page_title: Fresnel diffraction
section_title: Fresnel diffraction
hierarchical_section_title: Fresnel diffraction / The Fresnel diffraction integral / Fresnel diffraction
caption_attribution_description: English: Fresnel diffraction of circular aperture,plotted with Lommel functionsWith Maple Version 16.2 grid [200,200]
context_page_description: In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near field. It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed from relatively close to the object. In contrast the diffraction pattern in the far field region is given by the Fraunhofer diffraction equation.
  The near field can be specified by the Fresnel number, F of the optical arrangement. When the diffracted wave is considered to be in the near field. However, the validity of the Fresnel diffraction integral is deduced by the approximations derived below. Specifically, the phase terms of third order and higher must be negligible, a condition that may be written as
  where is the maximal angle described by, a and L the same as in the definition of the Fresnel number.
  The multiple Fresnel diffraction at closely spaced periodical ridges causes the specular reflection; this effect can be used for atomic mirrors.
context_section_description: The condition for validity is fairly weak, and it allows all length parameters to take comparable values, provided the aperture is small compared to the path length. For the r in the denominator we go one step further, and approximate it with only the first term, . This is valid in particular if we are interested in the behaviour of the field only in a small area close to the origin, where the values of x and y are much smaller than z. In general, Fresnel diffraction is valid if the Fresnel number is approximately 1.
  For Fresnel diffraction the electric field at point (x, y, z) is then given by :
  This is the Fresnel diffraction integral; it means that, if the Fresnel approximation is valid, the propagating field is a spherical wave, originating at the aperture and moving along z. The integral modulates the amplitude and phase of the spherical wave. Analytical solution of this expression is still only possible in rare cases. For a further simplified case, valid only for much larger distances from the diffraction source, see Fraunhofer diffraction. Unlike Fraunhofer diffraction, Fresnel diffraction accounts for the curvature of the wavefront, in order to correctly calculate the relative phase of interfering waves.
caption: Fresnel diffraction of circular aperture, plotted with Lommel functions
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