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

<math>d \sin{\theta} = n \lambda</math>

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

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

<math>y_n = \frac{nR \lambda}{d}</math>

where

<math>y_n</math> is the distance on the screen to the n^th maxima

<math>\lambda</math> is the wavelength

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

<math>y_n = \frac{(n+ \frac{1}{2})R \lambda}{d}</math>

for the n^th minima.

See also

Double-slit experiment