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The diffraction grating is an optical component used to spatially separate polychromatic light (white light) into its constituent optical frequencies. The simple grating consists of glass substrate with a series of parallel, equispaced lines on the front surface of the glass. Diffraction gratings are used in such diverse fields as spectroscopy, colorimetry, metrology and laser optics. The following technical discussion is intended to assist the potential customer in selecting the grating for their application.
The performance of a simple diffraction grating can best be shown with reference to Fig. 1. Notice that the optical beam enters the periodic pattern (spatial fringe pattern) with a particular angle of incidence. The beam then separates into one or more "orders" according to the grating equation:
d (sin a ± sin b) =ml
where
a = angle of incidence
b= angle of diffraction
d = distance between adjacent grooves,
m= order (integers 1,2,3 etc.)
l = wavelength of the incident beam
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Three characteristics of the simple diffraction grating stand out. As Fig. 1 shows, the zero order is not diffracted and therefore continues undisturbed (but has some loss of power). Also, for a given wavelength, the amount of beam turning is a function of the groove period and of the angle of incidence. Finally, no real diffracted beam exists when the wavelength is greater than twice the groove period. However, when the grove period is large compared to the wavelength many orders can exist. Indeed, echelle gratings often operate over hundreds of orders.
It is clear from the grating equation that the condition for the formation of a diffracted order depends on the wavelength of the incident light. To consider the formation of a spectrum we need to know how the angle of diffraction varies with the incident wavelength. This is found by differentiating the equation with respect to b, assuming that the angle of incidence is fixed:
db/dl = m/dcosb
The quantity db/dl is the change of the diffraction angle corresponding to a small change of wavelength. This is known as the angular dispersion on the grating. The linear dispersion of a grating is the product of this term and the effective focal length of the system.
For example: Calculation of the reciprocal linear dispersion at 500 nm wavelength of a grating system employing a 1200 groove/mm grating (period = d = 0.833 mm), 200 mm focal length imaging mirror, and operating in the first order with an angle of incidence of 25 degrees.
From the grating equation b = 10.23 degrees db/dl = 0.00122 rad/nm.
Linear dispersion=focal length x db /dl=0.2438mm/nm; reciprocal linear dispersion = 4.1 nm/mm
A particularly useful grating configuration is that of the "Littrow" case in which the diffracted beam returns along the incident beam. In this case a = b so that the grating equation reduces to: 2dsinb=ml
and the angular dispersion becomes:
db/dl = 2(tanb)/l
Since light of different wavelengths are diffracted at different angles , each order is drawn out into a spectrum. Each spectrum is composed of monochromatic images of the. incident bundle, with the blue image nearer to the central axis. However if monochromatic light is incident on the grating, several output beams will be generated. This type of device can be used for the generation of multiple lasers (i.e. a beamsplitter)
The number of orders that can be produced by a given grating is limited by the grating constant d because it cannot exceed 90 degrees. The highest order is given by d/l. Consequently a coarse grating (with large d) produces many orders while a fine grating may produce only one or two.
The free spectral range of a diffraction grating is defined as the largest bandwidth in a given order which does not overlap the same bandwidth in an adjacent order.
Referring to Fig .2, if l1 and l2 are the extremes of the spectrum band then overlapping will occur at the long wavelength end of the spectrum when l2 in order m is diffracted at the same angle as in order m+1. Conversely, overlapping will occur at the short wavelength end when l1 in order m coincides with l2 order m- 1. To avoid overlapping, the required conditions is:
l2-l1>=l1/m or l2-l1>=l2/(m-1)
however, since l1 <l2, we may say that the free spectral range is equal to the shortest wavelength in the allowed bandwidth divided by the order number.
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The various types of diffraction gratings have different efficiencies versus wavelength characteristics. Classical ruled gratings usually peak with very high efficiency at a certain wavelength and become rapidly less efficient as you deviate from that wavelength. Blazed holographic gratings have similar properties. On the other hand, standard holographic gratings have very little variation in efficiency over the spectral range.
Classically ruled gratings, those ruled mechanically by cutting grove after grove into a substrate with a diamond tool, are characterized by a blaze wavelength and a groove density. The blaze wavelength is the wavelength at which the grating is at its maximum efficiency. Groove density is the number of grooves per mm on the grating surface. The useful range of a grating can be described by the 2/3 - 3/2 rule which simply states that the range of a grating is approximately lower limit 2/3 lBlase; upper limit 3/2lBlase Thus for a grating with a blaze of 400 nm, its useful range is 266 to 600 nm. It is not unusual to be able to operate the grating with reasonable efficiency above the "magic" 3/2 value. However, this is not suggested on the short wavelength side below the "magic" 2/3 value.
