the aperture function as a step function The Fourier–Bessel transform for this function is given by the relationship where is the transform frequency which is equal to and . Thus, we get Intensity The intensity is given by: Form of the diffraction pattern This known as the Airy diffraction pattern The diffracted pattern is symmetric about the normal axis. Aperture with a Gaussian profile An aperture with a Gaussian profile, for example, a photographic slide whose transmission has a Gaussian variation, so that the amplitude at a point in the aperture located at a distance r''' from the origin is given by giving Solution using Fourier–Bessel transform The Fourier–Bessel or Hankel transform is defined as where is the Bessel function of the first kind of order with . The Hankel transform is giving and Intensity The intensity is given by: This function is plotted on the right, and it can be seen that, unlike the diffraction patterns produced by rectangular or circular apertures, it has no secondary rings. This can be used in a process called apodization - the aperture is covered by a filter whose transmission varies as a Gaussian function, giving a diffraction pattern with no secondary rings. Slits Two slits The pattern which occurs when light diffracted from two slits overlaps is of considerable interest in physics, firstly for its importance in establishing the wave theory of light through Young's interference experiment, and secondly because of its role as a thought experiment in double-slit experiment in quantum mechanics. Narrow slits right|thumb|Two slit interference using a red laserAssume we have two long slits illuminated by a plane wave of wavelength . The slits are in the plane, parallel to the axis, separated by a distance and are symmetrical about the origin. The width of the slits is small compared with the wavelength. Solution by integration The incident light is diffracted by the slits into uniform spherical waves. The waves travelling in a given direction from the two slits have differing phases. The phase of the waves from the upper and lower slits relative to the origin is given by and The complex amplitude of the summed waves is given by: Solution using Fourier transform The aperture can be represented by the function: where is the delta function. We have and giving This is the same expression as that derived above by integration. Intensity This gives the intensity of