# Physical Basis: Mie Theory

The main model has been updated to run using miepython for the Mie
theory computation. Miepython is an validated python package written by
Scott Prahl. Both miepython and the equivalent functions written by the
authors of PPOPS (contained in validation/mie_python.py) produce nearly
identical results. mie_modules.py is retained for illustrative and
testing purposes but is not used in the model.

## `mie_modules.py` — Mie Scattering Utilities

This module implements numerical routines for computing **Mie scattering** by spherical particles.  
It provides functions for calculating:

- Mie coefficients $a_n$ and $b_n$
- Angular functions $\pi_n(u)$ and $\tau_n(u)$
- Scattering amplitude functions $S_1(\theta)$ and $S_2(\theta)$

These quantities follow standard formulations from electromagnetic scattering theory and are commonly used in aerosol physics, atmospheric science, and optical particle characterization.

## Overview

### Functions Included

| Function | Description |
| --------- | ------------- |
| `mie_ab(m, x)` | Computes the complex Mie coefficients $a_n$ and $b_n$ describing electric and magnetic multipole contributions. |
| `mie_pt(u, n_max)` | Computes angular functions $\pi_n(u)$ and $\tau_n(u)$ derived from associated Legendre polynomials. |
| `mie_s12(m, x, u)` | Computes the scattering amplitude functions $S_1$ and $S_2$ for s- and p-polarized scattered light. |

These are the fundamental components needed to compute phase functions, scattering efficiencies, polarization ratios, and full intensity distributions from Mie theory.

## Dependencies

``` python
import numpy as np
from scipy.special import spherical_jn, spherical_yn
```

## Function Documentation

### `mie_ab(m: complex, x: float) -> np.ndarray`

Compute the Mie coefficients $a_n$ and $b_n$, which describe how a spherical particle scatters light.

**Parameters:**

- **`m`** (*complex*)  
  Complex refractive index of the particle, $m = n + ik$.

- **`x`** (*float*)  
  Size parameter, defined as, $x = \frac{\pi D}{\lambda}$

**Returns:**  

A 2×N NumPy array:

- `a_n` — electric multipole coefficients  
- `b_n` — magnetic multipole coefficients

**Key Implementation Notes:**

- Determines the required order $n_\text{max}$ via the approximation:
  $n_{\max} \approx 2 + x + 4x^{1/3}$

- Uses Riccati–Bessel functions:
  $\psi_n(x) = x j_n(x), \quad
  \chi_n(x) = -x y_n(x)$

- Computes the logarithmic derivative $D_n(z)$ using *backward recurrence*.

- Constructs the Mie coefficients from:
  $a_n = \frac{(\frac{D_n}{m} + \frac{n}{x})\psi_n - \psi_{n-1}}
               {(\frac{D_n}{m} + \frac{n}{x})\xi_n - \xi_{n-1}}$
  with an analogous expression for $b_n$.

---

### `mie_pt(u: float, n_max: int) -> np.ndarray`

Compute angular functions $\pi_n(u)$ and $\tau_n(u)$, where:

- $u = \cos\theta$
- $\theta$ is the scattering angle.

These functions capture the angular dependence of scattered light.

**Parameters:**

- **`u`** (*float*) — cosine of scattering angle  
- **`n_max`** (*int*) — maximum multipole order

**Returns:**  

A 2×N array: `[π_n(u), τ_n(u)]` .

**Internal Notes:**

- Initializes dipole (n=1) and quadrupole (n=2) terms.
- Higher-order terms computed via recurrence:
  $\pi_n = \frac{(2n-1)u\,\pi_{n-1} - n\,\pi_{n-2}}{n-1}, \quad
  \tau_n = n u \pi_n - (n+1)\pi_{n-1}$

---

### `mie_s12(m: complex, x: float, u: float) -> np.ndarray`

Compute the Mie scattering amplitude functions $S_1$ and $S_2$, which fully characterize scattering for perpendicular and parallel polarization states.

**Parameters:**

- **`m`** (*complex*) — refractive index
- **`x`** (*float*) — size parameter
- **`u`** (*float*) — $\cos(\theta)$

**Returns:**  
A 2-element array: `[S1, S2]` .

**Computation Outline:**

1. Determine `n_max` using the standard approximation.
2. Compute Mie coefficients `a_n`, `b_n` .
3. Compute angular functions `π_n(u)`, `τ_n(u)` .
4. Apply the weighting: $\frac{2n + 1}{n(n+1)}$
5. Form the scattering amplitudes:
   $S_1 = \sum_n \left(a_n \pi_n + b_n \tau_n\right), \quad
   S_2 = \sum_n \left(a_n \tau_n + b_n \pi_n\right)$

These can be further used to compute intensity patterns:
$I(\theta) \propto \lvert S_1 \rvert ^2 \quad \text{or} \quad  \lvert S_2 \rvert ^2$

## References

- **Bohren, C. F. & Huffman, D. R. (1983)**  
  *Absorption and Scattering of Light by Small Particles.*

- **C. Mätzler (2002)**  
  MATLAB codes for Mie scattering, IAP Research Report.