Editing Appendix/CGH/ParallelApertures (section)

===General Concept===
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  <th align="center">Figure 1</th>
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  <td align="center" bgcolor="lightgreen">[[File:Aperture3.gif|350px|Chapter1Fig1]]</td>
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Consider the amplitude (and phase) of light that is incident at a location <math>y_1</math> on an image screen that is located a distance <math>Z</math> from a slit of width <math>w</math>.  First, as illustrated in Figure 1, consider the contribution due only to two rays of light:&nbsp; one coming from location <math>~Y_1</math> at the top edge of the slit (a distance <math>~D_1</math> from point <math>~y_1</math> on the screen) and another coming from location <math>~Y_2</math> at the bottom edge of the slit (a distance <math>~D_2</math> from the same point on the screen).

The complex number, <math>~A</math>, representing the light amplitude and phase at <math>~y_1</math> will be,
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<math>~A(y_1)</math>
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<math>~=</math>
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<math>~
a(Y_1) e^{i(2\pi D_1/\lambda + \phi_1)} + 
a(Y_2) e^{i(2\pi D_2/\lambda + \phi_2)} \, ,
</math>
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where, <math>~\lambda</math> is the wavelength of the light, <math>~a(Y_j)</math> is the brightness of the light at point <math>~Y_j</math> on the aperture, and <math>~\phi_j</math> is the phase of the light as it leaves point <math>~Y_j</math>.

Now, if we consider for the moment that at all locations on the aperture, <math>~Y_j</math>, the light has a brightness <math>~a(Y_j) = 1</math> and a phase <math>~\phi_j = 0</math>, then,
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<math>~A(y_1)</math>
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<math>~=</math>
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<math>~A_0 \biggl[1 + e^{i(2\pi/\lambda)(D_2 - D_1)}  \biggr] \, ,</math>
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where, <math>~A_0 = e^{i(2\pi D_1/\lambda)}</math>.  So the question of whether the amplitude at <math>~y_1</math> on the image screen will be bright due to constructive interference or faint as a result of destructive interference comes down to a question of what the phase difference is between the two distances <math>~D_1</math> and <math>~D_2</math> where,
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<math>~D_j</math>
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<math>~\equiv</math>
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<math>~
[(Y_j - y_1)^2 + Z^2  ]^{1 / 2}
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&nbsp;
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<math>~=</math>
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<math>~
[Z^2 + y_1^2 - 2y_1 Y_j + Y_j^2 ]^{1 / 2}
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&nbsp;
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<math>~=</math>
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<math>~
L \biggl[1 - \frac{2y_1 Y_j}{L^2} + \frac{Y_j^2}{L^2} \biggr]^{1 / 2}
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and,
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<math>~L</math>
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<math>~\equiv</math>
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<math>~
[Z^2 + y_1^2  ]^{1 / 2} \, .
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