1. Atilla Ozgur Cakmak, PhD
Nanophotonics
Atilla Ozgur Cakmak, PhD

2. Lecture 23: An Exemplary Modeling Method-Introduction to FDTD
Unit 3
Lecture 23: An Exemplary Modeling Method-Introduction to FDTD

3. Outline Introduction to Modeling in EM Finite Difference Approximation
Finite Difference Time Domain
Absorbing Boundary
Stability and Discretization
Conclusion
1-D FDTD Problem

4. A couple of words…
This lecture aims to give the reader a feeling about the numerical solutions like we have employed using HFSS. Many of the problems in electromagnetics (EM) will not have exact analytical expressions. A lot of numerical methods are developed to handle such problems. Finite Difference Time Domain is only of those. It is heavily employed in the field of nanophotonics, though in order to model the nanostructures’ responses to light.
Suggested readings: Computational Electrodynamics: The Finite-Difference Time- Domain Method, 3rd Edition is a great book in the field by Allen Taflove. However, it is very much beyond the scope of this discussion. A quick review is suggested if possible rather than a detailed analysis.

5. Introduction to Modeling in EM
Exact solutions
Separation of variables
Series expansion
Numerical Methods
Finite Difference Time Domain
Finite Element Method (eg. HFSS)
Method of Moments
Transmission line matrix method
Only certain shapes will allow exact solutions like spheres, discs, wedges, cylinders.
EM (electromagnetic) problems are classified to be differential, integral and integrodifferential. We will only touch finite difference time domain in 1-D just as an introduction.

6. Finite Difference Approximation

7. Finite Difference Time Domain

8. Finite Difference Time Domain

9. Finite Difference Time Domainz
Step 1: Model the leapfrog scheme

10. Finite Difference Time Domainz
Step 2: Initialize the fields, such that for t<0 both E and H are zero

11. Finite Difference Time Domainz
Step 3: Define the boundaries such that Ex(-1/2,t)=f1(t) and Ex(5/2,t)=f2(t)

12. Finite Difference Time Domainz
Step 4: Run the exemplary leap form solely storing the previous time steps. Do not store more!

13. Absorbing Boundary
How should we terminate the boundaries to minimize the reflection back into the simulation domain?
Special care must be given to minimize such artificial reflections such that the simulation domain will act as if the waves are continuing to progress into free space. Here, we will discuss 1-D Absorbing Boundary Conditions (ABC)

14. Absorbing Boundary

15. Absorbing Boundary Backward Waves Forward Waves t
Black arrows show ABC …
Initialization z

16. Absorbing Boundary (problem)
Discretize for the backward waves at z=-L and attain a formula similar to the forward waves.

17. Absorbing Boundary (solution)
Discretize for the backward waves at z=-L and attain a formula similar to the forward waves.

18. Stability and Discretization

19. Stability and Discretization

20. Stability and Discretization
As it can be seen, people would want to work with a better mesh (smaller ∆z) and larger p value. But large p value makes FDTD unstable!
Plane wave in free space

21. Conclusions
So, we have discussed the stability and dispersion errors due to discretization. We are ready to apply our knowledge to a simple 1-D case with ABC.
But are we done with FDTD?
Far from it. There are 2-D and 3-D adaptations that we never touched. The main essence of this lecture has been to give a feeling of how the numerical tools are working and how multi purpose solvers are working for different geometries.
There are also Perfectly Matched Layers (PMLs) employed after mid-90s which work very well with different problems while introducing minimum errors (reflections) back into the simulation domain. For a radiation problem that I had solved before, the figure below shows the enhancement coming with PMLs for 2-D problems compared to ABCs to show how powerful PMLS are .
Errors introduced in
linear scale
Errors introduced in
log scale: 60 dB gain!

22. 1-D FDTD problem
Let us try to solve the same transmission problem through a slab that we had formulized for our TMM in the previous lecture with FDTD. ε1=1 and tslab=500nm with ε2=9. The slab is sandwiched between free space semi infinite layers. Please watch the video here.

23. 1-D FDTD problem
Let us try to solve the same transmission problem through a slab that we had formulized for our TMM in the previous lecture with FDTD. ε1=1 and tslab=500nm with ε2=9. The slab is sandwiched between free space semi infinite layers. Please watch the video here.
We will illuminate the slab with a pulse to get the whole frequency information. We will discuss how we will retrieve the data from the present excitation method with the help of fourier analysis.

24. 1-D FDTD problem
Let us try to solve the same transmission problem through a slab that we had formulized for our TMM in the previous lecture with FDTD. ε1=1 and tslab=500nm with ε2=9. The slab is sandwiched between free space semi infinite layers. Please watch the video here.

25. 1-D FDTD problem
Let us try to solve the same transmission problem through a slab that we had formulized for our TMM in the previous lecture with FDTD. ε1=1 and tslab=500nm with ε2=9. The slab is sandwiched between free space semi infinite layers. Please watch the video here.

26. 1-D FDTD problem
Let us try to solve the same transmission problem through a slab that we had formulized for our TMM in the previous lecture with FDTD. ε1=1 and tslab=500nm with ε2=9. The slab is sandwiched between free space semi infinite layers. Please watch the video here.

27. 1-D FDTD problem
Let us try to solve the same transmission problem through a slab that we had formulized for our TMM in the previous lecture with FDTD. ε1=1 and tslab=500nm with ε2=9. The slab is sandwiched between free space semi infinite layers. Please watch the video here.
Exact position is nm
Exact position is nm
source
monitor

28. 1-D FDTD problem
Let us try to solve the same transmission problem through a slab that we had formulized for our TMM in the previous lecture with FDTD. ε1=1 and tslab=500nm with ε2=9. The slab is sandwiched between free space semi infinite layers. Please watch the video here.