Scattering Module Overview

The Scattering module computes aerosol optical properties from Mie theory for user-defined size distributions and refractive indices. It supports:

• Siewert NAI-2 and Domke PCW Fourier decomposition methods

• scalar and polarized phase-matrix workflows (Stokes_I, Stokes_IQ, Stokes_IQU, Stokes_IQUV)

• automatic differentiation (AD) for core aerosol parameters (r_m, sigma, n_r, n_i)

• phase matrix reconstruction from Greek coefficients for downstream RT use

Primary reference: Sanghavi (2014)

Typical Workflow

1. Create an Aerosol from a size distribution and refractive index

2. Build a MieModel with make_mie_model

3. Compute bulk optical properties with compute_aerosol_optical_properties

4. Reconstruct scattering matrix elements with reconstruct_phase

For a runnable code walkthrough, see:

• Scattering example page

• Scattering tutorial

Core Outputs

compute_aerosol_optical_properties returns an AerosolOptics object containing:

• greek_coefs: alpha, beta, gamma, delta, epsilon, zeta

• omega_tilde: single scattering albedo

• k: bulk extinction cross-section

• f_t: truncation factor used by delta-M style workflows

• derivs (if AD enabled): Jacobian of outputs with respect to [r_m, sigma, n_r, n_i]

Theory Mapping (Sanghavi 2014)

The table below maps key Scattering APIs to equations/sections in: Sanghavi (2014).

vSmartMOM API	Theory anchor in paper	Notes
compute_aerosol_optical_properties(model::MieModel{NAI2})	Eq. (1), Fourier framework around Eq. (17), NAI-2 discussion in Secs. 3-4	Uses Mie-series cross-sections and numerical-angular-integration route to Greek coefficients
phase_function(aerosol, ...)	Eq. (1), Secs. 3-4	Returns scalar phase function with C_ext, C_sca, g
compute_aerosol_XS(aerosol, ...)	Eq. (1)	Cross-section-only path (no Greek reconstruction)
reconstruct_phase(greek_coefs, μ)	Fourier/Greek matrix representation around Eq. (17)	Rebuilds angle-space matrix elements from stored Greek moments
compute_aerosol_optical_properties(model::MieModel{PCW})	Eq. (22) for S_l, Eq. (24) for Greek coefficients	Direct Domke/PCW route using Wigner-symbol tables
compute_Sl(...)	Eq. (22)	Internal helper used by PCW implementation

Implementation notes:

• The code follows Sanghavi's corrected Domke/PCW formalism where applicable.

• Some engineering choices (memory layout, AD wrapping, truncation helpers) are implementation details and not direct one-to-one equation transcriptions.

API Map (for miepython Users)

If you are coming from miepython, this is the closest functional mapping:

• efficiencies(...) -> compute_aerosol_XS(...) or phase_function(...) for bulk cross-sections

• S1_S2(...) / phase_matrix(...) -> compute_aerosol_optical_properties(...) + reconstruct_phase(...)

• coefficients(...) -> low-level compute_mie_ab! and compute_mie_S1S2! routines

• normalization/RT-oriented outputs -> Greek coefficients + omega_tilde, k, f_t

This module is oriented toward atmospheric RT pipelines where Fourier moments (Greek coefficients) are the main interface.

Mathematical Background

Mie Cross-Sections

The extinction and scattering cross-sections for a single sphere are computed from the Mie coefficients and (Sanghavi 2014, Eq. 1):

where   and the sum is truncated at a size-parameter-dependent .

Scattering Matrix

The 4x4 scattering matrix for a single sphere has the form (Sanghavi 2014, Eq. 2):

For spherical particles,   and  , leaving four independent elements.

Greek Coefficient Matrix

The Greek matrix provides the Fourier-space representation of the scattering matrix (Sanghavi 2014, Eq. 16):

The six Greek coefficients are the compact representation stored in GreekCoefs and used by the RT kernels.

NAI2 vs PCW Performance

The two decomposition methods have different computational trade-offs:

"For polydispersions over a finite range of particle sizes, PCW is found to be at least 6–8 times faster for size parameters ranging from ~0 to beyond 900." – Sanghavi (2014)

NAI2 is the default due to its simplicity and lower memory footprint (no precomputed Wigner tables required). PCW becomes advantageous for large size parameters or when many repeated computations amortize the Wigner table precomputation.

Conventions

• Refractive index input uses n_r and n_i >= 0

• Length units for lambda, r, and r_max must be consistent

• phase_function(...) and compute_aerosol_XS(...) are convenience APIs when you do not need full Greek-coefficient reconstruction

Architecture

The design separates expensive Mie/Fourier computation from phase reconstruction:

• compute_aerosol_optical_properties computes and stores compact Fourier-space information

• reconstruct_phase evaluates angle-dependent scattering matrix elements at arbitrary angular grids

This separation is useful when RT or inversion workflows need repeated matrix reconstruction on different angle grids without recomputing the full Mie solution.