The Fourier transform formulation can be very useful in solving diffraction problems. Another form is: where represent the observation point and a point in the aperture respectively, and represent the wave vectors of the disturbance at the aperture and of the diffracted waves respectively, and represents the magnitude of the disturbance at the aperture. In polar coordinates When the diffracting aperture has circular symmetry, it is useful to use polar rather than Cartesian coordinates. A point in the aperture has coordinates giving: and The complex amplitude at is given by , and the area converts to ρ′ dρ′ dω′, giving Using the integral representation of the Bessel function: we have where the integration over gives since the equation is circularly symmetric, i.e. there is no dependence on . In this case, we have equal to the Fourier–Bessel or Hankel transform of the aperture function, Example Here are given examples of Fraunhofer diffraction with a normally incident monochromatic plane wave. In each case, the diffracting object is located in the plane, and the complex amplitude of the incident plane wave is given by where is the magnitude of the wave disturbance, is the wavelength, is the velocity of light, is the time = is the wave number and the phase is zero at time . The time dependent factor is omitted throughout the calculations, as it remains constant, and is averaged out when the intensity is calculated. The intensity at is proportional to the amplitude times its complex conjugate These derivations can be found in most standard optics books, in slightly different forms using varying notations. A reference is given for each of the systems modelled here. The Fourier transforms used can be found here. Narrow rectangular slit The aperture is a slit of width which is located along the -axis, Solution by integration Assuming the centre of the slit is located at , the first equation above, for all values of , is: Using Euler's formula, this can be simplified to: where . The sinc function is sometimes defined as and this may cause confusion when looking at derivations in different texts. This can also be written as: where is the angle between z-axis and the line joining to the origin and when . Fourier transform solution The slit can be represented by the rect function as: The Fourier transform of this function is given by where is the Fourier