the maxima and minima in the amplitude we must determine the path difference to the first slit and to the second one. In the Fraunhofer approximation, with the observer far away from the slits, the difference in path length to the two slits can be seen from the image to be Maxima in the intensity occur if this path length difference is an integer number of wavelengths. where is an integer that labels the order of each maximum, is the wavelength, is the distance between the slits, and is the angle at which constructive interference occurs. The corresponding minima are at path differences of an integer number plus one half of the wavelength: For an array of slits, positions of the minima and maxima are not changed, the fringes visible on a screen however do become sharper, as can be seen in the image. Mathematical description To calculate this intensity pattern, one needs to introduce some more sophisticated methods. The mathematical representation of a radial wave is given by where , is the wavelength, is frequency of the wave and is the phase of the wave at the slits at time t = 0. The wave at a screen some distance away from the plane of the slits is given by the sum of the waves emanating from each of the slits. To make this problem a little easier, we introduce the complex wave , the real part of which is equal to The absolute value of this function gives the wave amplitude, and the complex phase of the function corresponds to the phase of the wave. is referred to as the complex amplitude. With slits, the total wave at point on the screen is Since we are for the moment only interested in the amplitude and relative phase, we can ignore any overall phase factors that are not dependent on or . We approximate . In the Fraunhofer limit we can neglect terms of order in the exponential, and any terms involving or in the denominator. The sum becomes The sum has the form of a geometric sum and can be evaluated to give The intensity is given by the absolute value of the complex amplitude squared where denotes the complex conjugate of . Single slit As an example, an exact equation can now be derived for the intensity of the diffraction pattern as a function of angle in