Grating, diffraction resolving power

Apart from using the grating in zero order so that it works like a mirror, maintaining a (compact) round beam at the camera (A = 1) requires that the spectrograph adopt the Littrow configuration for which Tf = 0 and the incident and diffracted rays are parallel. From equations 6 and 9, the resolving power at blaze in the Littrow configuration is... 

Thus, dispersion of the grating increases as d decreases (i.e., as the grating contains more lines per cm). Also, dispersion is not a function of k, and the linear dispersion is therefore a constant, unlike in the case of a prism. The resolving power of a diffraction grating is proportional to the size of the grating and the order of the diffraction used. 

U Use Equation 7-13 for the resolving power of a grating monochromator to estimate the theoretical minimum size of a diffraction grating that would provide a profile of an atomic absorption line at 500 nm having a line width of 0.002 nm. Assume that the grating is to be used in the first order and that it has been ruled at 2400 grooves/mm. 

Here d is the line separation and a and p the angles of incidence and reflection, respectively. The resolving power of the grating is determined by the total number of illuminated lines N and by the diffraction order m, i.e.. 

Lord Rayleigh has somewhat arbitrary introduced a criterion of resolution in connection with prism and grating spectrometers, where the line profiles at the maximum attainable resolution are determined by diffraction and are of the form I(x - Xq) = I(XQ)[sin((x - Xq)/2)(x - Xq)/2] (see below). In this case two lines are considered to be just resolved if the central diffraction maximum of Ij(x - x ) coincides with the first minimum of l2(x - X2) [4.5a]. As shown below, the total intensity distribution I(X) exhibits in this case for I = I2 a dip between the two maxima which drops to (8/tt ) of the maximum intensity 1 (Fig.4.6). Generalizing this Rayleigh criterion to arbitrary line profiles, we may define the resolving power for any dispersing instrument by defining two lines with equal intensities to be just resolved if the dip between the two maxima drops to 2... 

The resolving power of a grating is proportional to the product Nm of the number of grooves times the diffraction order m (see Sect.4.1). The more grooves are hit by the laser beam, the better is the resolution and the smaller is the resulting laser linewidth. 

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