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?
Best Answer
The single source produces waves that travel towards the slit. The wavefronts represent peaks of the wave (in phase with each other). According to Huygens' principle, each point on the wave-front can be treated as an individual source. Thus, as this wavefront passes straight through the slit, all the points on it are individual coherent sources. So they cannot have any phase difference. Therefore it is incorrect to say that "points on the slit" have a phase difference. The constructive and destructive interference depends on the path difference only.