196
edits
Changes
Jump to navigation
Jump to search
Line 81:
Line 81: − + + + + +
Huygens Principle for a Planar Source (view source)
Revision as of 15:52, 9 July 2009
, 15:52, 9 July 2009→Special Case
This gives us the equation<math>\mathcal{J}(r')=2\pi\frac{e^{ikr'}}{r'^2}\int_0^a rdr \frac{d}{k\sin{\theta}'dr} J_1(kr\sin{\theta}')</math><br><br>
This gives us the equation<math>\mathcal{J}(r')=2\pi\frac{e^{ikr'}}{r'^2}\int_0^a rdr \frac{d}{k\sin{\theta}'dr} J_1(kr\sin{\theta}')</math><br><br>
We invoke the principle that the integral of a derivative is the function evaluated at the end points to give us the equation<br><br><math>\mathcal{J}(r')=2\pi\frac{e^{ikr'}}{k\sin{\theta}'r'^2}\left[aJ_1(ka\sin{\theta}')-0J_1(0k\sin{\theta}')\right]=2\pi\frac{e^{ikr'}}{k\sin{\theta}'r'^2}aJ_1(ka\sin{\theta}')</math><br><br> and <math>A(r')=\frac{z'\dot{A_0}}{2\pi c}e^{-ikct'} 2\pi\frac{e^{ikr'}}{k\sin{\theta}'r'^2}aJ_1(ka\sin{\theta}')=\frac{z'\dot{A_0}a}{c}\frac{e^{ikr'-ikct'}}{k\sin{\theta}'r'^2}J_1(ka\sin{\theta}')</math><br><br>
We invoke the principle that the integral of a derivative is the function evaluated at the end points to give us the equation<br><br><math>\mathcal{J}(r')=2\pi\frac{e^{ikr'}}{k\sin{\theta}'r'^2}\left[aJ_1(ka\sin{\theta}')-0J_1(0k\sin{\theta}')\right]=2\pi\frac{e^{ikr'}}{k\sin{\theta}'r'^2}aJ_1(ka\sin{\theta}')</math><br><br> and <math>A(r')=\frac{z'\dot{A_0}}{2\pi c}e^{-ikct'} 2\pi\frac{e^{ikr'}}{k\sin{\theta}'r'^2}aJ_1(ka\sin{\theta}')=\frac{z'\dot{A_0}a}{c}\frac{e^{ikr'-ikct'}}{k\sin{\theta}'r'^2}J_1(ka\sin{\theta}')</math><br><br>
To find the angle to the first diffraction minimum, we must find the first zero of this amplitude function. This will occur when <math>J_1(ka\sin{\theta}')=0 \quad</math>
To find the angle to the first diffraction minimum, we must find the first zero of this amplitude function. This will occur when <math>J_1(ka\sin{\theta}')=0 \quad</math><br><br>
[[Image:Bessels_J0.svg|thumb|300px|right|Plot of Bessel function of the first kind, J<sub>α</sub>(x), for integer orders α=0,1,2.]]
[[Image:Bessels_J0.svg|thumb|300px|right|Plot of Bessel function of the first kind, J<sub>α</sub>(x), for integer orders α=0,1,2.]]
To the right is a graph of three Bessel functions of the first order, specifically <math> J_0(x), J_1(x), and J_2(x) \quad</math>. It it is shown, the first zero of <math>J_1(x) \quad</math> will <br>
occur at <math>x=0 \quad</math>. This will correspond to the center of the pattern, at <math>\theta=0 \quad</math>. Here, we would expect a bright spot, so <math>A(r') \quad</math> sould be positive and finite. At <math>\theta=0 \quad</math> the term <math>\frac{J_1(ka\sin{\theta}')}{\sin{\theta}'}</math> is positive and finite, so this expression gives the correct tamplitude at <math>\theta=0 \quad</math>. The next zero of <math> J_1(x) \quad</math> corresponds to the first minumum of the diffraction pattern.
[[Image:Airy-pattern.svg|thumb|300px|right|Image of the Airy disk created at the lens surface]]