**5. Dispersion curves, spectral-gap**

Dispersion curves, describing how attenuation and phase vectors, solutions of dispersion equation (12), evolve, are a valid tool to study leaky waves.

As discussed previously, the radiation mechanism of higher order modes on microstrip LWA is attributed to a traveling wave instead of the standing wave as in patch antennas and above cutoff frequency, where the phase constant equals the attenuation constant (*c* = *<sup>c</sup>* ), it is possible to observe three different range of propagation: bound wave, surface wave and leaky wave [15]. At low frequency, below the cutoff frequency, we have the reactive region due to evanescent property of LWA.

From (3) we can observe that the leaky mode leaks away in the form of space wave when *K*<sup>0</sup> , therefore we can define the radiation leaky region from the cutoff frequency to the frequency at which the phase constant equals the free-space wavenumber 0 ( ) *K* . For ( ) *Ks* we have the bound mode region and for *K K* <sup>0</sup> *<sup>s</sup>* , exists a narrow frequency range ( <sup>1</sup> *<sup>s</sup> r K* ), in which we can have surface-wave leakage, where *Ks* is the surface

wavenumber.

Moreover the transition region between surface wave leakage and space wave leakage including a small range in frequency for which the solution is non-physical, and it therefore cannot be seen. For this reason, the transition region is called a spectral gap. Such a spectral gap occurs commonly (but not always) at such transitions in printed circuits, but it also occurs in almost all situations for which there is a change from a bound mode to a leaky

Fig. 14. The typical normalized attenuation constant, / <sup>0</sup> *k* , and phase constant / <sup>0</sup> *k* , in the direction of propagation of the first higher order mode, *TE*<sup>10</sup> . There are four frequency regions associated with propagation regimes: Reactive, Leaky, Surface, and Bound.

mode, or vice versa. For example, a spectral gap will appear when the beam approaches endfire in all leaky-wave antennas whose cross section is partly loaded with dielectric material. It is necessary to employ a greatly enlarged scale, on which the dispersion plot is sketched qualitatively. The transition region itself is divided into two distinct frequency ranges, one from point A to point B and the second from point B to point C. Before point B, a leaky wave occurs. As soon as frequency reaches 1*f* (point B), an improper superficial wave is solution of dispersion equation [9]. Because, both *<sup>z</sup>* and *<sup>z</sup>* cannot increase with frequency between 1*f* and 2*f* , their trend will change until point C, from which a confined superficial wave is an acceptable solution for increasing values of frequency.

To depict normalized constants behaviour around the spectral-gap, it's necessary a very precise numerical method since, leaving out particular structures, its width ( *f* ) is very small compared to working frequencies.

The dispersion characteristics for microstrip has been investigated by a number of authors using different full wave methods and evaluating different regimes of the dispersion characteristic.

The spectral domain analysis has proven to be one of the most efficient and fruitful techniques to study the dispersion characteristics of printed circuit lines [17]. As is explained in literature, the Galerkin method in conjunction with Parseval theorem can be used to pose the dispersion relation of an infinite printed circuit line as the zeros of the following equation:

*<sup>z</sup>* and

the direction of propagation of the first higher order mode, *TE*<sup>10</sup> . There are four frequency

mode, or vice versa. For example, a spectral gap will appear when the beam approaches endfire in all leaky-wave antennas whose cross section is partly loaded with dielectric material. It is necessary to employ a greatly enlarged scale, on which the dispersion plot is sketched qualitatively. The transition region itself is divided into two distinct frequency ranges, one from point A to point B and the second from point B to point C. Before point B, a leaky wave occurs. As soon as frequency reaches 1*f* (point B), an improper superficial wave

frequency between 1*f* and 2*f* , their trend will change until point C, from which a confined

To depict normalized constants behaviour around the spectral-gap, it's necessary a very

The dispersion characteristics for microstrip has been investigated by a number of authors using different full wave methods and evaluating different regimes of the dispersion

The spectral domain analysis has proven to be one of the most efficient and fruitful techniques to study the dispersion characteristics of printed circuit lines [17]. As is explained in literature, the Galerkin method in conjunction with Parseval theorem can be used to pose the dispersion relation of an infinite printed circuit line as the zeros of the following

superficial wave is an acceptable solution for increasing values of frequency.

precise numerical method since, leaving out particular structures, its width (

regions associated with propagation regimes: Reactive, Leaky, Surface, and Bound.

*k* , and phase constant / <sup>0</sup>

*k* , in

*<sup>z</sup>* cannot increase with

*f* ) is very

Fig. 14. The typical normalized attenuation constant, / <sup>0</sup>

is solution of dispersion equation [9]. Because, both

small compared to working frequencies.

characteristic.

equation:

$$F(k\_z) = \int\_{\mathbb{C}\_x} \tilde{\mathcal{G}}\_{zz}(k\_{x\cdot}k\_z) \tilde{T}^2(k\_x) dk\_x = 0 \tag{15}$$

where ( ) *T kx* , is the Fourier transform of the basis function ( ) *T kx* used to expand the longitudinal current density on the strip conductor as

$$J\_{sz}(\mathfrak{x}, z) = T(\mathfrak{x})e^{-jk\_z z}$$

The term ; , ( ) *G kk zz x z* is the zz component of the spectral dyadic Green's function, and *Cx* is an appropriate integration path in the complex *xk* plane to allow for an inverse Fourier transform non uniformly convergent function. The spectral dyadic Green's function has the following singularities in the complex *xk* plane: branch point, a finite set of poles on the proper sheet and a infinite set of poles on the improper sheet. For a fixed frequency, the function ( ) *F kz* is not uniquely defined because of the many possible different *Cx* integration paths that can be used to carry out the integral (15) [18].

Fig. 15. Transition region between leaky wave and confined superficial wave showing the spectral-gap occurring 1*f* and 2*f* .

The different *Cx* paths come from the different singularities of the spectral dyadic Green's function, that can be detoured around. For complex leaky mode solution, an integration path detouring around only the proper poles of the spectral dyadic Green's function is associated with an surface-wave leaky mode solution. If the path also detours around the branch points, passing trough the branch cuts and, therefore, lying partly on the lower Riemann sheet, the path will be associated with an space-wave leaky mode solution (see Fig. 16). This procedure, is not trivial. We shown in the next chapters how it is possible to extract the propagation constant of a microstrip LWA more simply using an FDTD code with UPML boundary condition, who directly solves the \_elds in the time domain using Maxwell's equations and with which the analysis is easy modifying the geometries of the LWAs. The results are in a good agreement with transverse resonance approximation (a full wave method) derived by Kuestner [19].

Fig. 16. Possible integration paths C . The three different are denoted as*C*0 for the real-axis path (bound mode solution), *C*1 for the path that detours around only the spectral dyadic Green's function poles (surface-wave leaky modes solution), and *C*2 for the path that also passes around the branch points (space-wave leaky modes solution).

### **6. Tapered leaky wave antennas**

Nowdays, some applications especially with regard to communication applications like the indoor wireless LAN(WLAN) actually are increasing the use of millimeterwave antennas like leaky-wave antennas (LWA), suited for more purpose. In detail, the transmitting/ receiving antennas with relatively broadbeam and broadband can be obtained from the curved and tapered leakywave structures. In fact, the microstrips (LWA), are very popular and widely used in applications thanks to their advantages of low-profile, easy matching, narrow beamwidth, fabrication simplicity, and frequency/electrical scanning capability. Is well know that the radiation mechanism of the higher order mode on microstrip LWAs is attributed to a traveling wave instead of the standing wave as in patch antennas. Moreover the symmetry of the structure along this physical grounding structure, thanks to the image theory, allows to design only half of an antenna with the same property of one in its entirety, and reducing up to 60% the antenna's dimensions. Using this tapered antenna we can obtained a quasi linear variations of the phase normalized constant and than a quasi linear variations of the its radiation angle. Moreover the profile of the longitudinal edges of the LWA, was designed, by means of the reciprocal slope of the cutoff curve, symmetrically to the centerline of the antenna, allows a liner started of leaky region.

Nevertheless the variation of the cross section of the antenna, allowing a non-parallel emitted rays, such as happens in a non-tapered LWA. In fact, using the alternative geometrical optics approach proposed in the tapering of the LWA, for a fixed frequency, involves the variation of the phase constant *β* and the attenuation constant *α*, obtained as a cut plane of 3D dispersion surface plot varying width and frequency. We can be determined a corresponding beam radiation interval with respect to endfire direction. As mentioned previously, for a tapered antenna with a curve profile (square root law profile) the radiation angle in the leaky regions, vary quasi linearly whit the longitudinal dimension, so it is possible to calculate the radiation angle of the antenna as a average of the phase constant using the simple formula.

Alternatively using the geometrical optics approach it is easy to determine the closed formula to predict the angle of main beam of a tapered LWA.
