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In the Finite Difference Time Domain (FDTD) method, a discretized form of Maxwell’s equations is solved numerically and simultaneously in both the 3D space and time. During this process, the electric and magnetic fields are computed everywhere in the computational domain and as a function of time starting at t = 0. From knowledge of the primary fields in space and time, one can compute other secondary quantities including frequency domain characteristics like scattering [[parameters]], input impedance, far field radiation patterns, radar cross section, etc.

Like every numerical technique, the FDTD method has disadvantages, too. Adding the fourth dimension, time, to the computations increases the size of the numerical problem significantly. Unfortunately, this translates to both larger memory capacity requirements and longer computation times. Note that the field data are generated in both the 3D space and time. [[EM.Cube]]'s [[FDTD Module]] uses a staircase "Yee" mesh to discretize the physical structure. This works perfectly well for rectangular objects that are oriented along the three principal axes. Difficulties start to appear for highly curved structures or slanted surfaces and lines. As a result, the quantization effect might compromise the geometrical fidelity of your structure. [[EM.Cube ]] provides a default adaptive FDTD mesher that can capture the fine details of geometric contours, slanted thin layers, surfaces, etc. to arbitrary precision. However, due to the stability criterion, smaller mesh cells lead to smaller time steps, hence longer computation times. Another disadvantage of the FDTD technique compared to naturally open-boundary methods like MoM is its finite-extent computational domain. This means that to model open boundary problems like radiation or scattering, absorbing boundary conditions are needed to dissipate the incident waves at the walls of the computational domain and prevent them from reflecting back into the domain. The accuracy of the FDTD simulation results depends on the quality of these absorbers and their distance from the actual physical structure. [[EM.Cube]]'s [[FDTD Module]] provides high quality perfectly match layer (PML) terminations at the boundaries, which can be placed only a quarter wavelength or less from your physical structure.

The simulated structure in FDTD usually consists of a number of objects that may have different material properties. [[EM.Cube’s Cube]]’s [[FDTD Module]] categorizes objects by their material composition. Several material types are currently offered: Perfect Electric Conductor (PEC), Perfect Magnetic Conductor (PMC), Isotropic & Homogeneous Dielectric, Uniaxial Anisotropic, Full Anisotropic, and three types of dispersive materials: Debye, Drude and Lorentz.

The computational domain is discretized using an appropriate meshing scheme. [[EM.Cube ]] uses a non-uniform, variable, staircase (pixelated) Yee mesh with a mesh density that you can customize. A fixed-cell mesh generator is also available, where you can set constant cell dimensions along the three principal axes for the entire computational domain. The variable mesh density is specified in terms of the effective wavelength inside material media. As a result, the mesh resolution and average mesh cell size differ in regions that are filled with different types of material. [[EM.Cube]]'s non-uniform mesher generates more cells in the areas that are occupied by dielectric materials, fewer cells in the free space regions and no cells inside (impenetrable) PEC regions. [[FDTD Module]]'s default "adaptive" mesh generator also refines the mesh around curved segments of lines, surface or solids to produce a far more accurate representation of your geometry. The example below illustrates a dielectric ellipsoid and a 3D view of its Yee mesh:

where i, j, k are the grid position indices along the X, Y, Z axes and n is the current time step. Similar expressions are obtained for the Y and Z components of the electric and magnetic fields. When your physical structure involves lossy materials with nonzero electric conductivity σ and/or nonzero electric conductivity σ_m, the above update equations become more complicated. In the case of anisotropic materials with tensorial constitutive [[parameters]], the electric displacement vector D and magnetic induction vector B need to be involved in the update of Maxwell's equations at every time step. This results in a total of twelve update equations at every time step. In the case of dispersive materials with time-varying constitutive [[parameters]], additional auxiliary differential equations are invoked and updated at every time step. Applying the proper boundary conditions for all the materials inside the computational domain and at the boundaries of the domain itself, [[EM.Cube ]] calculates and "updates" all the necessary field components at every mesh node, at every time step. The time marching loop continues in this way until it is terminated based on a certain criterion.

The FDTD method provides a wideband simulation of your physical structure. Frequency domain techniques often require a tedious frequency sweep to calculate the port characteristics (S/Y/Z [[parameters]]). By contrast, [[EM.Cube]]'s [[FDTD Module]] performs a discrete Fourier transform (DFT) of the time domain data to calculate these characteristics at the end of a single FDTD simulation run. In order to produce sufficient spectral information, an appropriate wideband temporal waveform is needed to excite the physical structure. The general form of [[EM.Cube]]'s default excitation waveform is a Modulated Gaussian Pulse given by:

The choice of the waveform, its bandwidth and time delay are important for the convergence behavior of the FDTD time marching loop. By default, [[EM.Cube ]] uses a modulated Gaussian waveform with optimal [[parameters]]: t = 0.966/Δf and t₀ = 4.5t, where Δf is the specified bandwidth of the simulation. The time delay t₀ is chosen so that the temporal waveform has an almost zero value at t = 0. Of the above waveforms, modulated Gaussian and sinusoidal waveforms are band pass with no DC content, while the Gaussian pulse is low pass with a frequency spectrum that is concentrated around f = 0. In a typical FDTD simulation, you set a center frequency for the structure of interest and then specify a bandwidth around this center frequency. These together determine the lowest and highest spectral contents of your FDTD waveform. Note that setting a bandwidth equal to 2f₀ sets the lowest frequency to DC (f_min = 0), which you may want to avoid in certain applications. On the other hand, using a Gaussian pulse waveform, you do want to set Δf = 2f₀. In contrast to the wideband, exponentially decaying, Gaussian pulse and modulated Gaussian waveforms, the sinusoidal waveform is extremely narrowband and single-frequency indeed. It does not decay over time and continues to oscillate indefinitely after reaching a steady state.

where c is the speed of light, and K_CFL is a constant. [[EM.Cube ]] uses a default value of K_CFL = 0.9. For a uniform grid with equal cell dimensions along the X, Y and Z directions, i.e. Δx = Δy = Δz = Δ, and the CFL condition reduces to:

As can be seen from the above criterion, a high resolution mesh requires a smaller time step. Since you need to let the fields in the computational domain fully evolve over time, a smaller time step will require a larger number of time steps to achieve convergence. [[EM.Cube ]] automatically chooses a time step that satisfies the CFL condition.

The CPML [[parameters ]] are chosen to be an increasing function of the distance from the boundaries of the computational domain. [[EM.Cube ]] uses a polynomial profile of degree n_PML. Given the interrelationships among these [[parameters]], one can write:

where r is the distance of field observation point inside the CPML layer from the edge of the computational domain. The [[parameters ]] σ_max, κ_max, α_min and α_max as well as n_PML can be modified by the user.

A periodic structure is one that repeats itself infinitely in one, two or three directions. [[EM.Cube]]'s [[FDTD Module]] allows you to simulate doubly periodic structures with periodicities along the X and Y directions. Many interesting structures such as frequency selective surfaces (FSS), electromagnetic band-gap (EBG) structures and metamaterial structures can be modeled using periodic geometries. In the case of an infinitely extended periodic structure, it is sufficient to analyze only a unit cell. In the FDTD method, this is accomplished by applying periodic boundary conditions (PBC) at the side walls of the computational domain. The application of the PBC is straightforward for the case of a normally incident plane wave source since the fields do not experience any delay as they travel across the unit cell. Obliquely incident plane waves, on the other hand, cause a time delay in the transverse plane. This delay requires knowledge of the future values of the fields at any time step.

A number of techniques have been proposed to solve this problem. [[EM.Cube ]] uses a recently developed novel technique that is known as Direct Spectral FDTD or Constant Transverse Wavenumber method. In this technique, the components of the transverse (horizontal) wavenumber are kept constant in the direction of periodicity. This technique shows a significant advantage over the other methods for simulation of the incident illuminations close to the grazing angles.

for TE_z polarization. Here, f₀ is the center frequency of the modulated Gaussian pulse waveform, t₀ is the time delay, and τ is the Gaussian pulse width. The choices of the Gaussian waveform [[parameters ]] are very critical in order to avoid possible resonances. For a fixed value of k_l, the horizontal resonance occurs at:

In [[EM.Cube]]'s [[FDTD Module]], a physical structure consists of one or several objects that are grouped and identified by their material types. Materials are divided into five categories that are listed under the '''Physical Structure''' node at the top of the Navigation Tree:

Under each material node, you can create new material groups of the same type/category but with different properties (color, texture, or electric and magnetic constitutive [[parameters]]). These material groups are used to organize the CAD objects that you draw in the project workspace or import from external model files. When you create a new geometrical object such as a Box or a Sphere, it is inserted under the currently active material type. There is only one material group that is active at any time. It is normally recommended that you first create material groups, and then draw new objects as part of the currently active material group. However, if you start a new [[FDTD Module]] project from scratch, and you start drawing a new object without having previously defined any material groups, a new default PEC group is created and added to the Navigation Tree to hold your new CAD object.

[[EM.Cube]]'s [[FDTD Module]] offers two types of perfect conductors:

PEC and PMC materials do not have any constitutive material properties that you can modify except for their color or texture. Note that [[FDTD Module]]'s PMC materials are different in nature than the PMC traces you will find in [[EM.Cube]]'s [[Planar Module]]. In the latter, the PMC term is applied to slot traces, which represent finite-sized cut-out objects (areas) from an infinite horizontal PEC ground plane. [[FDTD Module]]'s PMC objects, by contrast, can be arbitrarily oriented [[Surface Objects|surface objects]] or impenetrable [[Solid Objects|solid objects]] of finite extents.

In [[EM.Cube]]'s [[FDTD Module]], a dielectric material represents a general isotropic material with both electric and magnetic properties. The constitutive [[parameters ]] of a dielectric material include permittivity (ε), permeability (μ), electric conductivity (σ) and magnetic conductivity (σ_m):

* You may also choose from a list of preloaded material types. Click the button labeled '''Material''' to open [[EM.Cube]]'s Material List dialog. Select the desired material from the list or type the first letter of a material to find it. For example, typing '''V''' selects '''Vacuum '''in the list. Once you close the dialog by clicking '''OK''', the selected material properties fill the parameter fields automatically.

Figure 2: [[EM.Cube]]'s material list.

[[EM.Cube]]'s [[FDTD Module]] allows you to define a general anisotropic material, whose constitutive [[parameters]], i.e. permittivity ('''ε'''), permeability ('''μ'''), electrical conductivity ('''σ''') and magnetic conductivity ('''σ_m'''), are all tensorial in nature. Each constitutive parameter in this case is represented by a 3×3 matrix:

A "'''Uniaxial'''" material is a special case of an anisotropic material whose constitutive [[parameters ]] are all diagonal matrices. Specifying an anisotropic material as <u>'''Uniaxial'''</u> in the [[FDTD Module]] has a very important computational implication. There are six field update equations for uniaxial materials at each time steps: three for the electric field and three for the magnetic field. In this respect, a uniaxial material is similar to an isotropic dielectric material. On the other hand, a fully anisotropic material with non-zero off-diagonal constitutive matrix elements requires twelve update equations at each time step: three equations for the three components of each of the four vector fields '''E''', '''D''', '''H''' and '''B'''. As a result, the time loop for fully anisotropic materials takes much longer time than uniaxial materials.

[[#Perfect Conductors|PEC]], [[#Perfect Conductors|PMC]], [[#Dielectric Materials|dielectric]] and [[#Anisotropic Materials|anisotropic]] materials are non-dispersive. In other words, their constitutive [[parameters ]] do not vary with frequency. Most of the materials used in the design of RF and microwave circuits, antennas and systems fall into this frequency-independent category. However, there are other types of materials whose constitutive [[parameters ]] exhibit frequency-dependent behaviors. [[EM.Cube]]'s [[FDTD Module]] currently offers three types of dispersive material:

The FDTD simulation engine uses the Auxiliary Differential Equation (ADE) method to model dispersive materials. [[EM.Cube ]] allows you to define an arbitrary number of poles for each of the above dispersive material types. Keep in mind that all the objects belonging to the same dispersive material group have the same dispersion properties.

The following rules apply to the definition of materials and objects in [[EM.Cube]]'s [[FDTD Module]]:

[[EM.Cube]]'s [[FDTD Module]] allows overlapping objects, although it is generally recommended that object overlaps be avoided in favor of clearly defined geometries and object boundaries. If two or more objects of the same material type and group overlap, they are merged using the Boolean union operation during the mesh generation process.

In a similar way, you can move one or more objects from an FDTD material group to one of [[EM.Cube]]'s other modules. In this case, the sub-[[menus ]] of the '''Move To >''' item of the contextual menu will indicate all the [[EM.Cube ]] modules that have valid groups for transfer of the selected objects. You can also move one or more objects from [[EM.Cube]]'s other modules to a material group in the [[FDTD Module]]. This is especially useful when importing structure from external model files. Keep in mind that in [[EM.Cube ]] you can import external objects only to '''[[CubeCAD]]'''.

The FDTD method solves computational electromagnetic problems within a finite region of space. Thus, a finite-extent solution domain must be defined before the problem can be solved. This is rather straightforward in the case of a shielded structure, where a typical PEC enclosure box defines the computational domain. The difficulty arises when you try to model an open-boundary structure like an antenna. In that case, the computational domain must be truncated using appropriate termination boundary conditions. For open-boundary circuit-type structures like filters, couplers, etc., without a shielding, it is still possible to place a PEC enclosure box at a distance from the actual structure. However, the distance of the box walls from the structure must be large enough to eliminate any coupling effects. Furthermore, you must also watch for spurious box resonance effects, as you effectively place your structure inside a resonant cavity. The objective of termination boundary conditions is to eliminate the reflections from the walls of the domain box back to the computational domain. The previous releases of [[EM.Cube ]] featured various termination boundary conditions like absorbing boundary conditions (ABC) and uniaxial perfectly matched layer (UPML). In this release of [[EM.Cube]]'s [[FDTD Module]], we have phased out all the previous diverse boundary conditions in favor of Convolutional Perfectly Matched Layer (CPML), which has demonstrated a superior performance for a wide range of applications.

To solve an FDTD problem, proper boundary conditions must be enforced on the bounding walls of the solution domain. The current release of [[EM.Cube]]'s [[FDTD Module]] supports four types of boundary conditions:

By default, all the six sides of the computational domain box are set to CPML, representing a completely open-boundary structure. Different boundary conditions can be assigned to each of the six sides of the domain box. The periodic boundary conditions are special ones that are assigned through [[FDTD Module]]'s Periodicity Dialog and will be discussed later under modeling of periodic structures. The current release of [[EM.Cube ]] allows periodic boundary conditions only on the side walls of the computational domain, and not on the top or bottom walls.

[[EM.Cube ]] provides Perfectly Matched Layers (PML) as absorbing boundary conditions. PML's are layers of fictitious complex anisotropic materials designed to absorb any incident electromagnetic wave at all angles of incidence. The advantage of using a PML boundary condition over other types of ABC is that you do not need to care about what material or combinations of materials exist at the boundaries, since the PML can absorb anything! The disadvantage of using PML is that they require more computational resources and more sophisticated update field equations. Several types of PML have been proposed in the FDTD literature. [[EM.Cube]]'s [[FDTD Module]] uses the Convolutional Perfectly Matched Layer (CPML). This is based on a newer implementation of the complex frequency-shifted PML (CFS-PML) that uses recursive convolution. It has been shown that CPML is highly effective at absorbing evanescent waves and signals with a long time signature. Therefore, using CPML, you can place the boundaries much closer to the objects in the project workspace.

{{Note|[[EM.Cube]]'s default quarter wavelength offset for the domain box is a very conservative choice and can be reduced further in many cases. A offset equal to eight free-space grid cells beyond the largest bounding box usually give a more compact, but still valid, domain box.}}

You may occasionally want to use [[EM.Cube]]'s FDTD simulator to model planar structures. Although [[EM.Cube ]] provides the more computationally efficient [[Planar Module]] for this very purpose, there are many cases when an FDTD simulation might prove advantageous over a 2.5-D MoM simulation. Examples include examining the transient response of a planar structure, very wideband simulations, planar structures involving complex materials or 3D geometries embedded inside the substrate layers, to name a few.

A planar substrate usually consists of one or more dielectric layers, possibly with a PEC ground plane at its bottom. Unlike [[EM.Cube]]'s [[Planar Module]], where the substrate layers are defined implicitly in the "Stack-up Settings" dialog, in the finite-domain [[FDTD Module]], you need to draw each dielectric layer separately and then stack them up manually. The substrate of a planar layered structure extends laterally to infinity. In other words, the ±X and ±Y boundary walls must, in effect, retreat to infinity. This can be accomplished in the [[FDTD Module]] by setting up the CPML layers in a particular way. For this purpose, the lateral CPML layers need to move in and touch the sides of the dielectric layer stack-up.

{{Note|The current release of [[EM.Cube]]'s [[FDTD Module]] does not support anisotropic or dispersive layers of laterally infinite extents. In other words, You can only define anisotropic and dispersive material objects of finite size that do not touch the CPML boundaries.}}

The mesh generation process in [[EM.Cube]]'s [[FDTD Module]] involves four steps:

[[EM.Cube]]'s FDTD mesh is a rectangular Yee mesh that extends to the entire computational domain. It is primarily constructed from three mesh grid profiles along the XY, YZ and ZX principal planes. These projections together create a 3D rectangular (pixelated) mesh space. You have the option to choose one of the three FDTD mesh types:

According to the Courant-Friedrichs-Levy (CFL) stability criterion, the FDTD time step is determined by the smallest cell size in your FDTD mesh. Occasionally, [[FDTD Module]]'s adaptive mesh generator may create extremely tiny grid cells that would result in extremely small time steps. This would then translate into a very long computation time. [[EM.Cube ]] offers the "Regular" FDTD mesh generator, which is a simplified version of the adaptive mesh generator. In a regular FDTD mesh, the grid cell sizes stay rather the same in objects of the same material composition. The mesh resolution increases in materials of higher permittivity and/or permeability based on the effective wavelength in exactly the same way as the adaptive mesh. Finally, [[EM.Cube]]'s FDTD Modules offers a "Uniform" FDTD mesh generator. The uniform mesh consists of three uniform grids along the XY, YZ and ZX principal planes. In other words, the grid cell sizes &Delta;x, &Delta;y and &Delta;z are fixed throughout the entire computational domain. In this case, the uniform mesh generator has to fit your physical structure to the fixed mesh, rather than adapting the mesh to your physical structure.

Because a full 3D FDTD mesh is difficult to visualize everywhere in the computational domain, only the discretized objects are displayed in [[EM.Cube]]'s "'''Mesh View'''" mode. In particular, only the outer boundary cells on the surface of [[Solid Objects|solid objects]] are shown. However, you can view the mesh grid planes across the domain. You can even step these planes back and forth inside the domain and view different mesh profiles of your physical structure.

* First, click the '''Mesh Settings''' [[Image:mesh_settings.png]] button of the '''Simulate Toolbar''' or select '''Menu > Simulate > Discretization > Mesh Settings...''', or right click on the '''Yee Mesh''' item of the Navigation Tree and select '''Mesh Settings...''' from the contextual menu, or use the keyboard shortcut '''Ctrl+G'''. The Mesh Settings Dialog opens up, where you can set the values of the various mesh [[parameters ]] including the '''Mesh Density'''.

* To display the FDTD mesh, click the '''Show Mesh''' [[Image:mesh_tool.png]] button of the '''Simulate''' '''Toolbar '''or select '''Menu > Simulate > Discretization > Show Mesh''' or use the keyboard shortcut '''Ctrl+M'''. This takes [[EM.Cube ]] into its "Mesh View" mode, and the Yee mesh of the whole structure is displayed in the project workspace. While the mesh view is enabled, the '''Show Mesh''' [[Image:mesh_tool.png]] button remains depressed. To get back to [[EM.Cube]]'s "Normal View" mode, click this button one more time, or deselect '''Menu > Simulate > Discretization > Show Mesh''' to remove its check mark or simply hit the '''Esc Key''' of the keyboard.

In [[EM.Cube]]'s "Mesh View" mode, you can rotate or pan the view of the project workspace, but you cannot edit the objects. '''"Show Mesh"''' generates a new mesh and displays it if there is none in the memory, or it simply displays an existing mesh in the memory. This is a useful feature because generating an FDTD mesh may take a long time depending on the complexity of structure and the total size of the computational domain. If you change the structure or alter the mesh settings, a new mesh is always generated. You can ignore any mesh in the memory and force [[EM.Cube ]] to generate a fresh FDTD mesh from the ground up by selecting '''Menu > Simulate >Discretization > Regenerate Mesh''' or by right clicking on the '''Yee Mesh''' item of the Navigation Tree and selecting '''Regenerate''' from the contextual menu.

A volumetric FDTD mesh is overwhelming for visualization in the 3D space. For this reason, [[EM.Cube]]'s mesh view only shows the outline of the (staircased) meshed objects, skipping the outline of all the individual brick cells in the entire computational domain. The mesh grid planes provide a 2D profile of the mesh cells along the principal coordinate planes. Since the Yee cells are congruent along the coordinate axes, the three mesh grid planes together provide a complete picture of the entire FDTD mesh. To display a mesh grid planes, select '''Menu > Simulate > Discretization > Grid Planes >''' and pick one of the three options: '''XY Plane''', '''YZ Plane''' or '''ZX Plane'''. You may also right click on one of the '''XY Plane''', '''YZ Plane''' or '''ZX Plane''' items in the '''Discretization''' section of the Navigation Tree and select '''Show''' from the contextual menu.

* Using the [[Keyboard Shortcuts|keyboard shortcuts ]] '''">"''' or '''"<"'''.

Once the project structure is meshed in [[EM.Cube]]'s [[FDTD Module]], a second coordinate system becomes available to you. The mesh grid coordinate system allows you to specify any location in the computational domain in terms of node indices on the mesh grid. [[EM.Cube ]] displays the total number of mesh grid lines of an FDTD simulation domain (N_x × N_y × N_z) along the three principal axes on the '''Status Bar'''. Therefore, the number of cells in each direction is one less than the number of grid lines, i.e. (N_x-1)× (N_y-1) × (N_z-1). The minimum X, Y, and Z coordinates of the FDTD domain in the world coordinate system (Xmin, Ymin, Zmin), which represent the lower left front corner of the domain box, become the origin of the mesh grid coordinate system (0,0,0), The maximum domain coordinates, which represent the upper right back corner of the domain box, are therefore (N_x-1, N_y-1, N_z-1).

[[EM.Cube ]] allows you to navigate through the mesh grid and evaluate the grid points individually. Every time you display one of the three mesh grid planes, the "'''Grid Coordinate System (GCS)'''" is automatically activated. On the Status Bar, you will see [[Image:statusgrid.png]] instead of the default [[Image:statusworld.png]]. This means that the current coordinates reported on Status Bar are now expressed in grid coordinate system. The current grid point is displayed by a small white circle on the current mesh grid plane, and it always starts from (I= 0, J=0, K=0). Using the keyboard's '''Arrow Keys''', you can move the white circle through the mesh grid plane and read the current node's (I, J, K) indices on the status bar. You can switch back to the "'''World Coordinate System (WCS)'''" or change to the "'''Domain Coordinate System'''" by double-clicking the status bar box that shows the current coordinate system and cycling through the three options. The domain coordinate system is one that establishes its origin at the lower left front corner of the computational domain and measure distances in project unit just like the WCS.

[[EM.Cube]]'s [[FDTD Module|FDTD module]] discretizes objects using what is often referred to as the “staircase approximation”. In this mesh generation scheme, the structure is recreated using a large number of cubic cells carefully assembled in a way that approximates the shape of the original structure. By default, a carefully calculated, "<u>'''Adaptive'''</u>" mesh of your physical structure is generated in order to satisfy the following criteria:

Occasionally, you may prefer a more regular FDTD mesh with almost equal grid line spacing everywhere, but still with a frequency-dependent cell size. In that case, you can select the "<u>'''Regular'''</u>" option of the '''Mesh Type '''dropdown list in the FDTD Mesh Settings dialog. The regular FDTD mesh enforces only two of the above [[parameters]]: '''Minimum Mesh Density''' and '''Absolute Minimum Grid Spacing'''. Or you may opt for an absolutely "<u>'''Uniform'''</u>" mesh type, for which you need to specify the '''Cell Size '''along the X, Y, Z directions in project units.

When [[EM.Cube ]] generates an FDTD mesh, a large number of geometrical considerations are taken into account. These include the bounding box of each object and its corners, the ends of a line, the apex of a cone or pyramid, or the locations of lumped sources, field probes and sensors, vertices of plane wave or far field boxes, to name a few examples. These points are “locked” as fixed grid nodes in the FDTD mesh. [[EM.Cube ]] determines these points internally to generate a mesh that best approximates the original structure. As you saw earlier, you can use the FDTD mesh settings to control the shape and resolution of the mesh, for example, around the curved portions of your structure, or on slanted lines or faces, etc. These settings are global and apply to all the objects making up your physical structure.

In certain cases, you may wish to exert some level of local mesh control. For example, you may want to increase the mesh density at a very particular area of your structure. Or you may want to increase or decrease the mesh resolution inside certain types of materials independent of their permittivity and permeability. [[EM.Cube ]] provides two additional mechanisms for local control of the FDTD mesh: locking mesh of object groups and user defined fixed grid points. You can lock the mesh density of each material group. In this case, the areas inside and around the "Locked" material objects are meshed according to the locked mesh density, while the rest of the computational domain is meshed using the project's global mesh density. Locking mesh is done through the property dialog of each material group. At the bottom of the material property dialog, in the "Mesh Properties" section, there is a check box labeled '''Lock Mesh''', which is unchecked by default. If you check this box, it enables the box labeled '''Mesh Density''', which initially has a value equal to the project's global mesh density.

Before you can run an FDTD simulation, you have to define a source to excite your project’s physical structure. A physical source has a zero value at t = 0, but it rises from zero at t > 0 according to a specified waveform. [[EM.Cube]]'s [[FDTD Module]] currently offers three types of temporal waveform:

A sinusoidal waveform is single-tone and periodic. Its spectrum is concentrated around a single frequency, which is equal to your project's center frequency. A sinusoidal source does not have a finite energy and it does not decay as t &rarr; &infin;. A Gaussian pulse decays exponentially as t &rarr; &infin;, but it has a lowpass frequency spectrum which is concentrated around f = 0. A modulated Gaussian pulse decays exponentially as t &rarr; &infin;, and it does have a bandpass frequency spectrum concentrated around your project's center frequency. For most practical problems, a modulated Gaussian pulse waveform provides an adequate performance. That is why this type of waveform is chosen by [[EM.Cube ]] as your project's default waveform.

[[EM.Cube]]'s [[FDTD Module]] provides a number of sources or excitation schemes that have different applications. These source types will be described in the following sections. An [[#Ideal Sources|Ideal Source]] is the simplest way of exciting a structure in [[EM.Cube]]'s [[FDTD Module]]. It consists of an ideal voltage source connected between two consecutive nodes of your FDTD mesh. A [[#Lumped Sources|Lumped Source]] is an ideal voltage source in series with a resistor. You have to place a lumped source on a line object that is parallel to one of the three principal axes, and you can assign a port to it to calculate the circuit characteristics of your structure. A [[#Waveguide Sources|Waveguide Source]] is placed across a rectangular waveguide. In other words, it requires a hollow box object that is aligned along one of the three principal axes and has one or two open ends. A waveguide source can excite a certain TE_mn or TM_mn modal profile of the host rectangular waveguide. A [[#Distributed Sources|Distributed Source]] is defined on a finite-sized plane parallel to one of the three principal planes and with a prescribed field distribution profile on that plane. A [[#Plane Waves|Plane Wave Source]] is used to study the scattering characteristics of your structure and compute its radar cross section (RCS). A [[#Focused Gaussian Beams|Gaussian Beam Source]] is similar to a plane wave source but with a focused energy profile in the transverse direction and a beam-diverging profile in the longitudinal direction.

* By default, [[EM.Cube ]] creates a +Z-directed ideal source located at the origin of coordinates (0, 0, 0). You can change the location of the source by setting new values for the X, Y and Z coordinates. When you use the spin buttons to increment or decrement the source coordinates, you can see the source moving in the project workspace. You can also change the '''Direction''' of the source from a dropdown list in the Source Location section of the dialog that contains ±X, ±Y and ±Z options.

A '''Lumped Source ''' is the most commonly used way of exciting a structure in [[EM.Cube]]'s [[FDTD Module]]. A lumped source acts as a voltage source in series with an internal resistance that is placed between two adjacent mesh grid nodes on a line object. The line object must be parallel to one of the three principal axes.

Waveguide structures have many applications at microwave and millimeter wave frequencies. For example, a rectangular waveguide is used to feed a pyramidal horn antenna. A waveguide structure is usually excited using some type of strategically located probe mechanism. This can be modeled using a lumped source placed on a wire structure made up of line objects. Alternatively, use can use [[EM.Cube]]'s '''Waveguide Sources''', a special type of source that excites a prescribed modal field distribution in a rectangular waveguide structure. The scattering [[parameters ]] are calculated from knowledge of incident and reflected fields at designated waveguide ports. Waveguide sources typically provide more accurate results for scattering [[parameters ]] compared to lumped ports as they represent the actual dominant propagating modes at the transmission line ports.

[[EM.Cube ]] provides special waveguide sources that can excite either the TE_mn or TM_mn modes of a rectangular waveguide which is oriented along one of the three principal axes. In other words, the plane of the waveguide source must be parallel to one of the principal (XY, YZ or ZX) coordinate planes.

Waveguide sources are a special case of distributed sources in [[EM.Cube]]'s [[FDTD Module]]. A Distributed Source is defined in a rectangular plane of finite extents, parallel to one of the three principal coordinate planes. An impressed electric field component is assumed across the specified rectangular area, which pumps energy into the computational domain. The current version of [[EM.Cube ]] provides three spatial field profiles for a distributed source:

Ports are used to order and index sources for circuit parameter calculations like S/Y/Z [[parameters]]. That is why they are defined in the '''Observables''' section of Navigation Tree. In [[EM.Cube]]'s [[FDTD Module]], you can define ports at the location of '''Lumped Sources''', '''Waveguide Sources''' and '''Distributed Sources'''. In other words, ideal sources or other types of sources cannot be used to define ports or calculate port characteristics.

{{Note|In order to obtain correct results, the port impedance must equal the characteristic impedance of the transmission line on which the port is established. This is not done automatically in [[EM.Cube]].}}

Using simple lumped sources, you can simulate a variety of transmission line structures in [[EM.Cube’s Cube]]’s [[FDTD Module]] including filters, couplers or antenna feeds and you can calculate their scattering [[parameters]]. This approach may become less accurate at very high frequencies when the details of the feed structures become important and can no longer be modeled with highly localized lumped ports. In such cases, it is recommended to use “Distributed Sources”, which utilize accurate modal field distributions at the ports for calculation of the incident and reflected waves. This and next two sections explain how you can use simple lumped sources to model some popular transmission line feeds.

[[EM.Cube ]] sets the internal resistance of each of the two coupled lumped sources equal to twice the specified port (or line) impedance since the CPW port is a parallel connection of the two individual lumped sources. If you want to model a CPW with laterally infinite substrate and ground planes, make sure to set the domain offsets in the ±X and ±Y directions equal to zero. In the -Z direction, you need a nonzero offset to push the bottom CPML boundary down away outside the dielectric layer of finite thickness. Alternatively, you can terminate the bottom boundary by PEC and set its domain offset equal to zero to represent a CPW with a bottom PEC ground.

{{Note|[[EM.Cube ]] will set the internal impedance of each of the four coupled lumped sources equal to four times the specified port (or line) impedance since the coaxial port is a parallel connection of the four individual lumped sources.}}

In [[EM.Cube]]'s [[FDTD Module]] you can define simple lumped elements such as resistors, inductors, capacitors as well as nonlinear diodes. Although lumped loads are not sources and do not excite a structure, their properties are similar to lumped sources. Lumped Loads are incorporated into the FDTD grid across two adjacent nodes in a similar manner to lumped sources. Likewise, lumped loads are defined on Line objects. In order to create a lumped load, you must have at least one line object in your project.

In [[EM.Cube]]'s [[FDTD Module]], you can excite a structure with an arbitrary incident plane wave and compute its scattering pattern or bi-static radar cross section. A plane wave excitation is defined by its propagation vector indicating the direction of incidence and its polarization. [[EM.Cube]]'s [[FDTD Module]] provides the following polarization options:

[[EM.Cube ]] gives you an option to illuminate objects with a focused beam instead of a uniform plane wave. The focused beam is a Gaussian beam, which is a solution of the paraxial approximation to the Helmholtz equation. The fundamental Gaussian beam is rotationally-symmetric about its propagation axis, and its transverse field distribution follows a Gaussian function profile. The critical parameter is the beam radius w₀; it is the point where the field drops by 1/e from its value at the center. The beam opens up into a cone along the propagation direction, with a cone angle of tan &theta; = &lambda;₀/(&pi;.&omega;₀) (&lambda;₀ is the free-space wavelength). <font color="#a52a2a"><u>'''The beam radius has to be at least &lambda;₀/&pi;; otherwise, strong fields appear outside the excitation box.'''</u></font>

The FDTD method is one of the most versatile numerical techniques for solving electromagnetic modeling problems. Choosing the right settings and optimal values for certain numerical [[parameters ]] will have a significant impact on both accuracy and computational efficiency of an FDTD simulation. Below are a number of steps that you should typically follow by order when planning your FDTD simulation:

For certain problems, more than one combination or choice of settings and [[parameters ]] may still give acceptable results. In most cases, [[EM.Cube ]] tries to make these choices convenient for you by suggesting default settings or default parameter values. For example, [[EM.Cube ]] by default generated am "adaptive" type mesh with a default density of 20 cells per effective wavelength. The default computational domain features CPML walls placed a quarter free-space wavelength away from the large bounding box of the entire physical structure. A modulated Gaussian waveform with certain optimal [[parameters ]] is used to drive the project's excitation source by default. You can change most of these settings arbitrarily. For example, you can set up your own computational domain with different types of boundary conditions, customize the FDTD mesh by modifying a large number of mesh settings and use other types of excitation waveforms.

{{Note|Keep in mind that you are always responsible for the choice of excitation source and the project observables. In other words, [[EM.Cube ]] does not automatically provide a default excitation source or does not suggest default observables.}}

In [[EM.Cube]], project observables are the simulation data that are generated by the simulation engine at the end of each simulation run. [[EM.Cube]]'s FDTD simulation engine calculates all the six electric and magnetic field components (E_x, E_y, E_z, H_x, H_y and H_z) at every mesh grid node at all time steps from t = 0 until the end of the time loop. However, in order to save memory space, the engine has to destroy the temporal field data from each time step to the next and reuse the memory. Storage, manipulation and visualization of 3D data can become overwhelming for complex structures and larger computational domains. Furthermore, calculation of some field characteristics such as radiation patterns or radar cross section (RCS) can be sizable, time-consuming, post-processing tasks. That is why [[EM.Cube ]] asks you to define project observables to instruct why types of simulation data you seek in each simulation effort.

[[EM.Cube]]'s FDTD Modules currently offers the following types of observable:

* Port Characteristics (S/Y/Z [[Parameters ]] and VSWR)

Field probes monitor the field components at a certain point in the computational domain. They record the time-domain field data during the entire time loop and compute their frequency spectrum using a discrete Fourier transform. Field sensors are primarily intended for observation of near field maps on a certain cross section of the computational domain. The field sensor planes are parallel to one of the three principal XY, YZ or ZX planes. When you run a frequency sweep or parametric sweep, multiple maps are generated for each sample of your sweep variable, and you can animate these maps. You can also animate the evolution of the near fields in the time domain over the course of the simulated time loop. [[EM.Cube ]] can also keep track of the electric, magnetic and total energy of the computational domain as functions of the time step.

Using asymptotic near-to-far-field transformations, [[EM.Cube ]] calculates the far fields of your physical structure in the standard spherical coordinate system. The radiation patterns are indeed the spherical electric field components E_&theta; and E_&phi; expressed as functions of the observation angles &theta; and &phi; over a unit sphere. The far field data are calculated in the frequency domain at a specified frequency, which is equal to your project's center frequency by default. When your excitation source is a plane wave or a Gaussian beam, the far field data actually represent the scattering behavior of your "target". In the case of a plane wave source, the FDTD simulation engine can also compute the radar cross section of you target. If your structure is periodic, then the reflection and transmission coefficients of the periodic surface are also calculated over the entire bandwidth of your project.

You can define ports for lumped sources, waveguide sources and distributed sources. In that case, the FDTD simulation engine calculates the scattering (S) [[parameters ]] of your multiport network over the entire bandwidth specified in your project. From the scattering matrix, [[EM.Cube ]] determines the impedance and admittance matrices of your network over the operational bandwidth. You can plot the S/Y/Z [[parameters ]] in EM.Grid. If your project has more than one port, the FDTD time loop will be run as many times as the number of ports, N. In each time loop run j (j = 1, 2, ..., N), the source(s) associated with the jth port is (are) excited with a unit amplitude and all the other sources are turned off. In this run, all the S_ij parameters (i = 1, 2, ..., N) are calculated. At the end of the Nth run, the entire S matrix is completed.

An FDTD simulation involves a number of numerical [[parameters ]] that can be accessed and modified from the FDTD Engine Settings Dialog. To open this dialog, select '''Menu > Simulate > Simulation Engine Settings... '''or open the '''Run Dialog''', and click the '''Settings''' button next to the engine dropdown list.

In the " '''Convergence''' " section of the dialog, you can set the '''Termination Criterion''' for the FDTD time loop. The time loop must stop after a certain point in time. If you use a decaying waveform like a Gaussian pulse or a Modulated Gaussian pulse, after certain number of time steps, the total energy of the computational domain drops to very negligible values, and continuing the time loop thereafter would not generate any new information about your physical structure. By contrast, a sinusoidal waveform will keep pumping energy into the computational domain forever, and you have to force the simulation engine to exit the time loop. [[EM.Cube]]'s [[FDTD Module]] provides two mechanism to terminated the time loop. In the first approach, an energy-like quantity defined as U_n = &Sigma; [ &epsilon;₀|'''E_i,n'''|² + &mu;₀|'''H_i,n'''|² ].&Delta;V_i is calculated and recorded at a large random set of points in the computational domain. Here i is the space index and n is the time index. The quantity U_n has a zero value at t = 0 (i.e. n = 0), and its value starts to build up over time. With a Gaussian or Modulated Gaussian pulse waveform, U_n reach a maximum value U_max at some time step and starts to decline thereafter. The ratio 10.log( U_n/ U_max) expressed in dB is used as the convergence criterion. When its value drops below certain '''Power Threshold''', the time loop is exited. The default value of Power Threshold is -30dB, meaning that the FDTD engine will exit the time loop if the quantity U_n drops to 1/1000 of its maximum value ever. The second termination criterion is simply reaching a '''Maximum Number of Time Steps''' , whose default value set to 10,000. A third option, which is [[EM.Cube]]'s default setting (labeled "'''Both'''"), terminates the simulation as soon as either of the first two criteria is met first.

The serial CPU solver is [[EM.Cube]]'s basic FDTD kernel that run the time marching loop on a single central processing unit (CPU) of your computer. The default option is the multi-core CPU solver. This is a highly parallelized version of the FDTD kernel based on the Open-MP framework. It takes full advantage of a multi-core, multi-CPU architecture, if your computer does have one. The GPU solver is a hardware-accelerated FDTD kernel optimized for CUDA-enabled graphical processing unit (GPU) cards. If your computer has a fast NVIDIA GPU card with enough onboard RAM, the GPU kernel can speed up your FDTD simulations up to 50 times or more over the single CPU solver.

For structures excited with a plane wave source, there are two standard FDTD formulations: '''Scattered Field '''(SF) formulation and '''Total Field - Scattered Field''' (TF-SF) formulation. [[EM.Cube]]'s [[FDTD Module|FDTD module]] offers both formulations. The TF-SF solver is the default choice and is typically much faster than the SF solver for most problems. In two cases, when the structure has periodic boundary conditions or infinite CPML boundary conditions (zero domain offsets), only the SF solver is available. The other sections of the FDTD Simulation Engine Settings dialog will be described next in the context of Waveforms and Discrete Fourier Transforms.

Once you build your physical structure in the project workspace and define an excitation source, you are ready to run an FDTD simulation. The simulation engine will run even if you have not defined any observables. Obviously, no simulation data will be generated in that case. [[EM.Cube]]'s [[FDTD Module]] currently offers several different simulation modes as follows:

Figure 1: [[EM.Cube]]'s FDTD Simulation dialog.

To start the FDTD simulation, click the '''Run''' button at the bottom of this dialog. Once the simulation starts, the "'''Output Window'''" pops up and reports messages during the different stages of the FDTD simulation. During the FDTD time marching loop, after every 10th time step, the output window updates the values of the time step, elapsed time, the engine performance in Mega-cells per seconds, and the value of the convergence ratio U_n/U_max in dB. An [[EM.Cube ]] FDTD simulation is terminated when the ratio U_n/U_max falls below the specified power threshold or when the maximum number of time steps is reached. You can, however, terminate the FDTD engine earlier by clicking the '''Abort Simulation''' button.

The accuracy of the FDTD simulation results depends on the right choice of temporal waveform. The excitation waveform is the temporal source function that sets the initial conditions at t = 0 and provides an excitation signal afterwards. [[EM.Cube ]] currently offers three choices:

[[EM.Cube]]'s default waveform choice is a modulated Gaussian pulse. At the end of an FDTD simulation, the time domain field data are transformed into the frequency domain at your specified frequency. This is usually the same as the center frequency of your project. Different waveforms have different spectral behaviors. It is critical that you understand the time-domain-to-frequency-domain transformations in order to correctly interpret some simulation data.

From the "'''Excitation Waveform'''" section of the '''FDTD Simulation Engine Settings '''dialog, you can override [[EM.Cube]]'s default waveform and customize your own. The sinusoidal waveform has a frequency f₀ equal to the project's center frequency and does not have any parameter that you can modify. The delay in this case is set to t₀ = 1/(4f₀) so that the waveform vanishes at t = 0:

For Gaussian and modulated Gaussian waveforms you can set the pulse width and time delay. The pulse width should be chosen such that the Fourier transform of your selected waveform encompasses your project bandwidth. Rather than setting an arbitrary pulse width, [[EM.Cube ]] lets you set a parameter called '''Spectral Truncation Level '''or '''delta (&delta;)''', which is the normalized truncation threshold for the spectral content of the excitation pulse. This is explained as follows. Recall that the Fourier transform of a Gaussian pulse is also a Gaussian pulse. The Fourier transform of a modulated Gaussian pulse consists of two Gaussian pulses located at the two opposite sides of f = 0 (DC) shifted by the modulation frequency f₀:

Keep in mind that FDTD is a time domain algorithm. At the end of an FDTD simulation, a Discrete Fourier Transform (DFT) is performed on the time domain data to calculate frequency domain characteristics such as near fields, far field radiation patterns, RCS, S/Y/Z [[parameters]], etc.:

Of [[FDTD Module]]'s observables, the near fields, far fields and all of their associated [[parameters ]] like directivity, RCS, etc., are calculated at a certain frequency that is specified as part of the definition of the observable. On the other hand, port characteristics like S/Y/Z [[parameters]], VSWR and periodic characteristics like reflection and transmission coefficients, are calculated over the entire specified bandwidth of your project. In other words, you get a wideband frequency response at the end of a single time domain FDTD simulation run. The number of frequency point data over this bandwidth is equal to the number of DFT samples that are generated during the time marching loop. This number is set to 200 by default. In other words, 200 frequency data are generated at the end of an FDTD simulation. You can change this number through the box labeled '''No. DFT Samples''' in the "Discrete Fourier Transform" section of the FDTD Simulation Engine Settings dialog.

It can be seen that the discrete Fourier transform multiplies the samples of time-domain field quantities by the time step &Delta;t. This means that the resulting Fourier transforms of electric and magnetic field components now have units of V/m/Hz or A/m/Hz, respectively. Moreover, the Fourier transforms of the three waveform types have different spectral values at the observation frequency f₀. This makes it difficult to compare the FDTD simulation results with the results from [[EM.Cube]]'s other computational modules. For example, in the Planar, MoM3D and Physical Optics Modules, a plane wave source typically has a complex-valued functional form of exp(-jk₀'''k.r'''), which has a unit magnitude. In [[FDTD Module]], the time domain plane wave source has a functional dependence of the following form:

By computing the time domain fields at a certain location, you can examine the transient response of a system at that location. This is also very useful for monitoring the convergence of FDTD time marching loop. [[EM.Cube]]'s field probes allow you to save the temporal values of a field component at a specified point in the computational domain during the entire time marching loop. You can plot the time domain field components as a function of the time step index. You can also plot the spectral contents of those field components, i.e. their Fourier transform, over the project's specified frequency bandwidth. To define a new field probe, follow these steps:

* By default [[EM.Cube ]] creates a field probe located at the origin of coordinates (0,0,0). You can move the probe to any location by changing its X, Y and Z coordinates.

At the end of an FDTD simulation, the electric and magnetic field components along the specified probe direction are saved at the probe's location. Both the time domain fields from t = 0 to the last time step and their frequency domain spectrum are recorded. You can plot these data in EM.Grid, which can be accessed from [[EM.Cube]]'s Data Manager. To open data manager, click the '''Data Manager''' [[Image: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'''. In the Data manager Dialog, you see a list of all the data files available for plotting. These include the time-domain and frequency-domain probe data files with '''.DAT''' and '''.CPX''' file extensions, respectively. Select any data file by clicking and highlighting its row in the table and then click the '''Plot''' button to plot the graph. The time-domain field probe is plotted on a Cartesian graph showing the selected field component as a function of time step. The frequency-domain probe contains two Cartesian graphs: amplitude and phase of the selected field component over the project's frequency range.

In [[EM.Cube ]] you can visualize the near fields at a specific frequency in a specific plane of the computational domain. At the end of an FDTD simulation, all the time domain electric and magnetic field values are available at all mesh nodes. These temporal quantities are transformed into the frequency domain using discrete Fourier transforms to calculate the electric and magnetic fields on a specified sensor plane. To define a new Field Sensor, follow these steps:

* By default [[EM.Cube ]] creates a field sensor plane passing through the origin of coordinates (0,0,0) and coinciding with the XY plane. Note that the sensor plane extends across the entire computational domain. You can change the location of the sensor plane to any point by typing in new values for the X, Y and Z coordinates. Keep in mind that you can move a sensor plane only along the specified direction of the sensor. Therefore, only one coordinate can effectively be changed. As you increment or decrement this coordinate, you can observe the sensor plane moving along that direction in the project workspace.

You can plot frequency domain fields in EM.Grid on 2D Cartesian graphs. Using field probes, you can plot any frequency domain field component as a function of frequency over the specified bandwidth at any point within the computational domain. Using field sensors, you can plot the total frequency domain fields as a function of position (spatial coordinates) across the computational domain. Every field sensor has a crosshair made up of two perpendicular lines parallel to the boundaries of the sensor plane. When you define a field sensor for the first time, the crosshair passes through the origin of coordinates. You can change the location of the crosshair on the sensor plane using the other two coordinate boxes besides the one that moves the location of the sensor plane. At the end of an FDTD simulation, in addition to the 3D near field maps, [[EM.Cube ]] also generates 2D Cartesian graphs of the total electric and magnetic fields along the two perpendicular crosshair lines. A total of four Cartesian data files are generated, two for total E-field and two for total H-field along the two lines. You can plot these data in EM.Grid, which can be accessed from [[EM.Cube]]'s Data Manager. To open data manager, click the '''Data Manager''' [[Image: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'''. In the Data Manager dialog, you see a list of all the data files available for plotting including the frequency-domain sensor data files with a '''.DAT''' file extension. Select any data file by clicking and highlighting its row in the table and then click the '''Plot''' button to plot the graph. Frequency domain field sensor graphs show the total field as a function of cell index along one of the principal axes. If the FDTD mesh is uniform in that direction, the position is found by multiplying the cell index by the cell dimension and offsetting with respect to lower-front-left corner of the computational domain.

In the course of the FDTD time marching process, a tremendous amount of data are generated that include all the six E/H field components at every Yee cell and at every time step. The temporal field values at a sensor plane are of particular interest. Such plots show the evolution of the fields as a function of time starting from time t = 0, when all the fields are zero everywhere in the computational domain. [[EM.Cube ]] can record snapshots of the field sensor data as the time loop marches forward. When you define a field sensor for the first time, by default it displays the frequency domain near field data. In order to record and save the time domain data, you have to open the field sensor's property dialog by right clicking on the field sensor's name in the Navigation Tree and selecting '''Properties...'''from the contextual menu. In the section titled '''Sensor Domain''', select the radio button labeled '''Time Domain'''. Also, in the section titled "Field Display - Multiple Plots", select one of the two radio buttons labeled '''E-Field''' or '''H-Field'''. By default, the time domain field data are saved every 100 time steps. To change this setting, right click on the '''Field Sensors''' item in the Navigation Tree and select '''Time Domain Settings...''' from the contextual menu. In the Time Domain Settings Dialog, change the value of the box labeled '''Sampling Interval (in time steps)'''.

Time domain [[animation ]] is available only for FDTD simulations of "Analysis" type. It cannot be used in conjunction with sweep simulations. Once the FDTD Analysis is finished, you can click any of the field plots and visualize it in the main window or you can animate them by right clicking on the field sensor's name in the Navigation Tree and selecting '''[[Animation]]''' from the contextual menu. You can change the [[animation ]] settings from the '''[[Animation ]] Controls Dialog'''. Note that the [[animation ]] loop repeats itself indefinitely until you close the [[Animation ]] Controls dialog or hit the keyboard’s '''Esc Key'''.

Figure 1: (Left) Changing a field sensor's domain to generate time-domain field data for [[animation]], and (Right) [[FDTD Module]]'s Time Domain Settings dialog.

Figure 2: [[Animation ]] of E-field plots in time domain.

If your physical structure is excited by a Lumped Source or a Waveguide Source or a Distributed Source, and one or more ports have been defined, the FDTD engine calculates the scattering (S) [[parameters]], impedance (Z) [[parameters ]] and admittance (Y) [[parameters ]] of the selected ports. The S [[parameters ]] are calculated based on the port impedances specified in the project's "Port Definition". If more than one port has been defined in the project, the FDTD engine runs an internal port sweep. Each port is excited separately with all the other ports turned off. When the ''j''th port is excited, all the S_ij [[parameters ]] are calculated together based on the following definition:

where V_i is the voltage across Port i, I_i is the current flowing into Port i and Z_i is the characteristic impedance of Port i. The sweep loop then moves to the next port until all ports have been excited. After the FDTD simulation is finished, the S [[parameters ]] are written into output ASCII data files. Since these data are complex, they are stored as '''.CPX''' files. Every file begins with a header starting with "#". Besides the scattering [[parameters]], the admittance (Y) and impedance (Z) [[parameters ]] are also calculated and saved in complex data files with '''.CPX''' file extensions. The following relationships are used:

You can plot the port characteristics from the Navigation Tree. Right click on the '''Port Definition''' item in the '''Observables''' section of the Navigation Tree and select one of the items: '''Plot S [[Parameters]]''', '''Plot Y [[Parameters]]''', '''Plot Z [[Parameters]]''', or '''Plot VSWR'''. In the first three cases, another sub-menu gives a list of individual port [[parameters]]. Keep in mind that in multi-port structures, each individual port parameter has its own graph. You can also see a list of all the port characteristics data files in [[EM.Cube]]'s data manager. To open data manager, click the '''Data Manager''' [[Image: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 clicking and highlighting its row in the table and then click the '''Plot''' button to plot the graph in '''EM.Grid'''. By default, the S [[parameters ]] are plotted as dual magnitude-phase graphs, while the Y and Z [[parameters ]] are plotted as dual real-imaginary part graphs. The VSWR data are plotted on a Cartesian graph. You change the format of complex data plots. In general complex data can be plotted in three forms:

In particular, it may be useful to plot the S_ii [[parameters ]] on a Smith chart. To change the format of a data plot, select it and click the '''Edit '''button of Data Manager and select one of the available graph type options.<br />

Figure 1: [[EM.Cube]]'s data manager showing a list of complex data files available for plotting in EM.Grid.

For radiating structures or scatterers, the far field quantities are of primary interest. [[EM.Cube]]'s [[FDTD Module]] can calculate the far field radiation patterns of an antenna or the radar cross section (RCS) of a target. In general, by far fields we mean the electric fields evaluated in the far zone of a physical structure, which satisfies the following condition:

where r is the distance between the observation and source points, &lambda;₀ is the free space wavelength and D is the largest dimension of the radiating structure. In [[EM.Cube]], the far-zone electric fields '''E^ff'''(&theta;, &phi;) are functions of the spherical observation angles only and are defined as

In the FDTD method, the far fields are calculated using a near-field-to-far-field transformation of the field quantities on a given closed surface. [[EM.Cube ]] uses rectangular boxes to define these closed surfaces. You can use [[EM.Cube]]'s default radiation box or define your own. The far-zone electric field can be written as:

For any far field calculations in [[EM.Cube]], first you have to define a far field observable in the Navigation Tree. In [[FDTD Module]], defining a far field observable also initiates a far field box in the computational domain. This box is used to perform the near-to-far-field transformation at the end of an FDTD simulation. To insert a new far field box, follow these steps:

* At the end of an FDTD simulation, besides calculating the radiation data over the entire (spherical) 3D space, a number of 2D pattern graphs are also generated. These are indeed pattern cuts at certain planes, which include the three principal XY, YZ and ZX planes plus one additional constant f-cut. This latter cut is at &phi; = 45° by default. You can assign another phi angle in degrees in the box labeled '''Non-Principal Phi Plane'''. Also, the 2D radiation pattern graphs are normalized by default. You can instruct [[EM.Cube ]] to plot the 2D pattern graphs un-normalized (as calculated) by removing the check mark from the box labeled '''Normalize 2D Patterns'''.

At the end of an FDTD sweep simulation, other radiation characteristics are also computed as a function of the sweep variable (frequency, angle, or any other user defined variable). These include the '''Directivity (D0)''', '''Total Radiated Power (PRAD)''' and '''Directive Gain (DG)''' as a function of the &theta; and &phi; angles. Another radiation characteristic of interest especially in circularly polarized scenarios is the Axial Ratio. In [[EM.Cube]], the axial ratio is always defined in the LCP_z or RCP_z sense based on the X- and Y-components of the electric field. In order to calculate the directive gain or axial ratio, you have to check the boxes labeled '''Axial Ratio (AR)''' or '''Directive Gain (DG)''' in the "Additional Radiation Characteristics" section of the '''Radiation Pattern Dialog'''. Four 2D Cartesian graphs of the axial ratio as functions of the theta angle are generated in the three principal XY, YZ and ZX planes as well as the additional user defined phi plane cut. At the end of an FDTD sweep simulation, the directive gain and axial ratio can also be plotted as functions of the sweep variable. In that case, either quantity needs to be computed at a fixed pair of &theta; and &phi; angles. These angles are specified in degrees as '''User Defined Azimuth & Elevation''' in the "Output Settings" section of the '''Radiation Pattern Dialog'''. The default values of the user defined azimuth and elevation are both zero corresponding to the zenith.

Figure 1: [[EM.Cube]]'s Data Manager dialog showing a list of 2D polar and Cartesian radiation pattern graphs.

In other words, [[EM.Cube]]'s [[FDTD Module]] can calculate the far field radiation pattern of a structure in the presence of any of the above four background structure types. In the case of an infinite PEC or PMC ground plane at the bottom of the domain with a zero -Z offset, it is assumed that the PEC or PMC planes extend laterally to the infinity even though your structure has a finite domain. In this case, [[EM.Cube ]] automatically uses the correct dyadic Green's function and calculates the radiation pattern of a half-space structure. The far fields for observation in the lower half-space (0 &le; &theta; &le; 90°) are set equal to zero. In the fourth case, i.e. an infinite half-space ground (or a dielectric medium in general), your computational domain must end in a dielectric layer with a CPML -Z boundary. The -Z domain offset must be set equal to zero. You also have to make sure that the lateral domain offset values along the ±X and ±Y directions are set equal to zero, too. In addition, you have to specify the permittivity &epsilon;_r and electric conductivity &sigma; of the terminating medium. You can set these [[parameters ]] from [[FDTD Module]]'s Far Field Background Medium dialog. To access this dialog, open the radiation pattern dialog and click the button labeled '''Background...''' From this dialog, you can also set the Z-coordinate of the top of the terminating half-space medium. When the bottom of your computational domain is terminated by an infinite PEC or PMC plane, the Z-coordinates of the ground plane and the bottom face of the computational domain are identical. However, when your domain is terminated in a dielectric medium, you will typically have a dielectric layer beneath your physical structure. In that case, you may want to set the Z-coordinate of the top of that dielectric layer as the position of the interface between the free space and the lower dielectric half-space.

Note that the current version of [[EM.Cube]]'s [[FDTD Module]] does not calculate the far-field Green's function of a laterally infinite, conductor-backed, dielectric substrate with a finite layer thickness. For problems of this type, you should use [[EM.Cube]]'s [[Planar Module]]. In FDTD, either your PEC/PMC ground can be infinite or you can assume a dielectric half-space ground. Also, note that when infinite lateral dimensions are not required, like in the case of patch antennas with a finite substrate and finite ground, FDTD is the method of choice, as the planar MoM method cannot handle such cases.

When the physical structure is illuminated by a plane wave source, the calculated far field data indeed represent the scattered fields. In that case, the incident and scattered fields can be separated. [[EM.Cube ]] can calculate the radar cross section (RCS) of a target defined as:

At the end of an FDTD simulation, in the far field section of the Navigation Tree, you will have the &theta; and &phi; components of RCS as well as the total radar cross section: &sigma;_&theta;, &sigma;_&phi;, and &sigma;_tot. You can view a 3D visualization of these quantities by clicking on their entries in the Navigation Tree. The RCS values (&sigma;) are expressed in m². The 3D plots are normalized to the maximum RCS value, which is displayed in the legend box. The 2D RCS graphs can be plotted in '''EM.Grid '''exactly in the same way that you plot 2D radiation pattern graphs. A total of eight 2D RCS graphs are available: 4 polar and 4 Cartesian graphs for the XY, YZ, ZX and user defined plane cuts. at the end of a sweep simulation, [[EM.Cube ]] calculates some other quantities including the backscatter RCS (BRCS), forward-scatter RCS (FRCS) and the maximum RCS (MRCS) as functions of the sweep variable (frequency, angle, or any user defined variable). In this case, the RCS needs to be computed at a fixed pair of &phi; and &theta; angles. These angles are specified in degrees as '''User Defined Azimuth & Elevation''' in the "Output Settings" section of the '''Radar Cross Section Dialog'''. The default values of the user defined azimuth and elevation are both zero corresponding to the zenith.

{{Note|Unlike [[EM.Cube]]'s Planar, MoM3D and Physical Optics Modules, the [[FDTD Module]] currently does not support 3D mono-static RCS calculation due to the enormous amount of computational work needed. Only the bi-static RCS is calculated for a given plane wave source.}}

At the end of an FDTD simulation, an discrete Fourier transform (DFT) is performed on the time domain data, and frequency domain data like S/Y/Z [[parameters]], near field maps and radiation patterns are calculated. If you choose the right temporal waveform for the excitation source, the frequency domain results will have a good degree of accuracy over the specified project bandwidth. Alternatively, you can run a frequency sweep with the FDTD simulation engine, in which the engine is run as many times as the specified number of frequency samples. The center frequency of the FDTD excitation waveform is set equal to each frequency sample at each sweep run, and the frequency domain [[parameters ]] are calculated at each frequency sample. This is particularly useful if you want to visualize or animate near field sensors or 3D radiation patterns at different frequencies. To run a frequency sweep, open the FDTD '''Run Dialog''' and from the '''Simulation Mode '''dropdown list select the '''Frequency Sweep''' option. Click the '''Settings''' button next to this dropdown list to open up the Frequency Settings Dialog. Here you can set the values of the '''Start Frequency''' and '''End Frequency''' as well as the '''Number of Samples'''. The start and end frequencies are initially set to default values calculated based on the project's center frequency and bandwidth.<br />

At the end of an FDTD simulation, a number of computed quantities are designated as "Standard Output" [[parameters ]] and can be used for various post-processing data operations. For example, you can define design objectives based on them, which you need for [[optimization]]. The table below gives a list of all the currently available standard output [[parameters ]] in [[EM.Cube]]'s [[FDTD Module]]:

In the table above, SijM, etc. means the scattering parameter observed at port i due to a source excited at port j. Similar definitions apply to all the S, Z and Y [[parameters]]. If your planar structure has N ports, there will be a total of N² scattering [[parameters]], a total of N² impedance [[parameters]], and a total of N² admittance [[parameters]]. Additionally, there are four standard output [[parameters ]] associated with each of the individual S/Z/Y [[parameters]]: magnitude, phase (in radians), real part and imaginary part. The same is true for the reflection and transmission coefficients of a periodic planar structure excited by a plane wave source. Each coefficient has four associated standard output [[parameters]]. These [[parameters]], of course, are available only if your planar structure has a periodic domain and is also excited by a plane wave source incident at the specified &theta; and &phi; angles.

All the radiation- and scattering-related standard outputs are available only if you have defined a radiation pattern far field observable or an RCS far field observable, respectively. The standard output [[parameters ]] DGU and ARU are the directive gain and axial ratio calculated at the certain user defined direction with spherical observation angles (&theta;, &phi;). These angles are specified in degrees as '''User Defined Azimuth & Elevation''' in the "Output Settings" section of the '''Radiation Pattern Dialog'''. The standard output [[parameters ]] HPBWU, SLLU, FNBU and FNLU are determined at a user defined f-plane cut. This azimuth angle is specified in degrees as '''Non-Principal Phi Plane''' in the "Output Settings" section of the '''Radiation Pattern Dialog''', and its default value is 45°. The standard output [[parameters ]] BRCS and MRCS are the total back-scatter RCS and the maximum total RCS of your planar structure when it is excited by an incident plane wave source at the specified &theta;_s and &phi;_s source angles. FRCS, on the other hand, is the total forward-scatter RCS measured at the predetermined &theta;_o and &phi;_o observation angles. These angles are specified in degrees as '''User Defined Azimuth & Elevation''' in the "Output Settings" section of the '''Radar Cross Section Dialog'''. The default values of the user defined azimuth and elevation are both zero corresponding to the zenith.

If you are interested in calculating certain quantities at the end of a simulation, which you do not find among [[EM.Cube]]'s standard output data, you can define your own custom output. [[EM.Cube ]] allows you to define new custom output as any mathematical expression that involves the available standard output [[parameters]], numbers, [[variables]] and all of [[EM.Cube]]'s mathematical functions. For a list of legitimate mathematical functions, click the '''Functions [[Image:functions_icon.png]] '''button of the '''Simulate ''''''Toolbar''' or select '''Simulate > Functions... '''from the menu bar, or use the keyboard shortcut '''Ctrl+I''' to open the Function Dialog. Here you can see a list of all the available [[EM.Cube ]] functions with their syntax and a brief description. To define a custom output, click the '''Custom Output [[Image:custom_icon.png]] '''button of the '''Simulate ''''''Toolbar''' or select '''Simulate > Custom Output... '''from the menu bar, or use the keyboard shortcut '''Ctrl+K''' to open the Custom Output Dialog. This dialog has a list of all of your custom output [[parameters]]. Initially, the list empty. You can define a new custom output by clicking the '''Add '''button of the dialog to open up the '''Add Custom Output Dialog'''. In this dialog, first you have to choose a new label for your new parameter and then define a mathematical expression for it. At the bottom of the dialog you can see a list of all the available standard output [[parameters]], whose number and variety depends on your project's source type as well as the defined project observables. When you close the Add Custom Output dialog, it returns you to the Custom Output dialog, where the parameter list now reflects your newly defined custom output. You can edit an existing parameter by selecting its row in the table and clicking the '''Edit '''button, or you can delete any parameter from the list using the '''Delete '''button.

Figure 1: [[EM.Cube]]'s Custom Output dialog.

Figure 2: Defining a new custom output using the available standard output [[parameters]].

Parametric sweep is [[EM.Cube]]'s most versatile sweep type. During a parametric sweep, the values of one or more sweep [[variables]] are varied over their specified ranges, and the FDTD simulation is run for each combination of variable samples. If you define two or more sweep [[variables]], the process will then involve nested sweep loops that follow the order of definition of the sweep [[variables]]. The topmost sweep variable in the list will form the outermost nested loop, and the sweep variable at the bottom of the list will form the innermost nested loop. Note that you can alternatively run either a frequency sweep or an angular sweep as parametric sweeps, whereby the project frequency or the angles of incidence of a plane wave source are designated as sweep [[variables]]. Unlike [[optimization]] which will be discussed next, parametric sweeps are simple and straightforward and do not require careful advance planning.

Before you can run a parametric sweep, first you have to define one or more [[variables]] in your [[EM.Cube ]] project. A variable is a mathematical entity that has a numeric value. This numeric value can be changed at your discretion at any time. You can define a variable either directly as a number or as a mathematical expression that may involve other previously defined [[variables]]. Even in the latter case, an "expression" variable has a numeric value at any time. You can designate almost any numeric quantity or parameter in [[EM.Cube ]] as a variable. Or alternatively, you can associate a variable with almost anything in [[EM.Cube]]. This includes all the geometrical properties of CAD objects like coordinates, rotation angles, dimensions, etc. as well as material properties of object groups and background structure, source [[parameters]], project frequency, mesh density, and unit cell periods in the case of a periodic structure. You can define a variable either in a formal manner using [[EM.Cube]]'s Variable Dialog or directly from the project workspace or from the Navigation Tree. In the former "formal" manner, first you open the [[Variables]] Dialog by clicking the '''[[Variables]]''' [[Image:variable_icon_tn.png]] button of the '''Simulate Toolbar''' or selecting '''Menu > Simulate > [[Variables]]... '''or using the keyboard shortcut '''Ctrl+B'''. By default, the variable list is initially empty. To add a new variable, click the '''Add''' button to open the "Add Variable Dialog". Choose a '''Name '''for your new variable. In the box labeled '''Definition''', define your new variable either as an independent variable with a numeric value or as a dependent variable using a mathematical expression that involves previously defined [[variables]].

Figure 1: [[EM.Cube]]'s [[Variables]] dialog.

Once you finish the definition of a new variable, its name and syntax (Definition) are added to the "Variable List". You can also see the '''Current Value''' of every variable at any time in the [[Variables]] dialog. Note that at this stage, you have simply defined one or more [[variables]], but you have not yet associated them with actual objects or project properties. Wherever you see a numeric value for a parameter in a dialog, e.g. the length of a rect strip object in its property dialog, you can replace the numeric value with a variable name or a mathematical expression using the names of the currently available [[variables]]. From this moment on, that parameter or quantity becomes tied up with the associated variable. This means that every time you change the value of that variable, the value of the associated object parameter or project property will change accordingly. You can change the value of a variable directly from the [[Variables]] Dialog using the '''Edit '''button or indirectly during a parametric sweep. In the former case, you have to click the '''Update '''button of the [[Variables]] dialog to make the changes effective. Alternatively, you can define new [[variables]] directly from the property dialogs of CAD objects, material group dialogs, source dialogs or many other [[EM.Cube ]] dialogs. If you replace the numeric value of a parameter with a text-string name that has not already been defined as a variable, then a new variable by that name is created and added to the "Variable List". The numeric value of the associated parameter at the time of replacement is taken as the "current Value" of the newly created variable. In this way, you can easily and quickly define [[variables]] associated with the design [[parameters ]] that you intend to sweep in your project.

In a parametric sweep, you can vary the values of one or more "Independent" project [[variables]]. In other words, you designate one or more independent [[variables]] as sweep [[variables]] and specify how they should vary (be sampled) during the sweep simulation process. In [[EM.Cube ]] you can define three types of sweep [[variables]]: '''Uniform''', '''Discrete '''and '''Random'''. Each sweep [[variables]] can be defined as one of these three types, and you can mix sweep [[variables]] of different types in a multivariable parametric sweep. A uniform sweep variable is defined by a "Start" and "End" value and is incremented by a predetermined "Step" value during a sweep. A discrete sweep variable is defined by a discrete set of values and takes on these values by the order of their list during a sweep. A random sweep variable, on the other hand, takes on random values during a sweep according to a specified probability distribution. [[EM.Cube ]] currently offers two random distribution types. '''Uniform Distribution '''is defined by "Minimum" and "Maximum" values, while '''Normal (Gaussian) Distribution '''is defined by a "Mean" and "Standard Deviation". You need to specify the "Number of Samples" for both random variable types.

To define sweep [[variables]] and run a parametric sweep, open the '''Simulation ''''''Run Dialog''', and select the '''Parametric Sweep''' option from the dropdown list labeled '''Simulation Mode'''. When you choose the parametric sweep option, the '''Settings''' button next to the simulation mode dropdown list becomes enabled. Clicking this button opens the '''Parametric Sweep Settings''' dialog. The '''Sweep [[Variables]] List '''is initially empty. On the left side of the dialog you see the "'''Independent [[Variables]] Table'''", which lists all the available independent [[variables]] of your project. Select an independent variable from the table and use the right arrow ('''-->''') button of the dialog to move it to the Sweep [[Variables]] List. Before moving the variable to the new location, the "Define Sweep Variable Dialog" opens up, where you have to define the attributes of your new sweep variable. In this dialog, you need to choose the type of the sweep variable using the three radio buttons labeled '''Uniform''', '''Discrete '''and '''Random'''. Depending on your choice, the proper section of the dialog becomes enabled, where you can define the range of your sweep variable and other relevant [[parameters]].

The Parametric Sweep Settings dialog also features another useful button labeled '''Dry Run''', which runs a "fake" sweep [[animation]]. During a dry run, the sweep [[variables]] are varied sample by sample and all of their associated [[parameters ]] in the project workspace are updated at each run. However, the simulation engine is not called during a dry run, and no numerical computations take place at all. Also, the Variable Dry Run dialog appears on the screen which shows the changing values of all the [[variables]] at all times. This dialog works in a similar way as the [[Animation ]] Controls Dialog described earlier. You can change the speed of the updates or control them manually using the motion buttons. The dry run process continues forever until to stop it by clicking the close (X) button of the Dry Run dialog of simply hitting the keyboard's '''Esc Key'''. At ach update of a dry run, you can see how the CAD objects in your planar structure change. This is very useful to inspect the integrity of your structure and your defined [[variables]] before an actual simulation run. Once you are satisfied with the choice and definition of your sweep [[variables]], close the Parametric Sweep Settings dialog to return to the Simulation Run dialog, where you can start the planar MoM parametric sweep simulation by clicking the '''Run '''button.

Figure 3: [[EM.Cube]]'s Parametric Sweep Settings dialog.

Figure 5: [[EM.Cube]]'s Variable Dry Run dialog.

At the end of a sweep simulation (a frequency sweep, angular sweep, parametric sweep, etc.), if you have defined a field sensor or a far field observable (either radiation pattern or RCS) in your project, a number of near field plots or far field plots are generated for all samples of the sweep variable. Each field sensor normally has 14 field maps for the amplitude and phase of all the three components of E- and H-fields plus the magnitude of the total fields. Each far field observable has three spherical plots corresponding to the &theta; and &phi; components of the far field and the total far field. Generating all these 14 near-field intensity plots or 3 far-field spherical plots at each sweep sample would result in a very large number of graphs on the Navigation Tree. Instead, during a sweep simulation, [[EM.Cube ]] generates either a total E-field or a total H-field plot at each sweep run for the field sensors. By default, the total E-field plots are saved. You can change this setting from the field sensor's property dialog. Right click on the field sensor's name in the Navigation Tree and select '''Properties...'''from the contextual menu. In the section titled "Field Display - Multiple Plots", select one of the two radio buttons labeled '''E-Field''' or '''H-Field'''. As for far field observables, only the total field plot is generated at each sweep sample.

Once the sweep simulation is finished, you can click any of the near-field or far-field plots and visualize it in the main window. You can also animate these field plots. [[Animation ]] in [[EM.Cube ]] consists of sequential display of the plots in the main window at a preset speed. To animate the field sensor plots, right click on the field sensor's name in the Navigation Tree and select '''[[Animation]]''' from the contextual menu. The field plots start to animate beginning with the first sample, going through all the plots one by one until the last one and repeating the loop all over again. While the [[animation ]] proceeds in the main window, a dialog titled '''[[Animation ]] Controls Dialog''' pops up at the lower right corner of the screen. You can drag this dialog anywhere in the project workspace from its title bar. The controls dialog shows the title of each graph as it is reviewed. You can set the speed of [[animation ]] by typing in a value for '''Rate''', which is indeed the frame duration expressed in multiples of 100 milliseconds. The default frame duration is 300 msec. You can pause the [[animation ]] and resume at any time. You can rewind to the first sample or skip to the last sample. You can also step through the samples one at a time using the increment (forward) or decrement (backward) buttons. To stop [[animation ]] at any time, use the keyboard's '''Esc Key''' or click the '''Close (X)''' button of the [[animation ]] controls dialog.

Figure 2: [[EM.Cube]]'s [[Animation ]] Controls dialog.

Figure 3: [[Animation ]] of total E-field plots and far field radiation patterns at the end of an FDTD sweep simulation.

An [[optimization]] process in [[EM.Cube ]] involves several steps as follows:

# Define project (design) objectives using a combination of standard and custom output [[parameters]].

# Choose the [[optimization]] algorithm type and set its relevant [[parameters]].

# Examine the optimal values of the participating [[variables]] and the updated values of their associated [[parameters]].

To define an objective, open the '''Objectives Dialog''' either by clicking the '''Objectives''' [[Image:objective_icon.png]] button of the '''Simulate Toolbar''', or by selecting '''Menu > Simulate > Objectives...''' from the Menu Bar, or using the keyboard shortcut '''Ctrl+J'''. The objectives list is initially empty. To add a new objective, click the '''Add''' button to open up the '''Add Objective Dialog'''. At the bottom of this dialog, you can see a list of all the available [[EM.Cube ]] output [[parameters ]] including both standard and custom output [[parameters]]. This list may vary depending on the types of sources and observables that you have already defined in your project. You can enter any mathematical expressions in the two boxes labeled '''Expression 1''' and '''Expression 2'''. The Available Output Parameter List simply helps you remember the syntax of these [[parameters]]. You should also select one of the available options in the dropdown list labeled '''Logical Operator'''. The default operator is '''"== (Equal To)"'''. As soon as you finish the definition of an objective, its full logical expression is added to the Objective List. You can always modify the project objectives after they have been created. Select a row in the Objective List and click the '''Edit '''button of the dialog and change the expressions or the logical operator. You can also remove an objective from the list using the '''Delete '''button.

Figure 1: [[EM.Cube]]'s Objectives dialog.

Figure 2: Defining a new objective using a list of available output [[parameters]].

To define [[optimization]] [[variables]] and perform an [[optimization]], open the '''Simulation ''''''Run Dialog''', and select the '''[[Optimization]] '''option from the dropdown list labeled '''Simulation Mode'''. When you choose the [[optimization]] option, the '''Settings''' button next to the simulation mode dropdown list becomes enabled. Clicking this button opens the '''[[Optimization]] ''''''Settings''' dialog. This is a large dialog with several distinct sections. In the section titled "'''[[Optimization]] Algorithm'''", you can choose one of [[EM.Cube]]'s three currently available optimizers: '''Powell's Method''', '''Basic Genetic Algorithm '''and '''Fast Pareto Genetic Algorithm'''. For all three optimizers you have to set the '''Maximum Number of Iterations''', which has a default value of 5. For the two genetic algorithms, basic GA and Pareto, you also need to set the '''Population Size '''as a multiple of 5. The default population size is 50.

In the [[Variables]] section of the dialog, you designate the [[optimization]] [[variables]]. This is very similar to how you assign sweep [[variables]] in a parametric sweep as discussed earlier. The '''[[Optimization]] [[Variables]] List '''is initially empty. On the left side of the dialog you see the "'''Independent [[Variables]] Table'''", which lists all the available independent [[variables]] of your project. Select an independent variable from the table and use the right arrow ('''-->''') button of the dialog to move it to the [[Optimization]] Variable List. Before moving the variable to the new location, the "Define [[Optimization]] Variable Dialog" opens up, where you have to set the '''Minimum '''and '''Maximum '''values of your new [[optimization]] variable. Once you finish the definition of an [[optimization]] variable, its name and Min/Max values are added to the [[Optimization]] [[Variables]] List. Note that you can change your mind and remove an [[optimization]] variable from the list. To do so, select its name or row from the list and use the left arrow (<--) button to move it back to the Independent [[Variables]] Table. You can also change the Min/Max values of an [[optimization]] variable after it has been defined. Select the variable and click the '''Edit '''button of the dialog to change those value. If you check the box labeled "'''Update [[Variables]] with Optimal Values'''" (as it is always checked by default), [[EM.Cube ]] will automatically replace the definitions (and current values) of all the participating [[optimization]] [[variables]] in the '''[[Variables]] Dialog''' with their computed optimal values and will update all the associated [[parameters ]] in the project workspace. This, of course, will happen only if the [[optimization]] process successfully converges. Sometimes, the [[optimization]] process may get trapped in a local minimum. You may encounter this problem primarily when using the Powell method. You will notice that the values of the [[optimization]] [[variables]] soon get "saturated" and remain constant afterwards. [[EM.Cube ]] lets you exit such local traps if you check the box labeled "'''Exit Local Min Trap'''". There is a box underneath, labeled '''Tolerance''', that becomes enabled and has a default value of 0.01. This means that exit a local minimum trap when the value of the [[optimization]] variable stays within 1% error after successive [[optimization]] runs. In that case, the [[optimization]] process ends forcibly before having achieved convergence.

Finally, in the section titled "'''Error Function'''" of the [[Optimization]] Dialog you build the mathematical form of your objective function. Keep in mind that [[EM.Cube]]'s [[optimization]] is a numerical process. Therefore, you can rarely minimize your error function to zero literally. You need to set a "'''Maximum Error'''" value for the objective function, which terminates the process as "converged" when it is reached. The default value of Maximum Error is 0.01. In two tables labeled '''Goals '''and '''Constraints''', you see a list of all the project objectives that have been split between the two tables according to their types. Many [[optimization]] problems involve a single, straightforward goal. Others may involve multiple goals subject to multiple constraints. In those case, you have assign weights to your goals and constraints. If all of your goals and constraints have the same level of importance, then you assign equal weights to them. You can do this easily by clicking the '''Distribute '''button of the dialog. Or you may assign individual weights manually. However, you have to make sure that all the weight adds up to unity. The objective function of the [[optimization]] problem is constructed from the goals and constraints using the specified weights. This can be done in two different ways: as '''Linearly Weighted Goals '''or as '''Mean Square Weighted Goals.''' You set these options from the dropdown list labeled '''Weight Type'''. Once you are satisfied with the choice and definition of your [[optimization]] [[variables]], [[optimization]] algorithms and goals weights, close the [[Optimization]] dialog to return to the Simulation Run dialog, where you can start the planar MoM [[optimization]] process by clicking the '''Run '''button.

Figure 3: [[EM.Cube]]'s [[Optimization]] dialog.

Using [[EM.Cube]]'s [[FDTD Module]], you can simulate complex 3D periodic structures. A periodic structure is one that repeats itself infinitely along one, two or three directions. In this release of [[EM.Cube]]'s [[FDTD Module]], the periodicity is limited to the X-Y plane. In other words, the periodic structure repeats itself along the X- and Y-axes, but not along the Z-axis. By default, your physical structure is not periodic, and you have to instruct [[EM.Cube ]] to turn it into a periodic structure through [[FDTD Module]]'s 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.

In [[EM.Cube]]'s [[FDTD Module]], a periodic structure can be excited using various source types. Exciting the unit cell structure using a lumped source, a waveguide source, an ideal source or a distributed source, you can model an infinite periodic antenna array. For most practical antenna types, you will 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.Cube]]'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. At the end of the periodic FDTD simulation, the radiation pattern of the unit cell is calculated and stored in a radiation data file with a '''.RAD''' file extension. The 3D radiation patterns that you normally visualize in [[EM.Cube]], in this case, correspond to the single unit cell, not the infinite array. Therefore, they do not show the beam scanning even if you have entered nonzero values for the &theta; and/or &phi; scan angles. For this purpose, you have to define a finite-sized array factor. You do this in the "Impose Array Factor" section of the '''Radiation Pattern Dialog'''. In the case of a periodic structure, when you define a new far field item in the Navigation Tree, the values of element spacing along the X and Y directions are automatically set equal to the values of the periodic lattice spacing along those directions. Set the number of elements along the X and Y directions to any desired values. [[EM.Cube ]] will then compute the radiation pattern of the specified finite-sized periodic array, and the beam scanning will appear in the radiation pattern plots, if any.

Real antenna arrays have finite extents, that is, finite numbers of elements along the X and Y directions. Earlier, you saw how to excite an array of line objects using an array of lumped sources or an array of rectangular waveguides (hollow boxes) using an array of waveguide sources. Setting up array structures of this kind using [[EM.Cube]]'s '''Array Tool '''and exciting the individual elements using individual lumped or waveguide sources results in an accurate full-wave analysis of your antenna array. This type of simulation takes into account all the inter-element coupling effects as well as the finite edge and corner effects of the finite-sized array. At the end of the FDTD simulation of your antenna array, you can plot the radiation patterns and other far field characteristics of the array just like any other FDTD structure. However, depending on the total size of your array, a full-wave simulation like this may easily lead to a very large computational problem. As the number of elements grow very large, the array starts to look like an infinite periodic structure. In that case, it is possible to consider and analyze a periodic unit cell of the array structure and use an "Array Factor" representing the finite-extent topology of the array grid to calculate the radiation pattern of your antenna array. This approach works well for most large arrays. However, it ignores the finite edge and corner effects, which may be important for certain array architectures. In that case we recommend that you use [[EM.Cube]]'s [[Planar Module]]. Also, note that using an array factor for far field calculations, you cannot assign non-uniform amplitude or phase distributions to the array elements. For this purpose, you have to define an array object.

In the previous section, you saw how to excite a periodic unit cell using a lumped source or a waveguide source. You can specify the beam scan angles in the source dialogs. The finite array factor is defined in the radiation pattern dialog. At the end of the periodic FDTD simulation, you can visualize the 3D radiation patterns in the project workspace and plot the 2D Cartesian and polar pattern graphs in EM.Grid. [[EM.Cube ]] also calculates the '''Directive Gain (DG)''' as a function of the &theta; and &phi; angles. This is defined as:

The directivity D₀ is the maximum value of the directive gain. [[EM.Cube ]] generates four Cartesian graphs of directive gain in the three principal XY, YZ, ZX planes as well as in the user defined f-plane cut. The radiation patterns of antenna arrays usually have a main beam and several side lobes. Some [[parameters ]] of interest in such structures include the '''Half Power Beam Width (HPBW)''', '''Maximum Side Lobe Level (SLL)''' and '''First Null [[Parameters]]''' (i.e. first null level and first null beam width). You can have [[EM.Cube ]] calculate all such [[parameters ]] if you check the relevant boxes in the "Additional Radiation Characteristics" section of the '''Radiation Pattern Dialog'''. These quantities are saved into ASCII data files of similar names with '''.DAT''' file extensions. You can plot graphs of such data files at the end of a sweep simulation in''' '''EM.Grid. You can also plot the directive gain as a function of the sweep variable at the end of an FDTD sweep simulation. In that case, the directive gain is computed at a fixed pair of &theta; and &phi; angles. These angles are specified in degrees as '''User Defined Azimuth & Elevation''' in the "Output Settings" section of the Radiation Pattern dialog. The default values of the user defined azimuth and elevation are both zero corresponding to the zenith. The results are saved to an ASCII data file called "DGU.DAT". Note that DGU is also one of [[EM.Cube]]'s standard output [[parameters ]] and can be used to define custom output or design objectives.

Using a plane wave source to excite a periodic structure in [[EM.Cube]]'s [[FDTD Module]], you can model frequency selective surfaces, electromagnetic band-gap (EBG) structures, metamaterials, etc. Exciting periodic structures with plane wave sources requires careful attention. [[EM.Cube]]'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&deg; < &theta; < 180&deg;), 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&deg; < &theta; < 90&deg;), 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.Cube]]'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 '''TM_z''' and '''TE_z''' 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 &theta; > 0&deg;. [[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.

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''' [[Image: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.

{{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 &theta; > 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 &theta;₀ 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 &theta; 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".}}

Besides analyzing a periodic structure in a single-run simulation, [[EM.Cube]]'s [[FDTD Module]] offers a number of sweep simulations for periodic structures. These include '''Frequency Sweep''', '''Angular Sweep''', '''R/T Macromodel Sweep '''and '''Dispersion Sweep'''. These options are available from the '''Simulation Mode''' dropdown list of the [[FDTD Module]]'s '''Run Dialog'''. Of these, frequency sweep and angular sweep are similar to the non-periodic case as discussed earlier. Keep in mind that in this release of [[EM.Cube]]'s [[FDTD Module]], for oblique plane wave incidences, you need to run a frequency sweep to get wideband reflection/transmission coefficient data. Similarly, you need to run an angular sweep to plot R/T coefficients vs. the incident angle.

The '''R/T Macromodel Sweep''' option of the Simulation Mode dropdown list is only available for periodic structures. It is used to generate a lookup table model for the reflection and transmission coefficients of a periodic surface for both TM and TE polarizations. The results are written into a file named "PW_UserDefinedMacroData.mat". Through the Macromodel Settings dialog you can set the start and end value and number of samples for both the Theta (&theta;) and Phi (&phi;) angles of the incident plane wave. The R/T macormodels can be used by [[EM.Cube]]'s [[Propagation Module]] to calculate the reflection and transmission coefficients of incident rays at the facets of obstructing blocks with "non-standard" periodic surfaces.

The '''Dispersion Sweep '''option of the Simulation Mode dropdown list performs a sweep of constant k_l 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 k_li, you can find the reflection and transmission coefficients for all combinations of frequency f_j and incident angle &theta;_j such that (2&pi;/c) . f_j. sin &theta;_j = k_li. 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 k_l (or beta). The vertical axis represents frequency. Sometimes, the free space wave number k₀ = (2&pi;/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 f_max and k_l,max, respectively, where f_max = f₀ + &Delta;f/2, and &Delta;f 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 &phi; as specified in [[FDTD Module]]'s Plane Wave Dialog.