1. FDTD 1D-MAP Plane Wave TFSF Simulation for Lossy and Stratified Media
Tengmeng Tan and John Schneider*
Northwestern University, Biomedical Engineering Department, 2145 Sheridan Rd, Evanston, IL,
* Washington State University, School of Electrical Engineering and Computer Science, P.O. Box , Pullman, WA,
I. Objectives
II. 1D-MAP Methodology
(1D Multipoint Auxiliary Propagator)
IV. Proposed Boundary Value Solution
Step 2: Partition this dispersion relation into:
(7)
(8)
Free 1D Multipoint Auxiliary Propagator (1D-MAP) method from requiring propagation at rational angles.
Transform ONLY along the boundaries
f(xT,y,t)
f(x,yR,t)
f(x,yL,t)
f(xB,y,t) SF TF
1D-MAP makes use of a uniform r-grid SF TF r ir
Extend the 1D-MAP methodology to stratified media.
Preserve a perfectly matched TFSF solution.
Step 3: An FDTD solution along (xT,B, y) comes from (8):
II. Plane Wave Solution in Uniform Medium
(1)
Parallel to the interface / along y-direction
(5)
is considered excitation, this becomes 1D-plane wave problem propagating along y-direction. If
wavenumber vector
coordinate space vector
Perfectly matched plane wave source TFSF implementation.
RESULT:
Step 4: Solution along (x, yL,R) is obtained via phase matching:
The dispersion relation in each layer is:
Let (, ) be direction angles and define a directional vector:
Angles must align with a grid cell or propagate at rational angles
(because mx, my, and mz must be integers).
LIMITATIONS:
Has not been extended to a stratified media.
Perpendicular to the interface / along x-direction
(6)
is considered excitation, then the 1D-plane wave is propagating along x-direction. If
In particular, if  and  are function of only x, then these become transmission-line problems.
The dispersion along (x, yL,R) becomes:
(9)
III Challenges/Problems for Stratified Media
Maxwell’s equations in rm() space involve convolutions
Medium 1
Medium 2 x
 = 90o y
transmitted angle
depends on frequency
Frequency Dependent Coordinate Transform
Snell’s Law:
for all m
Mathematically, a plane wave can be expressed as a 1D problem:
where one defines a subspace coordinate transformation:
Step 1: Write FDTD dispersion relation as:
where Ls,1 = Ls,1 (µ, 1, t, ) is the lossy term.
Step 5: An FDTD solution along (x, yR,L) comes from (9)
(2)
This is independent of time and frequency.
For plane waves, Maxwell’s equations reduce to a 1D problem
V. Numerical Examples
1= 60o
(not a rational angle)
Stability value S =1/2
1  2  3  4 5
31 = 9 2 = 2 3 = 6 4 = 5 5
Gaussian Source
Lossy uniform   0
Sine Wave Source
( S/m)
VI. Discussions / Conclusions
(3)
One approach can be applied to solve both uniform and stratified media problems.
Field leakage from TFSF boundary is at machine precision (i.e., effectively zero).
Chain-Rule
Proposed technique introduces a spectrum of plane waves where angle of propagation is frequency dependent.
(4a)
Angular spreading can be quantified and represents an “opportunity” (e.g., sources along boundaries can be designed so that interior fields better match continuous-world fields). This is area of current study.
(4b)
Stability can be a concern but simple fix is to reduce Courant number.
VII. References
T. Tan and M. E. Potter, “1-D Multipoint Auxiliary source propagator for the total-field/scattered-field FDTD formulation,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 144–148, 2007.
T. Tan and E. M. Potter, “FDTD Discrete Planewave (FDTD-DPW) Formulation for a Perfectly Matched Source in TFSF Simulations,” IEEE Transactions on Antennas and Propagation, vol. 58, no. 8, pp , August 2010.
Valid for any finite difference scheme.
Discretizing along and can be made identical.
Possible to construct a true plane wave for FDTD scheme.
IMPORTANT IMPLICATIONS
Applied to the central difference scheme results in FDTD formulation.
( 10-4 – 10-3 S/m)