= Solution
Transport two connecting vectors between neighbouring rays so that they commute with $U$, and project them onto the screen. Their infinitesimal transverse lengths and angles evolve according to the symmetric part of the <optical tensor>. The trace part gives uniform fractional dilation, while the trace-free <null shear> stretches one transverse direction and contracts the other, changing a circular beam to an ellipse without changing its area to first order. With zero <null twist>, there is no antisymmetric rotation of the beam.
If $q_{AB}$ is the metric on an infinitesimal transverse element, then $dq_{AB}/d\lambda=2\widehat B_{(AB)}$. Its area $A$ therefore obeys
$$
\boxed{\frac1A\frac{dA}{d\lambda}=\frac12q^{AB}\frac{dq_{AB}}{d\lambda}=\theta.}
$$
Thus positive <null expansion> means a beam's cross-sectional area increases, and negative expansion means it decreases; <null shear> records the directional distortion rather than this overall area change.
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