# Mode Profile Shaping in Wire Media: Towards An Experimental Verification
## 3 Modulated Wires in A Supporting Waveguide or Cavity  
For either an experimental configuration or a technological application, we would not expect to be able to use an unsupported array of wires in free space. Instead, we would need to use a finite array of wires in a supporting structure such as a metal waveguide or metallic cavity. Further, we would need to arrange the wires within that support so that the longitudinal field structure and the ability to sculpt the profiles is not disrupted or lost. We choose to model a supporting structure that consists of either a metallic waveguide or cavity, aligned with a rectangular array of wires with varying radius. This leaves several design parameters to consider, (a) the (minimum) number of wires to place in the waveguide or cavity; (b) the distance of the waveguide or cavity side-walls parallel to the wires from the nearest wire, and; for a cavity, (c) the separation of the cavity end-walls perpendicular to the wires and their placement with respect to the modulation of the wires. Given that the original simulations [3] were for an infinite periodic array of wires, we might expect that a large array of many wires would be necessary. However, we will see below that good results can be achieved with only a \(2\times 2\) array; nevertheless in this article we concentrate on a \(4\times 4\) array as we expect it may offer experimental advantages. Given such a finite array, we now need to consider the positioning of the side-walls, which would ideally be placed where the longitudinal field is small. Please note that in Figure 3c we see there is an approximately circular region around the wire where the (longitudinal) \(E_{z}\) is zero. However, the \(E_{z}=0\) contour is not only curved in contrast to the planar side-walls of our waveguide or cavity, but its distance from the wire varies with the wire radius as shown on Figure 4. Nevertheless, a reasonable compromise is possible since it turns out only requiring \(E_{z}\simeq 0\) is sufficient. The first step when using our algorithm, detailed in [3], is to check that our proposed waveguide containing a \(4\times 4\) array of uniform wires does indeed support longitudinal modes. As can be seen on Figure 5, strong longitudinal fields are indeed present in the centre between the inner wires. Figure 4: How the radius of the wire affects the distance from the wire at which the longitudinal field \(E_{z}\) passes through zero. This result was calculated using periodic boundary conditions for an infinite rectangular array of uniform and infinitely long wires. Figure 3: Numerical results showing a longitudinal mode in a infinite array of uniform infinitely long wires; as depicted in Figure 1a. The distances between the wires are given by \(L_{x}=15.0\) mm and \(L_{y}=13.06\) mm. The single period of the longitudinal electromagnetic wave is given by \(L_{z}=70.71\) mm. We show a vector plot the electric field in the \((y,z)\) plane (**a**) cut through the wires and (**b**) cut between the wires. In (**c**) we show the longitudinal component \(E_{z}\) of the field in the \((x,y)\) plane. Observe there is a region about the wire, roughly circular in shape, where this longitudinal component is approximately zero. Next, we need to calculate the relationship between the wire radius \(r\) and the plasma frequency \(k_{p}^{2}\), and the results are shown on Figure 6 for the \(4\times 4\) case. Our exponential fit to that data is used simply because it provides a good match to the data - as yet we have no theoretical reasons for the behaviour, although the accuracy of the fit strongly suggests that an explanation could be found. Note, however, that as we are modelling dielectric wires, it is not straightforward to apply arguments based on the Drude model and free carrier concentrations.