# Enhancing Efficiency of Electromagnetic Simulation in Time Domain with Transformation Optics  
[PERSON]  
1 Department of Physics, National Cheng Kung University, 1 University Road, Tainan 70101, Taiwan; [EMAIL_ADDRESS] (J.-S.H. ); [EMAIL_ADDRESS] (M.-C.Y.) 2 Institute of Space, Astrophysical and Plasma Sciences, National Cheng Kung University, 1 University Road, Tainan 70101, Taiwan; [EMAIL_ADDRESS]  
3 Department of Photonics, National Cheng Kung University, 1 University Road, Tainan 70101, Taiwan; [EMAIL_ADDRESS] (R.-C.S. ); [EMAIL_ADDRESS] (Y.-C.L.) 4 Correspondence: [EMAIL_ADDRESS]; Tel. : +886-6-2080472  
computational resources. Non-uniform cell algorithms that have been developed for decades, such as the sub-gridding approach [10; 11], might be applied to lower the computational requirement; but, the implementation of the source codes is usually not simple. The novel methodology \"transformation optics\" (TO) [12; 13] is utilized for the coordinate transformation of light, and developed as a tailoring strategy of light for potential optical devices, such as invisible cloak [14; 15; 16; 17; 18; 19], electromagnetic rotator [20; 21], optical black hole or concentrator [22; 23; 24], planar focusing antennas [25], multi-layered plasmonic filter and coupler [26], panoramic lens [27], etc. The methodology is also applied to study plasmonics, such as the structured plasmonic cylindrical waveguide in a transformed Cartesian coordinate [28] or the frequency- and time-domain response of plasmonic particles under electron beam excitation [29]. For numerical purposes, a procedure for structured nonorthogonal discretization grids implemented in FDTD with the reconsideration of the EM field updating scheme is proposed [30]; the methodology is employed to generate non-uniform grids around the interfaces and openings of a metallic slit in a finite-difference frequency-domain scheme [31]; an additional cylindrical coordinate for the local mesh refinement is introduced in the Cartesian-coordinate-based FDTD simulation [32]. In this work, we propose a method that applies TO for the generation of non-uniform physical grids in the FDTD simulation. The complexity of implementing the non-uniformity will be transformed to the implementation of the derived spatially-varying numerical anisotropic medium (or the so-called optical transformation medium); therefore, the EM field updating scheme is conventional and does not need any further modification for the implementation. The physical coordinate that we study is Cartesian such that the implementation of the coordinate transformation is intuitive and simple. The mapping function between the physical and numerical coordinates can be designed according to the structure of interest. For demonstration, we simulate a rectangular sub-wavelength aperture in a nearly perfect electric conductor (PEC-like) for the transmission spectrum [33], where the Courant condition stability factor is re-considered. Fine cells are designed to be around the aperture while coarse cells are in other areas with the designed multiple linear and nonlinear mapping functions for the one-dimensional (1D) and the two-dimensional (2D) coordinate transformation. We will show that simulated transmission spectrums are in good agreement with that from the simulation in uniform grids, and in the meantime, the computational costs are remarkably reduced. In the optimized case from the 2D coordinate transformation, only a remarkable 5.31% of the computational cost is required. With this method, we can even utilize finer grids around the aperture. In this case, we obtain the more accurate spectrum while its cost is still less than that in uniform grids. Therefore, we are confident that the method should be helpful to the FDTD simulation study in sub-wavelength optics due to its verified accuracy and efficiency.