3 Lecture 1 Slide 5 Popular Numerical Techniques • Transfer matrix method • Scattering matrix method • Finite‐difference frequency‐domain • Finite‐difference time‐domain • Transmission line modeling method • Beam propagation
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DefinitionComputational electromagnetics (CEM) is the procedure we must follow to model and simulate the behavior of electromagnetic fields in devices or around structures.
Most often, CEM implies using numerical techniques to solve Maxwell’s equations instead of obtaining analytical solutions.
Why is this needed?
Very often, exact analytical solutions, or even good approximate solutions, are not available. Using a numerical technique offers the ability to solve virtually any electromagnetic problem of interest.
1. Golden Rule #1.2. Finite floating point precision introduces round ‐off errors.
Why?
Solution: MINIMIZE NUMBER OF COMPUTATIONS!!!
1. Take problems as far analytically as possible.2. Avoid unnecessary computations.
2 2
2exp
R x y R
g R
2 2
2
2exp
r x y
r g r
Never perform calculations.
Lecture 1 Slide 10
Golden Rule #3
1. It will run faster and more reliably.2. Easier to catch mistakes.3. Easier to troubleshoot.4. Easier to pick up again at a later date.5. Easier to modify.
Why?
Solution
1. Outline your code before writing it.2. Delete obsolete code.3. Comment every step.4. Use meaningful variable names.
There is a rhythm to computational electromagnetics
and it repeats itself constantly.
Starts with Maxwell’s equations and derives all the necessary equations to implement the algorithm in MATLAB.• Equations everywhere! Only a few are needed.• Implementation does not resemble the formulation.
Organizes the equations derived in the formulation and considers other numerical details.• Consider all numerical best practices.• Should end with a detailed block diagram.
Actually implements the algorithm in computer code.• Implementation should be simple and minimal.• The “art” of simulation begins here.• Practice, practice, practice!
Lecture 1 Slide 12
Don’t Be LazyA little extra time making your program more efficient or simulating a device in a more intelligent manner can save you lots of time, energy, and aggravation.
Physical Boundary ConditionsPhysical boundary conditions refer to the conditions that must be satisfied at the boundary between two materials. These are derived from the integral form of Maxwell’s equations.
Numerical Boundary ConditionsNumerical boundary conditions refer to the what is done at the edge of a grid or mesh and how fields outside the grid are estimated.
Tangential components are continuous
A
Lecture 1 Slide 22
Full Vs. Sparse Matrices
Full Matrices Sparse Matrices
A
Full matrices have all non ‐zero elements.
They tend to look banded with the largest numbers running down the main diagonal.
Sparse matrices have most of their elements equal to zero. They are often more than 99% sparse.
It is most memory efficient to store only the non ‐zero elements in memory.
They tend to “banded” matrices with the largest numbers running down the main diagonal.
Differential equations calculate a quantity at a specific point using only information from the local vicinity. They are usually written for points distributed throughout a volume and lead to formulations with sparse matrices. They require boundary conditions.
Integral equations calculate a quantity at a specific point using information from the entire domain. They are usually written around boundaries and lead to formulations with full matrices. They do not require boundary conditions.
Lecture 1 Slide 24
Frequency ‐Domain Vs. Time ‐Domain
This is what a frequency ‐domain code calculates.
This is what a time ‐domain code calculates.
Frequency ‐domain solutions are at a single frequency. Time ‐domain solutions look different because there are inherently a broad range of frequencies involved.
Virtually all numerical methods have some sort of “resolution”
parameter that when taken to infinity solves Maxwell’s equations exactly. In practice, we cannot this arbitrarily far because a computer will run out of memory and simulations will take prohibitively long to run.
There are no equations to calculate what “resolution” is needed to obtain “accurate” results. Instead, the user must look for convergence. There are, however, some good rules of thumb to make an initial guess at resolution.
Convergence is the tendency of a calculated parameter to asymptotically approach some fixed value as the resolution of the model is increased. A converged solution does not imply an accurate solution !!!
Tips About Convergence
• Make checking for convergence a habit that you always perform.
• When checking a parameter for convergence, ensure that it is the only thing about the simulation that is changing.
• Simulations do not get more “accurate” as resolution is increased. They only get more
Common Sense – Check your model for simple things like conservation of energy, magnitude of the numbers, etc.
Benchmark – You can verify your code is working by modeling a device with a known response. Does your model predict that response?
Convergence – Your models will have certain parameters that you can adjust to improve “accuracy” usually at the cost of computer memory and run time. Keep increasing “accuracy” until your answer does not change much any more.
When modeling a new device, benchmark your model using as similar of a device as you can find which has a known response. Compare your experimental results to the model. Do they agree? Reconcile any differences.
• Ray tracing• Geometric theory of diffraction• Physical optics• Physical theory of diffraction• Shooting and bouncing rays
Lecture 1 Slide 30
Classification by ApproximationsRigorous MethodsA method is rigorous if there exists a “resolution” parameter that when taken to infinity, finds an exact solution to Maxwell’s equations.• Finite ‐difference time ‐domain• Finite ‐difference frequency ‐domain• Finite element method• Rigorous coupled ‐wave analysis• Method of lines
Full Wave MethodsA method is full wave if it accounts for the vector nature of the electromagnetic field. A full wave method is not necessarily rigorous.• Method of moments• Boundary element method• Beam propagation method
Scalar MethodsA method is scalar if the vector nature of the field is not accounted for.• Ray tracing
Transmission through all the layers is described by multiplying all the individual transfer matrices.
1T
,2 ,12
,2 ,1
x x
x x
E E
E E
T
,trn ,2
3,trn ,2
x x
y x
E E
E E
T
global 3 2 1T T T T
,trn ,ref
global,trn ,ref
x x
y y
E E
E E
T
Transfer Matrix Method (2 of 2)
• Very fast and efficient• Rigorous• Near 100% accuracy• Unconditionally stable• Robust• Simple to implement• Thickness of layers can be anything• Able to exploit longitudinal periodicity• Easily incorporates material dispersion• Easily accounts for polarization and
angle of incidence• Excellent for anisotropic layered
materials
Lecture 1 Slide 36
• Limited number of geometries it can model.
• Only handles linear, homogeneous and infinite slabs.
• Cannot account for diffraction effects
• Inefficient for transient analysis
This method is good for…1. Modeling transmission and reflection from layered devices.2. Modeling layers of anisotropic materials.
This large set of equations is written in matrix form and solved to calculate the fields.
2 2, , , , y y z z z
E x y z E x y z E y y
x
y
z
e
x e
e
Finite ‐Difference Frequency ‐Domain (2 of 2)
• Accurate and robust• Highly versatile• Simple to implement• Easily incorporates dispersion• Excellent for field visualization• Error mechanisms are well
understood• Good method for metal devices• Excellent for volumetrically
complex devices• Good scaling compared to other
frequency ‐domain methods
Lecture 1 Slide 38
• Does not scale well to 3D• Difficult to incorporate
nonlinear materials• Structured grid is inefficient• Difficult to resolve curved
surfaces• Slow and memory innefficient
This method is good for…1. Modeling 2D devices with high volumetric complexity.2. Visualizing the fields.3. Fast and easily formulation of new numerical techniques.
Maxwell’s equations are enforced at each point at each time step.
Finite ‐Difference Time ‐Domain (2 of 2)
• Excellent for large ‐scale simulations. Easily parallelized.
• Excellent for transient analysis.• Accurate, robust, rigorous, and mature• Highly versatile• Intuitive to implement• Easily incorporates nonlinear behavior• Excellent for field visualization and learning
electromagnetics• Error mechanisms are well understood• Good method for metal devices• Excellent for volumetrically complex devices• Scales near linearly• Able to simulate broad frequency response in one
simulation• Great for resonance “hunting”
Lecture 1 Slide 40
• Tedious to incorporate dispersion• Typically has a structured grid
which is less efficient and doesn’t conform well to curved surfaces
• Difficult to resolve curved
surfaces• Slow for small devices• Very inefficient for highly
resonant devices
This method is good for…1. Modeling big, bad and ugly problems.2. Modeling devices with nonlinear material properties.3. Simulating the transient response of devices.
The plane wave expansion method (PWEM) calculates modes that exist in an infinitely periodic lattice. It represents the field in Fourier ‐space as the sum of a large set of plane waves at different angles.
Plane Wave Expansion Method (2 of 2)
• Excellent for all‐dielectric unit cells
• Fast even for 3D• Accurate and robust• Rigorous method
Lecture 1 Slide 50
• Scales poorly.• Weak method for high
dielectric contrast and metals.• Limited to modal analysis.• Cannot model scattering.• Cannot incorporate dispersion.
This method is good for…1. Analyzing unit cells2. Calculating photonic band diagrams and effective material properties.
Governing equation exists only at the boundary of a device so many fewer elements are needed.
400 elements5000 elements
The boundary element method (BEM) is also called the Method of
Moments, but is applied to 2D elements. The most famous element is the Rao‐Wilton ‐Glisson (RWG) edge element.
S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic Scattering by Surfaces of Arbitrary Shape,” IEEE Trans. Antennas and Propagation, vol. AP‐30, no. 3, pp. 409 ‐418, 1982.
Boundary Element Method (2 of 2)
• Highly efficient when surface to volume ratio is low
• Excellent representation of curved surfaces
• Unstructured grid is highly
efficient• Unconditionally stable• Can be hybridized with FEM• Domain can extend to infinity• Simpler meshing than FEM
Lecture 1 Slide 58
• Tedious to implement• Requires a meshing step• Not usually a rigorous method• Inefficient for volumetrically
complex geometries
This method is good for…1. Modeling large devices with simple geometries.2. Modeling scattering from homogeneous blobs.