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I want to calculate the net phase at a point on the screen in Fresnel diffraction. Attached below is a rough diagram if needed.

For this I considered hypothetical $n+1 $ small slits acting as sources within the original slit namely $ S_0 , S_1 … S_n$ equally spaced at a distance $x$. The thought was to calculate the net phase at a point at distance $y$ from the midpoint of the slit and extract the limit as $n$ tends to infinity with the constraint that $nx=a$ where $a$ is the width of the slit. Assuming the phase of the first slit $S_0$ is $\phi$ , the phase due to the $r^{th}$ slit will be $ \phi +\frac{2\pi}{\lambda} \frac{yrx}{D}$ under the assumption that the distance D is much much greater than the slit width. Therefore the net phase is $\sum_{r=0}^{n} \phi +\frac{2\pi}{\lambda} \frac{yrx}{D} = \lim_{n\to \infty}[ n\phi + \frac{\pi n(n+1)yx}{\lambda} ]$ . Upon putting $nx= a$ we get $ \lim_{n\to \infty}[ n\phi + \frac{\pi a(n+1)y}{\lambda}] $ which is clearly blowing to infinity. I think that this problem is because the interference pattern of the double slit actually involves the superposition of 2 diffraction patterns which is why it cannot be used to explain the single slit diffraction pattern. However I do not have clarity over this. Also, if there is a fairly easy proof of the single slit diffraction pattern which is understandable upto a high school (12th grade) level please do share it, I am highly intrigued! Thanks.

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  • $\begingroup$ Note that single slit diffraction interference only works for light .... matter waves do not show single slit diffraction interference. $\endgroup$ Mar 1 at 15:48

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See here for a simple solution using phasors. Most slit experiment patterns (double, n-slit, etc.) can be easily calculated using phasors.

P.S. Your solution is correct(i.e.doesn't have logical errors). Note that phases have a period of $2\pi$, so even if the sum mentioned in the solution goes to infinity the phase will have a value between 0 and $2\pi$.

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  • $\begingroup$ Note that single slit diffraction interference only works for light .... matter waves do not show single slit diffraction interference. $\endgroup$ Mar 1 at 15:49

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