Spherical harmonic transform accelerated spectral construction of the U-matrix in the invariant imbedding T-matrix method

The invariant imbedding T-matrix method (IITM) can solve electromagnetic scattering by inhomogeneous dielectric targets, but direct U-matrix integration is costly. We propose a fast U-matrix computation method based on the spherical harmonic transform (SHT). The permittivity distribution on each spherical shell is first expanded into spherical harmonics. The expansion coefficients are then combined with spin-weighted spherical harmonics and Clebsch-Gordan coefficients, replacing blockwise angular double integration with spectral summation. For general targets, the complexity is reduced from O(L⁶) to O(L⁵). Numerical results agree closely with direct integration and commercial software. At high sampling density, the method achieves 181.75-fold and 15.26-fold speedups for U-matrix and overall computations, respectively. The GPU implementation further improves efficiency for large targets.