, at the center of rotation of the platform, and the sensor scans in steps of a degree, recording, in each position, the scattered light intensity.

Scanning and measurement are performed with the photometer, which shows on the display the value of the rotation angle and the result of the measurement. The numerical data is transmitted via serial interface to a Raspberry PI used as a data logger.

The servo drive is easily implemented at the PSoC component level using the PWM module.
The image below shows the PWM component and the
c code  that performs the conversion between angle and the “ON time” of the PWM signal.

### Some Results

#### Mie Scattering

We measured some samples with colloidal solutions in order to highlight the phenomenon of scattering. The first tests were done with diluted milk, both whole milk and skimmed milk. Milk is a colloidal aqueous mixture containing numerous substances. In whole milk, in particular, fat globules (lipids) prevail with an average size of about 3μm, whole milk can therefore be considered a sort of emulsion.
Instead, skimmed milk (milk without fats) can be considered a colloidal dispersion of casein, casein micelles have an average size of about 300nm.

Considering that the wavelength of the radiation used is 650nm (semiconductor laser) it can be seen how the whole milk is placed in the category of optical scattering (d > 1μm), while the skimmed milk is more markedly placed in the category of scattering Mie (50nm < d < 500nm).

Naturally the diluted milk (both whole and skimmed) is characterized by the presence of colloids of the most varied dimensions and therefore we cannot expect results perfectly in line with the theory.

The two charts below show the measurements made at 650nm, for whole and skimmed milk, and at 450nm for whole milk only.

The results obtained do not correspond exactly to the theoretical data foreseen for this type of colloid, this due to both the type of rather rudimentary apparatus and the samples used which are very far from the ideal conditions of a fluid containing homogeneous particles.
The results are however appreciable : in the charts taken at 650nm a maximum can be seen in correspondence of the axial direction with some lateral “undulations” (relative maxima) for angles greater and lower than 90°. For skimmed milk, which falls into the Mie scattering category, the lateral maxima are more pronounced, the same phenomenon can also be seen in the scanning taken at 450nm.

#### Rayleigh Scattering

In addition to the samples described in the previous paragraph we have subjected to measurement a sheet of translucent nylon (polyamide) and a colloidal suspension of Quantum Dots (QD). For both these samples the dimensions of the diffusion centers are lower than the wavelength of the radiation, in particular the QDs have a size of only 4nm. In both cases we expect to obtain “Rayleigh” scattering. The graphs below show the results obtained.

In fact, we can see how the intensity of the scattering is maximum in the axial direction and gradually decreases by lateral angles, as theoretically predicted by Rayleigh scattering.

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