Diffraction

Diffraction occurs “when waves pass through small openings, around obstacles, or past sharp edges. When an opaque object is placed between a point source of light and a screen, no sharp boundary exists on the screen between a shadowed region and an illuminated region. The illuminated region above the shadow of the object contains alternating light and dark fringes. Such a display is called a diffraction pattern.”

Young's Double Slit Experiment

Consider two slits a distance $d$ apart and a screen a distance $R$ from the slits. To find the positions of the maxima (brightest points) on the screen, the path difference from each of the slits, $\Delta$ must be an integer number of wavelengths $n\lambda$. The the angle between a ray of light from one slit and the normal of that slit is $\theta$, then for constructive interference:

$d\sin {\theta }=n\lambda$

If $y$ is the distance from the centre of the screen to a maxima, then $y=R\tan {\theta }$.

If we assume the screen is much further away than the slits, such that $R>>d$ then using the small angle approximation, $\tan {\theta }\approx \sin {\theta }$, rearranging gives:

$y_{n}={\frac {nR\lambda }{d}}$

where

$y_{n}$ is the distance on the screen to the n^th maxima

$\lambda$ is the wavelength

For destructive interference, the path difference must be $(n+{\frac {1}{2}})\lambda$. This gives:

$y_{n}={\frac {(n+{\frac {1}{2}})R\lambda }{d}}$

for the n^th minima.

See also

Double-slit experiment