Huygens–Fresnel propagation

Write the physical electric component as the real part of a complex amplitude multiplied by $e^{-i\omega t}$. With this convention an outgoing spherical contribution carries $e^{ikr}/r$: surfaces of constant phase move outward as time increases. Here $k=2\pi/\lambda$ uses the wavelength in the propagation medium. Choosing the opposite time convention conjugates the propagation phases; measurable intensities stay unchanged if every formula is converted consistently.

A scalar field description assumes a narrow frequency band, a uniform isotropic medium, and polarization that can be treated as fixed. It works well for many apertures large compared with a wavelength and modest propagation angles. It does not satisfy every electromagnetic boundary condition at a metal edge. The boundary field immediately after a thin mask is often modeled as the incident field multiplied by a complex amplitude transmission; that is an approximation requiring its own physical justification.

Label points in the aperture plane by $(\xi,\eta)$ and an observation point by $(x,y,z)$ with positive $z$. Every contributing patch has a different propagation distance. Integrate the complex contributions before taking the absolute square, because a square taken patch by patch discards cancellation. A large collection of individually nonzero contributions can give a dark point when their phases wind around a closed polygon in the complex plane.

In the paraxial approximation the slowly changing denominator becomes $z$, while the rapidly changing phase retains the transverse distance squared. The factor $1/(i\lambda z)$ is essential for a field with the same units as the input: the area integral contributes square metres. An obliquity factor belongs to a more complete diffraction integral; it becomes approximately unity near the forward axis. Thus the simple kernel is a controlled forward propagation model, not an exact recipe for arbitrary boundaries.

The distance expansion contains a quadratic correction followed by a negative quartic correction. If $\rho$ denotes the transverse separation between a patch and the observation point, the omitted distance is approximately $-\rho^4/(8z^3)$. Multiplication by a large optical wavenumber can make that small geometrical error an appreciable phase error. Check the largest contributing separation, rather than only the observation angle or the aperture centre.

The field integral is linear, while intensity is quadratic. Doubling every incident amplitude doubles the downstream amplitude and quadruples intensity in this model. Moving a boundary or changing wavelength instead changes relative phases, so its effects cannot generally be inferred by a simple intensity scaling. That distinction explains both the usefulness of coherent superposition and the failure of the misconception that diffraction merely spreads a fixed brightness uniformly into the shadow.