Helmholtz equation

A light perturbation at point P and at instant t is represented by the scalar function u(P,t). For a monochromatic wave, we can explicitly express it as:

A(P) and φ(P) are respectively the amplitude and phase of the wave at point P; v is the temporal frequency. Using the complex notation:

U(P) is the complex amplitude.

If the real perturbation u(P,t) represents an optical wave, it must satisfy the following scalar wave equation at any space point where there is no source:

where Δ is the Laplace operator: and c is the speed of light.

Since the dependence with t is know a priori, the knowledge of the complex function U(P) is sufficient to describe the perturbation. By replacing (II-7) in (II-8), we deduce that the complex amplitude must obey the following equation:

The relation (II-9) is known as the Helmholtz equation. In the following, we will assume that the complex amplitude of any monochromatic optical wave propagating in free space must obey that relation