User:Tohline/Appendix/CGH/ParallelApertures2D - Revision history

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

2020-02-10T17:42:47Z

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

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	=CGH: 2D Rectangular ~~Appertures~~ that are Parallel to the Image Screen=		=CGH: 2D Rectangular Apertures that are Parallel to the Image Screen=
	This chapter is intended primarily to replicate [http://www.phys.lsu.edu/faculty/tohline/phys4412/howto/slit2d.html §I.B from the online class notes] — see also an updated [[User:Tohline/Appendix/Ramblings#Computer-Generated_Holography\|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 [[User:Tohline/Appendix/CGH/ParallelApertures\|§I.A]] on the one-dimensional aperture oriented parallel to the image screen.		This chapter is intended primarily to replicate [http://www.phys.lsu.edu/faculty/tohline/phys4412/howto/slit2d.html §I.B from the online class notes] — see also an updated [[User:Tohline/Appendix/Ramblings#Computer-Generated_Holography\|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 [[User:Tohline/Appendix/CGH/ParallelApertures\|§I.A]] on the one-dimensional aperture oriented parallel to the image screen.

Tohline: /* Analytic Results */

2020-02-10T17:02:55Z

Analytic Results

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	</table>		</table>

	It is worth noting that this derivation closely parallels the one presented in §8.5.1 (p. 393) of Born & Wolf (1980) (see the [[#See_Also\|reference below]]). Specifically, our double-integral expression is identical to the "Fraunhofer diffraction integral" written down by Born & Wolf at the beginning of their §8.5.1; and if we follow Born & Wolf's lead and position the origin of our coordinate system at the center of the rectangle, then <math>~\Theta_1 = \~~Phi_1~~ = 0</math> and the intensity <math>~I(P)</math> at point <math>~P</math> that is given by their equation (1) precisely matches the expression for the square of the amplitude, <math>~A^*A(x_1, y_1)</math>, that is obtained from our last expression.		It is worth noting that this derivation closely parallels the one presented in §8.5.1 (p. 393) of Born & Wolf (1980) (see the [[#See_Also\|reference below]]). Specifically, our double-integral expression is identical to the "Fraunhofer diffraction integral" written down by Born & Wolf at the beginning of their §8.5.1; and if we follow Born & Wolf's lead and position the origin of our coordinate system at the center of the rectangle, then <math>~\Theta_1 = \vartheta_1 = 0</math> and the intensity <math>~I(P)</math> at point <math>~P</math> that is given by their equation (1) precisely matches the expression for the square of the amplitude, <math>~A^*A(x_1, y_1)</math>, that is obtained from our last expression.

	=See Also=		=See Also=

Tohline at 02:37, 9 February 2020

2020-02-09T02:37:22Z

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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ A_0 a_0 (wh) e^{-i\Theta_1} ~ e^{-i\~~Phi_1~~} \mathrm{sinc}(\alpha_1) \mathrm{sinc}(\beta_1)		<math>~ A_0 a_0 (wh) e^{-i\Theta_1} ~ e^{-i\vartheta_1} \mathrm{sinc}(\alpha_1) \mathrm{sinc}(\beta_1)
	\, ,		\, ,
	</math>		</math>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\~~Phi_1~~</math>		<math>~\vartheta_1</math>
	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* See Also */

2020-02-09T00:21:16Z

Tohline: /* Analytic Results */

2020-02-09T00:18:32Z

Analytic Results

← Older revision		Revision as of 00:18, 9 February 2020
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	</table>		</table>

	It is worth noting that this derivation closely parallels the one presented in §8.5.1 (p. 393) of Born & Wolf. Specifically, our double-integral expression is identical to the "Fraunhofer diffraction integral" written down by Born & Wolf at the beginning of their §8.5.1; and if we follow Born & Wolf's lead and position the origin of our coordinate system at the center of the rectangle, then <math>~\Theta_1 = \Phi_1 = 0</math> and the intensity <math>~I(P)</math> at point <math>~P</math> that is given by their equation (1) precisely matches the expression for the square of the amplitude, <math>~A^*A(x_1, y_1)</math>, that is obtained from our last expression.		It is worth noting that this derivation closely parallels the one presented in §8.5.1 (p. 393) of Born & Wolf (1980) (see the [[#See_Also\|reference below]]). Specifically, our double-integral expression is identical to the "Fraunhofer diffraction integral" written down by Born & Wolf at the beginning of their §8.5.1; and if we follow Born & Wolf's lead and position the origin of our coordinate system at the center of the rectangle, then <math>~\Theta_1 = \Phi_1 = 0</math> and the intensity <math>~I(P)</math> at point <math>~P</math> that is given by their equation (1) precisely matches the expression for the square of the amplitude, <math>~A^*A(x_1, y_1)</math>, that is obtained from our last expression.

	=See Also=		=See Also=

Tohline: /* See Also */

2020-02-09T00:16:59Z

Tohline: /* Analytic Results */

2020-02-09T00:13:53Z

Analytic Results

← Older revision		Revision as of 00:13, 9 February 2020
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	</table>		</table>

			It is worth noting that this derivation closely parallels the one presented in §8.5.1 (p. 393) of Born & Wolf. Specifically, our double-integral expression is identical to the "Fraunhofer diffraction integral" written down by Born & Wolf at the beginning of their §8.5.1; and if we follow Born & Wolf's lead and position the origin of our coordinate system at the center of the rectangle, then <math>~\Theta_1 = \Phi_1 = 0</math> and the intensity <math>~I(P)</math> at point <math>~P</math> that is given by their equation (1) precisely matches the expression for the square of the amplitude, <math>~A^*A(x_1, y_1)</math>, that is obtained from our last expression.

	=See Also=		=See Also=

Tohline at 00:03, 9 February 2020

2020-02-09T00:03:23Z

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	=CGH: 2D Rectangular Appertures that are Parallel to the Image Screen=		=CGH: 2D Rectangular Appertures that are Parallel to the Image Screen=
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	<font color="red"><b>NOTE:</b></font> If x<sub>1</sub> and/or y<sub>1</sub> 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 A<sub>0</sub> cannot be moved outside of the double summation.		<font color="red"><b>NOTE:</b></font> If x<sub>1</sub> and/or y<sub>1</sub> 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 A<sub>0</sub> cannot be moved outside of the double summation.

			==Analytic Results==
			Now, as in [[User:Tohline/Appendix/CGH/ParallelApertures#Parallels_With_Example_.232\|§I.A]], we would like to consider the case where the rectangular aperture is divided into an infinite number of divisions in both the X and Y dimensions and convert the summations in this last expression into integrals whose limits in both directions are given by the edges of the aperture. If we specifically consider the case where the aperture is assumed to be uniformly bright (''i.e.,'' a<sub>jk</sub> = a<sub>0</sub> dX DY, and a<sub>0</sub> is the brightness per unit area), and the phase <math>~\phi_{jk} = 0</math> at all locations on the aperture, we can write,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~A(x_1, y_1)</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~ A_0 a_0 \iint \exp\biggl\{ -i \biggl[ \frac{2\pi(x_1 X_j + y_1Y_k}{\lambda L} \biggr] \biggr\} dX dY
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ A_0 a_0
			\int \exp\biggl\{ -i \biggl[ \frac{2\pi x_1 X_j }{\lambda L} \biggr] \biggr\} dX
			\cdot \int \exp\biggl\{ -i \biggl[ \frac{2\pi y_1Y_k}{\lambda L} \biggr] \biggr\} dY
			\, .
			</math>
			</td>
			</tr>
			</table>

			Both of these integrals can be completed in the same fashion as described in [[User:Tohline/Appendix/CGH/ParallelApertures#Parallels_With_Example_.232\|§I.A]] for the 1D slit, giving,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~A(x_1, y_1)</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~ A_0 a_0 (wh) e^{-i\Theta_1} ~ e^{-i\Phi_1} \mathrm{sinc}(\alpha_1) \mathrm{sinc}(\beta_1)
			\, ,
			</math>
			</td>
			</tr>
			</table>
			where,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\alpha_1</math>
			</td>
			<td align="center">
			<math>~\equiv </math>
			</td>
			<td align="left">
			<math>~\frac{\pi x_1 (X_1 - X_2)}{\lambda L}= \frac{\pi x_1 h}{\lambda L} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Theta_1</math>
			</td>
			<td align="center">
			<math>~\equiv </math>
			</td>
			<td align="left">
			<math>~\frac{\pi x_1 (X_1 + X_2)}{\lambda L} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\beta_1</math>
			</td>
			<td align="center">
			<math>~\equiv </math>
			</td>
			<td align="left">
			<math>~\frac{\pi y_1 (Y_1 - Y_2)}{\lambda L}= \frac{\pi y_1 w}{\lambda L} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Phi_1</math>
			</td>
			<td align="center">
			<math>~\equiv </math>
			</td>
			<td align="left">
			<math>~\frac{\pi y_1 (Y_1 + Y_2)}{\lambda L} \, .</math>
			</td>
			</tr>
			</table>


	=See Also=		=See Also=

Tohline: /* Utility of FFT Techniques */

2020-02-07T22:36:14Z

Utility of FFT Techniques

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	<td align="left">		<td align="left">
	<math>~		<math>~
	[Z^2 + y_1^2 + ~~x_1~~^2]^{~~1 /~~ 2} \, .		L \biggl[1 - \frac{2(x_1 X_j + y_1 Y_k ) }{L^2} \biggr]^{1 / 2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~
			L \biggl[1 - \frac{(x_1 X_j + y_1 Y_k ) }{L^2} \biggr] \, .
			</math>
			</td>
			</tr>
			</table>

			Hence, the double-summation expression for the amplitude at screen location (x<sub>1</sub>, y<sub>1</sub>) becomes,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~A(x_1, y_1)</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<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>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			where,
			<div align="center">
			<math>~A_0 \equiv e^{i(2\pi L/\lambda)}</math>.
			</div>
			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(x<sub>1</sub>, y<sub>1</sub>).

			<font color="red"><b>NOTE:</b></font> If x<sub>1</sub> and/or y<sub>1</sub> 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 A<sub>0</sub> cannot be moved outside of the double summation.

	=See Also=		=See Also=

Tohline: /* Utility of FFT Techniques */

2020-02-07T22:06:45Z

Utility of FFT Techniques

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	<td align="left">		<td align="left">
	<math>~		<math>~
	L \biggl[1 - \frac{~~2y_1 Y_k}{L^~~2} + ~~\frac{~~Y_k^2}{L^2} - \frac{~~2x_1~~ X_j~~}{L~~^2} + ~~\frac{X_j~~^2}{L^2} \biggr]		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>		</math>
	</td>		</td>
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	<td align="center">		<td align="center">
	<math>~\equiv</math>		<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			[Z^2 + y_1^2 + x_1^2]^{1 / 2} \, .
			</math>
			</td>
			</tr>
			</table>

			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<sub>jk</sub> and deduce that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~D_{jk}</math>
			</td>
			<td align="center">
			<math>~\approx</math>
	</td>		</td>
	<td align="left">		<td align="left">

← Older revision		Revision as of 00:21, 9 February 2020
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	=See Also=		=See Also=
	* Updated [[User:Tohline/Appendix/Ramblings#Computer-Generated_Holography\|Table of Contents]]		* Updated [[User:Tohline/Appendix/Ramblings#Computer-Generated_Holography\|Table of Contents]]
	* Born, M. and Wolf, E. (1980) ''Principles of Optics,'' ~~3rd~~ Edition. New York: Pergamon Press. [See especially their §§8.5 and 8.10.]		* Born, M. and Wolf, E. (1980) ''Principles of Optics,'' 3<sup>rd</sup> Edition. New York: Pergamon Press. [See especially their §§8.5 and 8.10.] A link to the 6<sup>th</sup> edition of this book can be found [https://www.sciencedirect.com/book/9780080264820/principles-of-optics here].


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← Older revision		Revision as of 00:16, 9 February 2020
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	=See Also=		=See Also=
	* Updated [[User:Tohline/Appendix/Ramblings#Computer-Generated_Holography\|Table of Contents]]		* Updated [[User:Tohline/Appendix/Ramblings#Computer-Generated_Holography\|Table of Contents]]
			* Born, M. and Wolf, E. (1980) ''Principles of Optics,'' 3rd Edition. New York: Pergamon Press. [See especially their §§8.5 and 8.10.]


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