Editing Appendix/CGH/ParallelApertures (section)

===Utility of FFT Techniques===
Let's rewrite the first of our above equations in a form that takes into account many more than just two points along the aperture.  That is,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A(y_1)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\sum_j
a_j e^{i(2\pi D_j/\lambda + \phi_j)} 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\sum_j
a_j \biggl[ \cos\biggl(\frac{2\pi D_j}{\lambda} + \phi_j \biggr) + i   \sin\biggl(\frac{2\pi D_j}{\lambda} + \phi_j \biggr) \biggr]
\, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="85%" cellpadding="8">
<tr><td align="left">
Note that this is identical to the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A(y_1)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\sum_j a_j
\biggl[\cos\phi_j + i \sin\phi_j \biggr] \biggl[ \cos\biggl(\frac{2\pi D_j}{\lambda} \biggr) + i   \sin\biggl(\frac{2\pi D_j}{\lambda}\biggr) \biggr]
\, .
</math>
  </td>
</tr>
</table>
</td></tr>
</table>

In the more restrictive case when we assume that everywhere along the aperture the phase <math>\phi_j = 0</math>, we have,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A(y_1)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\sum_j
a_j e^{i(2\pi D_j/\lambda)} 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\sum_j
a_j \biggl[ \cos\biggl(\frac{2\pi D_j}{\lambda} \biggr) + i   \sin\biggl(\frac{2\pi D_j}{\lambda} \biggr) \biggr]
\, ,
</math>
  </td>
</tr>
</table>

where, in each of these expressions, we have replaced <math>a(Y_j)</math> with <math>a_j</math>.  After acknowledging that the function, <math>A(y_1)</math>, is complex &#8212; with <math>\mathcal{R}e(A)</math> being given by the sum over cosine terms and <math>\mathcal{I}m(A)</math> being given by the sum over sine terms &#8212; it is clear that the brightness of each point on the image screen is given by (the square root of) <math>A</math> multiplied by its complex conjugate, <math>A^*</math>, that is, by the expression, <math>(A\cdot A^*)^{1 / 2}</math>.  Figures 2 &amp; 3 show how this brightness (amplitude) varies across the image screen for the case of a monochromatic light of wavelength, <math>\lambda = 500</math> nanometers, impinging on a single slit that is 1 mm in width; for all curves, the amplitude has been determined by summing over <math>j_\mathrm{max} = 51</math> equally spaced points across the slit.   The results displayed in Figure 2 are for an image screen that is placed <math>Z = 1</math> meter from the slit, while Figure 3 displays results for an image screen that is placed a distance, <math>Z = 10</math> m from the slit.

<div align="center" id="Figure2">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">Figure 2: &nbsp; <math>w = 1~\mathrm{mm}; Z = 1~\mathrm{m}; \lambda = 500~\mathrm{nm}; j_\mathrm{max} = 51</math></th>
</tr>
<tr>
  <td align="center">
[[File:SingleSlit01B.png|700px|First Plot of Single-Slit Diffraction Pattern]]
  </td>
</tr>
</table>
</div>

Two curves appear in both Figure 2 and Figure 3.  The solid green curve was obtained by plugging the precise, "nonlinear" determination of <math>D_j</math> &#8212; that is, the [[#Distance|mathematical expression for <math>D_j</math>, as given above]] &#8212; into the arguments of the trigonometric functions, while the dotted red curve was obtained by using the approximate, "linearized" expression for <math>D_j</math>, as will now be described.

<div align="center" id="Figure3">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">Figure 3: &nbsp; <math>w = 1~\mathrm{mm}; Z = 10~\mathrm{m}; \lambda = 500~\mathrm{nm}; j_\mathrm{max} = 51</math></th>
</tr>
<tr>
  <td align="center">
[[File:SingleSlit02B.png|700px|Second Plot of Single-Slit Diffraction Pattern]]
  </td>
</tr>
</table>
</div>


<span id="Simplify">If we assume that,</span> for all <math>j</math>,
<div align="center">
<math>\biggl| \frac{Y_j}{L}\biggr| \ll 1</math>
</div>
&#8212; in the present context, this generally will be equivalent to assuming that the width of the aperture is much much less than the distance <math>(Z)</math> separating the aperture from the image screen &#8212; we can drop the quadratic term in favor of the linear one in the above expression for <math>D_j</math> and deduce that,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>D_j</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
L \biggl[1 - \frac{2y_1 Y_j}{L^2}  \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
L \biggl[1 - \frac{y_1 Y_j}{L^2}  \biggr] \, .
</math>
  </td>
</tr>
</table>

<span id="Fourier">Hence, we have,</span>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A(y_1)</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>A_0 \sum_j
a_j e^{-i[2\pi y_1 Y_j/(\lambda L)]} 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>A_0 \sum_j
a_j \biggl[ \cos\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) - i  \sin\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \biggr]
\, ,
</math>
  </td>
</tr>
</table>

where, now, <math>A_0 = e^{i2\pi L/\lambda}</math>.  

Notice that this expression matches what we have [[Appendix/Ramblings/FourierSeries#Complex|referred to elsewhere as the ''Complex'' Fourier Series Expression]].  When written in this form, it should immediately be apparent why discrete Fourier transform techniques (specifically FFT techniques) are useful tools for evaluation of the complex amplitude, <math>A</math>.  See [[#Parallels_With_Standard_Fourier_Series|further elaboration below]].