**3.1 Simulation structures and conditions**

The near-field propagation analysis is implemented on the slot line and Vivaldi structures, as shown in **Figure 5**. A Rogers RT5880 (*ε<sup>r</sup>* ¼ 2*:*2 and tan *δ* ¼ 0*:*0009 at 10 GHz) dielectric substrate of width *W*, length *Ls* and thickness *h*ð Þ �*h*≤*z*≤0 is used for the two structures. The metallic objects of the structures are built on the copper layer with a thickness of *θ* ð Þ 0≤*z*≤*θ* .

The open-ended slot line consists of two copper patches with length *L* separated by a slot of width *s*. The symmetrical Vivaldi slot is built based on an exponential function:

**Figure 8.** *The MUSIC spectra and FDbM response signals at a point l* ¼ 30 *mm along the slot line edge.*

$$y = a(e^{\mu \underline{x}} - 1) + \frac{s}{2}, \quad 0 \le \underline{x} \le L,\tag{14}$$

where *a* is a *y*-axis scale factor, and

$$a = \frac{\left(\frac{W}{2} - \frac{s}{2}\right) - \varepsilon}{e^{pL} - 1},\tag{15}$$

where *ε* is the end segment width of the Vivaldi patch. The limits of the *x* variable and the *y* curve function are in the ranges of 0 ≤*x*≤*L* and *<sup>s</sup>* <sup>2</sup> ≤ *y*≤ *<sup>W</sup>* <sup>2</sup> � *ε*. **Table 2** shows the parameters of these structures.

The EM simulations were implemented in time domain with an open boundary condition. The distances from the structures to most of the boundaries are 5 mm, except at the radiation aperture boundary *xmax*, where the distance is 15 mm. Thus, the simulation volume is 85 � <sup>40</sup> � <sup>10</sup>*:*265 mm3. Differential waveguide ports with size of 4 � 4 mm<sup>2</sup> ð Þ *<sup>x</sup>* <sup>¼</sup> 0, �2 mm <sup>≤</sup>*y*<sup>≤</sup> 2 mm, �2 mm <sup>≤</sup>*z*≤2 mm excite these two structures, shown as red squares in **Figure 5**. An FMCW signal covering a band of 0–210 GHz and lasting 0.5 ns is the time-domain excitation signal. The observed signals are sampled at rate of 1 THz, and the sampled period is 1.25 ns.

#### **3.2 Propagation analysis on the conducting plane**

The *z* ¼ *θ* plane contains the most valuable field data because it includes the top conducting surface of the antenna structure. In this plane, the FDbM impulse response analysis with Gaussian windows for frequency limiting is implemented for all of the observed signals in different bands. The complex field data are postprocessed and presented in different forms, and the field vector magnitude quantities are normalized according to the maximum corresponding field vector magnitude at the excitation port position *<sup>x</sup>* <sup>¼</sup> 0, � *<sup>s</sup>* <sup>2</sup> ≤*y*≤ *<sup>s</sup>* 2 , *<sup>z</sup>* <sup>¼</sup> *<sup>θ</sup>* .

#### *3.2.1 Space distribution of field intensity*

The intensity distributions are examined based on the maximum (over time) of the impulse responses of EM field vector magnitudes in the two analysis bands as shown in **Figure 9**. Noticeably, the strongest EM field intensities distribute along the slot line and Vivaldi slot, especially at the slot conduction edges. **Figure 9(b)** and **(c)** shows that because of the discontinuity in the structure of the reconstructed


**Table 2.** *Parameters of the slot line and Vivaldi antenna.* *Near-Field Propagation Analysis for Traveling-Wave Antennas DOI: http://dx.doi.org/10.5772/intechopen.100856*

**Figure 9.**

*The maximum EM field vector magnitudes on the conducting plane of (a) a slot line and (b) and (c) a Vivaldi antenna in the analysis bands of (a) and (b) 0–30 GHz and (c) 0–60 GHz.*

meshed Vivaldi slot, as mentioned in the Section 2.1.1, at these discontinuous locations, there are abrupt changes in the spatial distribution of field intensities. This is an expression of scattering phenomena in the propagation along the tapered structure of the Vivaldi slot.

These plots also reveal other features of the propagation, for example while the E-field intensity increases at the endpoints ð Þ *x* ¼ 60 mm of the slot line and Vivaldi slot, the H-field intensity reduces at these endpoints. The dielectric material transition from RT5880 to vacuum at the substrate edge ð Þ *x* ¼ 65 mm in the radiating aperture region leads to a small abruption in the space distribution of the E-field intensity at the substrate edge. At the excitation port location, because of the limit in port width in the y direction, the H vectors change direction and their distribution is abrupt around the port edges ð Þ *x* ¼ 0, *y* ¼ �2 mm . This leads to EM energy of the excitation source being dispersed into multiple directions other than þ*x*, as per the port mode definition, and the port edges act like scattering sources.

## *3.2.2 Space distribution of first-ToA clusters and MUSIC spectra*

At each point in space, the impulse response analysis result is a time distribution of clusters, and this distribution reveals propagation path information such as the number of paths, time delay, and attenuation characteristic. If the time distribution of a certain cluster can be identified for each point in space, then the space distribution information of this cluster is determined. This information reveals the effects of the structure's spatial characteristics on the cluster's propagation.

In propagation characterization, the most important cluster is the earliest- or first-ToA cluster. This cluster is formed by the propagating EM energy flows from the source over the shortest path with the fastest velocity. In this analysis, the ToA of the first cluster of the EM field vector magnitudes is estimated for each point in the conducting plane based on a local peak-finding algorithm. Direction and magnitude information of the field vectors corresponding to the first clusters is derived as illustrated in **Figures 10** and **11**. These results reveal how the first clusters of EM fields depend on material and spatial characteristics of the structure. **Figure 12** presents the space distribution of firstcluster ToAs or propagation times from the source. ToA contour lines play the role of two-dimensional wavefront and present visually the effects of the material and spatial characteristics of the structure on the clusters' propagations.

However, due to spreading of the analysis impulse in time and the overlapping of multiple EM flows, the total response signals can be canceled, flat or not distinguishable as separate clusters. In this case, ToA of the clusters cannot be estimated *Antenna Systems*

**Figure 10.**

*The EM field vectors at the time of the first peaks on the conducting plane of (a) a slot line and (b) a Vivaldi antenna in the analysis bands of 0–30 GHz.*

#### **Figure 11.**

*The first-peak EM field vector magnitudes on the conducting plane of (a) a slot line and (b) and (c) a Vivaldi antenna in the analysis bands of (a) and (b) 0–30 GHz and (c) 0–60 GHz.*

#### **Figure 12.**

*The ToAs of the first clusters based on the first peaks of the EM field vector magnitudes on the conducting plane of (a) a slot line and (b) and (c) a Vivaldi antenna in the analysis bands of (a) and (b) 0–30 GHz and (c) 0–60 GHz.*

*Near-Field Propagation Analysis for Traveling-Wave Antennas DOI: http://dx.doi.org/10.5772/intechopen.100856*

**Figure 13.**

*The ToAs based on the MUSIC spectra of EM field vectors on the conducting plane of a Vivaldi antenna in the analysis band of 0–60 GHz.*

exactly, or the clusters cannot be distinguished in the time domain, and this can lead to significant estimation errors in the analysis results.

As described in Section 2.2.2, the MUSIC algorithm is used in this analysis for estimation of the ToA. The results of estimated ToAs are shown **Figure 13** based on the peaks of three MUSIC spectrum components summation of EM field vectors in the analysis band of 0–60 GHz and with MUSIC parameters *N*<sup>0</sup> ¼ 1 and *D* ¼ 35. Instead of presenting the ToA of the first cluster as the FDbM impulse response analysis, the result based on MUSIC spectra presents ToAs of the dominant component of the cluster with a certain correlation with the impulse. The comparison between **Figures 12(c)** and **13** shows that some dominant components are estimated by the MUSIC algorithm emerge in some regions, and their MUSIC ToAs are later than the FDbM impulse response ToAs. For example, at the lateral edges of the antenna, there are the scattered components from the Vivaldi endpoints, and these components cannot be shown in **Figure 12(c)** due to domination of the first clusters.

#### **3.3 Propagation analysis on the edge of the slots**

In this section, detailed near-field examinations of the slot line and Vivaldi slot conducting edge propagation are made along the *l*-axes from the sources as illustrated in **Figure 5**.

#### *3.3.1 Field intensity on the edges*

The stepped or staircase meshed structure of the Vivaldi antenna is different from the constant distance between the two parallel slot edges of the slot line. This structural feature is the cause of abrupt changes in intensity of the EM responses at transition positions along the Vivaldi edges as seen in **Figures 9** and **11**. In this section, this is investigated thoroughly based on the first-peak field vector magnitude analysis versus the distance from source *l*. The field vector magnitudes are normalized according to the first-peak vector magnitude of the corresponding field source.

**Figure 14** shows that the propagations on the slot line and the Vivaldi slot are similar in the first segment 0ð Þ ≤ *l*≤ 24 mm . The transition positions on the Vivaldi edge lead to scattering at these points, and both the E and H vectors magnitudes increase abruptly after the transition points. However, because of expansion in slot width after each transition position and this scattering, the concentration of EM energy flows on the edge in the þ*l* direction is reduced after each transition position. Faster reduction of the magnitudes versus *l* after each transition position

**Figure 14.**

*The first peak of the EM field vector magnitudes on the edge of the slot line and Vivaldi slot versus the distance from source l.*

demonstrates this spreading. Superposition between the incident first cluster from the source and the dominant scattering/reflecting cluster from the endpoint of the slot edges increases the E field vector magnitude and reduces the H field vector magnitude.
