diffractionelectromagnetic-radiationhuygens-principlemaxwell-equationssemiclassical
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?
The principle that every point on a wavefront can be thought of as an emitter of spherical (or, in 2D, circular) waves is applicable to any waves - see any introductory high school course on waves, where the demonstrations are typically done with water surface waves. Note that, as @ignacio pointed out, the construction is only exact in odd dimensions (in practice that means 3D), but even in 2D it is quite convincing:
Nice demonstration with water waves in 2D
And the link given by Ignacio:
Mathematical proof that Huygens construction is only valid in odd dimensions
In order to understand Huygens principle in this context clearly, one needs to resort to the mathematical formulation of the scalar diffraction theory for diffraction from an aperture. According to the Rayleigh-Sommerfeld formula, the diffracted field at a point in space in front of the aperture can be written as
$$U_P(x,y) = \frac{1}{j\lambda}\iint_{\text{aperture}}U_I(x',y')\frac{\exp{(jkr)}}{r}\cos \theta \,\,ds$$
As you see from the above equation, the observed field $U_P$ is a sum of diverging spherical waves in the form of $\dfrac{\exp{(jkr})}{r}$ located at each and every point in the aperture (as stated in the Huygens principle), multiplied by a factor of $\dfrac{1}{j\lambda}\,U_I(x',y') \cos \theta$. Therefore, the fictitious source located at $(x',y')$ has the complex amplitude proprtional to the incident field at that point, $U(x',y')$. This seems reasonable considering the linearity of the problem. (The presence of the remaining multiplicative factors $1/j\lambda$ and $\cos \theta$ may be explained in some other ways but not very intuitively.)
To summarize, your statement that "each point source on the wave front is spaced exactly one wavelength apart" is wrong. The single slit problem is usually treated within the scope of the Fraunhoffer (far-field) approximation of the more general formula above, where the observed diffraction pattern is the Fourier transofrm of the aperture. It means the width of of observed pattern is inversely proportional to the width of the aperture.
Best Answer
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:
In free space any component of the Maxwell field satisfies the (scalar) wave equation.
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.
The Huygen-Fresnel's equation is obtained as the stationary phase approximation of the Helmholtz formula.