## EECS 386 - Computational Electromagnetics and Photonics

**CATALOG DESCRIPTION: **Introduction to the finite-difference time-domain (FDTD) method in numerical modeling of electromagnetic and optical wave interactions with engineering structures. Topics: finite differences; Maxwell's equations; numerical dispersion and stability; free-space and waveguide field sources; absorbing boundary conditions; material dispersions and nonlinearities.

**REQUIRED TEXT:** A. Taflove and S. C. Hagness,*Computational Electrodynamics: The Finite-Difference Time-Domain Method,*Artech House, 3 rd edition (2005)

**REFERENCE TEXT:** A. Taflove,*Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method*, Artech House (1998)

**COURSE COORDINATOR:** Allen Taflove

**COURSE GOALS:** To provide the electrical engineering student with the foundation to numerically model electromagnetic wave interactions in modern electronic and optical circuits.

**PREREQUISITE: **EECS 308 or the equivalent.

**DETAILED COURSE TOPICS:**

**Week**** Lectures**** Topic**

**1)** 1 Introduction to contemporary problems in electromagnetic wave engineering and techniques in computational electromagnetics.

2 Development of finite differences from Taylor series; truncation error.

3 Application of finite differences to the 1-D scalar wave equation.

Assign Project 1 : Simulate the 1-D time-domain scalar wave equation; investigate its numerical dispersion, stability, and accuracy properties.

**2)** 4 Numerical dispersion of the 1-D scalar wave equation.

5 Numerical stability of the 1-D scalar wave equation.

6 Simple 1-D wave source and absorbing boundary conditions.

**3**) 7 Review of Maxwell's equations in differential and integral form; transverse electric (TE) and transverse magnetic (TM) polarizations.

8 Introduction to Yee's central differencing in 1-D space and time.

9 Numerical dispersion of the 1-D Yee algorithm.

Assign Project 2 : Program the 1-D Yee algorithm with wave-source and absorbing boundary conditions; examine its numerical dispersion, stability, and accuracy properties.

**4) **10 Numerical stability of the 1-D Yee algorithm.

11 Simple wave sources and absorbing boundaries for the Yee grid in 1-D.

12 Total-field / scattered-field zoning for the Yee grid in 1-D.

**5)** 13 Introduction to Yee's central differencing in 2-D space and time.

14 Numerical dispersion of the 2-D Yee algorithm.

15 Numerical stability of the 2-D Yee algorithm.

Assign Project 3 : Program the 2-D TM Yee algorithm to model the propagation of a cylindrical wave radiated by a line source. Examine the numerical dispersion and stability properties of the algorithm.

**6)** 16 Theory of analytical absorbing boundary conditions (ABC's) in 2-D.

17 Theory of 2-D analytical ABC's, continued.

18 Theory and numerical implementation of the Liao ABC.

**7)** 19 Introduction to Berenger's perfectly matched layer (PML) ABC.

20 Berenger's PML ABC, continued.

21 Introduction to the uniaxial PML (UPML) ABC.

Assign Project 4 : Incorporate the third-order Liao ABC in the grid of Project 3. Demonstrate outer-boundary reflectivity in the order of 1% or lower for an outgoing cylindrical wave.

**8)** 22 Propagation in metal-wall waveguides; cutoff and dispersion phenomena.

23 Propagation in dielectric slab waveguides; cutoff and dispersion phenomena.

24 Introduction to propagation in defect-mode photonic crystals.

**9)** 25 Introduction to the macroscopic physics of dispersive materials.

26 Auxiliary differential equation (ADE) modeling of dispersive materials.

27 ADE modeling of dispersive materials, continued.

Assign Project 5 : Use the grid of Project 4 to model a parallel-plate metal waveguide and a dielectric slab waveguide. Demonstrate cutoff and propagation phenomena for the lowest-order even and odd modes for each type of waveguide.

**10)** 28 Nonlinear materials; formation of temporal and spatial optical solitons.

29 Incorporation of optical gain; 1-D microcavity laser simulations.

30 Recent developments and research horizons.

**COMPUTER USAGE:** Five programming project assignments (see above).

**LABORATORY:** None

**GRADES:** No exams. Instead each student is assigned five electromagnetic wave simulation projects of progressive difficulty and sophistication weighted at 10%, 15%, 20%, 25%, and 30% of the final grade. Each project requires: (1) solution of the associated homework problems that spotlight the fundamental underlying theory; and (2) development of simulation software from the fundamental theory using Matlab, Fortran, C, or C++ as selected by each student. Grading factors include assessment of student understanding of the theory, success in programming, and effectiveness in displaying the results of the simulations.

**COURSE OBJECTIVES:** When a student completes this course, s/he should be able to:** **

1) Understand the scope of contemporary and emerging application areas in electromagnetic wave technology, especially high-speed electronic and optical communications.

2) Understand the concepts and analysis approaches for numerical dispersion and stability of FDTD electromagnetic wave simulations.

3) Understand means to source waves in free space and in waveguides in numerical FDTD simulations.

4) Understand the theory and numerical implementation of widely used analytical and PML absorbing boundary conditions for FDTD grids.

5) Understand the mathematical basis and numerical modeling of frequency-dispersive and nonlinear materials in FDTD simulations.

7) Construct working software that implements FDTD codes capable of solving real electromagnetic wave and optical engineering problems.

8) Begin to read the research literature in FDTD modeling for engineering electromagnetics.

**ABET CONTENT CATEGORY:** 100% Engineering (Design component)