Fraunhofer diffraction

In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens. In contrast, the diffraction pattern created near the object, in the near field region, is given by the Fresnel diffraction equation. W 2 L λ ≪ 1 {displaystyle {frac {W^{2}}{Llambda }}ll 1} λ {displaystyle lambda } – wavelength, L {displaystyle L} – distance from the aperture In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens. In contrast, the diffraction pattern created near the object, in the near field region, is given by the Fresnel diffraction equation. The equation was named in honor of Joseph von Fraunhofer although he was not actually involved in the development of the theory. This article explains where the Fraunhofer equation can be applied, and shows the form of the Fraunhofer diffraction pattern for various apertures. A detailed mathematical treatment of Fraunhofer diffraction is given in Fraunhofer diffraction equation. When a beam of light is partly blocked by an obstacle, some of the light is scattered around the object, and light and dark bands are often seen at the edge of the shadow – this effect is known as diffraction. These effects can be modelled using the Huygens–Fresnel principle. Huygens postulated that every point on a primary wavefront acts as a source of spherical secondary wavelets and the sum of these secondary wavelets determines the form of the wave at any subsequent time. Fresnel developed an equation using the Huygens wavelets together with the principle of superposition of waves, which models these diffraction effects quite well. It is not a straightforward matter to calculate the displacement given by the sum of the secondary wavelets, each of which has its own amplitude and phase, since this involves addition of many waves of varying phase and amplitude. When two waves are added together, the total displacement depends on both the amplitude and the phase of the individual waves: two waves of equal amplitude which are in phase give a displacement whose amplitude is double the individual wave amplitudes, while two waves which are in opposite phases give a zero displacement. Generally, a two-dimensional integral over complex variables has to be solved and in many cases, an analytic solution is not available. The Fraunhofer diffraction equation is a simplified version of the Kirchhoff's diffraction formula and it can be used to model the light diffracted when both a light source and a viewing plane (the plane of observation) are effectively at infinity with respect to a diffracting aperture. With the sufficiently distant light source from the aperture, the incident light to the aperture is a plane wave so that the phase of the light at each point on the aperture is the same. The phase of the contributions of the individual wavelets in the aperture varies linearly with position in the aperture, making the calculation of the sum of the contributions relatively straightforward in many cases. With a distant light source from the aperture, the Fraunhofer approximation can be used to model the diffracted pattern on a distant plane of observation from the aperture (far field). Practically it can be applied to the focal plane of a positive lens. When the distance between the aperture and the plane of observation (on which the diffracted pattern is observed) is large enough so that the optical path lengths from edges of the aperture to a point of observation differ much less than the wavelength of the light, then propagation paths for individual wavelets from every point on the aperture to the point of observation can be treated as parallel. This is often known as the far field and is defined as being located at a distance which is significantly greater than W2/λ, where λ is the wavelength and W is the largest dimension in the aperture. The Fraunhofer equation can be used to model the diffraction in this case.