Scattering from a pair of infinitely narrow slits
The scattering density of two infinitely narrow slits separated by spacing b can be described using delta functions:
$$\rho ({\rm{x}}) = \delta \left( {x - {b \over 2}} \right) + \delta \left( {x + {b \over 2}} \right)$$ |
therefore,
$${\rm{A(}}{\bf{S}}{\rm{) = }}{{\rm{A}}_0}\int\limits_{ - \infty }^\infty {\rho ({\rm{x}})\exp \left( {2\pi {\rm{i}}{{{\bf{x}} \cdot {\bf{S}}} \over \lambda }} \right){\rm{d}}x} $$ $$\eqalign{ & {\rm{ = }}{{\rm{A}}_0}\int\limits_{ - b/2}^{b/2} {\left( {\delta \left( {x - {b \over 2}} \right) + \delta \left( {x + {b \over 2}} \right)} \right)\exp \left( {2\pi {\rm{i}}{{{\bf{x}}\sin 2\theta } \over \lambda }} \right){\rm{d}}x} \cr & {\rm{ = 2}}{{\rm{A}}_0}\cos \left( {{{\pi b \sin 2\theta } \over \lambda }} \right) \cr} $$
This gives rise to a sinusoidal variation of amplitude with (sin 2θ) / λ. The intensity of the diffraction pattern is shown in the diagram below.