Flat-space anisotropic beam shear focusing ($A=1-{\lambda }^2,{\sigma }^2=(1-{\lambda }^2{)}^{-2}$)

In flat spacetime take a twist-free beam's two screen Jacobi factors to be $1+\lambda$ and $1-\lambda$. They solve the zero-curvature geodesic deviation equation. Their product is the displayed area, and subtracting the trace from their optical matrix gives the displayed nonnegative shear. Direct differentiation yields $(A^{1/2}{)}^{\prime \prime }=-(1-{\lambda }^2{)}^{-3/2}=-{\sigma }^2{A}^{1/2}$, an explicit check of the focusing sign.