In the paraxial approximation of the Helmholtz equation, the complex amplitude A is expressed as A ( r ) = u ( r ) e i k z where u represents the complex-valued amplitude which modulates the sinusoidal plane wave represented by the exponential factor. Then under a suitable assumption, u approximately solves ∇ ∇ ⊥ ⊥ 2 u + 2 i k ∂ ∂ u ∂ ∂ z = 0 , where ∇ ∇ ⊥ ⊥ 2 = def ∂ ∂ 2 ∂ ∂ x 2 + ∂ ∂ 2 ∂ ∂ y 2 is the transverse part of the Laplacian.