# Mode Profile Shaping in Wire Media: Towards An Experimental Verification
## 1 Introduction  
Spatial dispersion is a valuable tool, which can be used to customise the profiles of electric fields in metamaterial and photonic structures. As we have already shown [1; 2; 3], such field profile customisation is even possible at the sub-wavelength scale, and does not require exhaustive brute-force computation. Further, for our spatially dispersive wire medium [3], we show this system can support shaped _longitudinal_ electric fields. This field profiling is achieved by varying the radius of the wires in a carefully calibrated way. In [3] the system was numerically modelled as an infinite periodic array of wires, so that the computation only required a single wire with periodic boundary conditions. In addition to periodicity, the simulated wire material had a very high permittivity and concomitantly small radius, features that would be problematic in experiment, both in terms of manufacture and fragility. Nevertheless, we did show that such extreme structures were not necessarily required, since re-scaled calculations indicated the shaping effect would still persist even with lower permittivities and larger wire radii. Such field profiling has a variety of potential uses. The idea has been implemented by means of harmonic synthesis [4; 5], and suggested in the context of nonlinearity-induced carrier shocking [6; 7; 8]. Such shaped waveforms have been suggested as a means of enhancing ionisation in high harmonic generation [9; 10]. Other shapes could find uses, such as fields with locally high gradients but without a large peak-minimising nonlinear effects, or fields with pronounced peaks and low amplitude elsewhere to improve signal to noise ratios. In particular, we are interested in accelerator applications where the field profile might be used for electron bunch shaping, or as part of a laser wakefield accelerator [11; 12]. This requires us to improve on our initial idealised simulations and incorporate experimental features such as more realistic material choices and a supporting waveguide systems. In this article, we show that the field profiling of a longitudinal wave predicted in [3] can be numerically reproduced in an experimentally plausible system. In [3] we considered an infinite periodicarray of infinite wires as shown in Figure 1, where the wire radius varied from 0.2 mm to 0.4 mm with a relative permittivity of \(\epsilon=1600\). As longitudinal electromagnetic waves (EM waves) are known to exist in guided wave systems we consider a \(4\times 4\) array of finite-wires, a single period long, and contained in a metallic waveguide closed at both ends, as depicted in Figure 2. In the simulations reported here, our wire parameters are physically easier to realise than those used in [3], being based on a wire radius that varies from 0.5 mm to 0.7 mm, with a relative permittivity of \(\epsilon=50\). The simulation of this system was undertaken using the frequency domain solver of the commercial software CST Microwave Studio (2018 Update, Framingham, MA, US). As in [3] we concentrate on sculpting the electric field profile based on the periodic Mathieu function  
\[E_{z}(z)=\text{MathieuCE}\Big{(}1,0.8,\frac{2\pi z}{L}\Big{)} \tag{1}\]  
where \(L\) is the field wavelength, and the period of variation in the wire radius is \(L/2\). This field profile has a flatter maximum than a sinusoid, and a steeper gradient when passing though zero. The Mathieu functions are the solution to the differential equation  
\[\frac{d^{2}f}{d\sigma^{2}}+\big{(}a-2q\cos(2\sigma)\big{)}f=0\]  
The periodic solution \(f(\sigma)=\text{MathieuCE}\big{(}1,0.8,\sigma\big{)}\) in (1), is given when \(q=0.8\), \(a=1.712\) and \(\sigma=2\pi z/L\). 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.