Changes

A periodic structure is one that repeats itself infinitely in one, two or three directions. [[EM.Cube]]'s [[FDTD Module]] allows you to simulate doubly periodic structures with periodicities along the X and Y directions. Many interesting structures such as frequency selective surfaces (FSS), electromagnetic band-gap (EBG) structures and metamaterial structures can be modeled using periodic geometries. In the case of an infinitely extended periodic structure, it is sufficient to analyze only a unit cell. In the FDTD method, this is accomplished by applying periodic boundary conditions (PBC) at the side walls of the computational domain. The application of the PBC is straightforward for the case of a normally incident plane wave source since the fields do not experience any delay as they travel across the unit cell. Obliquely incident plane waves, on the other hand, cause a time delay in the transverse plane. This delay requires knowledge of the future values of the fields at any time step.

A number The details of techniques have been proposed to solve this problem. [[EM.Cube]] uses a recently developed novel technique that is known as Direct Spectral FDTD or Constant Transverse Wavenumber method. In this technique, the components of the transverse (horizontal) wavenumber are kept constant in the direction of periodicity. This technique shows a significant advantage over the other methods for simulation of the incident illuminations close to the grazing angles. The figure above shows a doubly periodic structure with periods S_x and S_y along the X and Y directions, respectively. The computational domain is terminated with PBC boundary condition used in both X and Y directionsEM. Along the positive and negative Z directions, it is terminated with CPML layers. Bear in mind that the PBC is also applied to the CPML layers. The computational domain is excited by a TM_z or TE_z plane wave incident at z = z₀. The plane wave incidence angles Temp are denoted by &theta; (elevation) and &phi; (azimuth) described in the spherical coordinate system. The constant wavenumber components k_x and k_y in this case are defined as: :<math>k_x = k_0 \sin\theta\cos\phi</math>:<math>k_y = k_0 \sin\theta \sin\phi</math><!--[[Image:FDTD85.png]]--> where <math>k_0 = \omega/c = 2\pi f/c = 2\pi/\lambda_0</math> is the free space propagation constant, f is the operational frequency, &omega; is the angular frequency, &lambda;₀ is the free space wavelength, c is the speed Time Domain Simulation of light in the free space. The constant transverse wavenumber k_l is then given by: :<math> k_l = \sqrt{k_x^2 + k_y^2} = k_0\sin\theta </math><!--[[Image:FDTD86.pngPeriodic Structures]]--> which depends only on &theta; and not on &phi;. On the excitation plane, the incident field adopts a modulated Gaussian waveform and a complex phase delay along the periodicity direction with the following form: :<math> H_x^{inc}(x,y,t) = -\frac{1}{\eta_0} \sin\phi \; \exp \left(-\frac{(t-t_0)^2}{\tau^2} \right) \exp(j2\pi f_0 t) \exp(-jk_x x) \exp(-jk_y y)</math>:<math> H_y^{inc}(x,y,t) = \frac{1}{\eta_0} \cos\phi \; \exp \left(-\frac{(t-t_0)^2}{\tau^2} \right) \exp(j2\pi f_0 t) \exp(-jk_x x) \exp(-jk_y y)</math><!--[[Image:FDTD87.png]]--> for TM_z polarization and :<math> E_x^{inc}(x,y,t) = \sin\phi \; \exp \left(-\frac{(t-t_0)^2}{\tau^2} \right) \exp(j2\pi f_0 t) \exp(-jk_x x) \exp(-jk_y y)</math>:<math> E_y^{inc}(x,y,t) = -\cos\phi \; \exp \left(-\frac{(t-t_0)^2}{\tau^2} \right) \exp(j2\pi f_0 t) \exp(-jk_x x) \exp(-jk_y y)</math><!--[[Image:FDTD88.png]]--> for TE_z polarization. Here, f₀ is the center frequency of the modulated Gaussian pulse waveform, t₀ is the time delay, and &tau; is the Gaussian pulse width. The choices of the Gaussian waveform [[parameters]] are very critical in order to avoid possible resonances. For a fixed value of k_l, the horizontal resonance occurs at: :<math> f_{res} = \frac{k_l c}{2 \pi} </math><!--[[Image:FDTD101.png]]--> For a fixed frequency <math>f_0</math> and a fixed incidence angle <math>\theta_0</math>, the resonant frequency is reduced to: <!-- Using "\mathbf" below because it tricks the wiki to rendering with PNG instead of html-->:<math> f_{res} = f_0 \mathbf{\sin} \theta_0 </math><!--[[Image:FDTD102.png]]--> The modulated Gaussian waveform must be chosen such that its effective bandwidth avoids the horizontal resonant frequency. Otherwise, the temporal response of the structure starts to oscillate, and the time marching loop will not converge. To avoid this problem, the modulation frequency and bandwidth of the waveform are chosen to satisfy the following condition: :<math> f_{mod} \ge f_{res} + \dfrac{1}{2}\Delta f = \dfrac{k_{l,fixed}\;c}{2\pi} + \dfrac{1}{2}\Delta f </math><!--[[Image:FDTD103.png]]-->