Grating

A repetitive array of diffracting elements, either apertures or obstacles, that has the effect of producing periodic alterations in the phase, amplitude, or both, of an emergent wave is said to be a diffraction grating [1]. One of the simplest of such arrangements is the multiple-slit configuration.

Figure 3: A transmission grating [1]

Figure 4: A reflection grating [1]

If one looks perpendicularly through a transmission grating at a distant parallel line source, the eye would serve as a focusing lens for the diffraction pattern. Recall the analysis of Section 1.3 and the expression

$$a \sin{\theta_{\mathrm{m}}}=m \lambda \quad \quad m=0,\pm1,\pm2,\pm3,\ldots$$ (12)

which is formally the same as equation 11 but now describes angular location of maxima and is known as the grating equation for normal incidence. The values of $m$ specify the order of the various principal maxima. Notice that the smaller $a$ becomes in Equation 12, the fewer will be the number of visible orders. It should be no surprise that the grating equation is in fact the equation which describes the location of the maxima in Young´s double-slit setup. The interference maxima, all located at the same angles, are now simply sharper. In the double-slit case where the point of observation is somewhat off from the exact center of an irradiance maximum, the two waves, one from each slit, will still be more or less in phase, and the irradiance, though reduced, will still be appreciable. Thus the bright regions are fairly broad. By contrast, with multiple-beam systems, although all the waves interfere constructively at the centers of the maxima, even a small displacement will cause certain waves to arrive out of phase by $\lambda/2$ with respect to others. For example, suppose $P$ is slightly off from $\theta_{1}$ so that $a \sin{\theta}=1.010\lambda$ instead of $1.000\lambda$ Each of the waves from successive slits will arrive at $P$ shifted by $0.01\lambda$ with respect to the previous one. Then 50 slits down from the first, the path length will have shifted by $\lambda/2$ and the light from slit 1 and slit 51 will essentially cancel. The same would be true for slit-pairs 2 and 52, 3 and 53, and so forth. The result is a rapid fall off in irradiance beyond the centers of the maxima. Consider next the somewhat more general situation of oblique incidence, as depicted in Figs 3 and 4 The grating equation, for both transmission and reflection, becomes

$$a(\sin{\theta_{\mathrm{m}}}-\sin{\theta_{\mathrm{i}}}) =m \lambda$$ (13)

This expression applies equally well, regardless of the refractive index of the transmission grating itself.

Figure 5: Part of reflection blazed phase grating [1]

In an article in the Encyclopaedia Britannica of 1888 Lord Rayleigh suggested that it was at least theoretically possible to shift energy out of the useless zeroth order into one of the higher-order spectra. So motivated, Robert Williams Wood (1868-1955) succeeded in 1910 to rule grooves with a controlled shape, as shown in Fig. 5. Most modern gratings are of this specially shaped or so called blazed variety [1].