Tutorial : reflection grating - spectral dispersion in the angular space
Diffraction by a grating
A grating is substrate whith a periodic succesion of diffractive elements. Thus, an incident light wave is scattered by each diffractive element and the emerging wave is the result of the interference between each scattered wave. A grating can be transmissive or reflective depending if the diffractive elements spread the light in the same sense or in the opposite sense than the incident wave. The most common gratings are plane gratings with a flat surface where the diffractive elements are arranged in parallel lines with a constant distance between them. The simplest diffractive elements are slits with a uniform transmittance. In the case of a plane monochromatic incident wave, the aforesaid intererferences are constructive in discret directions only. Therefore, the emerging wave is the superposition of plane waves with different directions corresponding to the constructive interference. These directions depend among other on the wavelength. Accordingly, a polychromatic plane wave is diffracted in as many directions as wavelengths. In other words, the grating allows to spread spatially the incident wave spectrum. Thats the reason why gratings are very common components in spectrometers. They are also used in many other devices as spectral filters in laser cavities, pulse compressors/stretchers in short pulse lasers amplifiers, etc.
Constructive interference occure when the phase difference Δφ induced between two consecutive elements is a multiple of 2π. The formulas below give the emerging direction id depending on the incident direction i , the wavelength λ (the wave incident wave is supposed plane and monochromatic), the pitch d (space between 2 diffractive elements) and the order k (The order k is the ratio Δφ/2π):
transmissive grating:
reflective grating:
k can be either positive or negative. The order 0 corresponds to the direction of the specular reflexion (reflective grating) or the direction transmitted by the substrate (transmissive grating). Note that depending on the grating efficiency, more or less intensity remains in the 0 order direction.
A more complete analysis of the diffracted wave can be made easily in the case where the diffractive elements are slits with a uniform transmittance. Each slit diffracts the incident wave according to the following formula :
where Φj is the complex amplitude of the slit number j and u is defined as follows :
, ie being the observation angle after diffraction. The complex amplitude Φ of the diffracted wave is obtained by summing the complex amplitudes of the waves diffracted by each single slit : 
.
The normalized intensity of the diffracted wave is then :
where N = 2.n+1 is the total number of slits and dx is the slits width. The normalized intensity is illustrated on the graphic nearby by the blue curve which is enveloped by the red dashed curve corresponding to the relative intensity obtained after diffraction by a single slit. Each pic corresponds to an order of diffraction, the order 0 (at the center of the curve) being in the present case the most efficient. Note that the envelope shape and its position depend on the diffractive elements that can be designed for maximizing the intensity diffracted in a given order.
The curve shows that the wider the grating, the thinner the diffracted order. Indeed, the Full Width Half Maximum (FWHM) is 
where D is the wave width (or the grating width if the wave is larger than the grating). Also, the thinner the slits, the flatter the envelope which means that the intensity is more uniformly distributed among the different orders.
In the general case, one can define the angular dispersion for a certain order k is 
.
One important parameter is the resolvance Res which is defined as the ratio between the wavelength λ and the smallest resolvable wavelength variation dλ.
dλ is defined according the Rayleigh criterion and corresponds to the FWHM (Full Width Half Maximum) of the diffracted pic. Therefore :
. The higher the order, the larger the resolvance. However, low diffraction orders are in general brighter than high diffraction orders.
Another important parameter is the free spectral range of the grating 
, λ being the smallest wavelength of the spectrum. It is the maximum spectral width of a plane wave for avoiding the spatial overlay of the considered diffraction order with the adjacent diffraction orders.
Littrow angle
There is a special angle named Littrow angle for which the angle of incidence is the same than the diffracted angle in the first order. Its value is given in the following formula :
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Gratings are usually used in spectrally tunable external cavity lasers. In these devices, a laser diode with an anti reflexion coating on its front end facet is used as an amplifier. The beam coming from the amplifier is collimated and hits a grating oriented with Littrow angle for a given wavelength λl. The laser cavity is made of the laser diode back end facet and the grating. Only the diffracted wave in the first order at λl is reflected back in the amplifier and can oscillate. by rotating the grating, the incidence angle corresponds to the Littrow angle of another wavelength which can then oscillate. It is then possible to tune the wavelength emitted by rotating the grating.