Fraunhofer Diffraction - The Fraunhofer Diffraction Equation

The 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 waves 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 the light source and the viewing plane are effectively at infinity with respect to the diffracting aperture. In this case, the incident light is a plane wave so that the phase of the light at each point in 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.

Strictly speaking, the Fraunhofer approximation only applies when the diffracted pattern is viewed at infinity, but in practice it can be applied in the far field, and also in the focal plane of a positive lens.