We have just seen that increasing the number of slits in an interference experiment (while keeping the spacing of adjacent slits constant) gives interference patterns in which the maxima are in the same positions, but progressively sharper and narrower, than with two slits. Because these maxima are so sharp, their angular position, and hence the wavelength, can be measured to very high precision. An array of a large number of parallel slits, all with the same width a and spaced equal distances d between centers, is called a diffraction grating. The first one was constructed by Fraunhofer using fine wires. Gratings can be made by using a diamond point to scratch many equally spaced grooves on a glass or metal surface, or by photographic reduction of a pattern of black and white stripes on paper. For a grating, what we have been calling slits are often called rulings or lines. In Fig. 38-14, GG’ is a cross section of a transmission grating;·the slits are perpendicular to the plane of the page, and an interference pattern is formed by the light that is transmitted through the slits. The diagram shows only six slits; an actual grating may contain several thousand. The spacing d between centers of adjacent slits is called the grating spacing. A plane monochromatic wave is incident normally on the grating from the left side. We assume far-field (Fraunhofer) conditions; that is, the pattern is formed
on a screen that is far enough away that all rays emerging from the grating and going to a particular point on the screen can be considered to be parallel. We found in Section 38-5 that the principal intensity maxima with multiple slits occur in the same directions as for the two-slit pattern. These are the directions for which the path difference for adjacent slits is an integer number of wavelengths. So the positions of the maxima are once again given by.
Intensity patterns for two, eight, and sixteen slits are displayed in Fig. 38-13 (Section 38-5), showing the progressive increase in sharpness of the maxima as the number of slits increases. When a grating containing hundreds or thousands of slits is illuminated by a beam of parallel rays of monochromatic light, the pattern is a series of very sharp lines at angles determined by Eq. (38-13). The m = ±llines are called the first-order lines, the m =±2lines the second-order lines, and so on. If the grating is illuminated by white light with a continuous distribution of wavelengths, each value of m corresponds to a continuous spectrum in the pattern. The angle for each wavelength is determined by Eq. (38-13); for a given value of m, long wavelengths (the red end of the spectrum) lie at larger angles (that is, are deviated more from the straight-ahead direction) than do the shorter wavelengths at the violet end of the spectrum. As Eq. (38-13) shows, the sines of the deviation angles of the maxima are proportional to the ratio AJd. For substantial deviation to occur, the grating spacing d should be of the same order of magnitude as the wavelength A.. Gratings for use with visible light (A.from 400 to 700 nrn) usually have about 1000 slits per millimeter; the value of d is the reciprocal of the number of slits per unit length, so d is of the order of I~ mrn = 1000 nm. In a reflection grating, the array of equally spaced slits shown in Fig. 38-14 is replaced by an array of equally spaced ridges or grooves on a reflective screen. The spectrum is always greater than the smallest angle (at the violet end) for the third-order spectrum, so the second and third orders always over.