A periodic structure is one that repeats itself infinitely along one, two or three directions. In this release of [[EM.Tempo]], the periodicity is limited to the X-Y plane. In other words, the periodic structure repeats itself along the X- and Y-axes, but not along the Z-axis. By default, your physical structure is not periodic, and you have to instruct [[EM.Cube]] to turn it into a periodic structure through [[FDTD Module]]'s Periodicity Dialog. By designating a structure as periodic, you enforce periodic boundary conditions (PBC) on the side walls of its computational domain. Your structure in the project workspace then turns into a periodic unit cell. The periodic side walls are displayed with dashed blues lines.
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[[Image:FDTD140(1).png|thumb|320px|Setting a custom plane wave source plane.]]
To define a periodic structure, follow these steps:
* Periodic boundary conditions (PBC) are established on the ±X and ±Y faces of the domain box. You still have to designate the boundary conditions on the ±Z faces of the computational domain. These are CPML by default. But you can change them to PEC or PMC.
[[Image:FDTD139.png|thumb|320px|Placing a field probe above a periodic structure excited by an obliquely incident plane wave source.]]
===Exciting Periodic Structures as Radiators===
Click here to learn more about [[Modeling Infinite Phased Arrays]].
[[Image:FDTD139.png|thumb|320px|Placing a field probe above a periodic structure excited by an obliquely incident plane wave source.]]
===Exciting Periodic Structures Using Plane Waves===
click here to learn more about [[Reflection & Transmission Characteristics of Periodic Structures]].
At the end of the FDTD simulation of a periodic structure with plane wave excitation, the reflection and transmission coefficients of the structure are calculated and saved into two complex ASCII data files with '''.CPX''' file extensions. These coefficients behave like the S<sub>11</sub> and S<sub>21</sub> [[parameters]] of a two-port network. You can think of the upper half-space as Port 1 and the lower half-space as Port 2 of this network. The reflection and transmission (R/T) coefficients can be plotted on 2D graphs in '''EM.Grid '''similar to the scattering [[parameters]]. You can plot them from the Navigation Tree. To do so, right click on the '''Periodic Characteristics''' item in the '''Observables''' section of the Navigation Tree and select '''Plot Reflection Coefficients''' or '''Plot Transmission Coefficients'''. The complex data files are also listed in [[EM.Cube]]'s data manager. To open data manager, click the '''Data Manager''' [[Image:data_manager_icon.png]] button of the '''Simulate Toolbar''' or select '''Simulate > Data Manager''' from the menu bar or right click on the '''Data Manager''' item of the Navigation Tree and select Open Data Manager... from the contextual menu or use the keyboard shortcut '''Ctrl+D'''. Select any data file by selecting its row in the table and then click the '''Plot''' button to plot the graph in EM.Grid.
{{Note|It is very important to keep in mind that only in the case of normal incidence does [[EM.Cube]] compute the reflection and transmission coefficients over the entire specified bandwidth of the project. At oblique incidences when θ > 0, the computed R/T coefficients after the discrete Fourier transformation are valid only at the center frequency of the project for the given value of the incident θ<sub>0</sub> angle. In other words, the computed R/T coefficients at all the other frequencies away from the center frequency correspond to different values of the incident θ angle. As a result, [[EM.Cube]] only saves the reflection and transmission coefficients at the center frequency into the output data files "reflection_coefficient.CPX" and "transmission_coefficient.CPX".}}
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===Periodic FDTD Simulation Types===
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[[Image:FDTD143.png|thumb|250px| [[EM.Tempo]]'s R/T Macromodel Settings Dialog.]]
[[Image:FDTD144.png|thumb|250px| [[EM.Tempo]]'s Dispersion Sweep Settings dialog.]]
Besides analyzing a periodic structure in a single-run simulation and other standard type sweeps, [[EM.Tempo]] offers a number of specialized sweep simulations for periodic structures. These include '''R/T Macromodel Sweep ''', '''Dispersion Sweep''' and '''Huygens Sweep'''. These options are available from the '''Simulation Mode''' dropdown list of the [[EM.Tempo]]'s '''Run Dialog'''.
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The '''R/T Macromodel Sweep''' option of the Simulation Mode dropdown list is only available for periodic structures. It is used to generate a lookup table model for the reflection and transmission coefficients of a periodic surface for both TM and TE polarizations. The results are written into a file named "PW_UserDefinedMacroData.mat". Through the Macromodel Settings dialog you can set the start and end value and number of samples for both the Theta (θ) and Phi (φ) angles of the incident plane wave. The R/T macormodels can be used by [[EM.Cube]]'s [[Propagation Module]] to calculate the reflection and transmission coefficients of incident rays at the facets of obstructing blocks with "non-standard" periodic surfaces.
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The '''Dispersion Sweep '''option of the Simulation Mode dropdown list performs a sweep of constant k<sub>l</sub> wavenumber values. This is a specialized sweep for the constant transverse wavenumber method that [[EM.Cube]]'s [[FDTD Module]] uses to model periodic structures illuminated by a plane wave source. The real advantage of a dispersion sweep is that through a one-dimensional sweep of k<sub>li</sub>, you can find the reflection and transmission coefficients for all combinations of frequency f<sub>j</sub> and incident angle θ<sub>j</sub> such that (2π/c) . f<sub>j</sub>. sin θ<sub>j</sub> = k<sub>li</sub>. This provides a complete picture of the dispersion behavior of your periodic structure. The sweep data can be graphed as a wavenumber-frequency intensity plot (also known as beta-k diagram) that projects the eigenvalues of the periodic structure. The horizontal axis represents the constant transverse wavenumber k<sub>l</sub> (or beta). The vertical axis represents frequency. Sometimes, the free space wave number k<sub>0</sub> = (2π/c).f is used as the vertical axis, hence, the term beta-k diagram. However, [[EM.Cube]] plots frequency vs. wavenumber. Both the horizontal and vertical axes start from 0 and extend to f<sub>max</sub> and k<sub>l,max</sub>, respectively, where f<sub>max</sub> = f<sub>0</sub> + Δf/2, and Δf is the specified bandwidth of the project. For this sweep option you have to specify the number of wavenumber samples. Note that the dispersion sweep is run for a fixed given value of the plane wave incident angle φ as specified in [[FDTD Module]]'s Plane Wave Dialog.
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{{isoimg|FDTD148.png|A typical dispersion diagram of a periodic structure}}
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