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[Z^2 + y_1^2  ]^{1 / 2} \, .
[Z^2 + y_1^2  ]^{1 / 2} \, .
</math>
</math>
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Notice that, in the definition of <math>~D_j</math>, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, <math>~Y_j/L</math>, as well as a term that depends linearly on this ratio.  As discussed below, our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored.  Via a related simplification, we will find that the various natural lengths of this problem are can be related via the expression,
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<math>~y_1\biggr|_{1^\mathrm{st} \mathrm{fringe}}</math>
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<math>~\approx</math>
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<math>~\frac{\lambda Z}{w} \, .</math>
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Revision as of 04:47, 28 December 2017

CGH: Philosophical Overview

Whitworth's (1981) Isothermal Free-Energy Surface
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Slit Diffraction

Single Aperture

Figure 1
Chapter1Fig1

As has been detailed in an accompanying discussion, we consider, first, 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 = (Y_1 - Y_2) = 2c</math>. The amplitude is given by the expression,

<math>~A(y_1)</math>

<math>~=</math>

<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>

where,

<math>~D_j</math>

<math>~=</math>

<math>~ L \biggl[1 - \frac{2y_1 Y_j}{L^2} + \frac{Y_j^2}{L^2} \biggr]^{1 / 2} \, , </math>

and,

<math>~L</math>

<math>~\equiv</math>

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

Notice that, in the definition of <math>~D_j</math>, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, <math>~Y_j/L</math>, as well as a term that depends linearly on this ratio. As discussed below, our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored. Via a related simplification, we will find that the various natural lengths of this problem are can be related via the expression,

<math>~y_1\biggr|_{1^\mathrm{st} \mathrm{fringe}}</math>

<math>~\approx</math>

<math>~\frac{\lambda Z}{w} \, .</math>

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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