Free-space diffraction

In a homogeneous medium, a paraxial envelope satisfies $E_x=i{\Delta }_{\perp }E/(2k)$. This free-space diffraction spreads transverse Fourier modes by the phase factor $\exp (-ix|\mathbf{p}{|}^2/(2k))$.

For $m_x=(i/k)m_{r_1{r}_2}$, the free-space fourth-moment propagator multiplies the two-dimensional Fourier transform by $e^{-ix{p}_1{p}_2/k}$. In physical coordinates its oscillatory kernel is