**3.2 Building of NWSIW walls**

Contrarily to classical a SIW where walls are created by the insertion of metallic vias, the NWSIW architecture exploits the nanoporous nature of AAO to grow arrays of metallic nanowires (MNW) inside pores. Instead of bulk metal for classical RW having bulk conductivity σ, each wall of the NWSIW formed by the nanowire array has a total equivalent conductivity reduced by a factor P [20]. As a consequence, the skin depth δNW of the nanowire array expresses as

$$
\delta\_{NW} = \frac{2}{a\mu P\sigma} \tag{1}
$$

The extinction thickness ET is here defined as the thickness of the nanowire array forming the MNW wall that ensures that 99.9% of the electromagnetic field is blocked/attenuated by the MNW wall:

*Challenges and Perspectives for SIW Hybrid Structures Combining Nanowires and Porous… DOI: http://dx.doi.org/10.5772/intechopen.105148*

$$\mathbf{0}.001 = e^{-ET/\delta\_{\text{MNN}}} \rightharpoonup \mathbf{ET} = \mathbf{7} \,\delta\_{\text{MNN}} \tag{2}$$

**Figure 3** shows the extinction thickness ET as a function of porosity, for different values of operating frequency, from DC up to 50 GHz, and for Copper (Cu) nanowires of conductivity given in **Table 1**. It is concluded that a thickness of 10 μm is sufficient to satisfy condition (2) for most values of porosity higher than 3% and frequency above 1 GHz.

Practically two rows of MNW are grown in the AAO template, having a 10 μm width and a length equal to the desired length of the NWSIW. The nanoporous AAO template imposes a close packing of the electrodeposited MNW array from which an efficient shield is obtained. This is different and easier than designing the diameter and spacing of metallic vias drilled in the dielectric substrate of classical SIWs.

#### **3.3 Propagation in NWSIW**

In this section the propagation of microwave signals is investigated through the description of the effective medium present in the waveguide, as influencing the propagation, and the formulation of the propagation constant and characteristic impedance of the waveguide.

### *3.3.1 Effective medium filling the NWSIW*

The porous medium resulting from alumina anodization was illustrated in **Figure 2**. The (complex) permittivity of the equivalent effective medium, noted εAAO, can be calculated using a simple volumetric law [20] involving porosity P of AAO and properties of bulk alumina reported in **Table 1**:

$$
\varepsilon\_{AAO} = P + (\mathbf{1} - P)\varepsilon\_r \left(\mathbf{1} - i \tan \delta\right) \tag{3}
$$

As illustrated in **Figure 4**, both dielectric constant (left figure) and loss tangent factor (right figure) of porous AAO show a linear dependence on porosity P.

**Figure 3.** *Extinction thickness (Eq. (2)) as a function of porosity P.*

#### **Figure 4.**

*Electrical properties of porous AAO depending on porosity P, according to Eq. (3). Left: Dielectric constant, right: Loss tangent factor.*

For P = 0, the dielectric constant is that of bulk alumina given in **Table 1**, while for P = 100% it corresponds to air since no more alumina is present.

The same is true for the loss tangent factor; it goes from 0.0125 for P = 0%, corresponding to the value in **Table 1** for bulk alumina, to zero since for P = 100% the substrate reduces to air having no significant losses.

It has to be noted however that for practical use, AAO templates having high porosity should be avoided since they are much more brittle.

#### *3.3.2 Propagation characteristics*

The propagation inside a NWSIW is very similar to that occurring in a classical MRW. Given its width noted W and height equal to the thickness T of AAO template given in **Table 1**, the complex propagation constant, noted γ, is given by the following general expression [23]:

$$\gamma = a + j\,\beta = \sqrt{\left(\frac{n\,\,\pi}{W}\right)^2 + \left(\frac{m\,\,\pi}{T}\right)^2 - \varepsilon\_{AAO} \left(\frac{2\,\,\pi f}{\varepsilon\_o}\right)^2} \tag{4}$$

In Eq. (4) m and n are indices associated to TEmn and TMmn modes of propagation in an MRW. In the case of NWSIW, the thickness of the AAO being much lower than the width W of the guide, the first modes of propagation are TEm0 modes. Eq. (4) indeed reveals that the propagation constant presents a cut-off phenomenon; propagation occurs (β > 0) only above a certain frequency named cut-off frequency and noted fc. Below fc, the signal is attenuated instead of propagating (β = 0, α > 0).

*Challenges and Perspectives for SIW Hybrid Structures Combining Nanowires and Porous… DOI: http://dx.doi.org/10.5772/intechopen.105148*

**Figure 5.**

*Cut-off frequency fc10 according to Eq. (5). Left: Function of porosity P, for W = 6 mm. Right: Function of NWSIW width W, for P = 50%.*

Derived from setting γ in Eq. (4) equal to zero, fc writes as:

$$f\_{c\ m0} = \frac{m\ c\_o}{2\ W\ c\_{AAO}}\tag{5}$$

where W is the width of the waveguide and co the light velocity in air. For the NWSIW the three lowest cut-of frequencies occur for TEm0 modes.

Two statements can be derived. At first, the permittivity of AAO filling the waveguide depends on porosity P. This obviously influences the cut-off, hence the propagation constant, by virtue of (4–5). This illustrated in **Figures 5** and **6**, left. As porosity P increases, the dielectric constant decreases so that cut-off moves to higher frequencies (**Figure 5**), and this is reflected in the behavior of propagation constant (**Figure 6**). For a proper operation, that is allowing propagation in the 10–60 GHz range, porosity P should not exceed 40%.

Secondly, similar conclusions can be drawn as concerns the influence of the width of the waveguide. The cut-off frequency decreases as the width of the NWSIW increases, and a width superior to 5 mm is necessary for obtaining propagation in the NWSIW starting at 10 GHz.

#### *3.3.3 Characteristic impedance of NWSIW*

The characteristic impedance of a NWSIW has an expression similar to an MRW, expressed here as function of geometrical parameters W and T of the waveguide [23]:

$$Z\_c = \frac{2\text{ T}}{W} \frac{j\text{ 2\pi f }\mu\_o}{\chi} \tag{6}$$

**Figure 6.**

*Frequency dependence of propagation coefficient β given by Eq. (4). Left: For different values of porosity P, and W = 6 mm. Right: For different values of NWSIW width W, and P = 50%.*

**Figure 7.**

*Frequency dependence of characteristic impedance Zc given by Eq. (6). Left: For different values of porosity P, and W = 6 mm. Right: For different values of NWSIW width W, and P = 50%.*

*Challenges and Perspectives for SIW Hybrid Structures Combining Nanowires and Porous… DOI: http://dx.doi.org/10.5772/intechopen.105148*

**Figure 8.** *Schematic representation of tapered microstrip-to-NWSIW transition for impedance matching.*

Introducing expression (4) of γ into (6), the characteristic impedance Zc can be represented in **Figure 7** as a function of frequency and for different values of porosity (left) and width of NWSIW (right). Resulting from definition (6) Zc has a singularity at cut-off frequency, since at this frequency γ present in the denominator of (6) tends to zero.

Another important feature is the low level of characteristic impedance far above cut-off. Even for the most favorable conditions, i.e. high porosity in order to decrease the effective permittivity inside the waveguide (P = 8O%), and moderate width of NWSIW (W = 4 mm), Zc does not exceed 10 Ω in the flat constant regime above 20 GHz. As a result, a mismatch occurs with respect to the conventional 50 Ω reference impedance used for measurements of microwave devices.
