The interference pattern from ## N ## equally spaced (narrow) slits has the formula for intensity ## I(\theta)=I_o \frac{sin^2(N \phi/2)}{sin^2(\phi/2)} ##, with ## \phi=\frac{2 \pi d \sin(\theta)}{\lambda} ##.## \\ ## When the denominator is zero, the expression is taken as the limit of where it approaches zero. In those cases, they are the primary maxima, with ## m \lambda =d \sin(\theta) ##, for integer ## m ##,and the intensity is ## I=N^2 I_o ##. ## \\ ## This result gets multiplied by the diffraction pattern for a single slit of width ## b ## when the slits have finite width: ## I_D(\theta)=\frac{\sin^2(\frac{\pi b \sin(\theta)}{ \lambda})}{(\frac{\pi b sin(\theta)}{\lambda})^2} ##.(Note the form of this expression: ## I_D(x)= \frac{sin^2x}{x^2} \, ## ). In the limit of a very narrow slit ## b ##, the diffraction factor is 1. Otherwise the interference pattern from ## N ## slits is reduced by the diffraction factor, which has zeros at ## m\lambda = b \sin(\theta) ## for non -zero integer ## m ##. When ## m=0 ##,##(\theta=0 )##, this is the central maximum of the diffraction pattern, and the diffraction factor is equal to 1. ## \\ ## And these results are not of quantum mechanical origin. They are classical Kirchhoff-Fresnel diffraction theory. ## \\ ## Editing: It should be mentioned that these intensity results are for the far-field=far from the slits, and the name Fraunhofer is often associated with these far-field results.
