🗯️MATLAB snippet

Mie scattering is more complex than Rayleigh scattering and occurs when the size of the scattering particles is comparable to the wavelength of light. Unlike Rayleigh scattering, which is limited to particles much smaller than the wavelength, Mie scattering can handle a broader range of particle sizes and is often used to model the scattering of larger particles, such as water droplets in clouds.

MATLAB Implementation Overview

MATLAB is well-suited for numerical computations and visualizations, making it ideal for simulating Mie scattering. Here’s how to approach this:

1. Mie Theory: Use the Mie solution to Maxwell’s equations to calculate scattering for spheres.
2. MATLAB Tools: Use built-in functions or implement custom functions to compute Mie coefficients and scattering cross-sections.

Example Code for Mie Scattering

Below is an example MATLAB script that calculates and plots the Mie scattering efficiency:

 % Define parameters
 wavelengths = linspace(400, 700, 100) * 1e-9; % Wavelengths in meters (400-700 nm)
 particleRadius = 100e-9; % Particle radius in meters (100 nm)
 refractiveIndex = 1.5 + 0.01i; % Complex refractive index of the particle
 mediumRefractiveIndex = 1.0; % Refractive index of the surrounding medium
 ​
 % Constants
 k = 2 * pi ./ wavelengths; % Wave number
 ​
 % Mie Scattering function (adapted from built-in functions or third-party code)
 mieScatteringEfficiencies = zeros(1, length(wavelengths));
 for i = 1:length(wavelengths)
     mieScatteringEfficiencies(i) = mieScattering(k(i), particleRadius, refractiveIndex, mediumRefractiveIndex);
 end
 ​
 % Plot the results
 figure;
 plot(wavelengths * 1e9, mieScatteringEfficiencies, 'LineWidth', 2);
 xlabel('Wavelength (nm)');
 ylabel('Scattering Efficiency');
 title('Mie Scattering Efficiency as a Function of Wavelength');
 grid on;
 ​
 % Mie Scattering calculation function
 function Q = mieScattering(k, a, m, n_medium)
     % Calculates Mie scattering efficiency Q
     % k - wave number, a - particle radius, m - refractive index of particle,
     % n_medium - refractive index of the surrounding medium

     % Mie coefficients (simplified for this example; a more thorough
     % implementation can include full summation series)

     x = k * a; % Size parameter
     m_rel = m / n_medium; % Relative refractive index

     % Example using MATLAB functions or numerical computation for efficiency
     % Replace this block with exact Mie calculations or external libraries
     Q = mie_analytic_solution(x, m_rel); % Placeholder function
 end
 ​
 % Note: Implement `mie_analytic_solution` based on Mie theory equations or use available MATLAB packages.

Explanation:

• Inputs:
  • wavelengths: The range of light wavelengths.
  • particleRadius: Radius of the particle.
  • refractiveIndex: Complex refractive index of the particle material.
• Mie Coefficients Calculation:
  • Use the size parameter ( $x = \frac{2 \pi a}{\lambda}$ ) where ( $a$ ) is the radius, and ( $\lambda$ ) is the wavelength.
  • The mieScattering function should compute Mie coefficients ( $a_n$ ) and ( $b_n$ ), which can be summed for scattering efficiencies ( $Q{\text{scat}}$ ) and ( $Q{\text{ext}}$ ).
• Visualization:
  • The plot shows the variation of scattering efficiency with wavelength, useful for visualizing how light interacts with different-sized particles.

Enhancements:

• Advanced Mie Calculation:
  • Implement a more detailed summation series for Mie coefficients ( $a_n$ ) and ( $b_n$ ) for a full solution.
  • Use external packages such as MiePlot or MATLAB toolboxes available for Mie scattering.
• 3D Visualization:
  • Use 3D plots or color mapping to visualize angular scattering patterns.
• Optimization:
  • Vectorize calculations to improve performance.

Available Resources:

• MATLAB Central: Contains user-contributed code for Mie theory implementations.
• Mie Theory Libraries: Open-source Mie scattering libraries are available and can be integrated with MATLAB for more comprehensive simulations.