1. Electromagnetic Wave and Particle Interaction:

Mie scattering theory solves Maxwell's equations to describe how an electromagnetic wave (such as light) interacts with a spherical particle. The key idea is that when light, described as an oscillating electromagnetic field, strikes a particle, the electric and magnetic fields in the light induce oscillations of electric charges inside the particle. This results in the scattering of the light in all directions.

The fundamental equations governing the interaction are Maxwell's equations:

Where:

• E is the electric field,

• B is the magnetic field,

• ρ is the charge density,

• J is the current density,

• μ0​ is the permeability of free space,

• ϵ0​ is the permittivity of free space.

When light interacts with a particle, the scattering problem becomes one of solving these equations in spherical coordinates around the particle.

2. Size Parameter and Scattering Regimes:

Mie scattering theory is most useful when the size of the particle is on the order of or larger than the wavelength of light. The relative size of the particle to the wavelength is expressed by the size parameter, x:

where r is the particle radius and λ is the wavelength of light.

Based on the size parameter, there are different scattering regimes:

• Rayleigh Scattering (x≪1): When the particle is much smaller than the wavelength, the scattering is weak and primarily in the forward and backward directions. This is the regime of Rayleigh scattering.

• Mie Scattering (x≈1): When the particle is comparable in size to the wavelength, the scattering is strong and occurs in all directions.

• Geometric Optics (x≫1): For very large particles compared to the wavelength, the scattering approaches the predictions of geometric optics, with clear refraction and reflection patterns.

3. Mathematical Formulation of Mie Scattering:

Mie scattering involves the solution of Maxwell’s equations for an electromagnetic wave interacting with a spherical particle. The solution is obtained by expanding the incident and scattered fields into series of spherical harmonics. This expansion leads to the computation of scattering coefficients, known as the Mie coefficients.

The incident plane wave is expressed as a sum of spherical wave functions:

Where:

• E0​ is the amplitude of the electric field,

• k=2π/λ is the wavenumber,

• z is the direction of propagation.

The scattered wave is expressed as a sum of spherical harmonics, which are functions of the scattering angle α and azimuthal angle ϕ:

Here, an​ and bn are the Mie scattering coefficients for electric and magnetic multipoles, respectively, and Ne1n​ and Mo1n are vector spherical harmonics representing the scattered electric and magnetic fields. These terms depend on the size parameter x and the relative refractive index m, which is the ratio of the refractive index of the particle to that of the surrounding medium:

Mie Scattering Coefficients:

 The Mie coefficients an​ and bn​ are derived from boundary conditions at the surface of the particle (continuity of the electric and magnetic fields). They are given by:

Where:

• ψn(x) and ξn(x) are Riccati-Bessel functions,

• The primes denote derivatives concerning the argument.

These coefficients describe how different multipole components (e.g., dipole, quadrupole) contribute to the scattering.

4. Scattering Cross-Section:

The scattering cross-section, Csca​, quantifies the total power scattered by a particle in all directions. It is derived from the scattered intensity distribution. The total scattering cross-section is the sum of the contributions from all the spherical harmonics (Mie terms):

This equation indicates that higher-order terms (multipoles) contribute to the scattering, and the strength of each term depends on the size parameter and refractive index contrast between the particle and the medium.

5. Intensity Distribution and Polarization:

The scattered light's intensity as a function of the angle θ\thetaθ (relative to the incident light direction) can be described by two components: parallel (S∥) and perpendicular (S⊥) polarization components relative to the scattering plane. These intensities are given by:

 where πn​ and τn​ are angle-dependent functions derived from the associated Legendre polynomials and spherical harmonics.

 The total scattered intensity is then:

This angular dependence explains the complex pattern of maxima and minima in the scattered light's intensity, which can be measured to infer the size and properties of the scattering particles.

6. Applications of Mie Theory in Your AIoTs Setup:

For your AIoTs Mie scattering experiment, the theory enables you to analyze the real-time scattering data using IoT-enabled sensors. By measuring the angular distribution of scattered light and comparing it to theoretical predictions, you can determine the particle size, shape, and refractive index. The AI algorithms can assist in automating the fitting process and provide real-time analysis of the scattering intensities.

Incorporating IoT allows continuous data acquisition from sensors placed at different angles, while AI techniques help in processing large datasets, fitting scattering patterns, and adjusting experimental conditions dynamically based on real-time feedback.

This detailed mathematical formulation of Mie scattering theory provides the basis for understanding and implementing the experiment in your AIoTs setup.

Due to the complexity of the calculations involved, we rely on Python to assist in generating theoretical simulation plots. We use the Python module miepython for these computations.

Overview of miepython

miepython is a pure Python module designed to compute light scattering by spheres that are non-absorbing, partially absorbing, or fully conducting. The code provides functions to calculate key scattering parameters, such as:

• Extinction efficiency: The fraction of incident light removed due to scattering and absorption by the particle.

• Scattering efficiency: The portion of incident light scattered by the particle.

• Backscattering efficiency: The fraction of light scattered in the backward direction.

• Scattering asymmetry: A measure of the directional bias of scattered light (whether more light is scattered forward or backward).

Additionally, the module allows you to input a set of angles to calculate the scattering intensity at those specific angles, making it a flexible tool for Mie scattering simulations.

Installation and Usage of miepython

To use miepython on your local machine, you can install it via pip by running the following command in your terminal:

pip install --user miepython

Application of miepython

Once installed, miepython can be used to simulate scattering for a variety of particle and light conditions. You can calculate the scattering intensities for a given particle size, wavelength, and refractive index. The module also allows for easy plotting of the results, enabling a detailed analysis of the scattering patterns at various angles. This is particularly useful for comparing experimental results with theoretical models.