# Enhancing Efficiency of Electromagnetic Simulation in Time Domain with Transformation Optics
## 3 Simulation System and Courant Condition  
A FDTD parallel computing package, Meep [35], is employed for our simulation. In this package, light speed in vacuum and metric scale are normalized to unity and 1 \(\upmu\)m, respectively. Thus, we can define the unit of time as \"Meep-time,\" a time interval that light in vacuum can travel 1 \(\upmu\)m. Accordingly, the relationship between the frequency \(f\) and wavelength \(\lambda\) of an EM wave in vacuum is simply as \(f=\lambda^{-1}\), where the units of \(\lambda\) are in micrometers. The system height in both physical and numerical coordinates is fixed to 0.2 \(\upmu\)m because we are interested in the transmission at the aperture exit. While \(L_{x}=L_{z}=8\)\(\upmu\)m, the system lengths \(L_{u}\) in the \(u\) and \(L_{w}\) in the \(w\) direction are estimated according to the mapping functions. A broadband incident EM wave source is located above the metallic film and extends in the entire \(xz\) plane for estimating the normalized transmittance spectra, and the incident electric field is polarized along the \(x\) axis. Its distribution in the wavelength domain is set to be a rectangular function ranged from \(\lambda_{a}=0.1\)\(\upmu\)m to \(\lambda_{b}=1.2\)\(\upmu\)m; from the inverse Fourier transform, the source in the time domain is given as:  
\[A(t)=\frac{1}{\pi(t-t_{R}/2)}\sin[2\pi f_{0}(t-t_{R}/2)]\sin(2\pi\Delta f(t-t_{ R}/2), \tag{9}\]where \(f_{0}=(1/\lambda_{a}+1/\lambda_{b})\) / 2, \(\Delta f=(1/\lambda_{a}-1/\lambda_{b})\), \(t_{R}\) is the running-time, and \(t_{R}/2\) is the time-delay. The singularity \(t=t_{R}/2\) in the denominator can be avoided since we use the \(E_{x}\) component as the wave source, and it is generated in every half integer time-step. The Fourier transform will be performed for the simulated electric and magnetic field in the time domain to the wavelength domain. According to our test, \(t_{R}\) should be equal to or larger than 16 Meep-time for the convergence of the transform. The perfect-matched-layer (PML) boundary condition is employed to the \(v\) direction with a thickness of 10 layers. In the \(u\) and \(w\) directions, however, the periodic boundary condition is applied. In the condition, the coupling due to the periodicity will be too weak to experience an influence because the system size is large enough, On the other hand, if the PML condition is used, then the edge effect at the boundaries due to the discontinuity of the source fields will generate non-physical waves that travel back and forth in the system, and thus cause unwanted resonance. The film is assumed to be PEC in [33]. However, for simulations with TO, the built-in PEC material is not suitable for the application because its permittivity is fixed to \(-\infty\), which does not meet the need of altering the properties of medium. Instead, an artificial dispersive medium simulated with the Drude mode is employed as the PEC-like material. In the Drude mode, the relative permittivity of the medium is:  
\[\varepsilon=\varepsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega^{2}+i\gamma_{D }\omega}, \tag{10}\]  
where \(\varepsilon_{\infty}\) is the relative permittivity at infinite frequency, \(\omega_{p}\) is the plasma frequency, and \(\gamma_{D}\) is the damping coefficient. To approximate the PEC, we let \(\varepsilon_{\infty}=1\), \(\omega_{p}\) be sufficiently high (e.g., \(6\pi\times 10^{17}\) rad/s) and \(\gamma_{D}=0\). In the FDTD simulation, the EM field updating scheme uses the auxiliary differential equation [8]:  
\[\frac{\partial\mathbf{J_{p}}}{\partial t}+\gamma_{D}\mathbf{J_{p}}=\varepsilon _{0}\omega_{p}^{2}\mathbf{E} \tag{11}\]  
in synchronism with [PERSON]'s curl equations to yield a composite self-consistent system, where \(\mathbf{J_{p}}\) is the polarization current associated with the wave interaction with the dispersive medium. According to our test, the numerical transmittance spectrums without coordinate transformation show no difference when compared with those using the built-in PEC material.