# Why are even number of point sources considered to explain Single Slit Diffraction Patterns?

So I was watching this Khan Academy video. For explaining the patterns formed on the screen they assumed $8$ point sources and said that if the point source number $1$ interferes destructively with point source number $5$ then the rest of the waves also mutually cancel out (destructive interference). I agree with the video upto this part. But my question is : Why should we consider 8 point sources and not an odd number of point sources like (say) 9 ? If we consider $9$ point sources the explanation is no longer valid. Suppose at the shown point of the screen wave from point source $1$ interferes destructively with wave from point source $6$ then waves from point sources $2,3,4,7,8,9$ also cancel out pairwise as before. But the wave from point $5$ still remains and does not mutually cancel out with another other light wave (from another point source). So if we consider odd number of point sources the shown point on the screen should not have a dark fringe (or minima).

To sum up my queries:

• Why should we take even number of point sources to explain single slit diffraction pattern ?

• Is there any method to resolve this discrepancy ?

• Why we have to use a number of points at all if behind a single edge fringes occur, too? Dec 26 '16 at 10:16

The problem is, taking an even or odd number of point sources is an approximation and therefore seems to lead to discrepancies in any case; as the guy says in the video, one should take into consideration an infinite number of point sources, but drawing them would be too long so he chooses to take eight.

The reasoning comes from Huygens' principle (what is it?): so to be completely correct, one should make all the calculations with an integral (thus including every one of the infinite, infinitesimally small point sources) and would indeed arrive to the same result for the interference pattern, obtained with rigorous means.

So to answer your question, the need to have an even number of point sources is a consequence of the approximate nature of the reasoning used in the video, and is not an intrinsic inconsistency of the theory (when you have an infinite number of point sources, it doesn't even make sense to wonder if they are even or odd).

• Is this youtube.com/… analysis correct? (It assumes infinite number of point sources) Dec 26 '16 at 9:54
• Yes, it is correct even if it uses a different approach to the one based on Huygens' principle. This one considers N sources and then takes the limit as N goes to infinity, keeping fixed the width of the slit, which is a standard method. Dec 26 '16 at 10:15

To really discribe single slit diffraction, we must assume, that every point of the slit acts like a point source. So really there are not 8, 9, or 10, but uncountably many of them, and the notion of odd or even does not make sense. We must add (integrate) contributions from all this points, and baceuse there are infinitely many of them, each contribution is infinitesimaly small. So choosing finite number of points is only an aproximation. Nevertheless if you would choose an odd number of points, than there would still be some places on a screen, where the intensity is equal to zero. This would be the places, where all the contributions, from let us say 11 sources add up to zero. In this case, there would not be this pairwise destructive interference you mentioned. But if you have three sine waves, 120 deg out of phase each, they will add up to zero.

The explanations given so far did not match what the author wanted. The reason why the author asked this specifically is 'cos the even number of point sources was the foundation to deriving eventually the destructive interference pattern, which is w x sin(theta) = m x lambda.

The correct explanation is that it need to be even number 'cos this is the only condition that allows a complete cancellation of waves resulting in a dark region to be formed on the screen. Therefore, w/2, w/4, w/6 etc. is key. this means 1-1, 2-2, 3-3, all equal to zero, thus perfect wave cancellation.

• Please use MathJax for typesetting mathematical expressions.
– user258881
Jun 20 '20 at 5:49