If we focus diffracted light on a screen, using a lens or concave mirror, the
Fraunhofer
diffraction pattern appears. The screen is placed at the focal plane of
the lens or mirror, so that parallel diffracted rays are brought
together to interfere. The fringes are brighter and of higher visibility
than Fresnel fringes, and much more useful for measurement and other
purposes. For this reason, Fraunhofer diffraction is of great importance
in optics. It is also easy to analyze for simple geometries--we will
not even need any explicit integrals!


Let's begin with a single slit of width a, illuminated by collimated
light. At an angle θ = 0, all the wavelets from a wavefront are brought
together at the focal point F, and the total amplitude may be denoted by
R
o, and the intensity will be I
o = R
o2.
At an angle θ, wavelets from the bottom of the slit will travel an
extra distance a sin θ with respect to those from the top. This
corresponds to a phase difference of 2β = 2πa sin θ/&lambda. The
plot of the resultant amplitude as we go from top to bottom of the slit
will be a circular arc, subtending an angle of 2β, and of total length R
o.
The vibration curve is a circle, because the phase difference is
proportional to the length of the curve, as in s = rθ. If R is the
length of the chord, the resultant amplitude for the whole slit, and r
is the radius of the arc, then R = 2r sin β. However, R
o = rβ, so r can be eliminated and we find R = R
o(sin β/β) or R = R
osinc β. This is the
single-slit Fraunhofer diffraction pattern.
As a becomes comparable to the wavelength, the diffracted light is
distributed in a broad cone. The first minimum of R occurs when the
vibration arc closes into a circle, at β = π.