# Mode Profile Shaping in Wire Media: Towards An Experimental Verification
## 3 Modulated Wires in A Supporting Waveguide or Cavity
For our chosen field profile, we use this information to calculate the generating radius variation, as seen on Figure 7. This produced a shaped longitudinal field shown in Figure 8, and which is compared with the desired profile on Figure 9.  
Figure 5: Longitudinal \(E_{z}\) component on the \((x,y)\) plane cutting across a \(4\times 4\) array of uniform infinitely long wires in a waveguide. The distances between the wires are given by \(L_{x}=15.0\) mm and \(L_{y}=13.06\) mm. The dimensions of the waveguide are given by \(S_{x}=52.5\) mm and \(S_{y}=45.71\) mm. Thus, the distances of the wires closest to the waveguide walls and the walls are \(\frac{1}{4}L_{x}\) and \(\frac{1}{4}L_{y}\). This is the data for our preferred longitudinal mode, which is only one of many. Figure 6: To design the required radius modulation, we need to know the variation of the effective plasma frequency of our \(4\times 4\) wire array in a waveguide, as a function of wire radius, which for our chosen parameters is shown here. This data is fitted with the exponential function \(k_{p}^{2}(r)=y_{0}+A\exp(-r/r_{0})\), where \(y_{0}=9.22\), \(A=2.23\) and \(r_{0}=2.69\). Figure 8: The field variation in our chosen longitudinal mode as present in a \(4\times 4\) array of infinitely long wires of varying radii in a waveguide. The distances between the wires are given by \(L_{x}=15\) mm and \(L_{y}=13.06\) mm. The single period of the longitudinal electromagnetic wave is given by \(L_{z}=80.60\) mm. The dimensions of the waveguide are given by \(S_{x}=52.5\) mm and \(S_{y}=45.71\) mm. Thus, the distance of the wires closest to the waveguide walls and the walls are \(\frac{1}{4}L_{x}\) and \(\frac{1}{4}L_{y}\). The electric field is shown as a vector on the \((y,z)\) plane as (**a**) cut through the wires, and (**b**) cut between the wires. In (**c**), the longitudinal component \(E_{z}\) is shown on the \((x,y)\). This plane is cut through the point \(z=48.36\) mm. Figure 7: By combining our desired field profile (the Mathieu function of (1)) with the plasma frequency response in Figure 6 we can predict the necessary radius variation for our \(4\times 4\) wire array in a waveguide. The result is shown here, with a radius that changes by about \(\pm 15\%\) around its average value. The next step was to consider the case of a cavity rather than the infinite wayguide which requires us to choose where to place the perpendicular end-walls. In this case we made the straightforward decision to place them where the field has a symmetry, i.e. at its maximum and minimum values. Thus the cavity contains an exact half wavelength of the field, which is the same as one full period of the wire variation. We see in Figure 10 that the longitudinal profile of the mode is unaffected and from Figure 11 that the simulated field profiles is still in very good agreement with the intended Mathieu profile. We also repeated the above procedure for the simpler cavity with only a \(2\times 2\) wire array, as shown on Figures 12 and 13. Even in this reduced situation there was still excellent agreement with the desired electric field profile. Figure 10: The field variation in our chosen longitudinal mode as present in a \(4\times 4\) array of wires with varying radii inside a metallic cavity, as depicted in Figure 4:0 The distances between the wires are given by \(L_{x}=15\) mm and \(L_{y}=13.06\) mm. The dimensions of the cavity are given by \(S_{x}=52.5\) mm, \(S_{y}=45.71\) mm and \(S_{z}=40.3\) mm. Thus, the distances of the wires closest to the waveguide walls and the walls are \(\frac{1}{4}L_{x}\) and \(\frac{1}{4}L_{y}\). The electric field is shown as a vector on the \((y,z)\) plane as (**a**) cut through the wires, and (**b**) cut between the wires. In (**c**), the longitudinal component \(E_{z}\) is shown on the \((x,y)\). This plane is cut through the point \(z=10.08\) mm. Figure 4:1 Field profile shaping in a metallic waveguide containing a \(4\times 4\) array of infinitely long wires of varying radii and with \(L=80.61\) mm.