The diffraction grating, a useful device for analyzing light sources, consists of a large number of equally spaced parallel slits. A transmission grating can be made by cutting parallel lines on a glass plate with a precision ruling machine. The spaces between the lines are transparent to the light and hence act as separate slits. A reflection grating can be made by cutting parallel lines on the surface of a reflective material. The reflection of light from the spaces between the lines is specular, and the reflection from the lines cut into the material is diffuse. Thus, the spaces between the lines act as parallel sources of reflected light, like the slits in a transmission grating. Gratings that have many lines very close to each other can have very small slit spacings. For example, a grating ruled with 5 000 lines/cm has a slit spacing d = (1/5 000) cm = 2.00 × 10^{-4} cm.
Figure 38.18 Side view of a diffraction grating. The slit separation is d, and the path difference between adjacent slits is d sin θ.

A section of a diffraction grating is illustrated in Figure 38.18. A plane wave is incident from the left, normal to the plane of the grating. A converging lens brings the rays together at point P. The pattern observed on the screen is the result of the combined effects of interference and diffraction. Each slit produces diffraction, and the diffracted beams interfere with one another to produce the final pattern.

The waves from all slits are in phase as they leave the slits. However, for some arbitrary direction θ measured from the horizontal, the waves must travel different path lengths before reaching point P. From Figure 38.18, note that the path difference δ between rays from any two adjacent slits is equal to d sin θ. If this path difference equals one wavelength or some integral multiple of a wavelength, then waves from all slits are in phase at point P and a bright fringe is observed. Therefore, the condition for maxima in the interference pattern at the angle θ is
We can use this expression to calculate the wavelength if we know the grating spacing and the angle θ. If the incident radiation contains several wavelengths, the mth-order maximum for each wavelength occurs at a specific angle. All wavelengths are seen at θ = 0, corresponding to m = 0, the zeroth-order maximum. The first-order maximum (m = 1) is observed at an angle that satisfies the relationship sin θ = λ/d; the second-order maximum (m = 2) is observed at a larger angle θ, and so on.
The intensity distribution for a diffraction grating obtained with the use of a monochromatic source is shown in Figure 38.19. Note the sharpness of the principal maxima and the broadness of the dark areas. This is in contrast to the broad bright fringes characteristic of the two-slit interference pattern (see Fig. 37.6). Because the principal maxima are so sharp, they are very much brighter than two-slit interference maxima. The reason for this is illustrated in Figure 38.20, in which the combination of multiple wave fronts for a ten-slit grating is compared with the wave fronts for a two-slit system. Actual gratings have thousands of times more slits, and therefore the maxima are even stronger.
A schematic drawing of a simple apparatus used to measure angles in a diffraction pattern is shown in Figure 38.21. This apparatus is a diffraction grating spectrometer. The light to be analyzed passes through a slit, and a collimated beam of light is incident on the grating. The diffracted light leaves the grating at angles that satisfy Equation 38.10, and a telescope is used to view the image of the slit. The wavelength can be determined by measuring the precise angles at which the images of the slit appear for the various orders.
Figure 38.21 Diagram of a diffraction grating spectrometer. The collimated beam incident on the grating is diffracted into the various orders at the angles θ that satisfy the equation d sin θ = mλ, where m = 0, 1, 2, . . . .

_{Resolving power of the diffraction grating}

The diffraction grating is most useful for measuring wavelengths accurately. Like the prism, the diffraction grating can be used to disperse a spectrum into its wavelength components. Of the two devices, the grating is the more precise if one wants to distinguish two closely spaced wavelengths. For two nearly equal wavelengths λ_{1} and λ_{2} between which a diffraction grating can just barely distinguish, the resolving power R of the grating is defined as
Where λ = (λ_{1} + λ_{2}) / 2 and Δλ= λ_{2} - λ_{1}.Thus, a grating that has a high resolving power can distinguish small differences in wavelength. If N lines of the grating are illuminated, it can be shown that the resolving power in the mth-order diffraction is
Thus, resolving power increases with increasing order number and with increasing number of illuminated slits.

Note that R = 0 for m = 0; this signifies that all wavelengths are indistinguishable for the zeroth-order maximum. However, consider the second-order diffraction pattern (m = 2) of a grating that has 5 000 rulings illuminated by the light source. The resolving power of such a grating in second order is R = 5 000 × 2 = 10 000. Therefore, for a mean wavelength of, for example, 600 nm, the minimum wavelength separation between two spectral lines that can be just resolved is Δλ = λ/R = 6.00 × 10^{-2} nm. For the third-order principal maximum, R =15 000 and Δλ = 4.00 × 10^{-2} and so on.

One of the most interesting applications of diffraction is holography, which is used to create three-dimensional images found practically everywhere, from credit cards to postage stamps.

# IIT JEE Physics Study Material

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