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Diffraction

Diffraction is the apparent bending and spreading of waves when they meet an obstruction. It can occur with any type of wave, including sound waves, water waves, and electromagnetic waves such as light and radio waves. Diffraction also occurs when any group of waves of a finite size is propagating; for example, a narrow beam of light waves from a laser must, because of diffraction of the beam, eventually diverge into a wider beam at a sufficient distance from the laser. It is the diffraction of "particles," such as electrons, which stood as one of the powerful arguments in favor of quantum mechanics.

Double-slit diffraction

Double-slit diffraction
(red laser light)

2-slit and 5-slit diffraction

The most conceptually simple example of diffraction is double-slit diffraction in which both slits have relatively narrow widths compared to the wavelength of the wave. Suppose, for the sake of visualization, that these are water waves. After passing through the slits, two overlapping patterns of semicircular ripples are formed, as shown in the first figure. Where a crest overlaps with a crest, a double-height crest will be formed; this is constructive interference. Constructive interference also occurs where a trough overlaps another trough. However, when a trough and a crest overlap, they cancel out; the interference is destructive. The second figure shows the result of this process with light waves of a single wavelength originating from a laser. The constructive-interference locations are called maxima, because they have maximum brightness. The destructive-interference locations are the minima. Historically, the first proof that light was a wave phenomenon came from the double-slit experiment of Thomas Young.

Several qualitative observations can be made:

• When the dimensions of the diffracting object are reduced, the angular spacing of the diffraction pattern is increased in inverse proportion. (More precisely, this is true of the sines of the angles.)
• The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to a dimension, d, of the diffracting object.
• When the diffracting object is repeated, the effect is to narrow each maximum, concentrating its energy within a narrower range of angles. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, d, between the center of one slit and the next.

It is mathematically easier to consider the case of far-field or Fraunhofer diffraction, where the diffracting obstruction is many wavelengths distant from the point at which the wave is measured. The more general case is known as near-field or Fresnel diffraction , and involves more complex mathematics. As the observation distance is increased the results predicted by the Fresnel theory converge towards those predicted by the simpler Fraunhofer theory. This article considers far-field diffraction, which is commonly observed in nature.

Quantitatively, the angular positions of the maxima in multiple-slit diffraction are given by the equation

$\sin \theta = \frac{\lambda}{d} m,$

where m is an integer that labels the order of each maximum. This is a form of Bragg's law (see below).

Graph and image

It is also possible to derive exact equations for the intensity of the diffraction pattern as a function of angle. One of the simplest analytic results occurs for single-slit diffraction. From monochromatic waves of wavelength λ incident on a slit of width d, the intensity I of the diffracted waves at an angle θ is given by:

$I(\theta) = {\left[ \operatorname{sinc} \left( \frac{\pi d}{\lambda} \sin \theta \right) \right] }^2$

where the sinc function is given by sinc(x) = sin(x)/x. If the aperture is circular, the pattern is similar to a radially symmetric version of this equation, representing a series of concentric rings surrounding a central Airy disc.

A wave does not have to pass through an aperture to diffract; for example, a beam of light of a finite size also undergoes diffraction and spreads in diameter. This effect limits the minimum size d of spot of light formed at the focus of a lens, known as the diffraction limit:

$d = 2.44 \lambda \frac{f}{D},\,$

where λ is the wavelength of the light, f is the focal length of the lens, and D is the diameter of the beam of light, or (if the beam is filling the lens) the diameter of the lens. (See Rayleigh criterion).

By use of Huygens' principle, it is possible to compute the diffraction pattern of a wave from any arbitrarily shaped aperture. If the pattern is observed at a sufficient distance from the aperture, it will appear as the two-dimensional Fourier transform of the function representing the aperture.

In the case of multiple slits, the resulting intensity pattern becomes:

$I(\theta) = I_0 \left(\frac{\sin\left[\frac{N\pi d}{\lambda}\sin(\theta)\right]}{\sin\left[\frac{\pi d}{\lambda}\sin(\theta)\right]}\right)^2$

where N is the number of slits, d is the slit width, λ is the wavelength of light, and θ is the angle of orientation of the screen.

Diffraction from multiple slits, as described above, is similar to what occurs when waves are scattered from a periodic structure, such as atoms in a crystal or rulings on a diffraction grating. Each scattering center (e.g., each atom) acts as a point source of spherical wavefronts; these wavefronts undergo constructive interference to form a number of diffracted beams. The direction of these beams is described by Bragg's law:

$m \lambda = 2 d \sin \theta,\,$

where λ is the wavelength, d is the distance between scattering centers, θ is the angle of diffraction and m is an integer known as the order of the diffracted beam. Bragg diffraction is used in X-ray crystallography to deduce the structure of a crystal from the angles at which X-rays are diffracted from it. Since the diffraction angle θ is dependent on the wavelength λ, diffaction gratings impart angular dispersion on a beam of light.

The most common demonstration of Bragg diffraction is the spectrum of colors seen reflected from a compact disc: the closely-spaced tracks on the surface of the disc form a diffraction grating, and the individual wavelengths of white light are diffracted at different angles from it, in accordance with Bragg's law.

In addition to diffraction of classical waves, it is also possible, due to wave-particle duality, to observe diffraction of particles such as neutrons or electrons. As the wavelengths of these particle-waves are so small they can be used as probes of the atomic structure of crystals. See electron diffraction and neutron diffraction.

Diffraction of light reflected off one transparent surface, through a narrow air gap, and back into another produces colored bands known as "Newton's bands". These can be seen if fragments of window glass are carefully cleaned, allowed to dry, and used to view the reflection of a light bulb. Newton's bands in a (nearly) monochromatic source like a laser are very noticeably light and dark. See Newton's rings.