3 Approximating the Time Derivative (1 of 3) An intuitive first guess at approximating the time derivatives in Maxwell’s equations is:This is an unstable formulation.
4 Approximating the Time Derivative (2 of 3) We adjust the finite‐difference equations so that each term exists at the same point in time.These equations will get messy if we includeinterpolations. Is there a simpler approach?
5 Approximating the Time Derivative (3 of 3) We stagger E and H in time so that E exists at integer time steps(0, Δt, 2 Δ t, …) and H exists at half time steps (0, Δ t/2, 3Δ t/2, 4Δ t/2 …).We will handle the spatial derivatives in curl nextlecture in a very similar manner.
6 Derivation of the Update Equations The “update equations” are the equations used inside the main FDTD loop to calculate the field values at the next time step.They are derived by solving our finite‐difference equations for the fields at the future time values.
7 Anatomy of the FDTD Update Equation (To speed simulation, we calculate these before iteration.)UpdatecoefficientField at the future time step.Field at the previous time step.Curl of the “other” field at an intermediate time step
10 Representing Functions on a Grid A grid is constructed by dividing space into discrete cellsRepresentation of what is actually stored in memoryExample physical(continuous) 2D functionFunction is known only at discrete points
11 Grid Unit Cell Whole Grid A function value is assigned to a specific point within the grid unit cell.
12 3D Grids A three‐dimensional grid looks like this: A unit cell from the grid looks like this:
13 Collocated GridWithin the unit cell, we need to place the field components Ex, Ey, Ez, Hx, Hy, and Hz.A straightforward approach would be to locate all of the field components within in agrid cell at the origin of the cell.
14 Yee GridInstead, we are going to stagger the position of each field component within the grid cells.
15 Reasons to Use the Yee Grid Scheme 3. Elegant arrangement to approximate Maxwell’s curl equations1. Divergence‐free2. Physical boundary conditions are naturally satisfied
16 Yee Cell for 1D, 2D, and 3D Grids 1D Yee Grid2D Yee Grids3D Yee Grid
17 Consequences of the Yee Grid • Field components are in physically different locations• Field components may reside in different materials even if they are in the same unit cell• Field components will be out of phase• Recall the field components are also staggered in time.
18 Visualizing Extended Yee Grids 2X2X2 Grid4X4 Grid (E Mode)
19 Finite‐Difference Approximation to Maxwell’s Equations
20 Normalize the Magnetic Field We satisfied the divergence equations by adopting the Yee grid scheme. We now only have to deal with the curl equations.The E and H fields are related through the impedance of the material they are in, so they are roughly three orders of magnitude different.This will cause rounding errors in your simulation and it is always good practice to normalize your parameters so they are all the same order of magnitude. Here we choose to normalize the magnetic field.
21 Curl Equations with Normalized Magnetic Field Using the normalized magnetic field, the curl equations becomeNote:Proof
24 Final Analytical Equations These are the final form of Maxwell’s equations from which we will formulate the FDTD method.Next, we will approximate these equations with finite‐differences in the Yee grid.
31 Summary of Finite‐Difference Equations Each equation is enforced separately for each cell in the grid. This is repeated for each time step until the simulation is finished. These equations get repeated a lot!!
33 Reduction to One Dimension We saw in Lecture 3 that some problems composed of dielectric slabs can be described in just one dimension. In this case, the materials and the fields are uniform in two directions. Derivatives in these uniform directions will be zero. We will define the uniform directions to be the x and y axes.
35 Maxwell’s Equations Decouple Into Two Independent Modes We see that the longitudinal field components Ez and Hz are always zero.We also see that Maxwell’s equations have decoupled into two sets of two equations.
36 Two Remaining Modes are the Same We see that the longitudinal field components Ez and Hz are always zero.We also see that Maxwell’s equations have decoupled into two sets of two equations.While these modes are physical and would propagate independently, the are numerically the same and will exhibit the same electromagnetic behavior. Therefore, it is only necessary to solve one. We will proceed with the Ey/Hx mode.
37 No Longer Need i and j Array Indices Ex/Hy ModeEy/Hz Mode
39 Update Equation for ExStart with the finite‐difference equation which has Ex in the time‐derivative:Solve this for Ex at the future time value.
40 Update Equation for EyStart with the finite‐difference equation which has Ey in the time‐derivative:Solve this for Ey at the future time value.
41 Update Equation for HxStart with the finite‐difference equation which has Hx in the time‐derivative:Solve this for Hx at the future time value.
42 Update Equation for HyStart with the finite‐difference equation which has Hy in the time‐derivative:Solve this for Hy at the future time value.
43 Implementation of the Basic Update Equations for the Ey/Hx Mode
44 Efficient Implementation of the Update Equations The update coefficients do not change their value during the simulation. They should be computed only once before the main FDTD loop and not at each iteration inside the loop.The finite‐difference equations in terms of the update coefficients are:Hx, Ey, εyy, μxx, mHx, and mEy are all stored in 1D arrays of length Nz.c0, Δt, and Δ z are single scalar numbers, not arrays.
45 The Basic 1D‐FDTD Algorithm Includes:• Basic FDTD engineExcludes:• Dirichlet BC’s• Calculate source parameters• Simple soft source• Perfectly absorbing BC’s• TF/SF source• Fourier transforms• Reflectance/Transmittance• Calculate grid parameters• Incorporate deviceCalculate the update coefficients before the main loop to save computations.
46 Equations → MATLAB Code Update CoefficientsUpdate EquationsYou will need to update the fields at every point in the grid so these equations are placed inside a loop from 1 to Nz.
47 Each Cell Has Its Own Set of Update Equations Each cell has its own update equation and its own update coefficients. They areimplemented separately for each cell. All of these equations have the same general form so it is more efficient to implement them using a loop. For a 1D grid with 10 cells, think of it this way…