User:Tohline/Appendix/CGH/ParallelApertures2D

CGH: 2D Rectangular Appertures that are Parallel to the Image Screen

This chapter is intended primarily to replicate §I.B from the online class notes — see also an updated Table of Contents — that I developed in conjunction with a course that I taught in 1999 on the topic of Computer Generated Holography (CGH) for a subset of LSU physics majors who were interested in computational science. This discussion parallels the somewhat more detailed one presented in §I.A on the one-dimensional aperture oriented parallel to the image screen.

Utility of FFT Techniques

Consider the amplitude (and phase) of light that is incident at a location (x₁, y₁) on an image screen that is located a distance Z from a rectangular aperture of width w and height h. By analogy with our accompanying discussion in the context of 1D apertures, the complex number, A, representing the light amplitude and phase at (x₁, y₁) will be,

<math>~A(x_1, y_1)</math>

<math>~=</math>

<math>~ \sum_j \sum_k a_{jk} e^{i(2\pi D_{jk} /\lambda + \phi_{jk})} \, , </math>

where, here, the summations are taken over all "j,k" elements of light across the entire 2D aperture, and now the distance D_jk is given by the expression,

<math>~D^2_{jk}</math>	<math>~\equiv</math>	<math>~ (X_j - x_1)^2 + (Y_k - y_1)^2 + Z^2 </math>
	<math>~=</math>	<math>~ Z^2 + y_1^2 - 2y_1 Y_k + Y_k^2 + x_1^2 - 2x_1 X_j + X_j^2 </math>
	<math>~=</math>	<math>~ L^2 \biggl[1 - \frac{2(x_1 X_j + y_1 Y_k ) }{L^2} + \frac{X_j^2 + Y_k^2}{L^2} \biggr] \, , </math>

and,

<math>~L</math>

<math>~\equiv</math>

<math>~ [Z^2 + y_1^2 + x_1^2]^{1 / 2} \, . </math>

If <math>~|X_j/L| \ll 1</math> and <math>~|Y_k/L| \ll 1</math> we can drop the quadratic terms in favor of the linear ones in the expression for D_jk and deduce that,

<math>~D_{jk}</math>	<math>~\approx</math>	<math>~ L \biggl[1 - \frac{2(x_1 X_j + y_1 Y_k ) }{L^2} \biggr]^{1 / 2} </math>
	<math>~\approx</math>	<math>~ L \biggl[1 - \frac{(x_1 X_j + y_1 Y_k ) }{L^2} \biggr] \, . </math>

Hence, the double-summation expression for the amplitude at screen location (x₁, y₁) becomes,

<math>~A(x_1, y_1)</math>

<math>~\approx</math>

<math>~ A_0 \sum_j \sum_k a_{jk} e^{i\phi_{jk} } \cdot \exp\biggl\{ -i \biggl[ \frac{2\pi(x_1 X_j + y_1Y_k}{\lambda L} \biggr] \biggr\} \, , </math>

where,

<math>~A_0 \equiv e^{i(2\pi L/\lambda)}</math>.

When written in this form, it should be apparent why discrete Fourier transform techniques — specifically, 2D-FFT techniques — are useful tools for evaluation of the complex amplitude, A(x₁, y₁).

NOTE: If x₁ and/or y₁ are ever comparable in size to Z — which may be the case for large apertures or for apertures tilted by nearly 90° to the image screen — then the variation of L with image screen position cannot be ignored and, accordingly, the coefficient A₀ cannot be moved outside of the double summation.

See Also

• Updated Table of Contents

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | 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