Computing Scattering Coefficients in Mie Scattering

In the study of particle scattering problems, the calculation of scattering coefficients an, bn, cn, and dn in Mie scattering represents a crucial computational step. Mie scattering, also known as Lorenz-Mie scattering or spherical scattering, describes elastic scattering phenomena where particles significantly exceed the wavelength of incident light. These coefficients quantitatively characterize scattered light intensity distributions across different angular directions, enabling investigation of various particle properties including size, morphology, and complex refractive index. Computational implementation typically involves recursive calculation of Bessel and Hankel functions using upward or downward recurrence relations to ensure numerical stability. Key algorithmic components include: - Riccati-Bessel function computations for calculating angular dependence - Logarithmic derivative implementations for efficient coefficient determination - Size parameter (x=2πa/λ) handling for different particle-wave interactions The coefficient calculations require careful handling of complex-number arithmetic and convergence checks for high-order terms. Modern implementations often incorporate continued fraction methods or Lentz's algorithm for stable computation of spherical Bessel function ratios. Accurate determination of these coefficients enables precise modeling of light scattering patterns, which finds applications in atmospheric sciences, biomedical particle analysis, and nanomaterials characterization.