CGH: Consolidate Expressions Regarding Parallel Apertures

One-dimensional Apertures

From our accompanying discussion of the Utility of FFT Techniques, we start with the most general expression for the amplitude at one point on an image screen, namely,

<math>~\sum_j a_j e^{i(2\pi D_j/\lambda + \phi_j)} \, , </math>

and, assuming that <math>~|Y_j/L| \ll 1</math> for all <math>~j</math>, deduce that,

<math>~\approx</math>

<math>~\sum_j a_j e^{i[ 2\pi L/\lambda + \phi_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>

where,

<math>~\equiv</math>

<math>~ Z \biggl[1 + \frac{y_1^2}{Z^2} \biggr]^{1 / 2} \, . </math>

Note that <math>~L</math> is formally a function of <math>~y_1</math>, but in most of what follows it will be reasonable to assume, <math>~L \approx Z</math>. Notice, as well, that this last approximate expression for the (complex) amplitude at the image screen may be rewritten in the form that will be referred to as our,

Focal-Point Expression
<math>~A(y_1)</math>	<math>~\approx</math>	<math>~ e^{i 2\pi L/\lambda } \sum_j a_j e^{i \phi_j} \cdot e^{-i \Theta_j } \, , </math>

where,

<math>~\Theta_j</math>

<math>~\equiv</math>

<math>~\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \, .</math>

Case 1

In a related accompanying derivation titled, Analytic Result, we made the substitution,

<math>~\rightarrow</math>

<math>~a_0(Y) dY = a_0(\Theta) \biggl[ \frac{w}{2\beta_1} \biggr] d\Theta \, ,</math>

where,

<math>~\equiv</math>

<math>~\frac{\lambda L}{\pi y_1w} \, ,</math>

and changed the summation to an integration, obtaining,

<math>~\approx</math>

<math>~ e^{i 2\pi L/\lambda }\biggl[ \frac{w}{2\beta_1} \biggr] \int a_0(\Theta) e^{i\phi(\Theta)} \cdot e^{-i \Theta } d\Theta \, . </math>

If we assume that both <math>~a_0</math> and <math>~\phi</math> are independent of position along the aperture, and that the aperture — and, hence the integration — extends from <math>~Y_2 = -w/2</math> to <math>~Y_1 = +w/2</math>, we have shown that this last expression can be evaluated analytically to give,

<math>~A(y_1)</math>	<math>~\approx</math>	<math>~ e^{i [2\pi L/\lambda + \phi] }\biggl[ \frac{a_0 w}{2\beta_1} \biggr] \int_{\Theta_2}^{\Theta_1} e^{-i \Theta } d\Theta </math>
	<math>~=</math>	<math>~ e^{i [2\pi L/\lambda + \phi] } \cdot a_0 w ~\mathrm{sinc}(\beta_1) \, . </math>

We need to explicitly demonstrate that an evaluation of our Focal-Point Expression with <math>~a_j = 1</math>, gives this last sinc-function expression, to within a multiplicative factor of, something like, <math>~j_\mathrm{max}</math>.

Case 2

In our accompanying discussion of the Fourier Series, we have shown that a square wave can be constructed from the expression,

<math>~f(x)</math>	<math>~=</math>	<math>~ \frac{c}{L} + \sum_{n=1}^{\infty} \biggl( \frac{2}{n\pi} \biggr) \sin \biggl( \frac{n\pi c}{L} \biggr) \cos \biggl(\frac{n\pi x}{L}\biggr) </math>
	<math>~=</math>	<math>~ \frac{2c}{L}\biggl\{\frac{1}{2} + \sum_{n=1}^{\infty} \mathrm{sinc} \biggl( \frac{n\pi c}{L} \biggr) \cos \biggl(\frac{n\pi x}{L}\biggr) \biggr\} \, . </math>

Can we make this look like our above, Focal-Point Expression?

Let's start by setting

<math>~\frac{j\cdot w}{(j_\mathrm{max}-1)} - \frac{w}{2} \, ,</math>

for <math>~0 \le j \le (j_\mathrm{max}-1)</math>, in which case,

<math>~\Theta_j</math>	<math>~\equiv</math>	<math>~ \frac{2\pi }{\lambda L} \biggl[ \frac{j\cdot w}{(j_\mathrm{max}-1)} - \frac{w}{2} \biggr] = \frac{2\pi }{\lambda L} \biggl[ \frac{j\cdot w}{(j_\mathrm{max}-1)} \biggr] - \frac{2\pi }{\lambda L} \biggl[ \frac{w}{2} \biggr] </math>
	<math>~=</math>	<math>~ j \biggl[ \frac{2\pi w}{(j_\mathrm{max}-1) \lambda L} \biggr] - \frac{\pi w }{\lambda L} = j \cdot \Delta\Theta - \frac{(j_\mathrm{max} -1)}{2} \Delta\Theta \, ,</math>

where,

<math>~\Delta\Theta \equiv \biggl[ \frac{2\pi w}{(j_\mathrm{max}-1) \lambda L} \biggr] \, .</math>

This means that <math>~\Theta_{i} = - \Theta_{( j_\mathrm{max} - 1 - i )}</math>.

To answer this, let's recognize that each term under the summation in our Focal-Point expression may be rewritten in a variety of ways. For example,

<math>~a_j e^{i \phi_j} \cdot e^{-i \Theta_j } </math>	<math>~=</math>	<math>~a_j\biggl[\cos\phi_j + i\sin\phi_j \biggr]\biggl[ \cos \Theta - i \sin\Theta \biggr] </math>
	<math>~=</math>	<math>~a_j \biggl\{ \biggl[\cos\Theta \cos\phi_j + \sin\Theta \sin\phi_j \biggr]+ i~\biggl[\cos\Theta \sin\phi_j - \sin\Theta \cos\phi_j \biggr] \biggr\}</math>
	<math>~=</math>	<math>~a_j \biggl\{ \biggl[\cos( \Theta - \phi_j ) \biggr] - i~\biggl[ \sin(\Theta - \phi_j) \biggr] \biggr\}</math>

User:Tohline/Appendix/CGH/ParallelAperturesConsolidate

Contents

CGH: Consolidate Expressions Regarding Parallel Apertures

One-dimensional Apertures

Case 1

Case 2

See Also

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