Harmonic morphisms, conformal foliations and shear-free ray congruences

Equivalences between conformal foliations on Euclidean 3-space, Hermitian structures on Euclidean 4-space, shear-free ray congruences on Minkowski 4-space, and holomorphic foliations on complex 4-space are explained geometrically and twistorially; these are used to show that 1) any real-analytic complex-valued harmonic morphism without critical points defined on an open subset of Minkowski space is conformally equivalent to the direction vector field of a shear-free ray congruence, 2) the boundary values at infinity of a complex-valued harmonic morphism on hyperbolic 4-space define a real-analytic conformal foliation by curves of an open subset of Euclidean 3-space and all such foliations arise this way. This gives an explicit method of finding such foliations; some examples are given.