Appendix/CGH/ParallelApertures - Revision history

Jet53man: /* Parallels With Example #2 */

2022-04-27T21:46:05Z

Parallels With Example #2

← Older revision		Revision as of 17:46, 27 April 2022
Line 1,015:		Line 1,015:
	===Parallels With Example #2===		===Parallels With Example #2===

	This "one dimensional aperture" analysis should exhibit features that strongly resemble the features that appear in our accompanying discussion of the Fourier series associated with a "[[~~User:Tohline/~~Appendix/Ramblings/FourierSeries#Example_.232\|square wave]]". In both cases — after performing both a Fourier transform ''and'' the inverse transform — the ultimate series expression that will represent the (square wave) amplitude across the aperture will take the form,		This "one dimensional aperture" analysis should exhibit features that strongly resemble the features that appear in our accompanying discussion of the Fourier series associated with a "[[Appendix/Ramblings/FourierSeries#Example_.232\|square wave]]". In both cases — after performing both a Fourier transform ''and'' the inverse transform — the ultimate series expression that will represent the (square wave) amplitude across the aperture will take the form,

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

Jet53man: /* Two Slits */

2021-07-26T17:42:52Z

Two Slits

← Older revision		Revision as of 13:42, 26 July 2021
Line 1,425:		Line 1,425:
	===Two Slits===		===Two Slits===

	Starting with just a pair of slits (as illustrated in frame ''b'' of [[#Figure5\|Figure 5]]) and setting the x-direction zero point exactly midway between the two, we can expand upon the [[~~User:Tohline/~~Appendix/Ramblings/FourierSeries#Example_.232\|accompanying ''Example #2'' discussion]] to obtain,		Starting with just a pair of slits (as illustrated in frame ''b'' of [[#Figure5\|Figure 5]]) and setting the x-direction zero point exactly midway between the two, we can expand upon the [[Appendix/Ramblings/FourierSeries#Example_.232\|accompanying ''Example #2'' discussion]] to obtain,

	<div align="center">		<div align="center">

Jet53man: /* Theory */

2021-07-26T17:42:05Z

Theory

← Older revision		Revision as of 13:42, 26 July 2021
Line 1,145:		Line 1,145:
	====Theory====		====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 [[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]],
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Parallels With Standard Fourier Series */

2021-07-26T17:41:10Z

Parallels With Standard Fourier Series

← Older revision		Revision as of 13:41, 26 July 2021
Line 594:		Line 594:
	===Parallels With Standard Fourier Series===		===Parallels With Standard Fourier Series===

	Let's draw parallels between the above discussion of the single-slit diffraction pattern and our [[~~User:Tohline/~~Appendix/Ramblings/FourierSeries#Standard_Setup\|separate presentation of the standard treatment of Fourier Transforms]].		Let's draw parallels between the above discussion of the single-slit diffraction pattern and our [[Appendix/Ramblings/FourierSeries#Standard_Setup\|separate presentation of the standard treatment of Fourier Transforms]].

	<div align="center">		<div align="center">
Line 600:		Line 600:
	<tr>		<tr>
	<th align="center" width="50%">		<th align="center" width="50%">
	[[~~User:Tohline/~~Appendix/Ramblings/FourierSeries#Standard_Setup\|Standard Fourier Transform Treatment]]		[[Appendix/Ramblings/FourierSeries#Standard_Setup\|Standard Fourier Transform Treatment]]
	</th>		</th>
	<td align="center">		<td align="center">
Line 794:		Line 794:
	----		----

	Drawing from an [[~~User:Tohline/~~Appendix/Ramblings/FourierSeries#ComplexExpression\|accompanying discussion]], the corresponding ''complex'' Fourier series expression is,		Drawing from an [[Appendix/Ramblings/FourierSeries#ComplexExpression\|accompanying discussion]], the corresponding ''complex'' Fourier series expression is,

	<div align="center" id="ComplexExpression">		<div align="center" id="ComplexExpression">
Line 963:		Line 963:
	<tr>		<tr>
	<th align="center" width="50%">		<th align="center" width="50%">
	[[~~User:Tohline/~~Appendix/Ramblings/FourierSeries#Standard_Setup\|Standard Fourier Transform Treatment]]		[[Appendix/Ramblings/FourierSeries#Standard_Setup\|Standard Fourier Transform Treatment]]
	</th>		</th>
	<td align="center">		<td align="center">

Jet53man: /* Utility of FFT Techniques */

2021-07-26T17:39:23Z

Utility of FFT Techniques

← Older revision		Revision as of 13:39, 26 July 2021
Line 336:		Line 336:
	where, now, <math>~A_0 = e^{i2\pi L/\lambda}</math>.		where, now, <math>~A_0 = e^{i2\pi L/\lambda}</math>.

	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 [[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===

Jet53man: /* Propagation of Light */

2021-07-26T17:38:49Z

Propagation of Light

← Older revision		Revision as of 13:38, 26 July 2021
Line 9:		Line 9:
	<math>~a(x) = a_0\cos\biggl(\frac{2\pi x}{\lambda} + \phi_0 \biggr) \, ,</math>		<math>~a(x) = a_0\cos\biggl(\frac{2\pi x}{\lambda} + \phi_0 \biggr) \, ,</math>
	</div>		</div>
	where:  the light-wave's peak-to-peak brightness is <math>~2a_0</math>; <math>~\phi_0</math> is the "phase" that the oscillatory wave displays at a reference location, <math>~x=0</math>, when the reference time, <math>~t = 0</math>; and <math>~x = ct</math>. As is demonstrated, for example, in our [[~~User:Tohline/~~Appendix/Ramblings/FourierSeries#ComplexExpression\|accompanying discussion of various, equivalent, Fourier series expressions]], this equation can be usefully rewritten in a number of other forms.		where:  the light-wave's peak-to-peak brightness is <math>~2a_0</math>; <math>~\phi_0</math> is the "phase" that the oscillatory wave displays at a reference location, <math>~x=0</math>, when the reference time, <math>~t = 0</math>; and <math>~x = ct</math>. As is demonstrated, for example, in our [[Appendix/Ramblings/FourierSeries#ComplexExpression\|accompanying discussion of various, equivalent, Fourier series expressions]], this equation can be usefully rewritten in a number of other forms.
	<!--		<!--
	<div align="center">		<div align="center">

Jet53man: Created page with " =CGH: Apertures that are Parallel to the Image Screen= This chapter..."

2021-07-26T17:38:26Z

Created page with "  =CGH: Apertures that are Parallel to the Image Screen= This chapter..."

Show changes