Diffraction

from his 1807 Lectures The effects of diffraction of light were first carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction, from the Latin diffringere, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in 1665. Isaac Newton studied these effects and attributed them to inflexion of light rays. James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating to be discovered. Thomas Young developed the first wave treatment of diffraction in 1800. In his model Young proposed that the fringes observed behind an illuminated sharp edge arose from interference between the direct transmitted plane wave and a cylindrical wave that appears to emitted from the edge. Augustin-Jean Fresnel revisited the problem and devised an alternative wave theory based on Huygens' principle. In this model, point sources of light are distributed up to the diffraction edge but not in the barrier. These point sources are driven by the incoming plane wave and they interfere beyond the barrier. Fresnel developed a mathematical treatment from his approach and Young's model was initially considered incorrect. Later work showed that Young's more physical approach is equivalent to Fresnel's mathematical one. In 1859 Hermann von Helmholtz and later in 1882 Gustav Kirchhoff developed integral equations for diffraction based on the concepts proposed by Fresnel as well as approximations needed to apply them. In general, all these approaches require formulating the problem in terms of virtual sources. Cases like those with an absorbing barrier require methods developed in the 1940s based on transverse amplitude diffusion. ==Basics==

Diffraction is a general phenomenon of waves, occurring whenever a wave encounters some form of obstruction. The obstruction may be solidly blocking the wave, or transparent and shifting the phase of the wave without any change in energy (elastic scattering). The waves beyond the obstacle interfere leading to a diffraction pattern. The diffraction pattern cannot be predicted by the straight line trajectories of geometrical optics. For a simple example, a strong light source blocked by a solid object does not show a crisp dark shadow when examined carefully. the Fraunhofer diffraction approximation of the Kirchhoff equation (applicable to the far field), as first detailed by Hans Bethe. The Fraunhofer and Fresnel limits exist for these as well, although they correspond more to approximations for the matter wave Green's function (propagator) for the Schrödinger equation. Multiple scattering models are required in many types of electron diffraction; in some cases similar dynamical diffraction models are also used for X-rays. The simplest descriptions of diffraction are those in which the situation can be reduced to a two-dimensional problem. For water waves, this is already the case; water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes, we have to take into account the full three-dimensional nature of the problem. File:Square diffraction.jpg|Diffraction pattern from a square aperture File:Two-Slit Diffraction.png|Interference pattern from two-slit diffraction File:Doubleslit.gif|Two-slit diffraction File:Optical diffraction pattern ( laser), (analogous to X-ray crystallography).JPG|Optical diffraction pattern File:Diffraction pattern in spiderweb.JPG|Colors in a spider web due to diffraction, ==Occurrence==

The effects of diffraction are often seen in everyday life. The most commonly encountered examples of diffraction are those that involve light; for example, the closely spaced tracks on a CD or DVD which act as a diffraction grating to form the familiar rainbow pattern seen when looking at a disc. This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the hologram on a credit card is an example. File:Screendiffraction.jpg|Pixels on smart phone screen acting as a diffraction grating File:Sunlight diffraction off of cd rom.jpg|Data as pits on CDs act as diffracting elements File:Lunar Halo .jpg|Lunar corona File:IMG 7474 solar glory.JPG|A solar glory, as seen from a plane Diffraction in the atmosphere by small particles can cause a corona—a bright disc and rings around a light source such as the Sun or the Moon. At the opposite point, one may observe a glory—bright rings around the shadow of the observer. In contrast to the corona, glory requires the particles to be transparent spheres (like fog droplets), since the backscattering of the light that forms the glory involves refraction and internal reflection within the droplet. Another frequently encountered example is diffraction spikes which are caused by a range of processes including a non‑circular aperture in a camera or by support struts in a telescope; in normal vision, diffraction through eyelashes may produce similar spikes. When deli meat appears iridescent, the effect is caused by diffraction from the meat fibres. While diffraction by light is the most common case encountered, diffraction can occur with any kind of wave, for instance ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, which is why one can still hear someone calling even when hiding behind a tree. Other examples of diffraction are considered in more detail below. ==Different cases==

The diffraction patterns depend upon the nature of the obstacles the wave encounters, both their physical dimensions as well as how the change the phase and/or direction of the wave. The simplest types of obstacles are slits or apertures which block of part of the wave {{gallery |align=center| noborder=yes| File:Diffraction sharp edge.gif|Diffraction on a sharp metallic edge | File:Diffraction softest edge.gif|Diffraction on a soft aperture, with a gradient of conductivity over the image width | Gratings A diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles θm which are given by the grating equation d \left( \sin{\theta_m} \pm \sin{\theta_i} \right) = m \lambda, where \theta_{i} is the angle at which the light is incident, d is the separation of grating elements, and m is an integer which can be positive or negative. The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a convolution of diffraction and interference patterns. The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different. General case for far field A more mathematical approach involves treating the problem as a summation over spherical waves derived from the relevant wave equation; see for instance Born and Wolf for details. The wave that emerges from a point source has an amplitude \psi at location \mathbf r that is given by the solution of the frequency domain wave equation for a point source (the Helmholtz equation), \nabla^2 \psi + k^2 \psi = \delta(\mathbf r), where \delta(\mathbf r) is the 3-dimensional delta function. By direct substitution, the solution to this equation can be shown to be the scalar Green's function, which in the spherical coordinate system (and using the physics time convention e^{-i \omega t}) is \psi(r) = \frac{e^{ikr}}{4 \pi r}.which is a spherical wave emanating from the origin, the mathematical form of Huygen"s wavelets in the Huygens-Fresnel appriach. This solution assumes that the delta function source is located at the origin. If the source is located at an arbitrary source point, denoted by the vector \mathbf r' and the field point is located at the point \mathbf r, then we may represent the scalar Green's function (for arbitrary source location) as \psi(\mathbf r | \mathbf r') = \frac{e^{ik | \mathbf r - \mathbf r' | }}{4 \pi | \mathbf r - \mathbf r' |}. In the far field, where r is large the Green's function simplifies to \psi(\mathbf{r} | \mathbf{r}') = \frac{e^{ik r}}{4 \pi r} e^{-ik ( \mathbf{r}' \cdot \mathbf{\hat{r}})}The expression for the far (Fraunhofer region) wave then becomes \Psi(r)\propto \frac{e^{ik r}}{4 \pi r} \iint\limits_\mathrm{aperture} \!\! E_\mathrm{inc}(x',y') e^{-ik ( \mathbf{r}' \cdot \mathbf{\hat{r}} ) } \, dx' \,dy'. with an electric field E_\mathrm{inc}(x, y) incident on the aperture for the case of an electromagnetic wave. In the far-field / Fraunhofer region, this becomes the spatial Fourier transform of the aperture. Huygens' principle when applied to an aperture simply says that the far-field diffraction pattern is the spatial Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields (see Fourier optics). In the far field, where is essentially constant, then the equation: \Psi = \int_{\mathrm{aperture}} \frac{i}{r\lambda} \Psi^\prime e^{-ikr}\,d\mathrm{aperture}is equivalent to doing a Fourier transform on the gaps in the barrier. Similar forms can be derived for matter and other types of waves. ==Matter wave diffraction==

According to quantum theory every particle exhibits wave properties and can therefore diffract. Diffraction of electrons and neutrons is one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a non-relativistic particle is the de Broglie wavelength \lambda=\frac{h}{p} \, , where h is the Planck constant and p is the momentum of the particle (mass × velocity for slow-moving particles). For example, a sodium atom traveling at about 300 m/s would have a de Broglie wavelength of about 50 picometres. Diffraction of matter waves has been observed for small particles, like electrons, neutrons, atoms, and even large molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic structure of solids, molecules and proteins. ==Bragg diffraction==

, each dot (or reflection) in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure. Diffraction from a large three-dimensional periodic structure such as many thousands of atoms in a crystal is called Bragg diffraction. It is similar to what occurs when waves are scattered from a diffraction grating. Bragg diffraction is a consequence of interference between waves reflecting from many different crystal planes. The condition of constructive interference is given by ''Bragg's law'': m \lambda = 2 d \sin \theta , where \lambda is the wavelength, d is the distance between crystal planes, \theta is the angle of the diffracted wave, and m is an integer known as the order of the diffracted beam. Bragg diffraction may be carried out using either electromagnetic radiation of very short wavelength like X-rays or matter waves like neutrons whose wavelength is on the order of (or much smaller than) the atomic spacing. The pattern produced gives information of the separations of crystallographic planes d, allowing one to deduce the crystal structure. For completeness, Bragg diffraction is a limit for a large number of atoms with X-rays or neutrons, and is rarely valid for electron diffraction or with solid particles in the size range of less than 50 nanometers. ==Importance of coherence==

The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent. The length over which the phase in a beam of light is correlated is called the coherence length. In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence, as it is related to the presence of different frequency components in the wave. In the case of light emitted by an atomic transition, the coherence length is related to the lifetime of the excited state from which the atom made its transition. If waves are emitted from an extended source, this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double-slit experiment, this would mean that if the transverse coherence length is smaller than the spacing between the two slits, the resulting pattern on a screen would look like two single-slit diffraction patterns. ==Main articles involving diffraction==

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