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Material Modeling
FDTD Solutions
© 2011 Lumerical Solutions, Inc.
Outline
� Dispersive materials in a time domain method
� Material model review
: How to choose the correct model
� Anisotropic materials
� Advanced tips
: Getting better material fits
: Understanding the mesh order
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© 2011 Lumerical Solutions, Inc.
Dispersive materials in the time domain
� The fields in FDTD are real
: exception for Bloch boundaries
� Well-known frequency domain relationship
� FDTD is a time domain technique: relationship?
)()()( ωωεω EDrr
=
∫ ′′−′=∗=
t
tdtttEtEttD
0
)()()()()( εεrrr
© 2011 Lumerical Solutions, Inc.
Dispersive Materials
� FDTD Solutions supports the following models: Dielectric
: Sampled Material
: PEC (Perfect Electrical Conductor)
: Analytic
: (n,k) Dielectric
: Conductive
: Plasma
: Debye
: Lorentz
: Kerr nonlinear
: PDLC (Plasma – Debye – Lorentz – Conductive)
• Backwards compatibility mode only
: Sellmeier
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© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Dielectric Material� There is no dependence on frequency!
� Restriction: n >= 1
constant)(2 == nr ωε
© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Sampled Material� There is experimental (or theoretical, or user’s own) data for (n,k) as a function of wavelength: From built-in material database: From your own data
� FDTD Solutions automatically fits the data over the wavelength range of your sources: Multi-coefficient model: You choose
• The number of coefficients• The fit tolerance
: More coefficients takes more time and memory
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© 2011 Lumerical Solutions, Inc.
� Example GaAs, 12 coefficients
Multi-coefficient model
GaAs, 200-800nm
© 2011 Lumerical Solutions, Inc.
Auto-fitting of materials
� Fitting your (proprietary) data
: Example, representative data of color filters
Red filter
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© 2011 Lumerical Solutions, Inc.
Auto-fitting of materials
� Fitting your (proprietary) data
: Example, representative data of color filters
Blue filter
© 2011 Lumerical Solutions, Inc.
Auto-fitting of materials
� Metals are not necessarily simple plasma materials
Chromium
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© 2011 Lumerical Solutions, Inc.
Some tips for material fitting� Check the imaginary data to avoid “fake” gain
� A fixed wavelength range for fitting can be specified
� Imaginary part of permittivity can be overweighted or underweighted
© 2011 Lumerical Solutions, Inc.
Auto-fitting of materials
� Built in material data with auto-fitting
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© 2011 Lumerical Solutions, Inc.
Working in simulations
� Simple tests: FDTD vs theory for a 50 nm thick span of Si
: Analytic result for R and T can be easily calculated
© 2011 Lumerical Solutions, Inc.
Working in simulations
� Simple tests: FDTD vs theory for a 50 nm thick span of Si
: multi-coefficient auto-fit to Si
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© 2011 Lumerical Solutions, Inc.
Working in simulations
� Simple tests: FDTD vs theory for a 50 nm thick span of Si
: Calculate the theoretical curve from the fit
: Average difference = 0.001
: Max difference = 0.008
© 2011 Lumerical Solutions, Inc.
Working in simulations� Simple tests: FDTD vs theory for a 50 nm thick span of Si
: Calculate the theoretical curve from the original material data
: Average difference = 0.0023
: Max difference = 0.031
: Results come from one simulation
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© 2011 Lumerical Solutions, Inc.
Working in simulations
� Compare Lorentz model with multi-coefficient model
Lorentz model Multi-coefficient model
© 2011 Lumerical Solutions, Inc.
Dispersive Materials(n,k) Dielectric� FDTD Solutions chooses the simplest dispersive model that can create the
correct permittivity (real and imaginary) at the center frequency of your simulation: Perfect for single wavelength simulations
� At other frequencies, the value of (n,k) will be different: In reality, physical materials with loss are also dispersive
• More accurate broadband results can be obtained using actual material data and a Sampled Material
: Use the Materials Explorer to see the difference between target (n,k) and actual (n,k) for broadband simulations
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© 2011 Lumerical Solutions, Inc.
Dispersive Materials
PEC (Perfect Electrical Conductor)
: Equivalent to a conductor with
0=Er
∞→σ
© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Analytic : The analytic material model allows the user to enter an equation for the real and imaginary part of the permittivity or refractive index which can depend on a set of variables.
: A common use example for the analytic material model is for materials such as AlxGa1-xAs where the refractive index is a function of x. The analytic material makes it easy to change x in between simulations.
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© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Conductive
f
ir
⋅=
+= ∞
πω
ωε
σεωε
2
)(0
© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Debye
( )f
ic
cdebye
r
⋅=
−
⋅+= ∞
πω
ων
νεεωε
2
)(
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© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Plasma
( )f
i c
p
r
⋅=
+−= ∞
πω
ωνω
ωεωε
2
)(
2
© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Lorentz
( )f
i
lorentzr
⋅=
−−
⋅+= ∞
πω
ωωδω
ωεεωε
2
2)(
2
0
2
0
2
0
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© 2011 Lumerical Solutions, Inc.
Dispersive Materials
PDLC (Plasma – Debye – Lorentz –Conductive): Combination of Plasma, Debye, Lorentz and Conductive models
: Backwards compatibility mode only • with FDTD Solutions 5.1 and before
: you cannot create one of these materials with FDTD 6.0 or above• For more these types of complex, dispersive materials, it is best to use Sampled Materials
© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Kerr nonlinear (instantaneous)
)()()(
)()()(
2)3(
2)3()1(
tEtEtD
tEtEtP
ro
o
rrr
rrr
+=
+=
χεε
χχε
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© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Sellmeier
: fs is center frequency of the sources in your simulation: The resulting material is not dispersive!: Should be used for single wavelength simulations only: Typically used in MODE Solutions to calculate fiber dispersion
s
s
s
s
s
s
s
s
f
c
C
B
C
B
C
BAconstn
=
−+
−+
−+===
λ
λ
λ
λ
λ
λ
λε
3
2
2
3
2
2
2
2
1
2
2
1
1
2
© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Check your material models before running simulations!
Adjust number of coefficients and tolerance if necessary
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© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Beware of errors in the data, and using too many coefficients
Removing noise from data and correcting errors will improve the fit
© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Example Mie Scattering, gold sphere
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© 2011 Lumerical Solutions, Inc.
Dispersive Materials
Example Mie Scattering
: mesh size 1 nm
© 2011 Lumerical Solutions, Inc.
Dispersive Materials
� What built-in materials are available?
� What material model should I use?
� How do I define my own dispersive materials?
� Cautions about divergence!!: Some models created by the Sampled Material auto-fit will diverge. Can be fixed by
• Reducing the “dt stability factor”
• Reducing “PML sigma” and increasing “PML Kappa” where materials intersect the PML boundary condition, or preventing materials from intersecting the PML
• See docs.lumerical.com/en/fdtd/user_guide_diverging_simulations.html for more details
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© 2011 Lumerical Solutions, Inc.
Anisotropic Materials
TOPICS
� Anisotropic materials
: introduction
: in FDTD Solutions
� Example
© 2011 Lumerical Solutions, Inc.
Anisotropic Materials
� Anisotropic materials have
� Where εij is a nine element tensor
=
333231
232221
131211
εεε
εεε
εεε
ε ij
jiji ED ε=
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© 2011 Lumerical Solutions, Inc.
Anisotropic Materials
� FDTD Solutions currently supports a diagonal permittivity tensor
=
z
y
x
ε
ε
ε
ε
00
00
00
© 2011 Lumerical Solutions, Inc.
Anisotropic Materials
� Set any material to anisotropic and you can enter values for each axis – or import data for each axis
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© 2011 Lumerical Solutions, Inc.
� Here’s how you enter it in the “index” field of any object
Anisotropic Materials
Index fieldnx ny nz
1.5 1.5 1.5 1.5
1.5;1.6 1.5 1.6 1.6
1.5;1.6;1.7 1.5 1.6 1.7
1.5+x/100;1.3;1.5+y/100 1.5+x/100 1.3 1.5+y/100
© 2011 Lumerical Solutions, Inc.
Anisotropic Materials
� Example: open the file anisotropy1.fsp
� n = 2;2;1.1
TE (Hz) TM (Ez)
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© 2011 Lumerical Solutions, Inc.
Anisotropic Materials
� What kind of anisotropy is available in FDTD Solutions?
� How do I define anisotropic materials?
© 2011 Lumerical Solutions, Inc.
Advanced tips
� Many fits with large numbers of coefficients will reduce numerical stability: Most issues can be resolved by carefully controlling the fit
: Sometimes the size of dt needs to be reduced by reducing the “dt stability factor”
• See
http://docs.lumerical.com/en/fdtd/user_guide_diverging_simulations.html for more details
� Tips
: Increase the weight of the imaginart part to get a better fit to imag(ε) if the absorption is critical for your simulation
: You may want to lock material fits to a particular wavelength range
• The fit will not change as you change the source bandwidth
: Unchecking “improve stability” may get a better fit but there is more chance of divergence
: If you uncheck “make fit passive” plot ε over the extended view range. If imag(ε)<0 your simulation will likely diverge.
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© 2011 Lumerical Solutions, Inc.
Advanced tips
� Tips
: If you combine results from several different bandwidth simulations, you may want to lock the simulation meshing algorithm to use a larger wavelength range that encompasses all the wavelengths you want to study
: This means that the FDTD mesh will not change as you change the source bandwidth
© 2011 Lumerical Solutions, Inc.
Advanced tips
� What happens when materials overlap?
� The mesh order determines the result
In this case, the order in the Objects Tree
determines the result.
This should be avoided since reordering
your objects will change the results!
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© 2011 Lumerical Solutions, Inc.
Advanced tips
� What happens at the interface where objects touch?
: Which material is used here?
� When conformal meshing is on, it does not matter!
� When conformal meshing is not used
: Set mesh order correctly for precise control
Silicon
Glass
•Set Silicon mesh order to 2
•Set Glass mesh order to 3
The interface point will be
Silicon!
Silicon
Glass
© 2011 Lumerical Solutions, Inc.
Questions and Answers…
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© 2011 Lumerical Solutions, Inc.
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