Fdtd Numerical Methods Lumerical

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

2

Application area overview


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


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.

3



Source = 0.55 um

4 um



45

n=1.5

n=1

Incident light

Snell’s Law gives

c: 41.8

R=100%

4



20 um

= 0.4 um = 4 um

rayvswave_700THz.mpg rayvswave_70THz.mpg



Conclusion: You need FDTD Solutions when feature sizes are on the order of a wavelength

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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


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.

6


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


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

http://en.wikipedia.org/wiki/Electric_field

http://en.wikipedia.org/wiki/Magnetic_field

http://en.wikipedia.org/wiki/Electric_displacement_field

http://en.wikipedia.org/wiki/Magnetic_field_density

http://en.wikipedia.org/wiki/Electric_charge

http://en.wikipedia.org/wiki/Current_density

7


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


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:

http://en.wikipedia.org/wiki/Permittivity

http://en.wikipedia.org/wiki/Permeability_(electromagnetism)

http://en.wikipedia.org/wiki/Permeability_(electromagnetism)

8


Maxwell equations on a mesh

Yee cell

E and H are discrete in space


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

http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1138693

http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1138693

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How are dielectric properties treated? The meshed structure The true structure

)(r

kji ,,

X

Y

Discrete

mesh

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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Finite difference methods cannot resolve interface positions or layer thicknesses to better than the mesh size



Staircasing can lead to unwanted effects

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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


Example: Yu-Mittra approach for PEC

Ex

Ex

Ey Ey

Bz

PEC

x

y

1C

C

z

ldE

ldEt

B

C

C1

12



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.



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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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


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

http://docs.lumerical.com/en/fdtd/user_guide_testing_convergence.html


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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


Tip

Use a coarse mesh for simulations

: Memory scales as 1/dx3

: Simulation time scales as 1/dx4

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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


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

16


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


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

http://www.lumerical.com/fdtd_online_help/ref_fdtd_units_units_and_normalization.php

17


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


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


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


Symmetric/Anti-Symmetric boundaries

Symmetric/Anti-Symmetric

• Can reduce memory/time for symmetric simulations

• Both structure AND fields must be symmetric

20



Non zero components of the electric and magnetic fields at symmetric/anti-symmetric boundaries



How the different electric and mangetic components behave under different symmetry conditions

21



Symmetry/Anti-Symmetry can be used for periodic structures

: Consider file from far field section, farfield4.fsp



Symmetry/Anti-Symmetry can be used for

periodic structures

: Image the near and far fields at 1.3 microns

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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


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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• 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



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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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


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

25


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 )()()(


Coherence

Temporal incoherence example

: Reflection of 50nm of silver on 500nm of silicon

CW or monochromatic response

http://docs.lumerical.com/en/fdtd/user_guide_spectral_averaging.html

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Coherence

From one simulation, we can calculate the incoherent results at many wavelengths

wwww dEWE22

0 )()()(


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

+ +……+

http://pre.aps.org/abstract/PRE/v74/i3/e036615

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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


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


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

29


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


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

30



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



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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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)


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


Review and tips

Questions and answers…

33


Getting help

Technical Support

: Email: [email protected]

: 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: [email protected]

Find an authorized sales representative for your region:

: www.lumerical.com and select Contact Us

mailto:[email protected]

http://docs.lumerical.com/en/fdtd/knowledge_base.html



mailto:[email protected]

http://www.lumerical.com/

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