Editing Fraunhofer diffraction (section)

=== Derivation of Fraunhofer condition ===
[[File:Fraunhofer Condition Derivation Geometry3.png|thumb|293x293px|A geometrical diagram used to derive Fraunhofer condition at which Fraunhofer diffraction is valid.]]
The derivation of Fraunhofer condition here is based on the geometry described in the right box.<ref>{{Cite book |last=Hecht|first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages=453 | chapter=Problem 9.21}}</ref> The diffracted wave path ''r''<sub>2</sub> can be expressed in terms of another diffracted wave path ''r''<sub>1</sub> and the distance ''b'' between two diffracting points by using the [[law of cosines]];

<math display="block">{r_2} = {\left( r_1^2 + b^2 - 2b{r_1}\cos \left( \frac{\pi }{2} - \theta \right) \right)}^{\frac{1}{2}} = {r_1}{\left( 1+\frac{b^2}{r_1^2} - 2\frac{b}{r_1} \sin \theta \right)}^{\frac{1}{2}}.</math>

This can be expanded by calculating the expression's [[Taylor series]] to second order with respect to <math>\frac{b}{r_1}</math>,
<math display="block">{r_2}={r_1}\left( 1-\frac{b}{r_1}\sin \theta +\frac{b^2}{2 r_1^2} \cos^2 \theta + \cdots \right) = {r_1} - b\sin \theta +\frac{b^2}{2 r_1} \cos^2 \theta + \cdots ~.</math>

The phase difference between waves propagating along the paths ''r''<sub>2</sub> and ''r''<sub>1</sub> are, with the wavenumber where λ is the light wavelength,
<math display="block">k{r_2}-k{r_1} = -kb\sin \theta +k\frac{b^2}{2r_1} \cos^2 \theta + \cdots .</math>

If <math>k\frac{b^2}{2{r_1}} \cos^2 \theta = \pi \frac{b^2}{\lambda r_1} \cos^2 \theta \ll \pi </math> so <math>\frac{b^2}{\lambda r_1} \cos^2 \theta \ll 1</math>, then the phase difference is <math>k r_2 - k r_1 \approx -kb\sin \theta </math>. The geometrical implication from this expression is that the paths ''r''<sub>2</sub> and ''r''<sub>1</sub> are approximately parallel with each other. Since there can be a diffraction - observation plane, the diffracted wave path whose angle with respect to a straight line parallel to the optical axis is close to 0, this approximation condition can be further simplified as <math>\frac{b^2}{\lambda }\ll L</math> where ''L'' is the distance between two planes along the optical axis. Due to the fact that an incident wave on a diffracting plane is effectively a plane wave if <math>\frac{b^2}{\lambda }\ll L</math> where ''L'' is the distance between the diffracting plane and the point wave source is satisfied, Fraunhofer condition is <math>\frac{b^2}{\lambda }\ll L</math> where ''L'' is the smaller of the two distances, one is between the diffracting plane and the plane of observation and the other is between the diffracting plane and the point wave source.