Single Slit Diffraction

" Physics is really nothing more than a search for ultimate simplicity, but so far all we have is a kind of elegant messiness.”
Bill Bryson, A Short History of Nearly Everything

Diffraction may be thought of as the "spreading out" of waves as they pass through or by an aperture or edge. We now investigate this phenomenon more closely for a single slit of width "a".

By considering point sources in pairs across the width of the slit we can use the ideas of double slit interference to show that minimum intensity is observed at point P on a screen when

fig1

fig4

However, in order to obtain the intensity profile as a function of θ we must perform a more complex analysis. Considering each point in the slit as a point source, for electromagnetic waves the electric field at a point P on the screen at right can be written,

where ω is the angular frequency of the wave and Δφ is the phase angle. The relationship between phase angle and path difference Δs is given by

From the diagram at right the path difference Δs, relative to the center, is given by ysinθ. In this case

so that the total electric field at a point on the screen is obtained by integration over the slit

Performing this integration we obtain

where

The intensity observed is proportional to the square of the electric field

eqn8

ssdifffig5

Using the intensity function above the minima condition is given by sinβ = 0 ( β ≠ 0 ), which means β = nπ where n = 1,2,3... Using the definition of β above gives the condition for minima

as expected.

For n = 0 above, θ = 0, but the intensity function leads to the central maximum.
Note that the width of the central maximum - 2λ/a - is double that of secondary maxima - λ/a. This is in contrast to the double slit interference pattern where all maxima have the same width.
The location of the secondary maxima are given approximately by

The exact position of these maxima is shifted slightly towards smaller θ.

In the above analysis we have (implicitly) assumed that the source and observation screen are infinitely far from the single slit (on opposite sides of the slit). This allows us to use the plane wave approximation leading to the intensity expression above. This is described as Fraunhofer diffraction. Practically we can approximate a Fraunhofer situation by using converging lenses to produce parallel rays.

Without assuming plane waves we must resort to a more complex analysis known as Fresnel diffraction.

The Official Unabashed Scientific Dictionary defines a transistor as a nun who's had a sex change

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu