Explanation of λ^−2 optical scattering and λ^−2 Strehl on-axis irradiance reduction

Abstract

The often-observed λ−2 total integrated scattering PTIS, the Strehl λ−2 on-axis irradiance reduction ΔI0, and the λ−2 specular-power reduction ΔPsp all require that the scatterers have large transverse dimensions ltr, kltr ≫ 1 in addition to the well-known requirement of small-amplitude phase distortion klopd ≪ 1. Here k and λ are the wave vector and the wavelength, respectively, and lopd is an optical path difference. All three λ−2 results are incorrect in the Rayleigh limit of kltr ≪ 1 and klopd ≪ 1, the correct results then being ΔI0 ~ PTIS ~ ΔPsp ~ λ−4. These incorrect λ−2 expressions result from the failure of the Kirchhoff diffraction theory and a resulting nonconservation of energy for kltr ≪ 1. The differential-scattering cross section dσ/dΩ is proportional to λ−4 in both limits. The scattering is into small angles, θ ≲ θdif ≅ λ/ltr for kltr ≫ 1 but into all angles I ~ 1 + cos2θ for kltr ≪ 1. Thus PTIS ~ Ωdσ/dΩ ~ θ2dσ/dΩ ~ λ2λ−4 ~ λ−2 for kltr ≫ 1, and P ~ ∫ dΩ(1 + cos2θ)λ−4 ~ λ−4 for kltr ≪ 1. Diffraction, rather than refraction, causes the λ−2 scattering. The common misconceptions that PTIS ~ λ−2 requires surface scattering, Gaussian or other distributions of lopd and or ltr, angle integration over Rayleigh scattering, or klopd ≅ 1 are shown to be incorrect.