# How can the Huygens-Fresnel principle be derived from the Maxwell equations?

The Huygens-Fresnel principle states that every point to which a luminous disturbance reaches becomes a source of a spherical wave. I have been trying to understand this considering a infinite screen with a microscopic hole $dS$ on which a plane wave is incident, but I see no obvious way to describe the resulting propagating wave.

How can it be derived from the Maxwell equations?

• What level of physics are you at? The rigorous way to solve this problem involves Greens' function solutions – Jerry Schirmer Jun 12 '12 at 13:54
• I'd say at the level of Jackson's Classical Electrodynamics. – Whelp Jun 13 '12 at 20:35

The Huygen's principle can be obtained from the Maxwell equations, please see Guillemin Sternberg's course Semi-classical analysis section 14.9.

The derivation is based upon the following:

1. In free space any component of the Maxwell field satisfies the (scalar) wave equation.

2. The solutions of the wave equation satisfy the Helmholtz formula obtained from the Green's theorem by substituting a spherical wave for one of the functions.

3. The Huygen-Fresnel's equation is obtained as the stationary phase approximation of the Helmholtz formula.

• Thanks for the reference. I see the general logic but I'm afraid the level of mathematical sophistication used in this derivation is somewhat too refined for me to feel comfortable with. – Whelp Jun 13 '12 at 20:35

My "Exact derivation of Kirchhoff's integral theorem and diffraction formula using high-school math" doesn't use Maxwell's equations, but it does note that a current element behaves like a simple monopole source if we describe it in terms of the magnetic vector potential. That makes it amenable to my simple math.