THE LITTLE PIGGY

Before kids had toys like Packman, computers and laser guns, they amused themselves by simple things like a hand mirror reflecting the sun into the shade and watching the bright spot dash around, preferably ending up in a distant pair of eyes. The bright spot, I am assured by friends who grew up in America, does not have a special name in English, but I will need it to explain a phased array, so I will make do with what Czech kids call it: "the piggy."

As explained last month, two waves, depending on their timing or phase, may result in the sum of the two, or they may cancel each other, or they may result in a wave with an intensity lying between these two extremes. For some time now we have had the benefit of an instrument that can reproduce incoming light waves, but giving them phases running the entire gamut from "in phase" to "antiphase [AtE Feb 87], corresponding to the two extremes of straight addition and mutual cancellation, respectively.

That fascinating device is called a mirror.

Why is it that, for a given position of the mirror, the piggy appears in only one spot? Why does the mirror not diffuse the incoming sunlight all over the place (as a rough surface would)?

Because, you may say, a mirror reflects incoming rays so that the angle of reflection equals the angle of incidence.

But that is only saying the same thing in more teamed words; why do mirrors act that way.?

Let's take a detailed look at what is going on. Ignore the glass, which is merely a mechanical carrier of the silver backing. That backing is a conductor and full of free electrons. The incoming light wave "excites' these electrons in much the same way as an incoming sea wave "excites" an empty bottle floating on the surface -- meaning that it makes it bop up and down.

But unlike a bottle, an electron bopping up and down becomes a transmitter of an electromagnetic wave; it re-radiates the incoming wave. It reradiates it not only with the same frequency (in this case, the frequency of the incident light), but also with the same phase. But that phase depends on its location, as shown on the next page. The row of circles is a row of bottles on the water (or electrons on the surface of the mirror silvering) and the waves (of water or light) are coming in at an angle from the left. The three short lines represent the crest of an incoming wave. That crest does not hit all the bottles at the same time: the bottle on the left end bops up first, the others follow successively as the wave crest reaches them.

From bottles to electrons. As each electron oscillates under the incoming wave, it will radiate in all directions; and it will radiate a crest when it is on the crest of the incoming wave. But now look at what happens to the combined waves from all these electrons. In a general direction, such as the one marked A, you get a jumble of

waves and no mutual reinforcement. The electron on the left radiated its crest earlier, the one on the right later; the crest (the three short lines) gets broken up, the waves are no longer in phase, and the result is a very weak wave, or even none at all, if the waves all cancel. But there is a special direction: the direction toward P, the one where the angle of reradiation equals the angle of incidence. Here, too, the leftmost electron gets the crest first and the other ones later; but its reradiated wave also has further to go toward a distant point, and the two effects will exactly cancel in that "direction of reflection." The crests of the individual waves will line up exactly, all the waves will be in phase, they will reinforce each other, and in that one direction the aggregate wave will be the simple sum of all the components, resulting in a very strong wave.

And that is why you get the piggy, and why it appears where the angle of reflection equals the angle of incidence.