Diffraction

Diffraction is a wave phenomenon in which the apparent bending and spreading of waves when they meet an obstruction. Diffraction occurs with electromagnetic waves, such as light and radio waves, and also in sound waves and water 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.

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 single-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.

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

<math> \sin \theta = \frac{\lambda}{d} m </math> ,

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

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:

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

where the sinc function is given by sinc(x) = sin(x)/x. If the aperture is circular, the pattern is given by 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:

<math> d = 2.44 \lambda \frac{f}{D} </math>,

where λ is the wavelength of the light, f is the focal length of the lens, and D is 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 arbitrary shapped aperture. If the pattern is observed at a sufficient distance from the aperture, it will appear as the two-dimentional Fourier transform of the function representing the aperture.

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:

<math> m \lambda = 2 d \sin \theta </math>,

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.

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.