In this tutorial you will learn how to define periodic structures in [[EM.Tempo]], excite them using plane wave sources and compute the reflection and transmission characteristics of the periodic surface. You will become familiar with the subtleties of modeling oblique incident angles.
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== Getting Started ==
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<td> [[Image:Tempo L8 Fig1.png|thumb|480px600px|The property dialog of the dielectric material.]] </td>
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*Keep the default domain settings.
*Keep the default mesh settings.
*Change the Termination Criterion to a Fixed '''Fixed Number of No. Time Steps''' equal to 2,500.
Now run a wideband Analysis of your periodic structure and visualize the near-field distributions as shown in the figure below. In particular, note the strong magnetic field intensity at the location of the field probe. This corresponds to the maximum of the electric surface current on the strip. Plot the graphs of the reflection and transmission coefficients of your periodic structure. This can be done conveniently by right clicking on the <b>Periodic Characteristics</b> item in the <b>Observables</b> section of the Navigation Tree and selecting either <b>Plot Reflection Coefficient…</b> or <b>Plot Transmission Coefficient…</b> from the contextual menu. You can also plot these graphs from the Data Manager through the output data files “reflection_coefficient.CPX” and “transmission_coefficient.CPX”. The plots show the variation of the reflection and transmission (R/T) coefficients with frequency over the specified bandwidth of your project. As you can see from the figures below, the FSS has a narrow transmission window slightly above 9GHz.
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[[Image:Tempo L8 Fig5.png|thumb|480px600px|Cartesian graph of the magnitude and phase of the reflection coefficient.]]
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[[Image:Tempo L8 Fig6.png|thumb|480px600px|Cartesian graph of the magnitude and phase of the transmission coefficient.]]
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==Examining a Case of Oblique Plane Wave Incidence==
In the case of oblique plane wave incidence on a periodic structure, [[EM.Tempo]] calculates the reflection and transmission coefficients only at the center frequency fc. As a result, you can reduce the bandwidth considerably. This helps the convergence of the FDTD marching loop in such cases. Open the frequency dialog either by double-clicking the small frequency box in the status bar or simply using the keyboard shortcut {{key|Ctrl+F}}. In the frequency dialog, set the bandwidth to 2GHz. Define an incident plane wave source with TEz Polarization and incident angles θ = 135° and φ = 0°.
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<td> [[Image:Tempo_L8_Fig4_135.png|thumb|480px|Setting an oblique plane wave source.]] </td>
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At a center frequency of 9GHz and a incident plane wave angle of θ = 135°, the transverse wavenumber is calculated to be:
<math> \frac{2\pi f_i}{c} \sin\theta_i = k_l = const. </math>
To make sure that a resonance doesn't happen, let's initiate a temporal field probe to monitor the fields as a function of time. A field probe is used to record all the six field components Ex, Ey, Ez, Hz, Hy, Hz, at a single point in the computational domain during the entire FDTD time marching loop. Right-click on the '''Temporal Field Probes''' item of the navigation tree and select '''Insert new Observable...''' from the contextual menu. In the field probe dialog, enter coordinated (1.5mm, 0, 6mm) for "Probe_1". Uncheck the "Compute Fields at the Center of Yee Cell" checkbox.
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Run a new FDTD analysis with a fixed maximum number <b>No. of time steps Time Steps</b> equal to 2,500 and wait until it is completed. Open the data manager and plot the data file "Probe_1_X_H_Time.CPX" in EM.Grid. Note that due to the special nature of [[EM.Tempo]]'s periodic FDTD formulation, the time-domain fields are complex-valued. By contrast, the time-domain fields of regular aperiodic structures are real-valued. The figure below shows the X-component of the magnetic field at the right edge of the metallic strip as a function of time. This is proportional to the surface current flowing at the edge of the strip. You can see from the figure that the waveform dies off over time. , so it converges and doesn't oscillate.
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If you view the contents of the data files “reflection_coefficient.CPX” and “transmission_coefficient.CPX” in Data Manager, you will find that both files contain only one frequency data point, which is the center frequency of the project:
R: -0.89 883362 + 0.217j216773j T: 0.24 106207 - 0.33j402591j
==Running a Parametric Frequency Sweep==
In order to examine the frequency variations of the reflection and transmission coefficients in the case of oblique plane wave incidence, you need to run a parametric sweep in [[EM.Tempo]] with the center frequency fc being your sweep variable. You already ran a parametric sweep of a patch antenna in Tutorial Lesson 3. Open the Run dialog and select '''Parametric Sweep''' from the <b>Simulation Mode</b> drop-down list. Click the <b>Settings</b> button next to this drop-down list to open up the sweep settings dialog. Select "fc" as your sweep variable with the start, stop and step values equal to 3e+9 (3GHz), 15e+9 (15GHz) and 2.5e+8 (250MHz). <u>Keep in mind that the project variables "fc" and "bw" are always expressed in Hz</u>.
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Start the sweep simulation and wait until all the frequency samples are computed. Then, plot the Cartesian graphs of the R/T coefficients in EM.Grid.
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[[Image:Tempo L8 Fig14.png|thumb|left|480px600px|Cartesian graph of the magnitude and phase of the reflection coefficient vs. frequency with an obliquely incident plane wave source.]]
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[[Image:Tempo L8 Fig15.png|thumb|left|480px600px|Cartesian graph of the magnitude and phase of the transmission coefficient vs. frequency with an obliquely incident plane wave source.]]
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