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EM.Picasso

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/* Simulating Infinite and Finite-Sized Periodic Planar Structures */
== Simulating Infinite and Finite-Sized Periodic Planar Structures ==
 
A periodic structure is made up of identical elements that exhibits a repeated geometric pattern and are arranged in the form of a periodic lattice. The spacing between the elements is denoted by Sx along the X direction and Sy along the Y direction. The number of elements is denoted by Nx along the X direction and Ny along the Y direction (i.e. a total of Nx.Ny elements). If Nx and Ny are finite numbers, you have a finite-sized periodic structure, which is constructed using an "'''Array Object'''". If Nx and Ny are infinite, you have an infinite periodic structure with periods Sx and Sy along the X and Y directions, respectively.
[[Image:PMOM99.png|thumb|300px|EM.Picasso's Periodicity Settings dialog.]]
=== The Infinite Periodic Lattice ===
A periodic structure is made up of identical elements that exhibits a repeated geometric pattern and are arranged in the form of a periodic lattice. The spacing between the elements is denoted by Sx along the X direction and Sy along the Y direction. The number of elements is denoted by Nx along the X direction and Ny along the Y direction (i.e. a total of Nx.Ny elements). If Nx and Ny are finite numbers, you have a finite-sized periodic structure, which is constructed using an "'''Array Object'''". If Nx and Ny are infinite, you have an infinite periodic structure with periods Sx and Sy along the X and Y directions, respectively. An infinite periodic structure in EM.Picasso is represented by a "'''Periodic Unit Cell'''".  To define a periodic structure, you must open EM.Picasso'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'''. 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.
Besides conventional rectangular lattices, EM.Picasso can also handle complex non-rectangular periodic lattices. For example, many frequency selective surfaces have skewed grids. In order to simulate skewed-grid periodic structures, the definition of the grid has to be generalized. Let us define a periodic structure as a repetition of a basic unit cell at pre-determined locations described by (x<sub>mn</sub>, y<sub>mn</sub>), where m and n are integers ranging from -&infin; to +&infin;. For a general skewed grid, x<sub>mn</sub> and y<sub>mn</sub> can be expressed as:
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