The grating is a system similar to the single and double slit, but with N parallel slits of width b and distance h.  The system observes the following characteristics:

multislit.gif (2880 bytes)      

(Left) Schematic for a grating with 4 parallel slits each of width b and separated by a distance h.  (Right)  Step function for the grating of five parallel slits.

       To find the Fraunhofer pattern of this system, the sum of the slits is taken for the Fresnel-Kirchhoff formula.  This results in the following equations:

     (Fowles, 122)

where , and  and N is the number of slits in the system.  The intensity distribution for the system then becomes:

        (Fowles, 123)

The factor  acts as the envelope for the diffraction pattern.

       When the above intensity equation was calculated in Mathematica, we received the following Fraunhofer pattern:

      

(Left)  Fraunhofer pattern for the grating in 2D.  The inner funtion is the fraunhofer pattern where as the outer function is the diffraction envelope.  (Right)  Fraunhofer pattern in 3D, birds eye view.

The Fraunhofer pattern for the grating is the inner pattern with N=3.  The outer function is the envelope defined by .  The width of the inner functions are determined by N.  For larger N, the width of the peaks decreases.