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

Tohline: /* Theory */

2020-03-15T22:22:01Z

Theory

← Older revision		Revision as of 22:22, 15 March 2020
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	</div>		</div>

			<table border="1" cellpadding="8" align="center" width="85%"><tr><td align="left">
			<div align="center"><font color="red">First (Misguided) Attempt</font></div>
	Next, let's set <math>~\phi_j = - 2\pi Z_0/\lambda</math>, which is independent of <math>~j</math> and therefore can be shifted outside of the summation. We have, then,		Next, let's set <math>~\phi_j = - 2\pi Z_0/\lambda</math>, which is independent of <math>~j</math> and therefore can be shifted outside of the summation. We have, then,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~A(y_1)</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~
			\mathrm{exp}\biggl\{i\biggl[
			\frac{2\pi (Z - Z_0)}{\lambda}+ \frac{\pi y_1 w}{Z \lambda}
			\biggr] \biggr\}
			\sum_j a_j
			\biggl\{ \cos\biggl( \frac{\pi j y_1 }{\mathfrak{L}} \biggr) - i \sin\biggl( \frac{\pi j y_1 }{\mathfrak{L}} \biggr) \biggr\} \, .
			</math>
			</td>
			</tr>

			</table>

			</td></tr></table>

			Let's set,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\phi_j</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{2\pi Z_0}{\lambda} - \frac{\pi y_1 w}{Z_0 \lambda} \, ,</math>
			</td>
			</tr>
			</table>

			which is independent of <math>~j</math> and therefore can be shifted outside of the summation. We have, then,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* More Attention to Detail */

2020-03-08T17:34:38Z

More Attention to Detail

← Older revision		Revision as of 17:34, 8 March 2020
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	===More Attention to Detail===		===More Attention to Detail===

			====Theory====

	Guided by our [[User:Tohline/Appendix/Ramblings/FourierSeries#One-Dimensional_Aperture\|accompanying discussion of the relationship between a one-dimensional aperture and the Fourier Series]], let's begin again with the [[#Utility_of_FFT_Techniques\|above general summation expression]],		Guided by our [[User:Tohline/Appendix/Ramblings/FourierSeries#One-Dimensional_Aperture\|accompanying discussion of the relationship between a one-dimensional aperture and the Fourier Series]], let's begin again with the [[#Utility_of_FFT_Techniques\|above general summation expression]],
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	Hopefully, the amplitude of this image will die off quickly as we insert values of <math>~Z \ne Z_0 \, .</math>		Hopefully, the amplitude of this image will die off quickly as we insert values of <math>~Z \ne Z_0 \, .</math>

			====Example #1====

			Let's try an example using the parameters listed at the top of Figure 4.

			<div align="center" id="Figure4">
			<table border="1" cellpadding="8" align="center">
			<tr>
			<th align="center">Figure 4:   <math>~w = 1~\mathrm{mm}; Z_0 = 10~\mathrm{m}; \lambda = 500~\mathrm{nm}; j_\mathrm{max} = 51</math></th>
			</tr>
			<tr>
			<td align="center">
			[[File:SingleSlit02B.png\|700px\|Second Plot of Single-Slit Diffraction Pattern]]
			</td>
			</tr>
			</table>
			</div>

	==Multiple One-Dimensional Apertures==		==Multiple One-Dimensional Apertures==

Tohline: /* More Attention to Detail */

2020-03-07T18:48:31Z

More Attention to Detail

← Older revision		Revision as of 18:48, 7 March 2020
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~[ Z^2 + y_1^2 ]^{1 / 2} \approx Z \biggl[1 + \frac{~~1}{~~2}~~\biggl( \frac~~{~~y_1}{Z}\biggr)~~^2 \biggr] \, ,</math>		<math>~[ Z^2 + y_1^2 ]^{1 / 2} \approx Z \biggl(1 + \frac{y_1^2}{2Z^2} \biggr) \, ,</math>
	</td>		</td>
	</tr>		</tr>
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	Now, let's adopt the following conventions:		Now, let's adopt the following conventions:
	<ol>		<ol>
	<li>Specify the values of the parameters, <math>~\lambda, w, c/\mathfrak{L},</math> and <math>~Z</math>, where,		<li>Specify the values of the parameters, <math>~\lambda, w, c/\mathfrak{L}_0,</math> and <math>~Z_0</math>, where,

	<table border="0" align="center">		<table border="0" align="center">
	<tr><td align="center"><math>~\mathfrak{L} \equiv ~~\frac{~~\lambda Z j_\mathrm{max}}{2w} \, .</math></td></tr>		<tr><td align="center"><math>~\mathfrak{L} \equiv \lambda Z j_\mathrm{max}/(2w) </math>     … hence …     <math>~\mathfrak{L}_0 = \lambda Z_0 j_\mathrm{max}/(2w) \, .</math></td></tr>
	</table>		</table>

	</li>		</li>
	<li>In specifying the properties of the light as it leaves the aperture, ensure that ''neither'' <math>~a_j</math> nor <math>~\phi_j</math> depends on the distance between the aperture and the image plane, <math>~Z</math>.</li>		<li>In specifying the properties of the light as it leaves the aperture, ensure that ''neither'' <math>~a_j</math> nor <math>~\phi_j</math> depends on the distance between the aperture and the image plane, <math>~Z</math>.</li>
	<li>It is best to evaluate the amplitude on the image screen over the range, <math>~-\mathfrak{L} \le y_i \le \mathfrak{L} \, .</math></li>		<li>It is best to evaluate the amplitude on the image screen over the range, <math>~-\mathfrak{L}_0 \le y_i \le \mathfrak{L}_0 \, .</math></li>
	<li>Expect the amplitude to be <math>~\sim 2c/\mathfrak{L} \, .</math></li>		<li>Expect the amplitude to be <math>~\sim 2c/\mathfrak{L}_0 \, .</math></li>
	</ol>		</ol>

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	<math>~		<math>~
	\mathrm{exp}\biggl\{i\biggl[		\mathrm{exp}\biggl\{i\biggl[
	\frac{2\pi ~~L_1~~}{\lambda} - \frac{2\pi y_1 ~~Y_0~~}{~~L_1~~ \lambda}		\frac{2\pi Z}{\lambda}\biggl(1 + \cancelto{0}{\frac{y_1^2}{2Z^2}} \biggr) + \frac{\pi y_1 w}{Z \lambda} \biggl( \cancelto{1}{\frac{Z}{L_1} } \biggr)
	\biggr] \biggr\}		\biggr] \biggr\}
	\sum_j a_j e^{i\phi_j}		\sum_j a_j e^{i\phi_j}
	\biggl[ \cos\biggl(\frac{2\pi y_1 j \~~Delta Y~~}{\~~lambda~~ L_1} \biggr) - i \sin\biggl(\frac{2\pi y_1 j \~~Delta Y~~}{\~~lambda L_1~~} \biggr) \biggr]		\biggl\{ \cos\biggl[ \frac{(\pi j y_1 ) 2w}{\lambda Z j_\mathrm{max}} \biggl( \cancelto{1}{\frac{Z}{L_1} } \biggr) \biggr] - i \sin\biggl[ \frac{(\pi j y_1 ) 2w}{\lambda Z j_\mathrm{max}} \biggl( \cancelto{1}{\frac{Z}{L_1} } \biggr) \biggr] \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~
			\mathrm{exp}\biggl\{i\biggl[
			\frac{2\pi Z}{\lambda}+ \frac{\pi y_1 w}{Z \lambda}
			\biggr] \biggr\}
			\sum_j a_j e^{i\phi_j}
			\biggl\{ \cos\biggl( \frac{\pi j y_1 }{\mathfrak{L}} \biggr) - i \sin\biggl( \frac{\pi j y_1 }{\mathfrak{L}} \biggr) \biggr\} \, .
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>

			Next, let's set <math>~\phi_j = - 2\pi Z_0/\lambda</math>, which is independent of <math>~j</math> and therefore can be shifted outside of the summation. We have, then,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~A(y_1)</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~
			\mathrm{exp}\biggl\{i\biggl[
			\frac{2\pi (Z - Z_0)}{\lambda}+ \frac{\pi y_1 w}{Z \lambda}
			\biggr] \biggr\}
			\sum_j a_j
			\biggl\{ \cos\biggl( \frac{\pi j y_1 }{\mathfrak{L}} \biggr) - i \sin\biggl( \frac{\pi j y_1 }{\mathfrak{L}} \biggr) \biggr\} \, .
			</math>
			</td>
			</tr>

			</table>

			Using this general expression, we should specifically expect the amplitude across the image plane at Z<sub>0</sub> to be a step function if we insert the expression,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~a_j</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{2}{j \pi} \sin\biggl( \frac{j\pi c}{\mathfrak{L}_0} \biggr) = \biggl(\frac{2c}{\mathfrak{L}_0} \biggr) \mathrm{sinc}\biggl( \frac{j\pi c}{\mathfrak{L}_0} \biggr) \, .</math>
			</td>
			</tr>
			</table>

			Hopefully, the amplitude of this image will die off quickly as we insert values of <math>~Z \ne Z_0 \, .</math>

	==Multiple One-Dimensional Apertures==		==Multiple One-Dimensional Apertures==

Tohline: /* One-Dimensional Aperture */

2020-03-07T17:36:46Z

One-Dimensional Aperture

Show changes

Tohline at 17:43, 10 February 2020

2020-02-10T17:43:17Z

← Older revision		Revision as of 17:43, 10 February 2020
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	<!-- __FORCETOC__ will force the creation of a Table of Contents -->		<!-- __FORCETOC__ will force the creation of a Table of Contents -->
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=CGH: ~~Appertures~~ that are Parallel to the Image Screen=		=CGH: Apertures that are Parallel to the Image Screen=
	This chapter is intended primarily to replicate [http://www.phys.lsu.edu/faculty/tohline/phys4412/howto/slit1d.html §I.A from the online class notes] — see also an associated [[User:Tohline/Appendix/CGH/Preface\|Preface]] and the original [http://www.phys.lsu.edu/faculty/tohline/phys4412/howto/ 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 chapter is intended primarily to replicate [http://www.phys.lsu.edu/faculty/tohline/phys4412/howto/slit1d.html §I.A from the online class notes] — see also an associated [[User:Tohline/Appendix/CGH/Preface\|Preface]] and the original [http://www.phys.lsu.edu/faculty/tohline/phys4412/howto/ 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.

Tohline: /* Analytic Result */

2020-02-09T22:27:19Z

Analytic Result

← Older revision		Revision as of 22:27, 9 February 2020
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~A_0 \int		<math>~A_0~a_0 \int
	~~a_0~~ e^{-i[2\pi y_1 Y/(\lambda L)]} dY		e^{-i[2\pi y_1 Y/(\lambda L)]} dY
	\, ,		\, ,
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\biggl[ A_0 ~a_0 ~w ~e^{-i\vartheta_1} \biggr]\frac{\sin\beta_1}{\beta_1} \, .		\biggl[ A_0 ~a_0 ~w ~e^{-i\vartheta_1} \biggr]\frac{\sin\beta_1}{\beta_1}
			= \biggl[ A_0 ~a_0 ~w ~e^{-i\vartheta_1} \biggr] \mathrm{sinc}(\beta_1) \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			The last two lines have been derived by realizing that, for any angle <math>~\chi</math>,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~e^{-i\chi} - e^{+i\chi}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-2i\sin(\chi) \, ;
			</math>
			</td>
			</tr>
			</table>
			and the "sinc" function,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\mathrm{sinc}(\chi)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\sin\chi}{\chi} \, .
			</math>
			</td>
			</tr>
			</table>
			Notice that the center of the aperture is used to define the origin of the Y (and y) coordinate axis so that Y<sub>2</sub> = -Y<sub>1</sub>, as illustrate in Figure I.1; then <math>~\vartheta_1 = 0</math> and the amplitude, A(y<sub>1</sub>), has no complex component. If the aperture is not centered as depicted in Figure I.1, the angle <math>~\vartheta_1</math> simple serves to introduce a phase shift in the evaluation of the amplitude.

			Although this problem was solved for a specific location, y<sub>1</sub>, on the image screen, we see that the sinc function solution can readily be evaluated for any other location on the screen without having to redo the integral.

	===Location of the Dark Fringe(s)===		===Location of the Dark Fringe(s)===

Tohline: /* Analytic Result */

2020-02-09T21:49:13Z

Analytic Result

← Older revision		Revision as of 21:49, 9 February 2020
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	</tr>		</tr>
	</table>		</table>
	This integral can be readily evaluated, giving,		This definite integral can be readily evaluated, giving,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(-i)^{-1} A_0 \biggl[ \frac{a_0 w}{2\beta_1} \biggr] \biggl[e^{-i\Theta_1} - e^{-\Theta_2} \biggr]		<math>~
	~~e^{-i \Theta} d\Theta~~		(-i)^{-1} A_0 \biggl[ \frac{a_0 w}{2\beta_1} \biggr] \biggl[e^{-i\Theta_1} - e^{-i \Theta_2} \biggr]
	~~\, ,~~
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	~~</table>~~


			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			i A_0 \biggl[ \frac{a_0 w}{2\beta_1} \biggr] e^{-i\vartheta_1} \biggl[e^{-i\beta_1} - e^{+i \beta_1} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ A_0 ~a_0 ~w ~e^{-i\vartheta_1} \biggr]\frac{\sin\beta_1}{\beta_1} \, .
			</math>
			</td>
			</tr>
			</table>

	===Location of the Dark Fringe(s)===		===Location of the Dark Fringe(s)===

Tohline: /* One-Dimensional Aperture */

2020-02-09T21:41:54Z

One-Dimensional Aperture

← Older revision		Revision as of 21:41, 9 February 2020
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>~\approx</math>
	</td>		</td>
	<td align="left">		<td align="left">
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	===Analytic Result===		===Analytic Result===

	If we consider the limit where the aperture shown in Figure I.1 is divided into an infinite number of divisions, then we can convert the summation in the above equations into an integral running between the limits, Y<sub>2</sub> and Y<sub>1</sub>. Specifically, ~~equation~~ (9) ~~becomes~~,		If we consider the limit where the aperture shown in Figure I.1 is divided into an infinite number of divisions, then we can convert the summation in the above equations into an integral running between the limits, Y<sub>2</sub> and Y<sub>1</sub>. Specifically, linearized expression immediately above becomes,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~A(y_1)</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~A_0 \int
			a_0 e^{-i[2\pi y_1 Y/(\lambda L)]} dY
			\, ,
			</math>
			</td>
			</tr>
			</table>

			where, again, we have returned to the case where the aperture is assumed to be uniformly bright and set a(Y) = a<sub>0</sub>dY. (It should be understood that a<sub>0</sub> is the brightness of the aperture per unit length.) If we now make the substitution,
			<div align="center">
			<math>~\Theta \equiv \frac{2\pi y_1 Y}{\lambda L} \, ,</math>
			</div>

			this integral expression takes the form,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~A(y_1)</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~A_0 \biggl[ \frac{a_0 w}{2\beta_1} \biggr] \int_{\Theta_2}^{\Theta_1}
			e^{-i \Theta} d\Theta
			\, ,
			</math>
			</td>
			</tr>
			</table>
			where the limits of integration are, respectively,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Theta_2</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			\frac{2\pi y_1 Y_2}{\lambda L} = \vartheta_1 - \beta_1
			\, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Theta_1</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			\frac{2\pi y_1 Y_1}{\lambda L} = \vartheta_1 + \beta_1
			\, ,
			</math>
			</td>
			</tr>
			</table>
			and,

			<table border="0" cellpadding="5" align="center">

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

			<tr>
			<td align="right">
			<math>~\vartheta_1</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			\frac{\pi y_1}{\lambda L} \biggl[ Y_1 + Y_2 \biggr]
			\, .
			</math>
			</td>
			</tr>
			</table>
			This integral can be readily evaluated, giving,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~A(y_1)</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~(-i)^{-1} A_0 \biggl[ \frac{a_0 w}{2\beta_1} \biggr] \biggl[e^{-i\Theta_1} - e^{-\Theta_2} \biggr]
			e^{-i \Theta} d\Theta
			\, ,
			</math>
			</td>
			</tr>
			</table>




	===Location of the Dark Fringe(s)===		===Location of the Dark Fringe(s)===

Tohline: /* Analytic Result */

2020-02-09T21:14:16Z

Analytic Result

← Older revision		Revision as of 21:14, 9 February 2020
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	===Analytic Result===		===Analytic Result===

			If we consider the limit where the aperture shown in Figure I.1 is divided into an infinite number of divisions, then we can convert the summation in the above equations into an integral running between the limits, Y<sub>2</sub> and Y<sub>1</sub>. Specifically, equation (9) becomes,

	===Location of the Dark Fringe(s)===		===Location of the Dark Fringe(s)===

Tohline: /* =Analytic Result */

2020-02-09T21:12:08Z

=Analytic Result

← Older revision		Revision as of 21:12, 9 February 2020
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	Notice that this expression matches what we have [[User:Tohline/Appendix/Ramblings/FourierSeries#Complex\|referred to elsewhere as the ''Complex'' Fourier Series Expression]]. When written in this form, it should immediately be apparent why discrete Fourier transform techniques (specifically FFT techniques) are useful tools for evaluation of the complex amplitude, <math>~A</math>. See [[#Parallels_With_Standard_Fourier_Series\|further elaboration below]].		Notice that this expression matches what we have [[User:Tohline/Appendix/Ramblings/FourierSeries#Complex\|referred to elsewhere as the ''Complex'' Fourier Series Expression]]. When written in this form, it should immediately be apparent why discrete Fourier transform techniques (specifically FFT techniques) are useful tools for evaluation of the complex amplitude, <math>~A</math>. See [[#Parallels_With_Standard_Fourier_Series\|further elaboration below]].

	===Analytic Result==		===Analytic Result===