11/25/2015
Instructor
Dr. Raymond Rumpf
(915) 747‐6958
rcrumpf@utep.edu
EE 5320
Computational Electromagnetics (CEM)
Lecture #1
Introduction to CEM
 These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited 
Lecture 1
Slide 1
Outline
•
•
•
•
•
What is CEM?
CEM wisdom
General concepts in CEM
Classification of methods
Overview of methods
Lecture 1
Slide 2
1
11/25/2015
What is CEM?
Lecture 1
Slide 3
Computational Electromagnetics
Definition
Computational 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.
Zc 
Lecture 1
 
2
r 
cosh 1  out 
 rin 
Zc  ?
Slide 4
2
11/25/2015
Popular Numerical Techniques
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Transfer matrix method
Scattering matrix method
Finite‐difference frequency‐domain
Finite‐difference time‐domain
Transmission line modeling method
Beam propagation method
Method of lines
Rigorous coupled‐wave analysis
Plane wave expansion method
Slice absorption method
Finite element analysis
Method of moments
Boundary element method
Spectral domain method
Discontinuous Galerkin method
Lecture 1
Slide 5
CEM Wisdom
Lecture 1
Slide 6
3
11/25/2015
The Key to Computation is Visualization
Is there anything wrong? If so, what is it?
i , j , k 1
Ezi , j 1,k  Ezi , j ,k E y

y 
 E yi , j ,k
z 
  xxi , j ,k H xi , j ,k


 xyi , j ,k H yi , j ,k   xyi 1, j ,k H yi 1, j ,k   xyi , j 1,k H yi , j 1,k   xyi 1, j 1,k H yi 1, j 1,k
4
 xzi , j ,k H zi , j ,k   xzi , j ,k 1 H zi , j ,k 1   xzi 1, j ,k 1 H zi 1, j ,k 1   xzi 1, j ,k H zi 1, j ,k
4
i 1, j , k  i 1, j , k
i , j ,k
i , j ,k
Hx
  yxi , j 1,k H xi , j 1,k   yxi 1, j 1,k H xi 1, j 1,k
Exi , j ,k 1  Exi , j ,k Ezi 1, j ,k  E zi , j ,k  yx H x   yx


4
z 
x 
  yyi , j ,k H yi , j ,k
 yzi , j ,k H zi , j ,k   yzi , j ,k 1 H zi , j ,k 1   yzi , j 1,k 1 H zi , j 1,k 1   yzi, j 1,k H zi, j 1,k
4
E yi 1, j ,k  E yi , j ,k Exi , j 1,k  E xi , j ,k  zxi , j ,k H xi , j ,k   zxi 1, j ,k H xi 1, j ,k   zxi 1, j ,k 1 H xi 1, j ,k 1   zxi , j ,k 1 H xi , j ,k 1


4
x
y 
 i , j ,k H yi , j ,k   zyi , j 1,k H yi , j 1,k   zyi , j 1,k 1 H yi , j 1, k 1   zyi, j ,k 1 H yi , j ,k 1
 zy
4
i , j ,k  i , j ,k
  zz H z
i, j,k
i , j , k 1
H zi , j ,k  H zi , j 1,k H y  H y

  xxi , j , k Exi , j ,k
y 
z 


Response

 xyi , j ,k E yi , j ,k   xyi , j 1,k E yi , j 1, k   xyi 1, j 1,k E yi 1, j 1, k   xyi 1, j , k E yi 1, j ,k
4
 xzi , j ,k Ezi , j ,k   xzi , j , k 1 Ezi , j ,k 1   xzi 1, j , k 1 Ezi 1, j ,k 1   xzi 1, j , k Ezi 1, j ,k
4
i , j 1, k i , j 1, k
i , j ,k i , j ,k
  yxi 1, j 1,k Exi 1, j 1, k   yxi 1, j ,k Exi 1, j ,k
Ex
H xi , j ,k  H xi , j ,k 1 H zi , j , k  H zi 1, j , k  yx Ex   yx


z 
x 
4
  yyi , j , k E yi , j ,k

 yzi , j ,k Ezi , j ,k   yzi , j , k 1 Ezi , j ,k 1   yzi , j 1, k 1 Ezi , j 1,k 1   yzi , j 1, k Ezi , j 1,k
4
H yi , j ,k  H yi 1, j ,k H xi , j , k  H xi , j 1, k  zxi , j , k Exi , j ,k   zxi 1, j ,k Exi 1, j , k   zxi 1, j ,k 1 Exi 1, j , k 1   zxi , j ,k 1 Exi , j ,k 1


x 
y 
4

 zyi , j ,k E yi , j ,k   zyi , j 1,k E yi , j 1, k   zyi , j 1,k 1 E yi , j 1,k 1   zyi , j ,k 1 E yi , j ,k 1
4
  zzi , j , k Ezi , j ,k
Lecture 1
Slide 7
Golden Rule #1
All numbers should equal 1.
Why?
(1.234567…) + (0.0123456…) = Lost two digits of accuracy!!
Solution: NORMALIZE EVERYTHING!!!
0 
0
1 m

 
E 0E
x  k0 x
or
y   k0 y
0

0 
H
H
0
Lecture 1
z   k0 z
Slide 8
4
11/25/2015
Golden Rule #2
Never perform calculations.
Why?
1. Golden Rule #1.
2. Finite floating point precision introduces round‐off errors.
Solution: MINIMIZE NUMBER OF COMPUTATIONS!!!
1. Take problems as far analytically as possible.
2. Avoid unnecessary computations.
r  x2  y 2
R  x2  y 2
 r2 
g  r   exp   2 
  
 R 
g  R   exp   2 
  
Lecture 1
Slide 9
Golden Rule #3
Write clean code.
• Well organized
• Well commented
• Compact
• No junk code
Why?
1.
2.
3.
4.
5.
It will run faster and more reliably.
Easier to catch mistakes.
Easier to troubleshoot.
Easier to pick up again at a later date.
Easier to modify.
Solution
1.
2.
3.
4.
Lecture 1
Outline your code before writing it.
Delete obsolete code.
Comment every step.
Use meaningful variable names.
Slide 10
5
11/25/2015
The CEM Process
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 11
Don’t Be Lazy
A 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.
Lecture 1
Slide 12
6
11/25/2015
Formulation Wisdom
• Derive equations as far and as simple as possible.
• Build big/complicated matrices from small/simple matrices.
– This usually requires converting to matrix form early in the formulation.
• Make your formulation documents very detailed.
• A good understanding of the formulation gives you the ability to modify your algorithm or to add/subtract features.
Lecture 1
Slide 13
Implementation Wisdom
• Make a detailed block diagram!
• Include only the equations you will code.
• Add all other steps
– Sources
– Building devices
– Extracting information
– Post‐processing data
– Etc.
Lecture 1
Slide 14
7
11/25/2015
Coding Wisdom
• Work hard to write clean, simple, and well commented code.
– Indent code inside loops, if statements, etc.
– Let linear algebra do the work for you.
– Use lots of comments.
• Match your code to your formulation.
– Try to use the same variables.
• Do not fix your code with incorrect equations.
– Changing signs arbitrarily is a common way to make things work, but you are hiding a problem and possibly creating more.
Lecture 1
Slide 15
Simulation Wisdom #1
Simulate devices in multiple steps using models of increasing complexity.
Avoid the temptation to jump straight to the big, bad, and ugly 3D simulation in all of its glorious complexity.
Model your device with slowly increasing levels of complexity.
You will get to your final answer much faster this way!
Lecture 1
R. C. Rumpf, “Engineering the dispersion and anisotropy in periodic electromagnetic structures,” Solid State Physics 66, 2015.
Slide 16
8
11/25/2015
Simulation Wisdom #2
It must be standard practice to ensure your results are converged.
w
Effective Refractive Index
w  2.0  m
h  0.6  m
a  0.25  m
nrib
a
nsup  1.0
nrib  1.9
h
nsup
ncore
nsub
ncore  1.9
nsub  1.52
neff  1.750, t  1.1 sec
neff  1.736, t  6.1 sec
Grid Resolution
Lecture 1
Slide 17
Simulation Wisdom #3
Those who simulate the most, trust the simulations the least.
Never trust your code or your results.
Benchmark. Benchmark. Benchmark.
Lecture 1
Slide 18
9
11/25/2015
Final Word of Wisdom
Do not EVER share your codes!
Only bad things will happen.
The best thing that can happen
is that you become useless. Instead, offer to simulate devices for
them and make yourself a collaborator.
Lecture 1
Slide 19
General Concepts in Computational EM
Lecture 1
Slide 20
10
11/25/2015
Physical Vs. Numerical Boundary Conditions
Physical Boundary Conditions
Tangential components are continuous
Physical 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 Conditions
Numerical boundary conditions refer to the what is done at the edge of a grid or mesh and how fields outside the grid are estimated.
Lecture 1
Slide 21
Full Vs. Sparse Matrices
Full Matrices







A







Sparse Matrices






















A






















Full matrices have all non‐zero elements.
Sparse matrices have most of their elements equal to zero. They are often more than 99% sparse.
They tend to look banded with the largest numbers running down the main diagonal.
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.
Lecture 1
Slide 22
11
11/25/2015
Integral Vs. Differential Equations (1 of 2)
Integral Equations
 f  x, x dx  g  x 
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.
Differential Equations
df  x 
 f  x  g  x
dx
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.
Lecture 1
Slide 23
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. Lecture 1
Slide 24
12
11/25/2015
Definition of “Convergence”
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!!!
Lecture 1
Slide 25
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 “converged.”
Lecture 1
Slide 26
13
11/25/2015
How Do You Know if Your Model Works?
In many cases, you may not know. 
1. BENCHMARK, BENCHMARK, BENCHMARK
2. CONVERGENCE, CONVERGENCE, CONVERGENCE
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.
Lecture 1
Slide 27
Classification of Methods
Lecture 1
Slide 28
14
11/25/2015
Classification by Size Scale
Low Frequency Methods
High Frequency Methods
0  a
0  a
Structural dimensions are on the order of the wavelength or smaller.
Structural dimensions much larger than the wavelength.
Polarization and the vector nature of the field is important.
Fields can be accurately treated as scalar quantities.
•
•
•
•
•
•
•
•
•
•
Finite‐difference time‐domain
Finite‐difference frequency‐domain
Finite element analysis
Method of moments
Rigorous coupled‐wave analysis
Method of lines
Beam propagation method
Boundary element method
Spectral domain method
Plane wave expansion method
•
•
•
•
•
Ray tracing
Geometric theory of diffraction
Physical optics
Physical theory of diffraction
Shooting and bouncing rays
Lecture 1
Slide 29
Classification by Approximations
Rigorous Methods
A 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 Methods
A 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 Methods
A method is scalar if the vector nature of the field is not accounted for.
• Ray tracing
Lecture 1
Slide 30
15
11/25/2015
Comparison of Method Types
Time‐Domain
Frequency‐Domain
+ resolves sharp resonances
+ handles oblique incidence
+ longitudinal periodicity
+ can be very fast
‐ scales at best NlogN
‐ can miss sharp resonances
‐ active & nonlinear devices
+ wideband simulations
+ scales near linearly
+ active & nonlinear devices
+ easily locates resonances
Fully Numerical
+ better convergence
+ scales better than SA
+ complex device geometry
‐ memory requirements
‐ long uniform sections
Semi‐Analytical
‐ slow for low index contrast
Fourier‐Space
+ moderate index contrast
+ periodic problems
+ very fast and efficient
Structured Grid
+ easy to implement
+ rectangular structures
+ easy for divergence free
‐ less efficient
‐ curved surfaces
‐ volume mesh
‐ spurious solutions
‐ field visualization
‐ formulation difficult
‐ resolving fine details
Unstructured Grid
+ most efficient
+ handles larger structures
+ conforms to curved surfaces
Differential Based
+ sparse matrices
+ easier to formulate
+ easier to implement
‐ convergence issues
‐ scales poorly
‐ complex device geometry
+ very fast & efficient
+ layered devices
+ less memory
Real‐Space
+ high index contrast
+ metals
+ resolving fine details
+ field visualization
‐ longitudinal periodicity
‐ sharp resonances
‐ memory requirements
‐ oblique incidence
‐ difficult to implement
‐ spurious solutions
Integral Based
‐ full matrices
+ surface mesh
+ Very efficient for many structures ‐ more difficult to formulate
‐ more difficult to implement
Lecture 1
Slide 31
Multiphysics Simulations
A multiphysics simulation is one that accounts for multiple simultaneous physical mechanisms at the same time.
•
•
•
•
•
•
•
Electromagnetic
Thermal
Fluids
Motion
Chemical
Acoustic
Optical
Lecture 1
Slide 32
16
11/25/2015
Any Method Can Do Anything
Any method can be made to do anything.
The real questions are:
• What devices and information is a particular method best suited for?
• How much of a “force fit” is it for that method?
Lecture 1
Slide 33
Overview of the Methods
Lecture 1
Slide 34
17
11/25/2015
Transfer Matrix Method (1 of 2)
Transfer matrices are derived that relate the fields present at the interfaces between the layers.
 E x ,trn 
 E x ,2 
 E   T3  E 
 x ,2 
 y ,trn 
T3
 Ex ,trn 
E 
 y ,trn 
 Ex ,2 
 Ex ,1 
 E   T2  E 
 x ,2 
 x ,1 
T2
 E x ,ref 
 Ex ,1 
 E   T1  E 
x
,1


 y ,ref 
 Ex ,2 
E 
 x ,2 
T1
Tglobal  T3T2 T1
 Ex ,1 
E 
 x ,1 
 Ex ,ref 
E 
 y ,ref 
Lecture 1
 Ex ,trn 
 Ex ,ref 
 E   Tglobal  E 
 y ,trn 
 y ,ref 
Transmission through all the layers is described by multiplying all the individual transfer matrices.
Slide 35
Transfer Matrix Method (2 of 2)
This method is good for…
1. Modeling transmission and reflection from layered devices.
2. Modeling layers of anisotropic materials.
Benefits
•
•
•
•
•
•
•
•
•
•
•
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
Drawbacks
• Limited number of geometries it can model.
• Only handles linear, homogeneous and infinite slabs.
• Cannot account for diffraction effects
• Inefficient for transient analysis
Slide 36
18
11/25/2015
Finite‐Difference Frequency‐Domain (1 of 2)
Space is converted to a grid and Maxwell’s equations are written for each point using the finite‐difference method. 


y
y
Ez E z x, y  2 , z  Ez x, y  2 , z

y
y

This large set of equations is written in matrix form and solved to calculate the fields.
Ez E y

  j H x
y
z
Ex Ez

  j H y
z
x
E y Ex

  j H z
x
y
H y
H z

 j Ex
y
z
H x H z

 j E y
z
x
H y H x

 j Ez
x
y
D Ey e z  D Ez e y   jμ xx h x
D Ez e x  D Ex e z   jμ yy h y
D Ex e y  D Ey e x   jμ zz h z
source
Ax  b
x  A 1b
D Hy h z  D Hz h y  jε xx e x
  jε e
D Hz h x  D Hx H
z
yy y
D Hx h y  D Hy h x  jε zz e z
e x 
x  e y 
 e z 
Lecture 1
Slide 37
Finite‐Difference Frequency‐Domain (2 of 2)
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.
Benefits
•
•
•
•
•
•
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
Drawbacks
• Does not scale well to 3D
• Difficult to incorporate nonlinear materials
• Structured grid is inefficient
• Difficult to resolve curved surfaces
• Slow and memory innefficient
Slide 38
19
11/25/2015
Finite‐Difference Time‐Domain (1 of 2)
Fields are evolved by iterating Maxwell’s equations in small time steps.
Maxwell’s equations are enforced at each point at each time step.
Reflection Plane
TF/SF Planes
Spacer Region
Unit cell of real device
Spacer Region
Transmission Plane
Lecture 1
Slide 39
Finite‐Difference Time‐Domain (2 of 2)
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.
Benefits
•
•
•
•
•
•
•
•
•
•
•
•
•
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
Drawbacks
• 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
Slide 40
20
11/25/2015
Transmission‐Line Modeling Method (1 of 2)
Space is interpreted as a giant 3D circuit.
Waves propagating through space are represented as current and voltage in extended circuits.
Also called transmission‐line matrix method (TLM).
Lecture 1
Slide 41
Transmission‐Line Modeling Method (2 of 2)
This method is good for…
1. Modeling big, bad and ugly problems.
2. Hybridizing models with microwave devices.
3. Representing digital waveforms.
Benefits
Drawbacks
• Essentially the same benefits at FDTD and FDFD.
• Excellent for large‐scale simulations. Easily parallelized.
• Excellent for transient analysis.
• No convergence criteria.
• Inherently stable.
• Time‐ and frequency‐domain implementations exist.
• Excellent fit with network theory in microwave engineering.
• Essentially the same drawbacks as FDTD and FDFD.
Lecture 1
Slide 42
21
11/25/2015
Beam Propagation Method (1 of 2)
The beam propagation method (BPM) is a simple method to simulate “forward” propagation through a device. It calculates the field one plane at a time so it does not need to solve the entire solution space at once.
2
A i  μ xx ,i D Hx μ zz1,i D Ex  μ xx ,i ε yy ,i  neff
I
1

 
j z 
j z  i
eiy1   I 
A i 1   I 
Ai  e y
4
4
n
neff
eff

 

Lecture 1
Slide 43
Beam Propagation Method (2 of 2)
This method is good for…
1. Nonlinear optical devices.
2. Devices where reflections and abrupt changes in the field are negligible (i.e. forward only devices)
Benefits
• Simple to formulate and implement (FFT‐BPM is easiest)
• Numerically efficient for faster simulations
• Well established for nonlinear materials (unique for frequency‐
domain method).
• Easily incorporates dispersion
• Excellent for field visualization
• Error mechanisms are well understood
• Well suited for waveguide circuit simulation
Lecture 1
Drawbacks
• Not a rigorous method
• Limited in the physics it can handle
• Typically uses paraxial approximation
• Typically neglects backward reflections
• FFT‐BPM is slower, less stable, and less versatile than FDM‐
BPM
Slide 44
22
11/25/2015
Method of Lines (1 of 2)
source
reflected
x
y
BCs
The method of lines is a semi‐analytical method.
BCs
Modes are computed in the transverse plane for each layer and propagated analytically in the z-direction.
Boundary conditions are used to matched the fields at the interfaces between layers.
BCs
Transmission through the entire stack of layers is then known and transmitted and reflected fields can be computed.
Lecture 1
transmitted
z
BCs
Slide 45
Method of Lines (2 of 2)
This method is good for…
1. Long devices.
2. Long devices with metals.
Benefits
• Excellent for longitudinally periodic devices
• Rigorous method
• Excellent for devices with high index contrast and metals
• Good for resonant structures
• Less numerical dispersion than fully numerical methods
• Easier field visualization than RCWA
Lecture 1
Drawbacks
• Scales very poorly in the transverse direction
• Cumbersome method for field visualization
• Less efficient than RCWA for dielectric structures.
• Rarely used in 3D analysis, but this may change with more modern computers
Slide 46
23
11/25/2015
Rigorous Coupled‐Wave Analysis (1 of 2)
Field in each layer is represented as a set of plane waves at different angles.
Plane waves describe propagation through each layer.
Layers are connected by the boundary conditions.
Lecture 1
Slide 47
Rigorous Coupled‐Wave Analysis (2 of 2)
This method is good for…
1. Modeling diffraction from periodic dielectric structures
2. Periodic devices with longitudinal periodicity
Benefits
•
•
•
•
•
•
•
•
Excellent for modeling diffraction from periodic dielectric structures.
Extremely fast and efficient for all‐dielectric structures with low to moderate index contrast
Accurate and robust
Unconditionally stable
Thickness of layers can be anything without numerical cost
Excellent for longitudinally periodic structures.
Excellent for structures large in the longitudinal direction.
Easily incorporates polarization and angle of incidence.
Lecture 1
Drawbacks
• Scales poorly in transverse dimensions.
• Less efficient for high dielectric contrast and metals due to Gibb’s phenomenon.
• Poor method for finite structures.
• Slow convergence if fast Fourier factorization is not used.
Slide 48
24
11/25/2015
Plane Wave Expansion Method (1 of 2)
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.
Lecture 1
Slide 49
Plane Wave Expansion Method (2 of 2)
This method is good for…
1. Analyzing unit cells
2. Calculating photonic band diagrams and effective material properties.
Benefits
• Excellent for all‐dielectric unit cells
• Fast even for 3D
• Accurate and robust
• Rigorous method
Lecture 1
Drawbacks
• Scales poorly.
• Weak method for high dielectric contrast and metals.
• Limited to modal analysis.
• Cannot model scattering.
• Cannot incorporate dispersion.
Slide 50
25
11/25/2015
Slice Absorption Method (1 of 2)
Virtually any method that converts Maxwell’s equations to a matrix equation can order the matrix to give it the following block tridiagonal form.
This allows the problem to be solved one slice at a time.
Lecture 1
Slide 51
Slice Absorption Method (2 of 2)
This method is good for…
1. Modeling structures with high volumetric complexity
2. Modeling finite size structures (i.e. not infinitely periodic)
Benefits
• Excellent for modeling devices with high volumetric complexity
• Easily incorporates dispersion
• Easily incorporate polarization and oblique incidence
• Potential for transverse devices
• Excellent for finite size devices
• Excellent framework to hybridize different methods.
• Transverse sources
• Stacking in three dimensions.
Lecture 1
Drawbacks
• New method and not well understood.
Slide 52
26
11/25/2015
Finite Element Method (1 of 2)
Step 1: Describe Structure
Step 2: Mesh Structure
This is a VERY important and involved step.
1  1.0
r  1.50
 2  2.5
Step 3: Build Global Matrix
Step 4: Solve Matrix Equation
Incorporate a source.
Iterate through each element to populate the global matrix.
Ax  b
Calculate field.
x  A 1b
Ax  0
Lecture 1
Slide 53
Finite Element Method (2 of 2)
This method is good for…
1. Modeling volumetrically complex structures in the frequency‐
domain.
Benefits
Drawbacks
• Very mature method
• Tedious to implement
• Excellent representation of • Requires a meshing step
curved surfaces
• Unstructured grid is highly efficient
• Unconditionally stable
• Scaling improved with domain decomposition
Lecture 1
Slide 54
27
11/25/2015
Method of Moments (1 of 2)
f   an v n
Lf  g
a
n
 a Lv
n
n
n
n
 v1 , Lv1

 v 2 , Lv1

Galerkin Method
Integral Equation
• Converts a linear equation to a matrix equation
•Usually uses PEC approximation
•Usually based on current
g
n
v m , Lv n  v m , g
v1 , Lv 2
v 2 , Lv 2
 a   v , Lg
 1   1
  a1   v 2 , Lg
    

   
 aN   v N , Lg






Ezinc 
j


L2
L 2

 2  e jkr
I z  z  k 2  2 
dz 
z  4 r

The Method of Moments
i1
v1
i2
v2
i3
v3
i4
v4
i5
i6
i7
v5
v6
v7
 z11
z
 21
 z31

 z41
 z51

 z61
z
 71
z12
z22
z13
z23
z14
z24
z15
z25
z16
z26
z32
z42
z33
z43
z34
z44
z35
z45
z36
z46
z52
z62
z72
z53
z63
z73
z54
z64
z74
z55
z65
z75
z56
z66
z76
z17   i1   v1 
z27  i2  v2 
z37  i3   v3 
   
z47  i4   v4 
z57  i5   v5 
   
z67  i6   v6 
z77  i7  v7 
Lecture 1
Slide 55
Method of Moments (2 of 2)
This method is good for…
1. Modeling metallic devices at radio frequencies
2. Modeling large‐scale metallic structures at radio frequencies
Benefits
• Extremely efficient analysis of metallic devices
• Full wave
• Very fast
• Excellent scaling using the fast multipole method
• No boundary conditions
• Simple implementation
• Mature method with lots of literature
• Can by hybridized with FEM
Lecture 1
Drawbacks
• Not a rigorous method
• Poor method for incorporating dispersion and dielectrics
• Long a tedious formulation
• Inefficient for volumetrically complex structures
Slide 56
28
11/25/2015
Boundary Element Method (1 of 2)
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.
Governing equation exists only at the boundary of a device so many fewer elements are needed.
5000 elements
400 elements
Lecture 1
Slide 57
Boundary Element Method (2 of 2)
This method is good for…
1. Modeling large devices with simple geometries.
2. Modeling scattering from homogeneous blobs.
Benefits
• 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
Drawbacks
•
•
•
•
Tedious to implement
Requires a meshing step
Not usually a rigorous method
Inefficient for volumetrically complex geometries
Slide 58
29
11/25/2015
Discontinuous Galerkin Method (1 of 2)
The discontinuous Galerkin method (DGM) combines features of the finite element and finite‐volume framework to solve differential equations. Lecture 1
Slide 59
Discontinuous Galerkin Method (2 of 2)
This method is good for…
1. Solving very complex equations.
2. Modeling very electrically large structures.
3. Time‐domain finite‐element method.
Benefits
• Mesh elements can have any arbitrary shape.
• Fields may be collocated instead of staggered.
• Inherently a parallel method.
• Easily extended to higher‐order of accuracy.
• Allows explicit time‐stepping
• Low memory consumption (no large matrices)
Lecture 1
Drawbacks
•
•
•
•
Tedious to implement
Requires a meshing step
Not usually a rigorous method
Inefficient for volumetrically complex geometries
Slide 60
30