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Fourier Optics
Published in James S. Walker, Fast Fourier Transforms, 2017
Fraunhofer diffraction theory is essential for understanding the diffraction of X-rays from crystals. X-ray diffraction photographs from crystals are used to determine the underlying crystal structure. For example, the structure of DNA as well as many protein structures have been determined from diffraction patterns. Another important application is Fraunhofer diffraction from diffraction gratings. This diffraction plays a role in physical chemistry, where it is used to identify chemical compounds. It is also used in astronomy, where it provides a way of identifying the chemical constituents of stars.
Diffraction
Published in Vasudevan Lakshminarayanan, Hassen Ghalila, Ahmed Ammar, L. Srinivasa Varadharajan, Understanding Optics with Python, 2018
Vasudevan Lakshminarayanan, Hassen Ghalila, Ahmed Ammar, L. Srinivasa Varadharajan
Fraunhofer diffraction corresponds to the case where the dimensions of the diffracting elements such as slits, diaphragms, etc., are very small and the distances from the light source and observation screen to the plane containing the diffracting element are large. For this part, and unlike the cases of Fresnel diffraction, the distance between any arbitrary point of the diaphragm and any arbitrary point of the observation screen is equal to the distance between both planes.
Diffraction
Published in Myeongkyu Lee, Optics for Materials Scientists, 2019
In the detailed treatment of diffraction, it is customary to distinguish between two general cases. Suppose that a small aperture is illuminated by a plane wave coming from a distant point source, as shown in Figure 5.6. Note that when the point source is located far away from the aperture, the incoming wavefront is nearly planar. If the screen of observation is far away from the aperture, the obtained diffraction pattern (i.e., the intensity distribution on the screen) bears little resemblance to the actual aperture. In this far-field region, moving the screen position changes only the size of the pattern without altering its shape. This is known as Fraunhofer or far-field diffraction. Fraunhofer diffraction occurs when the wavefronts arriving at the aperture and observation screen are effectively planar. This corresponds to the case where the distances from the source to the aperture and from the aperture to the screen are both large enough. In contrast, the diffraction pattern observed near the aperture is governed by Fresnel or near-field diffraction. In the near-field where Fresnel diffraction prevails, the curvature of the wavefront is significant. Therefore, both of the pattern size and shape vary with distance from the aperture. The fringe effects observed around shadows are examples of Fresnel diffraction. It is to be noted that Fresnel diffraction occurs when either the point source or the observation screen is close to the aperture. If the point source is moved toward the aperture, a spherical wave will impinge on the aperture. A Fresnel pattern then exists, even on a distant screen. Of course, there is no sharp boundary separating the near- and far-field regions. When the point source is far away from the aperture or the incident wave is a plane wave, the far-field approximation is valid if the distance R between the aperture and observation screen is sufficiently large so that () R≫a2/λ.
Study of the properties of non-integer order vortex beams at Fraunhofer zone
Published in Journal of Modern Optics, 2019
The well-known class of optical vortex is the LG beam, which has the complex amplitude given by where is the beam waist radius, is the topological charge or the vortex order and is the polar coordinate of a point in the incident plane, , . The complex amplitude on Fraunhofer diffraction plane can be expressed as (16) where is the polar coordinate of a point on the observation plane, represents the Fourier transform. It follows that where is the Hankel transform operator of order , here where is the Bessel function of the first kind.
Fresnel and Fraunhofer diffraction of (l,n)th-mode Laguerre–Gaussian laser beam by a fork-shaped grating
Published in Journal of Modern Optics, 2019
Furthermore, the analytical results are specialized for the following two particular cases: (a) when the incident LG beam has a zeroth radial mode number and azimuthal mode number l and (b) for incident LG beam with zeroth azimuthal mode number and radial mode number n. For the first specialized case, we arrive at results same as those presented in (26, 27) for Fresnel diffraction, i.e. as those given in (25) for Fraunhofer diffraction of (l,n = 0)th-mode LG beam by the FG. However, by the specialization in the second particular case, new results are obtained.
Pattern recognition using binary masks based on the fractional Fourier transform
Published in Journal of Modern Optics, 2018
Esbanyely Garza-Flores, Josué Álvarez-Borrego
The FT corresponds to the Fraunhofer diffraction and the FRFT to the Fresnel diffraction (18). The Fraunhofer diffraction or FT can be obtained at the focal plane of a converging lens. Therefore, the FRFT represents planes different than the focal (or Fourier) plane (19).