15.8 Diffraction Grating

15.8 Diffraction Grating

The property of bending of light around obstacles and spreading of light waves into the geometrical shadow of an obstacle is called diffraction.

If we increase 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 narrower, than with two slits. Because these maxima are so narrow, their angular position, and hence the wavelength, can be measured to very high precision. As we'll see, this effect has many important applications.

(Fig. 15.16: Diffraction Grating)

An array of a large number of parallel slits, all with the same width $d$ 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.

(Fig. 15.17: Orders of Pattern)

Fig. 15.16 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 five 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.

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.

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 this formula. The $n = \pm 1$ lines are called the first-order lines, the $n = \pm 2$ lines the second-order lines, and so on. If the grating is illuminated by white light with a continuous distribution of wavelengths, each value of $n$ corresponds to a continuous spectrum in the pattern. The angle for each wavelength is determined by Eq. (15.7) for a given value of $n$, 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 small angles (blue end of the spectrum).

As Eq. (15.7) shows, the sines of the deviation angles of the maxima are proportional to the ratio $\lambda/d$. For substantial deviation to occur, the grating spacing $d$ should be of the same order of magnitude as the wavelength $\lambda$. Gratings for use with visible light ($\lambda$ from $400$ to $700\,\mathrm{nm}$) 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 $1000\,\mathrm{nm}$.