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Diffraction Grating

 
[Gratings]

In optics, grating is a component with a surface covered by regular pattern, often just parallel lines. For practical applications, most gratings have grooves on their surface. Such gratings can be either transparent or reflective. Gratings can be flat or concave addionally they may be blazed or not, depending on the groove profile. For a given grating, light with a larger wavelength generally has a larger diffraction angle. Because of their ability of splitting light into different wavelengths, gratings are commonly used in monochromators and polychromtors. Gratings are characterised by their resolving power and angular dispersion.
The resolving power is given by

  Eq_opt_grating_1

where 'l'  is the wavelength, dl  is the difference in wavelength between two spectral lines of equal intensity, 'N'  stands for the total number of grooves on the grating and 'k'  is the diffraction order. Resolution is then the ability of the instrument to separate adjacent spectral lines. Two peaks are considered resolved if the distance between them is such that the maximum of one falls on the first minimum of the other. This is called the Rayleigh criterion. It may be shown that  

Eq_opt_grating_6

 with 'W'  as the illuminated width of the grating, 'a'  and 'b'  incident and exiting angles (angles of incidence and of diffraction). Consequently, the resolving power of a grating is dependent on the width of the grating, the centre wavelength to be resolved and the geometry of the used conditions. The numerical resolving power 'R'  should not be confused with the resolution or bandpass of an instrument system. Angular dispersion is the variation of the dispersion angle with wavelength and is defined as follows:

  Eq_opt_grating_11


In principle a diffraction grating is only marginally more complicated than Young's experiment.

A reflection diffraction grating consists of a flat or curved surface with a series of regular grooves. In holographic gratings the grooves in a master grating are formed by first exposing a photosensitive material on the surface to the pattern formed by the interference of light and then etching the surface.

Many thin slits separated by 'a' .
equation
 with 'n' the groove densitiy or the number of grooves/length unit.

Path difference:
equation


is In phase; and therefore leading to constuctive interference if equation
equation equation

Dispersion - Grating

This means that for a given diffraction angle i', constructive interference can occur for wavelengths: l, l/2, l/3, etc., corresponding to orders 1, 2, 3, etc. The spectrum produced by a grating is therefore complex, as it consists of several superimposed orders.

Further, there is a zero order, which is not scattered. For k = 0, the grating equation requires the angle of reflection to be equal and opposite to the angle of incidence. In zero order, the grating behaves as a mirror. Order zero represents about 40% of the total energy.

The rest of the energy is distributed amongst the various orders. Generally, the higher the order, the lower the brightness of its spectrum. The highest orders carry almost no energy. In practice, only the first and second orders are usable.

Further reading:

  • R Guenther, Modern Optics, John Wiley & Sons, New York (1990);
  • H Haken, Light, Vol. 1, Waves, Photons, Atoms, North Holland, Amsterdam (1981)
  • M Garbuny, Optical Physics, Academic Press, New York (1965).

First published on the web: 15 February 2000 by Richard Payling

revised and extended 19.11.2007

Authors: Aranka Derzsi and Giovanni Lotito. The text is based on a lecture given by Thomas Nelis at the first Gladnet training course in Antwerp Sept. 2007