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As has been detailed in an [[User:Tohline/Appendix/CGH/ParallelApertures#One-Dimensional_Aperture|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,
+
As has been detailed in an [[User:Tohline/Appendix/CGH/ParallelApertures#One-Dimensional_Aperture|accompanying discussion]], we examine, first, the amplitude (and phase) of light which, after passing through  an aperture of width <math>~w = (Y_1 - Y_2) = 2c</math>, is incident on an image screen that is parallel to, and located a distance <math>~Z</math> from, the aperture.  The amplitude at any location, <math>~y_1</math>, along the image screen is given by the expression,
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<math>~A(n\Delta y_1)</math>
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<math>~A(y_1)</math>
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<math>~\sum_j
<math>~\sum_j
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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]
+
\beta_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]
\, ,
\, ,
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where, <math>~a_j = a(Y_j)</math> is the specified brightness at each "j" point along the illuminated aperture,
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where, <math>~\beta_j = \beta(Y_j)</math> is the specified brightness at each "j" point along the illuminated aperture,
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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 also will find that the various natural lengths of this problem are can be related via the expression,
+
 
 +
The green solid curves in [[User:Tohline/Appendix/CGH/ParallelApertures#Figure2|Figures 2 &amp; 3 of our accompanying, more detailed discussion]] display &#8212; for a fixed set of values of the parameter-pair, <math>~(\lambda, w)</math> &#8212; how the amplitude of the incident light, <math>~(A\cdot A^*)^{1 / 2}</math>, varies across an image screen that is placed at two different distances from the aperture.
 +
 
 +
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 [[#Simplifications|discussed below]], our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored. (The red dotted curves in [[User:Tohline/Appendix/CGH/ParallelApertures#Figure2|Figures 2 &amp; 3 of our accompanying discussion]] illustrate how the image-screen diffraction pattern is altered as a result of this simplification.) Via a related simplification, we also will find that the various natural lengths of this problem can be related to one another via the expression,
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CAUTION:  Be sure that the "L" in these various expressions is, indeed, always the same quantity.
CAUTION:  Be sure that the "L" in these various expressions is, indeed, always the same quantity.
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===Simplifications===
 +
 +
 +
If we assume that everywhere along the aperture the phase <math>~\phi_j = 0</math>; and if we assume that, for all <math>~j</math>,
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<math>~\biggl| \frac{Y_j}{L}\biggr| \ll 1</math>
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</div>
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we can drop the quadratic term in favor of the linear one in the above expression for <math>~D_j</math> and [[User:Tohline/Appendix/CGH/ParallelApertures#Simplify|deduce that]],
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<math>~A(y_1)</math>
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<math>~\approx</math>
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  <td align="left">
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<math>~e^{i2\pi L/\lambda} \sum_j
 +
\beta_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]
 +
\, .
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</math>
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  </td>
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</table>
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</div>
=See Also=
=See Also=

Revision as of 15:19, 28 December 2017

Contents

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 examine, first, the amplitude (and phase) of light which, after passing through an aperture of width ~w = (Y_1 - Y_2) = 2c, is incident on an image screen that is parallel to, and located a distance ~Z from, the aperture. The amplitude at any location, ~y_1, along the image screen is given by the expression,

~A(y_1)

~=

~\sum_j
\beta_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]
\, ,

where, ~\beta_j = \beta(Y_j) is the specified brightness at each "j" point along the illuminated aperture,

~D_j

~=

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

and,

~L

~\equiv

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

The green solid curves in Figures 2 & 3 of our accompanying, more detailed discussion display — for a fixed set of values of the parameter-pair, ~(\lambda, w) — how the amplitude of the incident light, ~(A\cdot A^*)^{1 / 2}, varies across an image screen that is placed at two different distances from the aperture.

Notice that, in the definition of ~D_j, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, ~Y_j/L, 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. (The red dotted curves in Figures 2 & 3 of our accompanying discussion illustrate how the image-screen diffraction pattern is altered as a result of this simplification.) Via a related simplification, we also will find that the various natural lengths of this problem can be related to one another via the expression,

~y_1\biggr|_{1^\mathrm{st} \mathrm{fringe}} = \frac{L \Delta y_1}{c}

~\approx

~\frac{\lambda Z}{2c} \, .

CAUTION: Be sure that the "L" in these various expressions is, indeed, always the same quantity.

Simplifications

If we assume that everywhere along the aperture the phase ~\phi_j = 0; and if we assume that, for all ~j,

~\biggl| \frac{Y_j}{L}\biggr| \ll 1

we can drop the quadratic term in favor of the linear one in the above expression for ~D_j and deduce that,

~A(y_1)

~\approx

~e^{i2\pi L/\lambda} \sum_j
\beta_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]
\, .

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation

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