Let \(\pi\) : \(E\to B\) be a Lagrangian fibration. A hypersurface M in E is called layerwise convex if the intersection of M with each fiber is locally strictly convex in the fiber. For fixed M, a Lagrange submanifold L of E is called a ray manifold if L is contained in M. In geometrical optics it is natural to treat the ray manifolds, and in this paper a nontrivial restriction on the bifurcation of ''optical caustics'' for ray manifolds with respect to the projection \(\pi\) is given, comparing to the theory of Arnol'd on the classification of generic one-parameter bifurcations of caustics. Precisely, it is shown that if the singular locus \(\Sigma\) of a ray manifold with respect to \(\pi\) is nonsingular and compact, then the Euler characteristic of \(\Sigma\) is equal to zero, or equivalently, there exists a smooth direction field on \(\Sigma\). Some enumerative equalities are given also for the case \(\Sigma\) has a singularity. This paper is important for singularity theory and topology, and the reader is referred to \textit{V. I. Arnol'd}'s recent survey [Usp. Mat. Nauk 41, No.6, 3-18 (1986; Zbl 0618.58021)].