1. Making the Finite Difference Time Domain computational method better... stronger... faster (at a cost much much less than $6M) Rodolfo E. Díaz Michael Watts Igor Scherbatko Laboratory for Material Wave Interactions

2. Standard FDTD on the Yee grid is based on a Cartesian finite difference version of Maxwell’s Curl Equations The Yee lattice intercalates E and H in space, making the definitions of the curl operators straight-forward. The Yee unit cell: At the (i,j,k) point E is on the edges, H is on the faces.

3. The Fields are advanced in time by considering the Curl term to be the source over the timestep. Consider the x component of the Curl of H

4. Similarly, the x component of the Curl of H drives D x

5. This is an Initial Value PDE problem that can be solved from time = t to t+dt To solve the inhomogeneous PDE in discretized time, set up a leapfrog scheme: If H is evaluated at the half-integer steps while E is evaluated at the integer steps, the curl acts as a source term.

6. We therefore have a PDE with constant coefficients and a constant inhomogeneous term. We have two alternatives: Solve the initial value problem: So that (gives an exponential characteristic solution) with

7. Or turn the equation into finite difference form : The time derivative of E is clearly evaluated at the half-integer step. So is the curl of H. Therefore so must be E(?)=(E(t+dt)+E(t))/2 Now, call t+dt t time t t+dt t+dt/2 E (t+dt) E (t) E (t+dt/2)

8. The simulation proceeds from E to H, from H to E, satisfying BCs automatically What’s the problem? Because all of space must be discretized we have a problem with the computational volume. Small details (fine grid) or Large objects translate into Large Scenes Large scenes = –Large Memory requirement –Long Time of execution

9. What are the assets of FDTD? Full wave time domain solution lets you see the physics of the problem. Simple to program = Intuitive Eminently Stable. No theoretical limit on ability to solve a problem (no matrix inversion). Trade-off of CPU memory versus time.

10. The domain must be truncated in such a way that… The scatterer is nevertheless illuminated with the correct incident field. –Plane wave injection. The scattered energy exits the domain as if it were in an infinite space. –Absorbing Boundary Conditions (ABCs) The scattered energy is available for observation with minimized interference by the (blinding) incident wave. –Separation of Total field and Scattered field regions

11. Our Goal is to improve the performance of FDTD with the simplest possible fixes Performance: Greater accuracy for same domain size Faster execution for same domain size Same accuracy for minimized domain size Simplicity: Of derivation, implementation and execution Low computational overhead

12. Today’s Presentation Field Teleportation: Injection of True Plane Waves into Finite domains. Allows separation of total field and scattered field regions. With no apriori analytical solution of the environment. Field Teleportation of arbitrary time domain fields

13. Today’s Presentation (continued) Symmetrized FDTD Removes Yee asymmetry from material discontinuities. Reduces “staircasing”error. A new Radiation Boundary Condition Simple to program Beats the PML

14. Why do we need Plane wave injection? (Near) Plane Wave illumination is a fact of life in many applications: RCS, Remote Ground Penetrating Radar schemes, Laser inspection of semiconductor surfaces... Almost all optical scattering problems. In these, the background scattering is uniform (enough) and not of interest, we want to measure the response of the isolated defect.

15. To maximize SNR we don’t want our ABCs to have to deal with the strong incident plane wave. Previous authors have tried to generate these “local” plane wave regions with analytic Huygens sources. In the most challenging cases, the incident wave is >> the scatter ABC Huygens Sources Scattered Field Region Total Field Region

16. That is, without causing scattering from the FDTD grid which is itself an anisotropic dispersive object. What if there is no analytic solution? The Huygens sources must create and absorb a large number of waves transparently. ABC Huygens Sources Scattered Field Region Total Field Region

17. FDTD is not a discrete-time/discrete-space approximation of Maxwell’s equations. It is a discrete time simulation of the behavior of a medium where every element is only connected to its nearest neighbors, and obeys the same Hamiltonian as Maxwell’s Equations. G. F. Fitzgerald built a tabletop example of this medium in the late 1880’s. The Answer: Use FDTD to create the Huygen’s sources. Pulleys Rubber bands H=spin D=strain

18. In this discrete space, Schelkunoff’s currents have exact discrete analogs And they are trivial to implement: E total H total Sources E total H total E =0 H =0 K e= n x H total K m=- n x E total Sources

19. Similarly for H excess ; so that the fields inside a “Window”can be teleported from one domain to another

20. Since Plane waves are infinite, they cannot be created in a finite domain However, they can be created in a periodic domain... And copied onto another domain

21. Demo: Scattering from a cylinder Run FDTD302a.EXE (after compiling it) It should be inserted into this presentation as a hyperlink

22. The teleportation is perfect because it uses the FDTD update equations Typical leakage is –300dB and due only to round-off error Lossy dielectric half- space Free space  r =2.0  e =0.5 Sample point

23. Any FDTD field of arbitrary time dependence can be teleported… Wrap-around conditions

24. To determine how a defect in a complex environment alters the signature of the environment

25. Next subject: Yee’s grid assures us second order accuracy but it is asymmetric (i,j,k+1) This is inconvenient when modeling material objects – where is the real boundary? But even worse, how do you model a smooth surface? Staircasing is supposed to be a major source of “noise”in Cartesian FDTD.

26. The partially filled capacitor is equivalent to an effective medium Standard FDTD update equation Ex i,j,k Media #1 i,j,k Media #2 i+1,j,k Series sum of two caps Only applies to up and right This undos the asymmetry of the Yee grid

27. Symmetrizing removes the asymmetry error from the staircasing approximation Ordinary FDTD Symmetrized FDTD Run SymP1.EXE (after compiling it) It should be inserted into this page as a hyperlink Run SymP2.EXE (after compiling it) It should be inserted into this page as a hyperlink

28. This can improve dramatically the range of validity of coarse (fast) models Ordinary and Symmetrized FDTD compared to the exact solution for Cylinder RCS

29. So far… We can inject perfect plane waves into finite domains. Separates the Scattered Field from the Total field for maximum SNR Illuminates finite objects with true model of typical incident wave. We can create smooth material objects without having to reduce the grid size. Speeds up execution

30. The Next step: Minimize the domain size This is the job of the Absorbing Boundary Condition. These are traditionally derived by taking analytic solutions to one-sided wave equations (Mur) or ideal fictitious absorbing materials (PML) and discretizing into FDTD. But that was precisely the problem with Huygen’s sources…

31. So instead use Self-Teleportation The Radiation Boundary Condition: Recipe: Teleport the exiting field back into the source space with a minus sign. Repeat as needed Terminate with a simple one- cell absorber

32. Scheme of the 2D FDTD experiment (field termination in free space and dielectric media ) P – polarized, harmonic cylindrical wave (H x component) PEC field measurement plane PEC12 cells ABC

33. Comparison new ABC and Berenger’s PML* 12-cells ABC Data is taken from W.Yu, R. Mittra, et al. “FDTD modeling of an artificially synthesized absorbing medium”, IEEE Microwave and Guided Wave Lett., Vol. 9, N0.12, Dec.1999.

34. A more challenging problem:Field termination on the lossy dielectric/free space interface

35. The Radiation boundary Condition is impervious to material discontinuities 6 - cells ABC o:  r =4-j0.6,  :  r =4-j0.75, q :  r =4-j1.0

36. An even more challenging problem: How does this Radiation Boundary Condition fare with Surface Waves? Incident Wave Curvature scatter Trailing edge wrap-around and scatter The TM Radar Cross section of an airfoil is a surface wave dominated phenomenon. Creeping Wave Traveling Wave

37. Compare a large domain to one with the RBC at 3 cells from the creeping wave The large domain The small domain Run SurfP1.EXE (after compiling it) It should be inserted into this page as a hyperlink Run SurfP2.EXE (after compiling it) It should be inserted into this page as a hyperlink

38. The time domain history of the echoes differs only slightly The RBC dampens the source so compare in Freq. Domain Timestep Hz

39. The spectral content of the incident pulses show the effect of the RBC Input > -50dB below 3 cells/ point Large Small ds= /3 Grid cut-off Frequency

40. The frequency domain echoes are extremely close to each other ds= /3 Grid cut-off Frequency

41. Up to 0.66 f cutoff the typical deviation is less than 1dB ds= /3 Grid cut-off ds= /4 Frequency Note: In this region the RBC starts <<1 from the object

42. Conclusions We can inject perfect plane waves (or any field for that matter) into finite domains. We can create smooth material objects without having to reduce the grid size. We can truncate the FDTD domain extremely close to the scattering object (< 1 ) regardless of the complexity of the environment in which that object is submerged.

43. FDTD… Better, Stronger, Faster… All at less than $6M