How the Surface Waves are Excited on a Finite Array

Consider an infinite array exposed to an incident plane wave E with the direction of propagation being s, as shown in Fig. 4.4, top. From the basic theory for periodic structures we know that the reradiated (scattered) field consists of plane waves propagating in the directions s = xs ysy + zs and, eventually, a finite number of grating lobes, as also illusttated in Fig. 4.4, top. Furthermore, there will be an infinite number of evanescent waves that die out quickly as we move away from the array. 

Let us next consider a finite array as illustrated in Fig. 4.4, middle. When exposed to the same incident plane wave with the direction of propagation s, the element currents will again, to a first-order approximation, be the Floquet currents as given by (4.1). 

It is a simple matter to find the far field from these currents as a function of the continuous radiation direction r - We show a typical example in Fig. 4.4, bottom. However, rather than plotting it as a function of radiation angle, it is plotted as a function of rex, see Fig. 4.4 middle. This makes it more compatible with our choice of variable for the incident wave and the plane wave expansion in general. We obtain mainbeams at = Sx and we also note that the visible 

Note also that as we continue into the invisible space for 

However, the radiation pattern in Fig. 4.4 bottom is only a first-order approximation. To see what really goes on, consider Fig. 4.5. At the very top (Fig. 4.5a) we show an infinite periodic structure being exposed to an incident plane wave with direction of propagation s. The element currents will be of the Floquet type only as we saw earlier in Fig. 4.4. In Fig. 4.5b we create a finite array by 

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