Reflection & Transmission Characteristics of Periodic Structures

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Setting a custom plane wave source plane.

Defining a Periodic Plane Wave Source in EM.Tempo

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.

Calculating Reflection & Transmission Coefficients

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 data files with .CPX file extensions. These coefficients behave like the S11 and S21 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 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.

Attention icon.png 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".

Special Periodic FDTD Simulation Types

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.

A typical dispersion diagram of a periodic structure


Exciting Periodic Structures Using Plane Waves in EM.Picasso

When a periodic structure is excited using a plane wave source, it acts as a periodic surface that reflects or transmits the incident wave. You can model frequency selective surfaces, electromagnetic band-gap structures and metamaterials in this way. EM.Cube calculates the reflection and transmission coefficients of periodic surfaces or planar structures. If you run a single plane wave simulation, the reflection and transmission coefficients are reported in the Output Window at the end of the simulation. Note that these periodic characteristics depend on the polarization of the incident plane wave. You set the polarization (TMz or TEz) in the Plane Wave Dialog when defining your excitation source. In this dialog you also set the values of the incident Theta and Phi angles.

At the end of the planar MoM simulation of a periodic structure with plane wave excitation, the reflection and transmission coefficients of the structure are calculated and saved into two complex data files called "reflection.CPX" and "transmission.CPX". These coefficients behave like the S11 and S21 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. As a result, you can run an adaptive sweep of periodic structures with a plane wave source just like projects with gap or probe sources. The reflection and transmission (R/T) coefficients can be plotted in EM.Grid on 2D graphs similar to the S 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, where you can view or plot them.

Attention icon.png In the absence of any finite traces or embedded objects in the project workspace, EM.Cube computes the reflection and transmission coefficients of the layered background structure of your project.
A periodic planar layered structure with slot traces excited by a normally incident plane wave source.

You learned earlier how to use EM.Cube's powerful, adaptive frequency sweep utility to study the frequency response of a planar structure. Adaptive frequency sweep uses rational function interpolation to generate smooth curves of the scattering parameters with a relatively small number of full-wave simulation runs in a progressive manner. Therefore, you need a port definition in your planar structure to be able to run an adaptive frequency sweep. This is clear in the case of an infinite periodic phased array, where your periodic unit cell structure must be excited using either a gap source or a probe source. You run an adaptive frequency sweep of an infinite periodic phased array in exactly the same way to do for regular, aperiodic, planar structures.

EM.Cube's Planar Modules also allows you to run an adaptive frequency sweep of periodic surfaces excited by a plane wave source. In this case, the planar MoM engine calculates the reflection and transmission coefficients of the periodic surface. Note that you can conceptually consider a periodic surface as a two-port network, where Port 1 is the top half-space and Port 2 is the bottom half-space. In that case, the reflection coefficient R is equivalent to S11 parameter, while the transmission coefficient T is equivalent to S21 parameter. This is, of course, the case when the periodic surface is illuminated by the plane wave source from the top half-space, corresponding to 90°< θ = 180°. You can also illuminate the periodic surface by the plane wave source from the bottom half-space, corresponding to 0° = θ < 90°. In this case, the reflection coefficient R and transmission coefficient T are equivalent to S22 and S12 parameters, respectively. Having these interpretations in mind, EM.Cube enables the "Adaptive Frequency Sweep" option of the Frequency Settings Dialog when your planar structure has a periodic domain together with a plane wave source.

 

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