Extending the Debye-Török theory: modeling and simulation of light focusing through an absorbing interface

We develop a unified theoretical and numerical framework for vectorial focusing across a planar interface into a lossy medium. By embedding complex-angle refraction into the Debye-Török integral and enforcing the radiation-branch condition, we derive transmitted-field integrals that explicitly separate phase propagation from absorption-induced attenuation, yielding numerically stable evaluations at high NA. Simulations quantify how increasing loss reshapes the focal field: the Strehl ratio decreases, the axial maximum shifts toward the interface, the effective aperture NA_eff narrows, and the longitudinal-field fraction η_z decreases, while the lateral main lobe is largely preserved to first order. The framework clarifies energy-flow topology in inhomogeneous waves and provides actionable metrics (Strehl, Δz, NA_eff, η_z) for design and compensation in absorption-dominated settings (e.g., bioimaging, thermoplasmonics, and high-NA patterning). Transparent-limit and convergence checks establish physical consistency and numerical robustness, and the formulation is readily extensible to stratified and anisotropic media.