1. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering An Introduction to Computational Electromagnetics using FDTD R. E. Diaz

2. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering Finite Difference Time Domain is a Partial Differential Equation method The DDSURF/SUB/FILM family is an Integral Equation method. –Advantage: Need to dicretize only scatterer. –If: The Green function of the environment is known. –Disadvantage: Must invert a huge matrix. Unstable if  >>1. PDE methods: –Advantage: No Green function, no inversion. –If: You have time (computing power) if  >>1. –Disadvantage: Must discretize all of space.

3. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering FDTD encodes Maxwell’s curl equations over all space. Maxwell’s first curl equation: where  is in general an operator  Take its Fourier Transform: Thus, And:

4. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering 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.

5. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering We therefore have a PDE with constant coefficients and a constant inhomogeneous term. We have two alternatives: (a) solve the initial value problem (gives an exponential characteristic solution) or (b) turn the equation into finite difference form using the fundamental theorem of calculus: But at what time must E(?) be evaluated?

6. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering For the equation to be valid at the same instant everything must be evaluated at the half-integer time step. 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

7. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering In the same way, Maxwell’s second curl equation is reduced to an “update” equation. Now, since curl of E is evaluated at t, and the time derivative of H also occurs at t, H(?)=(H(t+dt/2)+H(t-dt/2))/2 And becomes

8. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering Thus, we have a leapfrog algorithm, from H to E and from E to H, and back again... Now let’s talk about discretization in space. 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.

9. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering Consider the x component of the curl of H

10. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering Similarly, the x component of the curl of E...

11. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering The FDTD algorithm marches E and H in time throughout the grid the way an EM wave propagates. All interactions in materials and satisfaction of boundary conditions across material boundaries occur automatically. Because the computational domain must be truncated, need Absorbing Boundary Conditions. The finite grid size introduces “grid dispersion” that limits the upper frequency at which a Fourier Transform of the time domain result is valid. The time domain solution is multi-frequency by nature. Not necessarily overkill because it can be used to discern phenomena.

12. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering Consider a 108 nm Si sphere sitting on a 75 nm SiO 2 film on top of Si

13. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering First illuminate the scene with a pulse in the absence of the scatterer. 1 2 3 4 5 6

14. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering 1 2 3 4

15. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering If the film were the higher dielectric constant material, then it would tightly bind the surface waves. Let the film be Si and the substrate SiO2. Plot Power.Notice how the surface waves now do not leak into the lower space. Incident Pulse Scattered Wave Time=150 dt Time=250 dt Time=350 dt

16. Consortium for Metrology of Semiconductor Nanodefects Mechanical Engineering Conclusions and Future Work FDTD is a very stable PDE computational electromagnetics tool that is rapidly becoming a standard in Optics as well as RF research. It serves well as a complement to the IE method family of codes DDSURF/SUB/FILM. The latter can be extremely fast for moderate scenes But when material parameters are extreme and/or the matrix just cannot be inverted, all that FDTD needs is memory and time. Ongoing work to improve its efficiency for optical scattering focuses on dispersive material modeling, the absorbing boundary conditions, and the input and output of incident plane waves.