The Infinite Array Case

The presence of surface waves on finite periodic structures is of concern primarily for two reasons  

If used as an FSS to reduce the backscattering, the reradiation from the surface waves can lead to an increase in the total RCS. 

If used as a phased array, surface waves can lead to a significant variation in scan impedance from element to element, making precise matching difficult (see Figs. 1.3 and 1.5). 

In this chapter we shall smdy the FSS case in more detail and in particular how to condol the surface waves. The phased array case will be discussed in Chapter 5. 

The finite array will be modeled by a finite number of infinitely long column arrays (also called stick arrays see Fig. 4.1). This approach has been widely used by several researchers [74-80]. One of them, Usoff, wrote as part of his dissertation [24] the computer program Scattering fi om a Periodic Array of Thin Wire Elements (SPLAT). The excitation can be either in the form of an incident plane wave propagating in the direction 5 = + ysy + zsz (passive case). Or 

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When analyzing the finite array, it is advisable first to review the infinite array case that is, the finite number of columns shown in Fig. 4.1 becomes infinite as shown earlier in Fig. 1.1. As discussed for example in reference 62 or Chapter 3, infinite arrays are significantly simpler to analyze than finite arrays. In particular, if exposed to an incident plane wave with direction of propagation equal to s, the element cnrrents are all related to each other by Floquet s Theorem ... 

Only in the infinite array case is it immaterial how the elements are fed. Whatever the type of generator, we will always obtain merely Floquet currents. 

So far we have merely tacitly approved of the standard practice, namely the use of infinite array theory to solve finite periodic structure problems, at least in the case of an FSS with no loads and no groundplane. However, even in that case we may encounter a strong departure from the infinite array approach. In short, we may encounter phenomena that shows up only in a finite periodic structure and never in an infinite as will be discussed next. 

However, while the adjustment of Zl can be done precisely in the Smith chart, this is not quite the case for F shown in Fig. 5.6. One reason being that the field reflected from the infinite array is a simple plane wave (we assume the evanescent waves have died out), while the fields from the triads are a combination of Hankel functions. Should we call this new chart a Hankel chart Somebody could work it out and cover himself with fame and glory. See also comments in Section 5.8. 

The second problem is actually more complex. When considering an infinite array, the terminal impedance will be the same from element to element in accordance with Floquet s Theorem. However, when the array is finite, it is well known that the terminal impedance will differ from element to element in an oscillating way around the infinite array value (sometimes denoted as jitter). We postulated that this phenomenon was related to the presence of surface waves of the same type as encountered in Chapter 4. However, there is a significant difference in amplitude of these surface waves in the passive and active cases. This is due to the fact that the elements in the former case in general are loaded with pure reactances (if any), while the elements in the latter case are (or should be) connected to individual amplifiers or generators containing substantial resistive components (as encountered when conjugate matched). 

Figure 18.1 Random walk of an object through an infinite array of discrete boxes numbered by m = 0, 1, + 2,.... At time t = 0 the object is located in box m = 0 (probability 1) and then moves with equal probability to the two adjacent boxes m = 1 (probabilities 1/2). The time steps are numbered by n. The resulting occupation probabilities, p(n,m), of being in box m after time step n are the Bernoulli coefficients (Eq. 18-1). Curve A shows a typical individual path. Curve B represents the unlikely case in which the object jumps six times in the same direction.

All of the other chapters in this book deal with the symmetries of finite (discrete) objects. We now turn to the symmetry properties of infinite arrays. The end use for the concepts to be developed here is in understanding the rules governing the structures of crystalline solids. While an individual crystal is obviously not infinite, the atoms, ions, or molecules within it arrange themselves as though they were part of an infinite array. Only at, or very close to, the surface is this not the case this surface effect does not, in practice, diminish the utility of the theory to be developed. 

The surface and bulk unit cell vectors represent the periodicity which allows a translation operation to generate an infinite array of atoms in the surface or bulk structure. The coefficients m, mu, m2, and m2i define a matrix which describes the transformation of the bulk unit vectors into the surface unit vectors. For example, the simplest surface structure occurs when the snrface maintains the same nnit cell as the bnlk. In this case the unit cell matrix would be... 

JJ Another array of orbitals with Ihe same sort or linear combinations is lhal of ihe tr system of cydopolyencs (benzene, naphthalene) with which you are probably already familiar. The molecular orbilals constructed from sets of parallel p orbitals are both bonding and anubonding at various energies. The system is particularly stable if HflckePs rule (2n + 2 electrons see also Chapter IS) is obeyed, tn Ihe case of an infinite array of hydrogen atoms (If1). Ihe situation is unstable (as you should have questioned immediately) it reverts to an array of H2 molecules ... 

Figure 23. (A) A schematic representation of a cross section of a two-dimensional electric dipole array. (B) Electrical potential profile across a two-dimensional dipole array of infinite dimension, where the dipole array is at the hydrocarbon/vacuum interface. (C) The same dipole array as in (B) except that the dipole array is at the hydrocarbon/aqueous interface in this case.

Consider next the finite x infinite array shown in Fig. 1.2. It consists, like the infinite x infinite case in Fig. 1.1, of columns that are infinite in the Z direction, however, there is only a finite number of these columns in the X direction. Such arrays have been investigated by numerous researchers [4-23]—in particular, by Usoff, who wrote the computer program SPLAT (Scattering fi om a Periodic Linear Array of Thin wire elements) as part of his doctoral dissertation in 1993 [24, 25]. 

Fig. 1.3 Various cases of a plane wave incident upon infinite as well as finite arrays at 45° from normal in the H plane. Element length 21=1.5 cm, load impedance Zl = 0 and frequencies as indicated, (a) Element currents for an infinite x infinite array at 10 GHz as obtained by the PMM program (close to resonance), (b) Element currents for a finite x infinite array of 25 columns at 10 GHz (close to resonance), (c) Element currents for a finite x infinite array of 25 columns at 7.8 GHz ( 25% below resonance).

Finally, we show a polar plot of the far field obtained by simple aperture integration. Comparing the infinite and finite cases we observe that the former has an infinite narrow beam (like a Dirac-Delta function) in the direction r (0,0) while the latter exhibits the well known far field pattern. At the array elements they both look similar except that the finite case shows ripples along the aperture as caused by the presence of surface waves as discussed in Chapters 1, 4 and 5. 

Note Sometimes the reader perceives the introduction of the two semi-infinite arrays with negative Floquet currents as an inadequate approximation. As discussed in detail in Section 4.19 (Common Misconceptions), this is not the case. However, even if some inaccuracies were present in our explanation, it really would not matter since the currents in Fig. 4.5f were obtained by direct calculation of the actual currents obtained from the SPLAT program applied to a finite array. 

Fig. 5.7 A plane wave incident upon an infinite array of active elements in front of an infinite groundpiane. By plotting the reflected field in a complex plane (/n this case a Smith chart) we can adjust the load impedances Zrofthe active dipoles such that the reflected field disappears in the backscatter direction.

A plot of the scan impedance Za for an infinite x infinite array without a ground-plane and with the same interelement spacings as used in Chapter 6 as obtained from the PMM code is shown in Fig. D.3. The impedance is plotted in a Smith chart normalized to 100 ohms. Furthermore, the impedance in Fig. D.3 as well as in later figures includes the matching of a short transmission line with characteristic impedance 200 ohms and length 0.13 cm. This was typically true in the cases shown in Chapter 6, which should make comparisons between the various curves more meaningful. We remind the reader that the purpose of this transmission line (also referred to as a pigtail ) was to better center the impedances as well as compress them (for details see Chapter 6 and Appendix B). 

Fig. D.3 The scan impedance for an infinite x infinite array without a groundplane obtained from the PMM program. The array dimensions are identical to the broadband array in Chapter 6 (see insert) but includes only the underwear" (see Dieteclric Profile above). Also included is a small transmission line matching section as was the case in Chapter 6 (see text).

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