Why( and how) does the intensity of the wave vary along a screen perpendicular to the optical axis?
The light arriving at the screen is the superposition (sum) of waves from the various parts of the slit. In general, these waves each travel slightly different distances to reach a particular point on the screen, and so they interfere with each other. The amount of interference varies from one point to another on the screen.
One way to derive the formula that gives the intensity at various points on the screen:
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