Editing Fraunhofer diffraction (section)

==== Single-slit diffraction using Huygens' principle ====
[[File:Singleslithuygens.jpg|thumb|Continuous broadside array of point sources of length ''a''.]]
We can develop an expression for the far field of a continuous array of point sources of uniform amplitude and of the same phase. Let the array of length ''a'' be parallel to the y axis with its center at the origin as indicated in the figure to the right. Then the differential [[Electric field|field]] is:<ref name=":0">{{Cite book|url=https://books.google.com/books?id=NRxTAAAAMAAJ|title=Antennas for all applications|last1=Kraus|first1=John Daniel|last2=Marhefka|first2=Ronald J.|date=2002|publisher=McGraw-Hill|isbn=9780072321036|language=en}}</ref>

<math display="block">dE=\frac{A}{r_1}e^{i \omega [t-(r_1/c)]}dy=\frac{A}{r_1}e^{i(\omega t-\beta r_1)}dy</math>

where <math>\beta=\omega/c=2\pi /\lambda</math>. However <math>r_1=r-y\sin\theta</math> and integrating from <math>-a/2</math> to <math>a/2</math>,

<math display="block">E \simeq A' \int_{-a/2}^{a/2} e^{i\beta y \sin\theta} dy</math>
where <math>A' = \frac{Ae^{i(\omega t-\beta r)}}{r}</math>.

Integrating we then get
<math display="block">E = \frac{2A'}{\beta \sin \theta} \sin\left(\frac{\beta a}{2} \sin \theta\right)</math>

Letting <math>\psi^'=\beta a \sin \theta = \alpha_r \sin \theta</math> where the array length in radians is <math>a_r=\beta a=2\pi a/\lambda</math>, then,<ref name=":0" />
<math display="block">E= A' a \frac{\sin(\psi^'/2)}{\psi^'/2}</math>