Editing Diffraction (section)

=== General aperture ===
The wave that emerges from a point source has amplitude <math>\psi</math> at location <math>\mathbf r</math> that is given by the solution of the [[frequency domain]] [[wave equation]] for a point source (the [[Helmholtz equation]]),
<math display="block">\nabla^2 \psi + k^2 \psi = \delta(\mathbf r),</math>
where <math> \delta(\mathbf r)</math> is the 3-dimensional delta function.  The delta function has only radial dependence, so the [[Laplace operator]] (a.k.a. scalar Laplacian) in the [[spherical coordinate system]] simplifies to
<math display="block">\nabla ^2\psi = \frac{1}{r} \frac {\partial ^2}{\partial r^2} (r \psi) .</math>

(See [[del in cylindrical and spherical coordinates]].) By direct substitution, the solution to this equation can be readily shown to be the scalar [[Green's function]], which in the [[spherical coordinate system]] (and using the physics time convention <math>e^{-i \omega t}</math>) is
<math display="block">\psi(r) = \frac{e^{ikr}}{4 \pi r}.</math>

This solution assumes that the delta function source is located at the origin.  If the source is located at an arbitrary source point, denoted by the vector <math>\mathbf r'</math> and the field point is located at the point <math>\mathbf r</math>, then we may represent the scalar [[Green's function]] (for arbitrary source location) as
<math display="block">\psi(\mathbf r | \mathbf r') = \frac{e^{ik | \mathbf r - \mathbf r' | }}{4 \pi | \mathbf r - \mathbf r' |}.</math>

Therefore, if an electric field <math>E_\mathrm{inc}(x, y)</math> is incident on the aperture, the field produced by this aperture distribution is given by the [[surface integral]]
<math display="block">\Psi(r)\propto \iint\limits_\mathrm{aperture} \!\! E_\mathrm{inc}(x',y') ~ \frac{e^{ik | \mathbf r - \mathbf r'|}}{4 \pi | \mathbf r - \mathbf r' |} \,dx'\, dy',</math>

[[Image:Fraunhofer.svg|upright=1.4|thumb|On the calculation of Fraunhofer region fields]]
where the source point in the aperture is given by the vector
<math display="block">\mathbf{r}' = x' \mathbf{\hat{x}} + y' \mathbf{\hat{y}}.</math>

In the far field, wherein the parallel rays approximation can be employed, the Green's function,
<math display="block">\psi(\mathbf r | \mathbf r') = \frac{e^{ik | \mathbf r - \mathbf r' |} }{4 \pi | \mathbf r - \mathbf r' |},</math>
simplifies to
<math display="block"> \psi(\mathbf{r} | \mathbf{r}') = \frac{e^{ik r}}{4 \pi r} e^{-ik ( \mathbf{r}' \cdot \mathbf{\hat{r}})}</math>
as can be seen in the adjacent <!-- "adjacent" is not the best description but is better than "to the right". The latter is affected by screen size (esp smartphones), aspect ratio and font size. -->figure<!-- to the right -->.

The expression for the far-zone (Fraunhofer region) field becomes
<math display="block">\Psi(r)\propto \frac{e^{ik r}}{4 \pi r} \iint\limits_\mathrm{aperture} \!\! E_\mathrm{inc}(x',y') e^{-ik  ( \mathbf{r}' \cdot \mathbf{\hat{r}} ) } \, dx' \,dy'.</math>

Now, since
<math display="block">\mathbf{r}' = x' \mathbf{\hat{x}} + y' \mathbf{\hat{y}}</math>
and
<math display="block">\mathbf{\hat{r}} =  \sin \theta \cos \phi \mathbf{\hat{x}} + \sin \theta ~ \sin \phi ~ \mathbf{\hat{y}} +  \cos \theta \mathbf{\hat{z}},</math>
the expression for the Fraunhofer region field from a planar aperture now becomes
<math display="block">\Psi(r) \propto \frac{e^{ik r}}{4 \pi r} \iint\limits_\mathrm{aperture} \!\! E_\mathrm{inc}(x',y') e^{-ik \sin \theta (\cos \phi x' + \sin \phi y')} \, dx' \, dy'.</math>

Letting
<math display="block">k_x = k \sin \theta \cos \phi </math>
and
<math display="block">k_y = k \sin \theta \sin \phi \,,</math>
the Fraunhofer region field of the planar aperture assumes the form of a [[Fourier transform]]
<math display="block">\Psi(r)\propto \frac{e^{ik r}}{4 \pi r} \iint\limits_\mathrm{aperture} \!\! E_\mathrm{inc}(x',y') e^{-i (k_x x' + k_y y') } \, dx' \, dy' ,</math>

In the far-field / Fraunhofer region, this becomes the spatial [[Fourier transform]] of the aperture distribution.  Huygens' principle when applied to an aperture simply says that the [[far-field diffraction pattern]] is the spatial Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields (see [[Fourier optics]]).