Leaky Wave Antennas for Exam - Advance Electromagnetic Waves | ECE 6341, Exams of Electrical and Electronics Engineering

Download Leaky Wave Antennas for Exam - Advance Electromagnetic Waves | ECE 6341 and more Exams Electrical and Electronics Engineering in PDF only on Docsity! 1 Chapter 11 Leaky-Wave Antennas Arthur A. Oliner Polytechnic University David R. Jackson University of Houston 2 11.1 Introduction 11-3 11.2 Design Principles for Uniform Leaky-Wave Antennas 11-6 11.3 Design Principles for Periodic Leaky-Wave Antennas 11-12 11.4 Specific Structures: Overview 11-14 11.5 Specific Structures Based on Closed Waveguides 11-16 11.6 Specific Structures Based on Periodic Open Waveguides 11-22 11.7 Specific Structures Based on Uniform Open Waveguides 11-29 11.8 Arrays that Scan in Two Dimensions 11-39 11.9 Narrow-Beam Antennas Based on a Partially Reflective Surface 11-44 5 In the second type of leaky-wave antenna, the periodic type, some periodic modulation of the guiding structure is introduced, and it is this periodicity that produces the leakage. The periodic modulation itself is uniform along the structure’s length, again except for the small taper of the periodic properties along the length to control the sidelobes. Again, a complex propagation wave number results, with β and α; large or small values of α are related to the beamwidth and the radiation efficiency in the same manner as that found for uniform leaky-wave antennas. A typical example of a periodic leaky-wave antenna is a dielectric rectangular rod on which a periodic array of metal strips is placed, as seen in Figure 11-2. FIGURE 11-2 A typical and important example of a periodic leaky-wave antenna: a rectangular dielectric rod (which may or may not be situated on a metal plane) on which is placed a periodic array of metal strips. This antenna can radiate into either the forward or backward quadrants. An important difference between uniform and periodic leaky-wave antennas is that the dominant mode on the former is a fast wave that therefore radiates whenever the structure is open. On the other hand, the dominant mode on a periodic leaky-wave antenna is a slow wave that does not radiate even though the structure is open. Introduction of the periodic array produces an infinity of space harmonics, some of which may be fast while the rest are slow; the fast space harmonics would radiate. Since you desire an antenna that radiates only a single beam, the structure is designed so that only the first space harmonic (n = −1) is fast. The relevant design considerations are presented in Section 11.3. It is also shown there that the scan range for this class of antennas is from backward end-fire through broadside into part of the forward quadrant, except for a narrow region around broadside, where an “open stop band” occurs. In general terms, therefore, we see that the scan range is completely different from that for the uniform leaky-wave antennas. There the beam scans in the forward quadrant only; also, it cannot approach broadside too closely, and sometimes it cannot be used too near to end fire. For the periodic leaky-wave antenna, you can scan over almost all the backward quadrant and into some of the forward quadrant as well. Relation to Surface-Wave Antennas and Slot Arrays Surface-wave antennas, leaky-wave antennas, and slot arrays are all members of the family of traveling- wave antennas, yet they are treated in this Handbook in separate chapters (Chapters 10, 11, and 9 respectively). They are similar to each other in some evident respects (e.g., the basic structure in each case is a waveguide of some sort), but they all differ from each other in important ways that lead to different design procedures and to different performance expectations. Surface-wave antennas are purely end-fire antennas, whereas leaky-wave antennas and slot arrays do not radiate well in the end-fire direction and, in fact, are designed either to radiate in some other direction or to scan over a range of angles. The basic guiding structure for surface-wave antennas is an open waveguide 6 (such as a dielectric rod) whose dominant mode (the surface wave) is purely bound, so that the surface wave will radiate only at discontinuities, such as the very end of the waveguide. It does not radiate along the length of the guide because the surface wave is a slow wave (except for some small leakage into the almost-end fire direction if the surface-wave antenna is tapered), whereas a uniform leaky-wave antenna, which supports a fast wave, leaks power all along the length of the waveguide. Periodic leaky-wave antennas, however, are often based on surface waveguides. There the basic waveguide can be an open structure whose dominant mode is a surface wave that is a slow wave, and the radiation is produced by placing a periodic array of discontinuities on the guide in such a way that the first space harmonic becomes fast. The resulting leaky wave may therefore be viewed as arising from a surface-wave- excited array. There is thus a strong kinship between the basic structures employed for surface-wave antennas and those for periodic leaky-wave (or surface-wave-excited) arrays, and, for this reason, periodic leaky-wave antennas were, in the first two editions of this Handbook, included within the same chapter as surface-wave antennas. On the other hand, since the performance properties of periodic leaky-wave antennas, and the design procedures to achieve those properties, are very different from those for surface- wave antennas but very similar to those for uniform leaky-wave antennas, the leaky-wave antennas of both types are currently incorporated into the present chapter. The differences between periodic leaky-wave antennas and slot arrays are more subtle but still very significant. A visually evident difference is that most slot arrays are fed from air-filled rectangular waveguides whose dominant mode is fast; to suppress the radiation from this fast dominant (n = 0) space harmonic while retaining that from the n = −1 space harmonic, it is necessary to place successive slots on alternate sides of the guide centerline or to alternately tilt the slots to produce phase reversals. Periodic leaky-wave antennas do not need such phase reversals because the (slow) n = 0 space harmonic does not radiate. However, this distinction is not fundamental, as may be noted when the slot-array rectangular waveguide is dielectric-filled and the alternation of slots is no longer needed. The structures then resemble each other in principle. The basic distinction between periodic leaky-wave antennas and slot arrays lies in the nature of the individual discontinuities, and, therefore, in the basic design approaches. The intention in the leaky-wave antennas is to produce a slow leakage per unit length; thus each discontinuity element in the periodic array of elements produces a small loading on the basic waveguide mode. The individual elements are intentionally made nonresonant. As a result, the design procedure views the leaky-wave antenna as an equivalent “smooth” structure with a complex propagation wave number, where the array of discontinuity elements is considered as a whole in the analysis. In contrast, the slots in a slot array are considered individually, and then mutual coupling effects are taken into account when the array itself is formed. Furthermore, the individual slots are usually resonant. Variations in the loading on the basic waveguide mode are achieved by rotating the slots or by moving them closer to the guide centerline. The design approach thus becomes quite different from that for leaky- wave antennas, thereby warranting inclusion in a separate chapter. The resonant loading in slot arrays, as opposed to the nonresonant loading in periodic leaky-wave antennas, also influences performance. For example, slot arrays are more frequency-dependent, which can be good if you wish to scan more rapidly with frequency, but the trade-off is that the scan range is narrowed. Most slot arrays are not intended for a large scan range, however. On the other hand, if slot arrays are built with nonresonant slots, their behavior would be very similar to that of periodic leaky-wave antennas. The question then would be whether the customary slot-array design or the leaky-wave design would be more suitable. One last observation relates to the fact that the treatment in Chapter 9 is restricted to slot arrays fed by rectangular waveguide. The design approach described for such arrays could be applied equally well to arrays of other resonant elements, e.g., the two-wire line with proximity-coupled dipoles.6 11.2 Design Principles for Uniform Leaky-Wave Antennas The physical structure of a leaky-wave antenna consists of a leaky waveguide with a length L along which the leakage occurs. The propagation characteristics of the leaky mode in the longitudinal (z) direction are 7 given by phase constant β and leakage constant α, where α is a measure of the power leaked (and therefore radiated) per unit length. The length L then forms the aperture of the line-source antenna, and the amplitude and phase of the traveling wave along the aperture are determined by the values of β and α as a function of z. When the leaky waveguide is completely uniform along its length, β and α do not change with z, and the aperture distribution has an exponential amplitude variation and a constant phase. Such an aperture distribution results in a high sidelobe level, so that the design of a practical leaky-wave antenna will include a variation of α with z in order to control the sidelobes in some specified fashion. More is said about this later in this section. The values of β and α will depend on the precise cross-sectional geometry of the leaky waveguide, and the determination of β and α, whether theoretically or experimentally, is in most cases the most difficult part of the design. Their knowledge, however, is essential to any systematic design procedure. Once β and α are known as a function of frequency and cross-sectional geometry, the principal behavioral features of a leaky-wave antenna follow very quickly. Such features include the beam direction, the beamwidth, the radiation efficiency, the variation of the scan angle with frequency, and the taper in α required to control the sidelobes. Beam Direction, Beamwidth, and Radiation Efficiency These major behavioral features follow directly once the value of β and α are known, and they are given to a good approximation by a set of very simple relations. We first consider the beam direction and the beamwidth: 0 sin km β θ ≈ (11-1) mL θλ θ cos)/( 1 0 ≈Δ (11-2) Here θm is the angle of the maximum of the beam, measured from the broadside direction (perpendicular to the leaky waveguide axis), L is the length of the leaky-wave antenna, Δθ is the beamwidth, and k0 is the free-space wave number (=2π/λ0). Both θm and Δθ are in radians in Eqs. 11-1 and 11-2. The beamwidth Δθ is determined primarily by the antenna length L, but it is also influenced by the aperture field amplitude distribution. It is narrowest for a constant aperture field and wider for sharply peaked distributions. Equation 11-2 is a middle-of-the-range result. For a constant aperture distribution, the unity factor in the numerator should be replaced by 0.88; for a leaky-wave structure that is maintained uniform along its length, consistent with 90 percent radiation, the factor should be 0.91; for a tapered distribution that is sharply peaked, the factor could be 1.25 or more (see Table 11-1 for examples). The antenna length L is usually selected, for a given value of α, so that 90 percent (or at most 95 percent) of the power is radiated, with the remaining 10 percent or so absorbed by a matched load. Attempting to radiate more than 90 percent or so creates two problems: First, the antenna must be made longer, and second, the variation in α(z) required to control the sidelobes becomes extreme. For 90 percent of the power radiated, we find 00 / 18.0 k L αλ ≈ (11-3) This simple but useful relation follows from writing )]/)(/(4exp[)2exp( )0( )( 00 λαπα LkLP LP −=−= (11-4) 10 distribution is exponentially decaying. If the antenna length is infinite, the radiation (power) pattern R(θ) is given by 2 0 2 0 2 )sin/()/( cos~)( θβα θθ −+ kk R (11-8) which does not exhibit any sidelobes. If the antenna length is finite, the expression for R(θ) becomes more involved, and the pattern possesses sidelobes that modify the basic shape for infinite length. The preceding comments are illustrated well in a paper on dielectric-grating leaky-wave antennas by Schwering and Peng.7 They present several examples of such patterns, two of which are shown in Figure 11-4a and b. The length of the antenna in Figure 11-4a, which clearly exhibits sidelobes, is 10λ0. As the antenna length increases, the amplitude of the sidelobe variations decreases. For the radiation pattern in Figure 11-4b, which shows a smooth pattern only, the antenna length is 150λ0, which is evidently effectively infinite. Although the leaky-wave antenna for which these calculations were made is periodic rather than uniform, the basic features are identical. Schwering and Peng7 contains an extended discussion of radiation-pattern considerations, including equations and other figures. FIGURE 11-4 Radiation patterns of dielectric grating leaky-wave antennas, showing the changes in the sidelobe behavior with antenna length. These antennas are not tapered to control the sidelobes. (a) Antenna length L = 10λ0 (sidelobes clearly present). (b) Antenna length L = 150λ0 (no sidelobes). (after Schwering and Peng7 © IEEE 1983) 11 The radiation pattern in Fig 11-4a is seen to possess first sidelobes that are only about 13 dB down, which is generally undesirable. To greatly reduce the sidelobe level and to control the pattern in other ways, it is customary to appropriately taper the amplitude of the aperture distribution, as is discussed next. Control of Aperture Distribution to Reduce Sidelobes The procedure to design the leaky-wave antenna so that it produces a final desired radiation pattern is straightforward, though somewhat complicated, involving the following steps. First, the final desired radiation pattern is specified, and then the corresponding aperture amplitude distribution is determined by standard antenna techniques. Then, by using the expression derived next, the values of α/k0 are computed as a function of position along the antenna length in accordance with the aperture amplitude distribution that was just determined. At the same time β/k0 must be maintained constant along the length so that the radiation from all parts of the aperture point in the same direction. Finally, from the theory that relates α and β to the geometry of the structure, we compute the tapered geometry as a function of position along the antenna length. When we change the local cross-sectional geometry of the guiding structure to modify the value of α at some point z, however, it is likely that the value of β at that point is also modified slightly. However, since β must not be changed, the geometry must be further altered to restore the value of β, thereby changing α somewhat as well. In practice, this difficulty requires a two-step process for most leaky-wave antennas, which is not bad. Because of this added complexity, we seek leaky-wave structures for which we can vary geometric parameters that change β and α essentially independently. The first design step mentioned earlier, i.e., determining the required aperture amplitude distribution for the selected desired radiation pattern, is a standard antenna procedure not specifically related to leaky-wave antennas. The second step, calculating the value of α(z) corresponding to the aperture amplitude distribution found from the first step, is directly pertinent to leaky-wave antennas, and we therefore present now a derivation of the expression needed for the second step. The power distribution along the antenna can be expressed in the form ⎥⎦ ⎤ ⎢⎣ ⎡−= ∫ z dPzP 0 )(2exp)0()( ζζα (11-9) where P(0) is the power at the input point, z = 0, and ζ is the integration variable. Upon differentiation of Eq. 11-9, we obtain )()(2)( zPz dz zdP α=− (11-10) Suppose now that the desired aperture distribution (which would achieve the specified radiation pattern) is A(z) exp (−jβz). We may then write |)(|)( zAc dz zdP =− (11-11) where c is a constant of proportionality. Comparison of Eqs. 11-10 and 11-11 yields )( |)(|)(2 2 zP zAcz =α (11-12) 12 Upon integration of Eq. 11-11, we obtain, corresponding to two sets of limits of integration, the following: 2 0 | ( ) | (0) ( ) L c A d P P Lζ ζ = −∫ (11-13) 2 0 | ( ) | (0) ( ) z c A d P P zζ ζ = −∫ (11-14) We next use Eq. 11-14 to substitute for P(z) in Eq. 11-12, and then we employ Eq. 11-13 to eliminate the proportionality constant c. In a straightforward fashion, we then obtain the desired result: 2 2 2 0 0 | ( ) |2 ( ) (0) | ( ) | | ( ) | (0) ( ) L z A zz P A d A d P P L α ζ ζ ζ ζ = − − ∫ ∫ (11-15) The units of α(z) in Eq. 11-15 are nepers per unit length. To obtain α(z) in decibels per unit length, we multiply by 8.68. If P(L), the power remaining at the end of the aperture, is allowed to approach zero, we note from Eq. 11-15 that α(z) then becomes very large for points near to the end of the aperture, i.e., for z approaching L. This is the main reason why it is common for P(L)/P(0) to be equal to 0.1 or so, but not much smaller, with the remaining power being absorbed in a matched load to avoid the presence of any backlobe. 11.3 Design Principles for Periodic Leaky-Wave Antennas As discussed in Section 11.1, periodic leaky-wave antennas differ from uniform ones in that the waveguiding structure is modulated periodically along its length instead of being completely uniform (again, except for the small taper for both types to control the sidelobes). The dominant mode on uniform antennas is fast relative to free-space velocity, whereas the one on periodic antennas is slow, so that the dominant mode itself does not radiate and it needs the periodic modulation to produce the radiation. Since the physical processes that produce the radiation are different, these two antenna types have different scan ranges. On the other hand, most of the design principles for the uniform leaky-wave antennas discussed in Section 11.2 also apply to the periodic ones. The treatment given next indicates in what ways changes in design are necessary. First, however, we summarize how the periodicity produces the leakage and, in that context, why the scan ranges are different for the two types. Effect of Periodicity on Scan Behavior To explain the source of the leakage and to understand the scan behavior as a function of frequency, we invoke the concept of space harmonics. Suppose we first take a uniform dielectric waveguide, and then we place an array of metal strips periodically along its length (as in Figure 11-2). Before the metal strips are added, we choose the guide dimensions and frequency so that only the dominant mode is above cutoff; furthermore, β > k0 for this mode, so it is purely bound. When the periodic array of strips is added, the periodicity introduces an infinity of space harmonics, each characterized by phase constant βn and related to each other by βn d = β0d + 2nπ (11-16) where d is the period and β0, the fundamental space harmonic, is simply the original β of the dominant mode of the uniform dielectric waveguide, but perturbed somewhat in value because of the addition of the strips. As seen from Eq. 11-16, βn can take on a large variety of values, so that these space harmonics can be forward or backward in nature, and be slow or fast. Since the structure is open, a space harmonic that is fast will radiate. To say it in another way, since the space harmonics are all tied together, and all of them together comprise the dominant mode of the loaded structure, the whole mode becomes leaky if one or more of the space harmonics becomes fast. 15 The data relating the wave-number behavior to the geometric parameters are usually plotted as β/k0 or λ0λg, where λg is the guide wavelength in the propagation direction (z), and as α/k0 or αλ0, where α is the leakage constant. In previous editions of this Handbook, plots were presented in the form of λ/λz and αzλ, where αz, λ, and λz are written instead of α, λ0, and λg. Where the discussion and graphical plots in the present chapter employ material taken from the previous edition, the previous notation is carried over. It is essential that such information be available for any specific antenna structure because the correct taper for low sidelobes cannot be designed without it. A last general remark to be made in connection with leaky- wave structures based on closed waveguides is that some of the structures employ a series of round holes or small (nonresonant) slots. These holes or slots, however, are closely spaced, so that the structures should be viewed as quasi-uniform rather than periodic, even though these holes are periodically spaced. The radiation produced by them radiates the n = 0 space harmonic and not the n = −1 space harmonic. The next stage in the development of specific leaky-wave antennas involved those based on open waveguides. Some of these antennas are uniform structures that employ open waveguides on which the dominant mode is initially purely bound, and others are periodic structures that are excited by surface waves and radiate via the n = −1 space harmonic. These two categories are considered separately in Sections 11.7 and 11.6 respectively. The best-known examples of surface-wave-excited periodic leaky-wave antennas are dielectric rectangular rods (or slabs), with or without ground planes, that have on their tops or on one of their sides a periodic array of grooves or a periodic array of metal strips. Another large group is based on microstrip line. These antennas have been studied rather extensively, both experimentally and, more recently, theoretically. Together with other, similar structures, they are discussed in Section 11.6. An important problem for this class of structures is the incorporation of a feed mechanism that does not contribute spurious radiation. Uniform leaky-wave antennas based on open waveguides offer a special challenge. Since the guide is already open, it cannot be cut to induce radiation, and other approaches are needed. The most common one is the appropriate introduction of asymmetry, but other mechanisms, such as the use of a leaky higher-order mode or some modification in the geometry, have also been found useful. The first (and only early) example of such a leaky-wave antenna based on an open waveguide was invented by W. Rotman in the late 1950s. This pioneering study10,11 involved several versions of a form of trough waveguide whose dominant mode is purely bound but was made leaky by introducing asymmetry. The design procedure for this antenna was successful and practical, but the approach was not pursued further until about 20 years later, in the context of a new need that emerged in connection with millimeter waves. As interest in millimeter waves revived during the 1970s, it was recognized that new forms of leaky-wave antennas were needed because of the smaller wavelengths involved and because the usual waveguides had higher loss at those higher frequencies. Since these smaller wavelengths caused fabrication problems due to small dimensions, simpler structures were sought; in fact, the latest structures are designed to permit the complicated portion of the structure, including the taper for sidelobe control, to be deposited photo- lithographically, in printed-circuit form, by using a mask. Because of the loss considerations, the new antennas are often based on new lower-loss waveguides designed for millimeter-wave applications. These include nonradiative dielectric (NRD) guide, groove guide, and rectangular dielectric rods, sometimes used in conjunction with microstrip in novel ways. Unfortunately, these last-mentioned structures were difficult to analyze theoretically, so their designs were empirical only; as a result, the experimental radiation patterns showed poor sidelobe performance, leading some people to conclude unfairly that poor sidelobe behavior was a necessary consequence of this class of antennas. However, many other leaky-wave antennas in this class were analyzed accurately, primarily by Oliner and his colleagues, and their results agreed very well with measurements. Some of the more promising of these new millimeter-wave antennas are described in Section 11.7. 16 A more recent development relating to leaky-wave antennas is their incorporation into arrays that permit scanning in two dimensions. The arrays are essentially a linear phased array of leaky-wave line-source antennas, where the scanning in elevation is obtained in leaky-wave fashion by varying the frequency, and the scanning in azimuth is achieved by varying the phase difference between the successive parallel leaky- wave line sources. The architecture underlying this approach is described in Section 11.8, and several examples are given of specific antenna structures in this category. A partial motivation for this approach is to achieve a lower-cost substitute for phased-array antennas in some applications. 11.5 Specific Structures Based on Closed Waveguides The earliest example of a leaky-wave antenna was the one for which W. W. Hansen was granted a patent.1 He had proposed during the late 1930s that an antenna could be created by cutting a rectangular waveguide longitudinally, as shown in Figure 11-1, thereby producing a long slit in the side of the initially closed guide, out of which power could leak away. His concept was not pursued at that time because of the success of slot arrays, but the simplicity of the structure remained attractive, and it was reexamined about a decade later. The 1950s, in fact, represented a very active period during which many leaky-wave antennas based on closed waveguides were proposed, analyzed, measured, and utilized. Several excellent references summarize in detail the state of the art in this class of antennas as of the middle 1960s, including a comprehensive book12 by C. H. Walter; a chapter by F. J. Zucker13 in the First Edition of this Handbook; a chapter by T. Tamir14 in Part II of the book Antenna Theory, edited by R. E. Collin and F. J. Zucker; and a summary by A. A. Oliner and R. G. Malech15 in Volume II of the book Microwave Scanning Antennas, edited by R. C. Hansen. All these specific structures based on closed waveguides are “uniform” leaky-wave antennas, so the principles for their design are those discussed earlier in Section 11.2. The remaining information required to complete the design involves the expressions for β/k0 (=λ/λz) and α/k0 (or αzλ) as a function of the frequency and geometric parameters of the specific structure. Since the period before the middle 1960s predates the computer era, theoretical expressions had to be simple to be considered practical. Fortunately, the structures themselves were simple, leading automatically to relatively simple expressions that were accurate, but, in addition, many of the expressions were further simplified by the use of perturbation relations. In some cases, these values of α and β were measured rather than calculated, and then employed in the design. Two experimental methods are outlined in Walter’s book.16 Long Slits in Rectangular Waveguide Radiation from long slits in rectangular waveguide can be accomplished in several ways, where the leakage rate can be adjusted by changing the slit width, and the polarization of the radiated beam can be selected by changing the waveguide mode. Some examples are illustrated in Figure 11-5. The best-known example in this well-known category is the narrow slit in the side wall of rectangular waveguide shown in Figure 11-1, or the same structure with a ground plane seen in Figure 11-5a. The antenna shown in Figure 11-5a also differs from the structure in Figure 11-1 in that it is rotated by 90° and the slit is shown tapered (in exaggerated fashion) to remind us that in the design the slit width is varied to control the sidelobes in the radiation pattern. (The rectangular waveguide dimensions h and w correspond, of course, to the usual a and b respectively.) 17 FIGURE 11-5 Several examples of leaky-wave antennas based on long slits in rectangular waveguide. (a) Narrow slit in the narrow wall, shown with an exaggerated taper. The remaining figures show the relative guide wavelength λ/λz and the relative leakage constant αzλ for (b) the channel-guide antenna, (c) the dielectric-filled channel, for εr = 2.56, and (d) a narrow slit in square guide. The dominant mode excites the slit in the first three cases, while the TM11 mode excites the slit in the last case. (after Goldstone and Oliner17 © IRE (now IEEE) 1959 and Hines et al.18 © IRE 1953) For this antenna, with the ground plane present but with the slit uniform, simple theoretical expressions are available for the relations between λ/λz and αzλ and the frequency and the geometry. These expressions were derived by Goldstone and Oliner17 using a transverse resonance approach together with a simpler perturbation procedure. The results agreed very well with measurements. The expressions (with the notation differing somewhat from that in Goldstone and Oliner17 to be consistent with Figure 11-5a) are ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + −= 22 2 0 0 12 1 p p hw z zz π λ λ λ λ λ (11-21) 20 The antenna employing a series of closely spaced round holes was proposed and measured at the Ohio State University,25 and it became known as the OSU “holey guide.” By varying the diameter d of the holes and, to a lesser extent, the hole spacing s and the guide width w, a very large range in the value of αλz was found experimentally. This structure was also analyzed by Goldstone and Oliner,17 by employing small aperture procedures for the series of holes, together with a transverse resonance approach. Using a perturbation form for the result, they obtained the following simple expressions (using the notation in Eqs. 11-21 and 11-22): ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ′+= X h z zz 2 2 0 0 4 1 π λ λ λ λ λ (11-25) and 2 0 2h Rz z ′ = λλ λα (11-26) where 2 2 2 2( ) ( ) ( ) ( ) B GX R G B G B ′ ′ ′ ′= = ′ ′ ′ ′+ + (11-27) G′ = πw/2h (11-28) B′ = 6hws/πd3 (11-29) 20 )2/(1/ hz λλλ −= (11-30) Calculations from these expressions agreed very well with measurements made both at the Ohio State University (Walter,12 Chapter 5) and the Polytechnic Institute of Brooklyn.17 When the frequency of operation is near to cutoff, these perturbation expressions become inaccurate, and we should instead solve the transverse resonance relation17 exactly. Array of Closely Spaced Wide Transverse Strips This array of transverse strips, sometimes called an inductive-grid antenna and due to R. C. Honey,26 is shown in Figure 11-8a. The antenna consists of a parallel-plate waveguide operated in its first higher-order (TE1) mode, with its upper plate composed of an array of closely spaced transverse strips, and fed from a reflector arrangement so as to fill the space with the field having the polarization shown in Figure 11-8a. The upper plate can be photoetched on a thin laminate and then be supported by polyfoam, or it can consist of a grid of transverse round wires. The structure was analyzed26 by using the transverse resonance method, which yielded simple and accurate expressions for the λ/λz and αzλ values. From these expressions we can compute the design curves presented in Figure 11-8b and c. The design procedure is to first select the desired λ/λz and αzλ and then to read from the curves in Figure 11-8b the corresponding abscissa value. The curves in Figure 11-8c then yield the value of d for the design wavelength corresponding to the abscissa value obtained from Figure 11- 8b. Now, in a design for low sidelobes in some specified fashion (see Section 11.2), αzλ must vary from point to point in a tapered fashion along the longitudinal direction, while λ/λz must remain the same at each point. The plot in Figure 11-8b then tells us how c can be varied, by changing the strip width t, to obtain the desired different values of αzλ while trying to maintain λ/λz constant. However, λ/λz will change somewhat as t is varied, since it is not independent of t, and the plot in Figure 11-8c then indicates how d can be modified to change λ/λz back to the desired constant value. 21 FIGURE 11-8 Inductive-grid antenna comprised of a parallel-plate guide fed in its first higher-order TE mode and with its upper plate consisting of a series of closely spaced wide transverse strips. (a) Structure. (b) Relative leakage constant with the relative guide wavelength as a parameter. (c) A plot to aid in the design procedure (see text). Quantity c, which appears in the abscissas of parts (b) and (c), is defined as c = 2πd/p ln [csc (πt/2p)]. (after Honey26 © IRE (now IEEE) 1959) In his final design, Honey26 found that it was necessary to flex the bottom plate slightly along the longitudinal direction, and he built his structure accordingly. He was also meticulous with respect to both the accuracy of his theory and the details of the structure to be measured. As a result, the correspondence between his theoretical and measured radiation patterns was remarkably good, down to almost −40 dB, as may be observed in Figure 11-9. 22 FIGURE 11-9 Theoretical (a) and measured (b) radiation patterns for the inductive-grid antenna shown in Figure 11-8a. The accurate theoretical design and the carefully fabricated experimental structure both took into account the taper for low sidelobes. The agreement between the two patterns is seen to be remarkable. (after Honey26 © IRE (now IEEE) 1959) 11.6 Specific Structures Based on Periodic Open Waveguides The design principles for leaky-wave antennas based on periodic open waveguides are presented in Section 11.3. The important points to recall are that the basic open waveguide supports a slow wave, which does not radiate, and that the period of the structural modulation is selected relative to the wavelength so that the n = −1 space harmonic, and only that one, radiates the power. In contrast, the periodic structures discussed in Section 11.5, such as the closely spaced series of round holes in rectangular waveguide, are based on a fast wave and are quasi-uniform so that only the n = 0 space harmonic radiates. Furthermore, the beam radiated from the periodically modulated slow-wave structures may be scanned throughout most of the backward quadrant and into part of the forward quadrant, whereas the beam radiated from the uniform or quasi-uniform structures is restricted to the forward quadrant only. The two most common open waveguides that support dominant modes and which serve as the basis for periodic leaky-wave antennas are rectangular dielectric rods, with or without ground plane, and microstrip line. The class of dielectric rods includes a variety of known waveguides, such as dielectric image guide, insular guide, inset guide, and so on. Since the leaky wave is fast and the basic surface wave (or microstrip dominant mode) is slow, you must be careful about the feed arrangement to make sure that little spurious radiation is introduced. Many of these periodic leaky-wave antennas have been known (measured and used) for some years, but accurate theories for β and α, suitable for careful design purposes, have become available only recently, and only for some structures. Early Structures A few pioneering examples in this class were proposed and studied as far back as the late 1950s, but the ideas behind them were not pursued then. When they did reemerge, a decade or two later and in a somewhat different form, most people did not recognize the relationship with the past. Two examples of this early novel thinking are presented here. 25 Design Theory for Wide Periodic Dielectric Antennas It is only within the last two decades that accurate theoretical analyses became available that permit the systematic design of this class of leaky-wave antennas. For antennas employing a grating of grooves, thorough and detailed studies were performed by Schwering and Peng,7,33,34 based in part on earlier analytical work by Peng and Tamir.44,45 The two best sources for systematic design information are Schwering and Peng7 and Schwering and Oliner.46 Corresponding, but less thorough, design information on metal-strip-grating antennas comes mostly from work by Guglielmi and Oliner,47 based on earlier analyses by them48–50 of scattering by metal strip gratings on a dielectric substrate. Another accurate method of analysis for these antennas was presented by Encinar.51 Many of the general conclusions appropriate to grooved antennas, however, apply as well to those with metal-strip gratings. The detailed expressions for β/k0 and α/k0 are different, of course. The theoretical design information referred to earlier is applicable directly to dielectric image guides, i.e., structures for which a ground plane is present under the rectangular dielectric layer. The procedure is readily extendable to structures without a ground plane, however, by some suitable, basically straightforward modifications. The numerical values presented in Schwering and Peng7 and Schwering and Oliner46 assume that the dielectric material has an εr = 12, corresponding to Si or GaAs, so that the antenna performance can be controlled, if desired, by semiconductor devices. For the antenna employing a grating of grooves, the groove depth must be chosen to lie within a certain range if we wish to optimize the leakage constant α. The reason for this can be understood physically in a simple way. Consider the structure shown in the inset in Figure 11-13, where the height of the uniform dielectric region is h and the groove depth is t. Suppose that we maintain the sum of h + t constant and we increase the groove depth t. Although the total antenna height is fixed, the effective dielectric constant εeff of the structure is decreased as t is increased, because the groove region is now partly air-filled. The value of εeff, combined with the height h + t, may be viewed as an “effective height” and is an important design parameter. When the effective height is small, most of the guided energy travels in the air region above the antenna, and the grooves would cause little radiation. When the effective height is large, on the other hand, the energy is confined primarily to the interior of the antenna, and again the grooves will have little effect. An intermediate effective height thus exists for which the energy density in the grooved region reaches a peak value. We would therefore expect that α could be maximized by an optimal combination of groove depth and effective height. The curve of αλ0 versus t/λ shown in Figure 11-13 illustrates precisely such behavior for the structure treated there. Similar qualitative reasoning applies to other periodically modulated open dielectric structures. 26 FIGURE 11-13 Calculations for the relative leakage constant and the radiation angle as a function of groove depth for the dielectric image guide with a grating of grooves shown in Figure 11-11a (after Schwering and Peng7 © IEEE 1983) Design Theory for Narrow Periodic Dielectric Antennas The theoretical results referred to in this subsection make the assumption that the dielectric structures have infinite width w. It has been found, however, that they apply quite accurately to “wide” structures, for which 0 / 1.effw λ ε> − If εeff = 2, for example, these results are applicable to antennas for which w > λ0. For εεff = 6, on the other hand, w need satisfy only w > 0.45λ0. Thus the theory for “infinite” width can actually be applied with good accuracy to structures that are fairly narrow. When the antennas are narrower than the criterion just mentioned will allow, a correction scheme is available that yields accurate results for them as well. For antennas of narrow width 0( / 1),effw λ ε< − the value of β can be derived in a simple way with good accuracy by using the EDC (equivalent dielectric constant) procedure, but α can no longer be approximated by that of an infinitely wide antenna with the same dielectric constant as the finite antenna. As w decreases, the phase velocity of the leaky mode increases, so that an increasing portion of the guided energy now travels in the air regions on both sides in the neighborhood of the perturbing mechanism, whether grooves or metal strips. As a result, the leakage constant α of the narrower antenna becomes smaller. A simple procedure has been developed52 that employs the EDC method to replace the antenna of finite width by an equivalent antenna of infinite width, but with a lower effective dielectric constant. As a result, a lower value of α is obtained. This procedure has produced numerical values for α and β that have agreed very well with measured results for a quite narrow antenna, with εr = 16 and w = 1.3 mm, in the frequency 27 range from 30 to 36 GHz.52 Some details regarding this procedure may also be found in Schwering and Oliner,46 pages 17-64 to 17-68. Periodic Leaky-Wave Antennas Based on Microstrip Line A wide variety of possible traveling-wave periodic array antennas can be achieved by employing microstrip line. As examples, you can employ a series of resonant patch antennas connected by the microstrip line, as seen in Figure 11-14a, or a series of array elements coupled by proximity to the microstrip line, as shown in Figure 11-14b. Top views of the structures are presented. Another wide class of possibilities is illustrated in Figure 11-15a through c, and these involve periodic meanderings of the microstrip line strip itself. These last structures are reminiscent of the sandwich-wire antenna29,30 described earlier in the subsection “Early Structures.” These and other traveling-wave arrays based on microstrip line are discussed in detail in the two comprehensive books53,54 on microstrip antennas by James et al. and by James and Hall respectively. The original references for the antennas in Figure 11-14a and b are Derneryd55 and Cashen et al.56 respectively. FIGURE 11-14 Top views of traveling-wave periodic array antennas based on microstrip line: (a) series of resonant patch antennas connected by microstrip line (b) series of elementary radiators, resonant or not, coupled by proximity to the microstrip line (after James et al53) No theory is available for most of these antennas. For the few structures, such as the one shown in Figure 11-14a, for which some theory is available, the theory is of the type used to describe the behavior of slot arrays. All the thinking, in fact, parallels that employed for slot arrays. For example, the array elements are usually assumed to be individually resonant, although there is no reason why they need to be. The theory then treats each element as loading the line individually, instead of viewing the structure in leaky-wave fashion. (See the discussion in Section 11.1 on the relation between leaky-wave antennas and slot arrays.) Furthermore, in many cases the arrays are designed to be resonant (standing wave rather than traveling wave). The array in Figure 11-14a is specified as a resonant one, but an interesting traveling-wave modification has been reported.57 This modification contains additional phase shift between successive elements to reduce the frequency change needed to cover a given range of scan angle. 30 Early Structure: Asymmetrical Trough Waveguide Antenna The first open waveguide that was made leaky by introducing asymmetry in the cross section was the trough waveguide. The antenna structure, shown in Figure 11-17 in full view and in cross section, was invented and measured by Rotman64 and analyzed by Oliner65 about 40 years ago. It was a practical antenna, widely used, and is still useful today. It appears to be the only early example of the class of uniform leaky-wave antennas based on open waveguides. FIGURE 11-17 The uniformly asymmetrical trough waveguide antenna: (a) full view; (b) cross section for zero-thickness center fin showing the electric field orientations (after Rotman and Oliner65 © IRE (now IEEE) 1959) The trough waveguide itself, when operated as a nonradiating transmission structure, is symmetrical about the center fin and is derived from symmetrical strip transmission line by placing a short-circuiting plate at its midplane. The dominant mode in the trough waveguide is therefore identical with the first higher-order mode in stripline. Trough waveguide therefore combines the mechanical simplicity of a stripline with the frequency characteristics of a waveguide, and its bandwidth for single-mode propagation is greater than that for rectangular waveguide by about 50 percent. It also can be coupled smoothly to a coaxial line, a feature that makes it convenient for use at lower frequencies. Despite the fact that trough waveguide is open on one side, it is nonradiating when the structure is symmetrical. The introduction of asymmetry, however, will produce radiation in a leaky-wave fashion. In fact, one virtue of this type of leaky-wave antenna is the simple means by which radiation can be controlled. The asymmetry can be varied, for example, by placing a metal insert in one of the halves of the line, as shown in Figure 11-17, and adjusting its thickness d. 31 A theoretical analysis65 employing transverse resonance together with a perturbation treatment yields the following simple expressions for the leaky-wave propagation characteristics: 0 0 0 Re( ) 2 g g g c λλ λ λ κ λ λ π λ = − Δ (11-31) and )Im( 0 0 κ λ λ α Δ= c g (11-32) where ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + −=Δ 2121 1 4 )Re( LL d LL dπκ (11-33) and )(8 )()(Im 2121 2 LLLL d + =Δ πκ (11-34) with 2ln211 π bsL += and 2ln222 π bsL += (11-35) The structural dimensions s1, s2, d, and b are indicated in Figure 11-17; λc0 and λg0 are the cutoff wavelength and guide wavelength respectively of the symmetrical and therefore nonradiating trough waveguide. These expressions have been found to give rather good agreement with measured values,65 especially for narrow radiated beams. For wider beams it is better to use the transverse resonance expression itself (without the perturbation simplifications), which is also given and derived in Rotman and Oliner.65 Numerical data for various parameter combinations, and corrections to be made for center fins of appreciable thickness, are also found in this reference. Foreshortened-Top NRD Guide Antenna Nonradiative dielectric (NRD) guide is a low-loss open waveguide for millimeter waves that was first proposed and described in 1981.66 It is a modification of H guide where the spacing between the metal plates is less than λ0/2 so that all junctions and discontinuities that maintain symmetry become purely reactive instead of possessing radiative content. The waveguide structure is shown on the left-hand side of Figure 11-18. The dielectric material in the center portion confines the main part of the field, and the field decays exponentially in the vertical direction in the air region away from the dielectric-air interfaces. Two new leaky-wave antennas are based on this waveguide: the foreshortened-top NRD guide antenna and the asymmetrical NRD guide antenna. The former antenna is discussed now, and the latter one is treated in the next subsection. 32 FIGURE 11-18 The nonradiative dielectric (NRD) guide, on the left, and the foreshortened-top leaky- wave antenna based on it, on the right (after Sanchez and Oliner67 © IEEE 1987) When the vertical metal plates in the NRD guide are sufficiently long, as shown on the left-hand side of Figure 11-18, the dominant mode field is effectively completely bound, since the field has decayed to negligible values as it reaches the upper and lower open ends. If the upper portion of the plates is foreshortened, as seen on the right-hand side of Figure 11-18, the field is still finite at the upper open end (but negligible at the lower open end). A traveling-wave field of finite amplitude then exists along the length of the upper open end, and, if the dominant NRD guide mode is fast (it can be fast or slow depending on the frequency), power will be radiated away at an angle from this open end. An accurate theory was developed by Sanchez and Oliner,67 in which an accurate transverse equivalent network was developed for the cross section of the antenna, and the dispersion relation for the values of α and β was obtained from the resonance of this network. All the elements of this dispersion relation are in closed form, thus permitting easy calculation. The leakage constant α is determined simply by the amount of foreshortening, measured by d, and β is essentially unaffected by changes in d unless d becomes very small. Thus α and β can be adjusted independently, to a great extent, which is very desirable because the procedure for sidelobe control is then simplified. Careful measurements were taken by Yoneyama68 at 50 GHz and by Han et al69 in the vicinity of 10 GHz on a scaled structure. Excellent agreement between measurement and theory was found over the range of parameter values examined, and results for one case are given on Figure 11-19. You also can see that α can be varied over an extremely wide range, permitting narrow beams or wide beams, simply by altering the value of d. FIGURE 11-19 Comparison between measurement and theory for the leakage constant as a function of the length d of the foreshortened top for the NRD guide leaky-wave antenna shown in Figure 11-18 (after Han et al69 © IEEE 1987) 35 Groove guide is a low-loss open waveguide for millimeter waves somewhat similar to the H guide or NRD guide; the dielectric central region is replaced by an air region of greater width. The fields are again strongest in the central region, and they decay exponentially vertically in the regions of narrower width above and below. Groove guide was popular some years ago, but it lost its glamour when it was realized that its higher-order modes would radiate, so that discontinuities in the guide would have some resistive content. This feature limited the range of applications for which groove guide was suitable, but it did not influence its application to leaky-wave antennas. This antenna has a remarkable set of unusual virtues. One of the virtues is its versatility, which may be understood from the following considerations. When the stub is centered, the structure becomes nonradiating; alternatively, it may be viewed then as a slotted section cut in rectangular waveguide. For small off-center positions of the stub, the leakage rate will be small, yielding radiated beams of narrow width. When the offset is increased, α will increase, and the beamwidth will increase. We therefore have a relatively simple leaky-wave antenna, easily fed from a rectangular waveguide, that permits great versatility with respect to beamwidth by simply adjusting the location of the stub guide. Even more important, it is found that the value of β changes very little as the stub is moved, and α varies over a very large range. This feature, namely, that β remains almost constant while α varies, makes it easy to taper the antenna aperture to control sidelobes. Thus it should be easy with this antenna to design beams with low sidelobe levels. An additional advantage follows from the fact that the antenna is filled with only one medium, namely, air. As was shown in Section 11.2, the beamwidth then remains constant when the beam is scanned in elevation as you change the frequency. The antenna was analyzed using a transverse equivalent network based on a new E-plane T-junction network.79 This T-junction network is notable in that the expressions for the network elements are all in simple closed form and yet are very accurate. The resulting transverse equivalent network for the antenna is seen in Figure 11-23. Since the network elements are in closed form, the resonance relation for the complex propagation wave number is also in closed form, making calculations quick and easy. FIGURE 11-23 The transverse equivalent network for the antenna structure in Figure 11-22. Closed- form expressions for the network elements are given in the text. The expressions for the elements of the network in Figure 11-23 may be written as 20 0 16 2 2 a x xB k a k aa J Y b π ′ ′′ ⎛ ⎞= − ⎜ ⎟ ⎝ ⎠ (11-36) 36 where J0 is the Bessel function of zero order, ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =+ 2 2 00 2 143.1ln1 2 1 ππ bk a bbk nY B Y B xx c aL (11-37) 2/ )2/sin( ak ak n x x c ′ ′ = (11-38) )/(22 bann ccs ′= (11-39) 2 2 0 2 0 1 1ln 1.43 2 32 2 2 x xL c x x k b k bB b Y an k a k aa J b π π π ⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞= +⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟′⎝ ⎠⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ ′ ′′ ⎛ ⎞ ⎛ ⎞+ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ((11-40) so that the dispersion relation for the transverse wave number kx becomes 0 2 cot 2 cot2 2 cot 2 cot 1 0 00 0 2 = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′+ ′ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ′ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′+ ′ −⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ′ − ++ dakdak Y B dak Y B dak Y B j Y B j n xx a x a x a L cs (11-41) where wave number kx is related to kz, the result that we seek, by 220 xz kkjk −=−= αβ (11-42) These expressions, and the transverse equivalent network in Figure 11-23, assume that the stub guide is infinite in length. In practice, of course, the stub length is finite, and it should only be long enough that the vertical electric field (represented in the stub guide by the below-cutoff TM1 mode, viewed vertically) can decay to negligible values, permitting essentially pure horizontally polarized radiation. Usually, the stub length need only be about a half wavelength or less if the stub is narrow. The finite stub length can be readily taken into account, and a detailed treatment of its effects is reported in Oliner,63 but it produces only a small change in the numerical values obtained from the procedure given earlier. FIGURE 11-24 Comparison of numerical results for the stub-loaded rectangular guide antenna, obtained via two completely different theoretical methods. Additionally, we see that by varying the offset d, we can change α over a wide range of values while affecting β very little. Such independence makes it easy to taper the aperture distribution in order to control the sidelobes. f = 28 GHz, a = 1.0 cm, a′ = 0.7 cm, b = 0.3 cm. (after Oliner, with Shigesawa62) As an independent check on the accuracy of these expressions, the values of α and β were calculated using an entirely different theoretical approach, that of mode matching. As shown in Figure 11-24, where the 37 dashed lines represent values obtained using the network and the solid lines represent those derived via the mode-matching procedure, the agreement is seen to be very good. We also can see from this figure that, as the stub is shifted laterally, β remains almost constant as α varies monotonically from zero to large values. FIGURE 11-25 Photograph of the cross section of a stub-loaded rectangular guide antenna that was measured at millimeter wavelengths. A 500-yen Japanese coin (26.5 mm in diameter) is shown for size comparison. (after Oliner, with Shigesawa63) Measurements were also taken of the values of α and β and of the radiation patterns over the frequency range from 40 to 60 GHz.78 A photograph of the cross section of the structure appears in Figure 11-25. The Japanese 500-yen coin (26.5 mm in diameter) is seen to dwarf the antenna cross section. The comparisons between the theoretical and measured values for both α and β are found to be very good. The stub-loaded rectangular guide leaky-wave line source antenna is thus an attractive structure for millimeter wavelengths, since it is simple in configuration, easily fed, versatile in beamwidth, suitable for low-sidelobe-level designs, and capable of furnishing essentially pure horizontally polarized radiation. In addition, a simple and accurate theory is available for it that has been verified by an alternative, totally different computational approach, as well as by measurements. Printed-Circuit Version of Stub-Loaded Rectangular Waveguide Antenna If the versatile leaky-wave antenna just described could be made in printed-circuit form, the fabrication process could make use of photolithography, and the taper design for sidelobe control could be handled automatically in the fabrication. That is, the location and width of the stub, and their variations along the antenna length in conformity with the sidelobe design requirements, could all be accomplished at the same time by either depositing the metal or etching some away to produce the gap. With this goal in mind, a printed-circuit version of the structure in Figure 11-22 took the shape shown in Figure 11-26. 40 FIGURE 11-28 Schematic of the recent approach to simpler 2D scanning, involving a linear phased array of leaky-wave line sources, with frequency scanning in elevation and phase scanning in azimuth (after Oliner, with Peng62) The radiation will therefore occur in pencil-beam form and will scan in both elevation and azimuth in a conical-scan manner. The spacing between the line sources is chosen such that no grating lobes occur, and accurate analyses show that no blind spots appear anywhere. The leaky-wave line-source antennas employed in the three examples of arrays described here are ones we have already discussed earlier or are modifications of them. The advantage of negligible cross polarization at all angles follows from the fact that the individual line-source antennas possess that feature and that the array arrangement does not introduce any cross-polarized components. In principle, a large variety of different leaky-wave line sources can be used in this array architecture. In fact, however, you must be very selective here, because the line sources must be integrated into the overall geometry if the resulting antenna is to remain simple in configuration. The list is further limited to those structures amenable to analysis because design requires a theoretical basis. If the radiating portion of a suitable structure in the class can be fabricated by photolithographic means, using a mask, the costs can also be kept down, and the method is amenable to mass-production techniques. Two of the three arrays to be described fall into this category. Analytical Approach The arrays to be described have been analyzed accurately by a unit-cell approach that takes into account all mutual coupling effects. Each unit cell incorporates an individual line-source antenna, but in the presence of all the others. These individual line-source antennas were analyzed using a transverse equivalent network in which the radiating open end was representative of the environment of the single lone line source. In the array of such line sources, the radiating environment is, of course, quite different, and it will change as the array is scanned in azimuth. The treatment of the periodic external environment by a unit-cell approach automatically accounts for all mutual coupling effects and provides information on all the effects of scan. The radiating termination on the unit cell modifies the transverse equivalent network, and the resonances of this transverse network yield the properties of the leaky wave guided along the line sources. A key new feature of the array analysis is therefore the determination of the active admittance of the unit cell in the 2D environment as a function of scan angle. This active admittance is the input admittance to the external radiating region, and it is appended to the remainder of the transverse equivalent network, the latter being different for each of the arrays to be discussed. Array of Asymmetrical NRD Guide Line Sources A cross-sectional view of the linear phased array of asymmetrical NRD guide leaky-wave line sources appears in Figure 11-29. By comparison with the individual line-source antenna in Figure 11-21, it is clear that the array consists of a number of these line sources placed directly next to each other. The line sources provide the elevation pattern, modified by their presence in the array, and the geometry in the plane shown 41 specifies the cross-plane behavior, with the angle of scan determined by the phase shift imposed between successive line sources. FIGURE 11-29 Cross section of linear phased array of asymmetrical NRD guide leaky-wave line sources. The individual line source appears in Figure 11-30. (after Oliner, with Xu63) The unit cell for this array is shown in Figure 11-30, where the phase-shift-wall properties depend on the scan angle in the cross-plane. Thus the discontinuity at the end of the stub of length c, and therefore the values of β and α, depend on that scan angle. The analysis proceeds, therefore, by knowing the imposed phase shift between line sources, and, from it and the unit-cell network, by finding β and α in the axial direction. Everything else follows directly from this information. FIGURE 11-30 Unit cell of the linear array of NRD guide line-source antennas shown in Figure 11-29. The phase-shift walls change with scan angle, and their use leads to an analytical approach that takes all mutual coupling effects into account automatically. (after Oliner, with Xu63) If the values of β and α did not change with phase shift, the scan would be exactly conical. However, it is found that these values change only a little, so that the deviation from conical scan is small. We next consider whether blind spots are present. Blind spots refer to angles at which the array cannot radiate or receive any power; if a blind spot occurred at some angle, therefore, the value of α would rapidly go to zero at that angle of scan. To check for blind spots, we would then look for any sharp dips in the curves of α/k0 as a function of scan angle. No such dips were ever found. Typical data of this type exhibit fairly flat behavior for α/k0 until the curves drop quickly to zero as they reach the end of the conical scan range, where the beam hits the ground. The theoretical analysis and many numerical results appear in Chapter IV of Oliner63 and a presentation was made.81 Array of Printed-Circuit Uniform Line Sources If the leaky-wave line sources in the array were in printed-circuit form, the fabrication process could make use of photolithography, and the taper design for sidelobe control could be handled automatically in the fabrication. The array structure shown in Figure 11-31 fits into this category. It may be seen that the line- 42 source elements in this array are exactly the printed-circuit version of the stub-loaded rectangular waveguide antenna appearing in Figure 11-26. The leakage rate, and therefore the beamwidth, can again be controlled by varying the width or location of the gap within each element. And again, the value of β remains almost constant as the gap width or location is varied, so that a taper design for sidelobe control is easy to implement. FIGURE 11-31 Cross section of the linear phased array of printed-circuit uniform leaky-wave line sources. The individual line source appears in Figure 11-26. (after Lampariello and Oliner82) The metal fins that project vertically, which may alternatively be called baffles or stubs, serve two purposes. The first purpose is to ensure essentially pure horizontally polarized radiation, with negligible cross polarization. As explained in the discussion associated with Figure 11-27, the fins form a stub guide that is below cutoff for the vertical electric field, thereby permitting only the horizontal electric field to radiate. The second purpose of the stubs is to eliminate blind spots. With the stubs present, we have never found any, and it is known that arrays of this sort with dielectric layers often exhibit them when there are no stubs. A careful examination shows that the stub length should be roughly a half wavelength, which means that the projection is actually rather small, particularly at millimeter wavelengths. The transverse equivalent network for the unit cell representing the array in Figure 11-31 is very similar to the one for the isolated line-source antenna and differs from it only in the terminating admittance placed on the end of the stub line. This difference is important, however, in that the terminating admittance is a function of scan angle and takes into account the mutual coupling effects of all the neighbors in the array. As in the case of the array of asymmetrical NRD guide line sources, the value of β/k0 changes very little with the phase shift between successive line sources, so that, as a result, there is little deviation from conical scan. Also, the curve of α/k0 as the phase shift is changed does not show any sharp dips but remains fairly flat until it drops as the beam approaches the ground at the end of the scan range. Thus no blind spots have ever been observed. Calculated curves for one set of geometric parameters that demonstrate this behavior are given in Figure 11-32. FIGURE 11-32 Variations of the normalized phase constant and leakage constant with cross-plane scan for the array appearing in Figure 11-31, showing conical scan and no blind spots. f = 50 GHz, εr = 2.56, a = 2.25 mm, a′ = 1.00 mm, b = 1.59 mm, c = 6.00 mm, d = 0.25 mm. (after Lampariello and Oliner82) 45 The radiation produced may be a narrow pencil beam pointing at broadside (θ0 = 0), or a conical beam pointing at any desired scan angle θ0 > 0. As discussed later, the thickness of the dielectric layer controls the scan angle. For a conical beam, the pattern is usually fairly omnidirectional (azimuthally independent) for small angles θ0, but the E- and H-plane beamwidths typically become more different as the scan angle increases. A vertical dipole source can only produce a conical beam at a scan angle θ0 > 0, while a horizontal dipole source may produce either a broadside pencil beam or a conical beam at any desired scan angle θ0 > 0. An illustration of these two types of patterns is shown in Figure 11-35. Basic Principles of Operation One of the main differences between this type of structure and the ones considered previously is that the leaky wave on this structure is a 2D cylindrical wave, which propagates outward radially from the source along the interface.84 The leaky wave then furnishes a large aperture that in turn produces the narrow radiation beam. As is true for all leaky-wave antennas, the narrow beam angle and the beamwidth are frequency sensitive. A vertical electric or magnetic dipole source launches only a TMz or TEz leaky wave respectively, which has no φ variation. This results in an omnidirectional conical beam.84 A horizontal electric or magnetic dipole source launches a pair of leaky waves, one TMz and one TEz. The TMz leaky wave determines the E-plane pattern, while the TEz leaky wave determines the H-plane pattern.84 For a broadside beam these two leaky waves have very nearly the same phase and attenuation constants, and hence an omnidirectional pencil beam is created. Interestingly, this is true even if the PRS is not similar in the E- and H-plane directions. For example, the PRS may consist of a periodic array of slots in a metal plate, with the slots being long in the x direction and narrow in the y direction, having very different periodicities in the two directions. As the scan angle increases, the pencil beam turns into a conical beam, similar to how the petals on a flower unfold. As the scan angle increases (by increasing the frequency or the layer thickness) the wavenumbers of the two leaky waves typically begin to differ, and this explains why the beamwidths often become different in the principal planes. (The exact nature of the beamwidth variation with scan angle depends on the particular type of PRS.) FIGURE 11-35 An illustration of the beam types that can be realized by using a PRS leaky-wave antenna excited by a horizontal dipole source. (a) A pencil beam at broadside. (b) A conical beam pointing at an angle greater than zero. The dipole source launches a radially propagating cylindrical leaky wave, the phase fronts of which are shown by the dashed lines. (a) (b) Beam Source Beam Source 46 The PRS is used to create a leaky parallel-plate waveguide region, and the leaky waves are leaky (radiating) versions of the parallel-plate waveguide modes that would be excited by the source in an ideal parallel-plate waveguide, which results if the PRS is replaced by a perfectly conducting metal plate. This point of view allows for a simple design formula for the thickness of the dielectric layer in order to obtain a beam at a desired angle θ0 (either a broadside or a conical beam). The parallel-plate waveguide modes are described by n = 1, meaning that there is one half-wavelength variation vertically inside the parallel-plate waveguide. (Although large values of n could be used, this would result in a design that has a thicker dielectric layer.) The radial wavenumber of the TMz and TEz parallel-plate waveguide modes for an ideal waveguide would be 2 2 1k k hρ πβ ⎛ ⎞= = − ⎜ ⎟ ⎝ ⎠ (11-43) where h is the thickness of the dielectric layer and k1 is the wavenumber of the layer, which may also be expressed as k1 = k0 n1, where n1 is the refractive index of the layer. Using the simple relation β = k0 sinθ0 that is valid for any leaky wave, we obtain the result 0 2 2 1 0 / 2 sin h n λ θ = − (11-44) The location of the source usually has little effect on the pattern shape, since this is dictated by the leaky- wave phase and attenuation constants. The phase constant is primarily determined by the thickness of the dielectric layer (see Eq. 11-43), while the attenuation constant is primarily determined by the properties of the PRS. However, the power density at the peak of the beam will occur when a horizontal electric dipole source is placed in the middle of the dielectric layer, or a horizontal magnetic dipole is placed on the ground plane. A vertical electric dipole source maximizes the peak power density when it is placed on the ground plane, while a vertical magnetic dipole source does so when it is placed in the middle of the layer. Changing the peak power density, and hence the overall power radiated by the source, directly affects the input resistance of the source. The PRS may be either uniform (e.g., one or more dielectric layers) or periodic in one or two dimensions (e.g., an array of closely spaced wires, or a 2D array of slots in a metal plate). However, it is important to note that for the periodic PRS structures, the radiation still occurs via the fundamental parallel-plate waveguide modes, and not from a space harmonic of these modes. That is, the PRS acts as a quasi-uniform reflective surface, where the reflection coefficient of the fundamental parallel-plate wave determines the characteristics of the antenna. The physical principle of operation is thus as a quasi-uniform leaky-wave antenna, and not a periodic leaky-wave antenna that radiates from a space harmonic. To our knowledge, the first use of a quasi-uniform PRS to improve the antenna gain was described by von Trentini85 in 1956. A sketch of that structure was presented by the author and is repeated here as Figure 11- 36, in which he placed a source at P on a ground plane and then located a PRS parallel to the ground plane a distance l in front of it. He views the performance in terms of multiple reflections between the ground plane and the PRS, and then derives an expression for the resonance condition that yields maximum radiated power at broadside. In this derivation, he assumes that the PRS consists of an array of closely spaced parallel conducting wires oriented parallel to the electric field. With this model, he calculated the radiation patterns for several sets of dimensions. He also built and measured several antennas based on this PRS, and on others, including one consisting of an array of closely spaced circular holes in a metal plate. The source employed was a rectangular waveguide aperture, and measurements were made at a wavelength of 3.2 cm. The measurements and calculations for the main lobe of the radiation pattern agreed well with each other. An improved version of the von Trentini antenna has been developed recently by Feresidis and Vardaxoglou.86 These authors followed the von Trentini ray theory analysis, assuming the structure to have infinite extent, and employed a feed consisting of a rectangular waveguide built into the ground plane. The 47 new contribution, and the “optimization” to which they refer, applies to their PRS structure. They note that the antenna would have greater bandwidth if the phase of its PRS were to linearly increase with frequency, in effect compensating somewhat for the path length that the rays must traverse (or equivalently for the change in the electrical thickness of the dielectric layer—see Eq. 11-44). They therefore investigated PRSs loaded with several different elements, such as crossed dipoles, patches, rings, and square loops. They found that dipoles, or square or circular patches (or their complementary structures), particularly with close packing of the elements in the array, produced less of a variation of the beam with frequency. This slower variation was not found for crossed dipoles, square loops, and rings, even for close packing of these elements. They therefore chose to use arrays of closely spaced dipoles in their PRS. Dielectric-Layer PRS Structures Printed-circuit antennas, such as microstrip antennas, have many advantages for conformal applications, but they have the major disadvantage of low gain. During 1984 and 1985, Alexopoulos and Jackson published a pair of papers87,88 showing that the gain could be enhanced significantly by placing a dielectric superstrate, or cover layer, over the original dielectric layer. By choosing the layer thicknesses and the dielectric constant values appropriately, a large gain can be obtained for radiation at any desired angle, in the form of a pencil beam at broadside or a conical beam pointing at an angle θ0. The first of these two papers showed that a properly designed two-layer (substrate/superstrate) structure could produce such beams. The later paper88 by Jackson and Alexopoulos examined in greater detail the quantitative relationships between the radiation properties of the two-layer dielectric structure and the parameter values (dimensions and permittivities) of the structure. Neither paper recognized the gain enhancement effect as due to leaky modes, however. The substrate-superstrate geometry of the structure, identifying the various parameters, is shown in Figure 11-37. A horizontal electric dipole source is shown located in the substrate and is parallel to the ground plane and the dielectric interfaces. l PRS P 0 1 2 FIGURE 11-36 A ray explanation for the PRS-based leaky-wave antenna, as originally introduced by von Trentini (after von Trentini85 © IEEE 1990) 50 0 30 60 90 12 0 15 0 18 0 21 0 24 0 27 0 30 0 33 0 -5 0 -4 0 -4 0 -3 0 -3 0 -2 0 -2 0 -1 0 -1 0 h=1.333 cmh=1.9 cm 0 90 90 60 60 30 30 The limitation on the beam angle due to the higher-order parallel-plate modes may be overcome by using a dielectric layer with εr > 4/3, which allows for the beam angle θ0 to approach 90o before radiation from a secondary beam occurs.93 However, when using a dielectric layer with the patch-PRS, it is observed that undesirable secondary beams arise from another source, namely from the –1 space harmonic of the (perturbed) TM0 surface wave that is supported by the grounded dielectric layer (which is perturbed by the metal patches).93 This problem is avoided when using the slot-PRS structure since the guiding structure is now a perturbed parallel-plate waveguide (perturbed by the apertures) rather than a perturbed grounded slab. A beam approaching 90o may be realized by using a slot-PRS structure with a dielectric layer.94 However, as the beam angle increases, the E- and H-plane patterns become increasingly different, with the E-plane pattern broadening and the H-plane pattern narrowing. Very recent work has shown that it is possible to overcome the problem of different beam behaviors in the E- and H-planes by the use of a wire PRS (an array of closely spaced conducting wires) together with a dielectric layer that is air or a low-permittivity material.95 Evidently, this is because the characteristics of the wire PRS change with the angle θ0 in a manner so as to compensate for the natural change in the different principal planes as the beam angle changes (see the discussion in the next subsection). General Design Formulas The far-field pattern of a PRS leaky-wave antenna structure may be calculated by reciprocity, in which the far field is determined by illuminating the structure with an incident plane wave and calculating the field at the source dipole location.96 The plane-wave calculation may be carried out by using a simple transverse equivalent network (TEN) model, which is a transmission-line model that represents the field behavior in the plane-wave problem. The PRS is assumed to be lossless and infinitesimally thin in the vertical direction, and therefore it is represented as a shunt susceptance BL. Based on this simple model, formulas may be derived for the beamwidth and pattern bandwidth in the E- and H-planes, in terms of BL.96 The pattern bandwidth is defined from the frequency range over which the power density radiated at the angle θ0 changes by less than 3 dB from the maximum value obtained at the center frequency (for which the structure is designed to radiate at angle θ0). Table 11-1 shows the beamwidth in the E- and H-planes for three separate cases: (1) broadside (θ0 = 0o), (2) a general beam angle 0 < θ0 < 90o, and (3) end fire (θ0 = 90o). The formulas are expressed in terms of the normalized shunt susceptance LB = BL η0, where η0 is the intrinsic impedance of free space. It is seen that the beamwidths are equal at broadside, consistent with the fact that the beam is nearly omnidirectional. However, as the beam angle increases, the H-plane pattern becomes narrower, while the E-plane pattern FIGURE 11-39 H-plane radiation patterns (in dB) at 12 GHz for a broadside beam and a 45o conical beam, for the patch-PRS structure of Figure 11-34b. The patches have lengths (x-dimension) of 1.25 cm and widths (y-dimension) of 0.01 cm, with periodicities of 1.35 cm in the x direction and 0.3 cm in the y direction. The thickness of the dielectric layer is h = 1.33 cm (0.534λ0) for the broadside beam and h = 1.90 cm (0.761λ0) for the conical beam. The structure is excited by an x-directed infinitesimal horizontal electric dipole in the middle of the layer. (after Zhao et al93 © IEEE 2005) 51 becomes broader. A narrow beam can be obtained at the horizon in the H-plane (at least in theory) but not in the E-plane. It is noted that the beamwidth is inversely proportional to BL2 for a nonzero beam angle, but inversely proportional to BL for a broadside beam. This means that the PRS must be much closer to a perfectly reflecting surface in order to obtain a very narrow beam at broadside, compared with a nonzero beam angle. An analysis shows that the pattern bandwidth in the E- and H-planes is inversely proportional to BL2 for both a broadside beam and a conical beam. This means that for the same beamwidth the broadside pattern will have a much smaller pattern bandwidth than the conical beam. TABLE 11-1 Expressions for Beamwidth E-plane H-plane Broadside 3 1 2 22 L n Bπ 3 1 2 22 L n Bπ General beam angle 2 2 2 1 1 0 2 2 0 0 2 sin sin cosL n n B θ π θ θ − ( )32 21 0 2 0 2 sin sinL n B θ π θ − End fire Narrow beam not possible ( ) 3 2 1 2 2 1 L n Bπ − REFERENCES 1 W. W. Hansen, Radiating Electromagnetic Waveguide (1940): U.S. Pat. 2,402,622. 2 L. O. Goldstone and A. A. Oliner, “Leaky-Wave Antennas—Part I: Rectangular Waveguides,” IRE Trans. Antennas Propagat., vol. AP-7 (October 1959): 307–319. See Section II for a simple discussion of the properties of leaky waves. 3 N. Marcuvitz, “On Field Representation in Terms of Leaky Modes or Eigenmodes,” IRE Trans. Antennas Propagat., vol. AP-4 (July 1956): 192–194. 4 T. Tamir and A. A. Oliner, “Guided Complex Waves—Part I: Fields at an Interface,” Proc. Inst. Elec. Eng. (London), vol. 110 (February 1963): 310–324. 5 T. Tamir and A. A. Oliner, “Guided Complex Waves—Part II: Relation to Radiation Patterns,” Proc. Inst. Elec. Eng. (London), vol. 110 (February 1963): 325–334. 6 C. J. Sletten and G. R. Forbes, “A New Antenna Radiator for VHF-UHF Communications,” AFCRC-TR-57-114, Air Force Cambridge Research Center, Bedford, MA, 1957. See also the discussion in Chap. 10. 7 F. Schwering and S. T. Peng, “Design of Dielectric Grating Antennas for Millimeter Wave Applications,” IEEE Trans. Microwave Theory Tech., vol. MTT-31 (February 1983): 199–209. 8 J. R. James and P. S. Hall, “Microstrip Antennas and Arrays—Part 2: New Array-Design Technique,” IEE J. Microwaves Optics Antennas, no. 1 (1977): 175–181. 9 K. Solbach and B. Adelseck, “Dielectric Image Line Leaky-Wave Antennas for Broadside Radiation,” Electron. Lett., vol. 19 (August 1983): 640–644. 10 W. Rotman and S. J. Naumann, “The Design of Trough Waveguide Antenna Arrays,” AFCRC-TR- 58-154, Air Force Cambridge Research Center, Bedford, MA, 1958. 11 W. Rotman and A. A. Oliner, “Asymmetrical Trough Waveguide Antennas,” IRE Trans. Antennas Propagat., vol. AP-7 (April 1959): 153–162. 52 12 C. H. Walter, Traveling Wave Antennas (New York: McGraw-Hill, 1965). 13 F. J. Zucker, “Surface- and Leaky-Wave Antennas,” Chap. 16, in H. Jasik (ed.), Antenna Engineering Handbook, 1st Ed. (New York: McGraw-Hill, 1961). 14 T. Tamir, “Leaky-Wave Antennas,” Chap. 20, in R. E. Collin and F. J. Zucker (eds.), Antenna Theory, Part II (New York: McGraw-Hill, 1969). 15 A. A. Oliner and R. G. Malech, “Radiating Elements and Mutual Coupling,” in R. C. Hansen (ed.), Microwave Scanning Antennas, Vol. II, Chap. 2 (New York: Academic Press, Inc., 1966): 132–157. 16 C. H. Walter, Traveling Wave Antennas, Secs. 4 and 5 (New York: McGraw-Hill, 1965): 156–169. 17 L. O. Goldstone and A. A. Oliner, “Leaky-Wave Antennas—Part I: Rectangular Waveguides,” IRE Trans. Antennas Propagat., vol. AP-7 (October 1959): 307–319. 18 J. N. Hines, V. H. Rumsey, and C. H. Walter, “Traveling-Wave Slot Antennas,” Proc. IRE, vol. 41 (November 1953): 1624–1631. 19 V. H. Rumsey, “Traveling Wave Slot Antennas,” J. Appl. Phys., vol. 24 (November 1953): 1358– 1365. 20 A. L. Cullen, “On the Channel Section Waveguide Radiator,” Phil. Mag., vol. 40 (April 1949): 417– 428. 21 W. Rotman, “The Channel Guide Antenna,” Proc. Natl. Electronics Conf., vol. 5 (1949): 190. 22 F. L. Whetten and C. A. Balanis, “Meandering Long Slot Leaky-Wave Waveguide Antennas,” IEEE Trans. Antennas Propagat., vol. AP-39 (November 1991): 1553–1560. 23 R. F. Harrington, “Propagation Along a Slotted Cylinder,” J. Appl. Phys., vol. 24 (November 1953): 1366–1371. 24 L. O. Goldstone and A. A. Oliner, “Leaky-Wave Antennas—Part II: Circular Waveguides,” IRE Trans. Antennas Propagat., vol. AP-9 (May 1961): 280–290. 25 J. N. Hines and J. R. Upson, “A Wide Aperture Tapered-Depth Scanning Antenna,” Ohio State Univ. Res. Found., Report 667-7, Columbus, OH, December 1957. 26 R. C. Honey, “A Flush-Mounted Leaky Wave Antenna with Predictable Patterns,” IRE Trans. Antennas Propagat., vol. AP-7 (October 1959): 320–329. 27 D. D. King, “Dielectric Image Line,” J. Appl. Phys., vol. 23 (1952): 669. 28 H. W. Cooper, M. Hoffman, and S. Isaacson, “Image Line Surface Wave Antenna,” IRE Conv. Rec., Part 1 (1958): 230. 29 W. Rotman and N. Karas, “The Sandwich Wire Antenna: A New Type of Microwave Line Source Radiator,” IRE Conv. Rec., Part 1 (1957): 166. 30 W. Rotman and N. Karas, “The Sandwich Wire Antenna,” Microwave J., vol. 2, no. 8 (1959): 29. 31 T. Itoh and B. Adelseck, “Trapped Image Guide for Millimeter-Wave Circuits,” IEEE Trans. Microwave Theory Tech., vol. MTT-28 (December 1980): 1433–1436. 32 T. Rozzi and L. Ma, “Mode Completeness, Normalization, and Green’s Function of the Inset Dielectric Guide,” IEEE Trans. Microwave Theory Tech., vol. MTT-36 (March 1988): 542–551. 33 S. T. Peng and F. Schwering, “Dielectric Grating Antennas,” R&D Tech. Rep., CORADCOM-78-3, Fort Monmouth, NJ, July 1978. 34 F. Schwering and S. T. Peng, “Design of Dielectric Grating Antennas for Millimeter-Wave Applications,” IEEE Trans. Microwave Theory Tech., vol. MTT-31 (February 1983): 199–209. 35 F. Schwering and S. T. Peng, “Design of Periodically Corrugated Dielectric Antennas for Millimeter-Wave Applications,” Proc. 1982 Antenna Appl. Symp. (1982). 36 K. L. Klohn, R. E. Horn, H. Jacobs, and E. Freibergs, “Silicon Waveguide Frequency Scanning Linear Array Antenna,” IEEE Trans. Microwave Theory Tech., vol. MTT-26 (October 1978): 764– 773. 37 R. E. Horn, H. Jacobs, E. Freibergs, and K. L. Klohn, “Electronic Modulated Beam-Steering Silicon Waveguide Array Antenna,” IEEE Trans. Microwave Theory Tech., vol. MTT-28 (June 1980): 647– 653. 38 R. E. Horn, H. Jacobs, K. L. Klohn, and E. Freibergs, “Single-Frequency Electronic-Modulated Analog Line Scanning Using a Dielectric Antenna,” IEEE Trans. Microwave Theory Tech., vol. MTT-30 (May 1982): 816–820. 39 S. Kobayashi, R. Lampe, R. Mittra, and S. Ray, “Dielectric-Rod Leaky-Wave Antennas for Millimeter-Wave Applications,” IEEE Trans. Antennas Propagat., vol. AP-29 (September 1981): 822–824.