Editing Fraunhofer diffraction (section)

==Equation==
{{main|Fraunhofer diffraction equation}}
[[File:Frounhofer02.webm|300px|thumb|right|alt=Video of how the far field diffraction changes with the aperture shape.|Example of far field (Fraunhofer) diffraction for a few aperture shapes.]]

When a beam of [[light]] is partly blocked by an obstacle, some of the light is scattered around the object, light and dark bands are often seen at the edge of the shadow – this effect is known as [[diffraction]].<ref>{{Cite book | last1 = Heavens | first1 = O. S. | url = https://www.worldcat.org/oclc/22114471 | title = Insight into Optics | date = 1991 | publisher = Longman and Sons | first2 = R. W. | last2 = Ditchburn | isbn = 0-471-92769-4 | location = Chichester | oclc = 22114471 | page = 62}}</ref> These effects can be modelled using the [[Huygens–Fresnel principle]]; Huygens postulated that every point on a wavefront acts as a source of spherical secondary wavelets and the sum of these secondary wavelets determines the form of the proceeding wave at any subsequent time, while [[Augustin-Jean_Fresnel|Fresnel]] developed an equation using the Huygens wavelets together with the principle of superposition of waves, which models these diffraction effects quite well.

It is generally not straightforward to calculate the wave amplitude given by the sum of the secondary wavelets (The wave sum is also a wave.), each of which has its own [[amplitude]], [[Phase (waves)|phase]], and oscillation direction ([[Polarization (waves)|polarization]]), since this involves addition of many waves of varying amplitude, phase, and polarization. When two light waves as [[Electromagnetic field|electromagnetic fields]] are added together ([[vector sum]]), the amplitude of the wave sum depends on the amplitudes, the phases, and even the polarizations of individual waves. On a certain direction where electromagnetic wave fields are projected (or considering a situation where two waves have the same polarization), two waves of equal (projected) [[amplitude]] which are in phase (same phase) give the amplitude of the resultant wave sum as double the individual wave amplitudes, while two waves of equal amplitude which are in opposite phases give the zero amplitude of the resultant wave as they cancel out each other. Generally, a two-dimensional integral over complex variables has to be solved and in many cases, an analytic solution is not available.<ref>{{harvnb|Born|Wolf|1999|p=425}}</ref>

The Fraunhofer diffraction equation is a simplified version of [[Kirchhoff's diffraction formula]] and it can be used to model light diffraction when both a light source and a viewing plane (a plane of observation where the diffracted wave is observed) are effectively infinitely distant from a diffracting aperture.<ref>{{harvnb|Jenkins|White|1957|loc=Section 15.1, p. 288}}</ref> With a sufficiently distant light source from a diffracting aperture, the incident light to the aperture is effectively a [[plane wave]] so that the phase of the light at each point on the aperture is the same. At a sufficiently distant plane of observation from the aperture, the phase of the wave coming from each point on the aperture varies linearly with the point position on the aperture, making the calculation of the sum of the waves at an observation point on the plane of observation relatively straightforward in many cases. Even the amplitudes of the secondary waves coming from the aperture at the observation point can be treated as same or constant for a simple diffraction wave calculation in this case. Diffraction in such a geometrical requirement is called ''Fraunhofer diffraction'', and the condition where Fraunhofer diffraction is valid is called ''Fraunhofer condition'', as shown in the right box.<ref>{{Cite book |last1=Lipson |first1=A. |url=https://www.worldcat.org/oclc/637708967 |title=Optical physics |date=2011 |publisher=Cambridge University Press | first2 = S. G. | last2 = Lipson | first3 = H. | last3 = Lipson | author3-link = Henry Lipson | isbn=978-0-521-49345-1 |edition=4th |location=Cambridge |oclc=637708967 | page = 203}}</ref> A diffracted wave is often called ''Far field'' if it at least partially satisfies Fraunhofer condition such that the distance between the aperture and the observation plane <math>L</math> is <math>L\gg \frac{W^2}{\lambda}</math>.
{{Quote box|width=22em|quote='''Fraunhofer diffraction''' occurs when:
<math>\frac{W^2}{L\lambda} \ll 1</math> (Fraunhofer condition)
<p><math>W</math> – The largest size of a diffracting aperture or slit, <math>\lambda</math> – Wavelength, <math>L</math> – The smaller of the two distances, one is between the diffracting aperture and the plane of observation and the other is between the diffracting plane and the point wave source.</p>
}}
For example, if a 0.5 mm diameter circular hole is illuminated by a laser light with 0.6 μm wavelength, then Fraunhofer diffraction occurs if the viewing distance is greater than 1000 mm.

=== 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.

===Focal plane of a positive lens as the far field plane===

[[File:Lens and wavefronts rotated.gif|right|thumb|Plane wave focused by a lens.]]In the far field, propagation paths for wavelets from every point on an aperture to a point of observation are approximately parallel, and a positive lens (focusing lens) focuses parallel rays toward the lens to a point on the focal plane (the focus point position on the focal plane depends on the angle of the parallel rays with respect to the optical axis). So, if a positive lens with a sufficiently long focal length (so that differences between electric field orientations for wavelets can be ignored at the focus) is placed after an aperture, then the lens practically makes the Fraunhofer diffraction pattern of the aperture on its focal plane as the parallel rays meet each other at the focus.<ref>{{harvnb|Hecht|2002|p=448}}</ref>