Ultrasubwavelength Ferroelectric Leaky Wave Antenna in a Planar Substrate-Superstrate Configuration

The possibility of achieving directive fan-beam radiation with planar Fabry-Pérot cavity antennas constituted by an upper ferroelectric thin film and a lower ground plane having ultrasubwavelength thickness is studied by means of a simple transverseequivalent-network approach and a cylindrical leakywave analysis, deriving simple design formulas. The performance of the proposed antenna is investigated in terms of power density radiated at broadside and directivity in the principal planes, pointing out the main limitations and tradeoffs associated with the reduced thickness.


Introduction
Planar antennas based on partially reflecting surfaces (PRSs) have been receiving increasing interest during the last decade, thanks to their attractive features of low profile, high directivity, and simple feeding.Since the first design proposed and realized by von Trentini in 1956 [1], in which the directivity of truncated waveguides opening in an infinite conducting ground plane was increased by means of periodic metal screens, a number of different PRS realizations have been proposed, based, for example, on dielectric substrate-superstrate configurations [2] or on two-dimensional periodic array of metal patches [3,4] or slots cut in a top plate [5].
The typical thickness of PRS-based planar antennas is of the order of one-half of the free-space wavelength.In fact, the directivity enhancement afforded by the PRS can be attributed to the resonant behavior of the partially open planar cavity formed by the ground plane and the PRS; the antenna can thus be viewed as a sort of Fabry-Pérot resonator (hence the name Fabry-Pérot cavity (FPC) antennas) [2,6].Alternatively, the antenna can be viewed as a twodimensional leaky wave antenna, in which the primary source excites a pair of cylindrical TM z and TE z leaky modes; these propagate along the structure and give the main contribution to the antenna aperture field, thus establishing the main features of the antenna radiation pattern [7,8].A leaky wave analysis of FPC antennas showed that the power density radiated at broadside is maximum when the thickness of the cavity is optimum, that is, close to half of the wavelength inside the substrate above which the PRS is placed; when such an optimum thickness is used to create a broadside beam, the leaky modes supported by the structure have nearly the same wavenumber with nearly equal low values of the phase and attenuations constants [9][10][11] thus producing a highly directive broadside beam.
In this paper we study PRS-based planar antennas in which the total thickness is considerably smaller than the values typical of the usual FPC antennas [2,7] and which has been the subject of many investigations [12][13][14].A preliminary investigation on this kind of structure has been presented in [15].Here we consider a planar substrate constituted by a conventional dielectric covered by a thin ferroelectric film.The structure is similar to that presented in [16] where the ferroelectric film was used to allow for an electrical tuning of the main parameters characterizing the radiation pattern of the antenna.Here the ferroelectric film is used to obtain an equivalent capacitive PRS which allows for dramatically reducing the total thickness of the antenna capable of obtaining a directive beam.The performance of the proposed antenna is investigated in terms of power density radiated at broadside and directivity in the principal planes, pointing out the main limitations and tradeoffs associated with the reduced thickness.

Structure Description
The reference structure, shown in Figure 1, is a dielectric slab (the substrate) of thickness  and relative permittivity  1 , bounded below by a perfectly conducting (PEC) ground plane and covered by a ferroelectric film of thickness  with permittivity  2 =   2 −   2 .The materials are assumed to be nonmagnetic (i.e.,  1 =  2 = 1).The structure is assumed to be excited by a unit-amplitude (1 V⋅m) horizontal magnetic dipole (HMD) directed along the -axis and placed over the ground plane, thus modeling a thin slot etched on the ground plane and excited by a suitable waveguide structure (e.g., a microstrip line).This kind of source is chosen instead of the more conventional horizontal electric dipole (HED) since in an ultrasubwavelength configuration the latter would radiate poorly due to the electrical closeness to the PEC ground plane.A time-harmonic dependence exp() of sources and fields will be assumed and suppressed throughout.

Antenna Radiation
3.1.Far-Field Pattern via Reciprocity.The far-field pattern radiated by the HMD in the presence of the planar structure described above can be calculated through a standard application of reciprocity theorem, by letting a plane wave impinge on the structure from the observation direction (, ) and calculating the reaction between the resulting total field and the HMD source [2,6].The field produced by the incident plane wave can in turn be readily calculated using the transverse equivalent network (TEN) representation of the structure shown in Figure 2(a), where the substrate and superstrate layers are modeled with lengths of transmission lines (TLs).
Assuming that the relative permittivity of the ferroelectric film is very large (| 2 | ≫ 1),  2 =  0 √ 2 − sin 2  ≅  2 =  0 √ 2 results and hence  2 ≅ √ 2 / 0 for both TM and TE polarizations (here  0 = 2/ 0 and  0 are the free-space wavenumber and characteristic impedance, resp.).Assuming further that the electrical thickness of the ferroelectric film is small, the relevant length of TL can be replaced by a shunt admittance   =   +   =  0  2 / 0 for both TM and TE polarizations [16,17], thereby obtaining the TEN in Figure 2(b).Using the latter TEN, the expressions of the farfield pattern are obtained as where and the TL parameters are (3)

Design Equations for Maximum Broadside Radiation.
From (1), at broadside we have where  1 = √ 1 , b = / 0 , and   =  0   .For very thin substrates (i.e.,  1 b ≪ 1), (4) can be approximated using a Taylor expansion of the trigonometric functions as Then, for a high-permittivity low-loss ferroelectric medium [18][19][20],   ≫ max{1,   } results and the absolute value of the expression in ( 5) is maximized when 2 b  − 1 = 0; that is, Therefore, very thin substrates require large values of the normalized (capacitive) susceptance   .In such a case, the power density at broadside is given by Therefore, the power density radiated at broadside is enhanced of a factor  2  with respect to the case of a magnetic dipole radiating on a PEC ground plane with free space above.(This may be contrasted with the case of a HED excitation [15], in which no such enhancement is found because the short-circuiting effect of the PEC ground plane compensates the resonant effect of the PRS.)To establish whether the enhancement of power density is accompanied with an enhancement of directivity as well, we have to determine if the structure can support leaky waves with small values of the phase and attenuation constants.

Leaky Wave Analysis
The dispersion equation coincides with the denominator in (1); that is, (neglecting   ≪   , [18][19][20]) We assume a solution for the normalized longitudinal propagation constant k of the form where   and   are "small" (|  | ≪ 1 and |  | ≪ 1), so that the normalized transverse propagation constants k0 and k1 in air and inside the slab, respectively, are approximated as International Journal of Antennas and Propagation while the cotangent function in ( 9) can be approximated as Now we have to consider separately the two cases of TE and TM modes.

TE Leaky Waves.
For TE modes the transverse characteristic impedances are Based on the above approximations, ( 9) can be written as That is, By neglecting all the terms ( 4 ), (15) is written as Under the optimum condition (4) (or (5)), ( 16) becomes That is, By equating the real and the imaginary parts of (18) we obtain Equations ( 19) contradict the assumption of "small"   and   , so that no TE leaky waves exist in this case with β = α and α ≪ 1.

TM Leaky Waves.
For TM modes the transverse characteristic impedances are and ( 9) can be approximated as By neglecting all the terms ( 4 ), ( 21) is rewritten as Under the optimum condition (4) (or (5)), ( 22) becomes from which . ( For thin substrates the susceptance   is large (see (6)) so that (24) can be approximated as By equating the real and the imaginary parts of (25) we obtain In the limit of large   , the RHS of the first of (26) can be neglected with respect to the RHS of the second.Finally, from the second we obtain Since the radiation pattern in the  plane is mainly determined by radiation from TM leaky modes, a directive broadside beam is expected in the  plane for sufficiently large values of the normalized shunt susceptance   .Alternatively, (27) can be written in terms of the normalized thickness as which shows that a directive broadside beam in the  plane requires very thin substrates.

Pencil-Beam Radiation
Based on the above analysis a fan shape of the radiation pattern can be expected, directive in the  plane and nondirective in the  plane.In fact, as it will be shown in Section 6, when the antenna is designed for maximum broadside power density the pattern in the  plane is almost constant.Since the pattern in ( 1) is of BOR 1 type [21], that is, this is equivalent to saying that   () is peaked and   () is almost constant.Under this condition, it is easy to show that the maximum directivity of the pattern cannot be larger than 2: (e.g., for a HMD in free space, it is equal to 3/2, as is well known).If a pencil beam is desired, with narrow patterns in both principal planes, a linear array of HMD sources may be employed.Since the element pattern is directive in the  plane (orthogonal to the HMD direction) and nondirective in the  plane (parallel to the HMD direction), the elements of the array should be aligned along the HMD direction (i.e., in the cases considered so far, along the -axis).Assuming for simplicity a uniform equispaced array with interelement spacing  and 2+1 elements, placed symmetrically with respect to the origin, the resulting array factor is (, ) = sin[(2+1)/2]/ sin[/2], with  =  0  sin  cos  [22].In this way the  plane array pattern is determined by the element pattern, whereas the  plane array pattern is determined by the array factor.In particular, for small  plane beam widths    3 dB and a small array spacing (i.e.,  0  ≪ 1), approximately results [21].On the other hand, since the  plane element pattern is due to radiation from a TM leaky wave, using (7) the  plane array beam width   3 dB can approximately be evaluated as [9] with the approximation being more accurate for small beam widths (i.e., large normalized PRS susceptances).The array parameters, that is, the number  and the spacing , can then be determined through (31) and (32), for example, by requiring that the 3 dB beam width of the array factor in the  plane be equal to that of the element pattern in the  plane.Finally, for narrow beams the resulting broadside directivity can be estimated as [22]  max ≃ 9.9 It is worth noting that, remaining within the class of directive planar antennas, a similar performance in terms of directivity could be achieved also with ordinary phased arrays of, for example, metal patch antennas.As already mentioned, these structures would require a significant larger overall thickness; moreover, by considering truncated structures, whereas the transverse dimensions would be comparable, for these conventional classes of antennas, a 2D instead of a 1D array would be required, with a considerably larger number of radiating elements and hence a higher cost.

Results and Discussion
In order to check the accuracy of the asymptotic expressions derived in the above sections, a grounded dielectric slab covered with a capacitive PRS consisting of a lowloss ferroelectric thin film is considered with  1 = 2.2 at the operating frequency  = 1GHz.The values of the phase and attenuation constants have been numerically calculated as a function of the normalized shunt susceptance   representing the ferroelectric thin film in the transverse equivalent network.For each value of   , the thickness  of the slab has been calculated according to the optimization condition in (6).In particular, in Figure 3, the exact results are compared with the approximate asymptotic ones for the TM leaky mode.The asymptotic approximate expression ( 28) is seen to be very accurate in a wide range of values of   , starting approximately from   = 10.As expected, no TE leaky waves are found.In Figure 4, the optimum subwavelength thickness is reported as a function of   .
We then consider the same substrate with a fixed thickness  equal to  0 /70 at the frequency  = 1GHz (i.e.,  = 4.28 mm); according to (5), in order to have maximum broadside radiation at  = 1 GHz a normalized susceptance   = 11.14 is required, which can be obtained, for example, with a ferroelectric film having  = 2 mil and  2 = 6.68 ⋅ 10 3 [20].In Figure 5, the dispersion curve of the TM leaky mode is reported as a function of frequency, together with the power density radiated at broadside by a horizontal magnetic dipole on the ground plane.It can be seen that, at  = 1GHz, βTM ≃ αTM ≃ 0.31 results which is the value predicted by the approximate formula (28).It can also be observed that the broadside power density is maximum at  = 1 GHz.
In order to investigate the radiation properties, radiation patterns in the  and  planes are reported in Figure 6  Parameters:  1 = 2.2;  = 1 GHz; for each value of   , the thickness  of the slab has been calculated according to the optimization condition in (6).frequency  = 1 GHz.It can be seen that the pattern is flat in the  plane (no TE leaky wave is excited), while it is quite directive (at broadside) in the  plane (due to the excitation of a TM leaky wave).
More directive beams can be obtained with thinner substrates (and, accordingly, larger values of the normalized susceptance).An example is shown in Figures 7 and 8, where the same structure of Figures 5 and 6 has been considered but with a thickness  =  0 /300 at the frequency  = 1 GHz, for example,  = 1mm (according to (5), a normalized susceptance   = 47.75 is required).
In Figure 7, the dispersion curve of the TM leaky mode is reported as a function of frequency, together with the power density radiated at broadside by a horizontal magnetic dipole over the ground plane.It can be seen that, at  = 1 GHz, βTM ≃ αTM ≃ 0.15 results which is the value predicted by the approximate formula (28).It can also be observed that the broadside power density is maximum at  = 1 GHz and, with respect to the structure of Figure 5, the broadside bandwidth has been reduced.
The radiation patterns in the  and  planes are reported in Figures 8(a) and 8(b) at the frequency  = 1 GHz.Again a flat pattern is radiated in the  plane, whereas a highly directive beam at broadside is radiated in the  plane.
In Figure 9 the patterns at different frequencies are reported in order to verify the scanning features typical of a leaky wave antenna.The radiation patterns in the principal planes are so different because in the  plane the pattern is mainly determined by a weakly attenuated TM leaky wave; hence it is directive; on the other hand, as shown in Section 4, no weakly attenuated TE leaky waves exist in the considered structure at the same frequencies; hence the pattern in the  plane is not directive.Please note that the radial scale in the plot of Figure 9(a) is linear (not in dB); hence both the  plane and the  plane patterns are very broad at 0.8 GHz.(The pattern in the  plane is however narrower than at 1 GHz; this may be due to the excitation at 0.   with a large attenuation constant, which is not significant at 1 GHz and whose existence cannot be excluded from the analysis presented in this study.)Note that in the scanned case the maximum power density in the  plane is much lower than the maximum in the  plane; hence the nondirective  plane pattern is not visible in Figure 9(b).
The presence of losses in the ferroelectric film negligibly affects the performance of the antenna since it has been shown that doped relaxor ferroelectric materials may have very low values of the loss tangent [20].
Finally, the case of a linear array of HMD sources is considered, in order to obtain broadside pencil-beam radiation.As explained in Section 5, by aligning the sources along the HMD direction the almost isotropic pattern in the  plane can be made directive; in particular, the same structure as in Figure 9 has been excited by an array of 21 HMD elements with interelement spacing  =  0 /10, in order to obtain the same 3 dB beam widths in both principal planes.The relevant radiation patterns are shown in Figure 10; it can be seen that the principal-plane beam widths are indeed equal (while the secondary lobes are higher in the  plane due to the uniformamplitude excitation of the array).

Conclusion
The possibility of achieving directive fan-beam radiation with planar antennas having ultrasubwavelength thickness has been demonstrated.The proposed antenna consists of highpermittivity superstrate (e.g., a low-loss ferroelectric thin film) above a ground plane; the primary source is a thin slot etched in the ground plane.In particular, it has been shown analytically that this kind of structure supports a weakly attenuated TM leaky wave which is responsible of a highly directive beam in the  plane of the structure.By using a uniform array of slot sources aligned along the  plane the radiation pattern can be made directive in the  plane as well, thus obtaining a pencil broadside beam.Numerical results are provided which confirm the conclusion theoretically derived.Future work will concern the issues related to the practical excitation of these structures, that is, the feed design and the relevant analysis of the antenna input impedance.

Figure 1 :
Figure 1: Planar substrate-superstrate configuration excited by a magnetic dipole over the ground plane with the relevant geometric and physical parameters. b

Figure 2 :
Figure 2: Transverse equivalent network of the structure in Figure 1.Exact representation (a) and approximate representation (b) valid for a high-permittivity thin-film superstrate.

Figure 3 :
Figure 3: TM leaky wave normalized phase / 0 and attenuation / 0 constants as a function of the normalized susceptance   representing the ferroelectric thin film for the proposed antenna.Parameters:  1 = 2.2;  = 1 GHz; for each value of   , the thickness  of the slab has been calculated according to the optimization condition in(6).

Figure 4 :Figure 5 :
Figure 4: Optimum subwavelength thickness for directive TM radiation as a function of   .

Figure 6 :
Figure 6: Radiation patterns in the  and  planes at  = 1 GHz for a structure with  =  0 /70 and   = 11.41.
8GHz of a TE leaky wave