Resolving Power of a Grating

The resolving power of a grating has the same dimensions as for a prism and use i again made of the Rayleigh criterion for resolution. The numerical value of the resolving power is given by X/AX, where AX meets the Rayleigh definition. 

The above formula also assumes use of an infinitely narrow entrance slit, as does the formula for theoretical resolving power of a prism. The relation between theoretical and attainable resolution can be illustrated from Michelson s measurement of the wavelength of the green mercury line at 5641 A. Operating in the sixth order on a 10-in. ruled surface Michelson 

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The resolving power of a grating thus is better in higher orders and for small groove spac-ings. It is also linear in the total number of grooves illuminated i.e., it is proportional to the width of the grating. 

It can be shown that Ihe resolving power of a grating is given by the e.vpression... 

The resolving power of a grating is a measure of its ability to separate to the baseline two adjacent spectral wavelengths X and Xz- The resolution R is expressed as... 

The theoretical resolving power of a grating monochromator is identical to that derived for a prism. Hence the smaller the intergroove distance, the greater the resolution. 

The resolving power of a grating depends on its width, the central wavelength to be resolved, and the geometry of the optical instrument. 

The resolution or chromatic resolving power of a grating describes its ability to separate adjacent spectral lines. Resolution is generally defined as... 

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. 

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. 

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. 

Let us consider the attainable spectral resolving power of a spectrometer. When passing the dispersing element (prism or grating), a parallel beam composed of two monochromatic waves with wavelengths X and X -f AA. is split into two partial beams with the angular deviations 9 and 6> + A6> from their initial direction (Fig. 4.8). The angular separation is... 

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.. 

Let us cite an example to help us judge the equivalence between Fourier and dispersive instruments. A grating spectrometer employing a four-passed 8 x 104-line grating in the first order has a resolving power of 4 x 8 x 104 = 3.2 x 105. At 3200 cm -1 in the near infrared, this instrument has a Rayleigh resolution of 10" 2 cm- L The same resolution can be achieved by a Fourier... 

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