The problem of uncoupled modes of the pump in a double-clad fibre amplifier is analysed. As long as such modes avoid the core of the fibre, we consider the Dirichlet Laplacian problem which neglects the core. In this approach, the boundary of the cladding is treated as an ideal mirror. The laws of conservation established for the paraxial equation of diffraction give certain integral relations for the derivatives of modes at the boundary. Such relations are formulated as theorems. These theorems show ways to non-traditional design of single-mode fibre amplifiers with a multimode pump. In particular, the conservation of momentum can be applied to the design of the slab-pumped fibre amplifier. The conservation of angular momentum predicts high efficiency of coupling of a pump into a doped core embedded in a spiral-shaped double-clad fibre. Such predictions agree with results of numerical experiments published recently and allow a geometric optics interpretation.