## Diffraction

Fresnel and Fraunhofer diffraction:  Diffraction occurs when light passes through an aperture or around an edge.  When the source and the observer are so far away from the obstructing surface that the outgoing rays can be considered parallel, it is called Fraunhofer diffraction.  When the source or the observer is relatively close to the obstructive surface, it is Fresnel diffraction.

Single-slit diffraction:  Monochromatic light sent through a narrow slit of width $a$ produces a diffraction pattern on a distant screen.  The condition for destructive interference (a dark fringe) at a point in the pattern at angle $\theta$: $\sin\theta=\frac{m\lambda}{a}$, $(m=\pm1,\pm2,\pm3,\ldots)$.  The intensity in the pattern as a function of $\theta$: $I=I_0\{\frac{\sin[\pi a(\sin\theta)/\lambda]}{\pi a(\sin\theta)/\lambda}\}^2$.

Diffraction gratings:  A diffraction grating consists of a large number of thin parallel slits, spaced a distance $d$ apart.  The condition for maximum intensity in the interference pattern is the same as for the two-source pattern, but the maxima for the grating are very sharp and narrow.  $d\sin\theta=m\lambda$ $(m=\pm1,\pm2,\pm3,\ldots)$.

X-ray diffraction:  A crystal serves as a three-dimensional diffraction grating for x rays with wavelengths of the same order of magnitude as the spacing between atoms in the crystal.  For a set of crystal planes spaced a distance $d$ apart, constructive interference occurs when the angles of incidence and scattering (measured from the crystal planes) are equal and when the Bragg condition is satisfied: $2d\sin\theta=m\lambda$, $(m=1,2,3,\ldots)$.

Circular apertures and resolving power:  The diffraction pattern from a circular aperture of diameter $D$ consists of a central bright spot, called the Airy disk, and a series of concentric dark and bright rings.  Equation $\sin\theta_1=1.22\frac{\lambda}D$ gives the angular radius $\theta_1$ of the first dark ring, equal to the angular size of the Airy disk.  Diffraction sets the ultimate limit on resolution (image sharpness) of optical instruments.  According to Rayleigh’s criterion, two point objects are just barely resolved when their angular separation $\theta$ is given by the equation.