**Diffraction Gratings**

One of the most useful tools in the study of light and of objects that emit and absorb light is the diffraction grating. This device is somewhat like the double-slit arrangement of but has a much greater number N of slits, often called rulings, perhaps as many as several thousand per millimeter. An idealized grating consisting of only five is represented in When monochromatic light is sent through the slits, it forms narrow interference fringes that can be analyzed to determine the wavelength of the light. (Diffraction gratings can also be opaque surfaces with narrow parallel grooves arranged like the slits in Light then scatters .

back from the grooves to form interference fringes rather than being transmit through open slits.) With monochromatic light incident on a diffraction grating, if we graduig increase the number of slits from two to a large number N, the intensity plot change from the typical double-slit plot of to a much more complicated one

then eventually to a simple graph like that shown in . The pattern would see on a viewing screen using monochromatic red light from, say, a helium neon laser, is shown in b. The maxima are now very narrow (and so called lines); they are separated by relatively wide dark regions.We use a familiar procedure to find the locations of the bright lines on viewing screen. We first assume that the screen is far enough from the gratin that the rays reaching a particular point P on the screen are approximately par hen they leave the grating . Then we apply to each pair of rulings the same reasoning we used for double-slit interference. The separati between rulings is called the grating spacing. (If N rulings occupy a total widt then d = wIN.) The path length difference between adjacent rays is again d , where (J is the angle from the central axis of the grating (and 0

diffraction pattern) to point P. A line will be located at P if the path length differ between adjacent rays is an integer number of wavelengths-that is, if

d sin (J = mA, for m = 0, I, 2, …

where A is the wavelength of the light. Each integer m represents a different hence these integers can be used to label the lines. as in The integer then called the order numbers. and the lines are called the zeroth-order line central line. with m = 0). the first order line, the second-order line, and so on.If we rewrite (J = sin-I(mA/d), we see that, for a given diffract grating, the angle from the central axis to any line (say, the third-order line) on the wavelength of the light being used. Thus, when light of an unknown length is sent through a diffraction grating. measurements of the angles to the hi order lines can be used in to determine the wavelength. Even Ii several unknown wavelengths can be distinguished and identified in this way cannot do that with the double-slit arrangement of Section even though same equation and wavelength dependence apply there. In double-slit interface the bright fringes due to different wavelengths overlap too much to be disting

larger angles) onto a focal plane FF’ within the telescope. When we look through eyepiece E, we see a magnified view of this focused image. By changing the angle ()of the telescope, we can examine the entire diffraction patten. For any order number other than m = 0, the original light is spread out according to wavelength (or color) so that we can determine, with E. just what wavelengths are being emitted by the source. If the source emits discrete wavelengths, what we see as we rotate the telescope horizontally through the angles corresponding to an order m is a vertical line of color for each wavelength, with the shorter-wavelength line at a smaller angle () than the longer-wavelength line. For example, the light emitted by a hydrogen lamp, which contains hydrogen gas, has four discrete wavelengths in the visible range. If our eyes intercept this light directly, it appears to be white. If, instead, we view it through a grating spectroscope, we can distinguish, in several orders, the lines of the four colors corresponding to

these visible wavelengths. (Such lines are called emission lines.) F our orders are represented in . In the central order (m = 0). the lines corresponding to all four wavelengths are superimposed. giving a single white line at () = O. The colors are separated in the higher orders. The third order is not shown in 2 for the sake of clarity; it actually overlaps the second and fourth orders. The fourth-order red line is missing because it is not formed by the grating used here. That is, when we attempt to solve Eq. 37- 22 for the angle () for the red wavelength when m = 4, we find that sin () is greater than unity. which is not possible. The fourth order is then said to be incomplete for

this grating: it might not be incomplete for a grating with greater spacing d, which will spread the lines less than in . is a photograph of the visible emission lines produced by cadmium.

**CHECKPOINT**

The figure shows lines of different orders produced by·a diffraction grating in monochromatic red light. (a) Is the center of the pattern to the left or right? (b) If we switch to monochromatic green light, will the half-widths of the then produced in the same orders be greater than, less than. or the same as the half-widths of the l shown?