# Mode Profile Shaping in Wire Media: Towards An Experimental Verification

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## Abstract

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## 1. Introduction

## 2. Modulated Wire Media

## 3. Modulated Wires in A Supporting Waveguide or Cavity

## 4. Frequency Isolation

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A wire medium is formed from an array of parallel metal or dielectric wires. In this work we use results based on rectangular arrays of (

**a**) wires with uniform radii to predict the parameters needed for customised (

**b**) wires with varying radii that generate a desired subwavelength field profile shaping.

**Figure 2.**In our proposed experimental system, the wire medium will not be in free space but will be confined by metallic walls. Here we represent this in two ways, each containing a finite array of wires with varying radii. First, we confine the array in a rectangular waveguide with metal side-walls (in blue-gray), and use periodic boundary conditions to treat wires of infinite length. Second, we add metallic end-walls (grey) perpendicular to the wires to change the waveguide into a closed box or cavity. In either case we can do our numerical computations for only one period of variation in the wire radii. It is very important to note that this variation only corresponds to half a period of the electric field.

**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 ${L}_{x}=15.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$ and ${L}_{y}=13.06\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. The single period of the longitudinal electromagnetic wave is given by ${L}_{z}=70.71\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. 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.

**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 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 ${L}_{x}=15.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$ and ${L}_{y}=13.06\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. The dimensions of the waveguide are given by ${S}_{x}=52.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$ and ${S}_{y}=45.71\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. 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 ${k}_{p}^{2}\left(r\right)={y}_{0}+Aexp(-r/{r}_{0})$, where ${y}_{0}=9.22$, $A=2.23$ and ${r}_{0}=2.69$.

**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.

**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 ${L}_{x}=15\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$ and ${L}_{y}=13.06\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. The single period of the longitudinal electromagnetic wave is given by ${L}_{z}=80.60\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. The dimensions of the waveguide are given by ${S}_{x}=52.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$ and ${S}_{y}=45.71\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. 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 $z=48.36\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$.

**Figure 9.**Field profile shaping in a metallic waveguide containing a $4\times 4$ array of infinitely long wires of varying radii and with $L=80.61\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. The simulated field profile ${E}_{z}$ itself is normalised and shown with solid black lines, and is compared with the ideal Mathieu function profile shown as a dashed red curve, with a sinusoid shown in blue as a reference. In the simulation, CST gave the maximum longitudinal field as $3.28\times {10}^{7}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Vm}}^{-1}$.

**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 2. The distances between the wires are given by ${L}_{x}=15\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$ and ${L}_{y}=13.06\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. The dimensions of the cavity are given by ${S}_{x}=52.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$, ${S}_{y}=45.71\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$ and ${S}_{z}=40.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. 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 $z=10.08\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$.

**Figure 11.**Field profile shaping in a metallic waveguide containing a $4\times 4$ array of wires with varying radii in a metallic cavity, and with $L=80.61\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. The simulated field profile ${E}_{z}$ itself is normalised and shown with solid black lines, and is compared with the ideal Mathieu function profile shown as a dashed red curve.

**Figure 12.**Field variation of a longitudinal modes in a $2\times 2$ array of uniform infinitely long wires in a waveguide. The distances between the wires are given by ${L}_{x}=15\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$ and ${L}_{y}=13.06\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. The single period of the longitudinal electromagnetic wave is given by ${L}_{z}=80.6\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. The dimensions of the waveguide are given by ${S}_{x}=52.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$ and ${S}_{y}=45.7\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. Thus, the distances of the wires closest to the waveguide walls and the walls are $\frac{1}{4}}{L}_{x$ and $\frac{1}{4}}{L}_{y$. Here the longitudinal component ${E}_{z}$ in the $(x,y)$ plane is shown.

**Figure 13.**Profile of ${E}_{z}$ component for a waveguide with $2\times 2$ infinitely long wires of varying radii (solid Black) in comparison with Mathieu function (dashed red) and sin function (blue). Here $L\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}127.4\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$. Both ${E}_{z}$ and the Mathieu function are normalised. However CST gave the maximum longitudinal field as $6.03\times {10}^{7}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Vm}}^{-1}$.

**Figure 14.**An example showing the electric field vectors on a slice in the $(x,y)$ plane perpendicular to the wires. Most of the field modes near our desired longitudinal modes are of this transverse type, and should not be excited by a carefully designed (longitudinal) excitation field. The dimensions of the waveguide are the same as in Figure 8.

**Table 1.**Modes of the modulated wire media in a cavity that are nearby to our chosen longitudinal mode (index 90, shown in bold). They are categorized into either transverse (‘Trans’) or longitudinal modes (‘Long’). This depends on the field patern of the electric field. The electric field in the longitudinal modes is principally parallel to the wires as in Figure 8 whereas in the transverse modes it is perpendicular to the wires, Figure 14.

Mode Number | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | |

Freq (GHz) | 12.088 | 12.098 | 12.158 | 12.159 | 12.175 | 12.250 | 12.274 | 12.388 | 12.430 | |

Type | Long | Long | Long | Trans | Long | Trans | Trans | Trans | Our Mode | |

Mode Number | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | |

Freq (GHz) | 12.433 | 12.460 | 12.539 | 12.556 | 12.565 | 12.577 | 12.583 | 12.723 | 12.831 | |

Type | Trans | Trans | Long | Long | Long | Long | Trans | Long | Trans |

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**MDPI and ACS Style**

Boyd, T.; Gratus, J.; Kinsler, P.; Letizia, R.; Seviour, R.
Mode Profile Shaping in Wire Media: Towards An Experimental Verification. *Appl. Sci.* **2018**, *8*, 1276.
https://doi.org/10.3390/app8081276

**AMA Style**

Boyd T, Gratus J, Kinsler P, Letizia R, Seviour R.
Mode Profile Shaping in Wire Media: Towards An Experimental Verification. *Applied Sciences*. 2018; 8(8):1276.
https://doi.org/10.3390/app8081276

**Chicago/Turabian Style**

Boyd, Taylor, Jonathan Gratus, Paul Kinsler, Rosa Letizia, and Rebecca Seviour.
2018. "Mode Profile Shaping in Wire Media: Towards An Experimental Verification" *Applied Sciences* 8, no. 8: 1276.
https://doi.org/10.3390/app8081276