= Solution
A <null geodesic congruence> is a smooth family of null geodesics filling a region without intersections there, with one tangent direction assigned smoothly at every point. Choose an <affine parameter> $\lambda$ and tangent $U^a$, so $U^aU_a=0$ and $U^b\nabla_bU^a=0$. Choose a second null vector $N^a$ with $U\cdot N=-1$. The <screen-space projector> is
$$
q_{ab}=g_{ab}+U_aN_b+N_aU_b.
$$
It annihilates $U,N$ and restricts to a positive-definite metric on the two-dimensional transverse screen. Project the <covariant derivative> of the tangent to obtain the <optical tensor>
$$
\widehat B_{ab}=q_a{}^cq_b{}^d\nabla_dU_c.
$$
Its decomposition defines the three optical quantities:
$$
\boxed{\theta=q^{ab}\widehat B_{ab},\qquad\widehat\omega_{ab}=\widehat B_{[ab]},\qquad\widehat\sigma_{ab}=\widehat B_{(ab)}-\frac12\theta q_{ab}.}
$$
The <null expansion> is the fractional rate of change of a transverse area, $\theta=A^{-1}dA/d\lambda$. The <null shear> changes a beam's shape without changing its area to first order. The <null twist> is its antisymmetric transverse deformation. The factor $1/2$ uses four spacetime dimensions.
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