FresnelBeugung

Fresnel diffraction, also known as Fresnel diffraction or Fresnel's diffraction, is a type of wave diffraction that occurs when a wave passes through an aperture or around an obstacle. It is named after the French physicist Augustin-Jean Fresnel, who made significant contributions to the understanding of diffraction. Fresnel diffraction is characterized by the presence of both the near field and the far field regions, which are separated by the Fresnel distance. The Fresnel distance is the distance from the aperture at which the phase difference between the wavefronts is π/2 radians, which is approximately 1.12 times the ratio of the aperture's width to the wavelength of the wave.

In the near field region, the diffraction pattern is dominated by the interference of the wavelets emanating from different parts of the aperture. This results in a complex pattern of bright and dark fringes that can be described using the Fresnel diffraction integral. The Fresnel diffraction integral is a mathematical expression that relates the amplitude of the diffracted wave at a point in the near field to the amplitude of the incident wave at the aperture. It is given by:

A(x, y, z) = (1/iλz) ∫∫ U(x', y') exp[ik/2z (x'² + y'²)] exp[ik/2z (x - x')² + (y - y')²] dx' dy'

where A(x, y, z) is the amplitude of the diffracted wave at the point (x, y, z), U(x', y') is the amplitude of the incident wave at the point (x', y') on the aperture, λ is the wavelength of the wave, k is the wavenumber, and i is the imaginary unit.

In the far field region, the diffraction pattern is dominated by the interference of the wavelets emanating from the aperture as a whole. This results in a simpler pattern of bright and dark fringes that can be described using the Fraunhofer diffraction integral. The Fraunhofer diffraction integral is a mathematical expression that relates the amplitude of the diffracted wave at a point in the far field to the amplitude of the incident wave at the aperture. It is given by:

A(x, y, z) = (1/iλz) ∫∫ U(x', y') exp[ik/2z (x'² + y'²)] exp[ik/2z (x - x')² + (y - y')²] dx' dy'

where A(x, y, z) is the amplitude of the diffracted wave at the point (x, y, z), U(x', y') is the amplitude of the incident wave at the point (x', y') on the aperture, λ is the wavelength of the wave, k is the wavenumber, and i is the imaginary unit.

Fresnel diffraction is an important phenomenon in many areas of physics and engineering, including optics, acoustics, and radio wave propagation. It is used in a wide range of applications, such as the design of lenses, the analysis of diffraction gratings, and the study of the propagation of electromagnetic waves in the atmosphere.