Diffraction Patterns: Does Distance Affect Diffraction?

[haha1234]

Homework Statement

Is it necessary that a single slit diffraction occur with the two source having the distance of $\frac{a}{2}$?Are there any diffractions occur when the point source having the distance of $\frac{a}{3}$,$\frac{a}{4}$ and so on?
And I would like to know whether are there only destructive interference occur?

Homework Equations

The Attempt at a Solution

 

[BvU]

Hi Ha,
There are no two sources in the drawing; there is only the one slit opening that is imagined to be subdivided into two "sources" that are deemed a/2 apart. As you correctly suspect, it is equally well possible to subdivide into 3, 4, 5, etc. sources (with successively smaller extension / width) that interfere constructively at greater and greater angles. In the resulting intensity pattern the angles in between have destructive interference.

The intensity at a given point on the screen is in fact an integral over the slit opening. See e.g. Fraunhofer diffraction or Single slit diffraction.

The resulting intensity pattern is ${\sin^2 x \over x^2}\$; $\ \ \ {\sin x \over x}$ has been found to be important enough to get its own name: ${\rm sinc}\ x$