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EM.Tempo

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/* Radiation Pattern Above a Half-Space Medium */

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==Modeling 3D Periodic Structures in EM.Tempo==

[[EM.Tempo]] allows you to simulate doubly periodic structures with periodicities along the X and Y directions. In the FDTD method, this is accomplished by applying periodic boundary conditions (PBC) at the side walls of the computational domain.

{{Note| [[EM.Tempo]] can handle regular, non-skewed periodic lattices only with no secondary offsets.}}

[[Image:Info_icon.png|40px]] Click here to learn more about the theory of '''[[Basic_FDTD_Theory#Time_Domain_Simulation_of_Periodic_Structures | Time Domain Simulation of Periodic Structures]]'''.

[[Image:FDTD134.png|thumb|360px|EM.Tempo's Periodicity Settings dialog]]

===Defining a Periodic Structure in EM.Tempo===

By default, your physical structure in the project workspace is not periodic, and you have to instruct [[EM.Tempo]] to turn it into a periodic structure using its 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.

To define a periodic structure, follow these steps:

* Select '''Menu > Simulate > Computational Domain > Periodicity Settings...''' or right click on the '''Periodicity''' item in the '''Computational Domain''' section of the Navigation Tree and select '''Periodicity Settings...''' from the contextual menu. This open up the Periodicity Settings Dialog.

* Check the box labeled '''Periodic Structure''' and click the '''Apply''' button of this dialog. The default domain box initially shrinks to the edges of the physical structure in the project workspace. The default periods along the X and Y axes appear in the dialog, which are equal to the dimensions of the structure's bounding box.

* Enter new values for '''X Spacing''' and '''Y Spacing '''in project units and close the dialog.

* Periodic boundary conditions (PBC) are established on the Â±X and Â±Y faces of the domain box. You still have to designate the boundary conditions on the Â±Z faces of the computational domain. These are CPML by default. But you can change them to PEC or PMC.

===Exciting Periodic Structures as Radiators in EM.Tempo===

In [[EM.Tempo]], a periodic structure can be excited using various source types. Exciting the unit cell structure using a lumped source, a waveguide source, or a distributed source, you can model an infinite periodic antenna array. For most practical antenna types, you excite your periodic structure with a lumped source or waveguide source. In this case, you can define a port for the lumped source or waveguide source and calculate the S11 parameter or input impedance of the periodic antenna array. You can also compute the near-field and far-field data.

[[EM.Tempo]]'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. To visualize the radiation pattern of the beam-steered array, you have to define a finite-sized array factor. You do this in the "Impose Array Factor" section of the '''Radiation Pattern Dialog'''.

{{Note|For large θ scan angles, the periodic FDTD time marching loop may take far more time steps to converge.}}

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<td> [[Image:Period1.png|thumb|350px|Setting periodic scan angles in EM.Tempo's Lumped Source dialog.]] </td>

<td> [[Image:Period2.png|thumb|350px|Setting the array factor in EM.Tempo's Radiation Pattern dialog.]] </td>

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<td> [[Image:Period3.png|thumb|360px|Radiation pattern of an 8Ã8 finite-sized periodic wire dipole array with 0° phi and theta scan angles.]] </td>

<td> [[Image:Period4.png|thumb|360px|Radiation pattern of a beam-steered 8Ã8 finite-sized periodic wire dipole array with 45° phi and theta scan angles.]] </td>

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===Exciting Periodic Structures Using Plane Waves in EM.Tempo===

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<td> [[Image:Period11.png|thumb|380px|Geometry of a periodic printed strip FSS in EM.Tempo.]] </td>

<td> [[Image:Period12.png|thumb|340px|Define a custom periodic plane wave box in EM.Tempo.]] </td>

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Using a plane wave source to excite a periodic structure in [[EM.Tempo]], you can model frequency selective surfaces, electromagnetic band-gap (EBG) structures, metamaterials, etc. Exciting periodic structures with plane wave sources requires careful attention. [[EM.Tempo]]'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.Tempo]]'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 '''TMz''' and '''TEz''' 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.

{{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 θ0 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".}}

=== Running a Dispersion Sweep in EM.Tempo ===

[[Image:FDTD144.png|thumb|250px| [[EM.Tempo]]'s Dispersion Sweep Settings dialog.]]

The '''Dispersion Sweep '''option of the Simulation Mode dropdown list performs a sweep of constant kl 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 kli, you can find the reflection and transmission coefficients for all combinations of frequency fj and incident angle θj such that (2π/c) . fj. sin θj = kli. 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 kl (or beta). The vertical axis represents frequency. Sometimes, the free space wave number k0 = (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 fmax and kl,max, respectively, where fmax = f0 + 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.

{{isoimg|FDTD148.png|A typical dispersion diagram of a periodic structure}}

Kazem Sabet

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