Fdtd Numerical Methods Lumerical
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1
Numerical Methods
FDTD Solutions
© 2012 Lumerical Solutions, Inc.
Outline
Application area review
Ray optics vs. wave optics
Finite Difference Time Domain method
: Yee cell
: Uniform, graded, and conformal meshing
: Obtaining frequency domain results from a time domain method
Boundary conditions
Coherence and polarization in FDTD
Accuracy and convergence testing
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Application area overview
© 2012 Lumerical Solutions, Inc.
Our products can accurately simulate many technologies
Photonic crystals Bandstructure Plasmonics CMOS Image sensors
Nanoparticles Solar cells Resonators LED/OLEDs
Grating devices Lithography Metamaterials Waveguides
© 2012 Lumerical Solutions, Inc.
Wave optics vs. ray tracing
Question: What features are common among these applications?
(When do you need to use FDTD Solutions?)
Answer: Feature sizes are on the order of the wavelength.
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Wave optics vs. ray tracing
Source = 0.55 um
4 um
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Wave optics vs. ray tracing
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n=1.5
n=1
Incident light
Snell’s Law gives
c: 41.8
R=100%
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Wave optics vs. ray tracing
20 um
= 0.4 um = 4 um
rayvswave_700THz.mpg rayvswave_70THz.mpg
© 2012 Lumerical Solutions, Inc.
Wave optics vs. ray tracing
Conclusion: You need FDTD Solutions when feature sizes are on the order of a wavelength
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© 2012 Lumerical Solutions, Inc.
Overview of FDTD method
TOPICS
Maxwell equations
Yee cell
Time domain technique
Fourier transform
Sources and Monitors
Computational requirements
2D vs. 3D
Advantages of the FDTD method
© 2012 Lumerical Solutions, Inc.
Maxwell’s equations
Name Differential form Integral form
Gauss’ law
Gauss' law for magnetism
(absence of magnetic monopoles):
Faraday’s law of induction:
Ampère’s law
(with Maxwell's extension):
Describe the behavior of both the electric and magnetic fields, as well as their interactions
with matter.
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© 2012 Lumerical Solutions, Inc.
Maxwell’s equations
Symbol Meaning SI Unit of Measure
electric field volt per meter
magnetic field
also called the auxiliary field
ampere per meter
electric displacement field
also called the electric flux density
coulomb per square meter
magnetic flux density
also called the magnetic induction
also called the magnetic field
tesla, or equivalently,
weber per square meter
free electric charge density,
not including dipole charges bound in a
material
coulomb per cubic meter
free current density,
not including polarization or magnetization
currents bound in a material
ampere per square meter
© 2012 Lumerical Solutions, Inc.
H
Wave Optics – Free space plane wave
In vacuum, without charges (=0) or currents (J=0)
Maxwell’s equations have a simple solution in terms of traveling sinusoidal plane waves
The electric and magnetic field directions are orthogonal to one another and the direction of travel k
The E, H fields are in phase, traveling at the speed c
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In linear materials, the D and B fields are related to E and H by:
where: ε is the electrical permittivity of the material, and μ is the permeability of the material In FDTD Solutions, we typically deal with the electrical
permittivity only μ=μ0 is the permeability of the free space
Wave Optics - Simple materials
HB
ED
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How the FDTD method works
E and H are discrete in time
12123
211
nnn
nnn
Et
HH
Ht
EE
0E
21H
1E
23H …
2nd order accurate in time: error ~ t2
tntn HtHEtE
)2
1()()(
The basic FDTD time-stepping relation:
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© 2012 Lumerical Solutions, Inc.
Maxwell equations on a mesh
Yee cell
E and H are discrete in space
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The Yee cell
Z
Y
X
Hy
Ez
Hz
Hx
Ey
Ex
(x,y,z)
Yee
cell
Kane Yee (1966). "Numerical solution of initial boundary value problems involving Maxwell's
equations in isotropic media". Antennas and Propagation, IEEE Transactions on 14: 302–
307.
Spatially stagger the vector components of the E-field and H-field about
rectangular unit cells of a Cartesian computational grid.
2nd order accurate in space
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© 2012 Lumerical Solutions, Inc.
How are dielectric properties treated? The meshed structure The true structure
)(r
kji ,,
X
Y
Discrete
mesh
Conformal mesh technology
© 2012 Lumerical Solutions, Inc.
Conformal mesh technology
Interfaces are a problem for Maxwell’s equations on a discrete mesh
: The fields can be discontinuous at interfaces
: The positions of the interface cannot be resolved to better than dx
: Staircasing effects
Solutions
: Graded mesh (reduce mesh size near interfaces)
: Conformal mesh technology
: Combination of both
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Conformal mesh technology
Finite difference methods cannot resolve interface positions or layer thicknesses to better than the mesh size
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Conformal mesh technology
Staircasing can lead to unwanted effects
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Conformal mesh technology
Conformal mesh technology uses an integral solution to Maxwell’s equations near interfaces
: Lumerical’s approach can handle arbitrary dispersive media
: More advanced than well known approaches such as the Yu-Mittra model for PEC
© 2012 Lumerical Solutions, Inc.
Example: Yu-Mittra approach for PEC
Ex
Ex
Ey Ey
Bz
PEC
x
y
1C
C
z
ldE
ldEt
B
C
C1
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© 2012 Lumerical Solutions, Inc.
Conformal mesh technology
Application Simulation mesh Best Solution
Multilayer Conformal mesh can allow you to use the default simulation mesh.
Mie scattering We combine conformal meshing with graded meshing, but the mesh is not as fine as with staircasing.
Waveguide couplers Conformal mesh can allow you to use the default simulation mesh.
© 2012 Lumerical Solutions, Inc.
Conformal mesh technology
There are 3 variants used
: Conformal variant 0 (default)
• Conformal mesh applied to all materials except metals and PEC (Perfect Electrical Conductor)
• Metals are materials with real() < 1
• This is the best setting without doing convergence testing
: Conformal variant 1
• Conformal mesh applied to all interfaces
: Conformal variant 2
• Yu-Mittra model for PEC applied to PEC and metals
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© 2012 Lumerical Solutions, Inc.
Conformal mesh technology
Simple rule of thumb
: Use the default conformal variant 0
Possible exceptions
: If the simulation diverges
• Test using staircasing
: If you are studying plasmonic effects
• Consider using conformal variant 1
• Do some careful convergence testing and read the page at http://docs.lumerical.com/en/fdtd/user_guide_testing_convergence.html
© 2012 Lumerical Solutions, Inc.
Computational Resource Requirements
3D 2D
Memory Requirements
~ (/dx)3 ~ (/dx)2
Simulation Time ~ (/dx)4 ~ (/dx)3
How do the required computational resources scale with grid size?
dx=/10
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Controlling the mesh
We provide a mesh accuracy slider that ranges from 1-8
The mesh algorithm then targets a minimum
: /dx = 6, 10, 14, 18, 22, 26, 30, 34
Note that =0/n, where n is the refractive index
Generally, /dx=10 (mesh accuracy 2) is considered reasonable for many FDTD simulations and /dx ~ 20 (mesh accuracy 4,5) is considered very high accuracy
It is still often worth running at /dx=6 (mesh accuracy 1) for initial simulations
The default is /dx=10 (mesh accuracy 2)
We will see later why /dx = 1/(k dx) is the right quantity to consider
The meshing can be fully customized : Typically, we use mesh override regions to force a particular mesh in a given region, which
we recommend
: We can fully customize the mesh algorithm details if desired but this is not recommended
© 2012 Lumerical Solutions, Inc.
Tip
Use a coarse mesh for simulations
: Memory scales as 1/dx3
: Simulation time scales as 1/dx4
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© 2012 Lumerical Solutions, Inc.
2D vs 3D
FDTD simulations can be run in 2D or 3D
2D: Structure is infinite in z direction 3D 2D: Structure is infinite in z direction
Z X
Y
3D
Z
X
Y
© 2012 Lumerical Solutions, Inc.
2D vs 3D
2D assumes
We get perfect separation of Maxwell’s equations into two independent sets of equations
: Transverse Electric (TE) : involves only Ex, Ey, Hz
: Transverse Magnetic (TM) : involves only Ez, Hx, Hy
The terms “TE” and “TM” are no longer used in FDTD Solutions. Use the blue arrows of sources to determine the Electric field polarization.
0
z
H
z
E
z
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© 2012 Lumerical Solutions, Inc.
FDTD is a time domain technique!
The simulation is run to solve Maxwell’s equations in time to obtain E(t) and H(t)
Most users want to know the field as a function of wavelength, E(), or equivalently frequency, E(w)
The steady state, continuous wave (CW) field E(w) is calculated from E(t) by Fourier transform during the simulation.
SimT
ti dttEeE0
)()(
ww
See section on Units and Normalization of Reference Manual for more details: http://www.lumerical.com/fdtd_online_help/ref_fdtd_units_units_and_normalization.php
© 2012 Lumerical Solutions, Inc.
FDTD is a time domain technique!
Normalize E(w) to the source spectrum and we can obtain the impulse response of the system!
Eimpulse is a response of the system : It is independent of the source pulse used
: It is the CW, or monochromatic response
Ideally s(w)=1, meaning that s(t) is a delta function : In practice, we use a very short pulse
SimT
ti
impulse dttEes
E0
)()(
1)(
w
ww
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Advantages of the FDTD method
Advantages
Few inherent approximations = accurate
A very general technique that can deal with many types of problems
Arbitrarily complex geometries
One simulation gives broadband results
© 2012 Lumerical Solutions, Inc.
Example, ring resonator
4.4 m
through
drop
~1.55m
n=2.915
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Example: waveguide ring resonator
time
1
2
Fourier
Transform
ring.mpg
© 2012 Lumerical Solutions, Inc.
Example: waveguide ring resonator
through
drop
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Boundary conditions
PML
• Absorbs incident fields
• Use PML when the fields are meant to propagate away from the simulation region
Metal
• Perfect metal boundary
• 100% reflection, 0% absorption
Periodic
• For periodic structures
• The structure AND fields must be periodic
• Typically used with the plane wave source
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Symmetric/Anti-Symmetric boundaries
Symmetric/Anti-Symmetric
• Can reduce memory/time for symmetric simulations
• Both structure AND fields must be symmetric
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Symmetric/Anti-Symmetric boundaries
Non zero components of the electric and magnetic fields at symmetric/anti-symmetric boundaries
© 2012 Lumerical Solutions, Inc.
Symmetric/Anti-Symmetric boundaries
How the different electric and mangetic components behave under different symmetry conditions
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Symmetric/Anti-Symmetric boundaries
Symmetry/Anti-Symmetry can be used for periodic structures
: Consider file from far field section, farfield4.fsp
© 2012 Lumerical Solutions, Inc.
Symmetric/Anti-Symmetric boundaries
Symmetry/Anti-Symmetry can be used for
periodic structures
: Image the near and far fields at 1.3 microns
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Symmetric/Anti-Symmetric boundaries
Symmetry/Anti-Symmetry can even be used for periodic structures
: We can get the same results but the simulation runs faster
: Far field projections can still be done
: The data is automatically “unfolded” so we see the full image
Actual simulation
GUI and script results
© 2012 Lumerical Solutions, Inc.
Bloch boundary conditions
Bloch
• For periodic structures (similar to periodic)
• Use Bloch when the plane wave source is at non-normal incidence
• Use Bloch for bandstructure simulations.
• Bloch boundaries conditions ensure that E(x+a) = exp(ika)*E(x)
a is the simulation span
k is the Bloch vector
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Bloch boundary conditions
• Periodic boundaries are just a special case of Bloch boundaries (k=0)!
• Bloch BC requires 2x memory and simulation time
• When using Bloch boundaries for non-normal plane waves, you must check the following
© 2012 Lumerical Solutions, Inc.
Bloch boundary conditions
Consider the difference between correct and incorrect k settings for a plane wave in free space
Correct
Incorrect
usr_bloch_movie_2.mpg
usr_bloch_movie_3.mpg
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© 2012 Lumerical Solutions, Inc.
Bloch boundary conditions
When the setting are correct, we can study periodic structure illuminated by plane waves at angles
: We can calculate far field projections • Assume periodicity the same as with periodic boundary conditions
: We can calculate grating order efficiencies, the same as with periodic boundary conditions
: We must be cautious about the PML performance when the angle of incidence is steep
• Sometimes, we need to increase the number of layers of PML to get accurate results
© 2012 Lumerical Solutions, Inc.
Coherence
Temporal incoherence : The phase, j, of the light shifts randomly over time, on a time
scale tc
: Even without random phase shifts, if the light is not
monochromatic, it is incoherent
: In either case, the coherence length of the system is often much longer than any standard simulation time (tc >> T)
• It is not efficiency in general to directly model incoherence
• It is not possible to perform near to far field projections of incoherent results in the near field
s
sT
ttEtE
c
11
15
0
10
102
))(cos()(
t
w
jw
1 fct
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Coherence
Temporal incoherence
: The frequency domain monitors of FDTD Solutions calculate the monochromatic response of the system
• There are some advanced features to directly extract incoherent results where the value of f ~ 1/tc can be specified, see http://docs.lumerical.com/en/fdtd/user_guide_spectral_averaging.html for details
: In general, the best approach is to calculate the monochromatic (or CW) response first, then calculate the incoherent result with
• Where W(w) is the spectrum of the physical source used
wwww dEWE22
0 )()()(
© 2012 Lumerical Solutions, Inc.
Coherence
Temporal incoherence example
: Reflection of 50nm of silver on 500nm of silicon
CW or monochromatic response
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Coherence
From one simulation, we can calculate the incoherent results at many wavelengths
wwww dEWE22
0 )()()(
© 2012 Lumerical Solutions, Inc.
Coherence
Spatial incoherence can be simulated using the ergodic principle
Each ensemble consists of dipoles with randomized phase, amplitude, position, orientation and pulse time
A large number of ensembles must be averaged : There is a statistical error associated that decreases with increased number of ensembles
Typically 50 to 100 simulations is a minimum requirement and more may be required : See Chan, Soljačić, and Joannopoulos, “Direct calculation of thermal emission for three-
dimensionally periodic photonic crystal slabs” http://pre.aps.org/abstract/PRE/v74/i3/e036615 for discussion
It is often erroneously assumed that one simulation is enough : This may or may not be true, but it depends on the details of what is being recorded
+ +……+
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Coherence
Spatial coherence
: The same results can be reconstructed from spatially coherent results
• There is no statistical error
• The total number of simulations is typically less than is required by ensemble averaging
: Example for an ensemble of incoherent dipole emitters
© 2012 Lumerical Solutions, Inc.
Coherence
Incoherent sources are dealt with from the coherent results.
For example, consider 2 dipoles
log(|E(w)|2) Time domain
dipole_coherent.mpg
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Coherence
Incoherent dipoles, 2 simulations
Simulation 1
Simulation 2
log(|E1|2)
log(|E2|2)
log(|E1|2+|E2|
2)
log(|E1+E2|2)
dipole_incoherent1.mpg
dipole_incoherent1.mpg
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Polarization
FDTD simulations have well defined polarization
Unpolarized results are obtained by adding the results of 2 orthogonal polarization simulations incoherently
½( |E1|2 + |E2|
2 ) = <|E|2>unpolarized
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Dipoles
Incoherent, isotropic dipoles require the sum of 3 orthogonal polarization states, summed incoherently
= 1/3 { 2
E 2
pyE2
pxE 2
pzE
+ + }
The above theory greatly simplifies LED/OLED simulations
© 2012 Lumerical Solutions, Inc.
Coherence
We often have to find incoherent results
: Temporal incoherence
: Spatial incoherence
: Unpolarized light
: Anisotropic dipole emitters
: and more...
It is generally most efficient to reconstruct the incoherent results from the coherent results
: There is no statistical error
: The total number of simulations is typically less than a brute force approach
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Accuracy and convergence testing
Dispersion relation in FDTD (in free space)
When we have
Typical target for accuracy might be
Courant stability limit
2222
2sin
1
2sin
1
2sin
1
2sin
1
zk
z
yk
y
xk
x
t
tc
zyxw
0
0
0
0
z
y
x
t
ckkkkc zyx 222w
103.0~20
2
2
x
xkx
3c
xt
© 2012 Lumerical Solutions, Inc.
Accuracy and convergence testing
In reality, many factor beyond the mesh size affect the accuracy of FDTD
For example
: The proximity of the PML
: PML reflections
: The multi-coefficient model fit
• How well do you know n,k for your materials? What is the experimental error? Do you expect the same n,k as other authors?
: and more...
Please read http://docs.lumerical.com/en/fdtd/user_guide_testing_convergence.html for a detailed list and steps on doing convergence testing
Do not waste time making the mesh size really small without considering the other factors
: And some of them can get worse as the mesh size gets small!
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© 2012 Lumerical Solutions, Inc.
Accuracy and convergence testing
With care and computing power, almost any accuracy can be achieved
The more important limit for real devices is often how well you know the geometry and the material properties (n,k)
© 2012 Lumerical Solutions, Inc.
Review and tips
What mesh size should I use? : “mesh accuracy” of 1 or 2 for initial setup (faster)
: Use “mesh accuracy” of 2-4 for final simulations
: “mesh accuracy” 5-8 is almost never necessary • Use mesh overrides instead for most applications
How long a simulation time should I use? : Start with longer simulations times and let the “auto-shutoff”
feature find out when you can stop the simulation
: Check with point time monitors
How do I check the memory requirements?
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Review and tips
What are some tricks for speeding up FDTD Solutions, and reducing the memory requirements?
Avoid simulating homogeneous regions with no structure : Use far field projections instead
Use symmetry where possible : Gain factors of 2, 4 or 8
Use periodicity where possible : Gain factors of 100s or 1000s
Use a coarse mesh (use a refined mesh for final simulations) : Start with “mesh accuracy” of 1 instead of 2
• gives 8 times faster simulation • 5 times less memory • within 10-20% accuracy in general
: User mesh accuracy of 2-4 for final simulation : Use mesh override regions for local regions of fine mesh
Reduce amount of data collected
: Down sample monitors spatially
: Record fewer frequency points in frequency monitors
: Record only the necessary field components
© 2012 Lumerical Solutions, Inc.
Review and tips
Questions and answers…
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Getting help
Technical Support
: Email: support@lumerical.com
: Online help: docs.lumerical.com/en/fdtd/knowledge_base.html
• Many examples, user guide, full text search, getting started, reference guide, installation manuals
: Phone: +1-604-733-9006 and press 2 for support
Sales information: sales@lumerical.com
Find an authorized sales representative for your region:
: www.lumerical.com and select Contact Us
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