For example, Maxwellâs equations for an isotropic, time-invariant and homogeneous medium without electric or magnetic losses are given by:
[[Image:FDTD60(<math>\dfrac{\delta \mathbf{H}}{\delta t} = -\dfrac{1).png]]}{\mu} \Delta \times \mathbf{E}</math>Â <math>\dfrac{\delta \mathbf{E}}{\delta t} = -\dfrac{1}{\epsilon} \Delta \times \mathbf{H}</math>
where '''E''' and '''H''' are the electric and magnetic fields, respectively, ε is the permittivity and μ is the permeability. Both time- and space-derivatives are approximated with central finite differences. This results in six differential equations, one for each field component. For the field components in x-direction, the field equations result in:
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>
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 (θ) and Phi (φ) 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.
[[Image:FDTD143.png]]