the combined waves as: Slits of finite width The width of the slits, is finite. Solution by integration The diffracted pattern is given by: Solution using Fourier transform The aperture function is given by: The Fourier transform of this function is given by where is the Fourier transform frequency, and the function is here defined as and We have or This is the same expression as was derived by integration. Intensity The intensity is given by: It can be seen that the form of the intensity pattern is the product of the individual slit diffraction pattern, and the interference pattern which would be obtained with slits of negligible width. This is illustrated in the image at the right which shows single slit diffraction by a laser beam, and also the diffraction/interference pattern given by two identical slits. Gratings A grating is defined in Born and Wolf as "any arrangement which imposes on an incident wave a periodic variation of amplitude or phase, or both". Narrow slit grating A simple grating consists of a screen with slits whose width is significantly less than the wavelength of the incident light with slit separation of . Solution by integration The complex amplitude of the diffracted wave at an angle is given by: since this is the sum of a geometric series. Solution using Fourier transform The aperture is given by The Fourier transform of this function is: Intensity thumb|200px|right|Detail of main maximum in 20 and 50 narrow slit grating diffraction patternsThe intensity is given by: This function has a series of maxima and minima. There are regularly spaced "principal maxima", and a number of much smaller maxima in between the principal maxima. The principal maxima occur when and the main diffracted beams therefore occur at angles: This is the grating equation for normally incident light. The number of small intermediate maxima is equal to the number of slits, and their size and shape is also determined by . The form of the pattern for =50 is shown in the first figure . The detailed structure for 20 and 50 slits gratings are illustrated in the second diagram. Finite width slit grating The grating now has N'' slits of width and spacing Solution using integration The amplitude is given by: Solution using Fourier transform The aperture function can be written as: Using the convolution theorem, which says that if we have two functions and