If the breadth of an aperture be a and that of a bar b, (a+b) is called the grating space, and the directions in which the various images of the slit appear are defined by the condition (a+b)sin0=mX where m=o, 1, 2. . . . The central image is represented by m = o, while m= 1, 2 . . . correspond to spectra of the first, sec ond and higher orders. The complete theory, in agreement with observation, shows that in directions other than those in which AB is an integral number of wave-lengths the luminosity is negligible if the number of apertures is large.
The explanation of dispersion by a grating readily follows, since the formula shows that the angle 0 increases or decreases with the wave-length. When composite light falls on the grating, it is clear that the shorter violet waves will give an image closer to the central image than the longer red waves. Thus, with white light, the appearances will be as roughly represented in fig. 14d. The dispersion increases and the intensity decreases with the order of the spectrum. The spectra of the different orders show a certain amount of overlapping, since mX may be given the same value in different ways. For example, X7,000 in the first order will be co incident with X3,5oo in the second order; and X4.000 in the third order would coincide with X6,000 in the second. The overlapping of spectra in part complicates the use of a grating, but in practice one or the other can usually be cut out by a colour filter.
When the light does not fall normally on the wire grating, but is inclined at an angle i to the normal, it is readily found that the condition for the production of a spectrum line of wave-length X is given by (a+b)(sini+ sin0) = mX.
It should be noted that when i=0 the grating is at the position of minimum deviation, the deviation then being (i+0). The condition for coincidences in different orders of the spectrum, as before, is mX=m'X' where m and m' are integers. It will be ob served also that the determination of wave-lengths by the use of a grating merely requires a knowledge of the grating space, and the measurement of angles.
The Reflection Grating.—As for the transmission grating, the angles of incidence and diffraction are connected with the wave length of the line observed by the relation: mX= (a + b) (sini-Fsin0), where i and 0 are each counted + or—according as they are measured in the conventional positive or negative direction from the normal to the grating, and m is similarly counted --I- or— with reference to the directly reflected light, for which m=o. In
using this formula for calculation, it is usually convenient to express the grating space in angstrom units. Thus, in a grating having 14,438 lines per inch (most frequently occurring in Row land gratings), there are 5,684 lines per centimetre, and since an angstrom unit is cm. (a+ b) is 17,600 angstrom units. The formula shows, for example, that at normal incidence (i=o9, the greatest possible wave-length observable in the first order (cor responding with 0=90°) is 17,600A; also represented by 8,800A in the second order, 5,866 in the third order, and so on. Greater wave-lengths, however, can be observed by changing the angle of incidence; thus, if i=3o° (sini=o.5), the limiting wave-length (0 = —9o° , sin 0 = I) will be 35,200A in the first order (m= 7,040A in the fifth order (m= —5), and so on.
Dispersion.—If 0 be the angle of diffraction, dispersion is defined as before by dO/dX, and since for a given position of the grating, the angle of incidence (i) is constant, the grating formula at once gives The equation shows quite clearly how the dispersion varies directly with the order of the spectrum, and inversely as the grating space ; the closer the rulings the greater the dispersion, irrespective of the total number. Again, it will be observed that the dispersion is smallest when 0=o, (cosh = I); i.e., when the spectrum is observed in a direction normal to the plane of the grating. In this position also the dispersion is most nearly uni form throughout the whole spectrum, since it varies with cose, and in the neighbourhood of 0=o this changes very slowly. The spectrum given by a grating, unlike that given by a prism, is accordingly "normal" in so far as cos0 can be considered con stant.
Resolving Power.—As in the case of the prism, the resolvin,g, power of a grating is a theoretical quantity expressing the separat ing power when the slit is indefinitely narrow, and is represented symbolically by R=X/dX. It is given in a simple form by: Rrnt N or, the resolving power is equal to the product of the order number of the spectrum and the total number of rulings. It should be remembered, however, that N in this formula involves the aperture and grating space.