Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

DoITPoMS Teaching & Learning Packages Diffraction and Imaging Scattering from a pair of infinitely narrow slits
Diffraction and Imaging AimsBefore you startIntroductionDiffraction patterns 1Diffraction patterns 2Two-dimensional diffractionImage formationInformation obtained from diffraction patternApplications of the theory of optical diffraction and imagingSummaryQuestionsGoing further
TLP creditsTLP contentsShow all content
Viewing and downloading resourcesAbout the TLPsTerms of useFeedbackCredits Print this page
PreviousNext

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:

$$\rho ({\rm{x}}) = \delta \left( {x - {b \over 2}} \right) + \delta \left( {x + {b \over 2}} \right)$$ Diagram of two slits

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.

Graph of intensity of diffraction pattern

PreviousNext