Grating lobes

Furthermore, the surface waves here are not related to the well-known surface waves that can exist on infinite arrays in a stratified medium next to the elements. These will readily show up in PMM calculations. These are simply grating lobes trapped in the stratified medium and will consequently show up only at higher frequencies, typically above resonance but not necessarily so in a poorly designed array. In contrast, the surface waves associated with finite arrays will typically show up below resonance (20-30%) and only if the interelement spacing is <0.5)t. 

We have demonstrated the presence of surface waves that can exist only on a finite periodic structure. It is quite different from the well-known types of surface waves that can exist in a stratified medium next to a periodic structure often referred to as Type 1. These merely represent grating lobes trapped inside the stratified medium. Thus, they will readily manifest themselves in computations based on infinite array theory at frequencies so high that grating lobes can be launched. 

In contrast, the new type of surface wave (Type 2) can exist only if the interelement spacing Dx is so small that no grating lobe can exist. In addition, the frequency must typically be 20-30% below the resonance frequency of the periodic structure. 

However, as also mentioned in Section 2.11.2, this lack of performance of a horn as soloist does not prevent it from sounding pretty good in an orchestra, namely, in an array of horns, as long as no grating lobes are encountered. 

Based on our discussion in Section 2.9, it should be obvious that the radiation pattern of the individual elements plays almost no role in the array RCS. In fact, as illustrated in Fig. 2.22, top, the total antenna pattern is for no grating lobes so completely dominated by the array factor that any change in the element... 

Fig. 2.22 Top The effect of the element pattern will usually drown" in the array factor as long as we stay far away from onset of grating lobes. Bottom Whether you make the array of slotted square waveguides (Case I) or triangular waveguides (Case II), it has very little effect on the scattering in-band. At much higher frequencies, Case II may be superior.

However, it was always the author s dream to work in the spectral domain similar to the PMM program that is very fast converging and gives us the real part of the impedance as one term if no grating lobes are present. Thus, if anyone ever pulls it off, the author would appreciate to hear about it. 

At what value of do we encounter the first grating lobe ... 

If Dj is so small that no grating lobes are encountered, how does the radiation resistance Ra vary with (constant frequency). 

Every time we are inside one of the unit circles, we are in visible space where we observe propagating modes. When outside, we are in invisible space where the modes are evanescent. For more about the grating lobe diagram, see reference 61. 

If Dx/k = 0.5 or smaller, we readily see that the unit circles will never touch each other that is, we will never have a grating lobe, but will have only one... 

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. 

The sidelobe level of the backscattered field is higher because we are getting closer to the onset of the first grating lobe ( 18 GHz). 

The interelement spacings Dx were not less than X/2 that is, grating lobes could occur as we move into imaginary space. As discussed in Section 4.9.3, that automatically rules out free surface waves. 

However, early onset of grating lobes can in this case be easily prevented by interlacing the elements as shown in Fig. 4.32b. Note that array dimensions remain the same in the two cases except that adjacent columns have been shifted with respect to each other as shown. 

Fig. 4.32 Modification of the array when changing from H-plane to E-plane scan, (a) The original array used for H-plane scan. If we scan this array in the E plane, grating lobes will start too early, namely at 9.38 GHz. (b) By interlacing adjacent columns as shown, the onset of grating lobes can be delayed to a much higher frequency, (c) To comply with the structure of the SPLAT program the array in (b) is rotated 90° and the x and z axis interchanged as explained in the text.

The first type (Type I) can be viewed merely as grating lobes happed inside the shatified dielechic medium. Therefore they typically occur at frequencies above resonance, namely such that grating lobes has started to emerge inside the shatified medium. 

One physical explanation for this fact is that the imaginary part of the mutual impedances start canceling each other as we move away from the reference element, provided that the interelement spacings are small (<0.4A,), while they add up (eventually to disastrous grating lobes) if Dx/k is larger than 0.5. For further discussion of this subject see Section 6.12 and also Appendix D. 

Finally, the designer should be reminded that the element spacing should always be sufficiently small to avoid grating lobes and surface waves in dielectric slabs adjacent to the dipole elements. 

It was already pointed out in Chapter 2, Section 2.6.2, that a large array with groundplane, uniform illumination, and physical area A also has a receiving area equal to A. Thus, if the array is designed to have no grating lobes and no significant surface waves, we may conclude from one of the most fundamental laws in antenna theory that the directivity D of the array is... 

Let us now look at the case where the element density is so large that no grating lobe can exist. In that case the radiation resistance Ra of each element is given by [103]... 

Thus, for no grating lobes and no surface waves we observe that Efar is completely independent of the total number of elements. Obviously so is the gain. That should come as no surprise. 

When no grating lobes fi om the subarrays exist, we have... 

So far we have assumed that the interelement spacings Dx and were so small that no grating lobes would emerge from the two subarrays. However, if this is not the case, the expressions above must be modified. More specifically, (9.11) becomes... 

Substituting (9.18), (9.19), and (9.20) into (9.10) yields the reflection coefficient r where grating lobe(s) can emerge from the two subarrays as evaluated by Hill [112]. 

However, in this chapter it will be sufficient to state the overall effect of grating lobes ... 

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