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SingleSlit Diffraction 
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#1
May210, 04:06 PM

P: 21

Hi everyone. I need help comprehending single slit diffraction. I am using the textbook: College Physics by Serway / Vuille 8th Edition. This is what it says:
"To analyze the diffraction pattern, it's convenient to divide the slit into halves, as in Figure 24.17 (picture below). All the waves that originate at the slit are in phase. Consider waves 1 and 3, which originate at the bottom and center of the slit, respectively. Wave 1 travels farther than wave 3 by an amount equal to the path difference [tex] \frac{a}{2}\sin\theta[/tex], where a is the width of the slit. Similarly, the path difference between waves 3 and 5 is [tex] \frac{a}{2}\sin\theta[/tex]. If this path difference is exactly half of a wavelength, the two waves cancel each other and destructive interference results. Therefore, waves from the upper half of the slit interfere destructively with waves from the lower half of the slit when [tex] \frac{a}{2}\sin\theta = \frac{\lambda}{2} [/tex] The question is: why is a/2 the hypotenuse? why not just use a? I realize the first sentence says, "it's convenient to divide the slit into halves," but why? 


#2
May410, 04:01 PM

P: 21

Though I'd bump this thread since I've gotten no reply in two days.



#3
May410, 05:01 PM

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P: 41,443




#4
May410, 06:12 PM

Sci Advisor
P: 5,517

SingleSlit Diffraction
I agree with Doc Al, but from my perspective, this is a case of trying to make a concept overly simplified.
The farfield diffraction pattern from an aperture illuminated by a monochromatic plane wave is simply the Fourier transform of the aperture. For a single slit, this pattern is sin(ka)/ka, where k is 2*pi/l (l is the wavelength of illumination) http://cnyack.homestead.com/files/af...iffraction.htm The usual results follow. 


#5
May410, 06:43 PM

P: 2,050

If they wanted a simple example, they could have just used two narrow slits. (If they want to go beyond that, why not proceed straight to full generality?) 


#6
May510, 05:57 AM

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Thanks
PF Gold
P: 12,135

Not everyone finds the Fourier transform cuddly and familiar. Whilst Fourier is, undoubtedly, the way to go in the end, the fact is that you can extend the idea of two slit interference to two halves of a wide slit without Fourier being invoked. He's introduced / hinted at the idea of infinitessimals which is a step forward.
The nice thing is that the two approaches give the same result for the null directions  good reinforcement for the student who is fresh to these matters, I'd have thought. I've seen worse! 


#7
May510, 06:22 AM

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#8
May510, 07:38 AM

Sci Advisor
P: 5,517

There's no need to *derive* the sinc function, but in the spirit of plugandchug, surely providing a table of simple transform pairs is within the ability of a Freshman to use. 


#9
May510, 09:37 AM

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Thanks
PF Gold
P: 12,135

2. As for providing students with a set of transform pairs and telling them that they just happen to apply in diffraction; can that really be the way for them to get to understand what's going on? Fourier transforms to go between frequency and time domains make it far more approachable as a start (as you say, you can show it and listen to the effects through a loudspeaker but there is a distinction between the Fourier Transform and Fourier Analysis which can be missed if not careful). I can't think of an equivalent way to show the same thing with diffraction  apart from the effect of blurring through small apertures. Peoples' brains don't always find plug and chug at all straightforward. You may be lucky enough not to have a problem but we're not all gifted in that way; it's a matter of confidence. If you are hell bent on having a go at oversimplification of Physics teaching there are many more deserving targets  like why people insist on trying to explain simply every phenomenon in terms of particles (e.g. Convection, for Chrissake) 


#10
May510, 06:57 PM

P: 2,050

Heck, you don't necessarily need to mention Fourier transforms (nor their "grammar"), just hand them the standard integral that happens to correspond to the particular Fourier transform they are performing, that way it's just as easy as any other problem they've done but it also lets them see a way to approach broader questions. (Edit: Having learnt the technique in the OP's text, how then might the student begin if asked to obtain the intensity pattern of a tripleslit?) 


#11
May510, 09:36 PM

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P: 5,517

As for introducing the Fourier transform in an introductory course, I do exactly that (algebrabased, Giancoli). Was it successful? Honestly, I don't know yet I have only one data point, so to speak. But I was able to present a unified treatment of sound and optics, among other things. Some of the students seemed to 'get it', some didn't but I have information about how to adapt the presentation. My relevant test had questions about resolution limits of 'perfect' vision v. Rayleigh limit, matching the pixel size of a camera to the magnification of the optical system, and comparison shopping for a telescope. Definitely not plugandchug. 


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