Another approach to modeling a finite-sized antenna array is to analyze one of its elements and use the "Array Factor" concept to calculate its radiation patterns. This method ignores any inter-element coupling effects. In other words, you can regard the structure in the project workspace as a single isolated radiating element. To define an array factor, open the '''Radiation Pattern Dialog''' of the project. In the section titled "'''Impose Array Factor'''", you will see a default value of 1 for the '''Number of Elements''' along the X and Y directions. This implies a single radiator, representing the structure in the project workspace. There are also default zero values for the '''Element Spacing''' along the X and Y directions. You should change both the number of elements and element spacing in the X and Y directions to define a finite array lattice. For example, you can define a linear array by setting the number of elements to 1 in one direction and entering a larger value for the number of elements along the other direction. Keep in mind that when using an array factor for far field calculation, you cannot assign non-uniform amplitude or phase distributions to the array elements. For that purpose, you have to define an array object with a source array.
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=== Defining A Periodic Domain ===
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In general, a planar structure in [[EM.Cube]]'s [[Planar Module]] is assumed to have open boundaries. This means that the structure has infinite dimensions along the X and Y directions. In other words, the layers of the background structure extend to infinity, while the traces and embedded object sets have finite sizes. Along the Z direction, a planar structure can be open-boundary, or it may be truncated by PEC ground planes from the top or bottom or both. You can define a planar structure to be infinitely periodic along the X and Y directions. In this case, you only need to define the periodic unit cell. [[EM.Cube]] automatically reproduces the unit cell infinitely and simulates it using a spectral domain periodic version of the Green's functions of your project's background structure.
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To define a periodic structure, you must open [[Planar Module]]'s Periodicity Settings Dialog by right clicking the '''Periodicity''' item in the '''Computational Domain''' section of the Navigation Tree and selecting '''Periodicity Settings...''' from the contextual menu or by selecting '''Menu''' '''>''' '''Simulate > 'Computational Domain > Periodicity Settings...''' from the Menu Bar. In the Periodicity Settings Dialog, check the box labeled '''Periodic Structure'''. This will enable the section titled''"''Lattice Properties". You can define the periods along the X and Y axes using the boxes labeled '''Spacing'''. You can also define values for periodic '''Offset''' along the X and Y directions, which will be explained later.
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In a periodic structure, the virtual domain is replaced by a default blue periodic domain that is always centered around the origin of coordinates. Keep in mind that the periodic unit cell must always be centered at the origin of coordinates. The relative position of the structure within this centered unit cell will change the phase of the results.
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[[File:PMOM99.png]]
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Figure 1: [[Planar Module]]'s Periodicity Settings dialog.
=== Interconnectivity Among Unit Cells ===
Figure 3: The PEC cross unit cell and its planar mesh. Notice the cell extensions at the unit cell's boundaries.
=== Defining A Periodic MoM Simulation Domain ===
In general, a planar structure in [[EM.Cube]]'s [[Planar Module]] is assumed to have open boundaries. This means that the case of an structure has infinite periodic dimensions along the X and Y directions. In other words, the layers of the background structure extend to infinity, while the traces and embedded object sets have finite sizes. Along the Z direction, a planar structurecan be open-boundary, or it may be truncated by PEC ground planes from the field equations top or bottom or both. You can define a planar structure to be written in infinitely periodic along the following form:X and Y directions. In this case, you only need to define the periodic unit cell. [[EM.Cube]] automatically reproduces the unit cell infinitely and simulates it using a spectral domain periodic version of the Green's functions of your project's background structure.
:<math> \mathbf{E(r) = E^{inc}(r)} + \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \biggTo define a periodic structure, you must open [ \iiint_V \mathbf{ \overline{\overline{G}}_{EJ}(r|r[Planar Module]]') \cdot J_{mn}(rs Periodicity Settings Dialog by right clicking the ') } \, d\nu' + \iiint_V \mathbf{ \overline{\overline{G}}_{EM}(r|r') \cdot M_{mn}(rPeriodicity') } \'' item in the '''Computational Domain''' section of the Navigation Tree and selecting '''Periodicity Settings...''' from the contextual menu or by selecting '''Menu''' '''>''' '''Simulate > 'Computational Domain > Periodicity Settings...''' from the Menu Bar. In the Periodicity Settings Dialog, d\nucheck the box labeled ' \bigg] </math>''Periodic Structure'''. This will enable the section titled''"''Lattice Properties". You can define the periods along the X and Y axes using the boxes labeled '''Spacing'''. You can also define values for periodic '''Offset''' along the X and Y directions, which will be explained later.
In a periodic structure, the virtual domain is replaced by a default blue periodic domain that is always centered around the origin of coordinates. Keep in mind that the periodic unit cell must always be centered at the origin of coordinates. The relative position of the structure within this centered unit cell will change the phase of the results.
:<math> \mathbf{H(r) = H^{inc}(r)} + \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \bigg[ \iiint_V \mathbf{ \overline{\overline{G}}_{HJ}(r|r') \cdot J_{mn}(r') } \, d\nu' + \iiint_V \mathbf{ \overline{\overline{G}}_{HM}(r|r') \cdot M_{mn}(r') } \, d\nu' \bigg] </math><!--[[File:PMOM94PMOM99.png]]-->
where :<math>\mathbf{J_{mn}(r) = J_{mn}}(x,y,z) = \mathbf{J_{00}}(x+m S_x, y+n S_y, z) e^{j(m k_{x00} S_x + n k_{y00} S_y)}</math> :<math>\mathbf{M_{mn}(r) = M_{mn}}(x,y,z) = \mathbf{M_{00}}(x+m S_x, y+n S_y, z) e^{j(m k_{x00} S_x + n k_{y00} S_y)}</math> and :<math> -\infty < m, n < \infty </math><!--[[File:PMOM95(Figure 1).png]]--> In the above equations, <math>\mathbf{J_{00}(r)}</math> and <math>\mathbf{M_{00}(r)}</math> are the periodic unit cell's electric and magnetic currents that are repeated everywhere in space on a rectangular lattice with periods S<sub>x</sub> and S<sub>y</sub> along the X and Y directions, respectively. <math>k_{x00}</math> and <math>k_{y00}</math> are the periodic propagation constants along the X and Y directions, respectively, and they are given by: :<math> k_{x00} = k_0 \sin\theta \cos\phi </math> :<math> k_{y00} = k_0 \sin\theta \sin\phi </math><!--[[File:PMOM96(1).pngPlanar Module]]--> where θ and φ are the beam scan angles in the case of periodic excitation of lumped sources, or they are the spherical angles of incidence in the case of a plane wave source illuminating the periodic structure. Using the infinite summations, one can define periodic dyadic Green's functions in the spectral domain in the following manner: :<math> \mathbf{ \overline{\overline{G}}_{\mu \nu}^{PER} (r|r') } = \frac{1}{S_x S_y} \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \mathbf{ \tilde{\overline{\overline{G}}}_{\mu \nu} } (k_x, k_y, z|z') e^{-j[k_{xm}(x-x') + k_{yn}(y-y')]} </math> where:<math> k_{xm} = k_{x00} + \frac{2\pi m}{S_x} \quad \text{and} \quad k_{ym} = k_{y00} + \frac{2\pi m}{S_y} </math><!--[[File:PMOM97.png]]--> The above doubly infinite periodic Green's functions are said to be expressed in terms of "Floquet Modes". The exact formulation involves an infinite set of these periodic Floquet modes. During the MoM matrix fill process for a periodic structure, a finite number of Floquet modes are calculated. By default, [[EM.Cube]]'s planar MoM engine considers M<sub>x</sub> = M<sub>y</sub> = 25. This implies a total of 51 modes along the X direction and a total of 51 modes along the Y direction, or a grand total of 51<sup>2</sup> = 2,601 Floquet modes. You can increase the number of Floquet modes for your project from the Planar MoM Engine Periodicity Settings Dialog. In the section titled "Periodic Simulation", you can change the values of '''Number of Floquet Modes''' in the two boxes designated X and Ydialog.
[[File:PMOM98.png]]
Figure: A periodic planar layered structure with slot traces excited by a normally incident plane wave source.
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=== Characterizing Periodic Surfaces Using Angular Sweeps ===
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The reflection and transmission characteristics of a period surface as functions of the incidence angle are often of great interest. For that purpose, you can run an angular sweep of your periodic structure, where you normally fix the φ angle and sweep the θ angle from 180 to 90 degrees for one-sided surfaces and from 180 to 0 degrees for two-sided surface. To run an angular sweep, open the [[Planar Module]]'s '''Simulation Run Dialog''' and select the '''Angular Sweep''' option from its '''Simulation Mode''' dropdown list. This enables the '''Settings''' button, which opens up the '''Angle Settings Dialog'''. First, you must choose either Theta or Phi as the '''Sweep Angle'''. Then you can set the '''Start''' and '''End''' values of the selected incidence angle as well as the '''Number of Samples'''. At the end of an angular sweep simulation, you can plot the reflection and transmission coefficients 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 reflection and transmission coefficients of the structure are saved into two complex data files called "reflection.CPX" and "transmission.CPX". These data files are also listed in [[EM.Cube]]'s '''Data Manager''', where you can view or plot them.
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[[File:PMOM103.png]]
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Figure 1: [[Planar Module]]'s Angle Settings dialog.
=== Modeling Periodic Structures Using Adaptive Frequency Sweeps ===