User:Tohline/Appendix/Ramblings/FourierSeries - Revision history

Tohline: /* One-Dimensional Aperture */

2020-03-04T17:14:36Z

One-Dimensional Aperture

← Older revision		Revision as of 17:14, 4 March 2020
Line 1,326:		Line 1,326:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\mathrm{exp}\biggl\{i\biggl[		<math>~
			\mathrm{exp}\biggl\{i\biggl[
	\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda}		\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda}
	\biggr] \biggr\}		\biggr] \biggr\}
Line 1,364:		Line 1,365:
	<td align="left">		<td align="left">
	<math>~Y_0 + j\Delta Y \, .</math>		<math>~Y_0 + j\Delta Y \, .</math>
			</td>
			</tr>
			</table>

			This is precisely the same as the [[#Standard\|above-defined]],
			<div align="center" id="StandardExpression">
			<table border="1" cellpadding="5" align="center">
			<tr>
			<th align="center">''Standard'' Fourier Series Expression</th>
			</tr>
			<tr><td align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~f(x)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{a_0}{2} + \sum_{n=1}^{\infty} \biggl[ a_n\cos \biggl(\frac{n\pi x}{L}\biggr) + b_n\sin \biggl(\frac{n\pi x}{L}\biggr) \biggr]
			</math>
			</td>
			</tr>
			</table>
			</td></tr></table>
			</div>
			if we accept the following associations:
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~n</math>
			</td>
			<td align="center">
			<math>~\leftrightarrow</math>
			</td>
			<td align="left">
			<math>~j</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\frac{x}{L}</math>
			</td>
			<td align="center">
			<math>~\leftrightarrow</math>
			</td>
			<td align="left">
			<math>~\frac{2 y_1 \Delta Y}{\lambda L_1}</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\frac{a_{n=0}}{2}</math>
			</td>
			<td align="center">
			<math>~\leftrightarrow</math>
			</td>
			<td align="left">
			<math>~\mathrm{exp}\biggl\{i\biggl[
			\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} + \phi_0
			\biggr] \biggr\} ~a_{j=0}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~a_n</math>
			</td>
			<td align="center">
			<math>~\leftrightarrow</math>
			</td>
			<td align="left">
			<math>~a_j ~ \mathrm{exp}\biggl\{i\biggl[
			\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} + \phi_j
			\biggr] \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~b_n</math>
			</td>
			<td align="center">
			<math>~\leftrightarrow</math>
			</td>
			<td align="left">
			<math>~-i a_j~\mathrm{exp}\biggl\{i\biggl[
			\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} + \phi_j
			\biggr] \biggr\}
			</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* One-Dimensional Aperture */

2020-03-04T16:31:53Z

One-Dimensional Aperture

← Older revision		Revision as of 16:31, 4 March 2020
Line 1,252:		Line 1,252:
	===General Concept===		===General Concept===

	~~Drawing~~ from our [[User:Tohline/Appendix/CGH/ParallelApertures#Utility_of_FFT_Techniques\|accompanying discussion of single-slit apertures]] in the context of computer-generated holography and, for the time being, ~~retaining~~ our option to specify a phase angle, <math>~\phi_j</math>, that varies with position across the aperture, we have,		Here we draw from our [[User:Tohline/Appendix/CGH/ParallelApertures#Utility_of_FFT_Techniques\|accompanying discussion of single-slit apertures]] in the context of computer-generated holography and, for the time being, retain our option to specify a phase angle, <math>~\phi_j</math>, that varies with position across the aperture. Recognizing from that discussion that <math>~D_j \approx L_1[1 - y_1 Y_j/L_1^2]</math>, we have,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 1,314:		Line 1,314:
	\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} + \phi_j		\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} + \phi_j
	\biggr] \biggr\} \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]		\biggr] \biggr\} \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]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\mathrm{exp}\biggl\{i\biggl[
			\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda}
			\biggr] \biggr\}
			\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] \, ,
	</math>		</math>
	</td>		</td>

Tohline: /* One-Dimensional Aperture */

2020-03-04T00:26:36Z

One-Dimensional Aperture

← Older revision		Revision as of 00:26, 4 March 2020
Line 1,251:		Line 1,251:
	==One-Dimensional Aperture==		==One-Dimensional Aperture==
	===General Concept===		===General Concept===
	<~~span id="Fourier"~~>~~Hence~~, we have,~~</span>~~
			Drawing from our [[User:Tohline/Appendix/CGH/ParallelApertures#Utility_of_FFT_Techniques\|accompanying discussion of single-slit apertures]] in the context of computer-generated holography and, for the time being, retaining our option to specify a phase angle, <math>~\phi_j</math>, that varies with position across the aperture, we have,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 1,263:		Line 1,264:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~A_0~~ \sum_j		<math>~\sum_j
	a_j e^{-i[2\pi y_1 Y_j/(\lambda ~~L)]~~}		a_j e^{i(2\pi D_j/\lambda + \phi_j)}
	\~~, ,~~		</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~\sum_j
			a_j \mathrm{exp}\biggl\{i\biggl[
			\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_j}{L_1 \lambda} + \phi_j
			\biggr] \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\sum_j
			a_j \mathrm{exp}\biggl\{i\biggl[
			\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} - \frac{2\pi j y_1 \Delta Y}{L_1 \lambda} + \phi_j
			\biggr] \biggr\}
	</math>		</math>
	</td>		</td>
Line 1,278:		Line 1,310:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~A_0~~ \sum_j		<math>~\sum_j
	a_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]		a_j \mathrm{exp}\biggl\{i\biggl[
	~~\, ,~~		\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} + \phi_j
			\biggr] \biggr\} \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]
	</math>		</math>
	</td>		</td>
	</tr>		</tr>

	</table>		</table>
	</div>		</div>
	where, ~~now,~~ <math>~~~A_0~~ = e^{~~i2\pi L~~/\~~lambda}~~</math>~~. When written in this form, it should immediately be apparent why [[User:Tohline~~/~~Appendix~~/~~CGH~~/~~ParallelApertures#Fourier\|~~discrete ~~Fourier transform techniques]] (specifically FFT techniques) are useful tools for evaluation of~~ the ~~complex amplitude~~, <math>~A</math>.		where,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~L_1</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~[ Z^2 + y_1^2 ]^{1 / 2} \, ,</math>
			</td>
			</tr>
			</table>
			and we have inserted the following expression in order to identify discrete locations along the aperture,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~Y_j</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~Y_0 + j\Delta Y \, .</math>
			</td>
			</tr>
			</table>

	=See Also=		=See Also=

Tohline: /* Complex */

2018-03-17T21:45:04Z

Complex

← Older revision		Revision as of 21:45, 17 March 2018
Line 325:		Line 325:
	</table>		</table>
	</div>		</div>

			<table border="1" align="center" cellpadding="8" width="70%">
			<tr>
			<th align="center" bgcolor="yellow">
			LaTeX mathematical expressions cut-and-pasted directly from
			<br />
			NIST's ''Digital Library of Mathematical Functions''
			</th>
			</tr>
			<tr>
			<td align="left">
			As an additional primary point of reference, note that according to [https://dlmf.nist.gov/1.8 §1.8(i) of NIST's ''Digital Library of Mathematical Functions''], this ''alternative form'' is,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~f(x)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\sum^{\infty}_{n=-\infty}c_{n}e^{inx},</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~c_n</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x)e^{-inx}\mathrm{d}x.</math>
			</td>
			</tr>
			</table>

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


	Let's demonstrate that this rewritten (complex) expression for <math>~f(x)</math> matches the standard Fourier series expression. First, we will refer to the above ''standard'' definitions of <math>~a_n</math> and <math>~b_n</math> as, respectively, <math>~a_{\|n\|}</math> and <math>~b_{\|n\|}</math>, and recognize that, as the summation is extended to negative numbers, the following mapping is appropriate:		Let's demonstrate that this rewritten (complex) expression for <math>~f(x)</math> matches the standard Fourier series expression. First, we will refer to the above ''standard'' definitions of <math>~a_n</math> and <math>~b_n</math> as, respectively, <math>~a_{\|n\|}</math> and <math>~b_{\|n\|}</math>, and recognize that, as the summation is extended to negative numbers, the following mapping is appropriate:

Tohline: /* Standard */

2018-03-17T21:35:55Z

Standard

← Older revision		Revision as of 21:35, 17 March 2018
Line 84:		Line 84:
	<tr>		<tr>
	<td align="left">		<td align="left">
	As an primary point of reference, note that according to [https://dlmf.nist.gov/1.8 §1.8(i) of NIST's ''Digital Library of Mathematical Functions''], a Fourier Series is defined as follows:		As an additional primary point of reference, note that according to [https://dlmf.nist.gov/1.8 §1.8(i) of NIST's ''Digital Library of Mathematical Functions''], a Fourier Series is defined as follows:
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Standard */

2018-03-17T21:35:13Z

Standard

← Older revision		Revision as of 21:35, 17 March 2018
Line 73:		Line 73:
	</table>		</table>
	</div>		</div>

			<table border="1" align="center" cellpadding="8" width="70%">
			<tr>
			<th align="center" bgcolor="yellow">
			LaTeX mathematical expressions cut-and-pasted directly from
			<br />
			NIST's ''Digital Library of Mathematical Functions''
			</th>
			</tr>
			<tr>
			<td align="left">
			As an primary point of reference, note that according to [https://dlmf.nist.gov/1.8 §1.8(i) of NIST's ''Digital Library of Mathematical Functions''], a Fourier Series is defined as follows:
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~f(x)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tfrac{1}{2}a_{0}+\sum^{\infty}_{n=1}\biggl[ a_{n}\cos\bigl(nx\bigr)+b_{n}\sin\bigl(nx\bigr) \biggr],</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~a_n</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\cos\bigl(nx\bigr)\mathrm{d}x,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~b_n</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\sin\bigl(nx\bigr)\mathrm{d}x.</math>
			</td>
			</tr>
			</table>

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

	===Alternate===		===Alternate===

Tohline: /* Example #3 */

2017-12-19T23:24:44Z

Example #3

← Older revision		Revision as of 23:24, 19 December 2017
Line 1,089:		Line 1,089:

	===Example #3===		===Example #3===
	Now, let's reverse the example provided as Example #2. Specifically, let's look for the Fourier series expression for the sinc(<math>~\alpha</math>)		Now, let's reverse the example provided as Example #2. Specifically, let's look for the Fourier series expression for the function,
			<div align="center">
			<math>~g(x) \equiv c~\mathrm{sinc}(\alpha) \, ,</math>
			</div>
			for <math>~\alpha \equiv \pi x/c</math>, over the range, <math>~-m \pi \le \alpha_n \le + m \pi</math>. Furthermore, we will assume that <math>~g(x)</math> is a periodic function of period <math>~2mc</math>, in which case it may be represented by the ''Fourier series'',
			<div align="center" id="StandardExpression">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~g(x)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{a_0}{2} + \sum_{n=1}^{\infty} \biggl[ a_n\cos \biggl(\frac{n\pi x}{L}\biggr) + b_n\sin \biggl(\frac{n\pi x}{L}\biggr) \biggr] \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~a_n</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{L} \int_{-L}^{L} g(x) \cos\biggl( \frac{n\pi x}{L} \biggr) dx
			</math>
			</td>
			<td align="center">   </td>
			<td align="right">for</td>
			<td align="left"><math>~n = 0, 1, 2, 3, \dots \, ;</math>
			</tr>

			<tr>
			<td align="right">
			<math>~b_n</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{L} \int_{-L}^{L} g(x) \sin\biggl( \frac{n\pi x}{L} \biggr) dx
			</math>
			</td>
			<td align="center">   </td>
			<td align="right">for</td>
			<td align="left"><math>~n = 1, 2, 3, \dots </math>
			</tr>
			</table>
			</div>

	==One-Dimensional Aperture==		==One-Dimensional Aperture==

Tohline: /* From Williams & Tohline (1987) */

2017-12-18T23:05:30Z

From Williams & Tohline (1987)

← Older revision		Revision as of 23:05, 18 December 2017
Line 1,087:		Line 1,087:
	</li>		</li>
	</ol>		</ol>

			===Example #3===
			Now, let's reverse the example provided as Example #2. Specifically, let's look for the Fourier series expression for the sinc(<math>~\alpha</math>)

	==One-Dimensional Aperture==		==One-Dimensional Aperture==

Tohline: /* Example #2 */

2017-12-15T22:41:58Z

Example #2

← Older revision		Revision as of 22:41, 15 December 2017
Line 1,082:		Line 1,082:
	</tr>		</tr>
	</table>		</table>
			</li>
			<li>
			The ''sinc'' function goes to zero each time its argument, <math>~\alpha_n</math>, is an integer multiple of π. This means that the first zero arises at <math>~n = L/c</math>, if <math>~L</math> is an integer-multiple of <math>~c</math>. This also means that the discrete array that marks the Fourier mode amplitude, <math>~a_n</math>, will resolve the central amplitude peak only if <math>~c/L \ll 1</math>.
	</li>		</li>
	</ol>		</ol>

Tohline: /* Example #2 */

2017-12-15T22:07:51Z

Example #2

← Older revision		Revision as of 22:07, 15 December 2017
Line 988:		Line 988:

	<div align="center" id="StandardExpression">		<div align="center" id="StandardExpression">

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

Line 1,045:		Line 1,044:
	<li>If we define,		<li>If we define,
	<div align="center">		<div align="center">
	<math>~\~~alpha~~ \equiv \frac{n\pi c}{L} \, ,</math>		<math>~\alpha_n \equiv \frac{n\pi c}{L} \, ,</math>
	</div>		</div>
	then, as a function of <math>~\alpha</math>, the Fourier mode amplitude is,		then, as a function of <math>~\alpha</math>, the Fourier mode amplitude is,
Line 1,053:		Line 1,052:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~a_n(\~~alpha~~)</math>		<math>~a_n(\alpha_n)</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,059:		Line 1,058:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl( \frac{2c}{L} \biggr) \frac{\sin \~~alpha~~}{\~~alpha~~} = \biggl( \frac{2c}{L} \biggr) \mathrm{sinc}(\~~alpha~~) \, .</math>		<math>~\biggl( \frac{2c}{L} \biggr) \frac{\sin \alpha_n}{\alpha_n} = \biggl( \frac{2c}{L} \biggr) \mathrm{sinc}(\alpha_n) \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
			</li>
			<li>
			The inverse transform may also then be rewritten as,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~f(x)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2c}{L}\biggl[ \frac{1}{2} +\sum_{n=1}^{\infty} \mathrm{sinc}(\alpha_n) \cos \biggl(\frac{n\pi x}{L}\biggr) \biggr] \, .
			</math>
			</td>
			</tr>
			</table>

	</li>		</li>
	</ol>		</ol>