Diffraction
Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Diffraction is the same physical effect as interference, but interference is typically applied to superposition of a few waves and the term diffraction is used when many waves are superposed.
Italian scientist Francesco Maria Grimaldi coined the word diffraction and was the first to record accurate observations of the phenomenon in 1660.
In classical physics, the diffraction phenomenon is described by the Huygens–Fresnel principle that treats each point in a propagating wavefront as a collection of individual spherical wavelets. The characteristic pattern is most pronounced when a wave from a coherent source encounters a slit/aperture that is comparable in size to its wavelength, as shown in the inserted image. This is due to the addition, or interference, of different points on the wavefront that travel by paths of different lengths to the registering surface. If there are multiple closely spaced openings, a complex pattern of varying intensity can result.
These effects also occur when a light wave travels through a medium with a varying refractive index, or when a sound wave travels through a medium with varying acoustic impedance – all waves diffract, including gravitational waves, water waves, and other electromagnetic waves such as X-rays and radio waves. Furthermore, quantum mechanics also demonstrates that matter possesses wave-like properties and, therefore, undergoes diffraction.
History
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 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 mathematical one.
In 1818, supporters of the corpuscular theory of light proposed that the Paris Academy prize question address diffraction, expecting to see the wave theory defeated. When Fresnel's presentation on his new theory based on wave propagation looked like it might take the prize, Siméon Denis Poisson challenged the Fresnel theory by showing that it predicted light in the shadow behind a circular obstruction. Dominique-François-Jean Arago proceeded to demonstrate experimentally that such light is visible, confirming Fresnel's diffraction model.
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 absorbing barrier require methods developed in the 1940s based on transverse amplitude diffusion.
Mechanism
In classical physics, diffraction arises because of how waves propagate; this is described by the Huygens–Fresnel principle and the principle of superposition of waves. The propagation of a wave can be visualized by considering every point on a wavefront as a source for a secondary spherical wave. The wave displacement at any subsequent point is the sum of these secondary waves. It is possible to obtain a qualitative understanding of many diffraction phenomena by considering how the relative phases of the individual secondary wave sources vary, and, in particular, the conditions in which the phase difference equals half a cycle in which case waves will cancel one another out. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima.In quantum mechanics, diffraction is also described in terms of a wave, but the wavefunction represents a probability amplitude which can be squared to give the probability of quantum detection. The light and dark bands are the areas where the quanta are more or less likely to be detected. The wavefunction is determined by the physical surroundings such as slit geometry, screen distance, and initial conditions.
Quantitative models allow the diffracted field to be calculated, including the Kirchhoff diffraction equation, the Fraunhofer diffraction approximation of the Kirchhoff equation, the Fresnel diffraction approximation and the Feynman path integral formulation. Most configurations cannot be solved analytically, but can yield numerical solutions through finite element and boundary element methods. In many cases it is assumed that there is only one scattering event, what is called kinematical diffraction, with an Ewald's sphere construction used to represent that there is no change in energy during the diffraction process. For matter waves a similar but slightly different approach is used based upon a relativistically corrected form of the Schrödinger equation, 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 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 will have to take into account the full three-dimensional nature of the problem.
Examples
The effects of diffraction are often seen in everyday life. The most striking examples of diffraction are those that involve light; for example, the closely spaced tracks on a CD or DVD 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.
Diffraction in the atmosphere by small particles can cause a corona - a bright disc and rings around a bright light source like the sun or the moon. At the opposite point one may also observe glory - bright rings around the shadow of the observer. In contrast to the corona, glory requires the particles to be transparent spheres, since the backscattering of the light that forms the glory involves refraction and internal reflection within the droplet.
A shadow of a solid object, using light from a compact source, shows small fringes near its edges.
Diffraction spikes are diffraction patterns caused due to non-circular aperture in camera or support struts in telescope; In normal vision, diffraction through eyelashes may produce such spikes.
The speckle pattern which is observed when laser light falls on an optically rough surface is also a diffraction phenomenon. When deli meat appears to be iridescent, that is diffraction off the meat fibers. All these effects are a consequence of the fact that light propagates as a wave.
Diffraction can occur with any kind of wave. 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.
Diffraction can also be a concern in some technical applications; it sets a fundamental limit to the resolution of a camera, telescope, or microscope.
Other examples of diffraction are considered below.
Single-slit diffraction
A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity, in accordance with the Huygens–Fresnel principle.An illuminated slit that is wider than a wavelength produces interference effects in the space downstream of the slit. Assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit interference effects can be calculated. The analysis of this system is simplified if we consider light of a single wavelength. If the incident light is coherent, these sources all have the same phase. Light incident at a given point in the space downstream of the slit is made up of contributions from each of these point sources and if the relative phases of these contributions vary by or more, we may expect to find minima and maxima in the diffracted light. Such phase differences are caused by differences in the path lengths over which contributing rays reach the point from the slit.
We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path difference is approximately so that the minimum intensity occurs at an angle given by
where is the width of the slit, is the angle of incidence at which the minimum intensity occurs, and is the wavelength of the light.
A similar argument can be used to show that if we imagine the slit to be divided into four, six, eight parts, etc., minima are obtained at angles given by
where is an integer other than zero.
There is no such simple argument to enable us to find the maxima of the diffraction pattern. The intensity profile can be calculated using the Fraunhofer diffraction equation as
where is the intensity at a given angle, is the intensity at the central maximum which is also a normalization factor of the intensity profile that can be determined by an integration from to and conservation of energy, and which is the unnormalized sinc function.
This analysis applies only to the far field, that is, at a distance much larger than the width of the slit.
From the intensity profile above, if the intensity will have little dependency on hence the wavefront emerging from the slit would resemble a cylindrical wave with azimuthal symmetry; If only would have appreciable intensity, hence the wavefront emerging from the slit would resemble that of geometrical optics.
When the incident angle of the light onto the slit is non-zero, the intensity profile in the Fraunhofer regime becomes:
The choice of plus/minus sign depends on the definition of the incident angle