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

Tohline at 03:29, 25 March 2020

2020-03-25T03:29:26Z

← Older revision		Revision as of 03:29, 25 March 2020
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	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=CGH: Consolidate Expressions Regarding Parallel Apertures=		=CGH: Consolidate Expressions Regarding Parallel Apertures=


			{{LSU_HBook_header}}


	==One-dimensional Apertures==		==One-dimensional Apertures==

Tohline: /* Case 2 */

2020-03-22T15:52:27Z

Case 2

← Older revision		Revision as of 15:52, 22 March 2020
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	As a result, we have,		As a result, we have,

			<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>~ \biggl[ \cos\biggl( \frac{\pi y_1^2}{\lambda Z} \biggr) + i \sin \biggl( \frac{\pi y_1^2}{\lambda Z} \biggr) \biggr] \biggl(\frac{\mathfrak{L}}{c} \biggr) \biggl\{ \biggl(\frac{c}{\mathfrak{L}} \biggr)
			~+~ \sum_{n~=~1}^{+(j_\mathrm{max} - 1)/2}
			\biggl(\frac{ 2 }{\pi n } \biggr) \sin \biggl(\frac{\pi n c}{\mathfrak{L} } \biggr)
			~ \cos \biggl( \frac{n \pi y_1 }{\mathfrak{L} } \biggr) \biggr\}
			\, .
			</math>
			</td>
			</tr>
			</table>
			Therefore, a clean square wave will appear only if <math>~[\pi y_1^2/(\lambda Z)] \ll 1</math>.

	=See Also=		=See Also=

Tohline: /* Case 2 */

2020-03-22T15:25:26Z

Case 2

← Older revision		Revision as of 15:25, 22 March 2020
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	</table>		</table>

	Finally, set <math>~e^{i\phi_j} = e^{-i2\pi Z/\lambda}</math>.		Finally, recalling that,
			<table border="0" cellpadding="5" align="center">

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

			let's set …
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~e^{i\phi_j}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			e^{-i2\pi Z/\lambda}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ e^{i2\pi L/\lambda} \cdot e^{i\phi_j}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			e^{i2\pi (L-Z)/\lambda} \approx e^{i\pi y_1^2/(\lambda Z)} = \cos\biggl( \frac{\pi y_1^2}{\lambda Z} \biggr) + i \sin \biggl( \frac{\pi y_1^2}{\lambda Z} \biggr) \, .
			</math>
			</td>
			</tr>
			</table>

			As a result, we have,

	=See Also=		=See Also=

Tohline: /* Case 2 */

2020-03-22T03:27:56Z

Case 2

← Older revision		Revision as of 03:27, 22 March 2020
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	</tr>		</tr>
	</table>		</table>

			Finally, set <math>~e^{i\phi_j} = e^{-i2\pi Z/\lambda}</math>.

	=See Also=		=See Also=

Tohline: /* Case 2 */

2020-03-22T03:06:12Z

Case 2

← Older revision		Revision as of 03:06, 22 March 2020
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	<math>~ e^{i 2\pi L/\lambda } e^{i \phi_{j=0} }		<math>~ e^{i 2\pi L/\lambda } e^{i \phi_{j=0} }
	~+~e^{i 2\pi L/\lambda } \sum_{n~=~1}^{+(j_\mathrm{max} - 1)/2} 2e^{i \phi_j} \cdot~		~+~e^{i 2\pi L/\lambda } \sum_{n~=~1}^{+(j_\mathrm{max} - 1)/2} 2e^{i \phi_j} \cdot~
	\mathrm{sinc} ~~\biggl[~~ \biggl( \frac{2n}{ j_\~~mathrm~~{~~max~~}-1} \biggr) \frac{\pi ~~w c~~}{\~~lambda~~ L} \biggr]		\mathrm{sinc} \biggl(\frac{\pi n c}{\mathfrak{L} } \biggr)
	~ \~~cos~~ \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi y_1 w }{\~~lambda~~ L} \biggr]		~ \cos \biggl( \frac{n \pi y_1 }{\mathfrak{L} } \biggr)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ e^{i 2\pi L/\lambda } \biggl(\frac{\mathfrak{L}}{c} \biggr) \biggl\{ e^{i \phi_{j=0} } \biggl(\frac{c}{\mathfrak{L}} \biggr)
			~+~ \sum_{n~=~1}^{+(j_\mathrm{max} - 1)/2} e^{i \phi_j} \cdot~
			\biggl(\frac{ 2 }{\pi n } \biggr) \sin \biggl(\frac{\pi n c}{\mathfrak{L} } \biggr)
			~ \cos \biggl( \frac{n \pi y_1 }{\mathfrak{L} } \biggr) \biggr\}
	\, .		\, .
	</math>		</math>

Tohline: /* Case 2 */

2020-03-22T02:52:35Z

Case 2

← Older revision		Revision as of 02:52, 22 March 2020
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	\mathrm{sinc} \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi w c}{\lambda L} \biggr]		\mathrm{sinc} \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi w c}{\lambda L} \biggr]
	\biggl\{ \cos \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi y_1 w }{\lambda L} \biggr] - i \sin \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi y_1 w }{\lambda L} \biggr]		\biggl\{ \cos \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi y_1 w }{\lambda L} \biggr] - i \sin \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi y_1 w }{\lambda L} \biggr]
	\biggr\} \, .		\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ e^{i 2\pi L/\lambda } e^{i \phi_{j=0} }
			~+~e^{i 2\pi L/\lambda } \sum_{n~=~1}^{+(j_\mathrm{max} - 1)/2} 2e^{i \phi_j} \cdot~
			\mathrm{sinc} \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi w c}{\lambda L} \biggr]
			~ \cos \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi y_1 w }{\lambda L} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ e^{i 2\pi L/\lambda } e^{i \phi_{j=0} }
			~+~e^{i 2\pi L/\lambda } \sum_{n~=~1}^{+(j_\mathrm{max} - 1)/2} 2e^{i \phi_j} \cdot~
			\mathrm{sinc} \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi w c}{\lambda L} \biggr]
			~ \cos \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi y_1 w }{\lambda L} \biggr]
			\, .
	</math>		</math>
	</td>		</td>

Tohline: /* Case 2 */

2020-03-22T02:38:57Z

Case 2

← Older revision		Revision as of 02:38, 22 March 2020
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ e^{i 2\pi L/\lambda } \~~sum_j~~ e^{i \phi_j} \cdot~		<math>~ e^{i 2\pi L/\lambda } \sum_{j=0}^{j_\mathrm{max}-1} e^{i \phi_j} \cdot~
	\mathrm{sinc} \biggl[ \biggl( \frac{2j}{j_\mathrm{max}-1} - 1 \biggr) \frac{\pi w c}{\lambda L} \biggr]		\mathrm{sinc} \biggl[ \biggl( \frac{2j}{j_\mathrm{max}-1} - 1 \biggr) \frac{\pi w c}{\lambda L} \biggr]
	\biggl\{ \cos \biggl[ \biggl( \frac{2j}{j_\mathrm{max} - 1} - 1 \biggr) \frac{\pi y_1 w }{\lambda L} \biggr] - i \sin \biggl[ \biggl( \frac{2j}{j_\mathrm{max} - 1} - 1 \biggr) \frac{\pi y_1 w }{\lambda L} \biggr]		\biggl\{ \cos \biggl[ \biggl( \frac{2j}{j_\mathrm{max} - 1} - 1 \biggr) \frac{\pi y_1 w }{\lambda L} \biggr] - i \sin \biggl[ \biggl( \frac{2j}{j_\mathrm{max} - 1} - 1 \biggr) \frac{\pi y_1 w }{\lambda L} \biggr]
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	This exhibits a very desirable feature: Both the sinc function and the sine function — and, hence, also their product — have reflection symmetry about the summation index, <math>~j = (j_\mathrm{max}-1)/2</math>. As a result, if the overall phase factor, <math>~e^{i \phi_j}</math>, behaves in an appropriately simple way — for example, if it is zero everywhere — then under the summation the sine term will sum to zero and leave only the desired — and ''real'' — product, <math>~\mathrm{sinc} \times \cos</math>. Try this out in Excel to see if it works!		This exhibits a very desirable feature: Both the sinc function and the sine function — and, hence, also their product — have reflection symmetry about the summation index, <math>~j = (j_\mathrm{max}-1)/2</math>. As a result, if the overall phase factor, <math>~e^{i \phi_j}</math>, behaves in an appropriately simple way — for example, if it is zero everywhere — then under the summation the sine term will sum to zero and leave only the desired — and ''real'' — product, <math>~\mathrm{sinc} \times \cos</math>. Try this out in Excel to see if it works!

			This could use a little more manipulation. Let's define the alternate summation index,

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

			<tr>
			<td align="right">
			<math>~n</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>\frac{1}{2} \biggl[ j_\mathrm{max}-1 \biggr] \biggl( \frac{2j}{j_\mathrm{max}-1} - 1 \biggr) \, ,</math>
			</td>
			</tr>
			</table>
			in which case we can write,

			<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>~ e^{i 2\pi L/\lambda } \sum_{n~=~-(j_\mathrm{max} - 1)/2}^{+(j_\mathrm{max} - 1)/2} e^{i \phi_j} \cdot~
			\mathrm{sinc} \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi w c}{\lambda L} \biggr]
			\biggl\{ \cos \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi y_1 w }{\lambda L} \biggr] - i \sin \biggl[ \biggl( \frac{2n}{ j_\mathrm{max}-1} \biggr) \frac{\pi y_1 w }{\lambda L} \biggr]
			\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>

	=See Also=		=See Also=

Tohline: /* Case 2 */

2020-03-20T21:36:11Z

Case 2

← Older revision		Revision as of 21:36, 20 March 2020
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	</table>		</table>

			This therefore gives the following,

			<table border="0" cellpadding="5" align="center">
			<tr>
			<th align="center" colspan="3">Focal-Point Expression for a Square Wave</th>
			</tr>

			<tr>
			<td align="right">
			<math>~A(y_1)</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~ e^{i 2\pi L/\lambda } \sum_j e^{i \phi_j} \cdot~
			\mathrm{sinc} \biggl[ \biggl( \frac{2j}{j_\mathrm{max}-1} - 1 \biggr) \frac{\pi w c}{\lambda L} \biggr]
			\biggl\{ \cos \biggl[ \biggl( \frac{2j}{j_\mathrm{max} - 1} - 1 \biggr) \frac{\pi y_1 w }{\lambda L} \biggr] - i \sin \biggl[ \biggl( \frac{2j}{j_\mathrm{max} - 1} - 1 \biggr) \frac{\pi y_1 w }{\lambda L} \biggr]
			\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>

			This exhibits a very desirable feature: Both the sinc function and the sine function — and, hence, also their product — have reflection symmetry about the summation index, <math>~j = (j_\mathrm{max}-1)/2</math>. As a result, if the overall phase factor, <math>~e^{i \phi_j}</math>, behaves in an appropriately simple way — for example, if it is zero everywhere — then under the summation the sine term will sum to zero and leave only the desired — and ''real'' — product, <math>~\mathrm{sinc} \times \cos</math>. Try this out in Excel to see if it works!

	=See Also=		=See Also=

Tohline at 21:14, 20 March 2020

2020-03-20T21:14:25Z

← Older revision		Revision as of 21:14, 20 March 2020
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	<math>~		<math>~
	j \biggl[ \frac{2\pi y_1 w}{(j_\mathrm{max}-1) \lambda L} \biggr] - \frac{\pi y_1 w }{\lambda L}		j \biggl[ \frac{2\pi y_1 w}{(j_\mathrm{max}-1) \lambda L} \biggr] - \frac{\pi y_1 w }{\lambda L}
	= j \cdot \Delta\Theta - \frac{(j_\mathrm{max} -1)}{2} \Delta\Theta		= \biggl( \frac{2j}{j_\mathrm{max} - 1} - 1 \biggr) \frac{\pi y_1 w }{\lambda L}
			\, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			j \cdot \Delta\Theta - \frac{(j_\mathrm{max} -1)}{2} \Delta\Theta
	\, ,</math>		\, ,</math>
	</td>		</td>
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	<math>~\Theta_0 \equiv \frac{(j_\mathrm{max} - 1)}{2} \cdot \pi y_1 \biggl[ \frac{2w}{(j_\mathrm{max}-1) \lambda L} \biggr] = \frac{\pi y_1 w}{\lambda L} \, .</math>		<math>~\Theta_0 \equiv \frac{(j_\mathrm{max} - 1)}{2} \cdot \pi y_1 \biggl[ \frac{2w}{(j_\mathrm{max}-1) \lambda L} \biggr] = \frac{\pi y_1 w}{\lambda L} \, .</math>
	</div>		</div>

			Now, what is the argument of the sinc function? By default, it needs to be something along the lines of,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{j \pi c}{\mathfrak{L}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~j \pi c \biggl[ \frac{2w}{(j_\mathrm{max}-1) \lambda L} \biggr] \, .</math>
			</td>
			</tr>
			</table>
			Then, as <math>~j</math> varies from <math>~0</math> to <math>~(j_\mathrm{max} - 1)</math>, the argument goes from <math>~0</math> to <math>~[2\pi w c/(\lambda L)]</math>. In an effort to make the function exhibit reflection symmetry as we move from one side of the aperture to the next, let's subtract half of this upper limit; that is, let's modify the argument of the sinc function to read,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{j \pi c}{\mathfrak{L}} - \frac{\pi w c}{\lambda L}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			j \pi c \biggl[ \frac{2w}{(j_\mathrm{max}-1) \lambda L} \biggr] - \frac{\pi w c}{\lambda L}
			= \biggl[ \frac{2j}{j_\mathrm{max}-1} - 1\biggr]\biggl[ \frac{\pi w c}{\lambda L} \biggr] \, .
			</math>
			</td>
			</tr>
			</table>
			This means that in our [[#FocalPoint\|above, Focal-Point Expression]] we want to set,
			<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>~
			\mathrm{sinc} \biggl[ \biggl( \frac{2j}{j_\mathrm{max}-1} - 1 \biggr) \frac{\pi w c}{\lambda L} \biggr] \, .
			</math>
			</td>
			</tr>
			</table>



	=See Also=		=See Also=

Tohline: /* One-dimensional Apertures */

2020-03-20T03:35:38Z

One-dimensional Apertures

← Older revision		Revision as of 03:35, 20 March 2020
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	This means that <math>~\Theta_{i} = - \Theta_{( j_\mathrm{max} - 1 - i )}</math>.		This means that <math>~\Theta_{i} = - \Theta_{( j_\mathrm{max} - 1 - i )}</math>.

	~~To answer this, let's recognize that each term~~ under the summation ~~in our Focal-Point expression may be rewritten in a variety of ways. For example~~,		The key expression under the summation therefore becomes,

	<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>~a_j~~\biggl[\cos\phi_j +~~ i~~\sin~~\phi_j \~~biggr]~~\biggl[ \cos ~~\Theta - i \sin\Theta \biggr] </math>~~		<math>~~a_j e^{i \phi_j} \cdot \biggl[ \cos \biggl( \frac{j\pi y_1}{\mathfrak{L}}- \Theta_0 \biggr) - i \sin \biggl( \frac{j\pi y_1}{\mathfrak{L}}- \Theta_0 \biggr) \biggr] \, ,</math>
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~a_j~~ \biggl\{ \~~biggl[~~\~~cos\Theta \cos\phi_j + \sin\Theta \sin~~\~~phi_j~~ \biggr]+ i~~~\biggl[\cos\Theta~~ \sin\~~phi_j -~~ \~~sin\Theta \cos\phi_j \biggr] \biggr~~\}~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~a_j \biggl~~\{ ~~\biggl[\cos( \Theta~~ - \~~phi_j )~~ \biggr~~] - i~\biggl[ \sin(\Theta - \phi_j~~) \biggr] \~~biggr\}~~</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			where,
			<div align="center">
			<math>~\Theta_0 \equiv \frac{(j_\mathrm{max} - 1)}{2} \cdot \pi y_1 \biggl[ \frac{2w}{(j_\mathrm{max}-1) \lambda L} \biggr] = \frac{\pi y_1 w}{\lambda L} \, .</math>
			</div>

	=See Also=		=See Also=