Click here to learn more about [[Modeling Infinite Phased Arrays]].
[[EM.Cube]]'s periodic FDTD simulator uses periodic boundary conditions (PBC) to model an infinite periodic array. All the periodic replicas of the unit cell structure are excited. In this case, you can impose a phase progression across the infinite array to steer its beam. You can do this from the property dialog of the lumped source or waveguide source. At the bottom of the '''Lumped Source Dialog''' or '''Waveguide Source Dialog''', there is a section titled '''Periodic Beam Scan Angles'''. This section is grayed out when the project structure is not periodic. You can enter desired beam scan angle values for both '''Theta''' and '''Phi''' in degrees. At the end of the periodic FDTD simulation, the radiation pattern of the unit cell is calculated and stored in a radiation data file with a '''.RAD''' file extension. The 3D radiation patterns that you normally visualize in [[EM.Cube]], in this case, correspond to the single unit cell, not the infinite array. Therefore, they do not show the beam scanning even if you have entered nonzero values for the θ and/or φ scan angles. For this purpose, you have to define a finite-sized array factor. You do this in the "Impose Array Factor" section of the '''Radiation Pattern Dialog'''. In the case of a periodic structure, when you define a new far field item in the Navigation Tree, the values of element spacing along the X and Y directions are automatically set equal to the values of the periodic lattice spacing along those directions. Set the number of elements along the X and Y directions to any desired values. [[EM.Cube]] will then compute the radiation pattern of the specified finite-sized periodic array, and the beam scanning will appear in the radiation pattern plots, if any. {{Note|For large θ scan angles, the periodic FDTD time matching loop may take far more time steps to converge.}} {{twoimg|FDTD137.png|Setting periodic scan angles in the lumped source dialog|FDTD138.png| Setting the array factor in radiation pattern dialog.}} {{twoimg|FDTD135.png|Radiation pattern of a 8Ã8 finite-sized periodic dipole array with scan angles with phi and theta equal to 0 degrees.|FDTD136.png| Radiation pattern of a 8Ã8 finite-sized periodic dipole array with scan angles theta equal to 45 degrees, and phi equal to 0 degrees.}} ===Analyzing Antenna Arrays=== Real antenna arrays have finite extents, that is, finite numbers of elements along the X and Y directions. Earlier, you saw how to excite an array of line objects using an array of lumped sources or an array of rectangular waveguides (hollow boxes) using an array of waveguide sources. Setting up array structures of this kind using [[EM.Cube]]'s '''Array Tool '''and exciting the individual elements using individual lumped or waveguide sources results in an accurate full-wave analysis of your antenna array. This type of simulation takes into account all the inter-element coupling effects as well as the finite edge and corner effects of the finite-sized array. At the end of the FDTD simulation of your antenna array, you can plot the radiation patterns and other far field characteristics of the array just like any other FDTD structure. However, depending on the total size of your array, a full-wave simulation like this may easily lead to a very large computational problem. As the number of elements grow very large, the array starts to look like an infinite periodic structure. In that case, it is possible to consider and analyze a periodic unit cell of the array structure and use an "Array Factor" representing the finite-extent topology of the array grid to calculate the radiation pattern of your antenna array. This approach works well for most large arrays. However, it ignores the finite edge and corner effects, which may be important for certain array architectures. In that case we recommend that you use [[EM.Cube]]'s [[Planar Module]]. Also, note that using an array factor for far field calculations, you cannot assign non-uniform amplitude or phase distributions to the array elements. For this purpose, you have to define an array object. [[Image:FDTD146(1).png|thumb|250px|Defining additional radiation characteristics in [[FDTD Module]]'s Radiation Pattern dialog.]] In the previous section, you saw how to excite a periodic unit cell using a lumped source or a waveguide source. You can specify the beam scan angles in the source dialogs. The finite array factor is defined in the radiation pattern dialog. At the end of the periodic FDTD simulation, you can visualize the 3D radiation patterns in the project workspace and plot the 2D Cartesian and polar pattern graphs in EM.Grid. [[EM.Cube]] also calculates the '''Directive Gain (DG)''' as a function of the θ and φ angles. This is defined as: <math>D(\theta,\phi) = \dfrac{4\pi [S(\theta,\phi)]}{P_{rad}} = \dfrac{4\pi \big| \mathbf{E}^{ff}(\theta,\phi) \big|^2} {\int\limits_0^{2\pi} \int\limits_0^{\pi} \big| \mathbf{E}^{ff}(\theta,\phi) \big|^2 \sin\theta \, d\theta \, d\phi}</math> The directivity D<sub>0</sub> is the maximum value of the directive gain. [[EM.Cube]] generates four Cartesian graphs of directive gain in the three principal XY, YZ, ZX planes as well as in the user defined f-plane cut. The radiation patterns of antenna arrays usually have a main beam and several side lobes. Some [[parameters]] of interest in such structures include the '''Half Power Beam Width (HPBW)''', '''Maximum Side Lobe Level (SLL)''' and '''First Null [[Parameters]]''' (i.e. first null level and first null beam width). You can have [[EM.Cube]] calculate all such [[parameters]] if you check the relevant boxes in the "Additional Radiation Characteristics" section of the '''Radiation Pattern Dialog'''. These quantities are saved into ASCII data files of similar names with '''.DAT''' file extensions. You can plot graphs of such data files at the end of a sweep simulation in''' '''EM.Grid. You can also plot the directive gain as a function of the sweep variable at the end of an FDTD sweep simulation. In that case, the directive gain is computed at a fixed pair of θ and φ angles. These angles are specified in degrees as '''User Defined Azimuth & Elevation''' in the "Output Settings" section of the Radiation Pattern dialog. The default values of the user defined azimuth and elevation are both zero corresponding to the zenith. The results are saved to an ASCII data file called "DGU.DAT". Note that DGU is also one of [[EM.Cube]]'s standard output [[parameters]] and can be used to define custom output or design objectives. ===Exciting A Periodic Surface With A Plane Wave=== Using a plane wave source to excite a periodic structure in [[EM.Cube]]'s [[FDTD ModuleTempo]], you can model frequency selective surfaces, electromagnetic band-gap (EBG) structures, metamaterials, etc. Exciting periodic structures with plane wave sources requires careful attention. [[EM.CubeTempo]]'s FDTD simulation engine uses the direct spectral domain FDTD or constant transverse wavenumber method for analyzing periodic structures. In this technique, instead of a plane wave box, one defines a plane wave surface parallel to the X-Y plane. If the plane wave source illuminates the periodic unit cell from the top (90° < θ < 180°), the excitation surface is placed above the structure's bounding box. If the plane wave source illuminates the periodic unit cell from the bottom up (0° < θ < 90°), the excitation surface is placed below the structure's bounding box. In either case, the plane wave must intercept the excitation surface before hitting the unit cell's physical structure. It is highly recommended that you accept [[EM.CubeTempo]]'s default settings for the plane wave box of periodic structures. Nevertheless, you can change the location of the excitation surface if you wish. To do so, you have to open the '''Plane Wave Dialog'''. In the Excitation Box section of the dialog, select the '''Size: Custom''' option. Only the '''Z Coordinate''' of '''Corner 1''' is available for editing. The rest of the coordinates are enforced by the periodic domain. You can enter the incidence angles '''Theta''' and '''Phi''' in degrees. For periodic structures, only the '''TM<sub>z</sub>''' and '''TE<sub>z</sub>''' polarization options are available.
One of the pitfalls of the direct spectral FDTD method is the possibility of horizontal resonances, which may lead to indefinite oscillation or even divergence of field values during the time marching loop. This happens in the case of oblique plane wave incidence when θ > 0°. [[EM.Cube]]'s FDTD engine automatically detects such cases and avoids those resonances by shifting the modulation frequency of the modulated Gaussian pulse waveform away from the resonant frequency. However, in some cases, the size of oscillations may still remain large after a large number of time steps. Occasionally, a late-time diverging behavior may appear. To avoid situations like these, it is highly recommended that you place a time-domain field probe above your structure and monitor the temporal field behavior during the time marching loop as shown in the figure below.