Mean path length inside non-scattering refractive objects

The paper derives the low-scattering relation ⟨L⟩ = ⟨L0⟩/(1 − PT) by modeling bulk scattering as a Poisson process and summing a geometric series for trapped episodes, explicitly explaining the discontinuity at zero scattering; this is stated in Eqs. (3)–(4) and surrounding text . The candidate solution reproduces the same argument and result. For the zero-scattering baseline ⟨L0⟩, the paper provides the reciprocity/etendue-weighted surface–angular integral (Eq. (5)) and closed-form evaluations for slab (Eq. (6)), sphere (Eq. (7)), and the cube in three regimes (Eqs. (8)–(10)) . The candidate’s computations for slab and sphere match exactly, and for the cube they identify the same three refractive-index regimes and the correct limiting expressions, but they omit the explicit closed form in the intermediate regime that the paper supplies. Methodologically, both rely on the reflection-cancellation insight (ignoring non-TIR Fresnel reflections) to reduce L to a chord-length calculation in the relevant cases . Overall, the model’s derivations align closely with the paper’s, with the only gap being the missing explicit cube formula in the middle regime.