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I understand that in the interference pattern of a double slit experiment, if the path difference is an integer multiplied by lambda, we have maxima. If it is 1/2 or 3/2 or 5/2 etc.. we get a minima.

However, what I don't get is: In a single slit experiment, we usually started out by assuming that at the point at the top of the slit, there is a light that has a path difference of 1 lambda with the one at the bottom of the slit. From there, we can derive the formulas: If slit width is a, then the distance between a point at the middle of the slit with a point at the top of the slit is only a/2. Thus, $\frac{a}{2} \sin(\theta) = \frac{\lambda}{2}$ since it will be destructive (because the path difference between the top and the bottom point is 1 lambda, then the path difference between a point at the top and at the middle should be lambda/2).

This is all clear to me.

But, all of this ONLY HOLDS IF the assumption is true: that the path difference between the point at the top of the slit and at the bottom of the slit is 1 lambda. Now, WHY exactlt do we assume this?

I mean, I can say that I assume the path difference to be a quarter of lambda, or 3/7 of lambda, or any arbitrary number for that matter. Thus, the path difference between the middle point of the slit and the top point of the slit will also become an arbitrary number of lambda, and thus the equation won't hold, right?

Is my logic mistaken?

user205891
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2 Answers2

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The wavefronts represent peaks of the wave and the points on a wavefront are in phase with each other. According to Huygens' principle, each point on a wavefront can be treated as an individual source. In the typical single-slit diffraction scenario, the light is incident normally on the slit which means that the points on the slit can be treated as individual sources, all of which are in phase with each other. The (subsequent) phase difference comes only from the path difference to the screen, which is proportional (in the small-angle limit) to the displacement from the center of the screen. Since the constant of proportionality is known, we can always find one value given the other. The assumption that you state is used to calculate the position of the first minimum. We assume that the path difference is $\lambda$ and then solve for the position on the screen.

Vincent Thacker
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at the point at the top of the slit, there is a light that has a phase difference of 1 lambda with the one at the bottom of the slit.

This is wrong. It becomes right if you talk about the phase difference at the screen at the position of the first minimum of light coming from the top vs. coming from the bottom of the slit.

The usual reasoning is that then the phase difference between top and middle is $\pi/2$, leading to destructive interference. The same holds for light from "slightly below top" together with light from "slightly below middle" and so on. For each point source along the slit, there is another one with phase difference $\pi/2$.

See also wikipedia.

Jasper
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