Changes

The simulated structure in FDTD usually consists of a number of objects that may have different material properties. EM.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.

Since FDTD is a finite domain numerical technique, the computational domain of the problem must be truncated. At the boundaries of the computational domain, proper boundary conditions must be enforced. In a shielded structure, all objects are enclosed within a perfect electric (or magnetic) conductor box. In an open boundary problem like an antenna, some kind of absorbing boundary conditions such as a perfectly matched layer (PML) must be used to emulate the free space.

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.

<!--[[Image:FDTD103.png]]-->

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:

Figure 1: [[FDTD Module]]'s Navigation Tree.

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

Figure 1: [[FDTD Module]]'s PEC and PMC material dialogs.

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 (&epsilon;), permeability (&mu;), electric conductivity (&sigma;) and magnetic conductivity (&sigma;_m):

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 ('''&epsilon;'''), permeability ('''&mu;'''), electrical conductivity ('''&sigma;''') and magnetic conductivity ('''&sigma;_m'''), are all tensorial in nature. Each constitutive parameter in this case is represented by a 3×3 matrix:

Figure 1: [[FDTD Module]]'s Anisotropic Material dialog.

[[#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:

Figure 1: [[FDTD Module]]'s Debye, Drude and Lorentz Material dialogs and their respective "Add Pole" dialogs.

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

Figure 1: Geometric construction of a dielectric-coated metallic cylinder.

If two or more material groups are defined in the Navigation Tree, only one of them can be active. By default, the last material defined is active. When you draw new objects, they are inserted under the active material node of the Navigation Tree, which is always listed in bold letters. Any material can be made active by right clicking on its name in the Navigation Tree and selecting the '''Activate''' item of the contextual menu.

=Computational Domain & FDTD Mesh Generation=

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.

Figure 1: The computational domain box enclosing a metallic sphere.

To set the solution domain of your [[FDTD Module]] project, follow these steps:

Figure 1: [[FDTD Module]]'s Domain Settings dialog.

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:

Figure 1: [[FDTD Module]]'s Boundary Conditions dialog.

In many electromagnetic modeling problems you need a boundary that simply absorbs all the incoming radiation. For problems of this nature, an absorbing boundary condition (ABC) is often chosen that effectively minimizes wave reflections at the boundary. Usually two or more ABC layers are placed at the boundaries of the structure to maximize wave absorption. The boundary ABC cells in the project workspace are transparent to the user. But, in effect, multiple rows of ABC cells are placed on the exterior side of each face of the visible domain box.<br />

Figure 2: [[FDTD Module]]'s CPML Settings dialog.

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.

Figure 1: Setting the ±X and ±Y and -Z domain offsets equal to zero for a laterally infinite planar structure with a PEC ground.

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

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. <font color="red"><u>'''When choosing a mesh type for your FDTD simulation, keep in mind that adaptive and regular mesh types are frequency-dependent and their density varies with the highest frequency of your specified bandwidth, while the uniform mesh type is always fixed and independent of your project's frequency settings. '''</u></font>

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 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.

Figure 2: The regular FDTD mesh of the human head model and the cellular phone handset.

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.

Figure 1:The grid cursor on the XY grid plane and its grid coordinates (I, J, K) displayed on the status bar.

Straight lines, boxes and rectangular plates whose edges are aligned with the three principal axes are the simplest objects to mesh in the [[FDTD Module]]. Such objects preserve their shapes exactly after discretization. All the objects with curved edges and curved surfaces or objects with straight edges and flat faces that are not parallel to the principal axes or principal planes need to be discretized using a staircase profile.

Figure 3. The geometry of a helix curve and its adaptive FDTD mesh.

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:

Figure 1: [[FDTD Module]]'s Mesh Settings dialog.

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.

Figure 2: Locking the mesh density of a material group in its property dialog.

Adding fixed grid points to an FDTD mesh increases its resolution locally. Each fixed grid point adds three grid lines along the three principal axes passing through that point. You can add as many fixed grid points as you desire and create dense meshes at certain regions. Fixed grid points appear as grey points in the project workspace. To insert a new fixed grid point, follow these steps:

Figure 2: User defined fixed grid points in an FDTD mesh. (Left) Regular mesh, (Right) Adaptive mesh.

<!-- ==Exciting The FDTD Computational Domain== -->

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.

An ideal source acts as a voltage source with a zero internal resistance that can be placed between any two adjacent mesh grid nodes anywhere in the computational domain. To create a new ideal source, follow these steps:

Figure 1: [[FDTD Module]]'s Ideal Source dialog.

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. <font color="red"><u>'''In order to create a lumped source, you must have at least one line object in the project workspace.'''</u></font>

Figure 1: [[FDTD Module]]’s Lumped Source dialog.

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.

Figure 1: [[FDTD Module]]'s Waveguide Source dialog.

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:

Figure 2: A distributed source placed between two horizontal rectangular strips.

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.

<font color="red"><u>'''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.'''</u></font> You can change the characteristic impedance of a port by selecting it from the Port List and clicking the '''Edit '''button of the dialog. This opens up the Edit Port dialog, where you can enter a new value in the box labeled '''Impedance'''.

Using simple lumped sources, you can simulate a variety of transmission line structures in EM.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.

Figure 1: A microstrip line port on a substrate terminated from the bottom by a PEC boundary plane.

Using lumped sources, you can define coplanar waveguide (CPW) ports either with or without a bottom ground plane. In order to build a CPW port with a lumped source, follow these steps:

Figure 1: A coplanar waveguide (CPW) port on a dielectric substrate.

Coaxial line is often used to feed various RF and microwave structures. In order to excite the dominant TEM mode of a coaxial line, it has to be fed symmetrically between its inner and outer conductors. Using lumped sources, you can define symmetrical coaxial line ports. To do so, follow these steps:

Figure 1: A coaxial line port using four symmetric 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.

Figure 1: [[FDTD Module]]'s Lumped Load dialog with the load type set to resistor, inductor, capacitor and diode.

If your project contains an array of line objects, you can also define an array of '''Lumped Sources''' to be placed on those lines. If you insert a new lumped source, all line array objects, if any, will be listed in the Lumped Source dialog as eligible objects for lumped source placement. A lumped source will be placed on each element of the array. All the lumped sources will have identical direction and offset. However, you can prescribe certain amplitude and/or phase distribution among the array elements. The available '''Weight Distributions''' include '''Uniform''', '''Binomial''', '''Chebyshev''' and '''Data File'''. In the last case, you need to set a value for maximum side lobe level ('''SLL''') in dB. You can also define a '''Phase Progression''' in degrees along each of the three principal axes.

If the project workspace contains an array of hollow box objects to model a rectangular waveguide array, you can also define an array of '''Waveguide Sources''' to be placed across those waveguides. If you insert a new waveguide source, all hollow box array objects, if any, will be listed as eligible objects for waveguide source placement. A waveguide source will be placed on each element of the array. All the waveguide sources will have identical direction and offset. However, you can prescribe certain amplitude and/or phase distributions. The available '''Weight Functions''' include '''Uniform''', '''Binomial''', '''Chebyshev''' and '''Data File'''. In the last case, you need to set a value for maximum side lobe level ('''SLL''') in dB. You can also define a '''Phase Progression''' in degrees along each of the three principal axes.

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:

A plane wave box placed around a PEC sphere object. The trident at the corner of the box shows the propagation vector as well as the E-field and H-field polarization vectors.

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>

A Gaussian beam box placed around a horizontal PEC plate. The trident at the corner of the box shows the propagation vector as well as the E-field and H-field polarization vectors. The titled transparent green circle shows the footprint of Gaussian beam at its focal (waist) point.

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. Keep in mind that you are always responsible for the choice of excitation source and the project observables. <font color="red"><u>'''In other words, EM.Cube does not automatically provide a default excitation source or does not suggest default observables.'''</u></font>

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.

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.

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 2: [[FDTD Module]]'s output window.

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:

[[Image:FDTD70.png]]

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:

Figure 2: The time domain and frequency domain field components plotted vs. time and frequency, respectively.

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:

Figure 3: Cartesian graph of total magnetic field vs. Y-index along the crosshair in the field senor plane.

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)'''.

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

<font color="red"><u>'''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.'''</u></font> 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:

Figure 2: Plot of S₂₁ of a filter 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:

Figure 1: [[FDTD Module]]'s Radiation Pattern dialog.

Once an FDTD simulation is finished, three far field items are added to the Far Field section of the Navigation Tree. These are the far-zone E-field component along &phi; direction, the far-zone E-field component along &phi; direction and the total far-zone E-field defined as:

Figure 1: The 3D total radiation pattern of a dipole antenna: (Left) polar type, and (Right) cone type.

At the end of an FDTD simulation, the radiation pattern data E_&theta;, E_&phi; and E_tot in the three principal XY, YZ and ZX planes plus one additional user defined phi plane cut are available for plotting on 2D graphs in '''EM.Grid'''. There are a total of eight 2D pattern graphs in the data manager: 4 polar graphs and 4 Cartesian graphs of the same pattern data. 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 will see a list of all the data files available for plotting. These include the four polar pattern data files with a '''.ANG''' file extension and the four Cartesian pattern data file with a '''.DAT''' file extension. Select any data file by highlighting its row in the table and then click the '''Plot''' button to plot the graph.

Figure 2: A 2D polar radiation pattern in the ZX plane cut.

As mentioned earlier when discussing boundary conditions and computational domain, you can use CPML boundary conditions with zero offsets to model a structure with infinite lateral extents. At the end of the FDTD simulation, the far fields are calculated using the near-field-to-far-field transformation. This calculation requires the dyadic Green's function of the background structure. By default, the FDTD engine uses the free space dyadic Green's function for the far field calculation. In general, the [[FDTD Module]] features dyadic Green's functions for four scenarios:

Figure 1: An example of the 3D radar cross section of a PEC plate.

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 />

Figure 2: [[FDTD Module]]'s Angle Settings dialog.

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]]:

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.

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.

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

Optimization is a process in which the values of one or more variables are varied in a systematic way until one or more design objectives are met. The design objectives are typically defined based on the output simulation data and are mathematically translated into an error (objective) function that is to be minimized. Running a successful optimization requires careful advance planning. First you have to make sure that your optimization problem does have a valid solution within the range of your optimization variables. In other words, the design objectives must be achievable for at least one combination of the optimization variable values within the specified ranges. Otherwise, the optimization process will not converge or will exhaust the maximum allowed number of iteration cycles and exit unsuccessfully.

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.

Figure 1: [[FDTD Module]]'s Periodicity Settings dialog.

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.

Figure 2: Radiation pattern of a 8×8 finite-sized periodic dipole array with scan angles: (Left) &theta; = &phi; = 0°, and (Right) &theta;= 45° and &phi; = 0°.

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.

Figure 1: Defining additional radiation characteristics in [[FDTD Module]]'s Radiation Pattern dialog.

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.

Figure 1: (Left) Setting a custom plane wave source plane for a periodic unit cell structure, and (Right) the visualization of the periodic plane wave surface along with a horizontal field probe placed above the structure.

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.

Figure 1: Magnitude and phase of the complex reflection and transmission coefficients of a periodic surface plotted vs. frequency.

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. <font color="red"><u>'''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.'''</u></font>